The basic principles of microbial destruction and their integration into the time temperature history of food products undergoing thermal sterilization were established in 1920 leading to the General (or graphical) Method for thermal process calculations (Bigelow et al., 1920).
Equation (1) is the governing equation for thermal process calculations, and it can be used to answer two types of problems; the first being the calculation of the F value of a thermal process and the second being the calculation of the heating time needed to achieve a target, required F value (Stoforos, 2015). F value calculation is rather straightforward and General Method is the method of choice for such calculations if time-temperature data are available. However, General Method lacks the predictive power required to answer the second type of problem, that is, the calculation of required heating time, as, with few exceptions, it relies on unrealistic assumptions.
To alleviate this problem, Charles Olin Ball (1893-1970) 100 years ago, on October1923, while working at the National Canners Association, introduced his mathematical method for thermal process calculations (Ball, 1923) in his classical publication (Figure 1). In brief, Ball used semi-empirical equations to describe the product temperature evolution during the heating and the cooling cycle of the thermal process (through the f and j parameters - Stoforos, (2015)) and used analytical and numerical or graphical techniques to solve the resulting equation, that is Eq. (2).
Note that in Eq. (2), the integral of Eq. (1) was divided into two integrals, the first one referring to the heating and the second one to the cooling cycle of the thermal process, in order for the time at the end of heating cycle, B, to be explicitly shown.
Through the solution of Eq. (2), Ball established a relationship between F value and heating time B, or actually between fh/U and log(g). Parameter U is defined as the F value calculated using the retort processing temperature (TRT) as reference temperature and g is the difference between retort and product temperature at the end of heating. Equation (3) was used to relate B with g.
To reduce the number of variables involved, Ball made assumptions such that his results could be presented in only a few tables or figures. Most noticeable assumptions were:
No product temperature rise after steam-off
jc = 1.41
fc = fh (Ball proposed a correction)
Constant retort temperature (Ball suggested to correct the heating time, by shifting the zero heating time axis by 0.58xCUT and apply his method, as for a process with CUT = 0, based on this corrected “zero” axis)
Lethality accumulation starts when product temperature reaches 80°F below TRT
Straight-line heating curve (Ball proposed a procedure for “broken” heating curves)
In its simplest application, the calculation of the heating time, B, for a given target F value, involves the following steps:
Calculation of the fh/U value based on the given F value
Determination of log(g) from the appropriate fh/U vs log(g) figure (for the proper TRT-TCW and z values)
Calculation of the heating time, B, from Eq. (3)
The same steps, in reverse order, are followed if the F value of a process, for a given heating time, B, is sought.
Following Ball’s Formula Method, a number of “formula” methods appeared in the literature (Stoforos et al., 1997). Nevertheless, Ball’s Formula Method remains a milestone in the thermal processing literature. Ball’s Formula Method became the standard in the food industry for many years and it is credited with the successful thermal processing industry. Ball’s method is a conservative method; the assumptions made were mainly towards the safe side, giving an additional safety margin to the calculations.
To facilitate the use of Ball’s Formula Method his figures or tabulated values were substituted by regression equations (Stoforos, 2010). Based on these equations, a user-friendly program was developed in MS EXCEL environment (Katsini and Stoforos, 2019) and can be provided to interested users. The program takes into account all Ball’s method limitations and warns the user when Ball’s assumptions are violated.
NOMENCLATURE
B | time at the end of heating (steam-off time), min |
CUT | retort come-up-time, i.e., time needed for the retort to reach heating cycle processing temperature, min |
DT | decimal reduction time or death rate constant: time at a constant temperature required to reduce by 90% the initial microbial load (or, in general, time required for 90% reduction of a heat labile substance), min (or simply F) time at a constant temperature, T, required to destroy a given percentage of microorganisms (or any other heat labile agent) whose thermal resistance is characterized by z, or, the equivalent processing time of a hypothetical thermal process at a constant, temperature, T, that produces the same effect (in terms of microorganisms or any other heat labile agent destruction) as the actual thermal process, min |
 | (or simply F) time at a constant temperature, T, required to destroy a given percentage of microorganisms (or any other heat labile agent) whose thermal resistance is characterized by z, or, the equivalent processing time of a hypothetical thermal process at a constant, temperature, T, that produces the same effect (in terms of microorganisms or any other heat labile agent destruction) as the actual thermal process, min |
f | time required for the difference between medium and product temperature to change by a factor of 10, min |
g | difference between retort and product temperature (at the critical point) at the end of heating (steam-off time), °C or °F |
I | difference between retort and initial product temperature, °C or °F |
j | lag constant, a parameter describing the deviation of the initial “curved” portion of the heating (or cooling) curve from the straight line that describes the late, straight, portion of the heating (or cooling) curve when plotted in a semi-logarithmic temperature difference (TRT-T) scale. |
N | number of spores/mL, (or microorganisms per container, or any other appropriate unit) |
T | (product) temperature, °C or °F |
t | time, min |
U | the F value using TRT as the reference temperature, min |
z | temperature difference required to achieve a decimal change of the DT value, °C or °F |
Subscripts
CW (water) cooling medium
c cooling phase
h heating phase
end end of processing
RT (retort) heating medium
ref reference value
o initial condition
T temperature
REFERENCES
Ball, C.O., 1923. Thermal Process Time for Canned Food, Bulletin of the National Research Council No. 37, Vol. 7, Part 1; Natl. Res. Council: Washington, DC, 76 pp.
Bigelow, W.D., Bohart, G.S., Richardson, A.C., and Ball, C.O., 1920. Heat Penetration in Processing Canned Foods. Bulletin 16-L, National Canners Association Research Laboratory: Washington, D.C., 128 pp.
Katsini, L. and Stoforos N.G., 2019. Development of a user-friendly computational tool in MS EXCEL environment for the design of thermal processes of canned foods with Ball’s Formula Method. In Proceedings of the 12th Greek National Scientific Congress of Chemical Engineering (# FE0238), May 29-31, 2019, Athens, Greece, (available in electronic form in Greek).
Stoforos, N.G., 2010. Thermal process calculations through Ball's original formula method: A critical presentation of the method and simplification of its use through regression equations. Food Engineering Reviews, 2(1): 1–16, doi: http://dx.doi.org/10.1007/s12393-010-9014-4.
Stoforos, N.G., 2015. Thermal Processing. In: Handbook of Food Processing: Food Preservation, Chapter 2, p. 27-56, T. Varzakas and C. Tzia, (Eds), CRC Press, Taylor & Francis Group, Boca Raton, FL, USA, http://dx.doi.org/10.1201/b19397-3.
Stoforos, N.G., Noronha, J., Hendrickx, M., and Tobback, P., 1997. A critical analysis of mathematical procedures for the evaluation and design of in-container thermal processes of foods. Critical Reviews in Food Science and Nutrition, 37(5): 411-441, doi: http://dx.doi.org/10.1080/10408399709527782.
Author
N.G. Stoforos retired from Agricultural University of Athens, Greece, where he served as professor in Food Engineering - Food Preservation.