Ball’s Formula Method Revisited
Ricardo J. Simpson1,2, Sergio F. Almonacid1,2, Marisol M. Sanchez1, Helena Nuñez1, and Arthur A. Teixeira3
1
Departamento de Ingeniería Química y Ambiental; Universidad Técnica Federico Santa María;
Valparaíso, Chile, [email protected]
2
Centro Regional de Estudios en Alimentos Saludables, Blanco 1623, Room 1402, Valparaíso, Chile.
3
Department of Agricultural and Biological Engineering; Frazier Rogers Hall, P. O. Box 110570;
University of Florida, Gainesville, Florida 32611-0570, USA.
ABSTRACT
This manuscript describes a review and analysis of the correction factor for come-up time (CUT),
introduced by Dr. C. Olin Ball in his famous Formula Method for thermal process calculations. This
correction factor has commonly been considered applicable only to the Ball Formula method. In the
alternative General Method, the effect of CUT is automatically included in the calculated lethality value as
long as numerical integration is carried out over the entire cold spot temperature-time profile from the point
when steam is turned on.
The hypothesis of this communication is that Ball’s formula method, just like the General Method,
also includes the effect of CUT in its calculations, regardless of where the zero time line is placed within the
come-up time.
Several computer simulation studies were carried out with the zero time shifted to different locations
within the come-up time, resulting in the calculated heating lethality determined by Ball’s method being
almost the same as that generated by the General Method. This work confirmed that it is not necessary to
shift the zero time in Ball’s formula method because the calculations will always reflect the effect of CUT
regardless of where the zero time is chosen.
Key words: Ball’s Formula Method, correction factor, come-up time effectiveness, thermal processing,
process calculations techniques.
INTRODUCTION
Thermal processing is an important method of food preservation in the manufacture of shelf stable
canned food. The basic function of a thermal process is to inactivate food spoilage microorganisms in sealed
containers of food using heat treatments at temperatures well above the ambient boiling point of water in
pressurized steam retorts (autoclaves).
The first procedure to calculate thermal processes was developed by W.D. Bigelow in the early part of
the 20th century, and is usually known as the General Method [1]. The General Method makes direct use of the
time-temperature history at the coldest point within a sealed food container to obtain the lethality value of a
thermal process. The lack of programmable calculators or personal computers until the latter part of the 20th
century made this method very time-consuming, tedious and impractical for most routine applications; and it
soon gave way to formula methods offering shortcuts. In response to this need, a semi-analytic method for
thermal process calculation was developed and proposed to the scientific community by [2]. This is the wellknown Ball Formula Method, and works in a different way from the General Method. It makes use of the fact
that the difference between retort and cold spot temperature decays exponentially over process time after an
initial lag period. Therefore, a semi-logarithmic plot of this temperature difference over time (beyond the initial
lag) appears as a straight line that can be described mathematically by a simple formula, and is related to lethality
requirements by a set of tables that must be used in conjunction with the formula.
However, several assumptions are made that cause the method to become less accurate in many
situations. According to [3], most Formula Methods have been applied to metallic cans or glass jars that can be
processed in pure steam or water-cook retorts with rapid come-up-times. The recent development of retortable
flexible pouches and semi-rigid bowls and trays has made it necessary to re-examine process calculation
methods. These packages are often processed with steam-air mixtures, and often require relatively slow come-up
times, which can introduce additional error with use of formula methods.
The hypothesis of this communication is that Ball’s Formula Method, just like the General Method,
also includes the effect of CUT in its calculations, regardless of where the zero time line is placed within the
come-up time. Then, there is no need for a correction factor.
The objectives are to offer a critical analysis of the correction factor for come-up time (CUT)
introduced by Dr. C. Olin Ball in his famous Formula Method, and show that operator’s process time (Pt) is
always the same, regardless of how much come-up-time is taken into account.
METHODOLOGY
Focus of the analysis
The F-value of a given thermal process (lethality) is the sum of lethality achieved during heating and
additional lethality delivered during cooling; it can be expressed as follows:
The focus of the analysis will be to evaluate the accuracy of Ball’s method in calculating only the
lethality during heating (FHeating) and its subsequent prediction of final cold spot temperature reached at end
of heating (Tg).
tg
FPr ocess   10
0
T Tref
z
t
dt   10
T Tref
z
dt
(1)
tg
FPr ocess  FHeating  FCooling
Ball’s Formula Method
Ball's formula method for calculating the process time at a given retort temperature is based on a
mathematical equation for the straight-line portion of the temperature-time profile at the can cold spot when
plotted on inverted semi-log graph paper. This method of data transformation is a straightforward
mathematical technique and allows Ball's formula to take on a simple expression that obeys standard heat
conduction and convection theory within certain constraints.
As was shown by [4], Ball's expression is valid not only for finite cylinders, but also for arbitrary
shapes (rectangular, oval shape, etc.). The main limitations are that for conduction heating foods, it is only
valid for heating times beyond the initial lag period (when the Fourier number > 0.6). Then, equation (1)
becomes useful when the heating rate (fh) and heating lag (jh) parameters are obtained experimentally.
Correction factor for come-up time according to the literature (CUT)
Given that Ball's expression considers that TRT (retort temperature) is instantly reached, the Ball
procedure introduced the famous correction factor (42% of the CUT) assuming that the contribution of CUT
in F-value was not taken into account. Figure 1 depicts graphically the classical way Dr. Ball incorporated
this CUT correction factor in his calculation of effective process time (B).
The process time at retort temperature, Pt, for a commercial operation is measured from the time
when the retort reaches processing temperature, TRT, to the time when the steam is turned off and the cooling
water is applied. However, significant time is often needed for the retort to reach processing temperature,
which makes a contribution to the total lethal effect; this is known as the “come-up time” or CUT (tc). [2]
determined a value of 0.42tc for this contribution to the lethal effect, making the effective process time (B)
equal to the sum of process time and 42% of come-up time:
B = Pt + 0.42tc
(2)
Figure 1. Graphical representation of correction factor utilized in Ball’s procedure
The factor of 42% is generally regarded as a conservative estimate and is really only applicable to
batch retorts with a linear heating profile. While the lethal effects of CUT at the product center of a container
are small for most canned food products, thin profile plastic packages processed under steam-air or water
spray could experience a more significant lethal effect from CUT. Merson and others (1978) mention that
one of the invalid assumptions is that the heating medium surrounding the can is suddenly raised to
processing temperature. As stated by several authors CUT effectiveness vary from 35–77%. Ramaswamy
(1993), using thin profile retort packages and two retort temperature profiles, one linear and the other
logarithmic, showed that the traditional 42% CUT was appropriate for the former, but for the latter the values
were twice as large. Apart from package thickness, other factors had only a small influence on the CUT. For
other types of retorts, initial conditions and venting procedures, abundant literature can be searched ([5], [6],
[7], [8], [9], [10]).
Correction factor under new perspective
The hypothesis of this communication is that Ball’s formula method, just like the General Method,
also includes the effect of CUT in its calculations, regardless of where the zero time line is placed within the
come-up time. Support for this argument is that Ball’s method requires a “curve fitting” of the data, so that
the linear regression of the straight-line segment of the heat penetration data fits the experimental data
independent of the location of the zero time. The real concern should be the accuracy of Ball's expression in
terms of “goodness of fit” during the heating stage.
Computer search
Heat penetration data were generated by computer software (C++) that executed an explicit finite
difference solution to the general heat conduction equation for a finite cylinder. The calculations for F-value,
according to the General Method, were performed at the geometric center using Simpson’s numerical
integration rule with a time interval of 30 s. In the case of Ball’s calculations, time-temperature data
generated by the software up to the end of heating were fitted with Ball's expression to obtain the heat
penetration parameters, fh and jh.
Accuracy of the Ball Method during the heating stage
Calculations of the lethality reached at the end of heating for different retort processes were carried
out with the Ball Formula method, and compared with those calculated by the General Method. A Cylindrical
container with an inner diameter of 83 [mm] and a height of 106 [mm] was used for computer experiments.
Three different shapes of CUT were analyzed: a) linear, b) convex and c) concave. In addition, two different
time-lengths were considered for a linear CUT: a) 5 [min] and b) 15 [min]. Finally, a process with timevarying (dynamic) retort temperature (TVT) was analyzed.
RESULTS AND DISCUSSIONS
As shown in Table 1, the Ball method was tested for its accuracy during the heating stage for
different CUT shapes and lengths. Three different types of CUT were considered: a) linear, b) concave and c)
convex. In addition, two time-lengths of CUT were analyzed. In each case in this study, the process time
predicted by the Ball formula method was essentially identical to that predicted by the General Method.
Table 1. Predictions of heating time by the Ball and General Methods for different CUT-shapes and CUT
times.
General Method
Case
Linear CUT
Concave CUT
Convex CUT
Non constant TRT
Linear CUT 5 min
Linear CUT 15 min
F min
B min
2,991
3,048
3,059
2,992
2,905
2,827
91
89
92
91
88
93
Ball Method
fh
55,404
55,398
55,41
55,42
55,407
55,401
jh
1,768
1,679
1,897
1,766
1,787
1,754
F min
B min
2,994
3,051
3,059
3,006
2,908
2,828
90,75
87,75
92,73
90,8
87,75
92,75
Tables 2, 3 and 4 show results for total heating time (Pt + tc) for different CUT contributions or zero
time location. In every case, independent of CUT shape (linear, concave or convex) or CUT contribution, the
total heating time was exactly the same, meaning that independent of the zero time location, the contribution
of CUT is taken into account in the F heating calculation. In addition, experiments for different container
sizes and geometries were carried out and showed the same results. Finally, experiments considering non
constant retort temperature with perturbations of ± 1°C were designed and evaluated, and showed the same
trends as before.
CUT effectiveness
This work has shown that Ball’s formula method can be as accurate as the General Method, and will
always take the come-up time completely into account. Thus, there should be no need for correction factors
or shifting of zero time. This is because the parameters in Ball's expression have been estimated from a
regression analysis (fitting a curve) of the experimental data through the straight line segment of the heat
penetration curve. The graphical location of those experimental data points is a direct result of the length and
shape of the temperature time profile during the come-up time. Thus, if the regression produces an adequate
goodness of fit, the F-value calculations will be as accurate as the General Method. This has been shown
repeatedly with both experimental temperature-time data as well as those generated by computer models.
None-the-less, prudence would dictate further testing of the goodness of fit of Ball's expression in new or
unusual cases, such as new packages (retort pouches, shallow trays) and/or new autoclaves with different
forms of heat exchange media and venting procedures.
Table 2. Prediction of total heating time (Pt + tc) using different correction factors for a linear CUT.
Linear CUT
CUT [min]
10
% of CUT
fh
jh
B [min]
Pt + tc [min]
100
55,404
2,250
98,85
98,85
70
55,404
1,986
95,85
98,85
42
55,404
1,768
93,05
98,85
20
55,404
1,613
90,84
98,84
0
55,404
1,484
88,84
98,84
Table 3. Prediction of total heating time (Pt + tc) using different correction factors for a concave CUT.
Concave CUT
CUT [min]
10
% de CUT
fh
jh
B [min]
Pt + tc [min]
100
55,398
2,137
97,6
97,6
70
55,398
1,887
94,6
97,6
42
55,398
1,680
91,8
97,6
20
55,398
1,533
89,6
97,6
0
55,398
1,411
87,6
97,6
Table 4. Prediction of total heating time (Pt + tc) using different correction factors for a convex CUT.
Convex CUT
CUT [min]
10
% de CUT
fh
jh
B [min]
Pt + tc [min]
100
55,410
2,414
100,55
100,55
70
55,410
2,131
97,55
100,55
42
55,410
1,897
94,75
100,55
20
55,410
1,731
92,55
100,55
0
55,410
1,593
90,55
100,55
CONCLUSIONS
Prediction of total heating time (Pt + tc) by the Ball Formula method was always the same regardless
of where time zero was chosen within the come-up time (or CUT contribution). The reason for this is that
linear regression of heat penetration data along the straight-line portion of the semi-log heat penetration curve
produces a mathematical expression (Ball formula) that predicts the same time-temperature history
independent of the zero time location. In addition, given that high correlations were obtained in all cases, the
calculation of the F-value from the regressed data (Ball procedure) was essentially identical to the F-value
calculated by the General Method, which is based directly on the experimental data points. Second, we
showed that temperature-time histories predicted by Ball's expression always has a high correlation (R2 over
0.99) with experimental data points that was independent of CUT shape and length, meaning that the F-value
at end of heating is well estimated. Thirdly, we also concluded that inaccuracies in the Ball method can be
attributed in almost 100% of cases to the cooling calculations. For example, the larger the containers size in a
slow conduction-heating food product the larger the error in F-value calculation. This is because the cooling
phase in large slow-heating food containers is more important in terms of F-value contribution.
Corollary
Independent of the correction factor established by Dr. C. Olin Ball, calculations carried out by the
Ball Formula method always take into account 100% of come-up time. Its accuracy on the heating stage,
normally high, depends on the accuracy of Ball's expression to fit experimental data.
REFERENCES
[1] Bigelow, W.D., Bohart, G.S., Richardson, A.C. and Ball, C.O. 1920. Heat penetration in processing canned foods.
Bull. No. 16-L Res. Lab. Natl. Canners Assn., Washington, D.C.
[2] Ball, C.O. 1923. Thermal processing time for canned foods. Bull. 7-1 (37), Natl. Res. council, Washington, D. C.
[3] Holdsworth, S.D. 1997. Thermal processing of packaged foods. Blackie Academic & Professional. London.
[4] Datta, A.K. 1990. On the theoretical basis of the asymptotic semi logarithmic heat penetration curves used in food
processing. J. Food Eng. 12: 177-190.
[5] Alstrand, D. V., & Benjamin, H. A. (1949). Thermal processing of canned foods in tin containers: Effect of retorting
procedures on sterilization values for canned foods. Food Res., 14, 253–257.
[6] Berry, M. R. 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.
[7] Ikegami, Y. (1974a). Effect of various factors in the come-up time on processing of canned foods with steam. Report
Tokyo Inst. Food Technol. Serial No. 11, 92–98 [in Japanese].
[8] Ikegami, Y. (1974b). Effect of `come-up' on processing canned food with steam. Canner's J., 53(1) : 79–84 [in
Japanese].
[9] Succar, J., & Hayakawa, K. (1982). Prediction of time correction factor for come-up heating of packaged liquid food.
J. Food Sci., 47(3), 614-618.
[10] 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: 1484–1487.
ACKNOWLEDGEMENTS
We kindly appreciate the contribution made by Dr. Alik Abakarov. Authors Ricardo Simpson and Sergio
Almonacid are grateful for the financial support provided by CONICYT through the FONDECYT project number
1090689.
NOMENCLATURE
B: Ball’s effective processing time
CUT: come-up time
Fo : sterilizing value at 121.1 °C
Fp : process sterilizing value
f: rate factor (related to slope of semi-log heat penetration curve)
f and f : heating and cooling rate factors (related to slope of semi-log heat penetration curve)
h
c
j : dimensionless lag factor
j and j : heating and cooling lag factors
h
c
Pt: operator process time (time that is measured from the time when the retort reaches processing temperature (TRT), until
the time when the steam is turned off).
TA: extrapolated initial can temperature obtained by linearizing entire heating curve of a can
Tg: temperature at the coldest point when cooling phase begins.
T: temperature
IT: initial temperature
TRT: retort temperature
Tref: reference temperature, 121.1 [ºC].
t: time
tc: come up time
tg: time in a thermal process corresponding to heat cutoff and initiation of cooling phase