# Introduction

This article provides access to a calculator and an introduction to the mathematics derived by Dr Tomas Skoglund that enables the z-value, a term used in microbial thermal death time calculations, to be corrected to comply with Arrhenius calculations. This value can be defined in several ways, including that z is the number of degrees that the temperature must be increased to achieve a tenfold (i.e.,1 log10) reduction in the decimal reduction value (D-value)**. **The D-value is the time required to reduce the number of organisms by 1 log cycle and is an indication of the heat resistance of an organism.

## Bigelow and the Arrhenius models

There are two main approaches to calculating the bactericidal effectiveness of a heat process namely the Bigelow and Arrhenius methods. Both methods require the use of constants to allow for the temperature dependence of the kinetics of thermal destruction. The Bigelow method uses z-value (SI unit K or °C) and the Arrhenius model uses activation energy (E_{a}, SI unit J/ mol).

With the Bigelow method z is assumed to be constant regardless of temperature. However, E_{a }varies with temperature. There has been debate for many years regarding the accuracy of both methods.

The Bigelow method is almost universally used and is the thermal processing method taught to most food science, food technology and food microbiology students globally and is also used by most processors and regulatory organisations. From a theoretical viewpoint it has been be argued that the Arrhenius method is likely to more validly represent the effect of temperature on microbial death.

Recently Skoglund (2022, 2023) has argued that the Arrhenius model should be used instead of the Bigelow method for high temperature thermal processes and in particular for direct UHT processing to avoid the under calculation of lethality and potential public health issues.

Skoglund (2022) has derived a mathematically valid approach (equation 1) to calculating the Arrhenius predicted change in value with temperature of z.

Equation 1. z = ln (10) x R x T_{r }x T x E_{a} which also equals z_{r} x T x T_{ r.}

_{Where }

R= gas constant

T_{r= Reference temperature e.g., 121.1°C }T= Processing temperature e.g. 140 °C

z_{r= z value at the reference temperature in }°C_{ z = Arrhenius predicted value of z at a designated temperature in }°C_{ }E_{a= activation energy at the reference temperature}

Knowing z from equation 1, lethal rate (LR) can be calculated using Equation 2 and integrated over the heat treatment process to give Fo.

Equation 2. LR = 10 ^{(T-T}_{r }^{/}^{z)} where the terms have been defined above.

I have prepared a spreadsheet to enable the differences in Fo obtained using the Bigelow and Arrhenius methods and also produced a free On-Line calculator (below) to enable users to see the difference in values obtained by using either method for UHT processing. Browsers can also use the calculator at https://www.dairyscience.info/newcalculators/validation/ to investigate the effects of using the Bigelow and Arrhenius methods to calculate the D-value value of microrganisms at different temperatures.

To use the calculator replace the values in the yellow cells with your values.

## Acknowledgement

I acknowledge the generous help of Dr Tomas Skoglund in helping me to understand the mathematics of calculating the temperature effects on z as predicted by his Arrhenius based derivations.

## Literature cited

Bigelow, W.G. (1921). The logarithmic nature of thermal death curves. The Journal of Infectious Diseases. 29, 528–538.

Skoglund, T. (2022) On the common misuse of a constant z-value for calculations of thermal inactivation of microorganisms Journal of Food Engineering 314, 110766

Skoglund, T. (2023) Standard microbiological approach to calculating z values, and consequences of approximations. Paper presented at the Society of Dairy Technology conference at Penrith, Cumbria, UK. on the 29^{th} March, 2023.