A Mixed Finite Element Formulation for Solving Phase Change Problems with Convection

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1 A Mixed Finite Element Formulation for Solving Phase Change Problems with Convection Youssef Belhamadia 1, Abdoulaye S. Kane 2, and André Fortin 3 1 University of Alberta, Campus Saint-Jean and Department of Mathematical and Statistical Sciences, Edmonton, AB, Canada. 2 University of Ottawa, Department of Mathematics and Statistics, Ottawa, ON, Canada. 3 Laval University, Department of Mathematics and Statistics, Québec, QC, Canada youssef.belhamadia@ualberta.ca ABSTRACT An enhanced mixed formulation, based on the general form of the enthalpy, is proposed for phase change problem with convection. This formulation presents the advantage to consider both equal and different liquid-solid physical properties. A fully implicit second order accurate temporal and spatial finite element discretizations is also employed. The numerical simulations for water solidification are presented and compared to the experimental data to illustrate the performance of the proposed numerical method. 1 INTRODUCTION Phase change problems with natural convection have enormous importance in many industrial precesses such as crystal growth, refinement of metals, and casting. The main challenge is to accurately compute the liquid-solid interface where phase change occurs. This phase change boundary is time dependent and its morphology can be affected with the melt flow. Mathematical models describing phase change problems with convection are based on the classical Stefan problem coupled with Navier Stokes equations for Newtonian and incompressible fluid. Considering the two phases, typically solid, s t), and liquid, l t), separated by an interface or free boundary), Γt), which is defined by the melting temperature T f, the set of equations can be written as: Conservation of momentum ρ l u ρ lu u = µ γu)) p ST ). 1) Conservation of mass u = 0 2) where u is the velocity, γu) = u ut ) is the symmetric part of the velocity gradient tensor, p is the pres- 2 sure and ST ) is the gravity source term assuming the Boussinesq approximation. Energy equation for the fluid domain ρ l c l T l ρ lc l u T l K l T l ) = 0 in l 3) Energy equation for the solid ρ s c s T s K s T s ) = 0 in s t) 4) Heat balance equilibrium condition must be enforced on the interface Γt) K s T ) n s K l T ) n l = ρ l LV Γ on Γt), 5) where K l,s is the thermal conductivity tensor, ρ l,s is the density, c l,s is the specific heat, L is the latent heat of fusion, and V Γ is the interface normal velocity. Numerical methods dealing with the system of equations 1)-5) can be classified into two main categories. The first is called multi-domain method, or front tracking method, that considers solving the momentum and energy equations in each phase domain separately and therefore it requires a continuous update of the two domains due to the time dependent interface position see [13] and [9]). The second is called single-domain method, or fixed grid method, where the system of momentum and energy equations is solved in the entire physical domain, = l t) s t) Γt). The main

2 advantage of this method is that the interface is not explicitly computed and the energy balance condition 5) is automatically satisfied at the interface see [12]). The enthalpy-porosity model is widely used as a single domain approach see [4] and [14]). This model is however mostly used when the thermophysical properties of solid and liquid are equal. In the more general case, the enthalpy-porosity model can be reformulated into a vorticity-velocity model that is based on averaging the physical variables velocity, density, and thermal conductivity, by using liquid-solid mass and volume fraction. The main goal of this paper is to present a mixed finite element formulation for both the momentum and energy equations. This formulation is in a more general form than the one in the previous work [2, 3]. It includes the presence of natural convection and is based on the enthalpy-porosity method. The proposed formulation presents the advantage to consider both equal and different liquid-solid physical properties. A fully implicit second order accurate temporal and spatial discretizations are also presented for solving the proposed formulation leading to accurate solutions. Numerical results will be performed to illustrate the accuracy of the proposed numerical methodology and will be compared with existing literature results. 2 MATHEMATICAL MODEL Under the assumption that the fluid is Newtonian and incompressible, the enthalpy-porosity governing equation see [4] and [14]) can be written as: Conservation of momentum ρ l u ρ lu u = 2µ γu)) p Au ST ) Conservation of mass Conservation of energy H u = 0 u H K T ) = 0 where u is the velocity, γu) = u ut ) is the symmetric part of the velocity gradient tensor, p is the pres- 2 sure, H is the enthalpy, T is the temperature, µ is the viscosity, ρ l is the liquid density, and K is the conductivity tensor. A is defined so that the momentum equations are forced to mimic the Carman-Kozeny equations, 1 ε)2 A = C ε 3 6) b where C is a constant accounting for the mushy-region morphology, ε is the porosity between 0 and 1 and b = 10 6 is a constant introduced to avoid division by zero. In this paper, only water solidification is considered and therefore a modified Boussinesq approximation is employed to include the non-linear variation of the water density and thus the gravity source term will have the form ST ) = ρt ) ρ 0 )g, where the water density function, ρt ), will be set in the numerical simulation section. We now consider the general case where the thermophysical properties of solid and liquid may be different. The formulation is derived from enthalpy-porosity model by considering the following general formula of the enthalpy H s = ρ s c s T if T < T f, H = H l = ρ l L ρ s c s T f ρ l c l T T f ) if T > T f. where L is the latent heat of fusion. As in [2, 3], a phase change variable φ is introduced as: { 0 in s, φ = 7) 1 in l. Now the energy equation H becomes see [2, 3]) u H K T ) = 0, αφ) T αφ)u T )ρ ll φ Kφ) T ) = 0, 8) where αφ) = ρ s c s φρ l c l ρ s c s ), Kφ) = K s φk l K s ), f φ) = f s φ f l f s ). and the function φ satisfies a simple algebraic equation of the form: { 0 if T < Tf, φ = FT ) = 1 if T > T f, Thus A can written as 1 φ)2 Aφ) = C φ 3 b In applications, phase change is not always instantaneous and may occur in a small temperature range

3 [T f ε,t f ε]. Relation for FT ) can thus be replaced by a regularized one F ε T )) and therefore the resulting curve is differentiable as long as ε 0. The functions α, K, and f are then automatically regularized in a similar way since they depend on φ. A mixed velocity-pressure) formulation for Navier- Stokes equations is used. The finite element discretization is based on quadratic polynomials written using a hierarchical basis) for the velocity and linear continuous) polynomials for the pressure. This discretization is second order accurate in space Oh 2 )) as shown in [5]. Similary, A mixed formulation is also employed for the energy equation see [2, 3]) and the finite element discretization is based on quadratic polynomials for both the temperature T and phase change variable φ and is therefore also second order accurate in space Oh 2 )). In all our numerical simulations, a fully implicit backward second order scheme is employed for time discretization and therefore the variational formulation or finite element method reads as: Find u n1), p n1),t n1),φ n1) such that and = 3u n1) 4u n) u n 1) ρ l v u )d 2 t ρ l U u n1)) ) v u 2µ γu) n1) : γv u ) d p n1) v u Au n1) v u ) d ST n) ) v u d, v p u n1) d = 0, αφ n1) ) 3T n1) 4T n) T n 1) v T )d 2 t αφ n1) ) u n1) T n1)) ) v T d ρ l L 3φn1) 4φ n) φ n 1) v T )d 2 t ) Kφ n1) ) T n1) ) v T d = 0, φ n1) F ε T n1) ))v φ d = 0. where U can be u n1) and therefore Newton s method must be used to linearize the system. A better and less expensive method consists in extrapolating U from previous time steps. In all cases, an efficient preconditioned iterative solver developed in [6] is employed for the quadratic discretization arising from the linearized incompressible Navier-Stokes equations. 3 NUMERICAL RESULTS Freezing of water is a very common phase change problem and is employed in this section to assess the reliability and accuracy of the proposed formulation and discretizations. A convective flow in a cubic box filled with pure distilled water is therefore considered. The inner dimension of the box is 38 mm. The right wall temperature is maintained at T c = 10 o C and the opposite wall is held at a temperature T h = 10 o C. The other four walls allow the entry of heat from the external fluid which surrounds the cavity and were made of 6mm thick Plexiglas. This problem has been widely investigated experimentally and numerically in the literature and the reader is refereed to [11], and [8, 7] and the references therein for more details. the non-linear variation of the water density is provided by ρt ) = T T T T 4 with ρ 0 = The remaining physical parameters are constant but different in each phase and are given in the following table: µ ) Ns/m 2 )), ρ s kg/m 3), ρ l kg/m 3), c s 2116J/kg C), c l 4202J/kg C), K s 2.26I W/m C), K l 0.56I W/m C), L J/kg), T f 0C). T 0 0.5C). 25C). T ext The corresponding non-dimensional numbers describing the investigated configuration are based on the fluid properties and their values are Pr = 13.3, Ra = and Ste = In all the numerical results presented in [10, 11, 8, 7, 1], important discrepancies between measurements and calculations in the ice front form are observed and

Figure 1: Velocity field for water without phase change Figure 3: 3000s Velocity field for water solidification at these differences are more pronounced when the freezing time exceeds 500s.

Thus, only two-dimensional case is employed and the initial velocity and temperature fields are considered from natural convection without phase change.

A sudden drop of temperature of the right wall to 0 o C causes buoyancy forces to arise and in this case the liquid-solid phase change is not present.

4 Figure 1: Velocity field for water without phase change Figure 3: 3000s Velocity field for water solidification at these differences are more pronounced when the freezing time exceeds 500s. In this work, we will compare our numerical results with the numerical and experimental data from [1]. Thus, only two-dimensional case is employed and the initial velocity and temperature fields are considered from natural convection without phase change. Indeed, we consider first a cavity filled with pure water at an initial temperature of 10 o C. A sudden drop of temperature of the right wall to 0 o C causes buoyancy forces to arise and in this case the liquid-solid phase change is not present. The steady state velocity and temperature fields are presented in figures 1 and 2. Now, the temperature of the right wall changes from 0 o C to 10 o C which causes ice creation near this wall. The liquid-solid interface and the velocity vectors at time t = 3000s are presented in figures 3. The temperature fields at time 3000s is also presented in figure 4. Figure 2: Temperature field for water without phase change Numerical results from [1] is presented in figure 5 where discrepancies between numerical and experiment results are clearly seen. A comparisons between our numerical simulations and experimental data is presented in figure 6 where the interface position at time 3000s is plotted. It is clear that the discrepancies as presented in [1] are reduced showing the performance of the proposed method.

Figure 6: Numerical results from this work and experimental results from [1] Figure 4: Temperature field for water solidification at 3000s 4 CONCLUSION In this paper, an enhanced mixed formulation

5 Figure 6: Numerical results from this work and experimental results from [1] Figure 4: Temperature field for water solidification at 3000s 4 CONCLUSION In this paper, an enhanced mixed formulation for phase change problem with convection was presented. Accurate temporal and spacial discretizations were also introduced for solving the proposed formulation. The overall technique has been tested for water solidification problem. The comparison between numerical and experimental presented in this work was very promising and it has the advantage to preserve the interface form. As mentioned in [1] some side wall effects are observed in the experiment and therefore it would be very interesting to investigate water solidification in the three-dimensional case. Our initial numerical results are also close to the experimental measurement which make our methodology very promising. ACKNOWLEDGEMENTS The authors wish to acknowledge the financial support of Natural Sciences and Engineering Research Council of Canada NSERC) and research funds of Campus Saint-Jean. Figure 5: Comparison between numerical and experimental results from [1] REFERENCES [1] J. Banaszek, Y. Jaluria, T. A. Kowalewski, and M. Rebow. Semi-implicit fem analysis of natural convection in freezing water. Num. Heat Transfer, Part A, 36: , [2] Y. Belhamadia, A. Fortin, and É. Chamberland. Anisotropic Mesh Adaptation for the Solution of

6 the Stefan Problem. Journal of Computational Physics, 1941): , [3] Y. Belhamadia, A. Fortin, and É. Chamberland. Three-Dimensional Anisotropic Mesh Adaptation for Phase Change Problems. Journal of Computational Physics, 2012): , [4] A. D. Brent, V. R. Voller, and K. J. Reid. Enthalpy-porosity technique for modeling convection-diffusion phase change: Application to the melting of a pure metal. Numerical Heat Transfer, 133): , [13] R. Viswanath and Y. Jaluria. A comparison of different solution methodologies for melting and solidification problems in enclosures. Numerical Heat Transfer, Part B: Fundamentals, 241):77 105, [14] V. R. Voller, M. Cross, and N. C. Markatos. An enthalpy method for convection/diffusion phase change. International Journal for Numerical Methods in Engineering, 241): , [5] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, [6] A. El Maliki and R. Guénette. Efficient Preconditioning Techniques for Finite-Element Quadratic Discretization Arising from Linearized Incompressible Navier Stokes Equations. International Journal for Numerical Methods in Fluids, 6312): , [7] M. Giangi, T. A. Kowalewski, F. Stella, and E. Leonardi. Natural convection during ice formation: Numerical simulation vs. experimental results. Computer Assisted Mechanics and Engineering Sciences, 7: , [8] M. Giangi, F. Stella, and T. A. Kowalewski. Phase change problems with free convection: Fixed grid numerical simulation. Comput Visual Sci, 2: , [9] S. C. Gupta. A moving grid numerical scheme for multi-dimensional solidification with transition temperature range. Computer Methods in Applied Mechanics and Engineering, 1892): , [10] T. A. Kowalewski and M. Rebow. An Experimental Benchmark for Freezing Water in the Cubic Cavity. In Advances in Computational Heat Transfer, Begel House Inc., New York, pages , [11] T. A. Kowalewski and M. Rebow. Freezing of water in a differentially heated cubic cavity. International Journal of Computational Fluid Dynamics, 113-4): , [12] M. A. Rady and A. K. Mohanty. Natural convection during melting and solidification of pure metals in a cavity. Numerical Heat Transfer, Part A: Applications, 291):49 63, 1996.

AN ENHANCED MATHEMATICAL MODEL FOR PHASE CHANGE PROBLEMS WITH NATURAL CONVECTION

INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 3, Number 2, Pages 192 206 c 2012 Institute for Scientific Computing and Information AN ENHANCED MATHEMATICAL MODEL FOR PHASE CHANGE