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SOP: Calculation of Process Time by the Formula Method — Straight-Line Heating Curve

GIFSQ — Global Institute for Food Safety & Quality

Principle

Ball’s Formula Method calculates the process mathematically rather than graphically, and — unlike the graphical methods — can be applied to any can size or retort temperature, provided the TDT and the heat penetration rates approximate straight lines on semi-log paper. The heating curve on semi-log paper is defined by the lag factor j and the slope fₕ; the process time Bₚ is found from the fₕ/U:g tables via the temperature deficit g.

Materials required

  • Heat penetration data (cold-point temperatures during heating and cooling)
  • Semi-log paper, usually having 3 cycles
  • Table 25-20 of the book (g values from fₕ/U for various m + g and z)
  • Table 25-21 of the book (Fᵣ values for various retort temperatures and z)

Procedure — Plotting the heating curve

Plot the heat penetration data on the semi-log paper: plot the time in minutes on the linear scale and the temperature difference on the logarithmic scale — plot the temperatures as retort temperature minus can temperature (turn the log paper upside down, start numbering the temperature divisions from the top, and denote to the top line a temperature 1 °F less than the retort temperature)

↓

Plot the temperatures for the corresponding times, starting with the initial temperature; draw a straight line through the points, ignoring the lag period, and position the line not more than 1 °F from any point above 210 °F for low-acid products and 140 °F for acid products

↓

Plot the cooling curve on a separate sheet (do not turn the semi-log paper upside down for cooling); mark the bottom line to represent 1 °F above the cooling water temperature

Procedure — Determining the heat penetration factors

Locate the corrected zero of the process: Corrected zero = Retort come-up time × 0.58 (this allows the heating value of the come-up period, conventionally taken as 0.42 of the come-up time — e.g., 10 min come-up gives a corrected zero 5.8 min from the time steam is turned on)

↓

Determine the lag factor j from the expression: j = (Retort temp. − Theoretical initial temp.) / (Retort temp. − Actual initial temp.), where the theoretical initial temperature is found by extending the straight-line portion of the heating curve to the corrected zero, and the actual initial temperature is the food temperature when steam begins to enter the retort

↓

Calculate I = RT − IT (retort temperature minus actual initial temperature) and jI — the intersection of the straight-line portion with the corrected zero

↓

Read the slope fₕ (minutes for the straight-line portion to traverse one log cycle) from the plotted heating curve; read the cooling-curve slope f꜀ similarly

Procedure — Calculation of process time (Bₚ)

Find Fᵣ from Table 25-21 (or from Fᵣ = antilog((250 − RT)/z)); calculate U = F × Fᵣ

↓

Calculate fₕ/U; find the corresponding g from Table 25-20

↓

Calculate the process time from the corrected zero to the end of the heating period:

Bₚ = fₕ(log jI − log g)

Calculation — Conversion of heat penetration data to other retort or initial temperatures

(i) When the initial temperature remains the same but the retort temperature is changed:

  (CT₁ − RT₁)       (CT − RT)
  ———————————— = ———————————————
  (RT₁ − IT)       (RT − IT)

Note: the printed layout of this equation is garbled in the available copy; the form above is reconstructed from the book’s symbol definitions, and should be checked against the printed book.

(ii) When the retort temperature remains the same but the initial temperature is changed:

            (RT − IT₁)
  CT₁ = RT − ——————————— × (RT − CT)
            (RT − IT)

Note: the printed layout of this equation is garbled in the available copy; the form above is reconstructed from the book’s symbol definitions, and should be checked against the printed book.

Where,

  • RT = original retort temperature; RT₁ = new retort temperature
  • IT = initial temperature; IT₁ = new initial temperature
  • CT = a can temperature of the original set; CT₁ = the new can temperature corresponding to CT

If both the retort temperature and the initial temperature are to be changed, apply the two formulae successively in either order. These equations hold for conduction as well as convection heating products. It is convenient to use a CT value of about 200 °F to determine the second point, since CT₁ plotted against CT gives a straight line.

Worked examples — Calculation of process time (Bₚ), simple heating curve

Problem 1: RT = 250 °F (note: the book’s data table prints “230 °F” at this entry, but the calculation row uses I = 250 − 160 = 90 and Fᵣ = 1 from Table 25-21, and the identical parameter set appears in nomogram Problem 6 with RT = 250 °F — the “230” is carried as printed but is an OCR misread of “250”), IT = 160 °F, j = 2, fₕ = 60 min, F = 12.1, z = 18, m + g = 130.

I = RT − IT = 250 − 160 = 90; jI = 2 × 90 = 180; log jI = 2.2553; Fᵣ = 1 (from Table 25-21); U = F × Fᵣ = 12.1 × 1 = 12.1; fₕ/U = 60/12.1 = 4.96; g corresponding to fₕ/U (from Table 25-20) = 5.1126; log g = 0.7086; Bₚ = fₕ(log jI − log g) = 60(2.2553 − 0.7086) = 92.8 min.

Problem 2: RT = 240 °F, IT = 160 °F, j = 1.25, fₕ = 40 min, F = 3, z = 18, m + g = 180.

I = 240 − 160 = 80; jI = 1.25 × 80 = 100; log jI = 2; Fᵣ = 3.594 (from Table 25-21); U = 3 × 3.594 = 10.782; fₕ/U = 40/10.782 = 3.71; g = 3.8652 (from Table 25-20); log g = 0.5872; Bₚ = 40(2 − 0.5872) = 56.5 min.

Worked example — Calculation of sterilizing value (F), simple heating curve

Problem 3: RT = 240 °F, IT = 160 °F, j = 1.25, fₕ = 40 min, z = 18, Bₚ = 56.5 min, m + g = 180.

I = 240 − 160 = 80; jI = 1.25 × 80 = 100; log g = log jI − Bₚ/fₕ = 2 − 56.5/40 = 0.5875; g = 3.8681; fₕ/U corresponding to g (from Table 25-20) = 3.7128; Fᵣ = 3.594 (from Table 25-21); F = U/Fᵣ where U = fₕ/(fₕ/U) = 40/3.7128 = 10.774, giving F = 10.774/3.594 = 3.0.

Note: the printed layout of Problem 3’s calculation rows is garbled in the available copy; the steps above are reconstructed from the legible values and verified by inversion against Problem 2 (Bₚ = 56.5 min at the same parameters gives F = 3.0, matching Problem 2’s F = 3).

Approximation for low g values

For processes where the z value is below 0.5 and the g value is below 0.1, Stumbo gives the following close approximation:

Bₚ = U + fₕ(log jI − 0.85)

Precautions

  • When heating is by convection, the semi-log plots of heat penetration data do not always fall on a straight line — excluding the initial lag period, apply the analysis of least squares to plot the data and find fₕ and f꜀.
  • The American Can Co. simplification of Ball’s method introduces small errors, not of a magnitude to influence the results appreciably.
  • In Ball’s development, fₕ was assumed equal to f꜀ and j to 1.41, which may not hold; Stumbo and co-workers revised the tables (the fₕ/U:g tables of Purohit and Stumbo give values very near to the graphical method for convection-heating tropical fruits processed in boiling water).

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.

Ball, C.O., Ind. Eng. Chem., 35, 71 (1943); Ball, C.O. & F.C.W. Olson, Sterilization in Food Technology, McGraw-Hill Book Co., New York (1957); Calculation of Process for Canned Foods, American Can Co., Barrington, Illinois, USA (1967); Steele, R.J., P.W. Board, D.J. Best & M.E. Willcox, J. Food Sci., 44, 954 (1979); Purohit & Stumbo — cited in the book.