Cyclic Melting/Solidification in a Cavity

 

 cygeo.gif cy.gif

 

An insulated rectangular cavity holds a pure material at it melting temperature. At time t = 0 the left hand wall is subjected to a cyclic temperature change which leads to a melting-solidification cycle. The simulation (using a uniform grid of 20x40 nodes) shows two cycles.

Note that

  1. during melting—the rising hot fluid leads to faster melting at the top of the cavity,
  2. when the solidification starts any temperature gradients in the liquid quickly dissipated and as a result the movment of solidification front is one-dimensional.

Voller’s research group have developed the so-called “Enthalpy Porosity” method for numerical simulation of this problem using a single fixed grid throughout the domain.

Approriate governing equations are

cyequ.gif

where (data is for gallium)

  • u is the x-velocity, v is the y-velocity—(m/s)
  • p is pressure (Pa)
  • T is temperature (K)
  • n=2.97 10-7 is the kinematic viscosity
  • b=1.3054 10-4 K-1 is the thermal expansion—The Boussinesq buoyancy treatment is invoked.
  • Tm=0 is the phase change temperature
  • a = 1.4145 10-5 m2/s is the thermal diffusivity
  • L = 80160 J/kg is the latent heat
  • c = 409.58 J/kg-K is the specific heat
  • f is the liquid fraction [0, 1]
  • Su=-K0(1-f)u -- similar for Sv-- is the “porosity” source term (Kv = 105). As a nodal liquid fraction approaches 0 this term becomes large and forces velocity calculations to zero—as required by a static solid phase.

    Typically the solution is based on a SIMPLE like routine for the pressure-velcoity coupling and an enthalpy solution for the heat transfer. The simulation shown here is taken from
     

    • V.R. Voller, P. Felix, and C.R. Swaminathan, Int. J. Num Meth. Heat and Fluid Flow, 6, 57-64, 1996.

    The original papers that mentions the term enthalpy porosity is
     

    • A.D. Brent, V.R. Voller and K.J. Reid, Numerical Heat Transfer 13, 297-318, 1988.

    This work was based on earlier ideas in
     

    • V.R. Voller, M. Cross and N. Markatos, Int. J. Num. Meth. Eng. 24, 271-284, 1987.
    • S.E. Hibbert, N.C. Markatos and V.R. Voller Int. J. Heat Mass Transfer 31, 1785-1795, 1988.
    • V.R. Voller and C. Prakash, Int J. Heat Mass Transfer 30, 1709-1719, 1987.

    Refinements and further discussions on the enthalpy-porosity methods can be found in
     

    • V.R. Voller, “Numerical Methods for Phase Change Problems,” Chapter 19 in HandBook for Numerical Heat Transfer, 2006.
    • V.R. Voller, Advances in Numerical Heat Transfer 1, 341-375, 1996.
    • V.R Voller and S. Peng, Computational Mechanics 14, 492-502, 1994.
    • V.R. Voller and C.R. Swaminathan, Numerical Heat Transfer B 24, 161-180, 1993.
    • C.R. Swaminathan and V.R. Voller, Int. J. Num. Meth. Heat and Fluid Flow 3, 233-244, 1993.
    • C.R. Swaminathan and V.R. Voller, Met. Trans. B 23, 651-664, 1992.
    • A.T. Chronopoulous, C. Swaminathan and V.R. Voller, J. Super. Comp, 5 74-91, 1991.
    • V.R. Voller and C.R. Swaminathan, Numerical Heat Transfer B 19, 175-189, 1991.
    • V.R. Voller, Numerical Heat Transfer B 17, 155-169, 1990.
    • V.R. Voller, C.R. Swaminathan and B.G. Thomas, Int. J. Num. Meth. Eng. 30, 875-898, 1990.
    • M. Lacroix and V.R. Voller, Numerical Heat Transfer 17, 25-42, 1990.