THERMAL-HYDRAULIC IN NUCLEAR REACTOR

GS. Tr n Đ i Phúc

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR Summary

1.Introduction 2.Energy from fission 3.Fission yield 4.Decay heat 5.Spatial distribution of heat sources 6.Coolant flow & heat transfer in fuel rod assembly 7.Enthalpy distribution in heated channel 8.Temperature distribution in channel in single phase 9.Heat conduction in fuel assembly 10.Axial temperature distribution in fuel rod 11.Void fraction in fuel rod channel 12.Heat transfer to coolant

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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I. Introduction I. Introduction  An important aspect of nuclear reactor core analysis involves the An important aspect of nuclear reactor core analysis involves the determination of the optimal coolant flow distribution and pressure determination of the optimal coolant flow distribution and pressure drop across the reactor core. On the one hand, higher coolant flow drop across the reactor core. On the one hand, higher coolant flow rates will lead to better heat transfer coefficients and higher Critical rates will lead to better heat transfer coefficients and higher Critical Heat Flux (CHF) limits. On the other hand, higher flows rates will also Heat Flux (CHF) limits. On the other hand, higher flows rates will also in large pressure drops across the reactor core, hence larger required in large pressure drops across the reactor core, hence larger required pumping powers and larger dynamic loads on the core components. pumping powers and larger dynamic loads on the core components. Thus, the role of the hydrodynamic and thermalhydraulic analysis is Thus, the role of the hydrodynamic and thermalhydraulic analysis is to find proper operating conditions that assure both safe and to find proper operating conditions that assure both safe and economical operation of the nuclear power plant. economical operation of the nuclear power plant.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

This chapter presents methods to determine the distribution of heat This chapter presents methods to determine the distribution of heat sources and temperatures in various components of nuclear reactor. In sources and temperatures in various components of nuclear reactor. In safety analyses of nuclear power plants the amount of heat generated in safety analyses of nuclear power plants the amount of heat generated in the reactor core must be known in order to be able to calculate the the reactor core must be known in order to be able to calculate the temperature distributions and thus, to determine the safety margins. temperature distributions and thus, to determine the safety margins. Such analyses have to be performed for all imaginable conditions, Such analyses have to be performed for all imaginable conditions, including operation conditions, reactor startup and shutdown, as well as including operation conditions, reactor startup and shutdown, as well as for removal of the decay heat after reactor shutdown. The first section for removal of the decay heat after reactor shutdown. The first section presents the methods to predict the heat sources in nuclear reactors at presents the methods to predict the heat sources in nuclear reactors at various conditions. The following sections discuss the prediction of such various conditions. The following sections discuss the prediction of such parameters as coolant enthalpy, fuel element temperature, void fraction, parameters as coolant enthalpy, fuel element temperature, void fraction, pressure drop and the occurrence of the Critical Heat Flux (CHF) in pressure drop and the occurrence of the Critical Heat Flux (CHF) in nuclear fuel assemblies nuclear fuel assemblies

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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I.1. Safety Functions & Requirements I.1. Safety Functions & Requirements

 The safety functions guaranteed by the thermalhydraulic design are The safety functions guaranteed by the thermalhydraulic design are following: following:  Evacuation via coolant fluid the heat generated by the nuclear fuel; Evacuation via coolant fluid the heat generated by the nuclear fuel;  Containment of radioactive products (actinides and fission products) Containment of radioactive products (actinides and fission products) inside the containment barrier. inside the containment barrier.  Control of the reactivity of the reactor core: no effect on the thermal Control of the reactivity of the reactor core: no effect on the thermal hydraulic design. hydraulic design.  Evacuation of the heat generated by the nuclear fuel: The aim of Evacuation of the heat generated by the nuclear fuel: The aim of thermalhydraulic design is to guarantee the evacuation of the heat thermalhydraulic design is to guarantee the evacuation of the heat generated in the reactor core by the energy transfer between the fuel generated in the reactor core by the energy transfer between the fuel

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Rods to the coolant fluid in normal operation and incidental Rods to the coolant fluid in normal operation and incidental conditions. conditions.  The thermalhydraulic design is not under specific design The thermalhydraulic design is not under specific design requirements. requirements.  However, the assured safety functions requires the application of a However, the assured safety functions requires the application of a Quality Assurance programme on which the main aim is to document Quality Assurance programme on which the main aim is to document and to control all associated activities. and to control all associated activities.  Preliminary tests: The basic hypothesis on scenarios adopted in the Preliminary tests: The basic hypothesis on scenarios adopted in the safety analyses must be control during the first physic tests of the safety analyses must be control during the first physic tests of the reactor core. Some of those tests, for example the measurements of reactor core. Some of those tests, for example the measurements of the primary coolant rate or the drop time of the control clusters, are the primary coolant rate or the drop time of the control clusters, are performed regularly. Other tests are performed in totality only on the performed regularly. Other tests are performed in totality only on the head of the train serial. head of the train serial.  For the following units, only the necessary tests performed to For the following units, only the necessary tests performed to guarantee that thermalhydraulic characteristics of the reactor core are guarantee that thermalhydraulic characteristics of the reactor core are identical to the ones of the head train serial. identical to the ones of the head train serial.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR The primary coolant rate and the drop time of the control rod clusters  The primary coolant rate and the drop time of the control rod clusters must be measured regularly. must be measured regularly. The main aim of the thermal-hydraulic design is principally to  The main aim of the thermal-hydraulic design is principally to guarantee the heat transfer and the repartition of the heat production guarantee the heat transfer and the repartition of the heat production in the reactor core, such as the evacuation of the primary heat or of in the reactor core, such as the evacuation of the primary heat or of the safety injection system (belong to each case) assures the respect the safety injection system (belong to each case) assures the respect of safety criteria. of safety criteria. I.2. Basis of thermal-hydraulic core analysis I.2. Basis of thermal-hydraulic core analysis The energy released in the reactor core by fission of enriched uranium  The energy released in the reactor core by fission of enriched uranium U235 and Plutonium 238 appears as kinetic energy of fission reaction U235 and Plutonium 238 appears as kinetic energy of fission reaction products and finally as heat generated in the nuclear fuel elements. products and finally as heat generated in the nuclear fuel elements. This heat must be removed from the fuel and reactor and used via This heat must be removed from the fuel and reactor and used via auxiliary systems to convert steam-energy to produce electrical power. auxiliary systems to convert steam-energy to produce electrical power.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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I.3. Constraints of the thermalhydraulic core design I.3. Constraints of the thermalhydraulic core design

 The main aims of the core design are subject to several important The main aims of the core design are subject to several important constraints. constraints.  The first important constraint is that the core temperatures remain The first important constraint is that the core temperatures remain below the melting points of materials used in the reactor core. This is below the melting points of materials used in the reactor core. This is particular important for the nuclear fuel and the nuclear fuel rods particular important for the nuclear fuel and the nuclear fuel rods cladding. cladding.  There are also limits on heat transfer are between the fuel elements There are also limits on heat transfer are between the fuel elements and coolant, since if this heat transfer rate becomes too large, critical and coolant, since if this heat transfer rate becomes too large, critical heat flux may be approached leading to boiling transition. This, in turn, heat flux may be approached leading to boiling transition. This, in turn, will result in a rapid increase of the clad temperature of the fuel rod. will result in a rapid increase of the clad temperature of the fuel rod.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The coolant pressure drop across the core must be kept low to minimize The coolant pressure drop across the core must be kept low to minimize pumping requirements as well as hydraulic loads (vibrations) to core pumping requirements as well as hydraulic loads (vibrations) to core components. components.  Above mentioned constraints must be analyzed over the core live, for all the Above mentioned constraints must be analyzed over the core live, for all the reactor core components, since as the power distribution in the reactor reactor core components, since as the power distribution in the reactor changes due to fuel burnup or core management, the temperature distribution changes due to fuel burnup or core management, the temperature distribution will similarly change. will similarly change.  Furthermore, since the cross sections governing the neutron physics of the Furthermore, since the cross sections governing the neutron physics of the reactor core are strongly temperature and density dependent, there will be a reactor core are strongly temperature and density dependent, there will be a strong coupling between thermalhydraulic and neutron behaviour of the strong coupling between thermalhydraulic and neutron behaviour of the reactor core. reactor core.

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II. Energy from nuclear fission II. Energy from nuclear fission

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Consider a monoenergetic neutron beam in which Consider a monoenergetic neutron beam in which n n is the neutron density is the neutron density is the number of is neutron speed then Snv nv is the number of (number of neutrons per m3). If v v is neutron speed then S (number of neutrons per m3). If neutron falling on 1 m2 of target material per second.. neutron falling on 1 m2 of target material per second

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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 Since s is the effective area per single nucleus, for a given reaction Since s is the effective area per single nucleus, for a given reaction and neutron energy, then S is the effective area of all the nuclei per m3 and neutron energy, then S is the effective area of all the nuclei per m3 gives the number of interactions of of target. Hence the product Snv nv gives the number of interactions of of target. Hence the product S nuclei and neutrons per m3 of target material per second. nuclei and neutrons per m3 of target material per second. , where ΣΣf f ==nv nv is In particular, the fission rate is found as: Σff nv nv = = ΣΣffФФ , where In particular, the fission rate is found as: Σ is the neutron flux (to be discussed later) and ΣΣff= = NσNσff , N being the , N being the the neutron flux (to be discussed later) and number of fissile nuclei/m3 and σσff m2/nucleus the fission cross m2/nucleus the fission cross number of fissile nuclei/m3 and section. In a reactor the neutrons are not monoenergetic and cover a section. In a reactor the neutrons are not monoenergetic and cover a wide range of energies, with different flux and corresponding cross wide range of energies, with different flux and corresponding cross section. section. In thermal reactor with volume V there will occur V ΣV Σff Ф Ф fissions, where fissions, where In thermal reactor with volume V there will occur ΣΣff and are the average values of the macroscopic fissions cross and ФФ are the average values of the macroscopic fissions cross section and the neutron flux, respectively. section and the neutron flux, respectively.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

168 168 7 7 5 5

1.11.1 7 7

6 6

 To evaluate the reactor power it is necessary to know the average To evaluate the reactor power it is necessary to know the average amount of energy which is released in a single fission. The table below amount of energy which is released in a single fission. The table below shows typical values for uranium235. shows typical values for uranium235.  Table II.1: Distribution of energy per fission of U235. Table II.1: Distribution of energy per fission of U235. 10101212 J = 1 MeV  J = 1 MeV  Kinetic energy of fission products 26.9 Kinetic energy of fission products 26.9  Instantaneous gammaray energy 1.11.1 Instantaneous gammaray energy  Kinetic energy of fission neutrons Kinetic energy of fission neutrons 0.80.8  Beta particles from fission products Beta particles from fission products  Gamma rays from fission products Gamma rays from fission products 1.01.0  Neutrinos Neutrinos  Total fission energy Total fission energy

1.6 1.6 3232 10 10 200 200

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 As can be seen, the total fission energy is equal to 32 pJ. It means that As can be seen, the total fission energy is equal to 32 pJ. It means that it is required ~3.1 101010 fissions per second to generate 1 W of the fissions per second to generate 1 W of the it is required ~3.1 10 thermal power. Thus, the thermal power of a reactor can be evaluated thermal power. Thus, the thermal power of a reactor can be evaluated as:as:

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1010 (W) (W)

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Σ Σ P (W) = V fФ / 3.1x10 P (W) = V fФ / 3.1x10 Thus, the thermal power of a nuclear reactor is proportional to the Thus, the thermal power of a nuclear reactor is proportional to the

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number of fissile nuclei, N, and the neutron flux f . Both these number of fissile nuclei, N, and the neutron flux f . Both these parameters vary in a nuclear reactor and their correct computation is parameters vary in a nuclear reactor and their correct computation is necessary to be able to accurately calculate the reactor power. necessary to be able to accurately calculate the reactor power.

Power density (which is the total power divided by the volume) in Power density (which is the total power divided by the volume) in nuclear reactors is much higher than in conventional power plants. Its nuclear reactors is much higher than in conventional power plants. Its typical value for PWRs is 75 MW/m3, whereas for a fast breeder reactor typical value for PWRs is 75 MW/m3, whereas for a fast breeder reactor cooled with sodium it can be as high as 530 MW/m3. cooled with sodium it can be as high as 530 MW/m3.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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thermal thermal

III. Fission yield III. Fission yield  Fissions of uranium235 nucleus can end up with 80 different primary Fissions of uranium235 nucleus can end up with 80 different primary fission products. The range of mass numbers of products is from 72 fission products. The range of mass numbers of products is from 72 (isotope of zinc) to 161 (possibly an isotope of terbium). The yields of (isotope of zinc) to 161 (possibly an isotope of terbium). The yields of fission of uranium233, uranium235, the products of fission of uranium233, uranium235, the products of plutonium239 and a mixture of uranium and plutonium are shown in plutonium239 and a mixture of uranium and plutonium are shown in following figure III.1. following figure III.1.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Figure III.1: Fission yield as a function of mass number of the fission Figure III.1: Fission yield as a function of mass number of the fission product. product.

 As can be seen in all cases there are two groups of fission products: a As can be seen in all cases there are two groups of fission products: a “light” group with mass number between 80 and 110 and a “heavy” “light” group with mass number between 80 and 110 and a “heavy” group with mass numbers between 125 and 155. group with mass numbers between 125 and 155.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

he 6 formula: Figure III.2: Illustration of the 6 formula: Figure III.2: Illustration of t

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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IV. Decay heat IV. Decay heat  A large portion of the radioactive fission products emit gamma rays, in A large portion of the radioactive fission products emit gamma rays, in addition to beta particles. The amount and activity of individual fission addition to beta particles. The amount and activity of individual fission products and the total fission product inventory in the reactor fuel products and the total fission product inventory in the reactor fuel during operation and after shutdown are important for several during operation and after shutdown are important for several reasons: namely to evaluate the radiation hazard, and to determine the reasons: namely to evaluate the radiation hazard, and to determine the decrease of the fission product radioactivity in the spent fuel elements decrease of the fission product radioactivity in the spent fuel elements after removal from the reactor. This information is required to evaluate after removal from the reactor. This information is required to evaluate the length of the cooling period before the fuel can be reprocessed. the length of the cooling period before the fuel can be reprocessed.  Right after the insertion of a large negative reactivity to the reactor Right after the insertion of a large negative reactivity to the reactor core (for example, due to an injection of control rods), the neutron flux core (for example, due to an injection of control rods), the neutron flux rapidly decreases according to the following equation, rapidly decreases according to the following equation,

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

/ /

/ l)t / l)t

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ρ β ρ ρ β ρ β β ρ β β ρ / / / / – – – –

β ρ β ρ – ( ))e ( – ))e

} (IV.1) } (IV.1) Φ Φ(t) = Ф (t) = Ф

λρ β ρ ( λρ β ρ )t ( – ( ) e )t ( – ) e

 Here f (

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{( 00{( ) is the neutron flux at time t after reactor shutdown, 0 f is the Here f (t t ) is the neutron flux at time t after reactor shutdown, 0 f is the neutron flux during reactor operation at full power, r is the step change neutron flux during reactor operation at full power, r is the step change of reactivity, ββ is the fraction of delayed neutrons, l is the prompt is the fraction of delayed neutrons, l is the prompt of reactivity, neutron lifetime and l is the mean decay constant of precursors of neutron lifetime and l is the mean decay constant of precursors of delayed neutrons. For LWR with uranium235 as the fissile material, delayed neutrons. For LWR with uranium235 as the fissile material, typical values are as follows: l = 0.08 s1, ββ = 0.0065 and l = 103s. = 0.0065 and l = 103s. typical values are as follows: l = 0.08 s1,  Assuming the negative stepchange of reactivity r = 0.09, the relative Assuming the negative stepchange of reactivity r = 0.09, the relative neutron flux change is given as: neutron flux change is given as:

+ 0.933 e 96.5t Ф(t) / Ф0 = 0.067 e 0.075t Ф(t) / Ф0 = 0.067 e

0.075t + 0.933 e

(IV2)) 96.5t (IV2

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The second term in Eq. (43) is negligible already after t = 0.01s and The second term in Eq. (43) is negligible already after t = 0.01s and only the first term has to be taken into account in calculations. As can only the first term has to be taken into account in calculations. As can be seen, the neutron flux (and thus the generated power) immediately be seen, the neutron flux (and thus the generated power) immediately jumps to ~6.7% of its initial value and then it is reduced efold during jumps to ~6.7% of its initial value and then it is reduced efold during = 1/0.075 = 13.3 s. period of time T T = 1/0.075 = 13.3 s. period of time  After a reactor is shut down and the neutron flux falls to such a small After a reactor is shut down and the neutron flux falls to such a small value that it may be neglected, substantial amounts of heat continue to value that it may be neglected, substantial amounts of heat continue to be generated due to the beta particles and the gamma rays emitted by be generated due to the beta particles and the gamma rays emitted by the fission products. FIGURE 42 shows the fission product decay heat the fission products. FIGURE 42 shows the fission product decay heat versus the time after shut down. The curve, which covers a time range versus the time after shut down. The curve, which covers a time range from 1 to 106 years after shut down, refers to a hypothetical from 1 to 106 years after shut down, refers to a hypothetical pressurized water cooled reactor that has operated at a constant pressurized water cooled reactor that has operated at a constant power for a period of time during which the fuel (with initial enrichment power for a period of time during which the fuel (with initial enrichment 4.5%) has reached 50 GWd/tU burnup and is then shut down 4.5%) has reached 50 GWd/tU burnup and is then shut down instantaneously. Contributions from various species which are present instantaneously. Contributions from various species which are present in the spent fuel are indicated. in the spent fuel are indicated.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure IV.1: Fission product decay heat power (W/metric ton of HM) Figure IV.1: Fission product decay heat power (W/metric ton of HM) versus time after shutdown.. versus time after shutdown

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure IV.2: Relative decay power versus relative time after reactor Figure IV.2: Relative decay power versus relative time after reactor shutdown for various operation periods from 1 month to 12 months. shutdown for various operation periods from 1 month to 12 months.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 q” / q”

t 0.20.2} (IV.3) } (IV.3) = 0.065 { t topop) ) 0.20.2 t

 The power density change due to beta and gamma radiation can be The power density change due to beta and gamma radiation can be calculated from the fllowing approximate equation [IV1], calculated from the fllowing approximate equation [IV1], q” / q”00 = 0.065 { t t  Here q”0 is the power density in the reactor at steady state operation Here q”0 is the power density in the reactor at steady state operation ” is the decay power density, t t is the time after is the time after before shut down, qq” is the decay power density, before shut down, is the time of reactor operation before reactor shut down [s] and top top is the time of reactor operation before reactor shut down [s] and shut down [s]. Equation (IV3) is applicable regardless of whether the shut down [s]. Equation (IV3) is applicable regardless of whether the fuel containing the fission products remains in the reactor core or it is fuel containing the fission products remains in the reactor core or it is removed from it. However, the equation accuracy and applicability is removed from it. However, the equation accuracy and applicability is limited and can be used for cooling periods from approximately 10 s to limited and can be used for cooling periods from approximately 10 s to less than 100 days. less than 100 days.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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 q”’ / q”

Equation (IV.3) can be transformed to: Equation (IV.3) can be transformed to:

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= 0.065 / topop 0.20.2{ 1 / (t – t { 1 / (t – topop / t / topop) ) 0.20.2 1 / (t / t 1 / (t / topop) ) 0.20.2} (IV.4) } (IV.4)

ʘ ʘ q”’ / q”00 = 0.065 / t = (t – t = (t – t Here Here

opop) / t

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is the relative time after reactor shut down. ) / topop is the relative time after reactor shut down. Equation (IV.4) is shown in FIGURE IV.2 for the reactor operation time Equation (IV.4) is shown in FIGURE IV.2 for the reactor operation time from 1 month to 1 year. top from 1 month to 1 year. top  V. Spatial distribution of the heat sources V. Spatial distribution of the heat sources  The energy released in nuclear fission reaction is distributed among a The energy released in nuclear fission reaction is distributed among a variety of reaction products characterized by different range and time variety of reaction products characterized by different range and time delays. Once performing the thermal design of a reactor core, the delays. Once performing the thermal design of a reactor core, the energy deposition distributed over the coolant and structural materials energy deposition distributed over the coolant and structural materials is frequently reassigned to the fuel in order to simplify the thermal is frequently reassigned to the fuel in order to simplify the thermal analysis of the core. The volumetric fission heat source in the core can analysis of the core. The volumetric fission heat source in the core can be found in general case as: be found in general case as:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 q”’ (r) = Σ

q”’ (r) = Σii w wff (i)(i) Ni (r) ƒ Ni (r) ƒ00

∞∞ dEσ dEσff((i)i) (E)Ф (r,E) (V.1) (E)Ф (r,E) (V.1)

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 Here (

is the recoverable energy released per fission event of ith f w is the recoverable energy released per fission event of ith

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is the number density of ith fissile material at i N is the number density of ith fissile material at

Here (i i ) ) f w fissile material, (r) i N fissile material, (r) location r and (EE) ) ii location r and ( s is its microscopic fission cross section for neutrons with energy E. f f s is its microscopic fission cross section for neutrons with energy E. Since the neutron flux and the number density of the fuel vary across Since the neutron flux and the number density of the fuel vary across the reactor core, there will be a corresponding variation in the fission the reactor core, there will be a corresponding variation in the fission heat source. heat source.  The simplest model of fission heat distribution would correspond to a The simplest model of fission heat distribution would correspond to a bare, bare,  homogeneous core. One should recall here the onegroup flux homogeneous core. One should recall here the onegroup flux distribution for such geometry given as: distribution for such geometry given as:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Ф(r, z) = Ф

are effective and HH are effective

π π π π Ф(r, z) = Ф00JJ00 {2.405r / R}cos{ z / H} (V.2) {2.405r / R}cos{ z / H} (V.2)  Here 0 is the flux at the center of the core and Here 0 is the flux at the center of the core and R R and (extrapolated) core dimensions that include extrapolation lengths as (extrapolated) core dimensions that include extrapolation lengths as for a reflected core. well as an adjustment to account for a reflected core. well as an adjustment to account  Having a fuel rod located at r = rf distance from the centerline of the Having a fuel rod located at r = rf distance from the centerline of the core, the core, the  volumetric fission heat source becomes a function of the axial volumetric fission heat source becomes a function of the axial coordinate, z, only: coordinate, z, only: q”’(z) = wf  q”’(z) = wf fФΣ 00JJ00{2.405rf / R}cos{ z / H} (V.3) fФΣ {2.405rf / R}cos{ z / H} (V.3)

 There are numerous factors that perturb the power distribution of the There are numerous factors that perturb the power distribution of the reactor core, and the above equation will not be valid. For example fuel reactor core, and the above equation will not be valid. For example fuel is usually not loaded with uniform enrichment. At the beginning of core is usually not loaded with uniform enrichment. At the beginning of core life, higher enrichment fuel is loaded toward the edge of the core in life, higher enrichment fuel is loaded toward the edge of the core in order to flatten the power distribution. Other factors include the order to flatten the power distribution. Other factors include the influence of the control rods and variation of the coolant density. influence of the control rods and variation of the coolant density.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 All these power perturbations will cause a corresponding variation of All these power perturbations will cause a corresponding variation of temperature distribution in the core. A usual technique to take care of temperature distribution in the core. A usual technique to take care of these variations is to estimate the local working conditions (power these variations is to estimate the local working conditions (power level, coolant flow, etc) which are the closest to the thermal limitations. level, coolant flow, etc) which are the closest to the thermal limitations. Such part of the core is called hot channel and the working conditions Such part of the core is called hot channel and the working conditions are related with socalled hot channel factors. are related with socalled hot channel factors.  One common approach to define hot channel is to choose the channel One common approach to define hot channel is to choose the channel where the core heat flux and the coolant enthalpy rise is a maximum. where the core heat flux and the coolant enthalpy rise is a maximum. Working conditions in the hot channel are defined by several ratios of Working conditions in the hot channel are defined by several ratios of These ratios, termed local conditions to coreaveraged conditions. local conditions to coreaveraged conditions. These ratios, termed the hot channel factors or power peaking factors will be considered in the hot channel factors or power peaking factors will be considered in more detail in coming Chapters. However, it can be mentioned already more detail in coming Chapters. However, it can be mentioned already here that the basic initial plant thermal design relay on these here that the basic initial plant thermal design relay on these

factors. factors.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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In thermal reactors it is assumed that 90% of the fission total energy is In thermal reactors it is assumed that 90% of the fission total energy is liberated in fuel elements, whereas the remaining 10% is equally liberated in fuel elements, whereas the remaining 10% is equally distributed between moderator and reflector/shields. . distributed between moderator and reflector/shields VI. Coolant flow and heat transfer in fuel rod assembly  VI. Coolant flow and heat transfer in fuel rod assembly  Rod bundles in nuclear reactors have usually very complex geometry. Rod bundles in nuclear reactors have usually very complex geometry. Due to that a thorough thermalhydraulic analysis in rod bundles Due to that a thorough thermalhydraulic analysis in rod bundles requires quite sophisticated computational tools. In general, several requires quite sophisticated computational tools. In general, several levels of approximations can be employed to perform the analysis: levels of approximations can be employed to perform the analysis: • • Simple onedimensional analysis of a single subchannel or bundle, Simple onedimensional analysis of a single subchannel or bundle, Analysis of a whole rod bundle applying the subchannelanalysis • • Analysis of a whole rod bundle applying the subchannelanalysis code, code, Complex threedimensional analysis using Computational Fluid • • Complex threedimensional analysis using Computational Fluid Dynamics Dynamics (CFD) codes.. (CFD) codes

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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In this chapter only the simples approach is considered. In this In this chapter only the simples approach is considered. In this approach, the single subchannel or rod bundle is treated as a one approach, the single subchannel or rod bundle is treated as a one dimensional pipe with a diameter equal to the hydraulic (equivalent) dimensional pipe with a diameter equal to the hydraulic (equivalent) diameter of the subchannel or bundle. The hydraulic diameter of a diameter of the subchannel or bundle. The hydraulic diameter of a channel of arbitrary shape is defined as: channel of arbitrary shape is defined as:  Dh = 4A / Pw (VI.1) Dh = 4A / Pw (VI.1)  where where A A is the channel crosssection area and is the channel is the channel crosssection area and Pw Pw is the channel wetted perimeter.figure VI.1shows typical coolant subchannels in wetted perimeter.figure VI.1shows typical coolant subchannels in infinite rod lattices. infinite rod lattices.  Figure VI.1: Typical coolant subchannels in fuel rods assembly. Figure VI.1: Typical coolant subchannels in fuel rods assembly.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Figure VI.1: Typical coolant subchannels in fuel rods assembly. Figure VI.1: Typical coolant subchannels in fuel rods assembly.

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 A = p  A = (3

The subchannel flow area is expressed as following: The subchannel flow area is expressed as following: A = p22 dπ dπ 22 / 4 for square lattice / 4 for square lattice A = (31/2 1/2 / 4)p / 4)p22 dπ dπ 22 / 4 (VI.2) / 4 (VI.2)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 And the wetted perimeter (part of the perimeter filled with heated walls) And the wetted perimeter (part of the perimeter filled with heated walls) is given by: is given by:

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for square lattice for square lattice

 Where

Pw = d π Pw = d π Pw = 1/2 dπ Pw = 1/2 dπ for triangular lattice (VI.3) for triangular lattice (VI.3)



is the diameter of fuel rods. The is the lattice pitch and dd is the diameter of fuel rods. The

π π

22 – 1} for square lattice – 1} for square lattice



(p / d) (p / d) π π / Where pp is the lattice pitch and hydraulic diameter is expressed as: hydraulic diameter is expressed as: Dh = d{4 / Dh = d{4 / Dh = d{2x31/21/2 / Dh = d{2x3 (p / d) (p / d)

22 – 1 } for triangular lattice – 1 } for triangular lattice



(VI.4) (VI.4) In case of fuel assemblies in Boiling Water Reactors (BWR), the In case of fuel assemblies in Boiling Water Reactors (BWR), the hydraulic diameter should be based on the total wetted perimeter and hydraulic diameter should be based on the total wetted perimeter and the total crosssection area of the fuel assembly. Assuming fuel the total crosssection area of the fuel assembly. Assuming fuel assembly as shown in FIGURE 45, the hydraulic diameter is as assembly as shown in FIGURE 45, the hydraulic diameter is as follows: follows:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR π – N dπ 22) / (4w + N d) (VI.5) π ) / (4w + N d) (VI.5)

Dh = 4A / Pw = (4w22 – N dπ

 Dh = 4A / Pw = (4w  Where NN is the number of rods in the fuel assembly, Where

is the width is the number of rods in the fuel assembly, ww is the width

of the box (m) and d is the diameter of fuel rods(m). of the box (m) and d is the diameter of fuel rods(m).  Figure VI.2: Crosssection of a BWR fuel assembly. Figure VI.2: Crosssection of a BWR fuel assembly. 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

, and an arbitrary, axiallydependent geometry, as q’’(z), and an arbitrary, axiallydependent geometry, as

 VII. Enthalpy distribution in heated channel VII. Enthalpy distribution in heated channel   Assume a heated channel with an arbitrary axial distribution of the Assume a heated channel with an arbitrary axial distribution of the heat flux, q’’(z) heat flux, shown in figure VII.1. The coolant flowing in the channel has a shown in figure VII.1. The coolant flowing in the channel has a constant mass flow rate WW.. constant mass flow rate  As follow As follow

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The energy balance for a differential channel length between z and z + The energy balance for a differential channel length between z and z + dz is given as follows: dz is given as follows:

(VII.1) (VII.1) H = W . ill (z) + q”(z).P (z) + q”(z).PHH(z).dz = W[i (z) + dill]] (z).dz = W[ill (z) + di

 ΔΔH = W . i  Which to the following differential equation for the coolant enthalpy: Which to the following differential equation for the coolant enthalpy: Δ ΔH = di H = di 

(VII.2) (VII.2)

 Where



(z) / dz = q”(z) PHH(z) / W(z) / W ll(z) / dz = q”(z) P is the heated perimeter of the channel. Integration of Eq. PH(z) is the heated perimeter of the channel. Integration of Eq.



+ 1/W ƒH/2H/2 zz q”(z).P (z)dz q”(z).PHH(z)dz

Where PH(z) yields: (413) from the channel inlet to a certain location z z yields: (413) from the channel inlet to a certain location ii l l(z) = il (VII.3) (z) = ili i + 1/W ƒ (VII.3)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 where



is the coolant ili is the coolant and ili is the coolant enthalpy at location z z and

where il(z) il(z) is the coolant enthalpy at location enthalpy at the inlet to the channel (z z = = H/2H/2).). enthalpy at the inlet to the channel (  VIII. Temperature distribution in channel in single phase VIII. Temperature distribution in channel in single phase   For low temperature and pressure changes the enthalpy of a single For low temperature and pressure changes the enthalpy of a single phase (nonboiling) coolant can be expressed as a linear function of phase (nonboiling) coolant can be expressed as a linear function of the temperature. Assuming a uniform axial distribution of heat sources the temperature. Assuming a uniform axial distribution of heat sources and a constant heated perimeter, Eq. (VII.3)) yields, and a constant heated perimeter, Eq. (VII.3)) yields,

(VIII.1) (VIII.1)

lbi + q”P

 TTlblb (z) = T

(z) = Tlbi (z + H/2) / CpW + q”PHH (z + H/2) / CpW

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Here

Tlb(z) is the coolant bulk temperature at location . The bulk is the coolant bulk temperature at location zz. The bulk

Here Tlb(z) temperature in a channel cross section is defined in such a way that it temperature in a channel cross section is defined in such a way that it can be obtained from the energy balance over a portion of the channel. can be obtained from the energy balance over a portion of the channel. For an arbitrary velocity, temperature and fluid property distribution For an arbitrary velocity, temperature and fluid property distribution across the channel crosssection, the bulk temperature is given by: across the channel crosssection, the bulk temperature is given by:

(VIII.2) (VIII.2) = ƒACAC lCρlCρ pl pl VV l ldA / ƒ dA / ƒAAρρllCCplplVVlldA dA

 TTlblb = ƒ  The temperature distribution along the channel is represented in figure The temperature distribution along the channel is represented in figure VIII.1. VIII.1.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure VIII.1: Bulk temperature distribution in a uniformly heated channel Figure VIII.1: Bulk temperature distribution in a uniformly heated channel with constant heated perimeter.. with constant heated perimeter

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



(VIII.3) (VIII.3) In nuclear reactor cores the axial power distribution may have various In nuclear reactor cores the axial power distribution may have various shapes. The cosineshaped power distribution is obtained in shapes. The cosineshaped power distribution is obtained in cylindrical homogeneous reactors, as previously derived using the cylindrical homogeneous reactors, as previously derived using the diffusion approximation for the neutron distribution calculation. diffusion approximation for the neutron distribution calculation.  Using Eq. (V.2) and the coordinate system as indicated in figure VIII.2, Using Eq. (V.2) and the coordinate system as indicated in figure VIII.2, the power distribution may be expressed as follows: the power distribution may be expressed as follows: π πcos( z / H) Q”(z) = q”00cos( z / H) Q”(z) = q”

π π (z)/W.cos( z / H) or (z) / dz = q”00PPHH(z)/W.cos( z / H) or

π π

 Equation (VII.2) becomes: Equation (VII.2) becomes:  DiDill(z) / dz = q”  dTlb(z) /dz = q”0PH(z) /W.Cp.cos( z /H) (VIII.4) dTlb(z) /dz = q”0PH(z) /W.Cp.cos( z /H) (VIII.4)  ..

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure VIII.2: Heated channel with cosines power distribution  Figure VIII.2: Heated channel with cosines power distribution

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 After intergration, the coolant enthalpy and temperature distribution After intergration, the coolant enthalpy and temperature distribution are as follows: are as follows:



π π

[sin ( [sin ( π π π ππz / H) + sin ( H /2H)] + ili or π z / H) + sin ( H /2H)] + ili or π ππz / H) + sin ( H /2H)] + T π z / H) + sin ( H /2H)] + T [sin ( [sin ( / W x H/ (z) = q”00PPHH / W x H/ iill(z) = q” TTlblb(z) = q” / W.Cp x H/ (z) = q”00PPHH / W.Cp x H/

lbi lbi



π π or H /2H) + ilili or π π .W.Cp.sin( H /2H) + T .W.Cp.sin( H /2H) + T

lbex = Tlb(H/2) = 2q”

(VIII.5) (VIII.5)  The channel exit temperature and enthalpy can be found by The channel exit temperature and enthalpy can be found by substituting z = H/2 into equation VIII.5: substituting z = H/2 into equation VIII.5: .H /ππW.sin(W.sin(ππH /2H) + i ilex(z) = il(H/2) = 2q”00.P.PHH.H / ilex(z) = il(H/2) = 2q” .H / = Tlb(H/2) = 2q”00.P.PHH.H / TTlbex i (VIII.6) lblbi (VIII.6)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure VIII.3 Represents the axial distribution of the coolant  Figure VIII.3 Represents the axial distribution of the coolant temperature with cosines heat flux distribution. temperature with cosines heat flux distribution.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 IX. Heat conduction in fuel assembly IX. Heat conduction in fuel assembly   Modern nuclear power reactors contain cylindrical fuel elements that Modern nuclear power reactors contain cylindrical fuel elements that are composed of ceramic fuel pellets located in metallic tubes (so are composed of ceramic fuel pellets located in metallic tubes (so called cladding). A crosssection over a square lattice of fuel rods is called cladding). A crosssection over a square lattice of fuel rods is shown in FIGURE 410. For thermal analyses it is convenient to shown in FIGURE 410. For thermal analyses it is convenient to subdivide the fuel rod assembly into subchannels. The division can subdivide the fuel rod assembly into subchannels. The division can be performed in several ways; however, most obvious choices are so be performed in several ways; however, most obvious choices are so called coolant centered subchannels and rodcentered subchannels. called coolant centered subchannels and rodcentered subchannels. Both types of subchannels are equivalent in terms of major Both types of subchannels are equivalent in terms of major parameters such as the flow crosssection area, the hydraulic parameters such as the flow crosssection area, the hydraulic diameter, the wetted perimeter and the heated perimeter. In diameter, the wetted perimeter and the heated perimeter. In continuation, the thermal analysis will be performed for a single sub continuation, the thermal analysis will be performed for a single sub channel. channel.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The stationary (time independent) heat conduction equation for an The stationary (time independent) heat conduction equation for an infinite cylindrical fuel pin, in which the axial heat conduction can be infinite cylindrical fuel pin, in which the axial heat conduction can be ignored is as follows: ignored is as follows:



 where

(IX.1) (IX.1)

q” = ( 1/r).d/dr[λFFr.dTr.dTFF /dr] q” = ( 1/r).d/dr[λ /dr] is the fuel temperature, [K], F F l is the thermal conductivity of where F T F T is the fuel temperature, [K], l is the thermal conductivity of ¢ is the density of heat sources, [W m the fuel material, [W m1 K1], q q ¢ is the density of heat sources, [W m the fuel material, [W m1 K1], is the radial distance. Here the angular dependence of the 3] and r r is the radial distance. Here the angular dependence of the 3] and temperature is omitted due to the assumed axial symmetry of the temperature is omitted due to the assumed axial symmetry of the temperature distribution. temperature distribution.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Assuming that q”’ is constant a crosssection equation (IX.1) could be Assuming that q”’ is constant a crosssection equation (IX.1) could be integrated to obtain: integrated to obtain:



(IX.2) (IX.2) λλFFrdTrdTFF / dr = r2 /2q”’ / dr = r2 /2q”’

rdr = ( r22 TF0TF0rdr = ( r If the fuel conductivity was constant, Eq. (IX.2) could be integrated and If the fuel conductivity was constant, Eq. (IX.2) could be integrated and the temperature distribution would be obtained. However in typical fuel the temperature distribution would be obtained. However in typical fuel materials the fuel thermal conductivity strongly depends on the materials the fuel thermal conductivity strongly depends on the temperature and this is the reason why the temperature distribution temperature and this is the reason why the temperature distribution can not be found from Eq. (IX.2) in a general analytical form. Instead, can not be found from Eq. (IX.2) in a general analytical form. Instead, Eq. (IX.2) is transformed and integrated as follows: Eq. (IX.2) is transformed and integrated as follows: ƒ→ TFcTFc ƒ→

TF0TF0λλFFdTdTFF = q” /2ƒ = q” /2ƒ00

/4)q”’ F0F0 /4)q”’

 λλFFrdTrdTFF = r2 /2q”’dr = r2 /2q”’dr

(IX.3) (IX.3)  where the integration on the lefthandside is carried out from the where the integration on the lefthandside is carried out from the , to the temperature on the fuel pellet temperature at the centerline, TFcTFc, to the temperature on the fuel pellet temperature at the centerline, ). Defining the average fuel conductivity as: surface TFoTFo==TF TF ((rForFo). Defining the average fuel conductivity as: surface

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

(IX.4) (IX.4)

TFcTFcλλFFdTdTFF

= 1 / (TFcFc – T – Tfofo)ƒ)ƒTFoTFo

(IX.5) (IX.5)

FoFo /4λ /4λFF



 λλFF = 1 / (T  The temperature drop across the fuel pellet can be found as follows: The temperature drop across the fuel pellet can be found as follows: TΔTΔ FF = T = TFcFc T TFoFo = q”’r = q”’r22  In the thermal analysis of reactor cores, the power is often expressed In the thermal analysis of reactor cores, the power is often expressed in terms of the linear power density, that is, the power generated per in terms of the linear power density, that is, the power generated per unit length of the fuel element: unit length of the fuel element:



Q’ = Q’ = rπ 22 rπ (IX.6) (IX.6)

FoFoq”’q”’

 By combining equations ( IX.5) &( IX.6), we have: By combining equations ( IX.5) &( IX.6), we have: 

= q’ /4 .π λFF TΔTΔ FF = q’ /4 .π λ

(IX.7) (IX.7)

 Equation (IX.7) reveals that the fuel temperature drop is a function of Equation (IX.7) reveals that the fuel temperature drop is a function of the linear power density and the averaged fuel thermal conductivity. the linear power density and the averaged fuel thermal conductivity.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



(IX.8) (IX.8) In a similar manner the temperature drop across the gas gap can be In a similar manner the temperature drop across the gas gap can be obtained. In particular, Eq. (IX.1) can be used to describe the obtained. In particular, Eq. (IX.1) can be used to describe the temperature distribution in the gas gap, however, unlike for the fuel temperature distribution in the gas gap, however, unlike for the fuel pellet, the heat source term is equal to zero and the gas thermal pellet, the heat source term is equal to zero and the gas thermal conductivity can be assumed constant, thus: conductivity can be assumed constant, thus: 1/2 (d/dr)λGGr(dTr(dTGG/dr) = 0 1/2 (d/dr)λ /dr) = 0 (r) = C11 /λ /λGGlnr +C lnr +C22 T→ GG(r) = C T→



The integration constant C, can be found from condition of the The integration constant C, can be found from condition of the

(IX.9) (IX.9) /dr) = C1 / rFoFo = q’ / 2 rπ = q’/2π C→ 11 = q’/2π C→ = q’ / 2 rπ FoFo

(IX.10) (IX.10) heat flux continuity at r = rFoFo:: heat flux continuity at r = r λλGG(dT(dTGG/dr) = C1 / r  And the temperature drop in the gap can be expressed as follows: And the temperature drop in the gap can be expressed as follows: TΔTΔ GG = T = TGG(r(rGiGi) –TG(r  ) = q’/2πλGGlnrGo/r lnrGo/rGiGi ) –TG(rGoGo) = q’/2πλ

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 where

(IX11) (IX11) ) = q’ /2πλπλccln r

) Tc (rcoco) = q’ /2 ln rcoco/r/rcici is the clad thermal is the outer clad radius and _C C is the clad thermal rCo is the outer clad radius and _



(IX.12) (IX.12)



, the temperature drop in the coolant boundary ¢ 2p r r , the temperature drop in the coolant boundary

(IX.13) (IX.13)

 Equation (IX.10) is applicable to the clad material as well, since the Equation (IX.10) is applicable to the clad material as well, since the assumptions on the heat generation and the thermal conductance are assumptions on the heat generation and the thermal conductance are valid in this case as well. Substituting the proper dimensions and valid in this case as well. Substituting the proper dimensions and property data yields, property data yields, TΔTΔ cc = T = Tcc(r(rcici) Tc (r where rCo conductivity. conductivity.  Heat transfer from the clad surface to the coolant is described by the Heat transfer from the clad surface to the coolant is described by the following following  Equation: Equation: q” = h(Tcoco – T – Tlblb))  q” = h(T  where is the convective heattransfer coefficient. Taking into account where h h is the convective heattransfer coefficient. Taking into account that that ¢¢ = qq¢ 2p ( ) Co qCo q¢¢ = ( ) layer is found as: layer is found as: TΔTΔ ll = T

= TCoCo T Tlblb = q’/2 rπ = q’/2 rπ CoCohh

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



the total temperature drop from the center of the fuel pellet to coolant the total temperature drop from the center of the fuel pellet to coolant is expressed as follows: is expressed as follows:



Δ Δ ΔT = T T = T Δ FF + TΔ + TΔ GG + TΔ + TΔ cc + TΔ + TΔ ll



= q’/2 [1/2π = q’/2 [1/2π λFF + 1/λ λ + 1/λGG lnr h] (IX.14) i + 1/rCoCoh] (IX.14) + 1/λCClnrlnrCoCo/r/rCCi + 1/r

lnrGoGo/r/rGiGi + 1/λ  The total temperature drop in a fuel rod crosssection is represented in The total temperature drop in a fuel rod crosssection is represented in following figure IX.2. following figure IX.2.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure IX.1: Cross section of a square fuel lattice. Figure IX.1: Cross section of a square fuel lattice.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 X. Axial temperature distribution in fuel rods X. Axial temperature distribution in fuel rods  In the previous section expressions for the axial distribution of coolant In the previous section expressions for the axial distribution of coolant temperature have been derived. It has been shown that the axial temperature have been derived. It has been shown that the axial distribution of coolant temperature varies with the shape of the axial distribution of coolant temperature varies with the shape of the axial heat flux distribution. heat flux distribution. In particular, substituting Eqs. (VIII.3) and (VIII.4) into (IX.12) gives the In particular, substituting Eqs. (VIII.3) and (VIII.4) into (IX.12) gives the following expression for the temperature of the clad outer surface: following expression for the temperature of the clad outer surface:

π π π π π π .W.Cp) x [sin( z /H) + sin( H /2H)] + q”0 /h.cos .W.Cp) x [sin( z /H) + sin( H /2H)] + q”0 /h.cos .H) /( (z) = (q”00.P.PHH.H) /(

(X.1) (X.1)

lbi lbi



 TTC0C0(z) = (q” π π( z / H) + T ( z / H) + T

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure X.1: Repartition of the temperature across the fuel rod. Figure X.1: Repartition of the temperature across the fuel rod.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure X.1: Represents the temperature of the cladding outer surface as Figure X.1: Represents the temperature of the cladding outer surface as function of axial distance. function of axial distance.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



at a certain location zCo,max



It should be noted that the temperature of the clad outer surface gets i/ It should be noted that the temperature of the clad outer surface gets i/ . This H)ts maximum value TCo,max zCo,max. This TCo,max at a certain location H)ts maximum value location can be found from Eq. (X.1) using the following condition: location can be found from Eq. (X.1) using the following condition:



(X.2) (X.2) (z) /dz = 0 dTdTC0C0(z) /dz = 0

It is convenient to represent the cladding outer temperature as: It is convenient to represent the cladding outer temperature as:

π π (X.3) (X.3) π π .cos( z /H) C0C0.cos( z /H)

π π

lbi lbi .W.Cp) .W.Cp)

(X.4) (X.4) (z) = A + Bsin ( z /H) + C  TTC0C0 (z) = A + Bsin ( z /H) + C  Where, Where, π π A = Bsin( z / 2H) + T  A = Bsin( z / 2H) + T .PH.H) / ( B = (q”00.PH.H) / (

= q”00 / h / h

 B = (q”  CCC0C0 = q”  By combining the equations (X.2) & (X.3), we have: By combining the equations (X.2) & (X.3), we have:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



/H) = 0 (X.5) Co,Max /H) = 0 (X.5) Bcos( zπ COCO,Max / H) C Bcos( zπ ,Max / H) CC0C0sin( zπsin( zπ Co,Max

(X.6) (X.6)

Co,Max / H)

 Which is equivalent to the following equation: Which is equivalent to the following equation: = B / C / H) Co Co = B / C  Tan( zπTan( zπ Co,Max

 Thus: Thus: 

π π )arctan(B /C )arctan(B /C (X.7) (X.7) = (H/ zzCo,Max Co,Max = (H/

C0C0))



It should be noted that a physically meaningful solution of the above It should be noted that a physically meaningful solution of the above equation should be positive and less than H. equation should be positive and less than H.  Noting that: Noting that:  ,Max /H))1/21/2 /H) = ± tan( zπ C0C0,Max /H) / (1 + tan sin( zπsin( zπ Co,Max

Co,Max /H) = ± tan( zπ = ± ((B/C

 andand 

,Max /H) / (1 + tan22( zπ( zπ CoCo,Max /H)) ) / (1 + (B /CC0C0))22))1/21/2 = ± ((B/CC0C0) / (1 + (B /C

C0,Max / H) = ± ( 1/ (1 + tan2( zπ



cos ( zπ C0,Max cos ( zπ / H) = ± ( 1/ (1 + tan2( zπ C0,Max

= ± (1 = ± (1 / (1 + (B /C / H) C0,Max / H) / (1 + (B /CC0C0))22))1/21/2

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The maximum temperature of the cladding outer surface becomes The maximum temperature of the cladding outer surface becomes (taking only + sign ): (taking only + sign ):



= A + (B22 + C (X.8) (X.8)

22))1/21/2

C0,Max = A + (B

 Using constants

TTC0,Max

+ CC0C0 given by Eq. (X.4), the maximum clad and CCCoCo given by Eq. (X.4), the maximum clad

Using constants AA, , B B and outer temperature is obtained as: outer temperature is obtained as: π π .W.Cp)sin( H / 2H) + T .H) / = (q”00.P.PHH.H) / .W.Cp)sin( H / 2H) + T

lbi + ((q” lbi

.H + ((q”00.P.PHH.H

 TTC0,Max C0,Max = (q” π π/ .W.Cp) / .W.Cp)

π π 22 + (q”0 /h)2) + (q”0 /h)2)1/21/2 (X.9) (X.9)

π π π π .W.Cp) / .W.Cp) /

lbi)) / q”

C0C0,Max T

,Max Tlbi .H = sin( H /2H) + ((1 + ( )) / q”00.P.PHH.H = sin( H /2H) + ((1 + (

.W.Cp(T .W.Cp(T .H.h)2)1/21/2

 oror π π( (  PPHH.H.h)2)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Since the clad maximum temperature is located on the inner surface, it Since the clad maximum temperature is located on the inner surface, it is of interest to find it as well. The axial distribution of the clad inner is of interest to find it as well. The axial distribution of the clad inner temperature can be obtained from Eqs. (IX.11) and X.1) as: temperature can be obtained from Eqs. (IX.11) and X.1) as:



π π π π π π .W.Cp).[sin( z /H) + sin ( H 2/H] + .W.Cp).[sin( z /H) + sin ( H 2/H] + .H / + (q”00.P.PHH.H / )lnrC0C0/r/rCi Ci + (q”

lbi lbi



π π .W.Cp).[sin( z /H) + sin ( H /2H] + .W.Cp).[sin( z /H) + sin ( H /2H] +



π π π π ) +1/h)cos( z/H) + T )ln(rC0/rCiCi) +1/h)cos( z/H) + T (X.10) lbi (X.10) lbi

π π (X.11) (X.11)

 TTCiCi = TΔ = TΔ CC + T + TC0C0(z)(z) = (q’/2λCC)lnr = (q’/2λ π π /h).cos( z /h) + T (q"(q"00 /h).cos( z /h) + T π π .H / = (q”00.P.PHH.H / = (q” q”0(rC0/λCC)ln(rC0/r q”0(rC0/λ  Equation (X.10) ca,n be expressed: Equation (X.10) ca,n be expressed: (z) = A + Bsin( z / H) + C  TTCiCi(z) = A + Bsin( z / H) + C

π πcos( z /H) CiCicos( z /H)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

(X.12) (X.12) = q”00(r(rC0C0/λ/λCC)ln(r

π π )arctyan(B/C )arctyan(B/C (X.13) (X.13)

CiCi))

Ci,Max = (q”

) + 1/h]2)1/2

 Where A & B are given by equation (X.3) and: Where A & B are given by equation (X.3) and: +1/h) )ln(rC0C0/r/rCiCi +1/h)  CCCiCi = q”  Using the same approach as in the case of the clad outer temperature, Using the same approach as in the case of the clad outer temperature, the location of the maximum temperature on the clad inner surface is the location of the maximum temperature on the clad inner surface is found as: found as:  zzCi,Max = (H/ Ci,Max = (H/  and the maximum corresponding temperature is: and the maximum corresponding temperature is: +  TTCi,Max lbi + lbi ((q” 

1/2 (X.14) (X.14)



.W.Cp) .W.Cp) π .H) / (ππ.W.Cp)sin( H/2H) + T π .W.Cp)sin( H/2H) + T + [q”00(r(rC0C0/λ/λCC)ln(r 22 + [q” = (q”00.P.PHH.H) / ( π π .H)/( ((q”00.P.PHH.H)/( )ln(rC0C0/r/rCiCi) + 1/h]2)



In a similar manner the fuel maximum temperature at the center of In a similar manner the fuel maximum temperature at the center of

the fuel pellet is given by: the fuel pellet is given by: π π (X.15) (X.15) (z) = A + Bsin ( z/H) + C TTFcFc(z) = A + Bsin ( z/H) + C

FcFccos( z/H)πcos( z/H)π

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 where where  CCFCFC = q” [(r

C0C0/r/rCiCi) + (r

) + (rC0C0/λ/λGG)ln(r ) + )ln(rG0G0/r/rGiGi) +



λ λ/ c)ln(r = q” [(rC00C00/ c)ln(r rrCoCo/2λ/2λFF + 1/h (X.16) + 1/h (X.16)

π π )arctan(B/C )arctan(B/C (X.17) (X.17)

FcFc))

π π .H/ = (q”00.P.PHH.H/

The maximum fuel temperature is located at: The maximum fuel temperature is located at: = (H/ zzFc,Max Fc,Max = (H/  and its value is: and its value is:  TTFC,Max FC,Max = (q”  π π .W.Cp).sin ( H/2H) + T .W.Cp).sin ( H/2H) + T π π

lbi + + lbi 00{(r{(rC0C0/λ/λCC)ln(r )ln(rC0C0/r/rCiCi) + (r

) + (rC0C0//λλGG)lnr )lnrG0G0/r/rGiGi



+ 1/h}]22))1/21/2.. .W.Cp)2 + [q” .W.Cp)2 + [q” (X.18) (X.18)

((q”0.PH.H/ ((q”0.PH.H/ +r+rC0C0/2λ/2λFF + 1/h}]  XI. Void fraction in boiling channels XI. Void fraction in boiling channels..   The characteristic feature of boiling channels is the presence of two The characteristic feature of boiling channels is the presence of two phases: the liquid and the vapor phase. phases: the liquid and the vapor phase.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Clearly, the presence of two phases changes the fluid flow and heat Clearly, the presence of two phases changes the fluid flow and heat transfer processes as compared to the nonboiling channels. In transfer processes as compared to the nonboiling channels. In addition, the density changes of coolant are more significant in boiling addition, the density changes of coolant are more significant in boiling channels due to the dramatic change of density once liquid transforms channels due to the dramatic change of density once liquid transforms into vapor. Thus, to be able to predict the local value of the coolant into vapor. Thus, to be able to predict the local value of the coolant density it is required to determine the local volume fraction of both density it is required to determine the local volume fraction of both phases. Typically, the void fraction (that is the volume fraction of the phases. Typically, the void fraction (that is the volume fraction of the vapor phase) is determined using various models, as described below. vapor phase) is determined using various models, as described below.  The various twophase flow and heat transfer regimes in a boiling The various twophase flow and heat transfer regimes in a boiling channel, such as BWR fuel assembly, is shown in figure( XI.1). channel, such as BWR fuel assembly, is shown in figure( XI.1).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure XI.1: Twophase flow & heat transfer regime in a boiling channel Figure XI.1: Twophase flow & heat transfer regime in a boiling channel ONB (Onset of Boiling), OSV (Onset of Significant Void, OAF (Onset of ONB (Onset of Boiling), OSV (Onset of Significant Void, OAF (Onset of Annular Flow). Annular Flow).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The heat exchange coefficient depends on the local properties of The heat exchange coefficient depends on the local properties of coolant flow, which evolve all along the hot channel. This coefficient is coolant flow, which evolve all along the hot channel. This coefficient is characterized by its FΔΔh and the coolant by: h and the coolant by: characterized by its F



Ti: the inlet temperature Ti: the inlet temperature g: the mass flow rate g: the mass flow rate Tsat: the saturation temperature Tsat: the saturation temperature X: the quality = steam mass/ mixture mass X: the quality = steam mass/ mixture mass αα: the void fraction = steam volume/mixture volume : the void fraction = steam volume/mixture volume * A first assumption is made that the channel is isolated & exchanges * A first assumption is made that the channel is isolated & exchanges

neither mass nor energy with neighboring channels. This hypothesis, neither mass nor energy with neighboring channels. This hypothesis, in fact highly penalizing, is not verified in a real PWR. in fact highly penalizing, is not verified in a real PWR.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 As the coolant rises along the channel (see following figure), its As the coolant rises along the channel (see following figure), its physical properties are modified because its temperature increases, physical properties are modified because its temperature increases, along with the temperature of the channel wall. The height of the along with the temperature of the channel wall. The height of the channel can be divided into a certain number of zones with different channel can be divided into a certain number of zones with different properties: properties: 1) A lower zone, in which the wall temperature and the coolant 1) A lower zone, in which the wall temperature and the coolant temperature are below the saturation temperature. In this zone, the temperature are below the saturation temperature. In this zone, the flow is single phase and the heat exchange regime is one of forced flow is single phase and the heat exchange regime is one of forced convection. convection.  The heat exchange between the cladding & the coolant is good & the The heat exchange between the cladding & the coolant is good & the temperature difference ΔΔT remains small, not exceeding tens of T remains small, not exceeding tens of temperature difference degrees. degrees.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 2) Starting from a certain length of the tube, the wall temperature 2) Starting from a certain length of the tube, the wall temperature exceeds the coolants saturate temperature, Tsat, whereas the coolant exceeds the coolants saturate temperature, Tsat, whereas the coolant remains at a temperature less than Tsat. Bubbles then begin to appear remains at a temperature less than Tsat. Bubbles then begin to appear along the cladding wall, while the coolant remains strongly under along the cladding wall, while the coolant remains strongly under saturation. These bubbles improve the thermal exchange, because saturation. These bubbles improve the thermal exchange, because they do not stuck to the wall but are carried along by coolant flow. they do not stuck to the wall but are carried along by coolant flow. Consequently they transmit calories from the wall to the coolant. Consequently they transmit calories from the wall to the coolant. 3) Since the coolant continues to heat up, the density & the size of 3) Since the coolant continues to heat up, the density & the size of the bubbles increase. Suddently, there is a coalescence of the bubbles the bubbles increase. Suddently, there is a coalescence of the bubbles and the creation of a stable vapor film along the cladding wall. From and the creation of a stable vapor film along the cladding wall. From this moment on, the heat exchange degenerates (h decreases, this moment on, the heat exchange degenerates (h decreases, Tcladding increases). Tcladding increases).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 This degraded heat exchange is explained, among other reasons, by This degraded heat exchange is explained, among other reasons, by the fact that steam has lower thermal conductivity than water. It occurs the fact that steam has lower thermal conductivity than water. It occurs when a certain value of thermal flux has been reached, and leads to when a certain value of thermal flux has been reached, and leads to « Departure from Nucleate Boiling » & burnout. « Departure from Nucleate Boiling » & burnout.  Burnout Burnout  corresponds to the forced convection regime. The fluid Zone 0 to 1: corresponds to the forced convection regime. The fluid Zone 0 to 1: is under the unsaturated liquid form (quality equal zero), its exchange is under the unsaturated liquid form (quality equal zero), its exchange coefficient is relatively constant at given mass flow. The temperature coefficient is relatively constant at given mass flow. The temperature difference between the surface and the center of the fluid flow is difference between the surface and the center of the fluid flow is proportional to calorific flux to be transferred. proportional to calorific flux to be transferred.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Zone 2

corresponds to nucleate boiling regime, which is interacting Zone 2: : corresponds to nucleate boiling regime, which is interacting because the sleep slope (large heatexchange coefficient) shows that a because the sleep slope (large heatexchange coefficient) shows that a considerable amount of heat is extracted by means of a small increase considerable amount of heat is extracted by means of a small increase in the wall temperature. The overheating of the liquid at the wall is in the wall temperature. The overheating of the liquid at the wall is sufficient to permit locally the creation of bubbles; the center of the sufficient to permit locally the creation of bubbles; the center of the fluid flow is being under the subsaturated liquid. fluid flow is being under the subsaturated liquid.  Theoretically, due to the forces of the superficial stresses, the over Theoretically, due to the forces of the superficial stresses, the over pressure and thus the overheating inside the bubbles of radius R is pressure and thus the overheating inside the bubbles of radius R is proportional to 1/R. proportional to 1/R.  When the R is nil or in order of value of intermolecular, the over When the R is nil or in order of value of intermolecular, the over heating is very great at which the prohibition of the bubble formation. heating is very great at which the prohibition of the bubble formation. Practically, on the wall there are germs permitting the bubbles to be Practically, on the wall there are germs permitting the bubbles to be created for the overheating very limited to some degrees. created for the overheating very limited to some degrees.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Since the apparition of the bubbles, there is stabilization of the Since the apparition of the bubbles, there is stabilization of the temperature at the wall which became less dependent of the heating temperature at the wall which became less dependent of the heating flux. There is nucleate boiling. However the mean temperature of the flux. There is nucleate boiling. However the mean temperature of the se fluid remains inferior to the saturation, these bubbles after being se fluid remains inferior to the saturation, these bubbles after being separated from the wall, are recondensate in the center of the fluid separated from the wall, are recondensate in the center of the fluid flow. flow. Cladding temperature = Tsat + ΔΔtsat tsat Φ Φ 0.250.25 e e p/p

p/p. .

 Cladding temperature = Tsat +  With With ΔΔtsat = k tsat = k  The difference between the cladding temperature and the fluid The difference between the cladding temperature and the fluid temperature does not vary linearily with the heat flux ΦΦ; thus allows ; thus allows temperature does not vary linearily with the heat flux the passing to higher flux ΦΦ without excessive increasing of the wall without excessive increasing of the wall the passing to higher flux temperature wich depends essentially to the pressure. Per example, at temperature wich depends essentially to the pressure. Per example, at fixed pressure, the increase of 100 % of the flux ΦΦ induces an increase induces an increase fixed pressure, the increase of 100 % of the flux of 20 % of the ΔΔtsat. tsat. of 20 % of the

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



; the mean temperature of the fluid has reach the In the zone 3, 4, 5; the mean temperature of the fluid has reach the In the zone 3, 4, 5 ˃ saturation (αα 0), we are in boiling phase. The precedent correlation ˃ 0), we are in boiling phase. The precedent correlation saturation ( still applicable because the wall temperature is always less dependent still applicable because the wall temperature is always less dependent of the heat flux ΦΦ.. of the heat flux Zone 3: This boiling is done always by the creation of the bubbles at Zone 3: This boiling is done always by the creation of the bubbles at the wall, with a flow in liquid phase containing the bubbles of vapor the wall, with a flow in liquid phase containing the bubbles of vapor more or less important. Starting at point 3 in the curve, which is the more or less important. Starting at point 3 in the curve, which is the critical temperature, the exchange of heat becomes less efficient. critical temperature, the exchange of heat becomes less efficient. Although the heat continues to be removed, the temperature rises Although the heat continues to be removed, the temperature rises quickly, which risks damaging the cladding material (as well as the quickly, which risks damaging the cladding material (as well as the oxide pellets). Furthermore, this zone is unstable, with oscillations of oxide pellets). Furthermore, this zone is unstable, with oscillations of the flow rate and the temperature. The regime in this zone is called the flow rate and the temperature. The regime in this zone is called « transition boiling » (region 3) « transition boiling » (region 3)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



There is some separation of liquid and vapor phase, the Zone 4: There is some separation of liquid and vapor phase, the Zone 4: vapor phase is concentrated at the center mixed to the droplets of vapor phase is concentrated at the center mixed to the droplets of liquid, the liquid phase forming a film wetting the wall. liquid, the liquid phase forming a film wetting the wall.

As the thickness of the vapor film decreases the exchange at the As the thickness of the vapor film decreases the exchange at the



wall is done by the direct vaporization of the film and the number of wall is done by the direct vaporization of the film and the number of the bubbles created at the wall is decreasing. the bubbles created at the wall is decreasing.



In that zone, there is essentially vaporization by nucleate boiling In that zone, there is essentially vaporization by nucleate boiling

by stabilization of temperature at the wall slightly above the saturation by stabilization of temperature at the wall slightly above the saturation temperature. temperature. Zone 5: The liquid at the wall is disappeared, inducing the brutal Zone 5: The liquid at the wall is disappeared, inducing the brutal degradation of the exchange coefficient. One passes by nucleate degradation of the exchange coefficient. One passes by nucleate boiling by liquid film (the difference temperature at the wall has boiling by liquid film (the difference temperature at the wall has tendency to decrease). tendency to decrease).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



: The liquid at the wall is disappeared, inducing the brutal Zone 6: The liquid at the wall is disappeared, inducing the brutal Zone 6 degradation of the exchange coefficient. One passes by liquid film to degradation of the exchange coefficient. One passes by liquid film to the vaporization of the vapor film. There is dryout of the wall. the vaporization of the vapor film. There is dryout of the wall. However, it remains liquid in gaseous core. The vapor is still very However, it remains liquid in gaseous core. The vapor is still very

Beyond the point 4, stable film boiling is achieved, with forced Beyond the point 4, stable film boiling is achieved, with forced convection in a single phase vapor flow. The value q’’c corresponding convection in a single phase vapor flow. The value q’’c corresponding to the point 3 is thus a threshold value, beyond which there is a risk of to the point 3 is thus a threshold value, beyond which there is a risk of dangerous high wall temperatures. This is therefore called « the dangerous high wall temperatures. This is therefore called « the Critical flux ». Theoretically this is a limiting value never to be Critical flux ». Theoretically this is a limiting value never to be exceeded under normal operating conditions. As we will see later, exceeded under normal operating conditions. As we will see later, however, in PWR a rule of always remaining well below this critical however, in PWR a rule of always remaining well below this critical value is imposed. value is imposed.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 The dryout and the burnout could induce the degradation of the The dryout and the burnout could induce the degradation of the cladding, or the partial fusion of the nuclear fuel pellets. cladding, or the partial fusion of the nuclear fuel pellets. In the simplest twophase flow model it is assumed that both phases In the simplest twophase flow model it is assumed that both phases are in the thermodynamic equilibrium and that they move with the are in the thermodynamic equilibrium and that they move with the same velocity. These assumptions are the basis of the Homogeneous same velocity. These assumptions are the basis of the Homogeneous Equilibrium Model (HEM), in which the local, channelaverage void Equilibrium Model (HEM), in which the local, channelaverage void fraction is determined from the corresponding local value of the fraction is determined from the corresponding local value of the equilibrium thermodynamic quality. equilibrium thermodynamic quality.  XI.1. Homogeneous Equilibrium Model (HEM) XI.1. Homogeneous Equilibrium Model (HEM)  The HEM expression for the void fraction takes the following form: The HEM expression for the void fraction takes the following form: for xe 0˂ for xe 0˂ 



α α ˂ ˂ ˂ ˂ for 0 xe 1 (XI.1) for 0 xe 1 (XI.1)

α α = 0 = 0 ρ ρ ρ ρ = 1/ 1 + g/ = 1/ 1 + g/ α α ˃ ˃ f x (1 – xe)/xe) f x (1 – xe)/xe) = 1 for xe 1 = 1 for xe 1

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Where



is the equilibrium thermodynamic quality, which is Where xexe is the equilibrium thermodynamic quality, which is determined from the energy balance of the coolant in the heated determined from the energy balance of the coolant in the heated channel. channel.  Equation (XI.1) strongly overpredicts the coolant density (that is it Equation (XI.1) strongly overpredicts the coolant density (that is it gives a higher value than the actual one) in the region of subcooled gives a higher value than the actual one) in the region of subcooled boiling, since it assumes only liquid, whereas in reality both the liquid boiling, since it assumes only liquid, whereas in reality both the liquid and the vapor coexist in that region. and the vapor coexist in that region.  XI.2. Driftflux model XI.2. Driftflux model  Once applying the Driftflux model, the void fraction is found as: Once applying the Driftflux model, the void fraction is found as: α α = Jv / C = Jv / C  (XI.2) (XI.2) J + Uvjvj 00J + U

Equation (XI.3) expresses the crosssection mean void fraction Equation (XI.3) expresses the crosssection mean void fraction α α

, total in terms of channel mean superficial velocity of gas, JvJv, total in terms of channel mean superficial velocity of gas, superficial velocity, JJ, and two parameters, superficial velocity, , and two parameters, C0C0 and and UvjUvj..

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 The first parameter is the socalled driftflux distribution parameter and The first parameter is the socalled driftflux distribution parameter and is simply a covariance coefficient for crosssection distributions of is simply a covariance coefficient for crosssection distributions of void fraction and total superficial velocity. The second coefficient is void fraction and total superficial velocity. The second coefficient is the socalled drift velocity and can be interpreted as crosssection the socalled drift velocity and can be interpreted as crosssection averaged difference between the gas velocity and the superficial averaged difference between the gas velocity and the superficial velocity, using local the void fraction as a weighting function. velocity, using local the void fraction as a weighting function. The driftflux parameters are not constant and depend on flow The driftflux parameters are not constant and depend on flow conditions. Table XI.1 gives expressions for driftflux parameters, conditions. Table XI.1 gives expressions for driftflux parameters, which are valid in a wide range of flow conditions. which are valid in a wide range of flow conditions.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Table XI.1: Distribution parameter & drift velocity used drift flux model.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 XI.3. Subcooled boiling region XI.3. Subcooled boiling region  It is commonly accepted that a significant void fraction in a boiling It is commonly accepted that a significant void fraction in a boiling channel appears at locations where bubbles depart from heated walls. channel appears at locations where bubbles depart from heated walls. The void fraction between that point, referred often as the Onset of The void fraction between that point, referred often as the Onset of Significant Void fraction (OSV) point, and the Significant Void fraction (OSV) point, and the  Onset of Nucleate Boiling (ONB) point is very small and can be Onset of Nucleate Boiling (ONB) point is very small and can be neglected. To establish the location of the OSV point it is neglected. To establish the location of the OSV point it is recommended to use a correlation proposed by Saha and Zuber (1974), recommended to use a correlation proposed by Saha and Zuber (1974), which states that OSV point is located at such position in a channel, which states that OSV point is located at such position in a channel, where the local equilibrium quality is as follows: where the local equilibrium quality is as follows:

˂Pe 70000 (XI.3) Pe 70000 (XI.3) ˃ ˃ for Pe 70000 for Pe 70000 = 0.0022 q”.Dh.Cpfpf / 1 / 1fgfg.λ.λff for for xxe,OSV e,OSV = 0.0022 q”.Dh.C = 154q”/G.ifgfg e,OSV = 154q”/G.i  xxe,OSV

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

where Pe is the Peclet number defined as follows:  where Pe is the Peclet number defined as follows: 

Pe = G.Dh.Cpff / λ / λff Pe = G.Dh.Cp (XI.4) (XI.4)

 For an uniform heat flux distribution, the location of the OSV point is For an uniform heat flux distribution, the location of the OSV point is given from the energy balance as: given from the energy balance as:

(XI.5) (XI.5) / q”.PHH]] = (xe,OSV )[W.ifgfg / q”.P

e,OSV x xee)[W.i

z=zNVG, where = 0 and also , where x x = 0 and also

α (XI.6) (XI.6)

[xe(z)/xe(zOSV) – 1] zOSV).e).e[xe(z)/xe(zOSV) – 1]

 zzOSVOSV = (x  Several models have been proposed to predict the flow quality Several models have been proposed to predict the flow quality downstream of the OSV point. Levy (1966) proposed a fitting downstream of the OSV point. Levy (1966) proposed a fitting relationship, which satisfy a condition at z=zNVG relationship, which satisfy a condition at which will predict the flow quality to approach the equilibrium quality which will predict the flow quality to approach the equilibrium quality is increasing downstream of the OSV point. The Levy’s when z z is increasing downstream of the OSV point. The Levy’s when relationship is as follows: relationship is as follows: αX (z) = x ee((zOSV  X (z) = x

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



β β (XI.7) (XI.7)

 Having the flow quality given by Eq. (457), one can apply the general Having the flow quality given by Eq. (457), one can apply the general drift flux model to calculate the void fraction distribution. The drift flux model to calculate the void fraction distribution. The recommended expression for thedistribution parameter for sub recommended expression for thedistribution parameter for sub cooled boiling is as follows: cooled boiling is as follows: β bb]] β ) [1 + (1/ = CC00 = ) [1 + (1/

 WhereWhere 



(XI.8) (XI.8) .(1 – xαα(z)/x (z)/xαα(z)(z)



(XI.9) (XI.9) Β ρgg/ρ/ρff.(1 – x Β ρ = 1/(1 + ( = 1/(1 + ( B = (ρgg/ρ/ρff))0.10.1 B = (ρ

The recommended by Lahey and Moody (1977) drift velocity is as The recommended by Lahey and Moody (1977) drift velocity is as



follows: follows:

22)})}0.250.25



(XI.10) (XI.10) UUvjvj = 2.9{σ = 2.9{σgg(ρ(ρff ρ ρgg) /ρ) /ρff

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 XII. Heat transfer to coolant XII. Heat transfer to coolant  XII.1. Single phase XII.1. Single phase  The heat transfer coefficient h for coolant flow in a rod bundle is The heat transfer coefficient h for coolant flow in a rod bundle is calculated from the Nusselt number Nu as follows,: calculated from the Nusselt number Nu as follows,:



H = Nu. H = Nu. (XII.1) (XII.1)



λ λ / Dh / Dh  where < is the fluid thermal conductivity and Dh is the bundle where < is the fluid thermal conductivity and Dh is the bundle hydraulic diameter. hydraulic diameter.  For laminar flow far from the inlet to a channel, the Nusselt number is For laminar flow far from the inlet to a channel, the Nusselt number is as follows: as follows:



Nu = 4.364 Nu = 4.364 (XII.2) (XII.2)



In the inlet region of the channel the following expression is valid: In the inlet region of the channel the following expression is valid:

ζ ζ ζ ζ ˂ ζ ˂ ˂ ζ ˂ Nu = 1.31 (1 + 2 ) /. 3/2 (0 Nu = 1.31 (1 + 2 ) /. 3/2 (0 0.4) 0.4) (XII.3) (XII.3)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 WhereWhere 



ζζ = (z / Dh)Pe = (z / Dh)Pe Pe = UDh/a Pe = UDh/a (XII.4) (XII.4) (XII.5) (XII.5)

For turbulent flow in a pipe the Nusselt number can be calculated For turbulent flow in a pipe the Nusselt number can be calculated



from the Dittus Boelter correlation: from the Dittus Boelter correlation:



 Where Pr is the Prandtl number; Pr =

Nu = 0.023Re0.80.8 Pr Prnn Nu = 0.023Re (XII.6) (XII.6)

, and is the kinematic viscosity Where Pr is the Prandtl number; Pr = /a/a, and is the kinematic viscosity = 0.4 for of liquid. The formula is valid for Re > 104 and 0.7 < Pr < 100, n n = 0.4 for of liquid. The formula is valid for Re > 104 and 0.7 < Pr < 100, = 0.3 for fluid cooling. Petukhov [46] proposed the fluid heating and n n = 0.3 for fluid cooling. Petukhov [46] proposed the fluid heating and following semiempirical expression for the Nusselt number for following semiempirical expression for the Nusselt number for turbulent flow in pipes: turbulent flow in pipes:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

(Pr2/32/3 1)) 1)) /2)RePr) /(1 +13.6Cf,pf,p + (11.7 +1.8Pr + (11.7 +1.8Pr 1/31/3)(C)(Cf,pf,p/2)/2)1/21/2 (Pr

 For rod bundles with triangular lattice and with 1.1 <

Nu = ((Cf,pf,p/2)RePr) /(1 +13.6C  Nu = ((C (XII.7) (XII.7) Where : C  Where : C (XII.8) (XII.8) = 0.25(1.82lg10Re – 1.64)22 f,pf,p = 0.25(1.82lg10Re – 1.64)

](p/d)0.150.15}Re}Re0.80.8 Pr Pr0.40.4

< 1.8 Ushakov For rod bundles with triangular lattice and with 1.1 < p/d p/d < 1.8 Ushakov (presented in [47]) proposed the following correlation: (presented in [47]) proposed the following correlation: Nu = {0.0165 + 0.02[1 – 0.91/(p/d)22](p/d)  Nu = {0.0165 + 0.02[1 – 0.91/(p/d) (XII.9) (XII.9)  where the correlation is valid for 5c103 < Re < 5c105 and 0.7 < Pr < 20. where the correlation is valid for 5c103 < Re < 5c105 and 0.7 < Pr < 20.  Similar correlation was derived by Weissman[414], Similar correlation was derived by Weissman[414],  Nu = CRe0.8Pr3/23/2 Nu = CRe0.8Pr (XII.10) (XII.10)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 WhereWhere 

= 0.026(p/d) < 1.5, – 0.024 for triangular lattices with 1.1 < p/d p/d < 1.5,

(p/d) – 0.024 for square lattices with 1.1 < C C = 0.026 = 0.042(p/d) (p/d) – 0.024 for triangular lattices with 1.1 < < 1.3. – 0.024 for square lattices with 1.1 < p/d p/d < 1.3.

 C C = 0.042  Subbotin et al. recommended for heat transfer to liquids flowing in a Subbotin et al. recommended for heat transfer to liquids flowing in a bundle with triangular lattice the following correlation: bundle with triangular lattice the following correlation:



Nu = A Re0.5 0.5 PrPr0.40.4 Nu = A Re

 Where: A = 0.0165 + 0.02[1 (0.91/(p/d)  The correlation is valid for 1.1 <

](p/d)0.150.15 (XII.11) (XII.11) (XII.12) (XII.12)

Where: A = 0.0165 + 0.02[1 (0.91/(p/d) 22](p/d) < 1.8, 1.0 < Pr < 20 and 5.103 < Re < The correlation is valid for 1.1 < p/d p/d < 1.8, 1.0 < Pr < 20 and 5.103 < Re < 5.105. 5.105.  For gas flow in tight rod bundles Ajn and Putjkov give: For gas flow in tight rod bundles Ajn and Putjkov give: 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Nu Nu bundle 

(XII.13) (XII.13)

= 1.184 + 0.35.lg(p/d 1) bundle/Nu/NuDBDB = 1.184 + 0.35.lg(p/d 1) < 2.4 and DB Nu is found The correlation is valid for 1.03 < p/d p/d < 2.4 and DB Nu is found The correlation is valid for 1.03 <



from the DittusBoelter correlation given by Eq. (XII.6). from the DittusBoelter correlation given by Eq. (XII.6).  Markoczy performed a study of experimental data obtained in 63 rod Markoczy performed a study of experimental data obtained in 63 rod bundles with different geometry details and proposed the following bundles with different geometry details and proposed the following relationship: relationship:

= 1 + 0.91Re 0.10.1 Pr Pr0.40.4(1 2e (1 2eBB)) (XII.14) (XII.14) NuNubundle

bundle/Nu/NuDBDB = 1 + 0.91Re

 WhereWhere 

B = (2x3 1/21/2//ππ)(p/d) B = (2x3 )(p/d)2 2 for triangular lattice for triangular lattice



(XII.15) (XII.15)

π π B = 4/ B = 4/ (p/d) (p/d)

22 1 1



for square lattice for square lattice

Here again DB Nu is found from the DittusBoelter correlation Here again DB Nu is found from the DittusBoelter correlation

given by Eq. (467). given by Eq. (467).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

ε ε ε ε ε ε ε ε

 The correlation is applicable in the following range of parameters: The correlation is applicable in the following range of parameters:  3c103 < Re < 106 3c103 < Re < 106  0.66 < Pr < 5 0.66 < Pr < 5  1.02 < < 2.5. 1.02 < p/d p/d < 2.5.  Another approach was proposed by Osmachkin [44], who Another approach was proposed by Osmachkin [44], who recommended to calculate the Nusselt number from correlations which recommended to calculate the Nusselt number from correlations which are valid for pipes, replacing however the hydraulic diameter with the are valid for pipes, replacing however the hydraulic diameter with the “effective”: “effective”: ε ε  Deff = 2 /(1 – )2{ /2 – 3/2 –ln /1 – }Dh Deff = 2 /(1 – )2{ /2 – 3/2 –ln /1 – }Dh

(XII.16) (XII.16)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



˂ ˂

 h = 5.5p  H = 0.577p

˂ ˂ ε ε  where is the fraction of the crosssection of the bundle which is where is the fraction of the crosssection of the bundle which is occupied by rods. occupied by rods.  The formula is applicable for rod bundles with triangular lattice and for The formula is applicable for rod bundles with triangular lattice and for > 1.3. p/d p/d > 1.3.  XIII. TwoPhase flow XIII. TwoPhase flow  Heat transfer coefficient for twophase boiling flow can be predicted Heat transfer coefficient for twophase boiling flow can be predicted from various correlations, for example from the Jens Lottes from various correlations, for example from the Jens Lottes (subcooled boiling) and the Chen (saturated boiling) correlations, (subcooled boiling) and the Chen (saturated boiling) correlations, described in [41]. described in [41].  A simple estimation of the boiling heat transfer coefficient can be A simple estimation of the boiling heat transfer coefficient can be obtained from a correlation proposed by Rasohin [48], obtained from a correlation proposed by Rasohin [48], h = 5.5p0.250.25(q”)2x(3) (q”)2x(3)1/21/2 H = 0.577p1.331.33(q”)2x(3) ˂ ˂ for 0.1 p 8 for 0.1 p 8 ˂ ˂ (q”)2x(3)1/21/2 for 8 p 20 (XIII.1) for 8 p 20 (XIII.1)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 where



is pressure [MPa] and is heat transfer coefficient [W/m2K], p p is pressure [MPa] and

where h h is heat transfer coefficient [W/m2K], is heat flux [W/m2]. q’’ q’’ is heat flux [W/m2].  XIV. Pressure drops XIV. Pressure drops   Calculation of pressure drops in a reactor core is important since they Calculation of pressure drops in a reactor core is important since they influence the flow distribution in subchannels and thus affect the local influence the flow distribution in subchannels and thus affect the local thermal margins. In addition, the total pressure drop over the coolant thermal margins. In addition, the total pressure drop over the coolant circulation loop has to be known in order to determine the needed circulation loop has to be known in order to determine the needed pumping power. pumping power.  XIV.1. Singlephase flows XIV.1. Singlephase flows  One can identify several mechanisms that will cause a pressure drop One can identify several mechanisms that will cause a pressure drop along the fuel assembly: along the fuel assembly:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



Δ Δ Δ Δ ρ ρ i}G2/2 + i}G2/2 + Σ ζ Σ ζ i i

is the Fanning friction coefficient, L L is the length of the is the length of the is the channel hydraulic diameter and is the massflux, Dh Dh is the channel hydraulic diameter and

* Friction losses from the fuel rod bundle * Friction losses from the fuel rod bundle * Local loses from the spacer grids * Local loses from the spacer grids *. Local loses at the core inlet and exit (contraction and expansion) *. Local loses at the core inlet and exit (contraction and expansion) *. Elevation pressure drop *. Elevation pressure drop  The total pressure drop in a channel with a constant crosssection The total pressure drop in a channel with a constant crosssection area can be area can be  calculated from the following equation: calculated from the following equation: Δ Δ Δ Δ  ptotal = pfric + ploc + pelev = {(4CfL / Dh) + ptotal = pfric + ploc + pelev = {(4CfL / Dh) + φ ρ φ ρ (XIV.1) L gsin (XIV.1) L gsin  Where Where Cf Cf is the Fanning friction coefficient, channel, G G is the massflux, channel, is the coolant density. is the coolant density.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

(XIV.2) (XIV.2)

are constants, which for the laminar flow in a pipe are and b b are constants, which for the laminar flow in a pipe are

 The friction coefficient for laminar flow can be written in a general form The friction coefficient for laminar flow can be written in a general form as,as,  Cf = a.Reb Cf = a.Reb  where where a a and equal to 16 and 1, respectively. For laminar flow in rod bundles, equal to 16 and 1, respectively. For laminar flow in rod bundles, Osmachkin proposed to use equations. (XIV.1) and (XIV.2)), where Osmachkin proposed to use equations. (XIV.1) and (XIV.2)), where is replaced with an “effective” diameter given hydraulic diameter Dh Dh is replaced with an “effective” diameter given hydraulic diameter as:as:



ε ε ε ε ε ε ε ε ε ε Deff = 2 /(1 – )2{ /2 – 3/2 –ln /1 – }Dh (XIV.3) Deff = 2 /(1 – )2{ /2 – 3/2 –ln /1 – }Dh (XIV.3)

ε ε

 where is the fraction of the crosssection of the bundle which is where is the fraction of the crosssection of the bundle which is occupied by rods. occupied by rods.  The formula is applicable for rod bundles with triangular lattice and for The formula is applicable for rod bundles with triangular lattice and for > 1.3.For turbulent flow the coefficients in equation. (XIV.2) are p/d p/d > 1.3.For turbulent flow the coefficients in equation. (XIV.2) are obtained experimentally. For flow in a rod bundle with triangular lattice obtained experimentally. For flow in a rod bundle with triangular lattice < 1.5, the Fanning friction coefficient can (see figure VI.1) and 1.0 < p/d p/d < 1.5, the Fanning friction coefficient can (see figure VI.1) and 1.0 < be calculated as: be calculated as:



THERMAL-HYDRAULIC IN NUCLEAR REACTOR

(XIV.4) (XIV.4)

f,b = {0.96(p/d) + 0.63)C

 CCf,b  where

= {0.96(p/d) + 0.63)Cf,pf,p



is the friction factor proposed by Filonenko [47], which is where Cf,p Cf,p is the friction factor proposed by Filonenko [47], which is valid for tubes and annuli for Re > 4000: valid for tubes and annuli for Re > 4000:

= 0.25(1.82lg10Re – 1.64)22(XIV.5) (XIV.5) CCf,pf,p = 0.25(1.82lg10Re – 1.64)

 For rod assemblies Aljoshin et al. [45] proposed a general correlation For rod assemblies Aljoshin et al. [45] proposed a general correlation as follows: as follows:



0.25 /Ar)mmReRe0.25

 where

(XIV.6) (XIV.6) = Ax Pw,ch w,ch /PW,r (A /PW,r (Achch /Ar)

Pw,ch and CCf f = Ax P and Pw,r

 The formula is valid for rod bundles with triangular lattice, for which

– are the wetted perimeters of the channel and Pw,r – are the wetted perimeters of the channel and – are the crosssection areas of the and Ar Ar – are the crosssection areas of the

< Re < 1055, and for rectangular lattice, for , and for rectangular lattice, for

= 0.45 and 1033 < Re < 5∙10 < Re < 5∙1055.. where Pw,ch rods, respectively; Ach Ach and rods, respectively; channel and rods, respectively. channel and rods, respectively. The formula is valid for rod bundles with triangular lattice, for which A A = 0.47, m m = 0.35 and 4∙10 = 0.47, which A A = 0.38, which = 0.35 and 4∙1033 < Re < 10 = 0.38, m m = 0.45 and 10



(XIV.7) (XIV.7)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Additional pressure losses are associated with spacer grids, the  Additional pressure losses are associated with spacer grids, the coolant inlet and exit of the bundle as well as the sudden area changes coolant inlet and exit of the bundle as well as the sudden area changes of the bundle cross-section area. Such losses are classified as the of the bundle cross-section area. Such losses are classified as the local pressure losses and are calculated according to the following local pressure losses and are calculated according to the following general expression: general expression: ΔpΔplocloc = ζ

/2ρ) = ζlocloc (G (G22 /2ρ) x is the local pressure loss coefficient. loc x is the local pressure loss coefficient.



where loc  where The local loss coefficient for grid spacers is in general dependent on  The local loss coefficient for grid spacers is in general dependent on the spacer geometry and is usually determined in an experimental way. the spacer geometry and is usually determined in an experimental way. Typical spacer loss coefficient is expressed as: Typical spacer loss coefficient is expressed as:

(XIV.8) (XIV.8)



a are constants determined experimentally. and c c a are constants determined experimentally.

= a + b.Re-c ζζspacer spacer = a + b.Re-c where aa, , b b and  where For sudden enlargement and contraction of the channel, the local  For sudden enlargement and contraction of the channel, the local pressure losses can be calculated according procedures described in pressure losses can be calculated according procedures described in [4-1[4-1].].

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



) + r44LρLρllgsinφ + r ((4L/D)(G22/2ρ/2ρll) + r (G2/ρl) gsinφ + r22(G2/ρl)

(XIV.9) (XIV.9)

NN Ф Ф22

loc,i



are two-phase pressure drop multipliers and r4 r4 are two-phase pressure drop multipliers

XIV.2. Two-phase flows  XIV.2. Two-phase flows Pressure drop in fuel assemblies with two-phase flow can be  Pressure drop in fuel assemblies with two-phase flow can be calculated according to the procedures described in [VI-1], using the calculated according to the procedures described in [VI-1], using the hydraulic diameter as described by equations. (VI.1) and (VI.4) with hydraulic diameter as described by equations. (VI.1) and (VI.4) with some modifications appropriate to the fuel assembly design. As some modifications appropriate to the fuel assembly design. As shown in [4-1], the total two-phase flow pressure drop in a channel shown in [4-1], the total two-phase flow pressure drop in a channel with a constant cross-section area can be calculated as: with a constant cross-section area can be calculated as: -Δp = r33CCf,loc f,loc ((4L/D)(G -Δp = r + {Σ+ {Σi=1 loc,i ζζii}} i=1 where r2r2, , r3 r3 and  where x are (acceleration, friction and gravitation, respectively) and 2 lolo,,i i f , f , iix are (acceleration, friction and gravitation, respectively) and 2 local loss multiplier and local pressure loss coefficient, respectively, at local loss multiplier and local pressure loss coefficient, respectively, at location i in the channel. location i in the channel.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



XV. Critical Heat Flux (CHF))  XV. Critical Heat Flux (CHF  The conditions at which the wall temperature rises and the heat  The conditions at which the wall temperature rises and the heat transfer decreases sharply due to a change in the heat transfer transfer decreases sharply due to a change in the heat transfer mechanism are termed as the mechanism are termed as the Flux (CHF) conditions. The nature of CHF, and thus the change of heat  Flux (CHF) conditions. The nature of CHF, and thus the change of heat transfer mechanism, varies with the enthalpy of the flow. transfer mechanism, varies with the enthalpy of the flow. At sub-cooled conditions and low qualities this transition At sub-cooled conditions and low qualities this transition corresponds to a change in boiling mechanism from nucleate to film corresponds to a change in boiling mechanism from nucleate to film boiling. For this reason the CHF condition for these circumstances is boiling. For this reason the CHF condition for these circumstances is usually referred to as the Departure from Nucleate Boiling (DNB). usually referred to as the Departure from Nucleate Boiling (DNB).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR At saturated conditions, with moderate and high qualities, the flow  At saturated conditions, with moderate and high qualities, the flow pattern is almost invariably in an annular configuration. In these pattern is almost invariably in an annular configuration. In these conditions the change of the heat transfer mechanism is associated conditions the change of the heat transfer mechanism is associated with the evaporation and disappearance of the liquid film and the with the evaporation and disappearance of the liquid film and the transition mechanism is termed as dry-out. Once dry-out occurs, the transition mechanism is termed as dry-out. Once dry-out occurs, the flow pattern changes to the liquid-deficient region, with a mixture of flow pattern changes to the liquid-deficient region, with a mixture of vapor and entrained droplets. It is worth noting that due to high vapor vapor and entrained droplets. It is worth noting that due to high vapor velocity the heat transport from heated wall to vapor and droplets is velocity the heat transport from heated wall to vapor and droplets is quite efficient, and the associated increase of wall temperature is not quite efficient, and the associated increase of wall temperature is not as dramatic as in the case of DNB. as dramatic as in the case of DNB.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The mechanisms responsible for the occurrence of CHF (DNB- and  The mechanisms responsible for the occurrence of CHF (DNB- and dry-outtype) are not fully understood, even though a lot of effort has dry-outtype) are not fully understood, even though a lot of effort has been devoted to this topic. Since no consistent theory of CHF is been devoted to this topic. Since no consistent theory of CHF is available, the predictions of CHF occurrence relay on correlations available, the predictions of CHF occurrence relay on correlations obtained from specific experimental data. LWR fuel vendors perform obtained from specific experimental data. LWR fuel vendors perform their own measurements of CHF in full-scale mock-ups of fuel their own measurements of CHF in full-scale mock-ups of fuel assemblies. Based on the measured data, proprietary CHF correlations assemblies. Based on the measured data, proprietary CHF correlations are developed. As a rule, such correlations are limited to the same are developed. As a rule, such correlations are limited to the same geometry and the same working conditions as used in experiments. geometry and the same working conditions as used in experiments. Most research on CHF published in the open literature has been  Most research on CHF published in the open literature has been performed for upward flow boiling of water in uniformly heated tubes. performed for upward flow boiling of water in uniformly heated tubes. The overall experimental effort in obtaining CHF data is enormous. It is The overall experimental effort in obtaining CHF data is enormous. It is estimated that several hundred thousand CHF data points have been estimated that several hundred thousand CHF data points have been obtained in different labs around the world. obtained in different labs around the world.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR More than 200 correlations have been developed in order to correlate  More than 200 correlations have been developed in order to correlate the data. the data. Discussion of all such correlations is not possible; however, some  Discussion of all such correlations is not possible; however, some examples will be described in this section. examples will be described in this section. XV.1. Departure from Nucleate Boiling (DNB)  XV.1. Departure from Nucleate Boiling (DNB) 



The usual form of a DNB correlation is as follows: The usual form of a DNB correlation is as follows: q”critical = q”critical (G,p,Dh, L, …) q”critical = q”critical (G,p,Dh, L, …) (XV.1) (XV.1)

, as well as the hydraulic diameter, , pressure, pp, as well as the hydraulic diameter,

of the heated channel. and length L L of the heated channel.

Which means that the main parameters that influence the occurrence  Which means that the main parameters that influence the occurrence of DNB are mass flux, GG, pressure, of DNB are mass flux, Dh Dh and length For upflow boiling of water in vertical 8-mm tubes with constant heat  For upflow boiling of water in vertical 8-mm tubes with constant heat flux, Levitan and Lantsman recommended the following correlation for flux, Levitan and Lantsman recommended the following correlation for DNB: DNB:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



](G/1000)1.2{[0.25(p – 98] – xe}e-1.5xe -1.5xe = [10.3 – 7.8(p/98) + 1.6(p/98)22](G/1000)

1.2{[0.25(p – 98] – xe}e

critical = [10.3 – 7.8(p/98) + 1.6(p/98)

q”q”critical



(XV.2) (XV.2) ¢¢ is the critical heat flux [MW m-2], p p is the pressure in where cr qcr q¢¢ is the critical heat flux [MW m-2], is the pressure in  where is the mass flux in [kg m-2 s-1]. The correlation is valid in [bar], G G is the mass flux in [kg m-2 s-1]. The correlation is valid in [bar], < 5000 [kg m-2 s-1] and is < 196 [bar] and 750 < G G < 5000 [kg m-2 s-1] and is ranges 29.4 < p p < 196 [bar] and 750 < ranges 29.4 < accurate to ±15%. accurate to ±15%. The correlation can be applied to channels with arbitrary diameters if  The correlation can be applied to channels with arbitrary diameters if the following correction factor is applied: the following correction factor is applied:

x (8 / D)0.50.5 (XV.3) (XV.3)

critical (8mm) x (8 / D)

 where

Q”critical = q”critical (8mm) Q”critical = q”



is the tube diameter in [mm] and cr mm q ¢¢ is the critical heat cr mm q¢¢ is the critical heat

where D D is the tube diameter in [mm] and from Eq. (XV.2). flux obtained from Eq. (XV.2). flux obtained There are several semi-empirical correlation used by reactor core There are several semi-empirical correlation used by reactor core designers such as: Westingouse (W3 & WRB1); GE; Babcock & designers such as: Westingouse (W3 & WRB1); GE; Babcock & Wilcock; CE,ect.. Wilcock; CE,ect..

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



XV.2. Dry-out  XV.2. Dry-out The usual form of a “dryout" correlation is as given:  The usual form of a “dryout" correlation is as given:



(XV.4) (XV.4) (G,p,Dh, L, …) critical (G,p,Dh, L, …) xxcritical critical = x = xcritical

, pressure, pp, hydraulic diameter

](G/1000)-0.5 -0.5 = [0.39 + 1.57(p/98) – 2.04(p/98)22 + 0.68(p/98) + 0.68(p/98)33](G/1000)

is the critical quality, p p is the pressure in [bar] and

which means that the main parameters that influence the occurrence  which means that the main parameters that influence the occurrence , boiling , hydraulic diameter DhDh, boiling of dryout are mass flux, GG, pressure, of dryout are mass flux, length (that it the distance from the beginning of saturated flow to the length (that it the distance from the beginning of saturated flow to the , and possibly other. dryout point), LBLB, and possibly other. dryout point), For dryout predictios in 8-mm pipes Levitan and Lantsman  For dryout predictios in 8-mm pipes Levitan and Lantsman recommended the following expression: recommended the following expression:  xxcritical (8mm) critical (8mm) = [0.39 + 1.57(p/98) – 2.04(p/98) (XV.5) (XV.5) is the is the pressure in [bar] and G G is the where xcr xcr is the critical quality,  where mass flux in [kg m-2 s-1]. The application region of the correlation is mass flux in [kg m-2 s-1]. The application region of the correlation is < 3000 [kg m-2 s-1] and the accuracy < 166.6 [bar] and 750 < G G < 3000 [kg m-2 s-1] and the accuracy 9.8 < p p < 166.6 [bar] and 750 < 9.8 < is ±0.05. of of xcr xcr is ±0.05.

) x (8/D)0.150.15 (XV.6) (XV.6)

critical(8mm) x (8/D)

 xxcritical  Here

= xcritical(8mm 8 is the critical quality obtained from Eq. (4-114) and D is cr mm x 8 is the critical quality obtained from Eq. (4-114) and D is

(XV.7) (XV.7)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The critical quality given by Eq. (4-114) can be used for other tube  The critical quality given by Eq. (4-114) can be used for other tube diameters with the following correction factor: diameters with the following correction factor: critical = x Here cr mm x diameter in [mm]. the tube diameter in [mm]. the tube For fuel rod bundles the following correlation was proposed by  For fuel rod bundles the following correlation was proposed by General Electric: General Electric: = ((A x L** critical = ((A x L

)/B + L** BB)/B + L )(1.24/Rf) BB)(1.24/Rf)

+ 0.907GR2 - 1.233GRR + 0. - 0.285GRR 33 907GR2- 0.285G

 xxcritical  WhereWhere B :LB/0.0254 (LB is boiling length I meter)  LL**B :LB/0.0254 (LB is boiling length I meter) Rf: radial peaking factor  Rf: radial peaking factor -600/400)22 - 1.233G A = 1.055 - 0.013{pRR -600/400)  A = 1.055 - 0.013{p – 35.464GRR B = 17.98 +78.873GRR – 35.464G 22

= G/1356.23 (G mass flux in (kgm-2-2ss-1-1))

 B = 17.98 +78.873G  GGRR = G/1356.23 (G mass flux in (kgm = p)/6894.757 (p: pressure in (Pa)  ppRR = p)/6894.757 (p: pressure in (Pa)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The correlation is valid for 7x7 bundles. It can also be applied to 8x8  The correlation is valid for 7x7 bundles. It can also be applied to 8x8 /1.12. bundles once replacing B B with with BB/1.12. bundles once replacing XV.3. Protection against boiling crisis  XV.3. Protection against boiling crisis The thermal-hydraulic design is such that the probability of non-  The thermal-hydraulic design is such that the probability of non- appearance of a boiling crisis during the normal operation, during the appearance of a boiling crisis during the normal operation, during the normal transients and during all transient conditions resulting normal transients and during all transient conditions resulting anomalies with moderated frequency (events classes 1 & 2), is at least anomalies with moderated frequency (events classes 1 & 2), is at least equal to 95% with a confidence level of 95%. equal to 95% with a confidence level of 95%.



By preventing the boiling crisis, one assures sufficient transfer of By preventing the boiling crisis, one assures sufficient transfer of heat between the fuel rod cladding to the primary coolant fluid, thus heat between the fuel rod cladding to the primary coolant fluid, thus The maximal temperature of assures the integrity of the fuel cladding. The maximal temperature of assures the integrity of the fuel cladding. the cladding could not be constituted as a criterion because it is the cladding could not be constituted as a criterion because it is situated at some degrees under the coolant fluid temperature under situated at some degrees under the coolant fluid temperature under nucleate boiling. The limits assured by the control systems, nuclear nucleate boiling. The limits assured by the control systems, nuclear limitations and protections are such that criterion will be respect for limitations and protections are such that criterion will be respect for the transients associated to events of class 2. the transients associated to events of class 2.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR There exists a supplementary margin to the DNBR in case of operating  There exists a supplementary margin to the DNBR in case of operating at nominal power and during the normal transients. at nominal power and during the normal transients. The utilization of a simplified calculation algorithm of the DNBR in the  The utilization of a simplified calculation algorithm of the DNBR in the protection system and in the surveillance system allow the respect of protection system and in the surveillance system allow the respect of the design criteria by defining an automatic shutdown of the reactor the design criteria by defining an automatic shutdown of the reactor core at low level DNBR (DNBR(shutdown)) and a limit condition of core at low level DNBR (DNBR(shutdown)) and a limit condition of operation (DNBR(lco)) associated to the DNBR. operation (DNBR(lco)) associated to the DNBR. The on-line calculation resulted on elaborated by systems which  The on-line calculation resulted on elaborated by systems which utilizes measurements to reconstitute the local conditions by mean of utilizes measurements to reconstitute the local conditions by mean of an algorithm and to apply a select experimental correlation of CHF to an algorithm and to apply a select experimental correlation of CHF to determine the DNBR. determine the DNBR.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 A.A. Fuel temperature Fuel temperature The thermal-hydraulic design is such that, in operation conditions  The thermal-hydraulic design is such that, in operation conditions associated to events classes 1 and 2, there is a minimal probability of associated to events classes 1 and 2, there is a minimal probability of 95% with confidence level of 95%, fuel rods having maximal power 95% with confidence level of 95%, fuel rods having maximal power density (W/cm) does not exceed the melting point of the nuclear fuel. density (W/cm) does not exceed the melting point of the nuclear fuel. The melting point admitted for UO2 is 2800°C for a non-irradiated fuel  The melting point admitted for UO2 is 2800°C for a non-irradiated fuel element, the melting point admitted for MOX fuel is 2737°C for non- element, the melting point admitted for MOX fuel is 2737°C for non- irradiated fuel element. These values decrease with the burn-up (- irradiated fuel element. These values decrease with the burn-up (- 32°C/10000 MWd/tHM). 32°C/10000 MWd/tHM). By preventing the fusion of the fuel, one preserves the geometry of the  By preventing the fusion of the fuel, one preserves the geometry of the later and eventual unfavourable effects of the fuel melting to the later and eventual unfavourable effects of the fuel melting to the cladding are eliminated. cladding are eliminated.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

B. Reactor core coolant mass flow rate  B. Reactor core coolant mass flow rate A minimal value of 94.5% of thermal-hydraulic design of the primary  A minimal value of 94.5% of thermal-hydraulic design of the primary circuit flows across different fuel elements and constituted effectively circuit flows across different fuel elements and constituted effectively the cooling of the fuel rods. The coolant fluid pass through the guide the cooling of the fuel rods. The coolant fluid pass through the guide tubes and the coolant mass rate leakage pass through reactor core tubes and the coolant mass rate leakage pass through reactor core baffles, are not be considered as efficiency for the heat evacuation. baffles, are not be considered as efficiency for the heat evacuation. The thermal-hydraulic studies utilize the thermal-hydraulic rate  The thermal-hydraulic studies utilize the thermal-hydraulic rate (minimal rate) in the inlet of the reactor pressure vessel. In the hot (minimal rate) in the inlet of the reactor pressure vessel. In the hot condition of the upper-plenum, a maximal of 5.5% for that value is condition of the upper-plenum, a maximal of 5.5% for that value is allocated to the by-pass flow. This included the cooling mass flow rate allocated to the by-pass flow. This included the cooling mass flow rate of the control clusters system, the cooling mass flow rate of the upper- of the control clusters system, the cooling mass flow rate of the upper- plenum, the leakages between baffles and leakages to the outlet of the plenum, the leakages between baffles and leakages to the outlet of the pressure-vessel. pressure-vessel.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

C. Hydro-dynamic stability of the reactor core  C. Hydro-dynamic stability of the reactor core  The operating modes associated to events of classes 1 and 2 do The operating modes associated to events of classes 1 and 2 do

not induce hydro-dynamic instability of the reactor core. not induce hydro-dynamic instability of the reactor core. D.Technology of the DNBR or Critical Heat Flux ratio and Mixing effect  D.Technology of the DNBR or Critical Heat Flux ratio and Mixing effect The minimal DNBR in the hot channel is located at the down-stream at is located at the down-stream at  The minimal DNBR in the hot channel point of maximal thermal flux (hot point) due to the elevation of the point of maximal thermal flux (hot point) due to the elevation of the enthalpy at down-stream of that point. enthalpy at down-stream of that point. D.1.Technology of the DNBR  D.1.Technology of the DNBR a) Critical Heat Flux (CHF) – Correlation  a) Critical Heat Flux (CHF) – Correlation The earlier CHF tests have been performed with a fluid pass through  The earlier CHF tests have been performed with a fluid pass through simple heated tubes and in annular configurations with one or two simple heated tubes and in annular configurations with one or two heated walls. The results obtained from such tests have been analyzed heated walls. The results obtained from such tests have been analyzed and correlated to different physical models to describe the CHF and correlated to different physical models to describe the CHF phenomenon. The correlations obtained are consequently, by nature, phenomenon. The correlations obtained are consequently, by nature, strongly empirical. strongly empirical.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The development of testing methods has conduct to the utilization of  The development of testing methods has conduct to the utilization of rod bundle instead of isolated channels, it is known that the mean rod bundle instead of isolated channels, it is known that the mean rates of the rods bundle could not be utilised in the CHF correlations. rates of the rods bundle could not be utilised in the CHF correlations. The results obtained from the tests have shown that the CHF The results obtained from the tests have shown that the CHF correlations could not be based on the mean conditions. Thus, it is correlations could not be based on the mean conditions. Thus, it is necessary to known the local conditions in the sub-channels. necessary to known the local conditions in the sub-channels. To determine the local conditions in the sub-channels, many thermal-  To determine the local conditions in the sub-channels, many thermal- hydraulic computer codes have been developed. In these computer hydraulic computer codes have been developed. In these computer codes, a rods bundle is considered as a sub-channels mesh, each of codes, a rods bundle is considered as a sub-channels mesh, each of them has have the passing surface the cross-section delimited by four them has have the passing surface the cross-section delimited by four adjacent fuel rods. adjacent fuel rods. The sub-channels are divided in axial meshes which defined the  The sub-channels are divided in axial meshes which defined the reference volumes. The local conditions of the fluid in each reference reference volumes. The local conditions of the fluid in each reference volume are calculated by resolving simultaneously the mass equation volume are calculated by resolving simultaneously the mass equation and energy equations and the quantity of movement. The predicted and energy equations and the quantity of movement. The predicted CHF is elaborated by utilizing the local conditions of the fluid in the CHF is elaborated by utilizing the local conditions of the fluid in the sub-channels calculated by the computer code and the correlation. sub-channels calculated by the computer code and the correlation.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Utilization of data obtained from CHF tests  Utilization of data obtained from CHF tests The experimental tests are based on results obtained from fuel  The experimental tests are based on results obtained from fuel assembly bundle. The tests have been performed under following assembly bundle. The tests have been performed under following conditions: conditions:

Axial distribution of uniform flux;  Axial distribution of uniform flux; Axial distribution of non-uniform flux;  Axial distribution of non-uniform flux; On typical cells;  On typical cells; On guide tubes cells;  On guide tubes cells;

The tests performed under following parameters:  The tests performed under following parameters:

Pressure (20.7< p < 170.6 bar);  Pressure (20.7< p < 170.6 bar); Mass rate (980 < G < 4790 kg/m2/s);  Mass rate (980 < G < 4790 kg/m2/s); Quality (-0.22 < X < 0.44 )  Quality (-0.22 < X < 0.44 )

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Those limits are representatives of the operating conditions of the  Those limits are representatives of the operating conditions of the PWR. PWR. C) The form of the correlation for axial uniform thermal flux  C) The form of the correlation for axial uniform thermal flux The CHF correlation is given under analytic form and function of:  The CHF correlation is given under analytic form and function of:

Thermal-hydraulic variables: pressure (p), mass flow rate (G)  Thermal-hydraulic variables: pressure (p), mass flow rate (G) and quality (X); and quality (X); Fuel assembly geometry (grid spacers distances and form);  Fuel assembly geometry (grid spacers distances and form); Cell types (typical or guide tube)  Cell types (typical or guide tube)

(p,G) – B(p,G)*X (p,G) – B(p,G)*X

The principal term of correlation of uniform flux does not depend on  The principal term of correlation of uniform flux does not depend on the fuel assembly geometry. It depends only on the thermal-hydraulic the fuel assembly geometry. It depends only on the thermal-hydraulic variables. It is supposed to depend to the linear variable X and variables. It is supposed to depend to the linear variable X and expressed as following: expressed as following: Ф (CHF) = Ạ  Ф (CHF) = Ạ The other terms associated to the fuel assembly depend on  The other terms associated to the fuel assembly depend on geometrical effects as following: geometrical effects as following:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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Distance between grid spacers;  Distance between grid spacers; Distance between the location of the CHF and the location of the  Distance between the location of the CHF and the location of the grid spacer at up-stream. grid spacer at up-stream.

ạ ạ

The CHF correlation is represented by: CHF(p, G, X, d(g), g(sp), r(g)),  The CHF correlation is represented by: CHF(p, G, X, d(g), g(sp), r(g)), with:with: CHF = (p,G,X,dg) + c(p,G,X, g(sp)) + d(p,X,g(sp),r(g)  CHF = (p,G,X,dg) + c(p,G,X, g(sp)) + d(p,X,g(sp),r(g) d) Form of the correlation for axial non-uniform thermal flux  d) Form of the correlation for axial non-uniform thermal flux The values of the CHF measured in the rods bundle with an axial  The values of the CHF measured in the rods bundle with an axial distribution non-uniform of the thermal flux are lowers than the ones distribution non-uniform of the thermal flux are lowers than the ones obtained with uniform distributions. The application of the CHF obtained with uniform distributions. The application of the CHF correlation has shown that the predicted flux is higher than the correlation has shown that the predicted flux is higher than the measured flux. The predicted value must be reset-up. To this effect, measured flux. The predicted value must be reset-up. To this effect, one applied the correction factor of non-uniform flux of L.S.Tong. The one applied the correction factor of non-uniform flux of L.S.Tong. The reset flux is expressed by following: reset flux is expressed by following:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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Ф = Фu/ F(nu)  Ф = Фu/ F(nu) Where :  Where : - Ф is the reset value of the flux: - Ф is the reset value of the flux:  - Фu is the predicted value of the flux for the axial distribution of the - Фu is the predicted value of the flux for the axial distribution of the uniform flux. uniform flux. - F(nu) is the non-uniform factor. - F(nu) is the non-uniform factor. e) Definition of the DNBR  e) Definition of the DNBR The DNBR, in typical cell and in tube guide cell with cold wall is  The DNBR, in typical cell and in tube guide cell with cold wall is defined as: defined as: DNBR = q”(CHF,N)/ q”(local)  DNBR = q”(CHF,N)/ q”(local) Where : q”(local) is the real thermal flux  Where : q”(local) is the real thermal flux q”(CHF,N) = q”(CHF(u)/ F  q”(CHF,N) = q”(CHF(u)/ F q”(CHF,u) is the thermal critical uniform flux which is predicted by the  q”(CHF,u) is the thermal critical uniform flux which is predicted by the correlation CHF. correlation CHF. F is Tong’s form factor for the non-uniform flux distribution.  F is Tong’s form factor for the non-uniform flux distribution.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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f) Mixing effect between sub-channels f) Mixing effect between sub-channels In a rods bundle, the channels formed by four adjacent fuel rods are In a rods bundle, the channels formed by four adjacent fuel rods are interconnected together by intermediary space between two interconnected together by intermediary space between two neighbouring fuel rods. There is cross-flow between the channels due neighbouring fuel rods. There is cross-flow between the channels due to their difference in pressure between them. to their difference in pressure between them. The effects of the cross decrease the enthalpy elevation in the hot  The effects of the cross decrease the enthalpy elevation in the hot channel. channel. In the energy balance equation of the computer code, a term permitting In the energy balance equation of the computer code, a term permitting the creation the turbulence enthalpy exchange model between the the creation the turbulence enthalpy exchange model between the neighbouring channels is taking into consideration. It is proportional neighbouring channels is taking into consideration. It is proportional to the enthalpy difference between the channels. In the proportional to the enthalpy difference between the channels. In the proportional factor which is defined as the turbulence exchange coefficient. The factor which is defined as the turbulence exchange coefficient. The value of this coefficient is determined after series of specific tests value of this coefficient is determined after series of specific tests performed with the corresponded spacer grids. performed with the corresponded spacer grids.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

g) Uncertanties relative to the manufacturing parameters  g) Uncertanties relative to the manufacturing parameters These uncertainties are taking into account the manufacturing  These uncertainties are taking into account the manufacturing variations in term of materials, fuel rods and assembly geometries. variations in term of materials, fuel rods and assembly geometries. There are two types of manufacturing uncertainties:  There are two types of manufacturing uncertainties:

Effect of the eccentricity of the fuel pellets and the ovality of the  Effect of the eccentricity of the fuel pellets and the ovality of the fuel rod cladding on the CHF; fuel rod cladding on the CHF; Effect on the manufacturing tolerances of the spacer grids on  Effect on the manufacturing tolerances of the spacer grids on the CHF. the CHF.

h) Effect of the eccentricity of the fuel pellets and the ovality of the  h) Effect of the eccentricity of the fuel pellets and the ovality of the cladding on the CHF. cladding on the CHF. Some fuel pellets could be eccentred with regard to the cladding in the  Some fuel pellets could be eccentred with regard to the cladding in the beginning of life. The cladding could be ovalized with the time. In these beginning of life. The cladding could be ovalized with the time. In these cases, there is a variation of the axial flux at a small axial distance. cases, there is a variation of the axial flux at a small axial distance.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR In the case of eccentric pellets, the local flux peak, under a given In the case of eccentric pellets, the local flux peak, under a given angle, will axially extend to a distance corresponding to maximal angle, will axially extend to a distance corresponding to maximal length of some pellets due to the random contact between the pellets length of some pellets due to the random contact between the pellets and the cladding at the beginning of life. This random character of the and the cladding at the beginning of life. This random character of the contact point is induced by the variations of the format pellet contact point is induced by the variations of the format pellet extremities and by their variations of diameters. extremities and by their variations of diameters. In the case of the ovality of the cladding, the local peak of the thermal In the case of the ovality of the cladding, the local peak of the thermal flux, under a given angle, will axially extend to a distance flux, under a given angle, will axially extend to a distance corresponding to maximal length of some pellets, due to the random corresponding to maximal length of some pellets, due to the random axial distribution of chips of the fissured fuel pellets. axial distribution of chips of the fissured fuel pellets. The uncertainties relatives to the hot channel are taking into account  The uncertainties relatives to the hot channel are taking into account of the non-perfect of geometries and materials of fuel rod and fuel of the non-perfect of geometries and materials of fuel rod and fuel assembly. assembly. One distinguishes the following uncertainties concerning the hot  One distinguishes the following uncertainties concerning the hot channel: channel:

into account a particular into account a particular

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Technological factor of hot point (F(q,E): this incertitude is utilized to  Technological factor of hot point (F(q,E): this incertitude is utilized to evaluate the maximal local power peak (the hot point) and it is evaluate the maximal local power peak (the hot point) and it is determined by the statistic combination of tolerances relative to the determined by the statistic combination of tolerances relative to the diameter, the density and the enrichment level of the fuel pellet. diameter, the density and the enrichment level of the fuel pellet. However, the CHF tests with the local peaks thermal flux have shown it However, the CHF tests with the local peaks thermal flux have shown it incertitude is not necessary to take is not necessary to take incertitude concerning the local flux. concerning the local flux. Nuclear factor of enthalpy elevation of the hot channel (FΔH, E): It is  Nuclear factor of enthalpy elevation of the hot channel (FΔH, E): It is determined by statistic combination of the effects on the enthalpy determined by statistic combination of the effects on the enthalpy elevation of manufacturing tolerances relative to the nuclear fuel elevation of manufacturing tolerances relative to the nuclear fuel density and enrichment and to the position of the fuel rods in the density and enrichment and to the position of the fuel rods in the reactor core. reactor core.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR i) Effect of the manufacturing tolerances of the spacer grids on CHF i) Effect of the manufacturing tolerances of the spacer grids on CHF This uncertainty represents directly the effect of manufacturing  This uncertainty represents directly the effect of manufacturing tolerances of the spacer grids on the CHF and, more precisely, effect tolerances of the spacer grids on the CHF and, more precisely, effect of tolerances relative to singular pressure-drops of the spacer grids on of tolerances relative to singular pressure-drops of the spacer grids on the redistribution the coolant mass flow rate in the reactor core. the redistribution the coolant mass flow rate in the reactor core. For the fuel assembly with the same design, the effect on the  For the fuel assembly with the same design, the effect on the dispersion of the values of CHF is negligible. dispersion of the values of CHF is negligible. j) Effect of the fuel rod bowing in the reactor on the CHF j) Effect of the fuel rod bowing in the reactor on the CHF The CHF could be influenced by the bowing phenomenon of the fuel  The CHF could be influenced by the bowing phenomenon of the fuel rods which has been detected on irradiated examinations. This rods which has been detected on irradiated examinations. This phenomenon consists to a displacement of the fuel rod with regard to phenomenon consists to a displacement of the fuel rod with regard to it nominal position in the channel. It depends strongly to the nuclear it nominal position in the channel. It depends strongly to the nuclear fuel rods. fuel rods.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The modification of the flow rate due to the fuel rods bowing induces a  The modification of the flow rate due to the fuel rods bowing induces a decrease of the CHF. The penalty which is resulted is quantified by two decrease of the CHF. The penalty which is resulted is quantified by two unfolding models: unfolding models: An envelope law defining the bowing order of the fuel rods, based on  An envelope law defining the bowing order of the fuel rods, based on the measurements of the bowing of the fuel rods in irradiated fuel the measurements of the bowing of the fuel rods in irradiated fuel assembly. assembly. A law defining the penalty to the CHF versus the closing of the channel  A law defining the penalty to the CHF versus the closing of the channel cross-section. The penalty law utilised is the one approved by the NRC cross-section. The penalty law utilised is the one approved by the NRC in 1979. It consists to differentiate the operating with nominal mass in 1979. It consists to differentiate the operating with nominal mass flow rate to the operating with reduce flowing rate. The resulting model flow rate to the operating with reduce flowing rate. The resulting model gives the penalty versus the burn-up of the fuel assembly. gives the penalty versus the burn-up of the fuel assembly. The experimental results obtained have shown that the bowing penalty  The experimental results obtained have shown that the bowing penalty is nil under burn-up of 16000 MWday/tHM. Upper that burn-up limit, the is nil under burn-up of 16000 MWday/tHM. Upper that burn-up limit, the penalty increase linearly, but could be limited if the burn-up of the fuel penalty increase linearly, but could be limited if the burn-up of the fuel rods increases. rods increases.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The anterior analyses have shown that beyond the burn-up of 35000  The anterior analyses have shown that beyond the burn-up of 35000 MWd/tHM, the fuel rods do not present maximal value of FΔH. MWd/tHM, the fuel rods do not present maximal value of FΔH. k) Correlation between the heat transfer and the void coefficient to the  k) Correlation between the heat transfer and the void coefficient to the radial distribution of the nuclear power radial distribution of the nuclear power The flowing model is based on the double-phase flowing by taking into  The flowing model is based on the double-phase flowing by taking into account the thermal unbalance of the liquid phase and the rate account the thermal unbalance of the liquid phase and the rate differences of the liquid and vapour phases. This model is obtained differences of the liquid and vapour phases. This model is obtained from the mass equation, the quantity of movement and the energy from the mass equation, the quantity of movement and the energy balance for the double-phase flowing. The equation of the enthalpy balance for the double-phase flowing. The equation of the enthalpy balance permits to the calculations of the local boiling. These balance permits to the calculations of the local boiling. These equations necessitated a physical model to describe the phase equations necessitated a physical model to describe the phase interactions, the turbulence mixing and interactions between fluid and interactions, the turbulence mixing and interactions between fluid and wall. Models using in these equations are: wall. Models using in these equations are:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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Friction model at the walls:  Friction model at the walls: Heat transfer model;  Heat transfer model; Slip model to take into account the mass flow rate different rates for  Slip model to take into account the mass flow rate different rates for the liquid and vapour phases: the liquid and vapour phases: Turbulent viscosity and diffusivity coefficients which are calculated  Turbulent viscosity and diffusivity coefficients which are calculated with an algebraic model permitting the description of the mixing effect. with an algebraic model permitting the description of the mixing effect. To effectuate the calculations of the thermal-hydraulic design of the To effectuate the calculations of the thermal-hydraulic design of the reactor core of the PWR or precisely to calculate the necessary local reactor core of the PWR or precisely to calculate the necessary local properties of the fluid for to predict the CHF margins, the computer properties of the fluid for to predict the CHF margins, the computer codes (THINC-IV, FLICA, etc.) with their proper models of heat transfer codes (THINC-IV, FLICA, etc.) with their proper models of heat transfer are utilized by the designer are utilized by the designer

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

E. Hydro-dynamic instability  E. Hydro-dynamic instability 

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The boiling flows could be subjected to the thermal-hydro- The boiling flows could be subjected to the thermal-hydro- dynamic instabilities. Those instabilities are not acceptable in the dynamic instabilities. Those instabilities are not acceptable in the reactor core because they could induce a modification of thermal- reactor core because they could induce a modification of thermal- hydraulic conditions which are leading to a reduction of the CHF with hydraulic conditions which are leading to a reduction of the CHF with regard to the one observed in the permanent flow conditions or to the regard to the one observed in the permanent flow conditions or to the undesirable vibrations of reactor components. Consequently, a undesirable vibrations of reactor components. Consequently, a thermal-hydraulic criterion has been established, to guarantee the thermal-hydraulic criterion has been established, to guarantee the operating modes in case of events classes 1 and 2 do not induce operating modes in case of events classes 1 and 2 do not induce thermal-hydro-dynamic instabilities. thermal-hydro-dynamic instabilities.

Two types of specific instabilities have been taking into account Two types of specific instabilities have been taking into account for the operation of the PWR. It is the permanent instability Ledinegg for the operation of the PWR. It is the permanent instability Ledinegg (rate oscillation) and dynamic instability with density of wave. (rate oscillation) and dynamic instability with density of wave. Ledinegg’s instability implies a sudden variation of the rate flow of a  Ledinegg’s instability implies a sudden variation of the rate flow of a permanent flow to another. permanent flow to another.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR This instability appears when the slope of the curve loss “pressure-  This instability appears when the slope of the curve loss “pressure- mass flow rate” (δP/δG internal) of the primary circuit become lower mass flow rate” (δP/δG internal) of the primary circuit become lower than of the curve loss “pressure- flow rate” (δP/δG external) of than of the curve loss “pressure- flow rate” (δP/δG external) of feedwater loop (load high). The stability criterion is: δP/δG internal) > feedwater loop (load high). The stability criterion is: δP/δG internal) > (δP/δG external). (δP/δG external).

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The mechanism of mass flow rate oscillation in a heated channel The mechanism of mass flow rate oscillation in a heated channel could be described as following: Briefly, an inlet mass flow rate could be described as following: Briefly, an inlet mass flow rate fluctuation produces a perturbation to the enthalpy. Thus perturbed fluctuation produces a perturbation to the enthalpy. Thus perturbed along the length and the loss-pressure of the single-phase zone and along the length and the loss-pressure of the single-phase zone and induces perturbations of the quality or void coefficient in the double- induces perturbations of the quality or void coefficient in the double- phase zone which rise up the channel with the mass flow. The phase zone which rise up the channel with the mass flow. The perturbations of the quality and the double-phase create the perturbations of the quality and the double-phase create the perturbations of the loss-pressure of the double-phase. perturbations of the loss-pressure of the double-phase.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR However, as the total loss-pressure of the reactor core is maintained  However, as the total loss-pressure of the reactor core is maintained by the characteristics of the external fluid circuit of the reactor core, by the characteristics of the external fluid circuit of the reactor core, thus results the perturbation of the loss-pressure of the double-phase thus results the perturbation of the loss-pressure of the double-phase rise up to the single-phase zone. The induced perturbations could be rise up to the single-phase zone. The induced perturbations could be attenuated or self-generated. attenuated or self-generated. The cooling mass flow rate passes through the adjustable inlet of the  The cooling mass flow rate passes through the adjustable inlet of the pressure-vessel. It goes down to the down-comer via the annular pressure-vessel. It goes down to the down-comer via the annular space formed by the pressure-vessel and an envelope of the reactor space formed by the pressure-vessel and an envelope of the reactor core and then goes up to upper- plenum of the reactor core. It gets out core and then goes up to upper- plenum of the reactor core. It gets out the pressure-vessel via adjustable pressure-vessel outlet. the pressure-vessel via adjustable pressure-vessel outlet. There are many by-pass ways:  There are many by-pass ways: • Cooling mass flow rate passes through the upper-plate, it Cooling mass flow rate passes through the upper-plate, it constitutes of water in annular down stream; the mass flow rate is constitutes of water in annular down stream; the mass flow rate is then directed from the pressure-vessel dome to the upper-plate. In then directed from the pressure-vessel dome to the upper-plate. In configuration “hot dome”, which is the retained design option, configuration “hot dome”, which is the retained design option,

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR * This mass flow rate is directed down to some guide tubes, in normal * This mass flow rate is directed down to some guide tubes, in normal operation. In other guide tubes there exist an up-stream circulation. operation. In other guide tubes there exist an up-stream circulation. * The mass flow rate gets in the guide tubes of control clusters to cool * The mass flow rate gets in the guide tubes of control clusters to cool down the control rods, the burnable poison (if utilized) or down the control rods, the burnable poison (if utilized) or instrumentation sources. instrumentation sources. * Leakage mass flow rate directly from the adjustable inlet to * Leakage mass flow rate directly from the adjustable inlet to adjustable outlet of the pressure-vessel via annular space between the adjustable outlet of the pressure-vessel via annular space between the pressure-vessel and core envelope. pressure-vessel and core envelope. * Mass flow rate passing between the hard reflector and core envelope * Mass flow rate passing between the hard reflector and core envelope and inside the hard reflector to cool down these components and is and inside the hard reflector to cool down these components and is not considered as available to refrigerate the reactor core. not considered as available to refrigerate the reactor core. * Mass flow rate flows between the peripheral fuel assemblies and the * Mass flow rate flows between the peripheral fuel assemblies and the adjacent hard reflector. adjacent hard reflector.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The maximal value of the by-pass fluid is 5.5% of the total rate. On the  The maximal value of the by-pass fluid is 5.5% of the total rate. On the total, 2.2% is associated to the reactor core, the rest is associated to total, 2.2% is associated to the reactor core, the rest is associated to internal components. The calculations have been effectuated by internal components. The calculations have been effectuated by utilizing of tolerances in the direction the most penalizing and by utilizing of tolerances in the direction the most penalizing and by taking into account uncertainties on pressure drops. taking into account uncertainties on pressure drops. F. Defect of distribution of the rate at pressure-vessel inlet  F. Defect of distribution of the rate at pressure-vessel inlet Generally, the distribution of the inlet mass flow rate is non-uniform.  Generally, the distribution of the inlet mass flow rate is non-uniform. Studies with thermal-hydraulic computer codes by decreasing the Studies with thermal-hydraulic computer codes by decreasing the mass flow rate of the fluid in a limited zone inlet of the reactor core mass flow rate of the fluid in a limited zone inlet of the reactor core have shown that a rapid readjustment of one third of reactor core high have shown that a rapid readjustment of one third of reactor core high has been observed, thus the defect on inlet mass flow rate distribution, has been observed, thus the defect on inlet mass flow rate distribution, in practice, has a negligible effect on the CHF of the hot channel. That in practice, has a negligible effect on the CHF of the hot channel. That redistribution of the mass flow rate is due to the readjustment of the redistribution of the mass flow rate is due to the readjustment of the fluid rate. Consequently, the defect of the distribution of inlet mass fluid rate. Consequently, the defect of the distribution of inlet mass flow rate induces no penalties on the CHF and its location; none of flow rate induces no penalties on the CHF and its location; none of penalty has been taking into account in the design. penalty has been taking into account in the design.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

G. Pressure drops in the pressure-vessel  G. Pressure drops in the pressure-vessel Pressure drops are due to the friction on walls and to changes of  Pressure drops are due to the friction on walls and to changes of geometry of walls guiding the fluid. One supposes that the mass flow geometry of walls guiding the fluid. One supposes that the mass flow rate is in single-phase, turbulence and the fluid is incompressible. rate is in single-phase, turbulence and the fluid is incompressible. These hypotheses apply to calculations of pressure drops in the These hypotheses apply to calculations of pressure drops in the reactor core and the pressure-vessel performed to evaluate the loss- reactor core and the pressure-vessel performed to evaluate the loss- pressure in the pressure-vessel because the mean void coefficient of pressure in the pressure-vessel because the mean void coefficient of the reactor core is negligible. the reactor core is negligible. The character double-phase of the mass flow rate has been taking into  The character double-phase of the mass flow rate has been taking into account in the thermal-hydraulic analyses of sub-channels. account in the thermal-hydraulic analyses of sub-channels. Due to the complexity of the geometry of the reactor core, one could  Due to the complexity of the geometry of the reactor core, one could not get the precise analytic values of the form and friction coefficients. not get the precise analytic values of the form and friction coefficients. The experimental values have been obtained on similar geometrical The experimental values have been obtained on similar geometrical model model The reactor core pressure drops have been determined during the the  The reactor core pressure drops have been determined during the the thermal-hydraulic tests on the fuel assembly. thermal-hydraulic tests on the fuel assembly.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR These tests have been performed in a loop test within large range of  These tests have been performed in a loop test within large range of Reynolds numbers, including the ones observed in the PWRs reactor Reynolds numbers, including the ones observed in the PWRs reactor core. core. The pressure-vessel pressure drops are obtained by combination of  The pressure-vessel pressure drops are obtained by combination of reactor core loss pressure observed on hydraulic moke-up tests on reactor core loss pressure observed on hydraulic moke-up tests on reduced scale of pressure-vessel and correlated to the pressure drops reduced scale of pressure-vessel and correlated to the pressure drops models. models. Measurements of primary circuit rate flow have been performed at the  Measurements of primary circuit rate flow have been performed at the start-up tests of the NPP to verify the design flow rate. start-up tests of the NPP to verify the design flow rate. H. Hydraulic forces  H. Hydraulic forces The maximal hydraulic forces exercised on the internal components of  The maximal hydraulic forces exercised on the internal components of the pressure-vessel are reached for the nominal mass flow rate. the pressure-vessel are reached for the nominal mass flow rate. In nominal operating condition, the hydraulic forces are determined In nominal operating condition, the hydraulic forces are determined with the mechanical design mass flow rate, by the taking into account with the mechanical design mass flow rate, by the taking into account the minimal value of the by-pass mass flow in the reactor core. the minimal value of the by-pass mass flow in the reactor core.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR At cold-shutdown, the hydraulic forces are determined with the same  At cold-shutdown, the hydraulic forces are determined with the same mass flow rates (pressure-vessel and core by-pass) but by taking into mass flow rates (pressure-vessel and core by-pass) but by taking into account the difference of density of the cooling water. Thus is account the difference of density of the cooling water. Thus is corresponded to the enveloped value for normal operating condition. corresponded to the enveloped value for normal operating condition. The transient conditions such as over-speed of primary pumps,  The transient conditions such as over-speed of primary pumps, capable to product a mass flow rate of 20% superior to the mechanical capable to product a mass flow rate of 20% superior to the mechanical design mass flow rate are utilized to determine the envelope of design mass flow rate are utilized to determine the envelope of hydraulic forces in the transient conditions. hydraulic forces in the transient conditions. The hydraulic tests have been performed to verify the hydraulic loads  The hydraulic tests have been performed to verify the hydraulic loads during the over-speed of the primary pumps to the calculated during the over-speed of the primary pumps to the calculated mechanical mass flow rate at hot and cold conditions. mechanical mass flow rate at hot and cold conditions. I. Hydraulic Dimensioning of the internal components I. Hydraulic Dimensioning of the internal components The dimensioning of the internal components is related to the specific  The dimensioning of the internal components is related to the specific design characteristics of the reactor core and the pressure-vessel design characteristics of the reactor core and the pressure-vessel , etc.). (structural components, guide tubes, baffles, reflectors, etc.). (structural components, guide tubes, baffles, reflectors

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Tests have been performed to determine the inlet and outlet mass flow  Tests have been performed to determine the inlet and outlet mass flow rate in the pressure-vessel. rate in the pressure-vessel. J. Thermal effects during the normal transients.  J. Thermal effects during the normal transients. The safety limits of the reactor with regard to DNBR are defined in  The safety limits of the reactor with regard to DNBR are defined in function of the cooling water temperature, of the pressure, of the function of the cooling water temperature, of the pressure, of the reactor core power and of the axial and radial distribution of the reactor core power and of the axial and radial distribution of the power. An operation conditions superior to these limits must power. An operation conditions superior to these limits must guarantee that the DNBR criteria are respected. guarantee that the DNBR criteria are respected. Preventions have been taken at operation by adopting the protection  Preventions have been taken at operation by adopting the protection chain at “low DNBR threshold” and by setting the automatic shutdown chain at “low DNBR threshold” and by setting the automatic shutdown of the reactor at “low DNBR threshold”. Thus assure sufficient in the of the reactor at “low DNBR threshold”. Thus assure sufficient in the same time the protection for the steady state and incidental transients same time the protection for the steady state and incidental transients which are sufficiently low with regard to the delay of fluid transport in which are sufficiently low with regard to the delay of fluid transport in the primary circuit. For the rapid transients and transients at hot the primary circuit. For the rapid transients and transients at hot condition with power equal zero, specific protection functions have condition with power equal zero, specific protection functions have been provided. been provided.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

K. Uncertainties  K. Uncertainties K.1/ Critical Heat Flux Ratio (DNBR)  K.1/ Critical Heat Flux Ratio (DNBR) Uncertainties treatment in the calculations  Uncertainties treatment in the calculations One utilizes a statistical approach to combine the uncertainties  One utilizes a statistical approach to combine the uncertainties affecting the DNBR. affecting the DNBR. The uncertainties representing a random character and a well  The uncertainties representing a random character and a well probability law are treated with statistic methods, the others are probability law are treated with statistic methods, the others are treated with the deterministic methods. treated with the deterministic methods. This approach is utilizes to guarantee the respect of the DNBR criteria  This approach is utilizes to guarantee the respect of the DNBR criteria for all transients excepted the transient of uncontrolled-withdrawal of for all transients excepted the transient of uncontrolled-withdrawal of control rod cluster in the case of reactor core un-critical or at low control rod cluster in the case of reactor core un-critical or at low power and for transient of steamline break on which the uncertainties power and for transient of steamline break on which the uncertainties are combined under deterministic manner. are combined under deterministic manner.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



b) Statistical approach  b) Statistical approach To establish the relation between the uncertainties affecting the DNBR  To establish the relation between the uncertainties affecting the DNBR and the variation of DNBR, one utilizes a variable defined by the and the variation of DNBR, one utilizes a variable defined by the following equation: following equation: Y = DNBR(r) /DNBR(c)  Y = DNBR(r) /DNBR(c) Where DNBR(r) is the real DNBR and DNBR(c) is the calculated value,  Where DNBR(r) is the real DNBR and DNBR(c) is the calculated value, determined by taking into account all the parameters associated to the determined by taking into account all the parameters associated to the calculation of the DNBR to their most probable value. calculation of the DNBR to their most probable value. DNBR(c) is the calculated DNBR in operating by an algorithm set in  DNBR(c) is the calculated DNBR in operating by an algorithm set in place on the I&C. place on the I&C. Prob (DNBR(r) > T) = 95% with a confidence level of 95% and is  Prob (DNBR(r) > T) = 95% with a confidence level of 95% and is equivalent to: Prob(DNBR(c) x Y > T) = 95% with a confidence level of equivalent to: Prob(DNBR(c) x Y > T) = 95% with a confidence level of 95%. 95%. If m and σ are the mean value and standard deviation of the If m and σ are the mean value and standard deviation of the distribution of probability for the random variable Y, the Prob(DNBR(c) distribution of probability for the random variable Y, the Prob(DNBR(c) x Y > T) = 95% with a confidence level of 95% is guaranteed if DNBR(c) x Y > T) = 95% with a confidence level of 95% is guaranteed if DNBR(c) > T/m(y)(1 – 1,645 V(y)). > T/m(y)(1 – 1,645 V(y)).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Ф(rc) is the real CHF Ф(rc) is the real CHF



DNBR(r) is a random variable could be decomposed in random  DNBR(r) is a random variable could be decomposed in random variable products as following: variable products as following: DNBR = Ф(rc)/Ф(cp) x Ф(cp)/Ф(LDC) x Ф(LDC)/Ф(rl) x P  DNBR = Ф(rc)/Ф(cp) x Ф(cp)/Ф(LDC) x Ф(LDC)/Ф(rl) x P Where:  Where: 

Ф(cp) is the predicted CHF determined by the CHF correlation Ф(cp) is the predicted CHF determined by the CHF correlation Ф(LDC) is the local CHF calculated by the computer code Ф(LDC) is the local CHF calculated by the computer code Ф(rl) is the local real CHF in the same thermal-hydraulic Ф(rl) is the local real CHF in the same thermal-hydraulic



conditions conditions

P is the penalty factor P is the penalty factor Ф(cp)/Ф(LDC) is DNBR(DC) is the DNBR caculated by the Ф(cp)/Ф(LDC) is DNBR(DC) is the DNBR caculated by the



computer code computer code

DNBR(DC) is a random variable which is function of variavles of DNBR(DC) is a random variable which is function of variavles of

the system (temperature, local power, etc.). the system (temperature, local power, etc.). DNBR(DC) could be decomposed as following:  DNBR(DC) could be decomposed as following: DNBR(DC) = DNBR(DC)/ DNBR(DCO) x DNBR(DCO)/ DNBR(AO) x  DNBR(DC) = DNBR(DC)/ DNBR(DCO) x DNBR(DCO)/ DNBR(AO) x DNBR(AO) DNBR(AO)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR DNBR(DCO) is the calculated DNBR by the computer code DNBR(DCO) is the calculated DNBR by the computer code



Where:  Where: to the most probable values. to the most probable values.

DNBR(AO) is the one-line calculated DNBR by the algoritm in the DNBR(AO) is the one-line calculated DNBR by the algoritm in the

I&C to the same most probable values. I&C to the same most probable values. Consequently:  Consequently: Y = DNBR(r)/DNBR(AO) = Ф(rc)/ Ф(cp) x DNBR(DC)/DNBR(DCO)  Y = DNBR(r)/DNBR(AO) = Ф(rc)/ Ф(cp) x DNBR(DC)/DNBR(DCO) xDNBR(DCO)/DNBR(AO)/ Ф(LDC)/ Ф(rl) x P xDNBR(DCO)/DNBR(AO)/ Ф(LDC)/ Ф(rl) x P Y is the product of P factor with the following variables:  Y is the product of P factor with the following variables: Ф(cp)/ Ф(LDC) : Distribution of probabilities of that variable, is  Ф(cp)/ Ф(LDC) : Distribution of probabilities of that variable, is provided by the correlation of CHF. It is a normal distribution provided by the correlation of CHF. It is a normal distribution characterized by a mean value m(c) and a standard deviation σ(c). characterized by a mean value m(c) and a standard deviation σ(c). DNBR(DC)/DNBR(DCO): This random variable is function of  DNBR(DC)/DNBR(DCO): This random variable is function of independent random variables: independent random variables:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR • Operating parameters of NPP and measured on site (temperature, Operating parameters of NPP and measured on site (temperature, reactor pressure, local power, relative measured primary mass flow reactor pressure, local power, relative measured primary mass flow rate); rate); • Parameters which are not detected but influencing the DNBR Parameters which are not detected but influencing the DNBR (uncertainties related to pellets enrichment, to diameter and to the (uncertainties related to pellets enrichment, to diameter and to the dishing) dishing)

The distribution of probabilities of that variable representative of the  The distribution of probabilities of that variable representative of the global uncertainties of the system is characterized by a value m(s) and global uncertainties of the system is characterized by a value m(s) and a standard deviation (σ(s)). a standard deviation (σ(s)). DNBR(DCO)/DNBR(CAO) : This random variable is taking into account  DNBR(DCO)/DNBR(CAO) : This random variable is taking into account the uncertainty of the computer code. The distribution of probabilities the uncertainty of the computer code. The distribution of probabilities is characterized by two parameters: m(DC) and σ(DC). is characterized by two parameters: m(DC) and σ(DC). * A supplementary uncertainty must be taking into account: transient * A supplementary uncertainty must be taking into account: transient

uncertainty with regard to the steady state. uncertainty with regard to the steady state.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR This uncertainty gives all the discordance introduced by the utilization  This uncertainty gives all the discordance introduced by the utilization of the local properties of the fluid resulting analyses of accidental of the local properties of the fluid resulting analyses of accidental transients for the determination of the DNBR in steady state. It is transients for the determination of the DNBR in steady state. It is independent of uncertainties mentioned above. independent of uncertainties mentioned above. The parameters characterizing the distribution of probabilities are:  The parameters characterizing the distribution of probabilities are: m(tss) and σ(tss). m(tss) and σ(tss). The P factor corresponds to the all uncertainties which are treated  The P factor corresponds to the all uncertainties which are treated under deterministic way: under deterministic way: • Absolute total mass flow rate; Absolute total mass flow rate; • Core by-pass mass flow rate; Core by-pass mass flow rate; • Fuel rods bowing effect ; Fuel rods bowing effect ; • Neutron data; Neutron data; • Set-point limit of automatic shutdown of the reactor, etc. Set-point limit of automatic shutdown of the reactor, etc.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

As all variables mentioned above (different measurements, I&C  As all variables mentioned above (different measurements, I&C algorithm, DNBR correlation, computer code, manufacturing algorithm, DNBR correlation, computer code, manufacturing uncertainties) are independent and as the perturbations with regard to uncertainties) are independent and as the perturbations with regard to the mean values are small, the coefficient taken into account the the mean values are small, the coefficient taken into account the distribution of uncertainty associated to the DNBR could be calculated distribution of uncertainty associated to the DNBR could be calculated as follow: as follow: V(r)expo2 = (σ(Y)/m(Y))expo2 = (σ(c)/m(c))expo2 + (σ(s)/m(s))expo2 +  V(r)expo2 = (σ(Y)/m(Y))expo2 = (σ(c)/m(c))expo2 + (σ(s)/m(s))expo2 + (σ(a)/m(a))expo2 (σ(tss)/m(tss))expo2 +(σ(DC)/m(DC))expo2 (σ(a)/m(a))expo2 (σ(tss)/m(tss))expo2 +(σ(DC)/m(DC))expo2 All the terms of the above equation are determined separately  All the terms of the above equation are determined separately excepted σ(s)/m(s) is determined by Monte Carlo method. excepted σ(s)/m(s) is determined by Monte Carlo method. On the other hand, the probability distribution function of Y is close to  On the other hand, the probability distribution function of Y is close to the normal distribution with: the normal distribution with: Mean value: m(Y) = m(c) x m(s) x m(a) x m(tss) x m(DC) x P  Mean value: m(Y) = m(c) x m(s) x m(a) x m(tss) x m(DC) x P



THERMAL-HYDRAULIC IN NUCLEAR REACTOR In consequence, the probability that the DNBR is superior to the In consequence, the probability that the DNBR is superior to the threshold T is 95% with the confidence of 95% if the DNBR is superior threshold T is 95% with the confidence of 95% if the DNBR is superior to the threshold of theoretical DNBR(th) defined as following: to the threshold of theoretical DNBR(th) defined as following:

DNBR(th) = T/m(Y)(1-1,645V(Y) DNBR(th) = T/m(Y)(1-1,645V(Y)

c) Deterministic approach  c) Deterministic approach All the uncertainties mentioned above are treated by deterministic way.  All the uncertainties mentioned above are treated by deterministic way. As the simplified calculation of the DNBR on site is utilized to protect  As the simplified calculation of the DNBR on site is utilized to protect the reactor against the low DNBR for the concerned transients by the reactor against the low DNBR for the concerned transients by deterministic approach, even for the action of the protection system or deterministic approach, even for the action of the protection system or by utilizing the surveillance of limit operating conditions (LCO) of the by utilizing the surveillance of limit operating conditions (LCO) of the DNBR, each parameter has effect on the DNBR must be controlled by a DNBR, each parameter has effect on the DNBR must be controlled by a specific limit operating condition (LCO) and, per consequence, its specific limit operating condition (LCO) and, per consequence, its uncertainty must be taking into consideration. uncertainty must be taking into consideration. Those parameters are:  Those parameters are:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



• • •

Mean temperature of the primary circuit; Mean temperature of the primary circuit; The reactor pressure; The reactor pressure; The local power. The local power. About the the power distribution, the analyses of the transients About the the power distribution, the analyses of the transients

will be effectuated by utilizing the one most unfavourable. will be effectuated by utilizing the one most unfavourable. d) Uncertainties relative to computer code and mixing coefficient  d) Uncertainties relative to computer code and mixing coefficient  The results obtained from a sensibility study with the computer The results obtained from a sensibility study with the computer



code has shown that the minimum DNBR in the hot channel is code has shown that the minimum DNBR in the hot channel is relatively less sensible to the variations of axial power distribution for relatively less sensible to the variations of axial power distribution for the hole reactor core. (for the same value of FΔH(N). the hole reactor core. (for the same value of FΔH(N).

Studies have been performed to determine the sensibility of Studies have been performed to determine the sensibility of minimum DNBR in the hot channel at axial and radial meshes, to the minimum DNBR in the hot channel at axial and radial meshes, to the inlet mass flow rate, to the loss-pressure, to the power distribution, to inlet mass flow rate, to the loss-pressure, to the power distribution, to the mixing coefficient and to the void model. the mixing coefficient and to the void model.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Results obtained have shown the minimum DNBR in the hot channel is  Results obtained have shown the minimum DNBR in the hot channel is sensible with the three of them: mixing coefficients, double-phase sensible with the three of them: mixing coefficients, double-phase model and the radial distribution of press-drop coefficients of the grid model and the radial distribution of press-drop coefficients of the grid spacers. spacers.



e) Justification of the statistic combination of the uncertainties.. e) Justification of the statistic combination of the uncertainties As explained above, one utilizes a statistic approach to combine the  As explained above, one utilizes a statistic approach to combine the following uncertainties which have effect of the DNBR: following uncertainties which have effect of the DNBR: • Uncertainty related to the CHF correlation (m(c), σ(c)); Uncertainty related to the CHF correlation (m(c), σ(c)); • Uncertainty related to the complete system (m(s), σ(s)); Uncertainty related to the complete system (m(s), σ(s)); • Uncertainty related to the algorithm (m(a), σ(a)); Uncertainty related to the algorithm (m(a), σ(a)); • Uncertainty related to the computer code (m(DC), σ(DC)); Uncertainty related to the computer code (m(DC), σ(DC)); • Uncertainty of transient regime in function of the steady state Uncertainty of transient regime in function of the steady state (m(tss), σ(tss). (m(tss), σ(tss).

The independent parameters on which the uncertainty presents a The independent parameters on which the uncertainty presents a

random character and a well know probability law are treated by the random character and a well know probability law are treated by the deterministic method. deterministic method.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



f) Uncertainties relative to the CHF correlation f) Uncertainties relative to the CHF correlation The evaluation of the characteristics of the CHF correlation on the  The evaluation of the characteristics of the CHF correlation on the basis of the comparison of the results obtained from the tests led to basis of the comparison of the results obtained from the tests led to the definition of the distribution of probabilities of the measured CHF the definition of the distribution of probabilities of the measured CHF to the predicted CHF. The latter presents a normal distribution. to the predicted CHF. The latter presents a normal distribution. g) Uncertainty relative to the whole system  g) Uncertainty relative to the whole system Two principal uncertainties are defined that each of them could be  Two principal uncertainties are defined that each of them could be divided in several uncertainties: divided in several uncertainties:

• Uncertainties related to the physical measured parameters in Uncertainties related to the physical measured parameters in operation: operation:

The following operating parameters of the reactor core are used to The following operating parameters of the reactor core are used to calculate the DNBR: the inlet temperature, the pressure of the calculate the DNBR: the inlet temperature, the pressure of the pressurizer, the relative measured primary mass flow and the local pressurizer, the relative measured primary mass flow and the local power.. power

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The inlet temperature is obtained by the pyrometric gage at the cold The inlet temperature is obtained by the pyrometric gage at the cold leg, the pressure of the pressurizer is obtained by the primary leg, the pressure of the pressurizer is obtained by the primary pressure gage, the primary mass flow obtained by the mass flow gage pressure gage, the primary mass flow obtained by the mass flow gage at the primary pumps and the power distribution of the hot channel is at the primary pumps and the power distribution of the hot channel is obtained directly from the in-core measurement of self-powered obtained directly from the in-core measurement of self-powered detectors. Each measurement is independent to each others. An detectors. Each measurement is independent to each others. An uncertainty, for example, on the pyrometric due to the scaling does not uncertainty, for example, on the pyrometric due to the scaling does not have any relation with the pressure gauge of the pressurizer neither to have any relation with the pressure gauge of the pressurizer neither to the mass flow rate gage of the primary pumps. the mass flow rate gage of the primary pumps. Between the gage and the utilized signal in the protection system,  Between the gage and the utilized signal in the protection system, certain dispositive are intercalated (for example for the temperature: certain dispositive are intercalated (for example for the temperature: convector ohms-amps, convector amps-volts, isolation module if convector ohms-amps, convector amps-volts, isolation module if these necessary and convector analogue/numerical), each of these necessary and convector analogue/numerical), each of dispositive has independent uncertainty and random, it is treated dispositive has independent uncertainty and random, it is treated statistically. statistically.



THERMAL-HYDRAULIC IN NUCLEAR REACTOR The power distribution of the hot channel induces uncertainties on the  The power distribution of the hot channel induces uncertainties on the accuracy of the aeroball measurement system (taking into account the accuracy of the aeroball measurement system (taking into account the accuracy of the activation rate, the reconstruction of the relative accuracy of the activation rate, the reconstruction of the relative density of power, the discretization of the burn-up and the number of density of power, the discretization of the burn-up and the number of instrumented fuel assembly) and on the accuracy of signals obtained instrumented fuel assembly) and on the accuracy of signals obtained from self-powered detectors (derivation, provision relative to the from self-powered detectors (derivation, provision relative to the burnable poisons). burnable poisons). The total uncertainty could be divided in several distributions of  The total uncertainty could be divided in several distributions of probabilities (uncertainties relative to the gage, to the scaling of gage- probabilities (uncertainties relative to the gage, to the scaling of gage- transmitters, etc.). The resulting distribution of probabilities of such a transmitters, etc.). The resulting distribution of probabilities of such a great number of random variables is a normal distribution, as generally great number of random variables is a normal distribution, as generally observed at the measurement uncertainties. observed at the measurement uncertainties. * Uncertainties relative to the manufacturing tolerances * Uncertainties relative to the manufacturing tolerances The FΔH(E) taking into account the manufacturing variables which  The FΔH(E) taking into account the manufacturing variables which affect the thermal power along the channel.* affect the thermal power along the channel.*



THERMAL-HYDRAULIC IN NUCLEAR REACTOR The FΔH(E) taking into account the manufacturing variables which  The FΔH(E) taking into account the manufacturing variables which affect the thermal power along the channel. affect the thermal power along the channel. Those variables are the diameter, the density and the enrichment rate  Those variables are the diameter, the density and the enrichment rate of U235 of the pellets. The uncertainties relatives to those variables are of U235 of the pellets. The uncertainties relatives to those variables are determined by sampling measurements on the fabrication. The determined by sampling measurements on the fabrication. The resulted uncertainty is independent of uncertainties notified, and it is a resulted uncertainty is independent of uncertainties notified, and it is a normal distribution. normal distribution. * Uncertainty relative to the algorithm * Uncertainty relative to the algorithm This uncertainty taking into account the difference between the  This uncertainty taking into account the difference between the calculations obtained from the design computer code and the calculations obtained from the design computer code and the calculations obtained from the algorithm of the DNBR implemented on calculations obtained from the algorithm of the DNBR implemented on the site in the same thermal-hydraulic conditions. The algorithm is the site in the same thermal-hydraulic conditions. The algorithm is adjusted to the calculations of the design computer code. adjusted to the calculations of the design computer code. A statistical analyse allows to determine the distribution of  A statistical analyse allows to determine the distribution of probabilities differences between the algorithm and the computer probabilities differences between the algorithm and the computer code. It is a normal distribution. code. It is a normal distribution.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



* Uncertainty relative to the design computer code * Uncertainty relative to the design computer code Uncertainty relative to the design computer code includes all the  Uncertainty relative to the design computer code includes all the effects of the complete core analysis by means a numerical computer effects of the complete core analysis by means a numerical computer code. As the analyse of thermal flux tests are conducted to define the code. As the analyse of thermal flux tests are conducted to define the correlation characteristics is performed by the design computer code, correlation characteristics is performed by the design computer code, the whole predicted thermal flux for each serial of experimental data the whole predicted thermal flux for each serial of experimental data includes the uncertainty relative to the calculation code and, includes the uncertainty relative to the calculation code and, consequently, also the parameters consequently, also the parameters * Uncertainty of the transient conditions versus the steady state * Uncertainty of the transient conditions versus the steady state conditions conditions This uncertainty taking into account all of discordance introduced by  This uncertainty taking into account all of discordance introduced by the calculation of the DNBR in steady state condition by utilizing local the calculation of the DNBR in steady state condition by utilizing local properties of the fluid obtained from the transient conditions; It is properties of the fluid obtained from the transient conditions; It is independent of the upper notified uncertainties. It is a normal independent of the upper notified uncertainties. It is a normal distribution. distribution.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

L. Temperatures of the fuel pellet and the cladding  L. Temperatures of the fuel pellet and the cladding  The temperature of the fuel pellet depends on the The temperature of the fuel pellet depends on the

•

thickness of the corrosion layer of zirconium dioxide (ZrO2) on the thickness of the corrosion layer of zirconium dioxide (ZrO2) on the cladding, on the pellet-cladding gap and the pellets conductance. The cladding, on the pellet-cladding gap and the pellets conductance. The uncertainties relative to the calculations of the fuel temperature are uncertainties relative to the calculations of the fuel temperature are essentially two types: essentially two types: • Manufacturing uncertainties (dimensional variations of pellets and Manufacturing uncertainties (dimensional variations of pellets and cladding); cladding); Uncertainties relative to the density and to pellet models Uncertainties relative to the density and to pellet models (conductivity and gap conductance variations). (conductivity and gap conductance variations).

These uncertainties have quantified by comparison of thermal model  These uncertainties have quantified by comparison of thermal model to measurements performed in the reactor core, and by results to measurements performed in the reactor core, and by results obtained from nuclear obtained from nuclear



in all the evaluations where in all the evaluations where

THERMAL-HYDRAULIC IN NUCLEAR REACTOR fuel and cladding during the fabrication. The obtained uncertainties are fuel and cladding during the fabrication. The obtained uncertainties are intervened the fuel then utilized then utilized intervened the fuel temperature. temperature. Other than the uncertainty relative to the temperature mentioned  Other than the uncertainty relative to the temperature mentioned above, uncertainty of measurement during the determination of local above, uncertainty of measurement during the determination of local power and the effect of the density variation and the enrichment rate power and the effect of the density variation and the enrichment rate on the local power are taking into account to establish the thermal flux on the local power are taking into account to establish the thermal flux factor of the hot channel. factor of the hot channel. Uncertainty affecting the determination of temperature of the cladding  Uncertainty affecting the determination of temperature of the cladding results the uncertainty relative to thickness of the zirconium dioxide results the uncertainty relative to thickness of the zirconium dioxide layer. Due to the excellent heat transfer between the surface of layer. Due to the excellent heat transfer between the surface of cladding and the cooling water, the temperature decrease on the cladding and the cooling water, the temperature decrease on the cladding does not give an appreciable contribution to that uncertainty. cladding does not give an appreciable contribution to that uncertainty.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

M. Hydraulic  M. Hydraulic a) Uncertainties related to the pressure-drops in the reactor core and  a) Uncertainties related to the pressure-drops in the reactor core and in the pressure-vessel in the pressure-vessel b) The pressure-drops are calculated on the basis of the most  b) The pressure-drops are calculated on the basis of the most probable mass flow rate. The attached uncertainties concern in the probable mass flow rate. The attached uncertainties concern in the same time the test results and the extrapolation of these values to the same time the test results and the extrapolation of these values to the reactor. reactor. The pressure-drops in the reactor and in the pressure-vessel have  The pressure-drops in the reactor and in the pressure-vessel have essential utilization in the determination of the primary mass flow rate. essential utilization in the determination of the primary mass flow rate. More than that, the tests will be performed on the primary pumps More than that, the tests will be performed on the primary pumps before the first initial start-up to verify the conservative value of the before the first initial start-up to verify the conservative value of the primary mass flow utilized during the design phase and the reactor primary mass flow utilized during the design phase and the reactor analysis. analysis. c) Uncertainties due to the defect of the repartition of the inlet mass  c) Uncertainties due to the defect of the repartition of the inlet mass flow rate flow rate

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The effects of the uncertainties relative to the distribution of the mass  The effects of the uncertainties relative to the distribution of the mass flow rate have been discussed above. flow rate have been discussed above. d) Uncertainties relative to the mass flow rate  d) Uncertainties relative to the mass flow rate The thermal-hydraulic mass flow rate is defined to be utilized in the  The thermal-hydraulic mass flow rate is defined to be utilized in the evaluations of the thermal-hydraulic characteristics of the reactor core evaluations of the thermal-hydraulic characteristics of the reactor core which are taking account on the uncertainties of the prediction and the which are taking account on the uncertainties of the prediction and the measurement. measurement. More however, one admits a maximal 5.5% of thermal-hydraulic mass  More however, one admits a maximal 5.5% of thermal-hydraulic mass flow rate is not efficient in term of heat evacuation capacity of the flow rate is not efficient in term of heat evacuation capacity of the reactor due to that part by-pass the reactor core in different ways in reactor due to that part by-pass the reactor core in different ways in the pressure-vessel mentioned earlier. the pressure-vessel mentioned earlier. e) Uncertainties relative to the hydraulic forces  e) Uncertainties relative to the hydraulic forces As explained earlier, the envelop of the hydraulic forces on the fuel  As explained earlier, the envelop of the hydraulic forces on the fuel assembly is evaluated in normal operation in cold-shutdown assembly is evaluated in normal operation in cold-shutdown conditions and in the transient conditions for a transient of over-speed conditions and in the transient conditions for a transient of over-speed of the primary pumps of 20% superior to the mechanical design mass of the primary pumps of 20% superior to the mechanical design mass flow rate. flow rate.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR f) Uncertainties relative to the hydraulic dimensioning of the internal f) Uncertainties relative to the hydraulic dimensioning of the internal components components The uncertainties are taking into account either by condering  The uncertainties are taking into account either by condering pessimistic hypotheses for the limit conditions of calculations, or with pessimistic hypotheses for the limit conditions of calculations, or with inherent concervatism to the computer code or to numerical schemas inherent concervatism to the computer code or to numerical schemas utilized to effectuate the calculations. utilized to effectuate the calculations. N. Methods of analysis and study data  N. Methods of analysis and study data N.1. Methods utilized to analyse the transients N.1. Methods utilized to analyse the transients  Set point of automatic shutdown of the reactor by the low level of  Set point of automatic shutdown of the reactor by the low level of DNBR and the limit conditions of operation (LCO). DNBR and the limit conditions of operation (LCO).

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In the cases of sollicitation of the low level DNBR protection In the cases of sollicitation of the low level DNBR protection system, it must define the low level DNBR protection threshold. In the system, it must define the low level DNBR protection threshold. In the second case, the main goal is to define a value of the criteria of DNBR second case, the main goal is to define a value of the criteria of DNBR which must be respected for each transient of that category.. which must be respected for each transient of that category

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Transients of type II

 Transients of type III:

The transients are classified in three categories:  The transients are classified in three categories: Transients of power of which the protection of low Transient of type I: Transients of power of which the protection of low  Transient of type I: level DNBR is effective; level DNBR is effective; : Transients of power of which the protection of Transients of type II: Transients of power of which the protection of low level DNBR is in-effective: low level DNBR is in-effective: Transients studied and based hot-shutdown Transients of type III: Transients studied and based hot-shutdown (withdrawal of a control rod cluster at nil power and stream line (withdrawal of a control rod cluster at nil power and stream line rupture). rupture). N.2. Transients of power (Set point of automatic shutdown threshold of  N.2. Transients of power (Set point of automatic shutdown threshold of the reactor by low level of DNBR and limit conditions of operation) the reactor by low level of DNBR and limit conditions of operation) As the the protection of the low level of DNBR and the surveillance of  As the the protection of the low level of DNBR and the surveillance of the LCO of the DNBR are based on the value of DNBR elaborated by the LCO of the DNBR are based on the value of DNBR elaborated by the simplified algorithm, the threshold of automatic shutdown of the the simplified algorithm, the threshold of automatic shutdown of the reactor at low level DNBR and the threshold of the LCO with regard to reactor at low level DNBR and the threshold of the LCO with regard to the DNBR(LCO) are defined by the DNBR(LCO) are defined by

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR taking into account the uncertainties related to the elaboration of the taking into account the uncertainties related to the elaboration of the DNBR and the accuracy of the measurements. DNBR and the accuracy of the measurements. The boiling crisis (DNBR criteria been reached) is avoided by  The boiling crisis (DNBR criteria been reached) is avoided by maintaining the calculated values of DNBR in the operation under the maintaining the calculated values of DNBR in the operation under the thresholds. thresholds. The uncertainties could be different for the two systems because they  The uncertainties could be different for the two systems because they are associated to the accuracy of the system to the relative operation are associated to the accuracy of the system to the relative operation conditions respectively to the latter. conditions respectively to the latter. To define the two thresholds, it is necessary to repartite the transients  To define the two thresholds, it is necessary to repartite the transients in two classes: in two classes: **The transients where the automatic shut-down of the reactor is The transients where the automatic shut-down of the reactor is effective (transients of type I): effective (transients of type I): They are characterized by the following conditions:  They are characterized by the following conditions: The parameters affecting the DNBR during the transient are utilized in  The parameters affecting the DNBR during the transient are utilized in the low level DNBR protection chain; the low level DNBR protection chain; The evolution of parameters is sufficiently low to be correctly  The evolution of parameters is sufficiently low to be correctly registered by the low level DNBR protection chain. registered by the low level DNBR protection chain.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The automatic shut-down of the reactor at low level DNBR is initiated  The automatic shut-down of the reactor at low level DNBR is initiated each time that the DNBR has reach the safety DNBR criterion. each time that the DNBR has reach the safety DNBR criterion. The variable Y = DNBR/ DNBR(PS) is characterised by a mean value  The variable Y = DNBR/ DNBR(PS) is characterised by a mean value (m(Y,PS) and a deviation standard (σ(Y,PS). (m(Y,PS) and a deviation standard (σ(Y,PS). DNBR(PS) is the value of the predicted DNBR by the algorithm of the  DNBR(PS) is the value of the predicted DNBR by the algorithm of the protection system. protection system. The automatic shut-down of the reactor at low level DNBR is then  The automatic shut-down of the reactor at low level DNBR is then initiated at the calculated value of the DNBR by the protection system initiated at the calculated value of the DNBR by the protection system reaches the threshold of the calculated DNBR(PS) as follow: reaches the threshold of the calculated DNBR(PS) as follow: DNBR(PS) = SC/ m(Y,PS)(1 – 1.645 V(Y,PS)  DNBR(PS) = SC/ m(Y,PS)(1 – 1.645 V(Y,PS) 

This means that the probability to avoid the boiling crisis is then This means that the probability to avoid the boiling crisis is then

95% with a confidence of 95%. 95% with a confidence of 95%.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



the surveillance the surveillance threshold of threshold of

The transients on which the protection at low level DNBR is not  The transients on which the protection at low level DNBR is not effective (transients type II): effective (transients type II): In this case, the automatic shut-down is obtained unically by utilizing In this case, the automatic shut-down is obtained unically by utilizing the specific parameters (in several cases, only one parameter, per the specific parameters (in several cases, only one parameter, per example low mass flow rate of the pump). Consequently, it is example low mass flow rate of the pump). Consequently, it is necessary to survey other parameters which are not taking into necessary to survey other parameters which are not taking into account in the elaboration of the automatic shut-down and to account in the elaboration of the automatic shut-down and to determine the threshold of the survey system of the limit of operation determine the threshold of the survey system of the limit of operation conditions. conditions. The most critical point is associated to the power distribution and the  The most critical point is associated to the power distribution and the best method is the surveillance of the initial DNBR. best method is the surveillance of the initial DNBR. In normal operation, the value of the calculated DNBR by the algorithm In normal operation, the value of the calculated DNBR by the algorithm the must be maintained above must be maintained above the DNBR(LCO) in order to avoid the boiling crisis during the considered DNBR(LCO) in order to avoid the boiling crisis during the considered transient. transient.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR The most characteristic transient of such an accident is the total loss  The most characteristic transient of such an accident is the total loss of the primary mass flow (4 pumps). of the primary mass flow (4 pumps). To dimension the surveillance chain of LCO with regard of the DNBR  To dimension the surveillance chain of LCO with regard of the DNBR for this type of transients, one calculates the maximal DNBR for all for this type of transients, one calculates the maximal DNBR for all transients of that category. transients of that category. The transient which lead to the maximal variation of DNBR permits the  The transient which lead to the maximal variation of DNBR permits the determination of the low threshold of DNBR of the surveillance of LCO determination of the low threshold of DNBR of the surveillance of LCO (DNBR(LCO)). (DNBR(LCO)). DNBR (LCO) = [SC + (ΔΔDNBR)max]/[m(Y,SS) (1 – 1.645V(Y,SS)] DNBR)max]/[m(Y,SS) (1 – 1.645V(Y,SS)]  DNBR (LCO) = [SC + ( Where m(Y,SS) and V(Y,SS) are the mean value and the standard  Where m(Y,SS) and V(Y,SS) are the mean value and the standard deviation of the vriable Y = DNBR/DNBR(SS) and DNBR(ss) is the value deviation of the vriable Y = DNBR/DNBR(SS) and DNBR(ss) is the value of DNBR calculated by the surveillance system. of DNBR calculated by the surveillance system. In normal operation, the DNBR must be always above the low In normal operation, the DNBR must be always above the low threshold DNBR (DNBR(LCO)), the DNBR criterion will be respected if threshold DNBR (DNBR(LCO)), the DNBR criterion will be respected if that type of transient occurs. that type of transient occurs.

Ỵ Ỵ

THERMAL-HYDRAULIC IN NUCLEAR REACTOR N.3. Transients at hot nil power (Design DNBR criterion)  N.3. Transients at hot nil power (Design DNBR criterion) The power distribution in the reactor core, largely determined at the  The power distribution in the reactor core, largely determined at the beginning of life of enrichment rate of the fuel, the loading plan and the beginning of life of enrichment rate of the fuel, the loading plan and the power level of the reactor core, depends equaly of variables of which power level of the reactor core, depends equaly of variables of which the efficiency and the position of the control rods cluster and the burn- the efficiency and the position of the control rods cluster and the burn- up nuclear fuel. The radial distributions of the enthalpy elevation in the up nuclear fuel. The radial distributions of the enthalpy elevation in the reactor core, which are determined by the integrated power of each reactor core, which are determined by the integrated power of each channel, are great importance in the analysis of the DNBR. Those channel, are great importance in the analysis of the DNBR. Those power distributions are characterized by the FΔH(N) and by the axial power distributions are characterized by the FΔH(N) and by the axial distribution of the thermal flux. distribution of the thermal flux. Due to the local power q’(W/cm) at a point x, y, z in a reactor core  Due to the local power q’(W/cm) at a point x, y, z in a reactor core composed with N fuel rods and with high (H), the nuclear enthalpy composed with N fuel rods and with high (H), the nuclear enthalpy elevation is expressed as: elevation is expressed as: FΔH(N) = Power of hot fuel rod/ mean power of the fuel =  FΔH(N) = Power of hot fuel rod/ mean power of the fuel = (maxƒ(0,H)q’(x(0), 0),Z)dz)/[1/N(all rods)∑ maxƒ(0,H)q’(x(0), 0), Z)dz Ỵ (maxƒ(0,H)q’(x(0), 0),Z)dz)/[1/N(all rods)∑ maxƒ(0,H)q’(x(0), 0), Z)dz Ỵ  Where x(0), 0) are coordinates of the position of hot fuel rod.  Where x(0), 0) are coordinates of the position of hot fuel rod.

ỵ ỵ

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The utilization modality of the FΔH(N) in the calculations of the DNBR  The utilization modality of the FΔH(N) in the calculations of the DNBR is important. is important. The minimal position of the DNBR depends on power axial shape and  The minimal position of the DNBR depends on power axial shape and the value of the DNBR depends on the enthalpy elevation at that point. the value of the DNBR depends on the enthalpy elevation at that point. Fundamentally, the maximal integrated power by the fuel rods in the Fundamentally, the maximal integrated power by the fuel rods in the core is utilized to identify the most probable fuel rod with the minimal core is utilized to identify the most probable fuel rod with the minimal DNBR. DNBR. One obtains an axial shape of the power which is, when normalized to  One obtains an axial shape of the power which is, when normalized to the FΔH(N) value represents the thermal axial flux all along the hot fuel the FΔH(N) value represents the thermal axial flux all along the hot fuel rod. The neighbouring fuel rods are supposed having the same axial rod. The neighbouring fuel rods are supposed having the same axial shape and the distributions of integrated power per fuel rod (radial shape and the distributions of integrated power per fuel rod (radial distribution) corresponds to a typical distribution of the hot fuel distribution) corresponds to a typical distribution of the hot fuel assembly. To this manner, the axial shapes corresponding to assembly. To this manner, the axial shapes corresponding to penalizing cases could be combined to the radial distributions penalizing cases could be combined to the radial distributions corresponding to the penalizing cases for the calculations of the corresponding to the penalizing cases for the calculations of the reference DNBR reference DNBR

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H(x,y)) H = max(xy)(PΔΔH(x,y))

THERMAL-HYDRAULIC IN NUCLEAR REACTOR It is convenience to note again that FΔH(N) is an integral and is utilized It is convenience to note again that FΔH(N) is an integral and is utilized such as in the determination of the DNBR. The local thermal flux are such as in the determination of the DNBR. The local thermal flux are obtained by utilizing the related power distributions of the hot fuel obtained by utilizing the related power distributions of the hot fuel rods and the adjacent fuel rods which are taking into account the rods and the adjacent fuel rods which are taking into account the variations of the power map forms for the whole reactor core. variations of the power map forms for the whole reactor core. The enthalpy elevation factor FΔH corresponds to:  The enthalpy elevation factor FΔH corresponds to:  FFΔΔH = max(xy)(P Where PΔH(x,y) is the radial enthalpy elevation of the channel (x,y) :  Where PΔH(x,y) is the radial enthalpy elevation of the channel (x,y) :  PPΔΔH(x,y) = ƒ(0,H) P(x,y,z)dz/ƒ(0,H)dz H(x,y) = ƒ(0,H) P(x,y,z)dz/ƒ(0,H)dz Where P(x,y,z)dz is the relative power in the channel (x,y) at level (z).  Where P(x,y,z)dz is the relative power in the channel (x,y) at level (z). For each channel, mean relative power is the mean of the relative  For each channel, mean relative power is the mean of the relative powers of fuel rod surrounding the channel, extrapolated to the heated powers of fuel rod surrounding the channel, extrapolated to the heated perimeter of the fuel part located at the considered channel. perimeter of the fuel part located at the considered channel.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

The design studies are performed with the power distributions  The design studies are performed with the power distributions calculated by the neutron computer codes. calculated by the neutron computer codes. For the transients of types I and II, the protection functions associated  For the transients of types I and II, the protection functions associated to I&C of the reactor core protect it against the penalizing power to I&C of the reactor core protect it against the penalizing power distributions. distributions. The power distribution of the hot channel is directly issued of the in-  The power distribution of the hot channel is directly issued of the in- core instrumentation by self-powered detectors. core instrumentation by self-powered detectors. Consequently, the two following uncertainties are associated to the  Consequently, the two following uncertainties are associated to the power distributions: power distributions: Uncertainty on the reconstruction of the power distribution;  Uncertainty on the reconstruction of the power distribution; The accuracy of the measurements.  The accuracy of the measurements. For the transient of type III, the uncertainty relative to the power  For the transient of type III, the uncertainty relative to the power distribution is the calculation uncertainty which affects the DNBR distribution is the calculation uncertainty which affects the DNBR determined by the design computer code. determined by the design computer code. For the analyses of accident such as LOCA, one utilizes a decoupling  For the analyses of accident such as LOCA, one utilizes a decoupling design value of FΔH(N) of 1.80. design value of FΔH(N) of 1.80.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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O. Hot channel factors  O. Hot channel factors The power distributions are characterized by “hot factors”. Such  The power distributions are characterized by “hot factors”. Such factors corresponded to the measurement of the peak power of the factors corresponded to the measurement of the peak power of the pellets in the reactor core and the total energy release in a cooled sub- pellets in the reactor core and the total energy release in a cooled sub- channel, and they are expressed in relative quantity to the nuclear channel, and they are expressed in relative quantity to the nuclear design or thermal. design or thermal. Once of the main goals of the thermal hydraulic core analysis is to  Once of the main goals of the thermal hydraulic core analysis is to ensure that the thermal limitations on the core behaviour are not ensure that the thermal limitations on the core behaviour are not exceeded. So far two such limitations have been discussed shortly: exceeded. So far two such limitations have been discussed shortly: To exclude melting of the fuel, the linear power density must be  To exclude melting of the fuel, the linear power density must be limited: limited:

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q’(r) < q”(max) (XVI.1) q’(r) < q”(max) (XVI.1)

Another limitation is dictated by the requirement that the surface Another limitation is dictated by the requirement that the surface

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heat flux of the fuel cladding always remains below the critical heat heat flux of the fuel cladding always remains below the critical heat flux limit, flux limit,

q’(r) < q”(CHF) (XVI.2) q’(r) < q”(CHF) (XVI.2)

Here r designs any location in the reactor core.  Here r designs any location in the reactor core.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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In addition to the two limitations, other issues like thermal and In addition to the two limitations, other issues like thermal and limit the core fission gas stresses on the fuel cladding can fission gas stresses on the fuel cladding can limit the core performance and thus can limit power generation. Furthermore, the performance and thus can limit power generation. Furthermore, the core stability consideration may be another limiting circumstance. core stability consideration may be another limiting circumstance.

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A thorough thermal-hydraulic analysis of the core requires a A thorough thermal-hydraulic analysis of the core requires a detailed, three-dimensional calculation of the core power distribution, detailed, three-dimensional calculation of the core power distribution, including the effects of fuel burn-up, fission product build-up, control including the effects of fuel burn-up, fission product build-up, control distributions, and moderator density variations over core life. distributions, and moderator density variations over core life.

This information is next used to determine the coolant flow and This information is next used to determine the coolant flow and temperature distribution throughout the reactor core. Even though temperature distribution throughout the reactor core. Even though such types of calculations are performed nowadays, they are quite such types of calculations are performed nowadays, they are quite Especially for fast transient expensive and time consuming. expensive and time consuming. Especially for fast transient applications they are prohibited expenses and are avoided. applications they are prohibited expenses and are avoided.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR To make the thermal-hydraulic analysis more practical, a common  To make the thermal-hydraulic analysis more practical, a common approach is to investigate how closely the “hot channel” in the reactor approach is to investigate how closely the “hot channel” in the reactor core approaches the operating limitations. Then if one can measure core approaches the operating limitations. Then if one can measure that thermal conditions of this channel remain below the reactor core that thermal conditions of this channel remain below the reactor core limitations, the remaining channels will presumably fall within design limitations, the remaining channels will presumably fall within design limitations. One usually defines the hot channel in the reactor core as limitations. One usually defines the hot channel in the reactor core as that coolant channel in which the reactor core feat flux and enthalpy that coolant channel in which the reactor core feat flux and enthalpy rise is maximal. Associated with this channel are various hot channels rise is maximal. Associated with this channel are various hot channels or “hot factors” relating the performance of this channel to the or “hot factors” relating the performance of this channel to the average behaviour of the reactor core. average behaviour of the reactor core.

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The fuel assembly having maximal power output is defined as the The fuel assembly having maximal power output is defined as the “hot assembly. The hot spot in the reactor core is the point of “hot assembly. The hot spot in the reactor core is the point of maximum heat flux or linear power density, while the hot channel is maximum heat flux or linear power density, while the hot channel is defined as the coolant sub-channel in which the hot spot occurs or defined as the coolant sub-channel in which the hot spot occurs or along which the maximal coolant enthalpy rise occurs. along which the maximal coolant enthalpy rise occurs.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The “nuclear hot channel” is defined to take into consideration the  The “nuclear hot channel” is defined to take into consideration the variation of the neutron flux and fuel distribution within the reactor variation of the neutron flux and fuel distribution within the reactor core. core. The “radial nuclear hot channel factor” is defined as:  The “radial nuclear hot channel factor” is defined as: F(N,R) = Average heat flux of the hot channel/Average heat flux of the  F(N,R) = Average heat flux of the hot channel/Average heat flux of the channels in the reactor core channels in the reactor core

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= ƒq”(r(hc))dz/(1/N(c)∑ƒq”(r)dz) = ƒq”(r(hc))dz/(1/N(c)∑ƒq”(r)dz)



(XVI.3) (XVI.3)

is the total number of channels in the reactor core. In a Where N(c)N(c) is the total number of channels in the reactor core. In a Where

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similar, the “axial nuclear hot channel factor” is defined as: similar, the “axial nuclear hot channel factor” is defined as: F(N,Z) = Maximal heat flux of the hot channel/Average heat flux of the  F(N,Z) = Maximal heat flux of the hot channel/Average heat flux of the hot channel = max[q”(r(hc)]/ [1/Lƒq”(r(hc))dz hot channel = max[q”(r(hc)]/ [1/Lƒq”(r(hc))dz (XVI.4) (XVI.4)

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The “total nuclear hot channel factor or nuclear heat flux factor” The “total nuclear hot channel factor or nuclear heat flux factor”

is then: is then:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR F(N,q) = Maximal heat flux in the reactor core/ Average heat flux in the  F(N,q) = Maximal heat flux in the reactor core/ Average heat flux in the reactor core = F(N,Z)F(N,R) reactor core = F(N,Z)F(N,R) (XVI.5) (XVI.5) The power distribution in a bar cylindrical core is described by the  The power distribution in a bar cylindrical core is described by the following expression: following expression:

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(XVI.6) Q’” (z) = w(f)∑(f)Ф(0)J(0)[(2.405r(f)/R]sin[[πz/L] (XVI.6) Q’” (z) = w(f)∑(f)Ф(0)J(0)[(2.405r(f)/R]sin[[πz/L]

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The radial factor is then: The radial factor is then:

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F(N,R) = [J(0)ƒsin(π(z)/l)dz]/[ƒJ(0)(2.405r/R)2πrdrƒsin(πz/L)dz F(N,R) = [J(0)ƒsin(π(z)/l)dz]/[ƒJ(0)(2.405r/R)2πrdrƒsin(πz/L)dz

(XVI.7) = 2.32 (XVI.7)

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= 2.32 And the axial factor is:  And the axial factor is: F(N,Z) = [J(0)sin(π/2)/J(0)1/Lƒsin(πz)/L)dz ~ 1.57  F(N,Z) = [J(0)sin(π/2)/J(0)1/Lƒsin(πz)/L)dz ~ 1.57 (XVI.8) (XVI.8)

This implies the overall hot channel factor: This implies the overall hot channel factor:

F(q,N) ~ 2.31 x 1.57 ~ 3.642  F(q,N) ~ 2.31 x 1.57 ~ 3.642

THERMAL-HYDRAULIC IN NUCLEAR REACTOR For a zone loaded PWR will have typically a nuclear hot channel factor  For a zone loaded PWR will have typically a nuclear hot channel factor F(q,N) = 2.6 F(q,N) = 2.6

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This is a quite conservative (e.g. high) estimation of the factor. A This is a quite conservative (e.g. high) estimation of the factor. A f(q,N) ~ zone-loaded PWR will typically have a nuclear hot channel of f(q,N) ~ zone-loaded PWR will typically have a nuclear hot channel of 2.6.2.6.

The nuclear heat flux hot channel is defined by assuming nominal The nuclear heat flux hot channel is defined by assuming nominal

 F(q,E) = Fq/F(q,N) 

(XVI.10) (XVI.10)

fuel pellet and rod parameters. In reality, however, there will be local fuel pellet and rod parameters. In reality, however, there will be local variation in fuel pellet density, enrichment and diameter, surface area variation in fuel pellet density, enrichment and diameter, surface area of fuel rod and eccentricity of the fuel-clad ding gap due to of fuel rod and eccentricity of the fuel-clad ding gap due to manufacturing tolerances and operating conditions. The more general manufacturing tolerances and operating conditions. The more general heat flux hot channel factor or total power peaking factor Fq is defined heat flux hot channel factor or total power peaking factor Fq is defined as the maximal heat flux in the hot channel divided by the average heat as the maximal heat flux in the hot channel divided by the average heat flux in the reactor core (allowing for above mentioned variability). Fq flux in the reactor core (allowing for above mentioned variability). Fq and F(q,N) are related by defining an “engineering heat flux hot- and F(q,N) are related by defining an “engineering heat flux hot- channel factor F(q,E): channel factor F(q,E): F(q,E) = Fq/F(q,N)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR is close to unity reflecting the fact that manufacturing F(q,E) is close to unity reflecting the fact that manufacturing

 Typically

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 F(ΔH) = Maximal coolant enthalpy rise/Average coolant enthalpy rise

Typically F(q,E) tolerances are quite small (in modern PWRs, F(q,E) ~ 1.03) tolerances are quite small (in modern PWRs, F(q,E) ~ 1.03)



One can also define an “enthalpy-rise hot channel factor”, as: One can also define an “enthalpy-rise hot channel factor”, as: F(ΔH) = Maximal coolant enthalpy rise/Average coolant enthalpy rise (XVI.11) (XVI.11)

This factor is a function of both variation in the power distribution This factor is a function of both variation in the power distribution

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and coolant flow. For example, some 3-10% of the coolant flow and coolant flow. For example, some 3-10% of the coolant flow bypasses the fuel assemblies, due to the leaks or the presence of bypasses the fuel assemblies, due to the leaks or the presence of other core components. This factor accounts for manufacturing other core components. This factor accounts for manufacturing tolerances and also structure displacement (box bow, rod bow, etc.) tolerances and also structure displacement (box bow, rod bow, etc.) caused by the operation conditions. caused by the operation conditions.



The factors used to characterize the power distributions are The factors used to characterize the power distributions are



ỵ ỵ

defined as following: defined as following: * Fx (z): Radial peak hot factor in the horizontal plane at elevation (z), * Fx (z): Radial peak hot factor in the horizontal plane at elevation (z), defines as the ratio of the maximal linear power on the average linear defines as the ratio of the maximal linear power on the average linear power of that plan. power of that plan.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR * P(z): Is the axial distribution of the relative average power, defined as * P(z): Is the axial distribution of the relative average power, defined as the ratio of of average linear power in the horizontal plane at elevation the ratio of of average linear power in the horizontal plane at elevation (z) on the average linear power of the fuel rods in the reactor core. (z) on the average linear power of the fuel rods in the reactor core. * Q(z): Is the relative maximal linear power at elevation (z), is defined * Q(z): Is the relative maximal linear power at elevation (z), is defined as the ratio of the local maximal linear power density of the fuel rods at as the ratio of the local maximal linear power density of the fuel rods at (z) to the average linear power density: (z) to the average linear power density:

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Q(z) = Fx (z) x P(z) ỵ Q(z) = Fx (z) x P(z) ỵ

* F(q): Is the hot factor, defined as the ratio of the local linear power * F(q): Is the hot factor, defined as the ratio of the local linear power density to the average linear power density: density to the average linear power density: F(q)max = MaxQ(z) without uncertainties and penalties  F(q)max = MaxQ(z) without uncertainties and penalties * The following uncertainties and penalties are applied to the design * The following uncertainties and penalties are applied to the design  values Fq: values Fq: * F(q,E): Technological uncertainty factor of hot point, is the heat flux * F(q,E): Technological uncertainty factor of hot point, is the heat flux provision to accommodate the manufacturing tolerances. This provision to accommodate the manufacturing tolerances. This technological factor permits to take into account the discrepancies of technological factor permits to take into account the discrepancies of manufacturing which are related to fuel enrichment, density, pellet manufacturing which are related to fuel enrichment, density, pellet diameter, fuel cladding surface and eccentricity of pellet-cladding gap. diameter, fuel cladding surface and eccentricity of pellet-cladding gap.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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* F(U,N): Nuclear incertitude factor, taking into account the * F(U,N): Nuclear incertitude factor, taking into account the uncertainties related to of calculated power distributions. uncertainties related to of calculated power distributions. * F(Xe): penalties taking into account the axial and radial oscillations * F(Xe): penalties taking into account the axial and radial oscillations of Xenon. of Xenon. * F(t): total uncertainty factor taking into account all of uncertainties * F(t): total uncertainty factor taking into account all of uncertainties mentioned above. mentioned above.

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The peak factor defined for the design can be expressed as: The peak factor defined for the design can be expressed as:

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F(q,D) = F(q) x F(t) F(q,D) = F(q) x F(t)

* F(ΔH, N), is the enthalpy rise factor, defined as the ratio between the * F(ΔH, N), is the enthalpy rise factor, defined as the ratio between the integrated linear power of the fuel rod (having the maximal integrated integrated linear power of the fuel rod (having the maximal integrated power) to the average power of fuel rod in the reactor core. power) to the average power of fuel rod in the reactor core.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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The manufacturing tolerances, the power distribution in the hot  The manufacturing tolerances, the power distribution in the hot channel and the power distributions of neighbouring channels are channel and the power distributions of neighbouring channels are treated in the calculation of the DNBR. treated in the calculation of the DNBR. XVI.1. Radial power distribution  XVI.1. Radial power distribution The radial power distributions in the horizontal planes of the reactor  The radial power distributions in the horizontal planes of the reactor core are function of: core are function of: Fuel assemblies loading scheme;  Fuel assemblies loading scheme; The place where is located the poisoned rods;  The place where is located the poisoned rods; On the fuel assembly burn-up;  On the fuel assembly burn-up; Level of nuclear power and effects of moderator density;  Level of nuclear power and effects of moderator density; On the concentration and the repartition of Xenon and Samarium.  On the concentration and the repartition of Xenon and Samarium. On the other hand, the effects of non-uniform distribution of the  On the other hand, the effects of non-uniform distribution of the primary coolant rate are negligible. primary coolant rate are negligible.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The hot channel power resulted of the superposition of the  The hot channel power resulted of the superposition of the macroscopic distribution of the power in the reactor core and the macroscopic distribution of the power in the reactor core and the power distribution per fuel rod in the fuel assembly. power distribution per fuel rod in the fuel assembly. As the position of the hot channel changes with the time, only one  As the position of the hot channel changes with the time, only one radial power distribution is choose as reference for the determination radial power distribution is choose as reference for the determination of the DNBR in the design. This reference distribution is choose as of the DNBR in the design. This reference distribution is choose as conservative manner by concentring the power in a zone of the reactor conservative manner by concentring the power in a zone of the reactor core, in order to minimize the gains due to the redistribution of the core, in order to minimize the gains due to the redistribution of the moderator coolant rate. The fuel assembly powers are normalized to moderator coolant rate. The fuel assembly powers are normalized to the core average power. the core average power. As the fine distribution of the power surrounding the hot channel  As the fine distribution of the power surrounding the hot channel changes with the time, one take as conservative manner in the changes with the time, one take as conservative manner in the analyses of the DNBR, a radial power distribution in the fuel assembly, analyses of the DNBR, a radial power distribution in the fuel assembly, by imposing artificially to the fuel rod the maximal integrated value of by imposing artificially to the fuel rod the maximal integrated value of FΔH(N), defined for the design. FΔH(N), defined for the design.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR One interests that, for the nuclear design, for all cycles and operating  One interests that, for the nuclear design, for all cycles and operating conditions, to guarantee a more plate distribution could not be conditions, to guarantee a more plate distribution could not be encountered with the FΔH(N) limits. encountered with the FΔH(N) limits. XVI.2. Axial power distribution  XVI.2. Axial power distribution The form of the axial power distribution depends principally:  The form of the axial power distribution depends principally: On the position of control clusters;  On the position of control clusters; On the nuclear power level;  On the nuclear power level; On the axial Xenon distribution;  On the axial Xenon distribution; The effects of feedback reactions of Doppler effects and the moderator  The effects of feedback reactions of Doppler effects and the moderator densities; densities; On the fuel assembly burn-up;  On the fuel assembly burn-up; On the axial design of the fuel assembly (to optimize the axial power  On the axial design of the fuel assembly (to optimize the axial power distribution).. distribution)

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR The theoretical distribution of neutron flux (or of the resulting thermal  The theoretical distribution of neutron flux (or of the resulting thermal power) is determined by core physics computations for the various power) is determined by core physics computations for the various reactor operating modes (normal or accidents), taking into account the reactor operating modes (normal or accidents), taking into account the fuel burn-up & subsequent refueling. fuel burn-up & subsequent refueling. As a first approximation, it can be assumed that the flux distribution  As a first approximation, it can be assumed that the flux distribution along the core axis (z) is the same at every (x,y) position. This average along the core axis (z) is the same at every (x,y) position. This average axial flux distribution (or axial flux shape) depends a certain number of axial flux distribution (or axial flux shape) depends a certain number of factors, such as the power level, the control rods positions and the factors, such as the power level, the control rods positions and the fuel burn-up. fuel burn-up. The axial flux shapes can be symmetrically (cosinus form per  The axial flux shapes can be symmetrically (cosinus form per example), offset towards the bottom (control rods inserted) or offset example), offset towards the bottom (control rods inserted) or offset towards the top (end of cycle, after the RCCA withdrawal), as shown in towards the top (end of cycle, after the RCCA withdrawal), as shown in the following figure. the following figure. The flux shape is characterized by its relative axial power unbalance or  The flux shape is characterized by its relative axial power unbalance or « axial offset » « AO », the value of which plays an important role in the « axial offset » « AO », the value of which plays an important role in the core design. core design.

(XVI.1) (XVI.1)

(XVI.2) (XVI.2)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The signals obtained from in-core and ex-core instrumentations. These  The signals obtained from in-core and ex-core instrumentations. These signals are used to control the reactor core in normal operation to signals are used to control the reactor core in normal operation to determine the average axial power distribution of the core, which is determine the average axial power distribution of the core, which is characterized by axial offset (AO) or axial unbalance ΔI: characterized by axial offset (AO) or axial unbalance ΔI: Axial offset = AO = (P(H) – P(B))/(P(H)+P(B))  Axial offset = AO = (P(H) – P(B))/(P(H)+P(B)) Where P(H) and P(B): fraction of power produced by the upper-half of  Where P(H) and P(B): fraction of power produced by the upper-half of the reactor core. the reactor core. ΔI = (P(h) – P(B))/(P(H) + P(B)nominal) = AO x Pr  ΔI = (P(h) – P(B))/(P(H) + P(B)nominal) = AO x Pr Where Pr: relative power level by comparing with the nominal power.  Where Pr: relative power level by comparing with the nominal power.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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Figure XVI.2.1: Representation of axial offset  Figure XVI.2.1: Representation of axial offset 

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVI.2.2: Evolution of the temperature in average channel & hot  Figure XVI.2.2: Evolution of the temperature in average channel & hot channel channel

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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XVI.3. Limitations of power distribution  XVI.3. Limitations of power distribution The fuel management of the fuel assembly and the positions of the  The fuel management of the fuel assembly and the positions of the control clusters are choose in manner to limit the perturbations of the control clusters are choose in manner to limit the perturbations of the radial power distribution during the normal operation. radial power distribution during the normal operation. The control rods worth are choose in manner to limit the perturbation  The control rods worth are choose in manner to limit the perturbation of the axial power distribution. of the axial power distribution. In order to limit the axial power oscillations due to Xenon, the axial In order to limit the axial power oscillations due to Xenon, the axial power distribution is control by maintaining the axial offset in the power distribution is control by maintaining the axial offset in the authorized zone of operating within the reference value. Thus authorized zone of operating within the reference value. Thus minimizes the effects of Xenon transients on the axial power form, minimizes the effects of Xenon transients on the axial power form, because the Xenon distribution remains in phase with the power because the Xenon distribution remains in phase with the power distribution. distribution. The more penalized power distribution or more limiting which could  The more penalized power distribution or more limiting which could encountered in normal operating, must be considered as the initial encountered in normal operating, must be considered as the initial state for the events classes 2 to 4. state for the events classes 2 to 4.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR These limiting power distribution are determined in a pessimistic  These limiting power distribution are determined in a pessimistic manner. However, they are taking into account of the surveillance of manner. However, they are taking into account of the surveillance of the maximal linear power density Q(z) and the DNBR, but by respecting the maximal linear power density Q(z) and the DNBR, but by respecting their value limits. Instruments and control functions are provided to their value limits. Instruments and control functions are provided to guarantee that those limits are respected. guarantee that those limits are respected. XVI.4. Reactivity coefficients  XVI.4. Reactivity coefficients The kinetic characteristics of the reactor core govern the response of  The kinetic characteristics of the reactor core govern the response of the reactor core to the variations of the states of the NPP or to the the reactor core to the variations of the states of the NPP or to the operator actions in normal operation, also to the response of the operator actions in normal operation, also to the response of the reactor core related to incidental or accidental conditions. The reactor core related to incidental or accidental conditions. The reactivity coefficients reflect the variations of the “multiplication reactivity coefficients reflect the variations of the “multiplication factor” of neutrons due to the changes of parameters of operating of factor” of neutrons due to the changes of parameters of operating of the reactor core (e.g. nuclear power, moderator temperature or fuel the reactor core (e.g. nuclear power, moderator temperature or fuel temperature). temperature).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR As the reactivity coefficients change during the cycle, one utilises the  As the reactivity coefficients change during the cycle, one utilises the zones of coefficients to analyse the transients, such a manner to zones of coefficients to analyse the transients, such a manner to determine the behaviour of the reactor for during all cycle. determine the behaviour of the reactor for during all cycle. XVI.5 Reactivity control  XVI.5 Reactivity control 

a) Safety requirements a) Safety requirements

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The safety function assured by the functional design of the control  The safety function assured by the functional design of the control systems of the reactivity is to control the reactivity of the reactor core systems of the reactivity is to control the reactivity of the reactor core in order to assure the shutdown of the chain reaction in any in order to assure the shutdown of the chain reaction in any circumstances and to the safe shutdown of the reactor core. circumstances and to the safe shutdown of the reactor core. The functional design of the reactivity control must assure the  The functional design of the reactivity control must assure the realization of such functions for the all reference operating conditions realization of such functions for the all reference operating conditions of classes 1 to 4. of classes 1 to 4. b) Functional criteria b) Functional criteria

The reactivity of the reactor core must controlled in all normal The reactivity of the reactor core must controlled in all normal

operating conditions, from the start-up to the shutdown, by two operating conditions, from the start-up to the shutdown, by two different functional means. different functional means.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

c) Design requirements  c) Design requirements The functional design of the control systems of the reactivity is not  The functional design of the control systems of the reactivity is not under the specific design requirements. However, the safety functions under the specific design requirements. However, the safety functions which assures require the application of quality assurance programme which assures require the application of quality assurance programme where the main objective is to record and control the associated where the main objective is to record and control the associated activities. activities. Thus, the systems which are realized all functions relative to the  Thus, the systems which are realized all functions relative to the control of the reactivity must respect the design requirements. Each control of the reactivity must respect the design requirements. Each system must respond to the design requirements described to the system must respond to the design requirements described to the section reporting to that system. section reporting to that system. d) Tests  d) Tests The conformity of the reactor core with regard to the functional design  The conformity of the reactor core with regard to the functional design of the reactivity control will be verified by the physical controls at of the reactivity control will be verified by the physical controls at certain moments and all along the reactor life. certain moments and all along the reactor life.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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e) Design basis  e) Design basis The safety criteria which are to be respected in the safety analyses.  The safety criteria which are to be respected in the safety analyses. They are defined in term of radiological limits. They are defined in term of radiological limits. Further more of such safety criteria, it is practice, to introduce some  Further more of such safety criteria, it is practice, to introduce some decoupling criteria which must be applied to the thermal-hydraulic and decoupling criteria which must be applied to the thermal-hydraulic and neutron calculations. neutron calculations. f) The maximal control insertion of the reactivity f) The maximal control insertion of the reactivity The maximal control insertion of reactivity is due to the withdrawal of  The maximal control insertion of reactivity is due to the withdrawal of the control rods clusters or the boron acid dilution is limited. For a the control rods clusters or the boron acid dilution is limited. For a normal operation in full power, the maximal variation of the reactivity normal operation in full power, the maximal variation of the reactivity in case of accident of withdrawal of the control rods clusters is defined in case of accident of withdrawal of the control rods clusters is defined as the thermal power peak and the DNBR do not exceeded the fixed as the thermal power peak and the DNBR do not exceeded the fixed limits in case of overpower. limits in case of overpower.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

g) Shutdown margin  g) Shutdown margin The sufficient shutdown margin and a certain subcritical are required  The sufficient shutdown margin and a certain subcritical are required in the full power operating conditions and in the shutdown conditions in the full power operating conditions and in the shutdown conditions respectively. respectively. Two independent reactivity control systems have been set up, such as  Two independent reactivity control systems have been set up, such as the control rods clusters system and the soluble boron acid in the the control rods clusters system and the soluble boron acid in the primary coolant. The control rods clusters system could compensate primary coolant. The control rods clusters system could compensate the reactivity effects due to the temperature variation of the fuel and the reactivity effects due to the temperature variation of the fuel and the coolant accompanying the variations of the power level during the the coolant accompanying the variations of the power level during the period of full power to zero power. More however, the control rods period of full power to zero power. More however, the control rods clusters system provide a minimal shutdown margin in case of clusters system provide a minimal shutdown margin in case of accident, and it is able to assure the sub-criticality of the core accident, and it is able to assure the sub-criticality of the core sufficiently rapid to avoid the exceeding of the acceptable limits of sufficiently rapid to avoid the exceeding of the acceptable limits of fuel damaging, by supposing the more efficiency remains blocked out fuel damaging, by supposing the more efficiency remains blocked out off the reactor core during the emergency shutdown. off the reactor core during the emergency shutdown.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR The borification systems could compensate all of variations of xenon  The borification systems could compensate all of variations of xenon concentrations and variations of density leading to the variations of concentrations and variations of density leading to the variations of the reactivity, and they allow the reactor to reach and to maintain in the reactivity, and they allow the reactor to reach and to maintain in the cold-shutdown. Thus, the sufficient shutdown margin is obtained the cold-shutdown. Thus, the sufficient shutdown margin is obtained by two means of control, one is the mechanical system, and another is by two means of control, one is the mechanical system, and another is the chemical burnable poison. the chemical burnable poison. h) Sub-criticality  h) Sub-criticality In shutdown states, the reactor core must be maintained in the sub- In shutdown states, the reactor core must be maintained in the sub-  criticality sufficient to assure the safety of the reactor in case off criticality sufficient to assure the safety of the reactor in case off accidental transients occurred in that state. accidental transients occurred in that state. When the fuel assemblies are in the reactor pressure-vessel and its  When the fuel assemblies are in the reactor pressure-vessel and its cover head has been removed, the considered transients are the cover head has been removed, the considered transients are the dilution of boron acid and the withdrawal of all control rods clusters. dilution of boron acid and the withdrawal of all control rods clusters. If the pressure–vessel is closed, in the cold-shutdown, the accidents If the pressure–vessel is closed, in the cold-shutdown, the accidents considered are the uncontrolled dilution and the control rods clusters considered are the uncontrolled dilution and the control rods clusters ejection; in the hot conditions, the vapor tube rupture must be ejection; in the hot conditions, the vapor tube rupture must be considered. considered.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR i) Description of the functional aspects of the design of the systems i) Description of the functional aspects of the design of the systems The functional design of the reactivity control has several  The functional design of the reactivity control has several consequences on a great number of systems. The principles of the consequences on a great number of systems. The principles of the specific design basis to those systems reflect the impact of the specific design basis to those systems reflect the impact of the functional design of the reactivity control. functional design of the reactivity control. j)The command system of the control rods clusters j)The command system of the control rods clusters The command system of the control rods clusters system responds to  The command system of the control rods clusters system responds to the activation signals which are could be generated by the control and the activation signals which are could be generated by the control and the protection systems of the reactor or by the intervention of the the protection systems of the reactor or by the intervention of the operator. These activation signals allow the displacement of the operator. These activation signals allow the displacement of the control rods clusters by the demagnetisation of the coils of control control rods clusters by the demagnetisation of the coils of control rods clusters mechanisms. rods clusters mechanisms. The command system of the control rods clusters allows the reactor  The command system of the control rods clusters allows the reactor core to reach the sub-critical state in a short period of time, what ever core to reach the sub-critical state in a short period of time, what ever the initial level power. For most of the accidental studies (except the the initial level power. For most of the accidental studies (except the LOCA) and Steam generator tube rupture) the control rods clusters LOCA) and Steam generator tube rupture) the control rods clusters provide sufficient margin to assure the safety state of the reactor. provide sufficient margin to assure the safety state of the reactor.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR k) Chemical & Volume Control System (CVCS) & Emergency Boration  k) Chemical & Volume Control System (CVCS) & Emergency Boration System (EBS) System (EBS) The CVCS responds to the activation signals generated by the  The CVCS responds to the activation signals generated by the command system of the reactor or by the intervention of the the command system of the reactor or by the intervention of the the operator. It allows to adjust the boron concentration to the required operator. It allows to adjust the boron concentration to the required value, in the shutdown states. value, in the shutdown states. The CVCS is not classified as a safety system, its function of the  The CVCS is not classified as a safety system, its function of the reactivity control is assured by the emergency boration system (EBS) reactivity control is assured by the emergency boration system (EBS) in case of accident. in case of accident. The EBS responds to the activation signals by the intervention of the  The EBS responds to the activation signals by the intervention of the operator, except the ATWS signal which is actuated automatically. operator, except the ATWS signal which is actuated automatically. The EBS allows the assurance of the reactor sub-criticality in the long  The EBS allows the assurance of the reactor sub-criticality in the long term phase of the accidents. Grace to him, it allows to reach the safe term phase of the accidents. Grace to him, it allows to reach the safe shutdown in most of cases of accidents (except the LOCA), and it is shutdown in most of cases of accidents (except the LOCA), and it is indispensable to reach the controlled state in case of steam generator indispensable to reach the controlled state in case of steam generator tube rupture. tube rupture.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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l) Emergency Core Coolant System (ECCS) l) Emergency Core Coolant System (ECCS)

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The ECCS responds to the activation signals which are could be The ECCS responds to the activation signals which are could be generated by the reactor protection system or by the intervention of generated by the reactor protection system or by the intervention of the operator. It allows to assure the reactor sub-criticality in long term the operator. It allows to assure the reactor sub-criticality in long term followed a LOCA and allows to reach the safe and controlled state. followed a LOCA and allows to reach the safe and controlled state. m) Information concerning the cumulated performances of reactivity  m) Information concerning the cumulated performances of reactivity control systems control systems

The emergency shutdown associated to the EBS guarantee the The emergency shutdown associated to the EBS guarantee the

reactor sub-criticality at controlled state and safe shutdown for reactor sub-criticality at controlled state and safe shutdown for situations classes 2 to 4 (except the LOCA & steam generator tube situations classes 2 to 4 (except the LOCA & steam generator tube rupture). When a LOCA has occurred with an emergency shutdown of rupture). When a LOCA has occurred with an emergency shutdown of the reactor, the ECCS is indispensable to reach the controlled the reactor, the ECCS is indispensable to reach the controlled shutdown state and safe shutdown; in case of steam generator tube shutdown state and safe shutdown; in case of steam generator tube rupture, the EBS is necessary to reach the controlled state. One must rupture, the EBS is necessary to reach the controlled state. One must be pointed out that the capacities of the boration system of the CVCS be pointed out that the capacities of the boration system of the CVCS are not taking into account in the analyses of the the transients. are not taking into account in the analyses of the the transients.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR The information relative to the capacities of the CVCS are The information relative to the capacities of the CVCS are described in the safety report. The untimely dilution possibilities of described in the safety report. The untimely dilution possibilities of boron acid due to CVCS are studied in the analyses of related boron acid due to CVCS are studied in the analyses of related accidents. One is supposing that the initial correct operating of the accidents. One is supposing that the initial correct operating of the CVCS was initial condition for the evaluation of the transients, and CVCS was initial condition for the evaluation of the transients, and appropriated technical specifications have been set-up to guarantee appropriated technical specifications have been set-up to guarantee the correct operation or the intervention of a corrective action. the correct operation or the intervention of a corrective action. n) Evaluation of the design  n) Evaluation of the design The individual evaluation of each system impacted by the functional  The individual evaluation of each system impacted by the functional design of the reactivity control is treated in the chapter related to these design of the reactivity control is treated in the chapter related to these systems. systems. The evaluation of situations classes 2 to 4, supposes the concomitant  The evaluation of situations classes 2 to 4, supposes the concomitant activation of the control rods clusters system and the emergency activation of the control rods clusters system and the emergency boration system by the protection system of the reactor and by the boration system by the protection system of the reactor and by the intervention of the operator intervention of the operator

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR The emergency shutdown signals of the reactor for these events are  The emergency shutdown signals of the reactor for these events are generated from the redundancy equipments of reactivity control. The generated from the redundancy equipments of reactivity control. The emergency shutdown of the reactor is obtained by the activation of the emergency shutdown of the reactor is obtained by the activation of the redundancy circuit breaker of emergency shutdown which cut-off the redundancy circuit breaker of emergency shutdown which cut-off the electric power supply of the control rods clusters mechanisms and electric power supply of the control rods clusters mechanisms and induces the falling of all the control rods clusters by gravity. induces the falling of all the control rods clusters by gravity. Following the emergency shutdown, the boration system is activated  Following the emergency shutdown, the boration system is activated manually by the operator conformity to the accidental procedure. It is manually by the operator conformity to the accidental procedure. It is constituted by 2 or 4 trains totally redundancy (2X100% or 4x100% of constituted by 2 or 4 trains totally redundancy (2X100% or 4x100% of functions). functions). For the evaluation of the LOCA, one supposes the concomitant of  For the evaluation of the LOCA, one supposes the concomitant of command system of the control rods clusters and the safety injection command system of the control rods clusters and the safety injection system by the reactor protection system. The signals of safety system by the reactor protection system. The signals of safety injection are generated by the gages and logic redundancy trains. The injection are generated by the gages and logic redundancy trains. The safety injection is constituted of 4 independent trains strictly safety injection is constituted of 4 independent trains strictly separated and are connected to primary circuit. separated and are connected to primary circuit.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

One must be pointed out that the emergency boration system is  One must be pointed out that the emergency boration system is activated automatically in case of shutdown without emergency activated automatically in case of shutdown without emergency shutdown (ATWS), in order to guarantee the sub- criticality of the shutdown (ATWS), in order to guarantee the sub- criticality of the reactor core, as the alternative to the command system of the control reactor core, as the alternative to the command system of the control rods clusters. Measures have been taken to limit the consequences of rods clusters. Measures have been taken to limit the consequences of the ATWS. the ATWS. XVI.6. Fuel temperature coefficient (Doppler effect)  XVI.6. Fuel temperature coefficient (Doppler effect) The fuel temperature coefficient (Doppler) is defined as the changing  The fuel temperature coefficient (Doppler) is defined as the changing of the reactivity induced by a variation of one degree Celsius of the of the reactivity induced by a variation of one degree Celsius of the effective temperature of fuel. Essentially, it measures the enlargement effective temperature of fuel. Essentially, it measures the enlargement of Doppler resonances of U-238 and of Pu-240. An increase of the fuel of Doppler resonances of U-238 and of Pu-240. An increase of the fuel temperature will increase the resonance absorption cross-section of temperature will increase the resonance absorption cross-section of the fuel, thus induced a reduction of the reactivity. the fuel, thus induced a reduction of the reactivity. When the power increases and reaches an un-negligible value, the  When the power increases and reaches an un-negligible value, the effective temperature of the fuel is not equal to the moderator effective temperature of the fuel is not equal to the moderator temperature but vary versus to the reactor core power. This effect has temperature but vary versus to the reactor core power. This effect has been taking into account in the Doppler-power coefficient. been taking into account in the Doppler-power coefficient.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR Integration of the differential curve of the Doppler-power coefficient Integration of the differential curve of the Doppler-power coefficient versus to the relative power, constituted to the contribution of the versus to the relative power, constituted to the contribution of the Doppler effect to the deficiency of the power which will be defined later Doppler effect to the deficiency of the power which will be defined later on (by considering that the moderator temperature vary according to on (by considering that the moderator temperature vary according to the operating diagram). the operating diagram). XVI.7. Moderator coefficient  XVI.7. Moderator coefficient The moderator coefficient measures the variation of the reactivity due  The moderator coefficient measures the variation of the reactivity due to a modification of specific parameters of coolant fluid such as the to a modification of specific parameters of coolant fluid such as the density and temperature. The coefficients obtained are the density and density and temperature. The coefficients obtained are the density and temperature coefficients of the moderator. temperature coefficients of the moderator. XVI.7a. Density moderator coefficient  XVI.7a. Density moderator coefficient The temperature (density) moderator coefficient is defined as  The temperature (density) moderator coefficient is defined as corresponding to variation of reactivity induced by a variation of one corresponding to variation of reactivity induced by a variation of one degree Celsius of the moderator temperature. In general, the effects of degree Celsius of the moderator temperature. In general, the effects of variation of moderator density and thus the temperature are taking into variation of moderator density and thus the temperature are taking into account together. account together.

temperature temperature increases, increases, the the

THERMAL-HYDRAULIC IN NUCLEAR REACTOR A decrease in the density induces a reduction of the moderation, thus  A decrease in the density induces a reduction of the moderation, thus give a negative moderator coefficient. An increase in the temperature give a negative moderator coefficient. An increase in the temperature of the coolant fluid with constant density (which could be obtained by of the coolant fluid with constant density (which could be obtained by an increase of the pressure), lead to a hard neutron spectrum resulting an increase of the pressure), lead to a hard neutron spectrum resulting an increase of the resonance absorption of the U-238, of Pu-238 and an increase of the resonance absorption of the U-238, of Pu-238 and others isotopes. These effects induce a moderator coefficient more others isotopes. These effects induce a moderator coefficient more negative. As the water density varies more rapidly than does the negative. As the water density varies more rapidly than does the the moderator temperature, when temperature, when the moderator temperature (density) coefficient will be more negative. temperature (density) coefficient will be more negative. The soluble boron used in the reactor to control the reactivity has also  The soluble boron used in the reactor to control the reactivity has also an effect on the density coefficient of the moderator in the way where an effect on the density coefficient of the moderator in the way where the soluble boron density decreases when the water density decreases the soluble boron density decreases when the water density decreases due to an increase in the temperature of the coolant fluid. The due to an increase in the temperature of the coolant fluid. The decrease in the concentration in soluble boron introduces a positive decrease in the concentration in soluble boron introduces a positive effect in the moderator coefficient. effect in the moderator coefficient.



THERMAL-HYDRAULIC IN NUCLEAR REACTOR Thus, if the boron concentration is sufficiently high, it could at moment  Thus, if the boron concentration is sufficiently high, it could at moment that the net value of the coefficient becomes positive. Thank to the that the net value of the coefficient becomes positive. Thank to the burnable poison, the initial concentration of the boron in hot situation burnable poison, the initial concentration of the boron in hot situation is sufficiently low thus the moderator temperature coefficient respects is sufficiently low thus the moderator temperature coefficient respects all indicated criteria. all indicated criteria. With the burn-up, the moderator coefficient becomes more negative With the burn-up, the moderator coefficient becomes more negative principally due to the dilution of boric acid, but also, one significant principally due to the dilution of boric acid, but also, one significant part, because the accumulation of plutonium and fission products. part, because the accumulation of plutonium and fission products. XVI.7b. Void moderator coefficient  XVI.7b. Void moderator coefficient The void moderator coefficient represents the variation of the The void moderator coefficient represents the variation of the  “multiplication factor” of neutrons due to the presence of the bubbles “multiplication factor” of neutrons due to the presence of the bubbles in the moderator. In PWR, this coefficient is not so important because in the moderator. In PWR, this coefficient is not so important because of the low proportion of the bubbles in the coolant fluid. The of the low proportion of the bubbles in the coolant fluid. The proportion in the reactor core is less than 0.5% and is due to local proportion in the reactor core is less than 0.5% and is due to local ebullition or statistic. ebullition or statistic.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



XVI.7c. Heat flux limitations  XVI.7c. Heat flux limitations If the wall heat flux exceeded a certain magnitude, there would be a If the wall heat flux exceeded a certain magnitude, there would be a  sudden deterioration of the heat transfer rate due to an unstable sudden deterioration of the heat transfer rate due to an unstable transition from nucleate boiling to film boiling for low quality (DNB), or transition from nucleate boiling to film boiling for low quality (DNB), or from evaporating liquid film to mist flow evaporation for high qualities from evaporating liquid film to mist flow evaporation for high qualities (dry-out) (figure VI.1). (dry-out) (figure VI.1). DNB type of critical heat flux is the primary concern for PWRs, since  DNB type of critical heat flux is the primary concern for PWRs, since such reactors operate with sub-cooled and low quality coolants. Even such reactors operate with sub-cooled and low quality coolants. Even for BWRs, which have a significantly bottom-peaked axial power for BWRs, which have a significantly bottom-peaked axial power profile, the DNB risks have to be taken into consideration. profile, the DNB risks have to be taken into consideration.

Dry-out type of critical heat flux Dry-out type of critical heat flux is of concern is of concern

in BWRs in BWRs exclusively, since it can occurs for high quality only, which does not exclusively, since it can occurs for high quality only, which does not appear under normal operation conditions in PWRs. appear under normal operation conditions in PWRs.

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to predict CHF, but to predict CHF, but

THERMAL-HYDRAULIC IN NUCLEAR REACTOR It is extremely important that CHF, whether of DNB or dry-out type, will It is extremely important that CHF, whether of DNB or dry-out type, will not occurs during the operation of the reactor and that the conditions not occurs during the operation of the reactor and that the conditions of CHF occurrence can be accurately determined. However, CHF of CHF occurrence can be accurately determined. However, CHF conditions depend on several different parameters and no method is conditions depend on several different parameters and no method is available to predict them with high accuracy. So various empirical available to predict them with high accuracy. So various empirical they correlations have been developed correlations have been developed they unavoidably have uncertainties and predict the CHF only to a limited unavoidably have uncertainties and predict the CHF only to a limited level of accuracy. level of accuracy. The core designer must use such correlations to ensure that CHF limit  The core designer must use such correlations to ensure that CHF limit is not exceeded during core operation. It is customary to define is not exceeded during core operation. It is customary to define minimum ratio of the theoretical heat flux to the heat flux existing in minimum ratio of the theoretical heat flux to the heat flux existing in the reactor core as the “Critical Power Ratio (CPR). For DNB this ratio the reactor core as the “Critical Power Ratio (CPR). For DNB this ratio is called as DNB (Departure of Nucleate Boiling) ratio and is also is called as DNB (Departure of Nucleate Boiling) ratio and is also defined as: defined as:



(XVI.6c.2) DNBR = q”(DNB)(z)/ q”(z) (XVI.6c.2) DNBR = q”(DNB)(z)/ q”(z)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The reactor core must be designed to keep the DNBR larger than the  The reactor core must be designed to keep the DNBR larger than the minimum allowable value, so called “Minimum DNBR”. Figure minimum allowable value, so called “Minimum DNBR”. Figure XVI.6c.1), shows typical relations shapes of heat flux and DNBR along XVI.6c.1), shows typical relations shapes of heat flux and DNBR along the axis of a nuclear fuel assembly. the axis of a nuclear fuel assembly. The aim is to avoid any damage to the fuel cladding due to an  The aim is to avoid any damage to the fuel cladding due to an excessive increase in temperature. We have seen that the heat flux q’’ excessive increase in temperature. We have seen that the heat flux q’’ must not exceed a certain value q’’c (critical flux). For this, a parameter must not exceed a certain value q’’c (critical flux). For this, a parameter that plays a fundamental role is used, the DNBR (Departure from that plays a fundamental role is used, the DNBR (Departure from Nucleate Boiling Ratio) which is equal to the ratio of the critical flux to Nucleate Boiling Ratio) which is equal to the ratio of the critical flux to the real flux at any time. the real flux at any time. The critical flux is experimentally determined. Correlations have been  The critical flux is experimentally determined. Correlations have been established that allow, for a given channel, the CHF q’’c to be established that allow, for a given channel, the CHF q’’c to be determined as a function of the flow characteristics (pressure, flow determined as a function of the flow characteristics (pressure, flow rate, inlet & oulet enthalpies, quality, etc..) and the geometrical rate, inlet & oulet enthalpies, quality, etc..) and the geometrical characteristics of the channel (hydraulic diameter, grid spacing & characteristics of the channel (hydraulic diameter, grid spacing & type). ). type

(B&W) BWCMV correlation, and (B&W) BWCMV correlation, and

THERMAL-HYDRAULIC IN NUCLEAR REACTOR These correlations make it possible to predict the CHF, under the  These correlations make it possible to predict the CHF, under the given conditions, within an uncertainty margin leads to never dropping given conditions, within an uncertainty margin leads to never dropping below a given DNBR value (typically 1.30), which corresponds to a CHF below a given DNBR value (typically 1.30), which corresponds to a CHF of about 185 W/cm2. The result for the reactor protection system is a of about 185 W/cm2. The result for the reactor protection system is a limitation on the power level, the pressure, the temperature & the limitation on the power level, the pressure, the temperature & the coolant flow rate in the core. coolant flow rate in the core. The behaviour of the DNB correlations  The behaviour of the DNB correlations Raw CHF data consist of measured (M) heat flux at corresponding sets  Raw CHF data consist of measured (M) heat flux at corresponding sets of hydraulic conditions for a given burn-up bundle-geometry & three- of hydraulic conditions for a given burn-up bundle-geometry & three- re always positive, dimensional power distribution. These data a dimensional power distribution. These data a re always positive, yet because their mean is not large compared to their deviation (the yet because their mean is not large compared to their deviation (the coefficient of variation is about 40 %), the data are sharply upskewed coefficient of variation is about 40 %), the data are sharply upskewed and thus approximately lognormal in appearance. and thus approximately lognormal in appearance. For example, the Westinghouse W3 & WRB1 correlations, the  For example, the Westinghouse W3 & WRB1 correlations, the the Babcock-Wilcox Company the Babcock-Wilcox Company Combustion Engineering CE-1 correlation, are all derived from Combustion Engineering CE-1 correlation, are all derived from normally distributed M/P data. normally distributed M/P data.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The tolerance limit must be defined so that the probability of  The tolerance limit must be defined so that the probability of experience DNB is « low », thus the tolerance limit on the normally experience DNB is « low », thus the tolerance limit on the normally distributed M/P ratio is: - ks distributed M/P ratio is: - ks



Limit (M/P) = (M/P) Limit (M/P) = (M/P)



Where (M/P) is the mean of M/P data and « s » is its standard deviation.  Where (M/P) is the mean of M/P data and « s » is its standard deviation. The multiplier « k » incorporates the uncertainties in both (M/P) and The multiplier « k » incorporates the uncertainties in both (M/P) and « s » due to the finite sample size & converges to 1.645 to provide one- « s » due to the finite sample size & converges to 1.645 to provide one- sided 95 % probability protection for a normal distribution. sided 95 % probability protection for a normal distribution. Ensuring that plant operations & accident responses occurs below the  Ensuring that plant operations & accident responses occurs below the limit fixed above means that the fuel is expected to avoid DNB at least limit fixed above means that the fuel is expected to avoid DNB at least of the time of 95 % & with 95 % of confidence. of the time of 95 % & with 95 % of confidence. However, since the tolerance limit of DNB is defined in terms of the  However, since the tolerance limit of DNB is defined in terms of the P/M ratio (the DNB ratio or DNBR) instead of its normally distributed P/M ratio (the DNB ratio or DNBR) instead of its normally distributed reciprocal, the DNB tolerance limit is generally takes the form: reciprocal, the DNB tolerance limit is generally takes the form:

Limit (P/M) = 1/(M/P) – ks Limit (P/M) = 1/(M/P) – ks



THERMAL-HYDRAULIC IN NUCLEAR REACTOR * k function of number of data sample, of degree of probability & * k function of number of data sample, of degree of probability & degree of confidence (see Owens statistical table) degree of confidence (see Owens statistical table) Figure XVI.7c.1: Non appearance of the DNB limit  Figure XVI.7c.1: Non appearance of the DNB limit

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure XVI.7c.2: Heat flux through cladding wall  Figure XVI.7c.2: Heat flux through cladding wall

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVI7c.3:Principles of DNBR monitoring in class 1 condition  Figure XVI7c.3:Principles of DNBR monitoring in class 1 condition 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVI7c.4:Principles of DNBR monitoring in class 1 condition  Figure XVI7c.4:Principles of DNBR monitoring in class 1 condition 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure XVI7c.5: Operation diagram  Figure XVI7c.5: Operation diagram 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure XVI7c.6: Operation diagram  Figure XVI7c.6: Operation diagram 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure XVI7c.7: Operation diagram  Figure XVI7c.7: Operation diagram 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR XVI.7d. Requirements relative to the instrumentation  XVI.7d. Requirements relative to the instrumentation A. operations at low level DNBR  A. operations at low level DNBR There are two I&C, associated to the DNBR:  There are two I&C, associated to the DNBR: The protection function at low level DNBR, which initiated the  The protection function at low level DNBR, which initiated the automatic shutdown of the reactor; automatic shutdown of the reactor; The surveillance function of the DNBR which limits the operation  The surveillance function of the DNBR which limits the operation conditions (LCO). conditions (LCO). The utilization of the algorithm for the calculations of the DNBR in one  The utilization of the algorithm for the calculations of the DNBR in one or several protection and surveillance systems allows the respect of or several protection and surveillance systems allows the respect of the DNBR criterion by defining an automatic shut-down chain at low the DNBR criterion by defining an automatic shut-down chain at low level DNBR and a surveillance chain of DNBR (LCO) utilizing directly level DNBR and a surveillance chain of DNBR (LCO) utilizing directly the reconstituted variable representative of phenomenon to be the reconstituted variable representative of phenomenon to be avoided. avoided.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The protection function at low level DNBR initiates the automatic shut-  The protection function at low level DNBR initiates the automatic shut- down of the reactor which protect the fuel against the boiling crisis down of the reactor which protect the fuel against the boiling crisis during the accidental transients, whatever of postulated event initiator during the accidental transients, whatever of postulated event initiator leading to a significant uncontrolled decrease of the DNBR. The leading to a significant uncontrolled decrease of the DNBR. The surveillance function of the DNBR (LCO) assures a sufficient margin to surveillance function of the DNBR (LCO) assures a sufficient margin to the DNBR criterion in normal operation to face the events leading to a the DNBR criterion in normal operation to face the events leading to a significant decrease of the DNBR. During events of class 1, the value significant decrease of the DNBR. During events of class 1, the value of the DNBR must be maintained above the DNBR(LCO) threshold in of the DNBR must be maintained above the DNBR(LCO) threshold in order, in case of apparition of an event on which the protection at low order, in case of apparition of an event on which the protection at low level DNBR is not effective, the DNBR could be avoided. level DNBR is not effective, the DNBR could be avoided. The over-passing of those LCO initiates the following corrective  The over-passing of those LCO initiates the following corrective actions: actions: At the first threshold, an alarm, the blocking of the withdrawal of the  At the first threshold, an alarm, the blocking of the withdrawal of the control rod cluster and the blocking of the increase of the load; control rod cluster and the blocking of the increase of the load;

THERMAL-HYDRAULIC IN NUCLEAR REACTOR At second threshold, a reduction of reactor power by insertion of the  At second threshold, a reduction of reactor power by insertion of the sub-groups of control rod clusters and, if necessary, an appropriated sub-groups of control rod clusters and, if necessary, an appropriated reduction of the load of the turbine. reduction of the load of the turbine. The protection and surveillance algorithm concerning the DNBR are all  The protection and surveillance algorithm concerning the DNBR are all founded on the same principles: founded on the same principles: The determination of the minimal DNBR utilises the following  The determination of the minimal DNBR utilises the following parameters: parameters: The power distribution of the hot channel: it is directly issued of in-  The power distribution of the hot channel: it is directly issued of in- core neutron instrumentation (self-powered detectors). The signals of core neutron instrumentation (self-powered detectors). The signals of detectors, installed in the reactor core, scaled per power unit, give in detectors, installed in the reactor core, scaled per power unit, give in the same time the local power and the integrated power along the hot the same time the local power and the integrated power along the hot channel via a polynomial development. channel via a polynomial development. The inlet temperature given by the pyrometric probe at cold leg;  The inlet temperature given by the pyrometric probe at cold leg;

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The core mass flow given by the mass flow gage of the primary  The core mass flow given by the mass flow gage of the primary pumps. pumps. The CHF is determined by the correlation by utilizing local thermal-  The CHF is determined by the correlation by utilizing local thermal- hydraulic parameters, such as: pressure, quality, mass flow. hydraulic parameters, such as: pressure, quality, mass flow. Those parameters are calculated by a simplified model at unique  Those parameters are calculated by a simplified model at unique channel representing the hot channel without taking into account the channel representing the hot channel without taking into account the exchange between the neighbouring channels. exchange between the neighbouring channels. At why it is adjusted on the thermal-hydraulic design computer code  At why it is adjusted on the thermal-hydraulic design computer code which is taking into account the mass and energy exchanges between which is taking into account the mass and energy exchanges between the channels. the channels. B. Operations at high linear power  B. Operations at high linear power 

There are two functions associated to high linear power: There are two functions associated to high linear power:

The protection function against high linear powers;  The protection function against high linear powers; The surveillance function of linear powers (LCO).  The surveillance function of linear powers (LCO).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The respect of the criterion on the melting at the central of the pellet  The respect of the criterion on the melting at the central of the pellet fuel is guaranteed if one satisfy the decoupling criterion concerning fuel is guaranteed if one satisfy the decoupling criterion concerning the linear power at hot point which is must be under certain limits. the linear power at hot point which is must be under certain limits. The systems of protection and surveillance permit the respect of the  The systems of protection and surveillance permit the respect of the safety criterion concerning the melting at the central of the pellet fuel safety criterion concerning the melting at the central of the pellet fuel by defining an automatic shut-down chain of the reactor at high level by defining an automatic shut-down chain of the reactor at high level linear power and a chain of surveillance of LCO at high linear power by linear power and a chain of surveillance of LCO at high linear power by taking into account directly as basis the reconstruction of the linear taking into account directly as basis the reconstruction of the linear power at hot point. power at hot point. The protection function against the high linear powers initiate the  The protection function against the high linear powers initiate the automatic shutdown of the reactor which prevents the melting at the automatic shutdown of the reactor which prevents the melting at the central of the fuel pellet during the accidental transients, whatever the central of the fuel pellet during the accidental transients, whatever the event leading to an uncontrolled increase of the linear power. event leading to an uncontrolled increase of the linear power. The surveillance function of high linear power (LCO) essentially  The surveillance function of high linear power (LCO) essentially assures the confirmative to the integrity of the reactor core in the case assures the confirmative to the integrity of the reactor core in the case of an LOCA or accidents such as “shaft rupture on the primary pump of an LOCA or accidents such as “shaft rupture on the primary pump and/or withdrawal of a control rod cluster”. and/or withdrawal of a control rod cluster”.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The over-passing of these LCO initiates the following corrective  The over-passing of these LCO initiates the following corrective measures: measures: At the first threshold, an alarm, the blocking of the control rod cluster  At the first threshold, an alarm, the blocking of the control rod cluster withdrawal or insertion of control rod cluster versus the form of the withdrawal or insertion of control rod cluster versus the form of the axial power and the blocking of the increase of the load; axial power and the blocking of the increase of the load; At the second threshold, a decrease of the reactor power by insertion  At the second threshold, a decrease of the reactor power by insertion of sub-groups of control rod clusters and, if it necessary, an of sub-groups of control rod clusters and, if it necessary, an appropriated decrease of the turbine load. appropriated decrease of the turbine load. A deformation of the shape of the power could be explained that the  A deformation of the shape of the power could be explained that the limit value has been reached. The limit value depends on the axial limit value has been reached. The limit value depends on the axial repartition of the flux (less on the half high level of the reactor core repartition of the flux (less on the half high level of the reactor core than the half low level of the reactor core); this means that the than the half low level of the reactor core); this means that the surveillance function control also the axial shape of the power. surveillance function control also the axial shape of the power. The calculation of the maximal linear power 5W/cm) is directly issued  The calculation of the maximal linear power 5W/cm) is directly issued by fixed in-core instrumentation constituted of self-powered detectors. by fixed in-core instrumentation constituted of self-powered detectors.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



impose a check impose a check

* Other criterion * Other criterion Let us also note the limits on the cladding stress, during class 2 power  Let us also note the limits on the cladding stress, during class 2 power transients likely to cause the cladding to crack (consequence of the transients likely to cause the cladding to crack (consequence of the fuel pellet cladding mechanical interaction). fuel pellet cladding mechanical interaction). Operating feedback showed this phenomenon may occur when the  Operating feedback showed this phenomenon may occur when the power quickly increases after a long period of operation at low power power quickly increases after a long period of operation at low power if the or zero power. The safety authorities if the or zero power. The safety authorities T » or the « high linear heat rate » protection channel are « overpower ΔΔT » or the « high linear heat rate » protection channel are « overpower able to avoid the risk of pellet-cladding interaction during class 2 able to avoid the risk of pellet-cladding interaction during class 2 accidents. accidents. To do so a complex calculations mixing nuclear & thermo-mechanical  To do so a complex calculations mixing nuclear & thermo-mechanical aspects were performed & have shown that it will be the case if the aspects were performed & have shown that it will be the case if the « extended low power operation » periods are limited. Acceptable « extended low power operation » periods are limited. Acceptable durations and sequences are indicated in the « Operating Technical durations and sequences are indicated in the « Operating Technical Specifications ». Specifications ».



THERMAL-HYDRAULIC IN NUCLEAR REACTOR Regarding to the LOCA which corresponds to breaks of different areas  Regarding to the LOCA which corresponds to breaks of different areas affecting the primary circuit pipes, the main associated criteria are: affecting the primary circuit pipes, the main associated criteria are: - temperature of the cladding < 1204 °C - temperature of the cladding < 1204 °C - cladding oxidation rate < 17 % - cladding oxidation rate < 17 % The corresponding calculations show that these criteria are met  The corresponding calculations show that these criteria are met provided that the maximum linear heat rate before the accident provided that the maximum linear heat rate before the accident expressed in W/cm is lower than a certain value close to 400W/cm expressed in W/cm is lower than a certain value close to 400W/cm (typical value). It is operator’s role to meet this value during the (typical value). It is operator’s role to meet this value during the operation by controlling the axial power distribution in the core operation by controlling the axial power distribution in the core through the axial flux difference (AFD) control & by keeping it inside through the axial flux difference (AFD) control & by keeping it inside the « class 1 Operating Diagram ». the « class 1 Operating Diagram ». XVIII. The corresponding core protection channels  XVIII. The corresponding core protection channels The above limits are never reached during normal operation but might  The above limits are never reached during normal operation but might be during accidents. In order to avoid the cladding damage due to the be during accidents. In order to avoid the cladding damage due to the above physical phenomena during accidents , two protection channels above physical phenomena during accidents , two protection channels are designed & installed to trip the reactor before the safety criteria are are designed & installed to trip the reactor before the safety criteria are met.met.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR These protection channels are called « Overpower & Over-temperature  These protection channels are called « Overpower & Over-temperature ΔΔT channels » dealing respectively with the « center fuel melting » & T channels » dealing respectively with the « center fuel melting » & DNB risks. By means analog circuits, these channels elaborate the DNB risks. By means analog circuits, these channels elaborate the T as function of different maximum acceptable power level or ΔΔT as function of different maximum acceptable power level or parameters such as the inlet temperature, the primary flow rate, the parameters such as the inlet temperature, the primary flow rate, the primary pressure and the AFD. If the ΔΔTTAA becomes higher than the becomes higher than the primary pressure and the AFD. If the a reactor trip signal is emitted, causing all the measured ΔΔTTMM a reactor trip signal is emitted, causing all the measured controlled banks to drop. controlled banks to drop.

to to

T channel set points are determined after complex calculations.  The The ΔΔT channel set points are determined after complex calculations. One of the calculation phases is illustrated of the following figure. The One of the calculation phases is illustrated of the following figure. The the saturation and DNB phenomena limits corresponding the saturation and DNB phenomena limits corresponding (mentioned above) are calculated & drawn up as a function of the (mentioned above) are calculated & drawn up as a function of the ) plan in order to determine the forbidden primary pressure in a (P, TMM) plan in order to determine the forbidden primary pressure in a (P, T zones (see following figure) zones (see following figure)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure XVII.1: Protection diagram  Figure XVII.1: Protection diagram

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVII.2: Protection diagram in class 1 situation.  Figure XVII.2: Protection diagram in class 1 situation.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



* Limiting conditions of operation * Limiting conditions of operation Whatever the technology used for the cited protection channels, their  Whatever the technology used for the cited protection channels, their response time is not compatible with fast accidents because of the response time is not compatible with fast accidents because of the signals used to measure the core power level, i.e. the hot leg signals used to measure the core power level, i.e. the hot leg temperature, is impacted by some lag due to the measurement of the temperature, is impacted by some lag due to the measurement of the hot leg temperature in the hot leg by-pass. hot leg temperature in the hot leg by-pass. This is the reason why it is necessary to design and install several  This is the reason why it is necessary to design and install several « specific » protection channels with fast response times. Each of « specific » protection channels with fast response times. Each of them related to one fast accident. In order to get fast response time, them related to one fast accident. In order to get fast response time, the specific protection channels are very simple: one parameter and the specific protection channels are very simple: one parameter and only one is measured & compared with a predetermined threshold in only one is measured & compared with a predetermined threshold in the case of overstepping, a reactor trip signal is emitted. the case of overstepping, a reactor trip signal is emitted. Theoretical calculations are necessary to demonstrate that the core  Theoretical calculations are necessary to demonstrate that the core melting and DNBR criterion values are not overstepped when the trip melting and DNBR criterion values are not overstepped when the trip signal is emitted by a specific protection channel occur is because all signal is emitted by a specific protection channel occur is because all the parameters needed to characterize the DNBR or the core hot spot the parameters needed to characterize the DNBR or the core hot spot factor FQ are not measured. factor FQ are not measured.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR For example, the primary pump speed signal which represents the  For example, the primary pump speed signal which represents the core primary flow rate, is not a correct picture of the DNBR since this core primary flow rate, is not a correct picture of the DNBR since this ratio depends upon several other parameters such as the temperature, ratio depends upon several other parameters such as the temperature, the primary pressure & the axial power distribution. the primary pressure & the axial power distribution. Assumptions dealing with the parameters which are not measured and  Assumptions dealing with the parameters which are not measured and the corresponding values before the accidents must be used & defined the corresponding values before the accidents must be used & defined to perform the calculations. In case of violation alarms are actuated in to perform the calculations. In case of violation alarms are actuated in the control room & operators must restore an acceptable situation as the control room & operators must restore an acceptable situation as soon as possible. soon as possible. Regarding the linear rate aspect, the most limiting initial condition to  Regarding the linear rate aspect, the most limiting initial condition to be met is governed by the LOCA which is a class 4 accident. be met is governed by the LOCA which is a class 4 accident. For NPP, the LHR monitoring during class 1 operation is performed  For NPP, the LHR monitoring during class 1 operation is performed through the AFD measured by the 2-section ex-core detectors. through the AFD measured by the 2-section ex-core detectors.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The measured AFD value must be kept inside boundaries which look  The measured AFD value must be kept inside boundaries which look like a trapezium (see figure) and constitute the class 1 operating like a trapezium (see figure) and constitute the class 1 operating domain (already mentioned above). But it is recommended to avoid domain (already mentioned above). But it is recommended to avoid strong variations in axial power distribution in order to avoid the strong variations in axial power distribution in order to avoid the generation of axial Xenon oscillation later and to keep the (P, AFD) generation of axial Xenon oscillation later and to keep the (P, AFD) operating close to its reference value. operating close to its reference value. For Advanced NPP, and taking profit from digital technology and from  For Advanced NPP, and taking profit from digital technology and from an improvement in the ex-core detections able to measure the real an improvement in the ex-core detections able to measure the real axial power distribution in the core and not only the AFD. LHR axial power distribution in the core and not only the AFD. LHR is directly performed by monitoring during class 1 operation monitoring during class 1 operation is directly performed by calculating the current value of the maximum LHR FQ and by calculating the current value of the maximum LHR FQ and by comparing this value with the LOCA limit (400 W/cm) corrected with comparing this value with the LOCA limit (400 W/cm) corrected with the adequate uncertainty (see following figure). the adequate uncertainty (see following figure). The new kind of ex-core detector is characterized by the presence of  The new kind of ex-core detector is characterized by the presence of six sections instead of two. Thanks to the mathematical process, it is six sections instead of two. Thanks to the mathematical process, it is possible to fit a curve through the six measuring points and to possible to fit a curve through the six measuring points and to reconstruct the current axial power distribution. reconstruct the current axial power distribution.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR It has to be noted that the LOCA limit determined during the LOCA It has to be noted that the LOCA limit determined during the LOCA analysis is not constant versus the core elevation Z. The 400 W/cm analysis is not constant versus the core elevation Z. The 400 W/cm typical value corresponds to the lower core half. typical value corresponds to the lower core half. For some advanced NPP, the ex-core detectors are replaced by new  For some advanced NPP, the ex-core detectors are replaced by new kind of neutron detectors which are permanently located at fixed kind of neutron detectors which are permanently located at fixed position inside the core. These detectors are made of Co59 and are position inside the core. These detectors are made of Co59 and are called « Self Powered Neutron Detectors » (SPND). called « Self Powered Neutron Detectors » (SPND). XVIII.1. PWR nuclear instrumentation  XVIII.1. PWR nuclear instrumentation In a PWR the neutron flux distributions are monitored by ec-core & in- In a PWR the neutron flux distributions are monitored by ec-core & in-  core instrumentations core instrumentations

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Figure XVIII.1.1: Location of neutron monitoring detectors  Figure XVIII.1.1: Location of neutron monitoring detectors

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conditions A. Ex-core instrumentation  A. Ex-core instrumentation Detectors for the routine monitoring of reactor power in a PWR are  Detectors for the routine monitoring of reactor power in a PWR are located outside the reactor pressure vessel and are characterized by located outside the reactor pressure vessel and are characterized by the following typical environmental conditions: neutron flux up to 10 1111 the following typical environmental conditions: neutron flux up to 10 temperature R/h, and irradiation up to 106 6 R/h, and .s, gamma n/cmn/cm22.s, gamma temperature irradiation up to 10 approximative of 100°C. approximative of 100°C. Ex-core detectors are the usual basis of reactor control and safety  Ex-core detectors are the usual basis of reactor control and safety channels in a PWR. In choosing specific detector types, consideration channels in a PWR. In choosing specific detector types, consideration must be to the expected neutron signal level compared with noise must be to the expected neutron signal level compared with noise sources, the speed of the detector, and the ability to discriminate sources, the speed of the detector, and the ability to discriminate against gamma induced signals. Each of these criteria assumes against gamma induced signals. Each of these criteria assumes different importance over various ranges of reactor power, and as a different importance over various ranges of reactor power, and as a result of multiple detector systems are usually provided, each result of multiple detector systems are usually provided, each designed to cover a specific subset of the power range. designed to cover a specific subset of the power range. The following figure illustrates a typical scheme for a PWR in which  The following figure illustrates a typical scheme for a PWR in which three set of detectors with overlapping operating ranges are used to three set of detectors with overlapping operating ranges are used to cover the entire power range of the reactor. The lowest range, usually cover the entire power range of the reactor. The lowest range, usually called the source start-up range, is encountered first when bringing up called the source start-up range, is encountered first when bringing up reactor power from shutdown conditions reactor power from shutdown

therefore possible, and therefore possible, and the the is is

level) associated level) associated to to

THERMAL-HYDRAULIC IN NUCLEAR REACTOR This range is characterized by conditions in which the gamma flux  This range is characterized by conditions in which the gamma flux from fission product inventory in the core may be large compared with from fission product inventory in the core may be large compared with the small neutron flux at low power levels. Under these conditions, the small neutron flux at low power levels. Under these conditions, good discrimination against gamma rays is at a premium. Also, the good discrimination against gamma rays is at a premium. Also, the expected neutron interaction rates will be relatively low in this range. expected neutron interaction rates will be relatively low in this range. Pulse mode operation of either fission chamber or BF3, proportional Pulse mode operation of either fission chamber or BF3, proportional required gamma-ray counters required gamma-ray counters discrimination can be accomplished by accepting only the much larger discrimination can be accomplished by accepting only the much larger amplitude of neutron pulses. amplitude of neutron pulses. Typical ranges covered by ex-core neutron detectors in a PWR.  Typical ranges covered by ex-core neutron detectors in a PWR. The signal of three levels of measurements (source level, intermediate  The signal of three levels of measurements (source level, intermediate the electronic of nuclear level, power level, power the electronic of nuclear instrumentation system, is utilized to limit the maximal power of the instrumentation system, is utilized to limit the maximal power of the reactor core in their respective domain; reactor core in their respective domain; The nuclear measurement gages are installed around the reactor, in  The nuclear measurement gages are installed around the reactor, in the primary shielding: the primary shielding:

THERMAL-HYDRAULIC IN NUCLEAR REACTOR 3 level source chambers installed in radial positions closer to the  3 level source chambers installed in radial positions closer to the reactor; reactor; 4 intermediate level chambers installed in radial positioning closer to  4 intermediate level chambers installed in radial positioning closer to the reactor and at half high upper of the reactor core; the reactor and at half high upper of the reactor core; 4 bi-section detectors of power level installed closer as to the  4 bi-section detectors of power level installed closer as to the pressure-vessel of the reactor. Axially, the detectors are distributed pressure-vessel of the reactor. Axially, the detectors are distributed along the active length of the core. along the active length of the core. The three levels of detectors are utilized as inputs of the neutron flux  The three levels of detectors are utilized as inputs of the neutron flux surveillance in the shutdown complete to 120% of nuclear power with surveillance in the shutdown complete to 120% of nuclear power with the possibility of the recording of highest excursions of the power. the possibility of the recording of highest excursions of the power. The output of level power chains could be utilized to:  The output of level power chains could be utilized to: To protect the reactor core against an uncontrolled feed of positive  To protect the reactor core against an uncontrolled feed of positive reactivity. It is utilized, essentially to allow the analyses of the safety of reactivity. It is utilized, essentially to allow the analyses of the safety of situations class 2 (for example rapid withdrawal of regulation groups) situations class 2 (for example rapid withdrawal of regulation groups) or class 4 (for example control rods cluster ejection); or class 4 (for example control rods cluster ejection);

THERMAL-HYDRAULIC IN NUCLEAR REACTOR To stress the operator that there is an excessive unbalance of the  To stress the operator that there is an excessive unbalance of the power between the quadrants of the reactor core: power between the quadrants of the reactor core: To realize a blocking of the regulation group  To realize a blocking of the regulation group Figure XVIII.1A.1: Typical ranges covered by ex-core neutron detectors  Figure XVIII.1A.1: Typical ranges covered by ex-core neutron detectors in a PWR. in a PWR.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVIII.1A.2: Geometry arrangement of ex-core neutron detectors  Figure XVIII.1A.2: Geometry arrangement of ex-core neutron detectors relative to core of a PWR. relative to core of a PWR.

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Figure XVIII.11.3: Axial location of SPNDs  Figure XVIII.11.3: Axial location of SPNDs

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR Because these instruments are often part of the reactor safety system,  Because these instruments are often part of the reactor safety system, there is a premium which also favors the uncompensated ion chamber there is a premium which also favors the uncompensated ion chamber construction. construction. In most PWR nuclear instrument systems the geometric arrangement In most PWR nuclear instrument systems the geometric arrangement shown above is employed. shown above is employed. Two BF3, proportional counters used in the source start-up range are  Two BF3, proportional counters used in the source start-up range are placed on opposite sides of the core and two CIC detectors used in the placed on opposite sides of the core and two CIC detectors used in the intermediate range are placed in the same location or on the two intermediate range are placed in the same location or on the two opposing sides. opposing sides. Four power range monitors are then located at 90°C intervals at  Four power range monitors are then located at 90°C intervals at positions between the BF3 and CIC detectors. Each of the four power positions between the BF3 and CIC detectors. Each of the four power ion chambers range monitors consists of two uncompensated range monitors consists of two uncompensated ion chambers arranged end-to-end, resulting in a total detector length of 3-4 m. arranged end-to-end, resulting in a total detector length of 3-4 m. This arrangement provides both radial and axial neutron flux data for  This arrangement provides both radial and axial neutron flux data for control and safety at full power as well as axial flux offset information control and safety at full power as well as axial flux offset information needed for control of xenon oscillations. needed for control of xenon oscillations.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Although PWR nuclear instrumentation employs ex-core gas filled  Although PWR nuclear instrumentation employs ex-core gas filled detectors (primarily ion chambers) for control and safety channels, detectors (primarily ion chambers) for control and safety channels, there remains a need for information on in-core spatial variations of there remains a need for information on in-core spatial variations of the neutron flux. the neutron flux. This information is necessary for fuel management and is provided by  This information is necessary for fuel management and is provided by various types of detectors placed within the reactor core. various types of detectors placed within the reactor core. B. In-core instrumentation  B. In-core instrumentation in-core in-core 

instrumentation system instrumentation system The The

is constituted by self- is constituted by self- powered detectors which are used to measure the local power powered detectors which are used to measure the local power generated by the nuclear fuels. generated by the nuclear fuels. The self-powered detector fingers are installed where they could give  The self-powered detector fingers are installed where they could give the maximal information on the power variations and their effects on the maximal information on the power variations and their effects on the essentials core parameters (FQ, FΔH), in particular is the perturbed the essentials core parameters (FQ, FΔH), in particular is the perturbed conditions. conditions. The self-powered detectors are radically distributed in homogenous  The self-powered detectors are radically distributed in homogenous manner in the reactor core in order their signals are representative of manner in the reactor core in order their signals are representative of the essential core parameters for different types of perturbations and the essential core parameters for different types of perturbations and different fuel managements differe nt fuel managements



THERMAL-HYDRAULIC IN NUCLEAR REACTOR Usually 12 fuel assemblies are instrumented, the core has been  Usually 12 fuel assemblies are instrumented, the core has been divided in 12 radial zones and each zone is controlled by a self- divided in 12 radial zones and each zone is controlled by a self- powered detector finger. powered detector finger. The self-powered detector fingers are distributed in such a manner  The self-powered detector fingers are distributed in such a manner that all the core surface is covered. that all the core surface is covered. The 12 self-powered detectors fingers contain each 6 self-powered  The 12 self-powered detectors fingers contain each 6 self-powered detectors. detectors. In each finger, three of them are installed in the upper half of the In each finger, three of them are installed in the upper half of the reactor core and the three others are in the lower half high of the reactor core and the three others are in the lower half high of the reactor core in order to detect the maximal power appeared in upper reactor core in order to detect the maximal power appeared in upper and lower high of the reactor core to cover all possible power and lower high of the reactor core to cover all possible power distributions (normal and accidental); the optimal axial positions will distributions (normal and accidental); the optimal axial positions will be determined by analyse of the axial form of the power in the be determined by analyse of the axial form of the power in the transient situations. transient situations. The axial positions are always located between the two grid spacers in  The axial positions are always located between the two grid spacers in order to avoid the flux depression at grids neighbouring. order to avoid the flux depression at grids neighbouring.



instruments must be instruments must be located, emphasis located, emphasis

THERMAL-HYDRAULIC IN NUCLEAR REACTOR In each finger is attributed a part of the core volume so called In each finger is attributed a part of the core volume so called “surveillance zone”. The fingers are scaled by the system of aeroball “surveillance zone”. The fingers are scaled by the system of aeroball measurement to reproduce the maximal power in the radial zones of measurement to reproduce the maximal power in the radial zones of which they control, the signals of the self-powered detectors have which they control, the signals of the self-powered detectors have been utilized to calculate the minimal DNBR. been utilized to calculate the minimal DNBR. There is provide often a need to place neutron detectors within the  There is provide often a need to place neutron detectors within the core of a nuclear reactor to information on the spatial variation of the core of a nuclear reactor to information on the spatial variation of the neutron flux. Because of the small size (1-7cm) of the channel in which neutron flux. Because of the small size (1-7cm) of the channel in which is placed on these is placed on these They may either be compactness and miniaturization in their design. They may either be compactness and miniaturization in their design. left in a fixed position or provided with a motorized drive to allow left in a fixed position or provided with a motorized drive to allow traverses through the reactor core. Some may provide a continuous traverses through the reactor core. Some may provide a continuous readout, whereas others are interrogated only at periodic intervals. readout, whereas others are interrogated only at periodic intervals. Typical operating conditions are:  Typical operating conditions are: .s, gamma fluw up to 108 R/h, - Neutron at full power of 5x101313 n/cm n/cm22.s, gamma fluw up to 108 R/h, - Neutron at full power of 5x10  operating temperature up to 300 °C, and operating pressure up to 17 operating temperature up to 300 °C, and operating pressure up to 17 Mpa.Mpa.

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR The gradual burn-up of neutron sensitive material is a serious The gradual burn-up of neutron sensitive material is a serious problem for the long term operation of in-core detectors. For example, problem for the long term operation of in-core detectors. For example, a fission chamber using 235U will shows a sensitivity decrease of a fission chamber using 235U will shows a sensitivity decrease of about 50% after exposure to an integrated neutron fluence of about about 50% after exposure to an integrated neutron fluence of about 1.7x102121 n/cm n/cm22. . 1.7x10 One method of reducing the effects of burn-up in fission chambers is  One method of reducing the effects of burn-up in fission chambers is to combine fertile and fissile material in the neutron-sensitive lining of to combine fertile and fissile material in the neutron-sensitive lining of the chamber. the chamber. Use of these « regenerative » chambers will gradually convert the  Use of these « regenerative » chambers will gradually convert the fertile isotopes to fissile nuclei to help compensate for the burn-up of fertile isotopes to fissile nuclei to help compensate for the burn-up of the original fissile material present in the lining. the original fissile material present in the lining. Fission ion chambers that have been operated for long periods in high  Fission ion chambers that have been operated for long periods in high neutron fluxes will show a residual current or « memory effect » due to neutron fluxes will show a residual current or « memory effect » due to the fission products within the chamber. the fission products within the chamber. These fission products emit beta and gamma-rays, which ionize the fill  These fission products emit beta and gamma-rays, which ionize the fill gas of the chamber and result in a significant current. gas of the chamber and result in a significant current.

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Figure XVIII.1B.1: In- core instrumentation pattern  Figure XVIII.1B.1: In- core instrumentation pattern

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVIII.1B.2/ Arrangement of in-core instrumentation components.  Figure XVIII.1B.2/ Arrangement of in-core instrumentation components.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR C. Aeroball system for monitoring power distribution in the reactor  C. Aeroball system for monitoring power distribution in the reactor core. core.



The The function of function of fundamental fundamental that movable nuclear that movable nuclear

fixed fixed instrumentation system in the reactor core is to measure the neutron instrumentation system in the reactor core is to measure the neutron flux in different spatial points of the reactor core. The preliminary flux in different spatial points of the reactor core. The preliminary locations of pneumatic propulsed ball probes are represented in figure locations of pneumatic propulsed ball probes are represented in figure 6.4.1. The flux map is utilized to obtain the axial distribution of the 6.4.1. The flux map is utilized to obtain the axial distribution of the neutron power of the hot channel of each fuel assembly (3D image of neutron power of the hot channel of each fuel assembly (3D image of the power distribution). the power distribution). From the mean axial distributions, it is possible to determine the the  From the mean axial distributions, it is possible to determine the the following core parameters: following core parameters: Axial neutron power of the hot channel of each fuel assembly and the  Axial neutron power of the hot channel of each fuel assembly and the FQ (hot point factor) maximal value of the core; FQ (hot point factor) maximal value of the core; The mean axial distribution of the neutron power of the core;  The mean axial distribution of the neutron power of the core;

THERMAL-HYDRAULIC IN NUCLEAR REACTOR The integrated power along the fuel assembly and the hot channel,  The integrated power along the fuel assembly and the hot channel, which allows to deduct the enthalpy rise of each fuel assembly and the which allows to deduct the enthalpy rise of each fuel assembly and the FΔH maximal value of the reactor core, utilized to calculate the FΔH maximal value of the reactor core, utilized to calculate the minimal DNBR by combining with the operating thermal-hydraulics; minimal DNBR by combining with the operating thermal-hydraulics; The power ratio between the two core quadrants.  The power ratio between the two core quadrants. The core parameters are utilized to accomplish the following tasks:  The core parameters are utilized to accomplish the following tasks: Verify the conformity of the reactor core at the first start-up and during  Verify the conformity of the reactor core at the first start-up and during the reloading; the reloading; Scaling of the fixed in-core instrumentation;  Scaling of the fixed in-core instrumentation; Justify the uncertainties of the measurements which are taken into  Justify the uncertainties of the measurements which are taken into account in the surveillance system; account in the surveillance system; Follow-up the burn-up of the fuel assembly;  Follow-up the burn-up of the fuel assembly; Effectuate researches and establish diagnostics in case of operating in  Effectuate researches and establish diagnostics in case of operating in particular and abnormal conditions. particular and abnormal conditions.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR An unique in-core monitoring system referred to as the computerized  An unique in-core monitoring system referred to as the computerized aeroball system is used in PWR. In the aeroball system, probes aeroball system is used in PWR. In the aeroball system, probes containing neutron sensitive isotopes are introduced into the reactor containing neutron sensitive isotopes are introduced into the reactor core and are subsequently activated. As shown in the following figure, core and are subsequently activated. As shown in the following figure, the probes are columns of 1.7 mm diameter steel balls. The ball the probes are columns of 1.7 mm diameter steel balls. The ball columns are piped into SS steel tubes incorporated in selected FA columns are piped into SS steel tubes incorporated in selected FA throughout the reactor core. The length of the columns corresponds to throughout the reactor core. The length of the columns corresponds to the height of the core. the height of the core.



The aeroball system permits: The aeroball system permits:



to take into account in the same manner the different zones of the core to take into account in the same manner the different zones of the core in the map of the flux (type of fuel assembly, local effects of control in the map of the flux (type of fuel assembly, local effects of control rods, radial surface). Thus when the measured fuel assembly represent rods, radial surface). Thus when the measured fuel assembly represent in the unique eighth of the core, nearly all the fuel assembly (except in the unique eighth of the core, nearly all the fuel assembly (except the ones located in the positions of the control rods clusters) are the ones located in the positions of the control rods clusters) are instrumented; instrumented; to realize the measurements in the the symmetrical fuel assemblies to realize the measurements in the the symmetrical fuel assemblies (with regard to the quarter of the core) in the different zones of the (with regard to the quarter of the core) in the different zones of the core, core,

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR to realize measurements uniformly distributed along the fissile length to realize measurements uniformly distributed along the fissile length of the fuel assembly. of the fuel assembly. Vanadium is added to the balls as a neutron flux indicator through 225  Vanadium is added to the balls as a neutron flux indicator through 225 V. The activation V activity induced by neutron capture in 5151V. The activation s half-life 5252V activity induced by neutron capture in s half-life of 51V and its characteristics are summarized in the above table. of 51V and its characteristics are summarized in the above table. The activity of the probes is measured, following removal from the  The activity of the probes is measured, following removal from the core, by using a set of silicon detectors. The count rate from each core, by using a set of silicon detectors. The count rate from each silicon detector is proportional to the relative integrated thermal silicon detector is proportional to the relative integrated thermal neutron flux at the point where the corresponding aeroball was located neutron flux at the point where the corresponding aeroball was located during the activation process. From many such measurements, the during the activation process. From many such measurements, the power density distribution of the reactor core can be determined. power density distribution of the reactor core can be determined.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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Figure XVIII.1C.1: Aeroball system for continous activation  Figure XVIII.1C.1: Aeroball system for continous activation measurements of the neutron distribution in a reactor core. measurements of the neutron distribution in a reactor core.

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Figure XVIII.1C.2: Aeroball system for continous activation  Figure XVIII.1C.2: Aeroball system for continous activation measurements & pneumatic system measurements & pneumatic system

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Figure XVIII.1c.3: Aeroball system.  Figure XVIII.1c.3: Aeroball system. 

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Figure XVIII.1c.5: Detail aeroball system & SPND probes.  Figure XVIII.1c.5: Detail aeroball system & SPND probes.

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Figure XVIII.1c.6: Instrumenation Lance & Nozzle arrangement  Figure XVIII.1c.6: Instrumenation Lance & Nozzle arrangement

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Figure XVIII.1c.7 : AMS system location  Figure XVIII.1c.7 : AMS system location

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Figure XVIII.1c.8: Time schedule of AMS measuring process  Figure XVIII.1c.8: Time schedule of AMS measuring process

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Figure XVIII.1c.9: AMS measurement table  Figure XVIII.1c.9: AMS measurement table

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVIII.1c.10: AMS measurement table used in KONVOI NPP.  Figure XVIII.1c.10: AMS measurement table used in KONVOI NPP.

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Figure XVIII.1c.11: Tapering port with semiconductor detector.  Figure XVIII.1c.11: Tapering port with semiconductor detector.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XVIII.1c.12: Semiconductor used in AMS table.  Figure XVIII.1c.12: Semiconductor used in AMS table.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



 1. Determination of set-point of each of chains

Tte in steady Top & ΔΔTte in steady

I. Protection chains ΔTe & ΔTOP I. Protection chains ΔTe & ΔTOP The protection systems must activate the scram since the core  The protection systems must activate the scram since the core parameters are out of the protection diagramm preveously defined. parameters are out of the protection diagramm preveously defined. This leads to the two chains of protections:  This leads to the two chains of protections: Tte high temperature; - the chain of ΔΔTte high temperature; - the chain of  Top (over-power) - the chain of ΔΔTop (over-power) - the chain of On which the thresholds are elaborated from primary parameters. The  On which the thresholds are elaborated from primary parameters. The determination of these chains has been done in five successive determination of these chains has been done in five successive stages: stages: 1. Determination of set-point of each of chains ΔΔTop & state & with axial offset A.O = 0; state & with axial offset A.O = 0; 2. Verification of the validity of the precedent calculation in transient  2. Verification of the validity of the precedent calculation in transient state & dimensioning the « forward-delay » boxes at ΔΔI = 0; I = 0; state & dimensioning the « forward-delay » boxes at 3. Taken into account the instrumentation & scalling errors;  3. Taken into account the instrumentation & scalling errors; 4. Taken into account the unbalance axial of the power in the  4. Taken into account the unbalance axial of the power in the protectrion chains; protectrion chains;

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 More however, to enhance the response time of the

I = 0 XIX.1.Steady state ΔΔI = 0  XIX.1.Steady state The establishing of the set-point constitutes of the linearization  The establishing of the set-point constitutes of the linearization descrbed preveously. It is effectuated with a power distribution in the descrbed preveously. It is effectuated with a power distribution in the H = 1.55 [ 1 + 0.2 ( 1 – form « truncked cosinus » (with Fz = 1.55 & FΔΔH = 1.55 [ 1 + 0.2 ( 1 – form « truncked cosinus » (with Fz = 1.55 & F p )]). p )]). XIX..2.Transient conditions  XIX..2.Transient conditions To compensate the transit time between the measured points & the  To compensate the transit time between the measured points & the reactor core, the delays due to the captors & to the scram chains, it is reactor core, the delays due to the captors & to the scram chains, it is necessary to introduce « forward & delay » boxes. necessary to introduce « forward & delay » boxes. The taken into account the uncontrolled withdrawal of a RCCA reactor  The taken into account the uncontrolled withdrawal of a RCCA reactor on power leads to the modification of the « k » coefficient , to avoid the on power leads to the modification of the « k » coefficient , to avoid the boiling crisis. boiling crisis. op, we introduce More however, to enhance the response time of the ΔΔop, we introduce a derivate term. a derivate term. XIX.3. Instrumentation & setting errors  XIX.3. Instrumentation & setting errors The taken into account of these errors leads to the modification of the  The taken into account of these errors leads to the modification of the nominal set point. nominal set point.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

XIX.4. Taken into account of the unbalance of the axial power  XIX.4. Taken into account of the unbalance of the axial power The taken into account of the unbalance of the axial power leads to the  The taken into account of the unbalance of the axial power leads to the introduction of a penalty . The following figure give the form of the introduction of a penalty . The following figure give the form of the penalty fte & fsp for the two chains. These penalties are given as penalty fte & fsp for the two chains. These penalties are given as indicative values, in effect, they vary versus of the cycle indicative values, in effect, they vary versus of the cycle

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure XIX.5.1: Penalty functions applied to ΔTop & ΔTe (cycle 1 &  Figure XIX.5.1: Penalty functions applied to ΔTop & ΔTe (cycle 1 & others others

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



XIX.5. Mass flow reductions  XIX.5. Mass flow reductions The possible domaine power-frequency in class 1 leads to the primary  The possible domaine power-frequency in class 1 leads to the primary mass flow between 93.8% to 102.2% of the nominal mass flow. mass flow between 93.8% to 102.2% of the nominal mass flow. op, the protections risk to Tte & ΔΔop, the protections risk to If we do not modify the thresholds ΔΔTte & If we do not modify the thresholds be not safisfied in case of uncontrolled withdrawal of a RCCA at Pn, be not safisfied in case of uncontrolled withdrawal of a RCCA at Pn, whereas the mass flow is reduced. This leads to the introduction of the whereas the mass flow is reduced. This leads to the introduction of the terms dependent on the mass flow: terms dependent on the mass flow: Top: a positive term when the mass flow decreases in order to - For ΔΔTop: a positive term when the mass flow decreases in order to - For obtaint a linear power at hot point independent of the mass flow; obtaint a linear power at hot point independent of the mass flow; Tte : a negative term when the mass flow decreases in order to - For ΔΔTte : a negative term when the mass flow decreases in order to - For assure the respect of the DNBR > 1.3 assure the respect of the DNBR > 1.3 Fundamental recalls of thermal-hydraulic & neuton aspects.  Fundamental recalls of thermal-hydraulic & neuton aspects. A.Thermal aspect  A.Thermal aspect The energy produced in a NSSS results from the release of heat due to  The energy produced in a NSSS results from the release of heat due to fission reaction in the assembly of fuel rods. The reactor coolant that fission reaction in the assembly of fuel rods. The reactor coolant that flows around these rods evacuate this heat energy & transfers it to the flows around these rods evacuate this heat energy & transfers it to the steam generator steam generator

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

The heat evacuated from the core by the reactor coolant is  The heat evacuated from the core by the reactor coolant is charactertized by a certain number of magnitudes. For a 1600 MWe charactertized by a certain number of magnitudes. For a 1600 MWe class NSSS these values are: class NSSS these values are:  Reactor Power Reactor Power Q = 4590 MWth Q = 4590 MWth

M = 83.380 t/h M = 83.380 t/h

72 KW72 KW

166 W/cm 166 W/cm



57 W/cm2 57 W/cm2 241 W/cm2 241 W/cm2

 Mass flow rate Mass flow rate  Increase in enthalpy Increase in enthalpy  Average power of: Average power of:  Fuel assembly 19 MW Fuel assembly 19 MW  One fuel rod One fuel rod  Average power density fuel rod: Average power density fuel rod:  Linear power q’ Linear power q’  Surface q’’ Surface q’’  Volume q’’’ Volume q’’’ 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



B. Neutron aspects  B. Neutron aspects The distribution of the power in the core is not uniform & large  The distribution of the power in the core is not uniform & large disparities exist betwwen the different zones, resulting from the disparities exist betwwen the different zones, resulting from the non_uniform of the neutron flux in the core due to nuclear physics. non_uniform of the neutron flux in the core due to nuclear physics. To recall, neutron flux is expressed in n/cm2 and is related at each  To recall, neutron flux is expressed in n/cm2 and is related at each point in the fuel to the thermal power release by the equation: point in the fuel to the thermal power release by the equation:

 Where :∑

x 3.2 x 10 -11-11 q’’’ = ∑ff Ø Øthth x 3.2 x 10 q’’’ = ∑

is the is the effective fission cross-section in cm-1 & the Ø thth is the

 Or : Or :

Where :∑ff is the effective fission cross-section in cm-1 & the Ø thermal flux in n/cm2.s. thermal flux in n/cm2.s.

) / 3.1x101010 P(W) = (VΣΣff ФФtthh) / 3.1x10 P(W) = (V

1. Shape of the neutron flux  1. Shape of the neutron flux the neutron flux, one distinguishes between In considering the form of the neutron flux, one distinguishes between In considering the form of  1) the radial distribution & 2) the axial distribution. 1) the radial distribution & 2) the axial distribution.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

a) Radial flux distribution::  a) Radial flux distribution For a homogeneous reactor (a theoretical case), the radial flux For a homogeneous reactor (a theoretical case), the radial flux  distribution is a Bessel function J0 (shape 1). Because of the presence distribution is a Bessel function J0 (shape 1). Because of the presence of a neutron reflector (the water surrounding the core or a metallic of a neutron reflector (the water surrounding the core or a metallic structure installed & because of different fuel enrichment zones loaded structure installed & because of different fuel enrichment zones loaded according to a certain fuel loading pattern the flux distribution as according to a certain fuel loading pattern the flux distribution as function of the radius is in reality more complex than the Bessel function of the radius is in reality more complex than the Bessel function as shown in the following figure. According to the fuel function as shown in the following figure. According to the fuel management strategies used in 80s & the fuel loading patterns which management strategies used in 80s & the fuel loading patterns which followed from these, the most enriched fuel assembly are loaded at the followed from these, the most enriched fuel assembly are loaded at the periphery of the core & the lowest at the centre in order to get a flatned periphery of the core & the lowest at the centre in order to get a flatned power distribution (shape 2). power distribution (shape 2). Modern fuel management strategies currently used require that the  Modern fuel management strategies currently used require that the highest fuel-enrichment zone be loaded at the centre and the lowest at highest fuel-enrichment zone be loaded at the centre and the lowest at the periphery in order to decrease the amplitude of the neutron the periphery in order to decrease the amplitude of the neutron leakage & then increase the initial core reactivity & consequently the leakage & then increase the initial core reactivity & consequently the fuel cycle length (shape 3). fuel cycle length (shape 3).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Obviously, the immediate consequence is an increase in the core  Obviously, the immediate consequence is an increase in the core peaking factors. The safety analysis must be reviewed to demonstrate peaking factors. The safety analysis must be reviewed to demonstrate that the safety margins are still acceptable. that the safety margins are still acceptable. Figure XX.1: Radial distribution of neutron flux  Figure XX.1: Radial distribution of neutron flux

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



level level

b) Axial neutron flux distribution.  b) Axial neutron flux distribution. When the core is homogenous & at the beginning of the life (BOL) &  When the core is homogenous & at the beginning of the life (BOL) & zero power (shape 1), the curve of the flux distribution has the shape zero power (shape 1), the curve of the flux distribution has the shape of a cosinus function. of a cosinus function. increasing to full power, the coolant The reactor power increasing to full power, the coolant  The reactor power temperature being higher towards the top of the core while being temperature being higher towards the top of the core while being roughly constant towards the bottom (according to the core average roughly constant towards the bottom (according to the core average temperature variation versus the power level), neutron moderation is temperature variation versus the power level), neutron moderation is more effective towards the bottom of the core where the water density more effective towards the bottom of the core where the water density is the highest (less effective towards the top where the density is the is the highest (less effective towards the top where the density is the lowest). lowest). This leads to an axial gradient of reactivity that induces a slight bulge  This leads to an axial gradient of reactivity that induces a slight bulge in the axial flux distribution towards the base of the core (shape 2). in the axial flux distribution towards the base of the core (shape 2). After few months of operation at full power, the fuel is burned up After few months of operation at full power, the fuel is burned up faster in the region where the neutron flux is higher i.e. a little bit lower faster in the region where the neutron flux is higher i.e. a little bit lower than the mid-axis. On the other hand, the fuel is burned up more slowly than the mid-axis. On the other hand, the fuel is burned up more slowly at the axial core extremities. The consequence is a so-called « double at the axial core extremities. The consequence is a so-called « double This is typically the case at the EOL. humped » axial curve (shape 3). This is typically the case at the EOL. humped » axial curve (shape 3).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

XX.2 : Axial distribution of the neutron flux  XX.2 : Axial distribution of the neutron flux



THERMAL-HYDRAULIC IN NUCLEAR REACTOR As indicated in the following figure, insertion of the control rods  As indicated in the following figure, insertion of the control rods modifies the shape of the axial flux distribution curve by moving the modifies the shape of the axial flux distribution curve by moving the position of maximum flux clearly into the lower half of the core. This position of maximum flux clearly into the lower half of the core. This movement is accompanied by an increase in the value of the maximum movement is accompanied by an increase in the value of the maximum flux in the same region. flux in the same region. It is therefore necessary, as we will see later on, to verify that the It is therefore necessary, as we will see later on, to verify that the thermal flux at this point does not exceed certain limiting values. thermal flux at this point does not exceed certain limiting values. The situation also makes it necessary to operate at normal rated  The situation also makes it necessary to operate at normal rated reactor power with minimum insertion of the control rods in the core, reactor power with minimum insertion of the control rods in the core, which allows more homogeneous fuel burn-up & ensures a greater which allows more homogeneous fuel burn-up & ensures a greater negative reactivity reserve in case of reactor trip. negative reactivity reserve in case of reactor trip. Nevertheless, the fast reactivity variations, related to rapid changes in  Nevertheless, the fast reactivity variations, related to rapid changes in the power level must be compensated for by the RCCA. That means a the power level must be compensated for by the RCCA. That means a compromise must be looked for & a certain insertion is acceptable. compromise must be looked for & a certain insertion is acceptable.



the boron concentration the boron concentration

THERMAL-HYDRAULIC IN NUCLEAR REACTOR To be able to keep the control rod clusters at the position indicated in  To be able to keep the control rod clusters at the position indicated in the figure, which is in the so-called « reference » zone, these must be the figure, which is in the so-called « reference » zone, these must be another means of reactor control available soluble boron. another means of reactor control available soluble boron. This element is present in the form of boric acid, diluted in the reactor  This element is present in the form of boric acid, diluted in the reactor coolant. Thus, slow reactivity variations, in particular those related to coolant. Thus, slow reactivity variations, in particular those related to progressive burn-up of the fuel, shift over from cold shutdown to hot- progressive burn-up of the fuel, shift over from cold shutdown to hot- shutdown, & scheduled slow reactor power changes are compensated shutdown, & scheduled slow reactor power changes are compensated for by causing the boron concentration to vary correspondinly. for by causing the boron concentration to vary correspondinly. This adjustment is quite slow (some 10 minutes is necessary to  This adjustment is quite slow (some 10 minutes is necessary to change the boron concentration but homogeneous i.e. does not change the boron concentration but homogeneous i.e. does not influence the power distribution. influence the power distribution. This drawback relates to the quantity of liquid effluents produced at  This drawback relates to the quantity of liquid effluents produced at each change in the boron concentration of the reactor coolant. each change in the boron concentration of the reactor coolant. Immediately after the first power escalation after reactor fueling, at full Immediately after the first power escalation after reactor fueling, at full is approximatively 1000 ppm power, is approximatively 1000 ppm power, according to the fuel management strategy and the fuel loading according to the fuel management strategy and the fuel loading pattern. pattern.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Normal fuel burn-up reduces the concentration by 3 to 4 ppm per day.  Normal fuel burn-up reduces the concentration by 3 to 4 ppm per day. Figure XX.3: Axial neutron flux distribution with RCCA insertion.  Figure XX.3: Axial neutron flux distribution with RCCA insertion.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR



c) The Xenon effect  c) The Xenon effect Some of the fission products formed strongly capture free neutrons &  Some of the fission products formed strongly capture free neutrons & are known as « poisons ». This particularly the case of the Xenon 135, are known as « poisons ». This particularly the case of the Xenon 135, which can cause considerable variations in the core reactivity within which can cause considerable variations in the core reactivity within the space of few hours, thus leading to difficulties in controlling the the space of few hours, thus leading to difficulties in controlling the reactor power (overall effect) & in controlling the power distribution reactor power (overall effect) & in controlling the power distribution (local effect). (local effect). The Xenon effect is responsible for the following phenomena:  The Xenon effect is responsible for the following phenomena: - Start-up: If one starts the reactor from a situation where no Xenon is - Start-up: If one starts the reactor from a situation where no Xenon is  present (after a shutdown lasting for several days), the progressive present (after a shutdown lasting for several days), the progressive formation of Xenon 135 makes it necessary to withdraw the RCCA or formation of Xenon 135 makes it necessary to withdraw the RCCA or to reduce the concentration of the soluble boron in order to maintain to reduce the concentration of the soluble boron in order to maintain the desired neutron flux or power level, until the Xenon has attained its the desired neutron flux or power level, until the Xenon has attained its equilibrium concentration. equilibrium concentration. - Power increase: Starting from a situation where the Xenon is in a - Power increase: Starting from a situation where the Xenon is in a state of equlibrium, it is necessary to temporarily increase the boron state of equlibrium, it is necessary to temporarily increase the boron concentration or to insert the control rod clusters, whereas power concentration or to insert the control rod clusters, whereas power increase is obtained by the opposite movement. increase is obtained by the opposite movement.



THERMAL-HYDRAULIC IN NUCLEAR REACTOR - Power decrease: Starting from a situation in which the Xenon is in - Power decrease: Starting from a situation in which the Xenon is in equilibrium, one obtains an opposite effect from that mentioned just equilibrium, one obtains an opposite effect from that mentioned just above, but one that is much more pronounced. This is called « the above, but one that is much more pronounced. This is called « the Xenon peak ». After a reactor trip, it can limlit the return to criticality Xenon peak ». After a reactor trip, it can limlit the return to criticality and then the full reactor power for periods of several hours. and then the full reactor power for periods of several hours. - Stable power distribution: The production of Xenon depends upon - Stable power distribution: The production of Xenon depends upon the disappearance of the iodine 135, the local quantity of the Xenon the disappearance of the iodine 135, the local quantity of the Xenon depends on the evolution of the flux at the point under consideration. depends on the evolution of the flux at the point under consideration. The distribution of Xenon in the core influence the power distribution The distribution of Xenon in the core influence the power distribution & can cause instabilities in the later. & can cause instabilities in the later. The following figure shows the evolution of the negative reactivity due  The following figure shows the evolution of the negative reactivity due to the Xenon over time, as function of the power level variations). to the Xenon over time, as function of the power level variations).

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

Figure XX.4: Evolution of the Xenon negative effect.  Figure XX.4: Evolution of the Xenon negative effect.

 XX1. 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Computer codes in Nuclear Reactor Thermal-Hydraulics XX1. Computer codes in Nuclear Reactor Thermal-Hydraulics From early days of the usage of nuclear technology, computer From early days of the usage of nuclear technology, computer codes have been used to support design and safety analyses. The codes have been used to support design and safety analyses. The raison for that is twofold: on the one hand nuclear engineering is an raison for that is twofold: on the one hand nuclear engineering is an excellent example of a multi-physics domain, where strong excellent example of a multi-physics domain, where strong interactions between different fields (e.g. neutron physics, multi-phase interactions between different fields (e.g. neutron physics, multi-phase flows, structural dynamics, chemistry, etc.) exists. On the other hand flows, structural dynamics, chemistry, etc.) exists. On the other hand nuclear industry applies very high standards as far as the operational nuclear industry applies very high standards as far as the operational safety of NPPs is concerned, which, in turn, requires high accuracy in safety of NPPs is concerned, which, in turn, requires high accuracy in estimation of the operational conditions. estimation of the operational conditions. The currently used codes can be divided in the following groups:  The currently used codes can be divided in the following groups: Reactor simulation codes: Such codes have well developed neutronic  Reactor simulation codes: Such codes have well developed neutronic modules (diffusion theory, transport theory, Monte Carlo theory) and modules (diffusion theory, transport theory, Monte Carlo theory) and somewhat more crude thermal-hydraulic modules. Examples of such somewhat more crude thermal-hydraulic modules. Examples of such codes are POLCA (Westinghouse, earlier ABB-Atom), SIMULATE, etc. codes are POLCA (Westinghouse, earlier ABB-Atom), SIMULATE, etc. Transport codes are used to obtain the macroscopic neutron cross- Transport codes are used to obtain the macroscopic neutron cross- sections which are later used as input to the diffusion theory codes. sections which are later used as input to the diffusion theory codes. Examples are PHOENIX, CASMO, DOT, ANISN, etc. Examples are PHOENIX, CASMO, DOT, ANISN, etc.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Reactor kinetics codes: Example such is the PARCS code that solves  Reactor kinetics codes: Example such is the PARCS code that solves the time dependent two-group neutron diffusion equation in three- the time dependent two-group neutron diffusion equation in three- dimensional Cartesian geometry using nodal methods to obtain the dimensional Cartesian geometry using nodal methods to obtain the transient neutron flux distribution. The code may be used in the transient neutron flux distribution. The code may be used in the analysis of reactivity initiated accidents in light water reactors were analysis of reactivity initiated accidents in light water reactors were spatial effects may be important. It may be run in the stand-alone mode spatial effects may be important. It may be run in the stand-alone mode or coupled to other codes such as RELAP5. or coupled to other codes such as RELAP5. Thermal-hydraulic system codes: Thermal-hydraulics codes are used  Thermal-hydraulic system codes: Thermal-hydraulics codes are used to analyse loss of coolant accidents, LOCAs, any system transients in to analyse loss of coolant accidents, LOCAs, any system transients in light water reactors. There is a variety of TH system codes used in light water reactors. There is a variety of TH system codes used in nuclear engineering. The best know are the RELAP5, CATHARE and nuclear engineering. The best know are the RELAP5, CATHARE and the TRAC codes, which primary goal are to predict small break Loss- the TRAC codes, which primary goal are to predict small break Loss- large break LOCA thermal- Coolant Accident (LOCA) and the large break LOCA thermal- Coolant Accident (LOCA) and the hydraulics, respectively.. hydraulics, respectively

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Thermal hydraulic fuel analysis codes: The most important group are  Thermal hydraulic fuel analysis codes: The most important group are so called sub-channel analysis codes, which are using flow averaging so called sub-channel analysis codes, which are using flow averaging on the sub-channel level and apply mixing models to account for the on the sub-channel level and apply mixing models to account for the mass, momentum and energy exchange between sub-channels. mass, momentum and energy exchange between sub-channels. Examples such codes are, THINC-IV, COBRA, VIPRE, COMETHE, Examples such codes are, THINC-IV, COBRA, VIPRE, COMETHE, THYC, FLICA, BUNGLE, MONA-3, etc. Typical application of such THYC, FLICA, BUNGLE, MONA-3, etc. Typical application of such codes I to predict void distributions, pressure drops and the margins codes I to predict void distributions, pressure drops and the margins to CHF. to CHF. Severe accident codes: Severe accidents codes are used to model the  Severe accident codes: Severe accidents codes are used to model the progression of accidents in light water reactor nuclear power plants. progression of accidents in light water reactor nuclear power plants. Three examples of such codes are MELCOR, SCDAP/RELAP5 and Three examples of such codes are MELCOR, SCDAP/RELAP5 and CATHARE. CATHARE.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

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THERMAL-HYDRAULIC IN NUCLEAR REACTOR Summary

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

II. Energy from nuclear fission II. Energy from nuclear fission

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

1010 (W) (W)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Figure III.1: Fission yield as a function of mass number of the fission Figure III.1: Fission yield as a function of mass number of the fission product. product.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

he 6 formula: Figure III.2: Illustration of the 6 formula: Figure III.2: Illustration of t

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

/ /

/ l)t / l)t

β ρ β ρ – ( ))e ( – ))e

λρ β ρ ­ ( λρ β ρ )t ( – ­ ( ) e )t ( – ) e

­0.075t + 0.933 e

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

opop) / t

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

∞∞ dEσ dEσff((i)i) (E)Ф (r,E) (V.1) (E)Ф (r,E) (V.1)

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Figure VI.1: Typical coolant sub­channels in fuel rods assembly. Figure VI.1: Typical coolant sub­channels in fuel rods assembly.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 And the wetted perimeter (part of the perimeter filled with heated walls) And the wetted perimeter (part of the perimeter filled with heated walls) is given by: is given by:

22 – 1} for square lattice – 1} for square lattice

22 – 1 } for triangular lattice – 1 } for triangular lattice

THERMAL-HYDRAULIC IN NUCLEAR REACTOR π – N dπ 22) / (4w + N d) (VI.5) π ) / (4w + N d) (VI.5)

 Dh = 4A / Pw = (4w  Where NN is the number of rods in the fuel assembly, Where

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 The energy balance for a differential channel length between z and z + The energy balance for a differential channel length between z and z + dz is given as follows: dz is given as follows:

 ΔΔH = W . i  Which to the following differential equation for the coolant enthalpy: Which to the following differential equation for the coolant enthalpy: Δ ΔH = di H = di 

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

lbi + q”P

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 TTlblb = ƒ  The temperature distribution along the channel is represented in figure The temperature distribution along the channel is represented in figure VIII.1. VIII.1.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR Figure VIII.1: Bulk temperature distribution in a uniformly heated channel Figure VIII.1: Bulk temperature distribution in a uniformly heated channel with constant heated perimeter.. with constant heated perimeter

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Equation (VII.2) becomes: Equation (VII.2) becomes:  DiDill(z) / dz = q”  dTlb(z) /dz = q”0PH(z) /W.Cp.cos( z /H) (VIII.4) dTlb(z) /dz = q”0PH(z) /W.Cp.cos( z /H) (VIII.4)  ..

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 After intergration, the coolant enthalpy and temperature distribution After intergration, the coolant enthalpy and temperature distribution are as follows: are as follows:

lbi lbi

lbex = Tlb(H/2) = 2q”

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

 Assuming that q”’ is constant a cross­section equation (IX.1) could be Assuming that q”’ is constant a cross­section equation (IX.1) could be integrated to obtain: integrated to obtain:

TF0TF0λλFFdTdTFF = ­ q” /2ƒ = ­ q” /2ƒ00

 λλFFrdTrdTFF = ­ r2 /2q”’dr = ­ r2 /2q”’dr

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

TFcTFcλλFFdTdTFF

FoFo /4λ /4λFF

FoFoq”’q”’

 By combining equations ( IX.5) &( IX.­6), we have: By combining equations ( IX.5) &( IX.­6), we have: 

 Equation (IX.7) reveals that the fuel temperature drop is a function of Equation (IX.7) reveals that the fuel temperature drop is a function of the linear power density and the averaged fuel thermal conductivity. the linear power density and the averaged fuel thermal conductivity.

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

THERMAL-HYDRAULIC IN NUCLEAR REACTOR

λρ β ρ ( λρ β ρ )t ( – ( ) e )t ( – ) e

0.075t + 0.933 e

 Figure VI.1: Typical coolant subchannels in fuel rods assembly. Figure VI.1: Typical coolant subchannels in fuel rods assembly.

 Assuming that q”’ is constant a crosssection equation (IX.1) could be Assuming that q”’ is constant a crosssection equation (IX.1) could be integrated to obtain: integrated to obtain:

TF0TF0λλFFdTdTFF = q” /2ƒ = q” /2ƒ00

 λλFFrdTrdTFF = r2 /2q”’dr = r2 /2q”’dr

 By combining equations ( IX.5) &( IX.6), we have: By combining equations ( IX.5) &( IX.6), we have: 

C0C0,Max T