Summary

This article discusses two methods for calculating the D-value of a microorganism, the Bigelow (D/z model) and Arrhenius models, and provides access to a free calculator that enables the magnitude of the differences between the methods to be determined along with the Arrhenius thermal constants activation energy (Ea) and k0/s.

Using the reference decimal reduction and Z-values to determine the heat resistance of a microorganism at an arbitrary temperature

The decimal reduction value, D, is an indicator of heat resistance, or how easy it is to kill organisms using heat. This is the time, in seconds or minutes, taken to decrease the number of microorganisms by a factor of 10 (or to reduce their number by 90%). With the Bigelow (1921) approach D is regarded as a constant at a constant temperature for a particular strain of microorganism and is usually referenced against the reference temperature, Tref as follows e.g., D 93.3°C where the reference temperature was 93.3°C and D was determined at this temperature.

Knowing the D-value say for a Listeria monocyotegenes strain at 72°C as 2 s  and the total equivalent processing time at this temperature, F, as 16 s we can determine the lethality of the process by dividing F by D.

No of log reductions = F  = 16   = 8
                                       D      2

A pasteurisation-type process giving 8 log reductions for this pathogen would be considered safe.

Food technologists and microbiologists generally have limited information on the effect of temperature on the D-values of pathogenic and spoilage organisms. It is often necessary to model the D-value at a higher or lower temperature than the reference value.

This can be done in two ways, using the Bigelow (1921) method, also known as the D/z model, or the Arrhenius equation. The most commonly used method is the Bigelow method.

Both methods discussed here require knowledge of the Z-value. The Z-value is measured in °C, and is the reciprocal of the slope of the thermal death curve for the target microorganism or spore; 10 °C is the value frequently used in Fo calculations performed on low-acid foods. Z is a thermal constant that expresses the increase in temperature necessary to obtain the same lethal effect in 1/10 of the time. The Z-value has a significant effect on the F value of a process. Note it is the decimal reduction value, D, that is a measure of the resistance of a microorganism to heat. The z-value provides information on how this heat resistance changes with temperature. 

Given that the  D value at 72°C for an organism is 14 seconds and the Z value is 7°C how can the D-value at 78°C be calculated using the Bigelow method?

Since 78°C is 7°C more than 71°C and the Z-value is also 7°C the D-value is reduced by 1-log cycle i.e. 14 seconds (the D-value at 72°C) is divided by 10 to give a predicted D-value at 78°C of 1.4 seconds. Given a D-value at a particular temperature, D values at other temperatures can also be calculated using equation 1.

 Equation 1. Dref  = 10^(Tref-T1/Z)  (Stumbo, 1973)
                     D1

Where:

 Dref  = Reference D-value in minutes or seconds.
D1    = Arbitrary D-value in minutes or seconds.
Z      = Thermal constant for the organism in °C
Tref   = Reference temperature in °C
T1   =  Arbitrary temperature in °C.

This equation gives acceptably accurate results providing that the Z value is linear over the temperature range. Regrettably, this is often not the case and consequently, this approach should only be used when the temperature difference between Dref and D1 is small.

 The alternative, a more mathematically complex approach and potentially more accurate, is to use the Arrhenius equation (equation 2). This equation has been extensively studied and it can be argued e.g., Skoglund (2022, 2023) amongst others that this is a more accurate way of calculating the lethal effects of heat on microorganisms particularly when high temperatures are involved e.g., >121 °C.

 

Equation 2. Dref  =  EXP  Ea x (Tref-T1)
                    D1                    R      T1xTref

Where:

Dref   = Reference D-value in minutes or seconds.
D1    = Arbitrary D-value in minutes or seconds.
EXP  =
 The exponential constant
Tref    = Reference temperature in K
T1     =  Arbitary temperature in K
Ea     =   Activation energy (J/mol)

 

Ea can be calculated using Equation 3.

 Equation 3. Ea = LN(10) x R x (Tref^2)/Z

Where:

R = Gas constant (J/Kmol)

Tref   = Reference temperature  in K

Z      = Thermal constant for the organism in °C

How do the D values calculated using the Bigelow and Arrhenius methods differ?

 Comparisons of the calculated values can be shown in Tables 2 and 3. At low temperatures and providing the temperature range is small the disparity between the Bigelow and Arrhenius methods is relatively small. However, as predicted by Skoglund (2022, 2023) the disparity at higher temperatures is significant.

However, commercial UHT and sterilisation processes add additional time to allow a margin of process safety and this combined with normally low concentrations of bacteria especially spore formers probably accounts for the safety of high-temperature processes modelled using the Bigelow method (Skoglund, 2022).

 

Table 1. Modelling the difference in D-value for Mycobacterium avium paratuberculosis when using a constant Z and Arrenhius adjusted Z  over 63 to 80 °C .

D-value, s

Temperature, °C

63

72

75

80

Constant-Z

179.2

 

12.45

5.12

1.16

Arrhenius adjusted -Z

13.34

5.78

1.48

% difference between values (rounded)

 

7

13

28

Notes: Pooled strains of M. paratuberculosis. Z=7.77 °C

 

Table 2. Modelling the difference in D-value  for Geobacillus stearothermophilus  strain when using a constant Z and Arrenhius adjusted  Z over 102 to 150 °C .

D-value, s

Temperature,  °C

102

125

130

150

D Constant-Z

1500

 

4.17

1.16

0.007

D Arrhenius

5.86

1.91

0.028

% difference between values (rounded)

 

41

65

300

Notes. Geobacillus stearothermophilus had a Z value of 9. 

Access the free calculator to compare the effects of temperature on D as predicted by the Bigelow and Arrhenius models.

Users can also download an Excel spreadsheet below to do these calculations. The cells are all unlocked and you are free to use it without copyright declarations.


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Acknowledgements

The generous help of Dr Tomas Skoglund in helping me to understand and appreciate the use of the Arrhenius equation and its applications in thermal processing is gratefully acknowledged.

 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 29th March 2023.

Stumbo, C. R. (1973). Thermobacteriology in food processing, 2nd ed. Academic Press, New York.

 
How to cite this article

Mullan, W.M.A. (2024). [On-line]. Available from: https://www.dairyscience.info/thermal-processing/451-arrhenius.html . Accessed: 23 February, 2024.