NUMERICAL INVESTIGATION OF THERMOCAPILLARY INDUCED MOTION OF A LIQUID SLUG IN A CAPILLARY TUBE
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1 Proceedings of the Asian Conference on Thermal Sciences 2017, 1st ACTS March 26-30, 2017, Jeju Island, Korea ACTS-P00786 NUMERICAL INVESTIGATION OF THERMOCAPILLARY INDUCED MOTION OF A LIQUID SLUG IN A CAPILLARY TUBE K Srinivasa Sagar 1, Arvind Pattamatta. 1, Sundararajan T 1 1 Department of Mechanical Engineering, IIT Madras, Chennai , India Presenting and Corresponding Author: arvindp@iitm.ac.in ABSTRACT A liquid slug in a microchannel or capillary tube can be propelled by setting up the temperature difference between the ends of the tube. The temperature difference induces a surface tension difference across the both ends causing a pressure difference to propel the drop. The current study uses an open source CFD solver OpenFOAM to investigate the dynamic behavior of liquid slug in a capillary driven by the thermocapillary effect under a transient temperature field. A Parametric study is conducted to ascertain the effect of temperature gradient, tube wall contact angle, liquid viscosity and the capillary tube diameter on the induced velocity of the slug in a capillary. This actuation concept has potential applications in the field of droplet based microfluidics. KEYWORDS: Thermocapillary, capillary, contact angle, surface tension, liquid slug, OpenFOAM 1. INTRODUCTION Liquid slugs are generally encountered in many multiphase systems such as heat pipes, pulsating heat pipes (PHP) etc. Apart from these systems, recent developments of lab on chip devices using liquid slugs as encapsulations which carry materials have been be analyzed for potential microfluidic applications [1]. Manipulations of these slugs are done through various methods [2] such as inclination to gravity field, pressure gradients, electro wetting, electro osmosis, thermocapillarity etc. in which the authors interest is in inducing a motion using thermocapillary forces. The method requires a very simple fabrication process and can be easily adapted on to a microfluidic device. Researchers such as Nguyen and Huang [3,4] have studied experimentally such a mechanism in an externally heated capillary and Sammarco and Burns [5] in a micro fabricated device. Analytical models are also available in the literature to explain the phenomenon under limiting conditions [2,4]. However, no comprehensive numerical studies are available in literature to study thermocapillary induced liquid slug motion to the best of the author s knowledge GOVERNING EQUATIONS 2. NUMERICAL MODEL The governing equations consider incompressible two-phase flows of immiscible Newtonian fluids with heat transport. Volume of fluid method with an iso-surface based reconstruction scheme in OpenFOAM is used to simulate the present scenario. The reconstruction scheme provides an advantage of smooth curvature and normal calculation and the formulation of governing equations are as follows. Conservation of mass. u = 0 (1) Conservation of momentum Conservation of Energy (ρu ) +. (u. ρu ) = p + μ( u + ( u ) T ) + f σ (2) (ρct) +. (ρcu T) =. (k T) (3) 1
2 Conservation of volume fraction α +. (u α) +. (u rα(1 α)) = 0 (4) The surface tension force is modelled using continuum surface force modelling as Where surface tension is modelled as a linear function of temperature. f σ = (σκn + (I n n) σ)δ s (5) σ = σ 0 + σ (T T ref) (6) Where ρ is the density, u is the velocity vector, p is the pressure, T is the temperature, k is the thermal conductivity, C is the specific heat capacity, α is the volume fraction of the phase, σ 0 is the interfacial tension at T ref, σ is the temperature coefficient of interfacial tension, κ is curvature of the interface, u r is the compression velocity, n is the unit normal to the interface,δ s is the Dirac distribution function at the interface. All the properties are calculated as volume fraction weighted average. More details about the numerical scheme are explained by Pattamatta et al.[6] 2.2. COMPUTATIONAL DOMAIN AND BOUNDARY CONDITIONS Two computational studies are carried out. The first considers simulating the transient motion of a liquid slug in an externally heated capillary while the latter study is conducted to ascertain the effect of various parameters on the motion of liquid slug. The fluid properties considered for the analysis are listed in Table 1. Table 1 : Properties of the two phases Liquid slug Bulk fluid Density (Kgm 3 ) Viscosity(Pa s) Thermal conductivity(wm 1 K 1 ) Specific heat (JKg 1 K 1 ) Surface tension(n m 1 ) Surface tension coefficient(n m 1 K 1 ) Computational domain-1: The modelling is based on the experiments done by Nguyen and Huang[4]. A capillary tube of radius 1.26mm and of length 14cm is considered. A small part of the tube is heated and the remaining part of the tube is exposed to ambient. Various grids were studied and the grids with mesh size of 10µm is selected for the analysis throughout the study. The thermocapillary motion of a liquid slug of length 2mm in a transient temperature field is studied. The heater is supplied with a constant heat flux of W m 2 while the remaining portion of the wall is having a convective heat transfer coefficient of 6.53W/m 2 K in ambient with temperature T =290K. Both ends of the capillary tube are exposed to ambient. The domain is modelled as axisymmetric and the computational 2
3 domain is as shown in the Fig 1. The inlet outlet velocity with zero gauge pressure boundary condition is specified at the inlet and outlet. A Constant contact angle of 25º is specified at the tube wall in the present simulations. Fig. 1 : Computational domain 1 schematic Computational domain-2: Though the computational domain-1captures evolution of thermocapillary motion in a transient temperature field, it is very difficult to ascertain the effects of individual parameters on the thermocapillary phenomenon. To simplify, the following computational domain is modelled in which the transient effects are minimized so that effect of other parameters can be focused. An isothermal inlet with atmosphere boundary conditions are given at both ends and the wall is insulated giving a fixed temperature gradient. The velocity profile evolves to a steady value and this value is considered for the analysis. The mesh size and contact angle boundary conditions are same as the computational model TRANSIENT CASE: Fig. 2 : Computational domain - 2 schematic 3. RESULTS AND DISCUSSION With the computational domain -1 specified and the above boundary conditions applied, the simulation was setup and the results obtained are shown in Fig.3 and 4. Fig. 3 : Transient velocity profile 3
4 The transient induced velocity of the liquid slug is found to be qualitatively agreeing with that of Nguyen and Huang[4] but exact comparison cannot be obtained because the authors did not specify all the material properties used in experiments. The increase and decrease in velocity with respect to slug position can be explained due to the temperature difference at the ends. At the initial stages, the slug accelerates due to newly setup temperature gradient by the heater but as the slug moves away from the heater, the temperature difference reduces over a period of time causing the slug to decelerate. Fig.4. shows the positions of the liquid slug in red color at the bottom and the temperature contours on the top. As time increases, the slug moves away from the heater simultaneously reducing the temperature difference at the both ends PARAMETRIC STUDY: Fig. 4 : Temperature and volume fraction contours at various time steps With the above study, we are able to get the transient evolution of the thermocapillary induced motion of the liquid slug. To analyze the effect of various parameters the system is modelled as in computational domain-2. For conducting the parametric study, the walls are insulated and a constant temperature difference is maintained between the ends of the tube. This constant temperature difference gives rise to a steady state velocity as shown in the Fig 5. Fig. 5 : Evolution of velocity profile in computational domain 2 The pressure difference across the liquid slug due to thermally induced surface tension differences can be described by the following relation given by Sammarco and Burns[5]. Here σ is surface tension and θ is the contact angle at the surface. The surface tension being inversely proportional to the local temperature; decreases at the hot end creating a pressure difference at both ends of the drop propelling it. P [( σcosθ ) d cold (σcosθ) ] (7) d hot 4
5 Effect of capillary diameter: To study the effect of capillary tube diameter, the size of the tube is varied keeping the slug size constant at a length of 2 mm and a temperature difference of 40K is induced at the both ends with a contact angle of 50º at the walls. Three tubes of different radii equal to 0.55mm, 0.65mm and 0.75mm are considered for the analysis. Large diameters are not considered for the study because the capillary effects may not be significant. From Eqn. 7 the pressure may appear to decrease with increasing capillary diameter. But the net thermocapillary force increases because it is proportional to the cross sectional area so the force scales pressure byd 2. This trend is observed in Fig. 6(a). Also, it is observed that with increasing radius the velocity shows a marginal increase. Therefore it can be concluded that the effect of capillary radius is not significant as far the thermocapillary motion is concerned. Effect of contact angle: The contact angle is the most sensitive parameter in this study. The slug length is taken as 2mm and the temperature difference is set to 40K with a capillary radius of 0.65mm. The contact angle at the wall is varied from 30º to 95º. It can be observed from the Fig.6 (b), that the thermocapillary velocity decreases with increasing contact angle in a hydrophilic channel (0º-90º) and becomes negative in a hydrophobic channel (90º-180º). It can also be observed that the velocity effect of contact angle variation at lower angles (30º-50º) did not seem to have as strong effect as at higher angles (50º-85º). This effect and the reversal of droplet motion in a hydrophobic channel can be attributed to the decreasing nature of cosine function in Eqn. 7. Fig. 6: (a) Effect of radius of capillary tube (b) Effect of contact angle (c) Effect of viscosity (d) Effect of temperature difference on thermocapillary velocity Effect of viscosity: Viscosity is a very important parameter in short scales of order of capillary diameter. Keeping the other parameters constant three different viscosities of Pa s, Pa s, and 0.012Pa s are considered. The contact angle is kept at 50º. From Fig.6(c), it is observed that at low viscosities the velocity is maximum and as 5
6 the viscosity increases the velocity shows a decreasing trend. At lower viscosities the slug responds quickly to the temperature gradient and moves faster but at higher viscosities for the same thermocapillary force the viscosity dampens the motion causing a considerable decrease in velocity. Therefore, it can be concluded that the viscosity effect is dominant on thermocapillary velocity. Effect of Temperature difference: The temperature gradient is the driving force in the thermocapillary motion. Four temperature gradients 20K, 40 K, 60 K and 80 K are considered for the analysis and keeping all other parameters as in previous studies. From Fig. 6(d) it can be observed that the velocity is a strong function of temperature gradient (i.e.) higher the temperature gradient higher the velocity. The behavior can be from Eqn. (7), because larger temperature gradient causes a larger surface tension difference. The increase creates a larger thermocapillary force propelling the slug faster. However the temperature gradient is limited by freezing and boiling points of liquid in the liquid slug. 4. CONCLUSIONS In this work, we had numerically investigated the thermocapillary motion of liquid slug in a capillary tube. The case is simulated using volume of fluid method in OpenFOAM considering all the thermocapillary effects. The evolution of thermocapillary velocity of liquid slug in a transient temperature field is studied and an extensive parametric study is done to ascertain the effects of capillary diameter, contact angle, viscosity and temperature difference on thermocapillary motion. It is observed that the capillary diameter and temperature difference are having progressive effect and contact angle and viscosity have adverse effect on the thermocapillary motion. In the future the work will be extended to cases involving contact angle hysteresis, effect of capillary wall conduction and non-circular geometries. REFERENCES [1] M.A. Burns, C.H. Mastrangelo, T.S. Sammarco, F.P. Man, J.R. Webster, B.N. Johnsons, B. Foerster, D. Jones, Y. Fields, A.R. Kaiser, D.T. Burke, Microfabricated structures for integrated DNA analysis., Proc. Natl. Acad. Sci. U. S. A. 93 (1996) [2] H.-Y. Kim, On Thermocapillary Propulsion of Microliquid Slug, Nanoscale Microscale Thermophys. Eng. 11 (2007) doi: / [3] Z. Jiao, N.T. Nguyen, X. Huang, Y.Z. Ang, Reciprocating thermocapillary plug motion in an externally heated capillary, Microfluid. Nanofluidics. 3 (2007) doi: /s [4] N.T. Nguyen, X. Huang, Thermocapillary effect of a liquid plug in transient temperature fields, Japanese J. Appl. Physics, Part 1 Regul. Pap. Short Notes Rev. Pap. 44 (2005) doi: /jjap [5] T.S. Sammarco, M. a Burns, Heat-transfer analysis of microfabricated thermocapillary pumping and reaction devices, J. Micromechanics Microengineering. 10 (2000) doi: / /10/1/307. [6] A. Pattamatta, A. Sielaff, P. Stephan, A numerical study on the hydrodynamic and heat transfer characteristics of oscillating Taylor bubble in a capillary tube, Appl. Therm. Eng. 89 (2015) doi: /j.applthermaleng
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