Multidimensional freezing time using a pseudo-one-dimensional method

Abstract

This study presents a method to calculate the freezing time of multidimensional objects using a pseudo-one-dimensional method. For example, the temperature of a rectangle in $\left(x,y,t\right)$ can be simulated from the two-dimensional heat conduction equation to obtain a pseudo-one-dimensional temperature $T\left(x,t\right)$, using the space grid $\Delta x=L_x/\left(n-1\right)$ (where $n=m$) and $\Delta y=\Delta x{L}_y/L_x$ as references. This procedure can be used to calculate the freezing time $\left(t_{\text{calc}}\right)$ at a selected point, such as the center of an object. A computer program with a runtime similar to that of a one-dimensional problem has been developed for the proposed model. The freezing times ($t_{\text{calc}}$) of 212 multidimensional objects (parallelepipeds, rectangles, and finite cylinders) were then compared with the experimental freezing times ($t_{\text{exper}})$. The calculations yielded the following parameters for all 212 objects: minimum error $E_{\text{min}}=-3.9\%$, mean error $E_{\text{mean}}=0.2\%$, maximum error $E_{\text{max}}=5.0\%$, standard deviation ${\sigma }_{n-1}=1.4\%$, and mean absolute error $E_{\text{abs}}=1.1\%$. The freezing times ($t_{\text{calc}}$) of 100 multidimensional objects (parallelepipeds and rectangles) were then compared with the freezing times of computational experiments (computational simulation) obtained from the literature using the finite element method ($t_{\text{comput}})$. The calculations yielded the following parameters for all 100 objects: $E_{\text{min}}=-2.8\%$, $E_{\text{mean}}=0.1\%$, $E_{\text{max}}=3.7\%$, ${\sigma }_{n-1}=1.4\%$, and $E_{\text{abs}}=1.1\%$.