A control volume capacitance method for solidification modelling with mass transport

Abstract

SUMMARYCapacitance methods are popular methods used for solidiÿcation modelling. Unfortunately, they su er from a major drawback in that energy is not correctly transported through elements and so providing a source of inaccuracy. This paper is concerned with the development and application of a control volume capacitance method (CVCM) to problems where mass transport and solidiÿcation are combined. The approach adopted is founded on theory that describes energy transfer through a control volume (CV) moving relative to the transporting mass. An equivalent governing partial di erential equation is established, which is designed to be transformable into a ÿnite element system that is commonly used to model transient heat-conduction problems. This approach circumvents the need to use the methods of Bubnov-Galerkin and Petrov-Galerkin and thus eliminates many of the stability problems associated with these approaches. An integration scheme is described that accurately caters for enthalpy uxes generated by mass transport. Shrinkage e ects are neglected in this paper as all the problems considered involve magnitudes of velocity that make this assumption reasonable. The CV approach is tested against known analytical solutions and is shown to be accurate, stable and computationally competitive.

A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

“…Results in figure 11 indicate that our algorithm also performs well for the case of advection with solidification, regardless of the direction of advection. Our results are also in agreement with those presented by [40] for a similar test case.…”

“…The reference solution for the computation of the error is the transient diffusion problem solved in each Eulerian mesh for the same initial condition (last column in figure 7). This benchmark aims to test the accuracy of our numerical scheme for both, isothermal and non-isothermal phase change problems ([39] [40]). The problem set up is displayed in figure 8 and the material properties in Table 2.…”

Numerical modelling of unsteady convective–diffusive heat transfer with a control volume hybrid method

“…The principal novelty of the method proposed here is the introduction of non-physical thermal conductivity k * and heat source q * in order to prevent the non-physical specific heat capacitance c * from becoming non-realisable (i.e. non-positive), which is possible for the methods proposed in references [14,15]. The approach proposed here couples transport control volume theory and the central difference scheme to ensure integral conservation of energy over each nodal-control volume.…”

“…Peclet number (the ratio between F and D) a nþ1 E the value of the east coefficient at time n + 1 a nþ1 W the value of the west coefficient at time n + 1 a n E the value of the east coefficient at time n a n W the value of the west coefficient at time n a n P coefficient with boundary condition and transient term at time n T b temperature at the boundary T 1 temperature at 1 T 0 initial temperature Subscripts f face of the control volume e east face of one-dimensional control volume w west face of one-dimensional control volume E east grid point W west grid point P at the unknown point considered by several users as a major weakness of the methods since the setting for upwind parameters is not known a priori [14]. The authors propose an approach which is called here the Control Volume Hybrid Method (CVHM) that is free from numerical oscillation regardless the magnitude of the Peclet number.…”

“…The authors propose an approach which is called here the Control Volume Hybrid Method (CVHM) that is free from numerical oscillation regardless the magnitude of the Peclet number. The work builds on the ideas underpinning the control volume capacitance method (CVCM) [14] applied to the FEM and those in Ref. [15] applied to central finite difference scheme.…”