Proceedings of the 24th National and 2nd International ISHMT-ASTFE Heat and Mass Transfer Conference (IHMTC-2017), December 27-30, 2017, BITS-Pilani, Hyderabad, India IHMTC2017-13-0160 EULER-EULER TWO-FLUID MODEL BASED CODE DEVELOPMENT FOR TWO-PHASE FLOW SYSTEMS Sanjeev Kumar Dept. of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India. sanjeevk@iitk.ac.in Arun K Saha Dept. of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India. aksaha@iitk.ac.in Prabhat Munshi Dept. of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India. pmunshi@iitk.ac.in ABSTRACT The flow regimes of gas-liquid two-phase flow in a vertical pipe depend upon the drag as well as non-drag (Lift, Virtual mass, turbulent dispersion, wall lubrication, etc.) forces. The effect of these forces in a code is incorporated by the closure models of the interphase forces for the momentum exchange between the continuous and dispersed phases. In the present work, different drag models with and without the effect of bubbles swarm and shear rate of continuous phase are investigated in the in-house code. The code is based on the Euler-Euler twofluid model. The code is validated against the experimental data of Monros-Andreu et al. (2013, EPJ Web Conf. 45, 01105). The radial distribution of the volume fraction, gas velocity and slip velocity has been presented for different gas and liquid flow rates. The predictions from the sets of inter-phase closure models presented in this paper yielded satisfactory results. It has been found that a set of Ishii-Zuber drag coefficient model with bubble swarm effect by Roghair and shear effect by Magnaudet model, Tomiyama lift coefficient model, Hosokawa and Tomiyama wall force model, and Lopez de Bertodano turbulent dispersion force model was found to provide the best agreement with the experimental data. NOMENCLATURE 2D Two-dimensional α Void fraction ε Turbulent dissipation rate σ Surface tension c/d Continuous/Dispersed phase d b Bubble diameter C D Overall Drag coefficient C D Drag coefficient of single bubble D Pipe diameter E o Eotvos number g Gravitational acceleration H Pipe height Corresponding Author k Turbulent kinetic energy p Pressure r/z Radial/Axial co-ordinates Re Pipe Reynolds number Re b Bubble Reynolds number t Time U Velocity vector Relative velocity U r INTRODUCTION A large number of flows encountered in nature and technology are a mixture of phases. Multi-phase processes are commonly used in many industrial fields, e.g. nuclear reactors and chemical reactors [1, 2]. The distribution of phases and how they interact with each other is still a field of research. Bubble columns are commonly used among the equipments for such type of processes. They are inexpensive reactors and easy to operate. An axis-symmetric code is developed to analyze the twophase pipe flow systems with bubbly flow regime. This code is based on Euler-Euler two-fluid model. Governing equations in 2D cylindrical co-ordinates system (r, z) are considered. Navier- Stokes equations have been solved using modified Marker and Cell (MAC) method [3]. Explicit schemes are used to solve the transport equations. Standard wall function k ε turbulence model is used to incorporate the turbulent phenomena. Model is validated against the experimental work performed by Andreu et al., 2013 [4]. They have studied the water temperature effect on upward air-water flow in a vertical pipe. They have presented the local void fraction distribution, interfacial velocity, turbulent kinetic energy and turbulent dissipation energy profiles for different flow conditions of bubbly flow regime and at different axial locations of the cylindrical pipe. In the present work, an in-house code is used to simulate the experimental work done by Andreu et al., 2013 [4]. Different correlations of drag and nondrag (lift, virtual mass, turbulent dispersion and wall lubrication) forces are tested and there effects are discussed. It is found that present model predicts the results with reasonable accuracy. In 1
future this code will be coupled with the nuclear kinetics code to analyze the core of the Pressurized Water Reactor (PWR) and Boiling Water Reactor (BWR) in detail. MATHEMATICAL MODEL Model equations rely on the Euler-Euler two-fluid methodology are described in this section. The local instantaneous equations of each phase are ensemble averaged to obtain an Euler- Euler two-phase flow description. The averaging process introduces the phase fraction α and unclosed terms M representing the interphase forces term. These unclosed terms must be modeled as these terms are crucial to the prediction of the two-phase flow. Both (continuous and dispersed) phases are assumed to be incompressible. For a monodisperse flow, i.e. for a flow with equally sized bubbles of diameter d B, and without mass transfer between the phases, these equations can be written as: U ϕ α ϕ ( τϕ + U ϕ U ϕ + R ϕ ) + α ϕ α ϕ ( τϕ ρ ϕ + R ϕ + (α ϕ U ϕ ) = 0, (1) + ρ ϕ ) = p ρ ϕ + g + M ϕ α ϕ ρ ϕ where U, τ, R, p, g, α and ρ are the velocity, laminar stress tensor, Reynolds stress tensor, pressure, gravity, volume fraction and density respectively. The subscript ϕ = c stands for the carrier phase (liquid) and ϕ = d for the dispersed phase (gas). The first term on the left side of Equation (2) is the transient term, second term represents the convective term, and the third and fourth terms represent the diffusion due to the laminar and turbulent stresses. The first, second and third terms on the right side of Equation (2) represents the pressure force, gravitational force and the inter phase momentum transfer terms respectively. Interfacial models The interphase force term or interfacial momentum transfer term is given by the sum of various (drag and non-drag) forces as given below M d = M c = M d,d + M d,v M + M d,l + M d,wl + M d,t D, (3) where M d,d, M d,v M, M d,l, M d,wl and M d,t D are the momentum exchange terms due to the drag force, the virtual mass force, the lift force, the wall lubrication force and the turbulent dispersion force respectively. The Basset force comes into the picture due to the formation of the boundary layer around the bubble. The Basset force is relevant only for the unsteady flows. In the present work, we have considered the cases with steady state flows and hence the magnitude of the Basset force is negligible and hence it is not considered in the present work. All these forces are described below in detail. (2) Drag force: The drag force exerted by the fluid on a bubble or droplet determines the relative velocity between the phases. The drag force is the function of slip velocity (U r = U d U c ). It can be calculated for spherical bubbles of uniform size by the following correlation [5]: M d,d = 3 4 α C D ρ c d U r U r, (4) d B where C D and d B are the drag coefficient and bubble diameter respectively. Hosokawa and Tomiyama (2009) [6] studied the effect of bubble swarm on the drag coefficient. They suggested a correction factor (C f,swarm ) with respect to the drag coefficient of single bubble (C D, ). Magnaudet and Legendre (1998) [7] suggested a correction factor (C f,shear ) for shear rate effect. The overall drag coefficient with the inclusion of both the above effects is given as: C D = C D, C f,swarm C f,shear, (5) Three different correlations of several distinct Reynolds number regions for individual bubbles proposed by Ishii and Zuber (1979), Tomiyama et al. (1998) [8] and Zhang and Banderheyde (2002) are tested in the present work. A drag correlation was derived by Ishii and Zuber (1979) for a spherical bubble as a function of the Eotvos number, C D, = 2 3 E0.5 o. (6) where E O = g(ρ c ρ d )db 2 σ is the Eotvos number. Tomiyama et al. (1998) [8] proposed another drag coefficient model. { [ 16 C D, = max mini (1 + 0.15Reb 0.687 ), 48 ] Re b Re b E O },, 8 3 E O + 4 (7) Zhang and Banderheyde (2002) proposed another expression for C D, as: C D, = 0.44 + 24 6 + Re b 1 + Re 0.5 b Roghair et al. (2013) [9] proposed a correlation to take into account the effect of bubble swarm on drag coefficient. They uses DNS simulation to see the effect of the Eotvos number. This correlation is valid for high volume fraction cases. The expression is given below: (8) [ ( ] 22 C f,swarm = (1 α d ) 1 + )α d. (9) E O + 0.4 Magnaudet and Legendre (1998) [7] studied the effect of shear rate on the drag coefficient and proposed the following correction coefficient: C f,shear = (1 + 0.55Sr 2 ), (10) 2
where Sr = d bω U is the shear rate. Sr attains a very high value, r usually in the first cell near the wall. It is limited to 2 in the absence of evidence about the effect of the drag force above this value (Hosokawa and Tomiyama, 2009) [6]. Lift force: The bubble in a continuous phase with velocity gradient feels a lateral force and moves laterally with relative velocity related to the velocity gradient. This lateral force acting on a bubble due to the velocity gradient is called the lift force. The effect of the lateral force due to the lift was first modeled by Auton et al. (1988) [10]: M d,l = α d ρ c C L U r U c, (11) where C L is the lift coefficient and it should be modeled. Tomiyama et al., 2002 [11] proposed a form of the lift coefficient that take into account the interaction between the distorted bubble and the shear field of the liquid phase and is given as: min[0.288 tanh(0.121re b ), f (E Ohd )], for E Ohd < 4 C L = f (E Ohd ), for 4 E Ohd 10 0.27, for E Ohd > 10 (12) where E Ohd is the modified Eotvos number, given in terms of the maximum horizontal dimension of the bubble d hb. It is given as: E Ohd = g(ρ c ρ d )dhb 2 3, with d hb = d B 1 + 0.163EO 0.757, (13) σ and f (E Ohd ) = 0.00105E 3 O hd 0.0159E 2 O hd 0.0204E Ohd + 0.474 (14) Virtual Mass force: The virtual mass (VM) force comes into picture when secondary (dispersed) phase accelerates relative to the primary (continuous) phase. The inertia of the primary phase mass encountered by the accelerating bubbles exerts a VM force on the particles. It is given as: M d,v M = α d ρ c C V M ( DUc Dt DU ) d, (15) Dt where C V M is the VM force coefficient and Dt D represents the phase material time derivative. The amplitude of the virtual mass force is very small at steady state in comparison with the amplitude of the other drag and non-drag forces and hence this force is not considered in the present work. Wall lubrication force: The liquid speed between bubble and the wall is lower than between the bubble and the main flow. This results in a hydrodynamic pressure difference. The force corresponding to this pressure difference is called the wall lubrication force. It drive bubble away from the wall. The expression for wall lubrication force was first modeled by Antal et al., (1991), for Re b < 1500 and α d < 0.1. Tomiyama et al., (1995b) [12] modified Antal et al. s (1991) wall force model: [ ] d b 1 M d,wl = α d ρ c C W 2 y 2 1 (D y) 2 U r n z 2 n r, (16) where C W is the wall lubrication force coefficient, n z is the unit vector parallel to the wall and n r is the unit vector normal to the wall. Hosokawa and Tomiyama (2009) [6] proposed the following model: [ max(6log10 M O + 24,4.4) C W = max Re 1.9 b where M O is the Morton number.,0.0217e O ], (17) Turbulent dispersion force: The turbulence of the phases helps to redistribute the dispersed phase from the regions of high concentration of void fraction to the regions of low concentration. This effect is called the turbulent dispersion effect and corresponding force is the turbulent dispersion force. There is a variety of models proposed by the researchers for the turbulent dispersion force. The model proposed by Antal et al., (1991b) [13] has been considered in this work. This model considers the effect of the turbulent fluctuations in the continuous phase on the dispersed phase and is given as: M d,t D = ρ c C T D k α d, (18) where k is the turbulent kinetic energy of the continuous phase and C T D is the turbulent dispersion force coefficient. Lopez de Bertodano (1998) [14] proposed a correlation for C T D : C T D = C1/4 µ St(1 + St), (19) St = τ d τ c, (20) where the coefficient C µ is 0.09, St is the Stokes number, τ d is the relaxation time of bubble and τ c is the relaxation time of flow. These relaxation times are given as [6]: τ d = 4d b 3C D U r ; τ c = Cµ 3/4 k ε where ε is the turbulent dissipation rate. (21) Turbulence model A standard wall function k ε turbulence model (Launder and Spalding, 1974 [15]) is used to model the effect of turbulent fluctuations in the continuous phase. The equations for the 3
turbulent kinetic energy kc and turbulent dissipation rate (εc ) for continuous phase are given as: kc + (Uc kc ) = νc + νt,c σk A proper drag coefficient determines the slip velocity of the two-phase flow system. The axial relative velocity profiles for Ishii-Zuber, Tomiyama and Zhang drag correlations are shown in Figure 1. It is clear that the slip velocity prediction by different models vary widely. kc + Pk εc + Sk, (22) νt,c εc + (Uc εc ) = νc + σε (23) εc εc + (C1ε Pk C2ε εc ) + Sε, kc The near wall region is modeled using the wall functions approach of Launder and Spalding, 1974 [15]. The bubble induced turbulence is modeled by the addition of a bubble-induced source term to the transport equations of the turbulence model. Lee et el., (1989) [16] proposed the following correlations for these source terms: Sk = αd C1k p Ur, z εc Sε = C3ε Sk, kc (24) Figure 1. SLIP VELOCITY COMPARISON WITH DIFFERENT DRAG CORRELATIONS. (25) where C1k = 0.03 + (0.243 0.344 10 5 Re), 1 + e(re 60000)/2000 The wall swarm effect and shear rate effect corrections are included in the drag correlation of Ishii-Zuber. The axial slip velocity with and without all these corrections are shown in Figure 2. It is clear that Ishii-Zuber drag model with the correction of bubble swarm and shear rate effect predicts the slip velocity which is close to the experimental data. (26) where Re is the continuous phase Reynolds number. The model constants Cµ, C1ε, C2ε, C3ε, σk and σε are 0.09, 1.44, 1.92, 1.92, 1.0 and 1.3 respectively. PROBLEM DESCRIPTION The experimental work done by Andreu et al., 2013 [4] is considered to validate the developed code. The test section was a 52mm diameter pipe of 5.5m length. The test section was maintained at adiabatic conditions with air and water at superficial velocity ranges of 0.05m/s-0.30m/s and 0.5m/s-2.0m/s respectively. The experiments have been performed at three different temperatures (T 15 C, 24 C and 36 C). The radial profiles are provided at three different heights (H/D = 22.4, 61 and 98.7). The radial porfiles have been compared at (H/D = 22.4) in the present work. SENSITIVITY ANALYSIS The strong dependence between the different closure models makes it difficult to consider each model separately. The approach adopted to first neglected all the non-drag forces and bubble induced turbulence force to predict a realistic slip velocity. Once the drag model is finalized, non-drag forces will also be considered. Figure 2. SLIP VELOCITY COMPARISON WITH BUBBLE SWARM AND SHEAR RATE EFFECT. 4
velocities (0.05m/s and 0.3m/s) are considered and the superficial liquid velocity is 1.0m/s. It is clear from Figure 3 that the predicted radial void fraction profiles by the present model is closely matching with the experimental profiles. The predicted data lies within the maximum error involved (±15%) in the experiment [4]. The near wall peak is resolved accurately, except the peak position is a little closer to the wall than the experimental profile peak. The radial gas velocity profiles at three different temperatures are shown in Figure 4. It is clear from Figure 4 that the predicted radial gas velocity profiles by the present model is closely matching with the experimental profiles, for the low superficial gas velocity (J G = 0.05m/s) case. It is over predicted in the case of high superficial gas velocity (J G = 0.3m/s). But, the predicted data lies within the maximum error involved (±15%) in the experiment [4]. CONCLUSIONS An Euler-Euler multiphase model based code is developed and validated against the experimental data available for airwater bubbly flow in adiabatic vertical pipe. The effect of different closure models is shown and a suitable set of closure models have been identified. The developed code successfully predicts the radial profiles of gas velocity and void fraction for different flow conditions. Figure 3. THE VOID FRACTION PROFILES COMPARISON WITH EXPERIMENTAL DATA AT H/D=22.4, J L =1.0m/s, J G =0.05, 0.3m/s AT TEMPERATURES (A) 15 C, (B) 24 C and (C) 36 C. RESULTS AND DISCUSSION The radial void fraction profiles at three different temperatures are shown in Figure 3. The two different superficial gas 5 REFERENCES [1] Fabris, G., and Hantman, R. G., 1976. Fluid dynamic aspects of liquid-metal gas two-phase magnetohydrodynamic power generators. Proceedings of the 1976 Heat Transfer and Fluid Mechanics Institute, pp. 92 113. [2] Gutierrez-Miravete, E., and Xiaole, X., 2011. A study of fluid flow and heat transfer in a liquid metal in a backwardfacing setup under combined electric and magnetic fields. Proceedings of the 2011 COMSOL Conference in Boston. [3] Harlow, F. H., and Welch, J. E., 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids, 12(8), pp. 2182 2189. [4] Monros-Andreu, G., C. S. M.-C. R. T. S. J. J. E. H. L., and Mondragon, R., 2013. Water temperature effect on upward air-water flow in a vertical pipe: Local measurements database using four-sensor conductivity probes and lda. EPJ Web of Conferences, 45(01105). [5] Ishii, M., and Mishima, K., 2013. Two-fluid model and hydrodynamic constitutive relations. Nuclear Engineering and Design, 82, pp. 107 126. [6] Hosokawa, S., and Tomiyama, A., 2009. Multi-fluid simulation of turbulent bubbly pipe flows. Chemical Engineering Science, 84, pp. 5308 5318. [7] Magnaudet, J., and Legendre, D., 1998. Some aspects of the lift force on a spherical bubble. Applied Scientific Research, 58, pp. 441 461. [8] Tomiyama, A., K. I. Z. I., and Sakaguchi, T., 1998. Drag coefficients of single bubbles under normal and micro gravity conditions. JSME International Journal, 41(2), pp. 472 479. [9] Roghair, I., A. M. V. S., and Kuipers, H. J. A. M., 2013.
Drag force and clustering in bubble swarms. American Institute of Chemical Engineers, 59(5), pp. 1791 1800. [10] Auton, T. R., H. J. C. R., and Prud Homme, M., 1988. The force exerted on a body in inviscid unsteady nonuniform rotational flow. Journal of Fluid Mechanics, 197, pp. 241 257. [11] Tomiyama, A., T. H. Z. I., and Hosokawa, S., 2002. Transverse migration of single bubbles in simple shear flows. Chemical Engineering Science, 57, pp. 1849 1858. [12] Tomiyama, A., S. A. Z. I. K. N., and Sakaguchi, T., 1995. Effects of eotvos number and dimensionless liquid volumetric flux on lateral motion of a bubble in a laminar duct flow. Advances in Multiphase Flow, pp. 3 15. [13] Antal, S. P., L. R. T., and Flaherty, J. E., 1991. Analysis of phase distribution and turbulence in dispersed particle/liquid flows. Chemical Engineering Communications, 174, pp. 85 113. [14] Lopez de Bertodano, M. A., 1998. Two fluid model for two-phase turbulent jets. Nuclear Engineering and Design, 179, pp. 65 74. [15] Launder, G. E., and Spalding, D. B., 1974. The numerical computation of turbulent flows. Computer Methods in Applied mechanics and Engineering, 3(2), pp. 269 289. [16] Lee, S. L., and Lahey, R. T., J. O. C., 1989. The prediction of two-phase turbulence and phase distribution phenomena using a k ε model. Japanese Journal of Multiphase Flow, 3(4), pp. 335 368. Figure 4. THE GAS VELOCITY PROFILES COMPARISON WITH EXPERIMENTAL DATA AT H/D=22.4, J L =1.0m/s, J G =0.05, 0.3m/s AT TEMPERATURES (A) 15 C, (B) 24 C and (C) 36 C. 6