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Thermal Process Calculations Through Ball’s Original Formula Method: A Critical Presentation of the Method and Simplification of its Use Through Regression Equations

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Ball’s formula method is a classical method used in the thermal processing industry, the basis and the precursor of a number of most recent methodologies. It has received a lot of attention, being reviewed, criticized, compared and evaluated by several investigators. The use of Ball’s method relies on appropriate diagrams which sometimes are difficult to use, find or reproduce. After presenting the principles involved in the development of Ball’s formula method, reviewing the literature related to that particular method, and discussing its limitations, some working, regression equations developed in order to facilitate the application and the use of the method are presented.

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a 1 a 7 :

Regression coefficients appearing in Eq. 27, dimensionless

b 1 b 7 :

Regression coefficients appearing in Eq. 28, associated with temperature differences, g and z, in °F

B :

Steam-off time (measured from corrected zero), min

C :

Concentration of a heat-labile substance, number of microorganisms/mL, spores per container, g/mL, or any other appropriate unit


Duration of retort come-up time, min

D T :

(Noted also as D) decimal reduction time or death rate constant; time at a constant temperature required to reduce by 90% the initial spore load (or, in general, time required for 90% reduction of a heat-labile substance), min

E1 :

Exponential integral defined by Eq. 15


Exponential integral defined by Eq. 16

\( F_{T}^{z} \) :

(Or simply F) time at a constant temperature, T, required to destroy a given percentage of microorganisms whose thermal resistance is characterized by z, or, the equivalent processing time of a hypothetical thermal process at a constant temperature that produces the same effect (in terms of spore destruction) as the actual thermal process, min

F i :

Factor defined by Eq. 14, which when multiplied F Tref by gives the F value at the retort temperature, dimensionless

f :

Time required for the difference between the medium and the product temperature to change by a factor of 10 min

g :

Difference between retort and product temperature (at the critical point) at steam-off time, °F


A small g value lower than or close to 0.1°F, after which product temperature is considered constant, °F

I :

Difference between retort and initial product temperature, °F

j :

A correction factor defined by Eqs. 5 and 8 for the heating and cooling curve, respectively, based on the intercept, with the temperature axis at time zero, of the straight line that describes the late, straight, portion of the experimental heating or cooling curve plotted in a semi-logarithmic temperature difference scale as shown in Figs. 1 and 2, respectively, dimensionless

Fig. 1
figure 1

Typical straight-line heating curve used for f h and j h parameter estimation

Fig. 2
figure 2

Typical cooling curve used for f c and j c parameter estimation

m :

Difference between product temperature at steam-off and retort temperature during cooling, defined by Eq. 12, °F

T :

(Product) temperature, °F

T A :

Extrapolated pseudo-initial product temperature at the beginning of heating defined as the intercept, with the temperature axis at time zero, of the straight line that describes the late, straight, portion of the experimental heating curve plotted as shown in Fig. 1, °F

T B :

Extrapolated pseudo-initial product temperature at the beginning of cooling defined as the intercept, with the temperature axis at zero cooling time, of the straight line that describes the late, straight, portion of the experimental cooling curve plotted as shown in Fig. 2, °F

T h :

Product temperature at the beginning of the cooling cycle, °F

t :

Time, min

U :

The F value at T RT , defined by Eq. 13, min

u :

Dummy variable

z :

Temperature difference required to achieve a decimal change of the D T value, °F

zc :

A correction temperature difference factor appearing in Eq. 27, °F

ρ :

The fraction of the total lethal value of a process (that is, the F value of the entire process) which is achieved during the heating cycle only of the thermal process, assuming that the slope of the heating curve is constant and equal to the slope of the cooling curve, dimensionless

1, 2:

Refers to a particular condition

a :

Initial condition

b :

Final condition

bh :

Condition at the time where the break, the change in the slope of the heating curve occurs

CW :

(Water) cooling medium

c :

Cooling phase

end :

End of cooling cycle

g :

Condition at steam-off time

h :

Heating phase

IT :

Initial condition (for product temperature only)

process :

Referring to process values

RT :

(Retort) heating medium

ref :

Reference value

required :

Referring to required values


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Stoforos, N.G. Thermal Process Calculations Through Ball’s Original Formula Method: A Critical Presentation of the Method and Simplification of its Use Through Regression Equations. Food Eng. Rev. 2, 1–16 (2010).

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