Báo cáo hóa học: " Heat transfer augmentation in nanofluids via nanofins Peter Vadasz1,2"

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Vadasz Nanoscale Research Letters 2011, 6:154 http://www.nanoscalereslett.com/content/6/1/154 NANO EXPRESS Open Access Heat transfer augmentation in nanofluids via nanofins Peter Vadasz1,2 Abstract Theoretical results derived in this article are combined with experimental data to conclude that, while there is no improvement in the effective thermal conductivity of nanofluids beyond the Maxwell’s effective medium theory (J.C. Maxwell, Treatise on Electricity and Magnetism, 1891), there is substantial heat transfer augmentation via nanofins. The latter are formed as attachments on the hot wire surface by yet an unknown mechanism, which could be related to electrophoresis, but there is no conclusive evidence yet to prove this proposed mechanism. Introduction succeeded to show a viable explanation. Jang and Choi [18] and Prasher et al. [19] show that convection due to The impressive heat transfer enhancement revealed Brownian motion may explain the enhancement of the experimentally in nanofluid suspensions by Eastman effective thermal conductivity. However, if indeed this is et al. [1], Lee et al. [2], and Choi et al. [3] conflicts apparently with Maxwell’s [4] classical theory of estimat- the case then it is difficult to explain why this enhance- ment of the effective thermal conductivity is selective ing the effective thermal conductivity of suspensions, and is not obtained in all the nanofluid experiments. including higher-order corrections and other than sphe- Alternatively, Vadasz et al. [20] showed that hyperbolic rical particle geometries developed by Hamilton and heat conduction also provides a viable explanation for Crosser [5], Jeffrey [6], Davis [7], Lu and Lin [8], Bonne- the latter, although their further research and compari- caze and Brady [9,10]. Further attempts for independent son with later-published experimental data presented by confirmation of the experimental results showed con- Vadasz and Govender [21] led them to discard this flicting outcomes with some experiments, such as Das possibility. et al. [11] and Li and Peterson [12], confirming at least Vadasz [22] derived theoretically a model for the heat partially the results presented by Eastman et al. [1], Lee conduction mechanisms of nanofluid suspensions et al. [2], and Choi et al. [3], while others, such as Buon- including the effect of the surface area-to-volume ratio giorno and Venerus [13], Buongiorno et al. [14], show in contrast results that are in agreement with Maxwell’s [4] of the suspended nanoparticles/nanotubes on the heat transfer. The theoretical model was shown to provide a effective medium theory. All these experiments were viable explanation for the excessive values of the effec- performed using the Transient-Hot-Wire (THW) tive thermal conductivity obtained experimentally [1-3]. experimental method. On the other hand, most experi- The explanation is based on the fact that the THW mental results that used optical methods, such as the “optical beam deflection” [15], “all-optical thermal len- experimental method used in all the nanofluid suspen- sing method” [16], and “forced Rayleigh scattering” [17] sions experiments listed above needs a major correction factor when applied to non-homogeneous systems. This did not reveal any thermal conductivity enhancement time-dependent correction factor is of the same order of beyond what is predicted by the effective medium the- magnitude as the claimed enhancement of the effective ory. A variety of possible reasons for the excessive thermal conductivity. However, no direct comparison to values of the effective thermal conductivity obtained in experiments was possible because the authors [1-3] did some experiments have been investigated, but only few not report so far their temperature readings as a func- tion of time, the base upon which the effective thermal Correspondence: peter.vadasz@nau.edu 1 Department of Mechanical Engineering, Northern Arizona University, P. O. conductivity is being evaluated. Nevertheless, in their Box 15600, Flagstaff, AZ 86011-5600, USA. article, Liu et al. [23] reveal three important new results Full list of author information is available at the end of the article © 2011 Vadasz; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Vadasz Nanoscale Research Letters 2011, 6:154 Page 2 of 12 http://www.nanoscalereslett.com/content/6/1/154 that allow the comparison of Vadasz’s [22] theoretical suspended solid particles by considering phase-averaged model with experiments. The first important new result equations will be presented only briefly without includ- presented by Liu et al. [23] is reflected in the fact that ing the details that can be obtained from [22]. The the value of “ effective thermal conductivity ” revealed phase-averaged equations are experimentally using the THW method is time depen- ∂Ts = h ( Tf − Ts ) dent. The second new result is that those authors pre- s (1) ∂t * sent graphically their time-dependent “effective thermal conductivity” for three specimens and therefore allow the comparison of their results with the theoretical pre- ∂Tf = k f ∇ * Tf − h ( Tf − Ts ) dictions of this study showing a very good fit as pre- f 2 (2) ∂t * sented in this article. The third new result is that their time dependent “ effective thermal conductivity” con- where t* is time, Tf (r*,t*), and Ts (r*,t*) are tempera- verges at steady state to values that according to our calculations confirm the validity of the classical ture values for the fluid and solid phases, respectively, Maxwell’s theory [4] and its extensions [5-10]. averaged over a representative elementary volume (REV) The objective of this article is to provide an explana- that is large enough to be statistically valid but suffi- tion that settles the conflict between the apparent ciently small compared to the size of the domain, and where r* are the coordinates of the centroid of the REV. enhancement of the effective thermal conductivity in In Equations (1) and (2), gs = εrscs and gf = (1 - ε)rfcp some experiments and the lack of enhancement in other experiments. It is demonstrated that the transient heat represent the effective heat capacity of the solid and fluid phases, respectively; with rs and rf are the densities conduction process in nanofluid suspensions produces results that fit well with the experimental data [23] and of the solid and fluid phases, respectively; cs and cp are validates Maxwell’s [4] method of estimating the effec- the specific heats of the solid and fluid phases, respec- tively; and ε is the volumetric solid fraction of the sus- tive thermal conductivity of suspensions. The theoretical results derived in this article are combined with experi- pension. Similarly, k f is the effective thermal mental data [23] to conclude that, while there is no conductivity of the fluid that may be defined in the   form k f = f ( ,  )k f , where k f is the thermal conductivity improvement in the effective thermal conductivity of  nanofluids beyond the Maxwell’s effective medium the- of the fluid,  = k s / k f is the thermal conductivity ratio, and ε is the solid fraction of suspended particles in the ory [4], there is nevertheless substantial heat transfer augmentation via nanofins. The latter are formed as suspension. In Equations (1) and (2), the parameter h, carrying units of W m -3 K -1 , represents an integral attachments on the hot wire surface by a mechanism that could be related to electrophoresis and therefore heat transfer coefficient for the contribution of the heat such attachments depend on the electrical current pas- conduction at the solid-fluid interface as a volumetric sing through the wire, and varies therefore amongst dif- heat source/sink within an REV. It is assumed to be ferent experiments. Also since the effective thermal independent of time, and its general relationship to conductivity does not increase beyond the Maxwell’s [4] the surface-area-to-volume ratio (specific area) was derived in [22]. Note that T s ( r * , t * ) is a function of effective medium theory, the experiments using optical methods, such as Putnam et al. [15], Rusconi et al. [16] the space variables represented by the position vector ∧ ∧ ∧ and Venerus et al. [17], are also consistent with the con- r* = x* e x + y* e y + z* e z , in addition to its dependence on time, because Ts(r*,t*) depends on Tf(r*,t*) as expli- clusion of this study. In this article, a contextual notation is introduced to dis- citly stated in Equation (1), although no spatial deriva- tinguish between dimensional and dimensionless variables tives appear in Equation (1). There is a lack of and parameters. The contextual notation implies that an macroscopic level conduction mechanism in Equation (1) asterisk subscript is used to identify dimensional variables representing the heat transfer within the solid phase and parameters only when ambiguity may arise when the because the solid particles represent the dispersed phase asterisk subscript is not used. For example t* is the dimen- in the fluid suspension, and therefore the solid particles sional time, while t is its corresponding dimensionless can conduct heat between themselves only via the neigh- bouring fluid. When steady state is accomplished ∂Ts/∂t* counterpart. However, kf is the effective fluid phase ther- = ∂ T f / ∂ t * = 0, leading to local thermal equilibrium mal conductivity, a dimensional parameter that appears between the solid and fluid phases, i.e. Ts(r) = Tf(r). without an asterisk subscript without causing ambiguity. For the case of a thin hot wire embedded in a cylind- Problem formulation rical container insulated on its top and bottom one can The theoretical model derived by Vadasz [22] to investi- assume that the heat is transferred in the radial direc- gate the transient heat conduction in a fluid containing tion only, r*, rendering Equation (2) into
Vadasz Nanoscale Research Letters 2011, 6:154 Page 3 of 12 http://www.nanoscalereslett.com/content/6/1/154 and t 2* respectively, the thermal conductivity can be 1 ∂ ⎛ ∂Tf ⎞ ∂Tf ⎟ − h ( Tf − Ts ) f = kf ⎜ r* approximated using Equation (4) in the form: (3) ∂t * r* ∂r* ⎝ ∂r* ⎠ ⎡ ⎛ t 2* ⎞ ⎤ iV k≈ ⎢ ln ⎜ ⎟⎥ In a homogeneous medium without solid-suspended (5) 4 ( T2 − T1 ) l* ⎢ ⎣ ⎝ t 1* ⎠ ⎥ ⎦ particles, Equation (1) is not relevant and the last term in Equation (3) can also be omitted. The boundary and Equation (5) is a very accurate way of estimating the initial conditions applicable are an initial ambient con- thermal conductivity as long as the validity condition is stant temperature, T C , within the whole domain, an fulfilled. The validity condition implies the application ambient constant temperature, TC, at the outer radius of of Equation (5) for long times only. However, when the container and a constant heat flux, q0, over the fluid- evaluating this condition to data used in the nanofluid wire interface that is related to the Joule heating of the suspensions experiments, one obtains that t0* ~ 6 ms, wire in the form q0 = iV/(πdw*l*), where dw* and l* are the and the time beyond which the solution (5) can be used diameter and the length of the wire respectively, i is the reliably is therefore of the order of hundreds of millise- electric current and V is the voltage drop across the wire. conds, not so long in the actual practical sense. Vadasz [22] showed that the problem formulated by Equations (1) and (3) subject to appropriate initial and Two methods of solution boundary conditions represents a particular case of Dual- While the THW method is well established for homoge- Phase-Lagging heat conduction (see also [24-28]). neous fluids, its applicability to two-phase systems such as An essential component in the application of the fluid suspensions is still under development, and no reliable THW method for estimating experimentally the effective validity conditions for the latter exist so far (see Vadasz [30] thermal conductivity of the nanofluid suspension is the for a discussion and initial study on the latter). As a result, assumption that the nanofluid suspension behaves basi- one needs to refer to the two-equation model presented by cally like a homogeneous material following Fourier law Equations (1) and (3), instead of the one Fourier type equa- for the bulk. The THW method is well established as tion that is applicable to homogeneous media. the most accurate, reliable and robust technique [29] for Two methods of solution are in principle available to evaluating the thermal conductivity of fluids. A very solve the system of Equations (1) and (3). The first is thin (5-80 μm in diameter) platinum (alternatively tanta- the elimination method while the second is the eigen- lum) wire is embedded vertically in the selected fluid vectors method. By means of the elimination method, and serves as a heat source as well as a thermometer one may eliminate Tf from Equation (1) in the form: (see [22] for details). Because of the very small diameter and high thermal conductivity of the platinum wire, it  s ∂Ts Tf = + Ts can be regarded as a line heat source in an otherwise (6) h ∂t * infinite cylindrical medium. The rate of heat generated per unit length ( l * ) of platinum wire due to a step and substitute it into Equation (3) hence rendering the  change in voltage is therefore q l* = iV l* W m-1 . Sol- two Equations (1) and (3), each of which depends on ving for the radial heat conduction due to this line heat both Ts and Tf, into separate equations for Ts and Tf, source leads to an approximated temperature solution respectively, in the form: in the wire’s neighbourhood in the form ⎡ 1 ∂ ⎛ ∂Ti ⎞ ∂ 2Ti ∂Ti q = e ⎢ + q⎡ ⎞⎤ ⎜ r* ⎟ ⎛ 4 t  ∂t * ⎢ r* ∂r* ⎝ ∂r* ⎠ 2 ∂t * T (r* , t *) ≈ l* ⎢ − 0 + ln ⎜ 2 * ⎟⎥ ⎣ (4) ⎜r ⎟ 4 k ⎣ (7)  ∂ ⎛ ∂ 2Ti ⎞ ⎤ ⎢ ⎥ ⎝* ⎠⎦ +T ⎟ ⎥ for i = s, f ⎜ r* r* ∂r* ⎜ ∂r*∂t * ⎟ ⎥ ⎝ ⎠⎦ provided a validity condition for the approximation is enforced, i.e. t * >> t 0* = rw* 4 , where rw* is the radius 2 where the index i takes the values i = s for the solid  of the platinum wire,  = k f /  f c p is the fluid’s thermal phase and i = f for the fluid phase, and the following notation was used: diffusivity, and g 0 = 0.5772156649 is Euler ’ s constant. Equation (4) reveals a linear relationship, on a logarithmic  s f kf q = ; e = time scale, between the temperature and time. Therefore, ; h(s + f ) (s + f ) one way of evaluating the thermal conductivity is from the (8)  sk f  T = =s slope of this relationship evaluated at r* = rw*. For any two h (  s +  f )e h readings of temperature, T1 and T2, recorded at times t1*
Vadasz Nanoscale Research Letters 2011, 6:154 Page 4 of 12 http://www.nanoscalereslett.com/content/6/1/154 In Equation (8), τq and τT are the heat flux and tem- Second, one may use Equation (6) and taking its deri- perature-related time lags linked to Dual-Phase-Lagging vative with respect to r* yields [22,24-27,31], while ae is the effective thermal diffusivity  s ∂ ⎛ ∂Ts ⎞ ∂Ts ∂Tf of the suspension. The resulting Equation (7) is identical ⎟+ = ⎜ (14) h ∂t * ⎝ ∂r* ⎠ ∂r* ∂r* for both fluid and solid phases. Vadasz [22] used this equation in providing the solution. The initial conditions applicable to the problem at hand are identical for both In Equation (14), the spatial variable r* plays no active phases, i.e. both phases’ temperatures are set to be equal role; it may therefore be regarded as a parameter. As a to the ambient temperature TC result, one may present Equation (14) for any specified value of r * . Choosing r * = r w* where the value of t * = 0 : Ti = TC = constant , for i = s, f (9) ( ∂Tf ∂r* ) r is known from the boundary condition w* The boundary conditions are (11), yields from (14) the following ordinary differential equation: r* = r0* : Tf = TC (10)  s d ⎛ ∂T s ⎞ ⎛ ∂Ts ⎞ q0 ⎟ +⎜ ⎟ =− ⎜ (15) h dt * ⎝ ∂r* ⎠ r ∂r* ⎠ r kf ⎝ ⎛ ∂T ⎞ q r* = rw* : ⎜ f ⎟ =− 0 w* w* (11) ⎝ ∂r* ⎠ r* = rw* kf At steady state, Equation (15) produces the solution where r 0* is the radius of the cylindrical container. ⎛ ∂Ts ,st ⎞ q0 ⎟ =− ⎜ Equation (7) is second-order in time and second- (16) ∂r* ⎠ r kf ⎝ order in space. The initial conditions (9) provide one w* such condition for each phase while the second-order where Ts,st is the steady-state solution. The transient Equation (7) requires two such conditions. To obtain solution Ts,tr = Ts - Ts,st satisfies then the equation: the additional initial conditions, one may use Equations (1) and (3) in combination with (9). From  s d ⎛ ∂Ts,tr ⎞ ⎛ ∂Ts,tr ⎞ (9), it is evident that both phases’ initial temperatures ⎟ +⎜ ⎟ =0 ⎜ (17) h d t * ⎝ ∂r* ⎠ r ⎝ ∂r* ⎠ rw* at t * = 0 are identical and constant. Therefore, w* = ( Ts ) t ( Tf ) t = TC = constant , leading to * =0 * =0 subject to the initial condition ( Tf − Ts ) t = 0 and ⎡ ∂ ∂r* ( r* ∂Tf ∂r* ) ⎤ ⎦ t * =0 = 0 to ⎣ * =0 ⎡ ⎛ ∂T ⎞ ⎤ ⎡ ⎛ ∂T ⎞ ⎤ = ⎢⎜ s ⎟ ⎥ ⎢ ⎜ s,tr ⎟ ⎥ =0 be substituted in (1) and (3), which in turn leads to (18) ⎢ ∂r ⎢ ⎝ ∂r* ⎠ r ⎥ ⎥ ⎦ t * =0 ⎣ ⎝ * ⎠ rw* the following additional initial conditions for each ⎦ t * =0 ⎣ w* phase: [ ∂Ts / ∂r* ]t =0 = 0 because for all values of ⎛ ∂T ⎞ * t* = 0 :⎜ i ⎟ = 0 for i = s, f (12) r* ∈ [ rw* , r0* ] given that according to (9) at t * = 0: ⎝ ∂t * ⎠ t * =0 ( Ts ) t =0 = ( Tf ) t =0 = TC = constant . Equation (17) can The two boundary conditions (10) and (11) are suffi- * * cient to uniquely define the problem for the fluid phase; be integrated to yield however, there are no boundary conditions set for the solid phase as the original Equation (1) for the solid ⎛h ⎞ ⎛ ∂Ts,tr ⎞ ⎟ = A exp ⎜ − t⎟ ⎜ (19) phase had no spatial derivatives and did not require ⎝ s ∂r* ⎠ r ⎝ ⎠ boundary conditions. To obtain the corresponding w* boundary conditions for the solid phase, which are which combined with the initial condition (18) pro- required for the solution of Equation (7) corresponding duces the value of the integration constant A = 0 and to i = s, one may use first the fact that at r* = r0* both therefore the transient solution becomes phases are exposed to the ambient temperature and therefore one may set ⎛ ∂T s ,tr ⎞ ⎟ =0 ⎜ (20) ⎝ ∂r* ⎠ rw* r* = r0* : Ts = TC (13)
Vadasz Nanoscale Research Letters 2011, 6:154 Page 5 of 12 http://www.nanoscalereslett.com/content/6/1/154 into a dimensionless form by introducing the following The complete solution for the solid temperature gradi- dimensionless variables: ent at the wire is therefore obtained by combining (20) with (16) leading to ( T − TC ) k f , r = r* , t =  et * q* , i = i q= ⎛ ∂Ts ⎞ (22) q0 2 ⎟ =− q0 q 0r0* r0* r0* ⎜ (21) ⎝ ∂r* ⎠ rw* kf where the following two dimensionless groups producing the second boundary condition for the solid emerged: phase, which is identical to the corresponding boundary  e q  e T condition for the fluid phase. One may therefore con- Fo q = ; Fo T = (23) clude that the solution to the problem formulated in 2 2 r0* r0* terms of Equation (7) that is identical to both phases, subject to initial conditions (9) and (12) that are identi- representing a heat flux Fourier number and a tem- cal to both phases, and boundary conditions (10), (11), perature Fourier number, respectively. The ratio and (13), (21) that are also identical to both phases, between them is identical to the ratio between the time should be also identical to both phases, i.e. Ts (t*,r*) = Tf lags, i.e. (t*,r*). This, however, may not happen because then Tf - Fo T  T  s +  f Ts = 0 leads to conflicting results when substituted into = = = (24) Fo q  q f (1) and (3). The result obtained here is identical to Vadasz [32] who demonstrated that a paradox revealed by Vadasz [33] can be avoided only by refraining from Equations (1) and (3) expressed in a dimensionless using this method of solution. While the paradox is form using the transformation listed above are revealed in the corresponding problem of a porous med- ∂ s = (f − s ) ium subject to a combination of Dirichlet and insulation (25) Fh s ∂t boundary conditions, the latter may be applicable to fluids suspensions by setting the effective thermal con- ductivity of the solid phase to be zero. The fact that in ∂ f ⎛ ∂ f ⎞ 1 1∂ ⎟ − (f − s ) = the present case the boundary conditions differ, i.e. a Fh f ⎜r (26) ∂t Ni f r ∂r ⎝ ∂r ⎠ constant heat flux is applied on one of the boundaries (such a boundary condition would have eliminated the where the following additional dimensionless groups paradox in porous media), does not eliminate the para- emerged: dox in fluid suspensions mainly because in the latter case the steady-state solution is identical for both  e s  +f Fo q =  Fo q Fh s = = Fo T = s (27) phases. In the porous media problem, the constant heat f 2 hr0* flux boundary condition leads to different solutions at steady state, and therefore the solutions for each phase s +f   even during the transient conditions differ. Fo T Fh f = Fo q = e 2f = = Fo (28) (  −1) (  −1) q The elimination method yields the same identical equa- s hr0* tion with identical boundary and initial conditions for both phases apparently leading to the wrong conclusion hr0* (  − 1 ) 2 that the temperature of both phases should therefore be Ni f = =2 (29) the same. A closer inspection shows that the discontinu-  Fo q kf ity occurring on the boundaries’ temperatures at t = 0, when a “ramp-type” of boundary condition is used, is the where Nif is the fluid phase Nield number. The solu- reason behind the occurring problem and the apparent tions to Equations (25) and (26) are subject to the fol- paradox. The question that still remains is which phase lowing initial and boundary conditions obtained from temperature corresponds to the solution presented by (9), (10) and (11) transformed in a dimensionless form: Vadasz [22]; the fluid or the solid phase temperature? t = 0 : i = 0 for i = s,f By applying the eigenvectors method as presented by (30) Vadasz [32], one may avoid the paradoxical solution and obtain both phases temperatures. The analytical solution The boundary conditions are to the problem using the eigenvectors method is r = 1 : f = 0 (31) obtained following the transformation of the equations
Vadasz Nanoscale Research Letters 2011, 6:154 Page 6 of 12 http://www.nanoscalereslett.com/content/6/1/154 where ⎛ ∂ ⎞ r = rw : ⎜ f ⎟ = −1 (32) ⎝ ∂r ⎠ r = rw (  −1) ; 1 − − a = −Fh s 1 = − ; c = Fh f 1 =  Fo q  Fo q No boundary conditions are required for θs. The solu- (42) ( ) = − 2 (  −1) tion to the system of Equations (25)-(26) is obtained by + Ni f n 2 =− − a superposition of steady and transient solutions θi,st(r) dn n  Fo q Ni f Fh f and θi,tr (t,r), respectively, in the form:  i ( t , r ) =  i ,st ( r ) +  i ,tr ( t , r ) for i = s, f and where the separation constant  n represents the 2 (33) eigenvalues in space. Substituting (33) into (25)-(26) yields to the following Equation (38) is the Bessel equation of order 0 produ- equations for the steady state: cing solutions in the form of Bessel functions (  f,st −  s,st ) = 0 Ron (  n , r ) = Y0 (  n ) J 0 (  n r ) − J 0 (  n ) Y0 (  n r ) (34) (43) Where J0(nr) and Y0(nr) are the order 0 Bessel func- 1 1 d ⎛ d f,st ⎞ ⎟ − (  f,st −  s,st ) = 0 tions of the first and second kind, respectively. The ⎜r (35) solution (43) satisfies the boundary condition (39) as Ni f r dr ⎝ dr ⎠ can easily be observed by substituting r = 1 in (43). leading to the following steady solutions which satisfy Imposing the second boundary condition (40) yields a transcendental equation for the eigenvalues n in the the boundary conditions (31) and (32): form:  f,st ( r ) =  s,st ( r ) = −rw ln r (36) J 0 (  n ) Y1 (  n rw ) − Y0 (  n ) J 1 (  n rw ) = 0 (44) The transient part of the solutions θ i ,tr ( t , r ) can be obtained by using separation of variables leading to the where J 1 ( n r w ) and Y 1 ( n r w ) are the order 1 Bessel following form of the complete solution: functions of the first and second kind, respectively, eval- uated at r = r w . The compete solution is obtained by ∞ ∑S ( t ) Ron ( r ) for  i = −rw ln r + i = s, f substituting (43) into (37) and imposing the initial con- (37) in ditions (30) in the form n =1 Substituting (37) into (25)-(26) yields, due to the ∞ (  i ) t =0 = −rw ln r + ∑ Sin ( 0 ) Ron ( r ) = 0 for i = s, f (45) separation of variables, the following equation for the unknown functions Ron (r): n =1 At t = 0, both phases’ temperatures are the same lead- 1 d ⎛ dRon ⎞ ⎟ +  n Ron = 0 2 ⎜r (38) ing to the conclusion that r dr ⎝ dr ⎠ S sn ( 0 ) = S fn ( 0 ) = S no (46) subject to the boundary conditions r = 1 : Ron = 0 (39) Multiplying (45) by the orthogonal eigenfunction Rom (m ,r) with respect to the weight function r and inte- grating the result over the domain [ r w ,1], i.e. ⎛ dR ⎞ r = rw : ⎜ on ⎟ =0 (40) 1 ∫r (•)R om( m , r ) r dr yield ⎝ dr ⎠ r = rw w and the following system of equations for the 1 1 ∞ ∑ S ∫ rR ∫ r ln rR om (  m , r ) dr = (  n , r ) Rom (  m , r ) dr unknown functions Sin (t), (i = s,f), i.e. (47) rw no on n =1 rw rw ⎧ ⎪ dS sn = aS − aS The integral on the right-hand side of (47) produces ⎪ dt sn fn (41) ⎨ the following result due to the orthogonality conditions ⎪ dS fn ⎪ dt = cS sn + d nS fn for Bessel functions: ⎩
Vadasz Nanoscale Research Letters 2011, 6:154 Page 7 of 12 http://www.nanoscalereslett.com/content/6/1/154 which upon substituting a , c and d n from Equation 1 for n ≠ m ⎧0 ⎪ ∫ r Ron (  n , r ) R om (  m , r ) dr = ⎨ (42) yields (48) N (  n ) for n = m ⎪ ⎩ rw ⎤ ( 1 +  Fo  ) ⎡ 1 + 2 4Fo q n 2 ⎥ ⎢ qn 1n = − where the norm N(n) is evaluated in the form: 1− (55) ⎥ ⎢ ( 1 +  Fo  ) 2 2Fo q 2 ⎥ ⎢ qn ⎦ ⎣ 2 ⎡ J 1 (  n rw ) − J 0 (  n ) ⎤ 1 2 2 ⎣ ⎦ ∫ N (n ) = rR on (  n , r ) dr = 2 (49)  n J 1 (  n rw ) 2 22 ⎤ ( 1 +  Fo  ) ⎡ 1 − rw 2 4Fo q n 2 ⎥ ⎢ qn  2n = − 1− (56) ⎥ ⎢ ( 1 +  Fo  ) 2 2Fo q The integral on the left-hand side of (47) can be eval- 2 ⎥ ⎢ qn ⎦ ⎣ uated using integration by parts and the equation for the eigenvalues (44) to yield The following useful relationship is obtained from (55) and (56): 1 ∫ r ln rR 1 (  n , r ) dr = ⎡ J 0 (  n ) Y 0 (  n rw ) − Y 0 (  n ) J 0 (  n rw ) ⎤ (50) n ⎣ ⎦ on 2 n2 1n 2n = rw (57) Fo q Substituting (48) and (50) into (47) yields the values of Sin at t = 0, i.e. Sno = Ssn(0) = Sfn(0) The corresponding eigenvectors υ1n and υ2n are evalu- ated in the form: rw ⎡ J 0 (  n ) Y0 (  n rw ) − Y0 (  n ) J 0 (  n rw ) ⎤ = S no N (  n ) n ⎣ ⎦ 2 ⎤ ⎤ ⎡ ⎡ ⎥ ⎥ ⎢ ⎢ 1 1 ⎥ and v1n = ⎢ ⎥ =⎢ v1n (58) that need to be used as initial conditions for the solu- ⎢ ( −1n + a ) ⎢ ( − 2n + a ) ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ tion of system (41) ⎦ ⎦ ⎣ ⎣ a a  2rw J 1 (  n rw ) ⎡ J 0 (  n ) Y0 (  n rw ) − Y0 (  n ) J 0 (  n rw ) ⎤ 2 leading to the following solution: ⎣ ⎦ S no = (51) 2 ⎡ J 1 (  n rw ) − J 0 (  n ) ⎤ 2 2 ⎣ ⎦ S n = v 1nC1ne 1nt + v 2 nC 2ne 2 nt (59) to produce the explicit solutions in time. With the and explicitly following the substitution of (58) and initial conditions for Sin evaluated (i = s,f), one may turn the initial conditions Sin (i = s,f), at t = 0, i.e. Ssn(0) = to solving system (41) that can be presented in the fol- Sfn(0) = Sno with the values of Sno given by Equation (51) lowing vector form: S no dS n ⎡  2ne 1nt − 1ne 2 nt ⎤ S sn = = AS n (60) (52) (  2n − 1n ) ⎣ ⎦ dt where the matrix A is explicitly defined by ( ) ( ) S no ⎡  2n 1 +  Fo q 1n e 1nt − 1n 1 +  Fo q  2n e 2 nt ⎤ S fn = (61) (  2n − 1n ) ⎣ ⎦ a −a A= (53) c dn Substituting (57) into (60) and (61) and the latter into the complete solution (37) yields with the values of a,c and dn given by Equation (42), and the vector S n defined in the form S n = [ S sn, S fn ]T . ∞ ∑B ⎡  2ne 1nt − 1ne 2 nt ⎤ Ron ( r )  s = −rw ln r + (62) The eigenvalues ln corresponding to (52) are obtained ⎣ ⎦ n n =1 as the roots of the following quadratic algebraic equa- tion: ∞ ∑ B ⎡⎣ (  ) ( ) +  n e 1nt − 1n +  n e 2 nt ⎤ Ron ( r )  f = −rw ln r + 2 2 (63) − ( a + d n ) n + a ( d n + c ) = 0 ⎦ n 2n n 2 (54) n =1 where Bn is leading to S no a + dn 1 Bn = ( a − d n ) 2 − 4ac (  2n − 1n ) 1n = + and 2 2 (64)  2rw J 1 (  n rw ) ⎡ J 0 (  n ) Y0 (  n rw ) − Y0 (  n ) J 0 (  n rw ) ⎤ 2 ⎣ ⎦ a + dn 1 = ( a − dn ) 2  2n = − − 4ac 2 (  2n − 1n ) ⎡ J 1 (  n rw ) − J 0 (  n ) ⎤ 2 2 ⎣ ⎦ 2 2
Vadasz Nanoscale Research Letters 2011, 6:154 Page 8 of 12 http://www.nanoscalereslett.com/content/6/1/154 C omparing the solutions obtained above with the where the temperature difference [Tw(t) - TC] is repre- solution obtained by Vadasz [22] via the elimination sented by the recorded experimental data, and the value method, one may conclude that the latter corresponds of the heat flux at the fluid-platinum-wire interface q0 is to the solid phase temperature θs. evaluated from the Joule heating of the hot wire. In ( ) The Fourier solution is presented now to compare the ∞ f ( t ) = ∑ n =1 C n Ron ( rw ) exp − n t , 2 Equation (72) solution obtained from the Dual-Phase-Lagging model to the former. The Fourier solution is the result where the coefficient C n is defined by (70) and the eigenvalues n are defined by Equation (44). Note that obtained by solving the thermal diffusion equation the definition of Cn here is different than in [22]. The 1 ∂ 1 ∂ ⎛ ∂ ⎞ results obtained from the application of Equation (72) = ⎜r (65) ⎟  ∂t r ∂r ⎝ ∂r ⎠ fit extremely well the approximation used by the THW method via Equation (5) within the validity limits of the subject to the boundary and initial conditions approximation (5). Therefore, the THW method is extremely accurate for homogeneous materials. t = 0 : = 0 (66) On the other hand, for non-homogeneous materials, by means of the solutions (62) and (63) applicable to r =1:  = 0 (67) fluid suspensions evaluated at r = rw, one obtains q 0*r0* [ Tsw − TC ] = ⎡ −rw ln ( rw ) + g s ( t ) ⎤ ⎛ ∂ ⎞ (73) k f,act ⎣ ⎦ r = rw : ⎜ = −1 ⎟ (68) ⎝ ∂r ⎠ r = rw q 0*r0* [ Tfw − TC ] = where the same scaling as in Equation (22) was ⎡ −rw ln ( rw ) + g f ( t ) ⎤ (74) k f,act ⎣ ⎦ applied in transforming the equation into its dimension- less form, hence the reason for the coefficient 1/b in the where kf,act is the actual effective thermal conductivity, equation. The Fourier solution for this problem has Tsw (t) and Tfw (t) are the solid and fluid phases tem- then the form [34] peratures “felt” by the wire at the points of contact with ∞ each phase, respectively, and the functions gs (t) and gf ∑C e Ron ( r ) −  n t 2  = −rw ln r + (69) (t) obtained from the solutions (62) and (63) evaluated n n =1 at r = rw take the form where ∞ ∑B R ( rw ) ⎡  n2 exp ( n1t ) − n1 exp ( n2t ) ⎤ gs ( t ) = (75) ⎣ ⎦ n on  2rw J 1 (  n rw ) ⎡ J 0 (  n ) Y0 (  n rw ) − Y0 (  n ) J 0 (  n rw ) ⎤ n =1 2 ⎣ ⎦ =S Cn = (70) no 2 ⎡ J 1 (  n rw ) − J 0 (  n ) ⎤ 2 2 ⎣ ⎦ ∞ ( rw ) ⎡ (  n2 +  n ) exp (  n1t ) ∑B R gf ( t ) = and the eigenvalues n are the solution of the same 2 ⎣ n on transcendental Equation (44) and the eigenfunctions Ron (76) n =1 ( ) (r) are also identical to the ones presented in Equation exp (  n2t ) ⎤ −  n1 +  n 2 ⎦ (43). The relationship between the Fourier coefficient Cn and the Dual-Phase-Lagging model’s coefficient Bn is When the wire is exposed partly to the fluid phase C n = (  2n − 1n ) B n and partly to the solid phase, there is no justification in (71) assuming that the wire temperature is uniform: on the contrary the wire temperature will vary between the regions exposed to the fluid and solid phases. Assuming Correction of the THW results that some solid nanoparticles are in contact with the When evaluating the thermal conductivity by applying wire in a way that they form approximately “solid rings” the THW method and using Fourier law, one obtains around the wire, then the “effective” wire temperature for the effective thermal conductivity the following rela- can be evaluated as electrical resistances in series. By tionship [22]: defining the relative wire area covered by the solid nanoparticles as a s = A s / A tot = A s /2 π r w* l * its corre- q 0r0* ⎡ −rw ln ( rw ) + f ( t ) ⎤ k f,app = ⎡ T w ( t ) − TC ⎤ ⎣ ⎦ sponding wire area covered by the fluid is af = Af/Atot = (72) ⎣ ⎦ 1 - as, then from the relationship between the electrical
Vadasz Nanoscale Research Letters 2011, 6:154 Page 9 of 12 http://www.nanoscalereslett.com/content/6/1/154 r esistance and temperature accounting for electrical compared with the experimental results presented by resistances connected in series, one obtains an expres- Liu et al. [23]. sion for the effective wire temperature (i.e. the tempera- ture that is evaluated using the wire’s lumped electrical Results and discussion The results for the solid and fluid phases’ temperature resistance in the THW Wheatstone bridge) Tw in the form: at r = rw as a function of time obtained from the solu- tions (62) and (63) are presented in Figures 1, 2 and 3 [ Tw − TC ] = a s ( Tsw − TC ) + ( 1 − a s ) ( Tfw − TC ) (77) in comparison with the single-phase Fourier solution (69) for three different combinations of values of Fo q Substituting (73) and (74) into (77) yields and as, and plotted on a logarithmic time scale. While the quantitative results differ amongst the three fig- q 0*r0* [ Tw − TC ] = ⎡ −rw ln ( rw ) + a s g s ( t ) + ( 1 − a s ) g f ( t ) ⎤ (78) ures, there are some similar qualitative features that k f,act ⎣ ⎦ are important to mention. First, it is evident from these figures that the fluid phase temperature is One may then use (78) to evaluate the actual nano- fluid’s effective thermal conductivity kf,act from (78) in almost the same as the temperature obtained from the single-phase Fourier solution. Second, it is also evi- the form dent that the solid phase temperature lags behind the q 0*r0* ⎡ −rw ln ( rw ) + a s g s ( t ) + ( 1 − a s ) g f ( t ) ⎤ fluid phase temperature by a substantial difference. k f,act = (79) ( Tw − TC ) ⎣ ⎦ They become closer as steady-state conditions approach. It is therefore imperative to conclude that When using the single phase Fourier solution (72) the only way, an excessively higher effective thermal applicable for homogeneous materials to evaluate the conductivity of the nanofluid suspension as obtained effective thermal conductivity of non-homogeneous by Eastman et al. [1], Lee et al. [2] and Choi et al. [3] materials like nanofluid suspensions instead of using could have been obtained even in an apparent form, is Equation (79), one obtains a value that differs from the if the wire was excessively exposed to the solid phase actual one by a factor of temperature. The latter could have occurred if the ⎡ −rw ln ( rw ) + f ( t ) ⎤ electric current passing through the wire created elec- k f,app ⎣ ⎦ = = (80) ⎡ −rw ln ( rw ) + a s g s ( t ) + ( 1 − a s ) g f ( t ) ⎤ tric fields that activated a possible mechanism of elec- k f,act ⎣ ⎦ trophoresis that attracted the suspended nanoparticles towards the wire. Note that such a mechanism does where kf,app is the apparent effective thermal conduc- not cause agglomeration in the usual sense of the tivity obtained from the single phase Fourier conduction word, because as soon as the electric field ceases, the solution while kf,act is the actual effective thermal con- agglomeration does not have to persist and the ductivity that corresponds to data that follow a Dual- Phase-Lagging conduction according to the derivations presented above. The ratio between the two provides a correction factor for the deviation of the apparent effec- tive thermal conductivity from the actual one. This cor-  rection factor when multiplied by the ratio k f,act / k f  )=k , produces the results for  (k f,act / k f f,app / k f  where k f is the thermal conductivity of the base fluid without the suspended particles, and kf,act is the effective thermal conductivity evaluated using Maxwell’s [4] the- ory, which for spherical particles can be expressed in the form: 3 (  − 1 ) k f,act =1+ (81) (  + 2 ) −  ( −1)  kf where kf,act is Maxwell’s effective thermal conductivity,   = k s k f is the ratio between the thermal conductivity of the solid phase and the thermal conductivity of the Figure 1 Dimensionless wire temperature. Comparison between base fluid, and ε is the volumetric solid fraction of the the Fourier and Dual-Phase-Lagging solutions for the following dimensionless parameters values Foq = 1.45 × 10-2 and as = 0.45.  suspension. Then, these results of k f,app k f can be
Vadasz Nanoscale Research Letters 2011, 6:154 Page 10 of 12 http://www.nanoscalereslett.com/content/6/1/154 Figure 2 Dimensionless wire temperature. Comparison between Figure 3 Dimensionless wire temperature. Comparison between the Fourier and Dual-Phase-Lagging solutions for the following the Fourier and Dual-Phase-Lagging solutions for the following dimensionless parameters values Foq = 1.1 × 10-2 and as = 0.55. dimensionless parameters values Foq = 6 × 10-3 and as = 0.35. p articles can move freely from the wire ’ s surface. From the figure, it is evident that the theoretical results Therefore, testing the wire ’ s surface after such an match very well with the digitized experimental data. experiment for evidence of agglomeration on the Furthermore, the steady-state result for the ratio wire ’ s surface may not necessarily produce the between the effective thermal conductivity and that of required evidence for the latter. the base fluid was estimated from the digitized data to  be k f,act k f = 1.003 ± 0.001 clearly validating Maxwell’s Liu et al. [23] used a very similar THW experimental method as the one used by Eastman et al. [1], Lee et al. [4] predicted value. The results applicable to specimen [2] and Choi et al. [3] with the major distinction being No. 5 in Liu et al. [23] and corresponding to values of Foq = 1.1 × 10-2 and as = 0.55 in the theoretical model in the method of producing the nanoparticles and a cylindrical container of different dimensions. They used are presented in Figure 5. The very good match between water as the base fluid and Cu nanoparticles as the sus- the theory and the digitized experimental data is pended elements at volumetric solid fractions of 0.1 and again evident. In addition, the ratio between the effective 0.2%. Their data that are relevant to the present discus- sion were digitized from their Figure 3 [23] and used in the following presentation to compare our theoretical results. Three specimen data are presented in Figure 3 [23] resulting in extensive overlap of the various curves, and therefore in some digitizing error which is difficult to estimate when using only this figure to capture the data. The comparison between the theoretical results pre- sented in this article with the experimental data [23] is presented in Figures 4, 5 and 6. The separation of these results into three different figures aims to better distin- guish between the different curves and avoid overlap- ping as well as presenting the results on their appropriate scales. Figure 4 presents the results that are applicable to specimen No. 4 in Liu et al. [23] and cor- responding to values of Foq = 1.45 × 10-2 and as = 0.45 Figure 4 Comparison of the present theory with experimental in the theoretical model. Evaluating Maxwell’s [4] effec- data of Liu et al. [23] (here redrawn from published data) of the tive thermal conductivity for specimen No. 4 leads to a effective thermal conductivity ratio for conditions compatible with specimen No. 4, leading to a Fourier number of Foq = 1.45 × 10-2 value of 0.6018 W/mK, which is higher by 0.3% than  and a solid particles to total wire area ratio of as = 0.45. that of the base fluid (water), i.e. k f,act k f = 1.003.
Vadasz Nanoscale Research Letters 2011, 6:154 Page 11 of 12 http://www.nanoscalereslett.com/content/6/1/154 Maxwell’s [4] effective thermal conductivity for this spe- cimen leads to a value of 0.6036 W/mK, which is higher by 0.6% than that of the base fluid (water), i.e.  k f,act k f = 1.006. The steady-state result for the ratio between the effective thermal conductivity and that of the base fluid was estimated from the digitized data to  be k f,act k f = 1.0059 ± 0.002 validating again Max- well’s [4] predicted value. It should be mentioned that Liu et al. [23] explain their time-dependent effective thermal conductivity by claiming that it was caused by nanoparticle agglomera- tion, a conclusion that is consistent with the theoretical results of this study. Conclusions Figure 5 Comparison of the present theory with experimental The theoretical results derived in this article combined data of Liu et al. [23] (here redrawn from published data) of the with experimental data [23] lead to the conclusion that, effective thermal conductivity ratio for conditions compatible with while there is no improvement in the effective thermal specimen No. 5, leading to a Fourier number of Foq = 1.1 × 10-2 and conductivity of nanofluids beyond the Maxwell’s effec- a solid particles to total wire area ratio of as = 0.55. tive medium theory [4], there is nevertheless the possibi- lity of substantial heat transfer augmentation via t hermal conductivity and that of the base fluid was nanofins. Nanoparticles attaching to the hot wire by a estimated from the digitized data to be mechanism that could be related to electrophoresis  k f,act k f = 1.004 ± 0.001 again validating Maxwell’s [4] depending on the strength of the electrical current pas-  predicted value of k f,act k f = 1.003. The last result is sing through the wire suggests that such attachments presented in Figure 6, which corresponds to specimen can be deliberately designed and produced on any heat No. 9 in Liu et al. [23] and to values of Foq = 6 × 10-3 transfer surface to yield an agglomeration of nanofins and as = 0.35 in the theoretical model. The results are that exchange heat effectively because of the extremely presented on an appropriately scaled vertical axis and high heat transfer area as well as the flexibility of such show again a very good match between the theory pre- nanofins to bend in the fluid ’ s direction when fluid sented in this article, and the experimental data as digi- motion is present, hence extending its applicability to tized from Liu et al. [23]. Since the volumetric solid include a new, and what appears to be a very effective, fraction for this specimen was 0.2%, its corresponding type of heat convection. A quantitative estimate of the effectiveness of nanofins requires, however, an extension of the model presented in this article to include heat conduction within the nanofins. Abbreviations REV: representative elementary volume; THW: transient-hot-wire. Author details 1 Department of Mechanical Engineering, Northern Arizona University, P. O. Box 15600, Flagstaff, AZ 86011-5600, USA. 2Faculty of Engineering, University of KZ Natal, Durban 4041, South Africa. Authors’ contributions PV conceived and carried out all work reported in this paper. Competing interests The author declares that they have no competing interests. Received: 11 September 2010 Accepted: 18 February 2011 Published: 18 February 2011 Figure 6 Comparison of the present theory with experimental data of Liu et al. [12] (here redrawn from published data) of the effective thermal conductivity ratio for conditions compatible with References 1. Eastman JA, Choi SUS, Li S, Yu W, Thompson LJ: “Anomalously increased specimen No. 9, leading to a Fourier number of Foq = 6 × 10-3 and effective thermal conductivities of ethylene glycol-based nanofluids a solid particles to total wire area ratio of as = 0.35. containing copper nanoparticles”. Appl Phys Lett 2001, 78:718-720.
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Int J Heat Mass Transfer 7 Convenient online submission 2002, 45:1065-1071. 7 Rigorous peer review Xu M, Wang L: “Thermal oscillation and resonance in dual-phase-lagging 27. heat conduction”. Int J Heat Mass Transfer 2002, 45:1055-1061. 7 Immediate publication on acceptance Cheng L, Xu M, Wang L: “From Boltzmann transport equation to single- 28. 7 Open access: articles freely available online phase-lagging heat conduction”. Int J Heat Mass Transfer 2008, 7 High visibility within the ﬁeld 51:6018-6023. 7 Retaining the copyright to your article Hammerschmidt U, Sabuga W: “Transient Hot Wire (THW) method: 29. Uncertainty assessment”. Int J Thermophys 2000, 21:1255-1278. Submit your next manuscript at 7 springeropen.com