The Use Of CFD To Simulate Capillary Rise And Comparison To Experimental Data Hong Xu, Chokri Guetari ANSYS INC. Abstract In a micro-gravity environment liquid can be pumped and positioned by cohesion and adhesion forces. On earth this effect is covered by the hydrostatic pressure, so it can only be observed in very fine tubes (socalled capillaries). The final liquid height depends on the balance of the capillary force and the weight of the liquid that has been lifted up. This paper reports on the CFD simulations performed to simulate the capillary rise of a liquid in a 0. mm diameter tube. This model is validated against published experimental data for water arising height vs. time. Good agreement between the CFX results and experimental data has been observed. Introduction Capillary rise is what we see when we put a small glass tube down into a pool of water and the water rises in the tube. The phenomenon is caused by a combination of adsorption and capillarity. First the water is absorbed onto to the surface of the glass. This forces the air-water surface to become curved. Because the tube is round, the curvature inside the tube is nearly spherical. Since radius curvature is in the air, the pressure on the water-side of the curved interface is less than the air pressure. So to maintain the same total head throughout the water, the water must rise in the tube to reach hydrostatic conditions. This is part of what causes water to rise in the tube above the water table. Many natural processes and human activities largely rely on the phenomenon of capillary, i.e. the ability of liquids to penetrate into fine pores and cracks with wettable walls and be displaced from those with non-wettable walls. It is the capillarity that brings water to the upper layer of soils, drives sap in plants, and is the basis for operation of pens. Knowledge of capillary laws is important in fuel cells research, oil recovery, civil engineering, dyeing of textile fabrics, ink printing and a variety of other fields. The issue of liquid transport in capillary systems dates back a long time. Basic understanding of the laws of capillarity was gained almost a century ago. The early work by Hagen and Poiseuille, Lucas and Washburn laid much of the theoretical foundation for describing capillary flow phenomena. Unfortunately, as often happens with classical equations, the underlying assumptions this equation rests on are not always kept in mind in its applications. Also, the spreading of the fluid into a structured micro-channel cannot be described sufficiently in an analytical way due to the complexity of the structure and the resultant shape of the fluid/fluid interface between the two fluids. So it is necessary to use simulation methods, which calculates the fluid/fluid interface with finite volume methods (FVM) or finite difference methods (FDM). Fluid rises in capillary tubes to a height, which is inversely proportional to the inner diameter of the tube. In the present investigation, we will use CFX software to simulate the fluid height changing with time in a long, small pipe. Also, since the flow behavior in this structure could be described analytically, we will compare the simulation results with the analytical solutions and experimental data. Procedure Fundamental equation of Dynamics Figure 1 shows a schematic of a typical capillary example with relevant dimensions. For this case, the Newton dynamics equation can be written for clean, viscous, incompressible fluids in a long cylindrical capillary as follows (1) :
'' ' 8 ' ρ[ zz + ( z ) ] = γ cosθ ηzz ρgz [1] r r where ρ is the density, η is the viscosity, γ is the surface tension coefficient, θ is the wetting angle of the liquid, z is the height of capillary rise, r is the capillary radius, and g is the acceleration of gravity. It is important to point out that equation [1] assumes a Poiseuille flow profile in the capillary. Usually this is not the case especially near the liquid-gas interface, where the profile is necessarily different from parabolic to achieve the convex or concave shape, which is dictated by the wall wetting properties. Figure 1(a) Figure 1. Figure 1(b) (a): Schematic of a typical capillary case with relevant dimensions (b): Mesh used for numerical solution As the liquid rises in the capillary, the velocity increases. For certain conditions, the flow will become turbulent. In this regime, the drag is a non-linear function of velocity. It was observed that the predictions of equation [1] were inaccurate. Therefore, an additional term can be added to improve the predictive capability of equation [1]. The modified form is shown in equation []. where '' ' 8 ' Φ ρ [ zz + ( z ) ] = γ cosθ ηzz ρgz [] r r r
0 Φ = q z ( z ' ) ( z ' < v ) c r ( z ' > v ) c r v c r is the critical velocity at which turbulence becomes important, and q is a coefficient that can be determined by matching experimental data. Lucas-Washburn Equation The Lucas-Washburn equation assumes quasi-steady state. This is valid when the capillary force is balanced by viscous and gravity forces. In this case, equation [1] can be reduced to (1) : dz dt γ r cosθ r ρ g 4η z 8η = [3] The long-time predictions of the above equation are in good agreement with experimental data. This is due to the fact that at long times the height of the capillary has started to stabilize and the quasi-steady state assumption becomes more appropriate. Numerical Methods Computational fluid dynamics (CFD) relies on solving the full Navier-Stokes equations, [4] & [5], using numerical techniques (3). The governing equations for this model were solved using a commercial CFD code CFX from ANSYS. ( r t ( r ρ ) + ( r ρu ) = S MS [4] t ρu ) + ( r ( ρu U ) = r P + ( r µ ( U + ( U ) T )) + S M + M where is denoted for different phase, r is the volume fraction for phase, S MS is for user specified mass sources. S M is for momentum sources due to external body forces and user defined momentum sources, M is for the interfacial forces acting on phase due to the presence of other phases. Analysis CFX supports a wide range of multiphase applications through the Eulerian-Lagrangian and the Eulerian- Eulerian models. These models have been implemented and tested rigorously against available data. The Eulerian-Eulerian approach requires the solution of individual transport equations for each phase. The interaction between phases is accounted for by including inter-phase transport terms for momentum, energy and mass. In this case, the homogeneous Eulerian-Eulerian model is used to simulate the capillary flow. Both the liquid and gas phases are treated as continuous. The homogeneous multiphase model implies that a common velocity field is shared by all fluids. However, the interface between fluids is tracked by solving separate volume fraction equations. For a given transport process, the homogeneous model assumes that the transported quantities (with the exception of volume fraction) for that process are the same for all phases. i.e. [5]
Φ = Φ 1 N p By summing the individual transport equations over all phases, a single transport equation can be obtained for Φ: ( ρφ ) + ( ρuφ Γ φ) = S [6] t where: Np Np Np 1 ρ = r ρ, U = r ρ U, Γ = r Γ [7] ρ = 1 = 1 = 1 In particular, the momentum transport equations can be reduced to equation [5], when in addition to [7] the viscosity is formulated as: Np µ = r [8] µ = 1 The surface tension forces, which are dominant in capillary flows, can be included by using the Continuum Surface Force model of Brackbill et al (). The surface tension force is included as an extra body force in the momentum equation. F s = σκ r [9] Where σ is the surface tension coefficient, r is the volume fraction, and κ is the surface curvature. Analysis Results & Discussion In this study, the experimental setup of B.V.Zhmud et al (1) was modeled numerically. Figure shows the dimensions and the initial conditions of the liquid and the gas. Both fluids were assumed to be initially stagnant. Based on the symmetry in the geometry, an axi-symmetric two-dimensional wedge was used. The resulting mesh was on the order of 58,000 cells.
Figure. Dimensions as well as the initial conditions used for the CFD analysis The fluids were assumed to be incompressible and isothermal and to have constant fluid properties. A laminar, transient simulations for a total of 1.5 seconds was performed. Small time steps were required to achieve good agreement with experimental data. Figure 3 shows liquid height vs. time predicted by CFX and how the results compare to experimental data as well as the Lucas-Washburn equation [3].
0.01 0.01 Liquid Height (m) 0.008 0.006 0.004 CFX Results 0.00 Experimental Lucas-Washburn 0 0 0.05 0.1 0.15 Time (s) Figure 3. Comparison of CFX results to experimental data and the Lucas-Washburn equation Details of the velocity field as well as the shape of the free surface between the liquid and the gas are shown in Figures 4 & 5. The vector plot highlights that the flow behavior is complicated at the interface due to surface tension forces. Figure 4. Vector plot showing the velocity profile at the liquid-gas interface.
Figure 5. Result at t=0.15 s. Conclusion CFX was used to simulate the flow in a capillary tube. Comparison of the results obtained using CFX to experimental and analytical solutions showed good agreement. The CFD results are in better agreement with experimental data than the Lucas-Washburn equation. References 1. Zhmud, B. V, Tiberg, F., Hallstensson, K., Dynamics of Capillary Rise, Journal of Colloid and Interface Science, 8, pp 63-69 (000).
. Brackbill, Kothe, Zemach, A Continuum Method for Modeling Surface Tension, J. Comp. Phys. 100, p 335 (119). 3. CFX Solver Manual.