Estimating the temperature evolution of foodstuffs during freezing with a 3D meshless numerical method

Freezing processes are characterised by sharp changes in specific heat capacity and thermal conductivity for temperatures close to the freezing point. This leads to strong nonlinearities in the governing PDE that may be difficult to resolve using traditional numerical methods. In this work we present a meshless numerical method, based on a local Hermite radial basis function collocation approach in finite differencing mode, to allow the solution of freezing problems. By introducing a Kirchhoff transformation and solving the governing equations in Kirchhoff space, the strength of nonlinearity is reduced while preserving the structure of the heat equation. In combination with the high-resolution meshless numerical method, this allows efficient and stable numerical solutions to be obtained for freezing problems using 3D unstructured datasets. We demonstrate the proposed numerical formulation for the freezing of foodstuffs. By approximating the shape of the thermal conductivity and heat capacity curves in a piecewise linear fashion the temperature-dependent material curves may be described using eight independent parameters. We consider the optimisation of these parameters to match experimental data for the freezing of a hemispherical sample of mashed potato by using a simple manual procedure that requires a minimal number of simulations to be performed. Working in this way, a good approximation is obtained to the temperature profile throughout the sample without introducing instability into the numerical results.