Computational Technology Resources - CCP

Keywords: finite elements, phase-change, mushy zone, enthalpy method, Newton-Raphson, tangent matrix.

Summary

In a large number of problems of engineering interest the transition of the material from one phase to another is of vital importance in describing the overall physical behaviour. Common applications include metal casting, freezing and thawing of foodstuffs and other biological materials, ground freezing and solar energy storage.
The phase-change problem is characterized by an abrupt change in enthalpy per unit temperature in a narrow temperature range around the freezing point. Indeed, it is often assumed that the enthalpy increases by some, usually rather large, amount at one characteristic temperature as shown in Figure 18.1(a). This assumption gives rise to a moving boundary problem which in quite a few cases can be solved analytically or otherwise by approximate hand calculation methods.
For many real problems such as the ones listed in the above, the change of phase does not take place at one characteristic temperature but over a temperature range as shown in Figure 18.1(b). One speaks of a mushy zone where at a given temperature both fluid and solid material may exist. This problem is no longer a moving boundary problem but rather a non-linear diffusion problem.
The governing equation for the general heat conduction problem, possibly including phase change, can be written as

$\displaystyle \rho \frac{\partial H}{\partial t} = \nabla \cdot (k\nabla T)$

(18.1)

With the present method a mixed formulation of this equation is used, i.e. the continuous enthalpy and temperature variables are interpolated separately by finite element functions to obtain a discrete system of equations on the form

$\displaystyle \mathbf M \mathbf H_{n+1} + \Delta t \mathbf K \mathbf T_{n+1} + \Delta t\mathbf f_{n+1} - \mathbf M \mathbf H_{n} = \mathbf 0$

(18.2)

This system is then solved iteratively by linking the nodal enthalpies and temperatures by the relevant enthalpy-temperature relation. In order to handle very narrow mushy zones a special iterative procedure has been developed.
Through examples and comparisons with analytical and other numerical solutions [1,2,3] the proposed method is shown to be accurate as well as efficient and robust. Other merits of the methods include a very simple implementation and easy generalization to arbitrary one, two and three dimensional elements. Furthermore, the non-physical temperature oscillations sometimes seen in numerical solutions are avoided.

Figure 18.1: Enthalpy-temperature relations for isothermal(a) and non-isothermal phase change(b).


(a)	(b)

References

1: L.A. Crivelli and S.R. Idelsohn. A temperature-based finite element solution for phase-change problems. International Journal for Numerical Methods in Engineering, 23:99-119, 1986. doi:10.1002/nme.1620230109
2: M. Storti, L.A. Crivelli, and S.R. Idelsohn. An efficient tangent scheme for solving phase-change problems. International Journal for Numerical Methods in Engineering, 66:65-86, 1988. doi:10.1016/0045-7825(88)90060-6
3: V.D. Fachinotti, A. Cardona, and A.E. Huespe. A fast convergent and accurate temperature model for phase-change heat conduction. International Journal for Numerical Methods in Engineering, 44:1863-1884, 1999. doi:10.1002/(SICI)1097-0207(19990430)44:12<1863::AID-NME571>3.0.CO;2-9

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