An Analytical Approximation of the Exponential Integral Function Applied to Food Thermal Process Calculations for Simple and Accurate Mathematical Modelling of Ball’s Tables

For the calculation of thermal processes of canned food, the original formula method of Ball is still widely used for its accuracy and safety. However, it requires the consultation of tables that Ball prepared, and the relative interpolation of the data. This is due to the exponential integral function (Ei) resulting after the integration of the differential equations obtained by combining the bacteriological laws on thermal death with those of non-stationary heat transfer. Mathematical modelling that replaces the Ball tables is useful for speeding up the thermal process calculations and for the control of the process by implementing it in PLC. Some mathematical models based on the regression of the data from the tables have already been proposed, among which the Stoforos’ equations stand out for accuracy and simplicity. However, these regression equations do not contain the influence of the temperature difference between the steam and the cold-water (m+g) when this is different from the two values ​​of the tables (180 and 130°F) for which it must be accepted that sometimes an over sterilization occurs. To overcome these limitations, in this work a nonlinear regression of the values ​​of the exponential integral function (Ei) has been developed, however using the expedient of performing the regression on the ratio between the function and its derivative. Two polynomial equations resulted, one of 5° for the negative values ​​of the domain and one of 6° for positive values. Furthermore, the hyperbola of the initial cooling imposed by Ball has been replaced with an appropriate exponential function so that after the integration there was the Ei function. The overall mean relative error MRE vs Ball’s tables was 0.955%, to be compared with that of Stoforos’ equations, equal to 1.035%. The equations of the proposed method are more numerous than those of Stoforos, but they are more general, allowing not only to cover all the values ​​of m+g, but also to be used in the future for the modelling of Stumbo’s tables, a method also present in the canning industry.