Karan SURANA, Aaron JOY, Luis QUİROS, Jn REDDY

7514

Mathematical Models and Numerical Solutions of Liquid-Solid and Solid-Liquid Phase Change

Anahtar Kelimeler:

-

Mathematical Models and Numerical Solutions of Liquid-Solid and Solid-Liquid Phase Change

This paper presents numerical simulations of liquid-solid andsolid-liquid phase change processes using mathematical models inLagrangian and Eulerian descriptions. The mathematical modelsare derived by assuming a smooth interface or transition region between the solid and liquid phases in which the specific heat, density,thermal conductivity, and latent heat of fusion are continuous anddifferentiable functions of temperature. In the derivations of themathematical models we assume the matter to be homogeneous,isotropic, and incompressible in all phases. The change in volumedue to change in density during phase transition is neglected in allmathematical models considered in this paper. This paper describesvarious approaches of deriving mathematical models that incorporate phase transition physics in various ways, hence results in different mathematical models. In the present work we only considerthe following two types of mathematical models: (i) We assume thevelocity field to be zero i.e. no flow assumption, and free boundaries i.e. zero stress field in all phases. Under these assumptionsthe mathematical models reduce to first law of thermodynamics i.e.the energy equation, a nonlinear diffusion equation in temperatureif we assume Fourier heat conduction law relating temperature graNomenclature
Keywords:

liquid-solid, solid-liquid, phase change, Lagrangian, Eulerian, mathematical models, space-time methods time-marching,

PDF

___

  • K. R. Rajagopal and L. Tao. Mechanics of Mixtures. World Scientific, River Edge, NJ, 1995.
  • M. Massoudi and A. Briggs and C. C. Hwang. Flow of a dense particulate mixture using a modified form of the mixture the- ory. Particulate Science and Technology, 17:1–27, 1999.
  • Mehrdad Massoudi. Constitutive relations for the interac- tion force in multicomponent particulate flows. International Journal of Non-Linear Mechanics, 38:313–336, 2003.
  • John W. Cahn and John E. Hilliard. Free Energy of a Nonuni- form System. I. Interfacial Free Energy. The Journal of Chem- ical Physics, 28(2):1015–1031, 1958.
  • Lev D. Landau and Evgenij Michailoviˇc Lifˇsic and Lev P. Pitaevskij. Statistical Physics: Course of Theoretical Physics. Pergamon Press plc, London, 1980.
  • K.S. Surana and J.N. Reddy. Mathematics of computations and finite element method for initial value problems. Book manuscript in progress, 2014.
  • T. Belytschko and T.J.R. Hughes. Computational Methods in Mechanics. North Holland, 1983.
  • B.C. Bell and K.S. Surana. A space-time coupled p-version LSFEF for unsteady fluid dynamics. International Journal of Numerical Methods in Engineering, 37:3545–3569, 1994.
  • Surana, K. S., Reddy, J. N. and Allu, S. The k-Version of Fi- nite Element Method for IVPs: Mathematical and Computa- tional Framework. International Journal for Computational Methods in Engineering Science and Mechanics, 8(3):123– 136, 2007. [10] J. Stefan.
  • Ober einige Probleme der Theorie der
  • Sitzungsber. Akad. Wiss. Wien, Math.- Warmeleitung.
  • Naturwiss. Kl., 1889.
  • L.I. Rubinstein. The Stefan Problem. American Mathematical Society, Providence, Twenty Seventh edition, 1994.
  • H.S. Carslaw and J.S. Jaeger. Conduction of Heat in Solids. Oxford University Press, New York, second edition, 1959.
  • K. Krabbenhoft and L. Damkilde and M. Nazem. An Implicit Mixed Enthalpy-Temperature Method for Phase- Change Problems. Heat Mass Transfer, 43:233–241, 2007.
  • Sin Kim and Min Chan Kim and Won-Gee Chun. A Fixed Grid Finite Control Volume Model for the Phase Change Heat Conduction Problems with a Single-Point Predictor-Corrector Algorithm. Korean J. Chem. Eng., 18(1):40–45, 2001.
  • V.R. Voller and M. Cross and N. C. Markatos. An enthalpy method for convection/diffusion phase change. International Journal for Numerical Methods in Engineering, 24(1):271– 284, 1987.
  • R. A. Lambert and R. H. Rangel. Solidification of a super- cooled liquid in stagnation-point flow. International Journal of Heat and Mass Transfer, 46(21):4013–4021, 2003.
  • Nabeel Al-Rawahi and Gretar Tryggvason. Numerical Sim- ulation of Dendritic Solidification with Convection: Two- Dimensional Geometry. Journal of Computational Physics, 180(2):471–496, 2002.
  • E. Pardo and D. C. Weckman. A fixed grid finite element tech- nique for modelling phase change in steady-state conduction– advection problems. Methods in Engineering, 29(5):969–984, 1990.
  • D. M. Anderson and G. B. McFadden and A. A. Wheeler. A phase-field model of solidification with convection. Physica D: Nonlinear Phenomena, 135(1-2):175–194, 2000.
  • Y. Lu and C. Beckermann and J.C. Ramirez. Three- dimensional phase-field simulations of the effect of convec- tion on free dendritic growth. Journal of Crystal Growth, 280(1-2):320–334, 2005.
  • C. Beckermann and H. J. Diepers and I. Steinbach and A. Karma and X. Tong. Modeling Melt Convection in Phase- Field Simulations of Solidification. Journal of Computational Physics, 154(2):468–496, 1999.
  • Curtis M. Oldenburg and Frank J. Spera. Hybrid model for solidification and convection. Numerical Heat Transfer Part B: Fundamentals, 21:217–229, 1992.
  • Surana, K. S. Advanced Mechanics of Continua. CRC/Tayler and Francis (in print), 2014.
  • M. Fabbri and V.R. Voller. The Phase-Field Method in Sharp- Interface Limit: A Comparison between Model Potentials. Journal of Computational Physics, 130:256–265, 1997.
  • Caginalp, G. Stefan and Hele-Shaw type models as asymp- totic limits of the phase-field equations. 39:5887–5896, 1989. Phys. Rev. A,
  • Caginalp, G. An analysis of a phase field model of a free boundary. 92:205–245, 1986.
  • G. Caginalp and J. Lin. A numerical analysis of an anisotropic phase field model. 39:51–66, 1987.
  • G. Caginalp and E.A. Socolovsky.
  • Computation of sharp
  • phase boundaries by spreading: The planar and spherically
  • symmetric cases. Journal of Computational Physics, 95:85– 100, 1991.
  • Surana, K. S., Ma, Y., Romkes, A., and Reddy, J. N. Devel- opment of Mathematicasl Models and Computational Frame- work for Multi-physics Interaction Processes. Mechanics of Advanced Materials and Structures, 17:488–508, 2010.
  • Surana, K. S., Blackwell, B., Powell, M., and Reddy, J. N. Mathematical Models for Fluid-Solid Interaction and Their Numerical Solutions. Journal of Fluids and Structures, (in print) 2014.
  • K.S. Surana and J.N. Reddy. Mathematics of computations and finite element method for boundary value problems. Book manuscript in progress, 2014.
  • Surana, K. S., Ahmadi, A. R. and Reddy, J. N. The k- Version of Finite Element Method for Self-Adjoint Operators in BVPs. International Journal of Computational Engineer- ing Science, 3(2):155–218, 2002.
  • Surana, K. S., Ahmadi, A. R. and Reddy, J. N. The k-Version of Finite Element Method for Non-Self-Adjoint Operators in BVPs. International Journal of Computational Engineering Sciences, 4(4):737–812, 2003.
  • Surana, K. S., Ahmadi, A. R. and Reddy, J. N. The k-Version of Finite Element Method for Non-Linear Operators in BVPs. International Journal of Computational Engineering Science, 5(1):133–207, 2004.
Journal of Thermal Engineering-Cover
  • Başlangıç: 2015
  • Yayıncı: YILDIZ TEKNİK ÜNİVERSİTESİ
Arşiv
Sayıdaki Diğer Makaleler

Nanofluid heat transfer and applications

Godson LAZARUS

Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview

Patrice ESTELLÉ, Salma HALELFADL, Thierry MARÉ

Performance optimization of Brayton heat engine at maximum efficient power using temperature dependent specific heat of working fluid

Rajesh Kumar, S C Kaushik, Raj KUMAR

Mathematical Models and Numerical Solutions of Liquid-Solid and Solid-Liquid Phase Change

Karan SURANA, Aaron JOY, Luis QUİROS, Jn REDDY