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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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Abstract

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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Abbreviations

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

CUT:

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

Ei:

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

g′:

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

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Fig. 2
figure 2

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

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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

References

  1. Afaghi MK (2000) Application of artificial neural network modeling in thermal process calculations of canned foods. MSc Thesis, Dept Food Science and Agr Chem, McGill University, Montreal, Canada

  2. Alstrand DV, Benjamin HA (1949) Thermal processing of canned foods in tin containers. V. Effect of retorting procedures on sterilization values in canned foods. Food Res 14:253–257

    CAS  Google Scholar 

  3. Ball CO (1923) Thermal process time for canned food. Bulletin of the National Research Council No. 37., 7, Part 1Natl Res Council, Washington, DC

    Google Scholar 

  4. Ball CO (1928) Mathematical solution of problems on thermal processing of canned food. Univ Calif Pubs Public Health, vol 1, No 2. University of California Press, Berkeley, CA (with supplement in 1936)

    Google Scholar 

  5. Bal CO (1949) Process evaluation. Food Technol 3:116–118

    Google Scholar 

  6. Ball CO, Olson FCW (1957) Sterilization in food technology. Theory practice and calculations. McGraw-Hill Book Co., New York

    Google Scholar 

  7. Berry MR Jr (1983) Prediction of come-up time correction factors for batch-type agitating and still retorts and the influence on thermal process calculations. J Food Sci 48:1293–1299

    Article  Google Scholar 

  8. Berry MR Jr, Bush RC (1987) Establishing thermal processes for products with broken heating curves from data taken at other retort and initial temperatures. J Food Sci 52(4):958–961

    Article  Google Scholar 

  9. Bigelow WD, Bohart GS, Richardson AC, Ball CO (1920) Heat penetration in processing canned foods. Bull. No. 16-L, Res. Lab. Natl Canners Assoc, Washington, DC

    Google Scholar 

  10. Cleland AC, Robertson GL (1985) Determination of thermal processes to ensure commercial sterility of foods in cans. In: Thorne S (ed) Developments in food preservation, vol 3. Elsevier Applied Science Publishers, London

    Google Scholar 

  11. Dail R (1985) Calculation of required time of aseptically processed low acid foods containing particulates utilizing the Ball method. J Food Sci 50:1703–1706

    Article  Google Scholar 

  12. Esty JR, Meyer KF (1922) The heat resistance of the spores of B. botulinus and allied anaerobes. J Infectious Dis 34(1):650–663

    Google Scholar 

  13. Flambert CMF, Deltour J (1971) Calcul des températures dans les aliments en boîtes cylindriques transmettant la chaleur par conduction. Indust Alim Agr 88(9):1279–1287

    Google Scholar 

  14. Flambert F, Deltour J (1972) Localization of the critical area in thermally-processed conduction heated canned foods. Lebensm Wiss Technol 5(1):7–13

    Google Scholar 

  15. Flambert CMF, Deltour J, Dickerson RW, Hayakawa K (1977) Lethal effect of food temperature on linear portion of a heating or cooling curve. J Food Sci 42:545–546

    Article  Google Scholar 

  16. Gautschi W, Cahill WF (1964) Exponential integral and related functions. In: Abramowitz M, Stegun IA (eds) Handbook of mathematical functions. Natl Bureau of Standards, Appl Math Ser 55, U.S. Covt Print Office, Washington, DC

    Google Scholar 

  17. Ghazala S, Ramaswamy HS, Smith J, Simpson B (1991) Thermal process time calculations for thin profile packages: comparison of formula methods. J Food Process Eng 13:269–282

    Article  Google Scholar 

  18. Griffin RC, Jr, Herndon DH, Ball CO (1969) Use of computer derived tables to calculate sterilizing processes for packaged foods. 2. Application to broken line heating curves. Food Technol 23(4):519–524

    Google Scholar 

  19. Griffin RC Jr, Herndon DH, Ball CO (1971) Use of computer derived tables to calculate sterilizing processes for packaged foods. 3. Application to cooling curves. Food Technol 25(2):134–143

    Google Scholar 

  20. Hayakawa K (1970) Experimental formulas for accurate estimation of transient temperature of food and their application to thermal process evaluation. Food Technol 24(12):1407–1418

    Google Scholar 

  21. Hayakawa K (1978) A critical review of mathematical procedures for determining proper heat sterilization processes. Food Technol 32(3):59–65

    Google Scholar 

  22. Hayakawa K, Ball CO (1971) Theoretical formulas for temperatures in cans of solid food and for evaluating various heat processes. J Food Sci 36:306–310

    Article  CAS  Google Scholar 

  23. Herndon DH, Griffin RC Jr, Ball CO (1968) Use of computer derived tables to calculate sterilizing processes for packaged foods. Food Technol 22(4):473–484

    Google Scholar 

  24. Holdsworth D, Simpson R (2008) Thermal Processing of Packaged Foods, 2nd edn. Springer, New York

    Google Scholar 

  25. Jen Y, Manson JE, Stumbo CR, Zahradnik JW (1971) A procedure for estimating sterilization of and quality factor degradation in thermally processed foods. J Food Sci 36:692–698

    CAS  Google Scholar 

  26. Kao J, Naveh D, Kopelman IJ, Pflug IJ (1981) Thermal process calculations for different z and jc values using a hand held calculator. J Food Sci 46:193–197

    Article  Google Scholar 

  27. Kim KH, Teixeira AA, Bichier J, Tavares J (1993) STERILIMATE: software for designing and evaluating thermal sterilization processes. ASAE paper No 93–4051. St. Joseph, MI

    Google Scholar 

  28. Larkin JW (1989) Use of a modified Ball’s formula method to evaluate aseptic processing of foods containing particulates. Food Technol 43(3):124–131

    Google Scholar 

  29. Larkin JW (1990) Revision of the hyperbolic equation used by the Ball formula method of thermal process evaluations. Paper No 157, 50th Annual Meeting of Institute of Food Technologists, June 16–20, Anaheim, CA

  30. Larkin JW, Berry RB (1991) Estimating cooling process lethality for different cooling j values. J Food Sci 56(4):1063–1067

    Article  Google Scholar 

  31. Merson RL, Singh RP, Carroad PA (1978) An evaluation of Ball’s formula method of thermal process calculations. Food Technol 32(3):66–72

    Google Scholar 

  32. NCA (1968) Laboratory manual for food canners and processors, Vol. 1. Microbiology and processing. Natl Canners Assoc Res Labs The Avi Pub. Co., Inc., Westport, CT

    Google Scholar 

  33. Noronha J, Hendrickx M, Van Loey A, Tobback P (1995) New semi-empirical approach to handle time-variable boundary conditions during sterilisation of non-conductive heating foods. J Food Eng 24:249–268

    Article  Google Scholar 

  34. Olson FCW, Jackson JM (1942) Heating curves. Theory and practical application. Ind Eng Chem 34(3):337–341

    Article  CAS  Google Scholar 

  35. Olson FCW, Stevens HP (1939) Thermal processing of canned foods in tin containers. II. Nomograms for graphic calculation of thermal processes for non-acid canned foods exhibiting straight-line semi-logarithmic heating curves. Food Res 4:1–20

    Google Scholar 

  36. Pham QT (1987) Calculation of thermal process lethality for conduction heated canned foods. J Food Sci 52(4):967–974

    Article  Google Scholar 

  37. Pham QT (1990) Lethality calculation for thermal processes with different heating and cooling rates. Int J Food Sci Technol 25(2):148–156

    Google Scholar 

  38. Purohit KS, Stumbo CR (1973) Refinement and extension of fh/U:g parameters for process calculation. J Food Sci 38:726–728

    Article  Google Scholar 

  39. Ramaswamy HS (1993) Come-up time effectiveness for process calculations involving thin-profile packages. J Food Eng 19:109–117

    Article  Google Scholar 

  40. Sasseen DM (1969) Computer program for process calculation by the simplified Ball formula method. Natl Canners Assoc, West Br Lab, Berkeley, CA

    Google Scholar 

  41. Simpson R, Almonacid S, Teixeira A (2003) Bigelow’s general method revisited: development of a new calculation technique. J Food Sci 68(4):1324–1333

    Article  CAS  Google Scholar 

  42. Smith T, Tung MA (1982) Comparison of formula methods for calculating thermal process lethality. J Food Sci 47:626–630

    Article  Google Scholar 

  43. Steele RJ, Board PW (1979) Amendments to Ball’s formula method for calculating the lethal value of thermal processes. J Food Sci 44:292–293

    Article  Google Scholar 

  44. Steele RJ, Board PW (1979) Thermal process calculations using sterilizing ratios. J Food Technol 14:227–235

    Article  Google Scholar 

  45. Steele RJ, Board PW, Bert DJ, Willcox ME (1979) Revision of the formula method tables for thermal process evaluation. J Food Sci 44:954–957

    Article  Google Scholar 

  46. Stoforos NG (1991) On Ball’s formula method for thermal process calculations. J Food Proc Eng 13(4):255–268

    Article  Google Scholar 

  47. Stoforos NG (1995) Thermal process design. Food Control 6(2):81–94

    Article  Google Scholar 

  48. Stoforos NG (2003) Heating and cooling lag constants. In: Heldman DR (ed) Encyclopedia of agricultural, food and biological engineering. Marcel Dekker, Inc., New York

    Google Scholar 

  49. Stoforos NG, Taoukis PS (2006) Heat processing: Temperature–time combinations. In: Hui YH (ed) Handbook of food science, technology, and engineering, vol 3. CRC Press, Taylor & Francis Group, Boca Raton, FL

    Google Scholar 

  50. Stoforos NG, Noronha J, Hendrickx M, Tobback P (1997) A critical analysis of mathematical procedures for the evaluation and design of in-container thermal processes for foods. Crit Rev Food Sci Nutr 37(5):411–441

    Article  CAS  Google Scholar 

  51. Stumbo CR (1973) Thermobacteriology in food processing, 2nd edn. Academic Press, Inc., New York

    Google Scholar 

  52. Stumbo CR, Longley RE (1966) New parameters for process calculation. Food Technol 20(3):341–345

    Google Scholar 

  53. Succar J, Hayakawa K (1982) Prediction of time correction factor for come-up heating of packaged liquid food. J Food Sci 47:614–618

    Article  Google Scholar 

  54. Teixeira AA, Dixon JR, Zahradnik JW, Zinsmeister GE (1969) Computer determination of spore survival distributions in thermally-processed conduction-heated foods. Food Technol 23(3):78–80

    Google Scholar 

  55. Teixeira AA, Balaban MO, Germer SPM, Sadahira MS, Teixeire Neto RO, Vitali AA (1999) Heat transfer model performance in simulation of process deviations. J Food Sci 64(3):488–493

    Article  CAS  Google Scholar 

  56. Townsend CT, Reed JM, McConnell J, Powers MJ, Esselen WB Jr, Somers II, Dwyer JJ, Ball CO (1949) Comparative heat penetration studies on jars and cans. Food Technol 3(6):213–226

    Google Scholar 

  57. Uno J, Hayakawa K (1980) Correction factor of come-up heating based on critical point in a cylindrical can of heat conduction food. J Food Sci 45(4):853–859

    Article  Google Scholar 

  58. Uno J, Hayakawa K (1981) Correction factor of come-up heating based on mass average survivor concentration in a cylindrical can of heat conduction food. J Food Sci 46:1484–1497

    Article  Google Scholar 

  59. van Boekel MAJS (2002) On the use of the Weibull model to describe thermal inactivation of microbial vegetative cells. Int J Food Microbiol 74:139–159

    Article  Google Scholar 

  60. Vinters JE, Patel RH, Halaby GA (1975) Thermal process evaluation by programmable computer calculator. Food Technol 29(3):42,44,46,48

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  1. Department of Food Science and Technology, Agricultural University of Athens, Iera Odos 75, 11855, Athens, Greece

    Nikolaos G. Stoforos

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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). https://doi.org/10.1007/s12393-010-9014-4

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