# American Institute of Mathematical Sciences

June  2015, 35(6): 2325-2347. doi: 10.3934/dcds.2015.35.2325

## Numerical simulation of two-phase flows with heat and mass transfer

 1 AM III, Department Math, Cauerstr. 11 91058 Erlangen, Germany, Germany 2 Reinbeckstrasse 7, 12459 Berlin, Germany

Received  January 2014 Revised  May 2014 Published  December 2014

We present a finite element method for simulating complex free surface flow. The mathematical model and the numerical method take into account two-phase non-isothermal flow of an incompressible liquid and a gas phase, capillary forces at the interface of both fluids, Marangoni effects due to temperature variation of the interface and mass transport across the interface by evaporation/condensation. The method is applied to two examples from microgravity research, for which experimental data are available.
Citation: Eberhard Bänsch, Steffen Basting, Rolf Krahl. Numerical simulation of two-phase flows with heat and mass transfer. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2325-2347. doi: 10.3934/dcds.2015.35.2325
##### References:
 [1] E. Bänsch, Simulation of instationary, incompressible flows, Acta Math. Univ. Com., 67 (1998), 101-114. [2] E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface, Numer. Math., 88 (2001), 203-235. doi: 10.1007/PL00005443. [3] J. Brackbill, D. Kothe and C. Zemach, A continuum method for modeling surface tension, Journal of Computational Physics, 100 (1992), 335-354. doi: 10.1016/0021-9991(92)90240-Y. [4] M.-O. Bristeau, R. Glowinski and J. Pariaux, Numerical methods for the Navier-Stokes equations. applications to the simulation of compressible and incompressible viscous flow, Computer Physics Report, 6 (1987), 73-187. doi: 10.1007/978-3-322-87873-1. [5] A. N. Brooks and T. J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199-259, URL http://www.sciencedirect.com/science/article/pii/0045782582900718. doi: 10.1016/0045-7825(82)90071-8. [6] S. Das and E. Hopfinger, Mass transfer enhancement by gravity waves at a liquid-vapour interface, International Journal of Heat and Mass Transfer, 52 (2009), 1400-1411, URL http://www.sciencedirect.com/science/article/pii/S0017931008005176. doi: 10.1016/j.ijheatmasstransfer.2008.08.016. [7] J. Donea, S. Giuliani and J. P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comp. Meth. App. MechEng., 33 (1982), 689-723. doi: 10.1016/0045-7825(82)90128-1. [8] M. E. Dreyer, Free Surface Flows under Compensated Gravity Conditions, no. 221 in Springer Tracts in Modern Physics, Springer, 2007. [9] G. Dziuk, An algorithm for evolutionary surfaces, Numerische Mathematik, 58 (1991), 603-611. doi: 10.1007/BF01385643. [10] C. Eck, M. Fontelos, G. Grün, F. Klingbeil and O. Vantzos, On a phase-field model for electrowetting, Interfaces Free Bound., 11 (2009), 259-290. doi: 10.4171/IFB/211. [11] E. Fuhrmann and M. Dreyer, Description of the Sounding Rocket Experiment SOURCE, Microgravity Science and Technology, 20 (2008), 205-212. doi: 10.1007/s12217-008-9017-4. [12] E. Fuhrmann and M. Dreyer, Heat transfer by thermo-capillary convection, Microgravity Science and Technology, 21 (2009), 87-93. doi: 10.1007/s12217-009-9125-9. [13] E. Fuhrmann, M. Dreyer, S. Basting and E. Bänsch, Free surface deformation and heat transfer by thermocapillary convection, 2013,, Submitted for publication., (). [14] J. Gerstmann, Numerische Untersuchung zur Schwingung freier Flüssigkeitsoberflächen, no. 464 in Fortschritt-Berichte VDI, Reihe 7: Strömungsmechanik, VDI-Verlag, Düsseldorf, 2004. [15] J. Gerstmann, M. Michaelis and M. E. Dreyer, Capillary driven oscillations of a free liquid interface under non-isothermal conditions, PAMM, 4 (2004), 436-437. doi: 10.1002/pamm.200410199. [16] F. Gibou, L. Chen, D. Nguyen and S. Banerjee, A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change, J. Comp. Phys., 222 (2007), 536-555. doi: 10.1016/j.jcp.2006.07.035. [17] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer, 1986. doi: 10.1007/978-3-642-61623-5. [18] S. Gross and A. Reusken, Numerical Methods for Two-phase Incompressible Flows, vol. 40 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-19686-7. [19] M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981. [20] C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comp. Phys., 39 (1981), 201-225. doi: 10.1016/0021-9991(81)90145-5. [21] C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics, 135 (1997), 203-216. doi: 10.1006/jcph.1997.5702. [22] B. Höhn, Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren, Dissertation, Albert-Ludwigs-Universität Freiburg i.Br., Mathematische Fakultät, 1999. [23] T. J. R. Hughes, W. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Computer Methods in Applied Mechanics and Engineering, 29 (1981), 329-349. doi: 10.1016/0045-7825(81)90049-9. [24] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comp. Phys., 155 (1999), 96-127. doi: 10.1006/jcph.1999.6332. [25] D. Jamet, O. Lebaigue, N. Coutris and J. M. Delhaye, The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change, J. Comp. Phys, 169 (2001), 624-651. doi: 10.1006/jcph.2000.6692. [26] E. Kennard, Kinetic theory of gases: with an introduction to statistical mechanics, International series in pure and applied physics, McGraw-Hill, 1938. [27] R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics, Eng. Math., 39 (2001), 261-343. doi: 10.1023/A:1004844002437. [28] R. Krahl, M. Adamov, M. Lozano Avilés and E. Bänsch, A model for two phase flow with evaporation, in Two-Phase Flow Modelling and Experimentation 2004 (eds. G. P. Celata, P. Di Marco, A. Mariani and R. K. Shah), vol. 4, Edizioni ETS, Pisa, 2004, 2381-2387. [29] R. Krahl, J. Gerstmann, P. Behruzi, E. Bänsch and M. E. Dreyer, Dependency of the apparent contact angle on nonisothermal conditions, Physics of Fluids, 20 (2008), 042101. doi: 10.1063/1.2899641. [30] R. Krahl and E. Bänsch, Impact of marangoni effects on the apparent contact angle - a numerical investigation, Microgravity Science and Technology, 17 (2005), 39-44. doi: 10.1007/BF02872086. [31] R. Krahl and E. Bänsch, On the stability of an evaporating liquid surface, Fluid Dynamics Research, 44 (2012), 031409. doi: 10.1088/0169-5983/44/3/031409. [32] R. Krahl and J. Gerstmann, Non-isothermal reorientation of a liquid surface in an annular gap, in $4^{th}$ International Berlin Workshop - IBW 4 on Transport Phenomena with Moving Boundaries (ed. F.-P. Schindler), no. 883 in Fortschritt-Berichte VDI, Reihe 3: Verfahrenstechnik, VDI-Verlag, Düsseldorf, (2007), 227-241. [33] N. Kulev, S. Basting, E. Bänsch and M. Dreyer, Interface reorientation of cryogenic liquids under non-isothermal boundary conditions, Cryogenics, 62 (2014), 48-59, URL http://www.sciencedirect.com/science/article/pii/S0011227514000794. doi: 10.1016/j.cryogenics.2014.04.006. [34] N. Kulev and M. Dreyer, Drop tower experiments on non-isothermal reorientation of cryogenic liquids, Microgravity Science and Technology, 22 (2010), 463-474. doi: 10.1007/s12217-010-9237-2. [35] D. Meschede (ed.), Gerthsen Physik, 22nd edition, Springer, 2004. [36] M. Michaelis, Kapillarinduzierte Schwingungen Freier Flüssigkeitsoberflächen, no. 454 in Fortschritt-Berichte VDI, Reihe 7: Strömungsmechanik, VDI-Verlag, Düsseldorf, 2003. [37] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [38] S. Ostrach, Low-gravity fluid flows, Ann. Rev. Fluid. Mech., 14 (1982), 313-345. doi: 10.1146/annurev.fl.14.010182.001525. [39] L. M. Pismen and Y. Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics, Phys. Rev. E, 62 (2000), 2480-2492. doi: 10.1103/PhysRevE.62.2480. [40] M. Rumpf, A variational approach to optimal meshes, Numerische Mathematik, 72 (1996), 523-540. doi: 10.1007/s002110050180. [41] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Sci. Comp., 7 (1986), 856-869. doi: 10.1137/0907058. [42] R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow, Ann. Rev. Fluid Mech., 31 (1999), 567-603. doi: 10.1146/annurev.fluid.31.1.567. [43] J. Schlottke and B. Weigand, Direct numerical simulation of evaporating droplets, J. Comp. Phys., 227 (2008), 5215-5237. doi: 10.1016/j.jcp.2008.01.042. [44] L. E. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids, Chem. Eng. Sci., 12 (1960), 98-108. doi: 10.1016/0009-2509(60)87003-0. [45] J. A. Sethian and P. Smereka, Level set methods for fluid interfaces, Ann. Rev. Fluid Mech., 35 (2003), 341-372. doi: 10.1146/annurev.fluid.35.101101.161105. [46] G. Son and V. K. Dhir, Numerical simulation of film boiling near critical pressures with a level set method, J. Heat Transfer, 120 (1998), 183-192. doi: 10.1115/1.2830042. [47] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114 (1994), 146-159. doi: 10.1006/jcph.1994.1155. [48] S. Tanguy, T. Ménard and A. Berlemont, A level set method for vaporizing two-phase flows, J. Comp. Phys., 221 (2007), 837-853. doi: 10.1016/j.jcp.2006.07.003. [49] M. Tenhaeff, Computation of Incompressible, Axisymmetric Flows in Electrically Conducting Fluids Under Influence of Rotating Magnetic Fields (in German), Diploma thesis, Albert-Ludwigs-Universität Freiburg i.Br., Mathematische Fakultät, 1997. [50] T. Tezduyar and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements, J. Applied Mechanics, 70 (2003), 58-63. [51] S. W. J. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change, J. Comp. Phys, 160 (2000), 662-682. doi: 10.1006/jcph.2000.6481. [52] T. Wick, Fluid-structure interactions using different mesh motion techniques, Computers & Structures, 89 (2011), 1456-1467. doi: 10.1016/j.compstruc.2011.02.019. [53] Y. F. Yap, J. C. Chai, K. C. Toh, T. N. Wong and Y. C. Lam, Numerical modeling of unidirectional stratified flow with and without phase change, J. Int. Heat Mass Transfer, 48 (2005), 477-486. doi: 10.1016/j.ijheatmasstransfer.2004.09.017.

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##### References:
 [1] E. Bänsch, Simulation of instationary, incompressible flows, Acta Math. Univ. Com., 67 (1998), 101-114. [2] E. Bänsch, Finite element discretization of the Navier-Stokes equations with a free capillary surface, Numer. Math., 88 (2001), 203-235. doi: 10.1007/PL00005443. [3] J. Brackbill, D. Kothe and C. Zemach, A continuum method for modeling surface tension, Journal of Computational Physics, 100 (1992), 335-354. doi: 10.1016/0021-9991(92)90240-Y. [4] M.-O. Bristeau, R. Glowinski and J. Pariaux, Numerical methods for the Navier-Stokes equations. applications to the simulation of compressible and incompressible viscous flow, Computer Physics Report, 6 (1987), 73-187. doi: 10.1007/978-3-322-87873-1. [5] A. N. Brooks and T. J. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199-259, URL http://www.sciencedirect.com/science/article/pii/0045782582900718. doi: 10.1016/0045-7825(82)90071-8. [6] S. Das and E. Hopfinger, Mass transfer enhancement by gravity waves at a liquid-vapour interface, International Journal of Heat and Mass Transfer, 52 (2009), 1400-1411, URL http://www.sciencedirect.com/science/article/pii/S0017931008005176. doi: 10.1016/j.ijheatmasstransfer.2008.08.016. [7] J. Donea, S. Giuliani and J. P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comp. Meth. App. MechEng., 33 (1982), 689-723. doi: 10.1016/0045-7825(82)90128-1. [8] M. E. Dreyer, Free Surface Flows under Compensated Gravity Conditions, no. 221 in Springer Tracts in Modern Physics, Springer, 2007. [9] G. Dziuk, An algorithm for evolutionary surfaces, Numerische Mathematik, 58 (1991), 603-611. doi: 10.1007/BF01385643. [10] C. Eck, M. Fontelos, G. Grün, F. Klingbeil and O. Vantzos, On a phase-field model for electrowetting, Interfaces Free Bound., 11 (2009), 259-290. doi: 10.4171/IFB/211. [11] E. Fuhrmann and M. Dreyer, Description of the Sounding Rocket Experiment SOURCE, Microgravity Science and Technology, 20 (2008), 205-212. doi: 10.1007/s12217-008-9017-4. [12] E. Fuhrmann and M. Dreyer, Heat transfer by thermo-capillary convection, Microgravity Science and Technology, 21 (2009), 87-93. doi: 10.1007/s12217-009-9125-9. [13] E. Fuhrmann, M. Dreyer, S. Basting and E. Bänsch, Free surface deformation and heat transfer by thermocapillary convection, 2013,, Submitted for publication., (). [14] J. Gerstmann, Numerische Untersuchung zur Schwingung freier Flüssigkeitsoberflächen, no. 464 in Fortschritt-Berichte VDI, Reihe 7: Strömungsmechanik, VDI-Verlag, Düsseldorf, 2004. [15] J. Gerstmann, M. Michaelis and M. E. Dreyer, Capillary driven oscillations of a free liquid interface under non-isothermal conditions, PAMM, 4 (2004), 436-437. doi: 10.1002/pamm.200410199. [16] F. Gibou, L. Chen, D. Nguyen and S. Banerjee, A level set based sharp interface method for the multiphase incompressible Navier-Stokes equations with phase change, J. Comp. Phys., 222 (2007), 536-555. doi: 10.1016/j.jcp.2006.07.035. [17] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer, 1986. doi: 10.1007/978-3-642-61623-5. [18] S. Gross and A. Reusken, Numerical Methods for Two-phase Incompressible Flows, vol. 40 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-19686-7. [19] M. E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981. [20] C. W. Hirt and B. D. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comp. Phys., 39 (1981), 201-225. doi: 10.1016/0021-9991(81)90145-5. [21] C. Hirt, A. Amsden and J. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics, 135 (1997), 203-216. doi: 10.1006/jcph.1997.5702. [22] B. Höhn, Numerik für die Marangoni-Konvektion beim Floating-Zone Verfahren, Dissertation, Albert-Ludwigs-Universität Freiburg i.Br., Mathematische Fakultät, 1999. [23] T. J. R. Hughes, W. Liu and T. K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Computer Methods in Applied Mechanics and Engineering, 29 (1981), 329-349. doi: 10.1016/0045-7825(81)90049-9. [24] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comp. Phys., 155 (1999), 96-127. doi: 10.1006/jcph.1999.6332. [25] D. Jamet, O. Lebaigue, N. Coutris and J. M. Delhaye, The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change, J. Comp. Phys, 169 (2001), 624-651. doi: 10.1006/jcph.2000.6692. [26] E. Kennard, Kinetic theory of gases: with an introduction to statistical mechanics, International series in pure and applied physics, McGraw-Hill, 1938. [27] R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics, Eng. Math., 39 (2001), 261-343. doi: 10.1023/A:1004844002437. [28] R. Krahl, M. Adamov, M. Lozano Avilés and E. Bänsch, A model for two phase flow with evaporation, in Two-Phase Flow Modelling and Experimentation 2004 (eds. G. P. Celata, P. Di Marco, A. Mariani and R. K. Shah), vol. 4, Edizioni ETS, Pisa, 2004, 2381-2387. [29] R. Krahl, J. Gerstmann, P. Behruzi, E. Bänsch and M. E. Dreyer, Dependency of the apparent contact angle on nonisothermal conditions, Physics of Fluids, 20 (2008), 042101. doi: 10.1063/1.2899641. [30] R. Krahl and E. Bänsch, Impact of marangoni effects on the apparent contact angle - a numerical investigation, Microgravity Science and Technology, 17 (2005), 39-44. doi: 10.1007/BF02872086. [31] R. Krahl and E. Bänsch, On the stability of an evaporating liquid surface, Fluid Dynamics Research, 44 (2012), 031409. doi: 10.1088/0169-5983/44/3/031409. [32] R. Krahl and J. Gerstmann, Non-isothermal reorientation of a liquid surface in an annular gap, in $4^{th}$ International Berlin Workshop - IBW 4 on Transport Phenomena with Moving Boundaries (ed. F.-P. Schindler), no. 883 in Fortschritt-Berichte VDI, Reihe 3: Verfahrenstechnik, VDI-Verlag, Düsseldorf, (2007), 227-241. [33] N. Kulev, S. Basting, E. Bänsch and M. Dreyer, Interface reorientation of cryogenic liquids under non-isothermal boundary conditions, Cryogenics, 62 (2014), 48-59, URL http://www.sciencedirect.com/science/article/pii/S0011227514000794. doi: 10.1016/j.cryogenics.2014.04.006. [34] N. Kulev and M. Dreyer, Drop tower experiments on non-isothermal reorientation of cryogenic liquids, Microgravity Science and Technology, 22 (2010), 463-474. doi: 10.1007/s12217-010-9237-2. [35] D. Meschede (ed.), Gerthsen Physik, 22nd edition, Springer, 2004. [36] M. Michaelis, Kapillarinduzierte Schwingungen Freier Flüssigkeitsoberflächen, no. 454 in Fortschritt-Berichte VDI, Reihe 7: Strömungsmechanik, VDI-Verlag, Düsseldorf, 2003. [37] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [38] S. Ostrach, Low-gravity fluid flows, Ann. Rev. Fluid. Mech., 14 (1982), 313-345. doi: 10.1146/annurev.fl.14.010182.001525. [39] L. M. Pismen and Y. Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics, Phys. Rev. E, 62 (2000), 2480-2492. doi: 10.1103/PhysRevE.62.2480. [40] M. Rumpf, A variational approach to optimal meshes, Numerische Mathematik, 72 (1996), 523-540. doi: 10.1007/s002110050180. [41] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Sci. Comp., 7 (1986), 856-869. doi: 10.1137/0907058. [42] R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow, Ann. Rev. Fluid Mech., 31 (1999), 567-603. doi: 10.1146/annurev.fluid.31.1.567. [43] J. Schlottke and B. Weigand, Direct numerical simulation of evaporating droplets, J. Comp. Phys., 227 (2008), 5215-5237. doi: 10.1016/j.jcp.2008.01.042. [44] L. E. Scriven, Dynamics of a fluid interface equation of motion for Newtonian surface fluids, Chem. Eng. Sci., 12 (1960), 98-108. doi: 10.1016/0009-2509(60)87003-0. [45] J. A. Sethian and P. Smereka, Level set methods for fluid interfaces, Ann. Rev. Fluid Mech., 35 (2003), 341-372. doi: 10.1146/annurev.fluid.35.101101.161105. [46] G. Son and V. K. Dhir, Numerical simulation of film boiling near critical pressures with a level set method, J. Heat Transfer, 120 (1998), 183-192. doi: 10.1115/1.2830042. [47] M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114 (1994), 146-159. doi: 10.1006/jcph.1994.1155. [48] S. Tanguy, T. Ménard and A. Berlemont, A level set method for vaporizing two-phase flows, J. Comp. Phys., 221 (2007), 837-853. doi: 10.1016/j.jcp.2006.07.003. [49] M. Tenhaeff, Computation of Incompressible, Axisymmetric Flows in Electrically Conducting Fluids Under Influence of Rotating Magnetic Fields (in German), Diploma thesis, Albert-Ludwigs-Universität Freiburg i.Br., Mathematische Fakultät, 1997. [50] T. Tezduyar and R. Benney, Mesh moving techniques for fluid-structure interactions with large displacements, J. Applied Mechanics, 70 (2003), 58-63. [51] S. W. J. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change, J. Comp. Phys, 160 (2000), 662-682. doi: 10.1006/jcph.2000.6481. [52] T. Wick, Fluid-structure interactions using different mesh motion techniques, Computers & Structures, 89 (2011), 1456-1467. doi: 10.1016/j.compstruc.2011.02.019. [53] Y. F. Yap, J. C. Chai, K. C. Toh, T. N. Wong and Y. C. Lam, Numerical modeling of unidirectional stratified flow with and without phase change, J. Int. Heat Mass Transfer, 48 (2005), 477-486. doi: 10.1016/j.ijheatmasstransfer.2004.09.017.
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