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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 03, March 2019, pp. 1326–1338, Article ID: IJMET_10_03_134
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=3
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
A NUMERICAL SIMULATION AND
VALIDATION STUDY OF THE
MATHEMATICAL MODEL OF DROPLET
FORMATION IN DROP ON DEMAND INKJET
PRINTER AND THE EFFECT OF
RHEOLOGICAL PROPERTIES OF
POLYMERINK FOR AUTOMOBILE LIGHTING
APPLICATION
Rajesh.P.K., and Aravindraj.S
Department of Automobile Engineering, PSG College of Technology,
Coimbatore, Tamilnadu. India.
ABSTRACT
A majority of the modern inkjet printers utilise drop on demand devices because of
its precision in terms of time and easy control. The time-dependent fluid interface
disruption renders the fluid dynamics process during droplet ejection complex. The
current work attempts to provide an idea of the drop ejection behaviour based on the
computation of energies required for droplet formation and splat formation. The
simulation results for various nozzle diameters with different polymer inks are examined
and it is validated with computational model. Further attempt is made to analyse the
effect of rheological properties like viscosity and surface tension in the droplet
formation.
Key words: Drop on demand; inkjet; droplet ejection; viscosity; surface tension.
Cite this Article: Rajesh.P.K. and Aravindraj.S, A Numerical Simulation and
Validation Study of the Mathematical Model of Droplet Formation in Drop on
Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application, International Journal of Mechanical Engineering
and Technology 10(3), 2019, pp. 1326–1338.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=3
1. INTRODUCTION
There is a tremendous efforts in the search for new means of further improving the quality and
reducing device cost of ink jet printing due to its rapid growth over manufacturing of
automotive electronics. Knowledge in fluid dynamic process of drop formation and drop
ejection takes precedence in research and development of new ink jet print heads. There are
Rajesh.P.K. and Aravindraj.S
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two main types of ink jet devices, namely, continuous–jet type and drop-on demand type. [3 -
4]. In a continuous-jet device, there is a disintegration into a train of drops of the liquid,
emerging from the nozzle continually in the form of a jet. The amount of electric charge on
each individual drop and direction of motion of each drop from the continuous jet require
sophisticated electrical signals. A drop-on-demand device, on the other hand, uses electrical
signal to control the actuation at the instance of the ejection of an individual drop. Due to its
basic simplicity, the drop-on-demand type is common in the most modern ink jet printers. This
work focuses on the basic drop ejection process in drop-on-demand devices.
A drop-on-demand inkjet printer consists of a fluid chamber with a nozzle which is actuated
to eject the droplet. The actuation pushes a certain amount of the liquid out of the fluid chamber
through the nozzle. A drop is ejected when the liquid pushed out of the nozzle gains enough
forward momentum to overcome the surface tension restoring effect, [5]. The generation and
behaviour of liquid droplets [6] is effected by the Surface tension, inertia and viscosity. Surface
tension is a contractive tendency of the surface of a liquid that allows it to resist an external
force. Liquid atoms or molecules at a free surface have higher energy than those inside the
liquid body. Therefore, the shape of liquid with the lowest surface tension energy is sphere. For
the generation of a droplet, a liquid must necessarily have the tendency to form a shape with
lowest energy. The attraction of water molecules to each other is greater than the molecules in
the air, when the droplet is generated. As a result, an inward force at its surface makes water to
behave as if its surface was covered by a stretched elastic membrane. This is also the primary
cause of pinch‐off effect [7]. The surface tension of most of the liquids used in inkjet printing
have the order of tens of dyn/cm (or mN/m). The importance and influence of the above
parameters can characterized by three essential dimensionless numbers: 1. Reynolds number,
2.Weber number and 3.Ohnesorge number [8].
Figure 1 Stages of droplet formation process [9]
The ink inside the nozzle stays at equilibrium state, before the nozzle gets actuated. Ink
velocity and pressure are zero at initial stage (A). A high pressure is generated inside the nozzle
when the nozzle gets actuated, and the liquid start to flow out from the nozzle orifice (B).
Kinetic energy is transported from the actuator walls to outflow and it undergoes an attenuation
process, in order to overcome the resistance from surface tension (C). The droplet is then
connected with the liquid inside the nozzle by a skinny fluid filament (D). When the liquid
column momentum is large enough, the droplet will escape from the nozzle (E). Surface tension
acts as a force to pinch off the ligament. The meniscus retracts inside the nozzle (F).
In the inkjet printing applications, a single droplet is invariably desired but due to surface
tension additional satellite droplets which are usually smaller than intended primary droplet,
are formed and they cause several problems in printing [10]. When the unexpected satellite
droplets land on places other than where primary droplets do, they result in the degradation of
print quality, which further leads amination or failure. This is most clearly indicated by the
blurring of the trailing edge of a printed area [11].
A Numerical Simulation and Validation Study of the Mathematical Model of Droplet Formation in
Drop on Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application
http://www.iaeme.com/IJMET/index.asp 1328 editor@iaeme.com
The drop formation phenomenon is theoretically, governed by Navier–Stokes equations
with appropriate boundary conditions describing fluid interface motions [12]. The conventional
methods cannot be used to obtain the desired mathematical solutions because of the prevalence
of the nonlinearities arising from inertia, capillarity, and coupling of the free surface kinematics
to the flow field. Hence, a Computational Fluid Dynamics (CFD) package FLOW-3D V 10.0
[13] is utilised to simulate the complex fluid dynamic process during drop ejection and it is
validated with energy model.
Some other researches on droplet formation process in Drop-On-Demand inkjet printing
need to be analysed to overcome the problem. A quantum of research work has been carried
out in the mechanism of droplet formation, especially on Newtonian fluids, subsequent to the
invention of the first DOD inkjet printer in the 1940s. In the recent years, as more Non‐
Newtonian fluids have been widely used in manufacturing of automobile electronics and a bulk
of research on the droplet formation of Non- Newtonian fluids are carried effectively. Due to
the tens of micron length and the time scale of less than a hundred micro‐second, micro scale
droplet generation differs from macro scale droplet generation. Shin et al. [14] and Verkouteren
et al. [15] analysed the transient process of droplet generation using a charge coupled device
(CCD). Dong [16] and Carr [17] studied the dynamics of drop‐on‐demand (DOD) droplet
formation using an imaging system with an inter‐frame time of 1 μs. The experiment was
conducted on a viscosity range of (1.0 ‐ 5.0 cP) and surface tensions (35 – 73 mN/m). They
investigated the stages of droplet formation including the ejection and stretching of liquid and
the pinch‐off of liquid thread from the nozzle exit. Lopez et al. [18] studied the combination
effect of ink rheological behaviour focusing on the dynamics of filament break‐up and effect of
rheological properties on droplet formation. Cittadino et al. [19] developed a non‐linear model
to predict the velocity of the ejected droplet, based on a balance of forces, showing that the
ejection velocity is a strong function of the applied voltage. Feng and James [20] proposed an
comparatively simpler approach based on a series of numerical calculations on Flow3D. This
reference has given an idea about the droplet ejection behaviour for establishing nozzle head
design and shows that the volume of ejected droplet is very close to the volume of fluid pushed
through the nozzle by an actuation pulse.
2. NUMERICAL SIMULATION FOR DROPLET FORMATION BASED
ON ENERGY APPROACH
The energy required to form the droplet are equated in energy approach, to find the diameter of
the droplet. Based on the law of conservation of momentum, the energies before and after
impact are equated to find the splat diameter (i.e.) Diameter of the spread after fall on the
substrate [21].
2.1. Energy Required to Eject Single Droplet from the Nozzle
These energies are required to eject the droplet is as follows:
Energy [E1] required to deflecting the membrane.
Frictional energy [E2] in the orifice.
Kinetic energy [E3] of droplet at the outlet of print head.
Surface tension energy [E4] of the droplet at outlet.
The total energy [E] required to eject the droplet from nozzle should be greater or equal to
the sum of these energies.
Rajesh.P.K. and Aravindraj.S
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E=E1+ E2+ E3+ E4
2.2. Volume of the Droplet Eject from the Nozzle
The volume of the droplet ejected from the nozzle is equal to the maximum volume displaced
by the deflecting membrane. The driving voltage on the piezoelectric device is converted into
required force [F] acting on the centre of the disc.
The maximum deflection in the disc is given by the formula [22]:
x= r2F
8πEIln2r
D+ F(D2−4r2)
64πEI (1)
The maximum deflection is obtained by substituting
r=0, ϵ = 0.3 and I=Et3
12(1−∈2), we get
δ= FD2
18.40 E t3 (2)
By integrating the volume displaced in small layer we get the total volume of the ink
displaced (i.e.) dv =2πrdr.x
Total volume of the ink displaced is given by
V= ∫2πr[Fr2
8πEI
D/2
0ln2r
D+ F (D2−4r2)
64πEI ]dr (3)
On integrating Equation 3, we get
V= F D4
1024EI (4)
On simplifying the equation 4, we get
V= D2
3 δ (5)
2.3. Diameter of the Droplet Formed from the Nozzle
The diameter of droplet is calculated by equating the volume of the sphere and the volume of
the ink displaced (i.e.)
D2
3 δ= 4
24 π d3 (6)
By simplifying the equation 6, we get
d= √2D2δ
π
3 (7)
Equation 7 gives the diameter of the spread which depends on the ink chamber diameter
and the maximum defelection of the membrane.
2.4. Energy Required for Deflecting the Membrane [E1]:
The energy required for deflecting the membrane is equated to the product of force [F] and the
maximum deflection (i.e.)
E1=Fxδ
E1= 18.40Et3
D2δ (8)
2.5. Frictional Energy Required at the Orifice [E2]:
A Numerical Simulation and Validation Study of the Mathematical Model of Droplet Formation in
Drop on Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application
http://www.iaeme.com/IJMET/index.asp 1330 editor@iaeme.com
Friction loss (or skin friction) is the loss of pressure or “head” that occurs in nozzle flow due to
the effect of the fluid's viscosity near the surface of the orifice [23]. The fictional loss is given
by the Hagen-poissoulle equation (i.e.)
hf= 32μu0l
ρgd0
2 (9)
Frictional energy[E2] is given by
E2 = ρ x g x hf (10)
2.6. Kinetic Energy of Droplet at the Outlet of Print Head [E3]:
When the droplet moves with velocity ub, it possess kinetic energy [E3]
E3= 1
2 m ub
2= 1
2 ρ V ub
2 (11)
Substituting the value of V from Equation 5 to Equation 11, we get
E3= [D2ρub
2
6]δ (12)
2.7. Surface Tension Energy Of Droplet At The Outlet [E4]:
The surface tension of the liquid is given by
σ=pd
4 (13)
The surface tension energy is given by
E4=pV= [4σD2
3d ]δ (14)
2.8. Total Energy Required for Eject the Droplet [E]:
E=E1+ E2+ E3+ E4
E= 18.40Et3
D2 δ + ρ∗g∗hf + [D2ρub
2
6]δ + [4σD2
3d ]δ (15)
Equation 15 refers to the total energy required to eject the droplet from the nozzle. The
energy must be greater or equal to the sum of all four energies in order to actuate the nozzle to
eject the droplet.
2.9. Tail and Pinch off Velocity
The pinch-off time coincides with the zero crossing of the ejection velocity (slender-jet
approximation) at the instance of the droplet ejection. In the slender-jet approximation with
neglected radial momentum, the stretching rate tends to become infinity at the nozzle when the
ejection velocity decreases through zero [24]. An instantaneous pinch off is indicated by an
infinite stretching rate. This pinch-off is, therefore, imposed through the boundary condition
and the approximations in the mathematical model. The imbalance in the capillary tension at
the end of the tail causes the formation of the tail droplet when the tail pinches off at the
meniscus. The capillary tension pulls the tail droplet toward the head droplet [25-26].
Simultaneously, the tail droplet mass increases as it sweeps up the ink in the tail, slowing down
the droplet. As a consequential combined effect, the velocity of the tail droplet relative to the
ink in the tail is independent of both viscosity and the size of the tail droplet. To calculate this
velocity, this problem is to be considered in a frame of reference in which the tail droplet is
stationary.