Pham's Method for Thermal Process Evaluation

The Pham Method Calculator is an Excel-based implementation of Pham's formula method for thermal process evaluation of conduction-heated canned foods (Pham, 1987), including its extension to different heating and cooling rates (Pham, 1990).

The calculator can determine, at the container center or for the whole container:

•       the process lethality (F) delivered by a specified process time, or

•       the process time (B_b) required to achieve a specified process lethality.

Method Basis

Pham's method expresses the parameters governing product heating, process conditions, and microbial thermal resistance using the following dimensionless variables:

·      Sterilizing value W = U/f_h, the reciprocal of the f_h/U parameter used in the Ball and Stumbo methods

·      Normalized temperature approach at the end of heating, N₃ = g/z

·      Process-temperature ratios N₁ = z/(T_h − T_i) and N₂ = z/(T_h − T_c)

Here U is the sterilizing value at retort temperature, g is the difference between retort and product temperature at the end of heating, and T_h, T_i, and T_c are the retort, initial product, and cooling-water temperatures.  Algebraic equations relate these quantities to the heating (j_h) and cooling (j_c) lag factors. The effect of initial product and cooling-water temperatures on the lethality is included directly through N₁ and N₂.

Two sets of equations are used: one for high sterilizing values (W ≥ 1), in which the heating and cooling contributions are calculated separately, and one for low sterilizing values (W < 1), in which a single correlation relates g/z to W and j_c.

Pham (1990) showed that the discrepancy between the low- and high-W equations is negligible over 1.0 ≤ W ≤ 1.5, and that any value within this range may be used as the changeover value. Within this interval, the calculator linearly blends the results from the two equations to provide a continuous transition. This blending is an implementation choice and is not part of Pham’s published method.

When f_c differs from f_h, the calculator applies Pham's (1990) correction, weighting the heating and cooling contributions by f_h and f_c, respectively.

Whole-container sterilization is calculated as a mass-average sterilizing value. The local sterilizing value is determined at the container center and at a representative point where g and the heating and cooling lag factors are one-half their center values. These two sterilizing values are then combined using the mass-average calculation described by Jen et al. (1971).

Although Pham’s equations are dimensionless, this calculator requires temperatures and z values to be entered in °F because its input validation limits are defined for °F units.

Pham’s method assumes pure conduction heat transfer in a finite cylindrical container whose surface reaches the heating or cooling medium temperature instantly. Pham reported the following ranges for the low-sterilizing-value correlations:

·      0.04\le W\le1.0

·      j_c ≥ 0.4

·      0.05 ≤ N₁ ≤ 1

·      0.05 ≤ N₂/N₁ ≤ 1

No corresponding numerical ranges were specified for the high-sterilizing-value equations.

Heat-Penetration Parameters

The heating and cooling parameters are taken from asymptotes fitted to the linear portions of the semilogarithmic heating and cooling curves.

Intended Use

This calculator is provided for educational, research, and technical purposes. It implements Pham's method and should be interpreted within the assumptions and limitations of that method.

It is not intended to replace the establishment, validation, or review of commercial thermal processes by a qualified thermal process authority. Users are responsible for determining the suitability of the calculations for their intended application and for compliance with applicable food-safety and regulatory requirements.

Macro Requirement

This calculator uses VBA macros to perform the thermal process calculations. Macros must be enabled for the calculator to function. Depending on Microsoft Excel security settings, users may need to unblock the downloaded workbook or designate its location as trusted before macros can be enabled.

Timothy A. Haley, Ph.D.

Department of Food Science

University of Tennessee

References

Jen, Y., Manson, J. E., Stumbo, C. R., & Zahradnik, J. W. (1971). A procedure for estimating sterilization of and quality factor degradation in thermally processed foods. Journal of Food Science, 36, 692–698.

Pham, Q. T. (1987). Calculation of thermal process lethality for conduction-heated canned foods. Journal of Food Science, 52, 967–974.

Pham, Q. T. (1990). Lethality calculation for thermal processes with different heating and cooling rates. International Journal of Food Science and Technology, 25, 148–156.

Custom Applications

For modifications tailored to a particular application, contact Timothy A. Haley, Ph.D. at THaley2@utk.edu.

Software © 2026 Timothy A. Haley, Ph.D.