1. 1
Ball Formula & General Methods
Historical Perspectives and Proper Use
Nikolaos G. Stoforos
Agricultural University of Athens, Department of Food Science and Human Nutrition,
Laboratory of Food Engineering, Iera Odos 75, 118 55 Athens, GREECE, stoforos@aua.gr
39th IFTPS Annual Conference: March 3 – 5, 2020
2. 2
Thermal Process Design
Heat Penetration
Data
Quality Factors
Kinetics
Estimate a Thermal
Process
Calculate Quality
Retention
Optimize
Microbiological Validation
Microbiological
Data
3. 3
Microbiological Data
Thermodacteriological approach:
Decimal reduction time, D value (min)
Temperature coefficient, z value (°F, °C)
and
Chemical reaction kinetics:
Reaction rate, k (s-1)
Activation energy, EA (J/mol)
Thermal inactivation kinetics
4. 4
Decimal Reduction Time, D value
10
100
1 000
10 000
100 000
0 2 4 6 8 10 12 14 16
Number
of
spores
per
unit
volume
.
Heating time (min)
Thermal Death Rate Curve (Survivor Curve)
T=115°C
T=121°C
D115 C = 5.6 min
D
121
C
=1.3
min
5. 5
1
10
100
1000
90 95 100 105 110 115 120 125
D
T
value
(min).
Temperature (°C)
Phantom Thermal Death Time Curvre
z =10.0 C
z value
6. 6
Microbiological Data
και
Chemical reaction kinetics:
Thermodacteriological approach:
T
D
t
o
N
N
10
z
T
T
T
T
ref
ref
D
D
/
)
(
10
and
k
ref
k
k
ref
k
g
a
ref
k
k
T
T
T
T
R
E
T
T k
k
)
(
303
.
2
10
t
k
o
T
e
C
C
8. 8
Equivalent Processes –variable temperature
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180
Time (min)
Temperature
(°C)
Medium
Product
9. 9
Equivalent Processes –variable temperature
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160 180
Time (min)
Temperature
(°C)
Medium
Product
Typical temperature profile at the geometric center of a conduction heated
canned product during the heating and cooling cycles of a thermal process.
The approximation of product temperature, used for the F value derivation, is
illustrated at two time steps with the rectangular drawings.
10. 10
Equivalent Processes –variable temperature
)
log(
)
log(
10
)
log(
)
log(
10
)
log(
)
log(
10
)
log(
)
log(
10
)
log(
)
log(
10
1
1
2
1
1
1
2
1
1
2
1
2
2
1
1
1
1
1
1
2
1
n
n
Tref
n
z
T
T
n
z
T
n
n
Tref
n
z
T
T
n
z
T
i
i
Tref
i
z
T
T
i
z
T
Tref
z
T
T
z
T
o
Tref
z
T
T
z
T
N
N
D
t
F
N
N
D
t
F
N
N
D
t
F
N
N
D
t
F
N
N
D
t
F
ref
n
ref
ref
n
ref
ref
i
ref
ref
ref
ref
ref
11. 11
Equivalent Processes –variable temperature
Adding in parts the equations in the previous set, for equal
spaced time intervals, we end up with:
)
log(
)
log(
10
1
1
n
o
Tref
n
i
z
T
T
n
i i
z
T
N
N
D
t
F
ref
i
ref
12. 12
Equivalent Processes –variable temperature
The first summation is equal to the F value of the entire
process. Taking the limit as i goes to infinity, the second
summation of the equation becomes a definite integral:
p ref
ref
i t
t
t
z
T
t
T
n
i
z
T
T
dt
t
0
)
(
1
10
10
lim
n
for tp being the total processing time.
13. 13
Equivalent Processes –variable temperature
Using N instead of Nn, the previous equation reduces to its
final form:
)
log(
)
log(
10
0
)
(
N
N
D
dt
F o
T
t
t
t
z
T
t
T
z
T ref
p ref
ref
14. 14
F value
)
log(
))
(
log( g
T
T
j
f
B IT
RT
h
h
)
log(
)
log(
10
0
)
(
N
N
D
dt
F o
T
t
t
t
z
T
t
T
z
T ref
p ref
ref
The equivalent processing time of a hypothetical thermal
process at a constant, temperature that produces the same
effect (in terms of spore or any heat labile substance
destruction) as the actual thermal process (min),
or equivalently
Time at constant temperature, required to reduce, to a given
value, the level of an attribute of a heat labile substance or
device whose thermal resistance is characterized by z (min)
15. 15
Equivalent Processes for Fo = 10 min
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140 160 180 200 220
Time (min)
Temperature
(°C)
16. 16
Thermal Process Design
• Calculate the heating time, B, (steam on to steam off
time) for achieving a required F value
end
c
c
ref
h
h
ref
z
ref
t
t
t
c
z
T
T
B
t
t
h
z
T
T
T dt
dt
F
0
0
10
10
)
parametres
,
(B
f
F z
Tref
)
parametres
,
( z
Tref
F
f
B
• Calculate the F value of a given process
17. 17
Thermal Process Design
1. General or Graphical Method (Bigelow et al., 1920)
2. Ball’s Formula Method (Ball, 1923)
19. 19
General or Graphical Method
Graphical (or numerical) calculation of the integral
i
z
T
T
t
t
z
T
t
T
process
z
T
ref
i
b
a
ref
ref
t
dt
F
/
)
(
/
)
)
(
(
10
10
As far as process F value calculation is concerned:
• Accurate method (accuracy of numerical integration)
• No assumptions about product temperature are made
• Reference method (for all subsequently proposed methods)
20. 20
General or Graphical Method
Weakness of the general method lies in the calculation of
the required heating time to achieve a target F value
General method suggests a “geometric similarity,” that is, it
assumes that the shape of the integrand (1/TDT) versus
time curve during cooling is geometrically the same,
irrespective of the total heating time
21. 21
General or Graphical Method
0,000
0,010
0,020
0,030
0,040
0,050
0,060
0,070
0 10 20 30 40
Χρόνος Κατεργασίας (min)
1/TDT
(min
-1
)
Steam off: 34 min; Area: 624.8 squares
Steam off: 30 min; Area: 520.7 squares
Steam off: 28 min; Area: 468.6 squares
Unit Sterilization Area (USA): 500 squares
USA=1/(1 min/dx × 0.010 min-1
/5 dy) =500 dx dy
Lethality of USA: 20 min x 0.050 min-1
= 1
USA
Processing time (min)
22. 22
General or Graphical Method
0,000
0,010
0,020
0,030
0,040
0,050
0,060
0 10 20 30 40 50 60 70 80 90 100
Χρόνος Κατεργασίας (min)
1/TDT
(min
-1
)
Steam off: 60 min; Area: 663.7 squares
Steam off: 54 min; Area: 518.8 squares
Steam off: 51 min; Area: 453.1 squares
Unit Sterilization Area (USA): 500 squares
USA=1/(1 min/dx × 0.010 min-1
/5 dy) =500 dx dy
Lethality of USA: 20 min x 0.050 min
-1
= 1
USA
Processing time (min)
23. 23
Ball’s Formula Method
Due to the limitations of the general method, a number of
thermal process calculation methodologies appeared in the
literature.
The common characteristic of these methodologies, termed
formula methods, was the incorporation of an equation, a
formula, to relate product temperature with time, so that one
could transform time–temperature data to different
conditions.
24. 24
Charles Olin Ball (1893-1970)
Charles Olin Ball was a pioneer in this
approach (1893–1970); while in graduate
school at George Washington University
(1919–1922), he did research on
sterilization of canned foods for the
National Canners Association (Valigra,
2011).
Soon after the introduction of the general
method, Ball developed his method (Ball,
1923), which became the most widely used
method in the United States for establishing
thermal processes and the basis for all
subsequently developed formula methods.
25. 25
Charles Olin Ball (1893-1970)
A pioneer in Thermal Death-
Time calculations, C. Olin Ball
used mathematical formulae to
determine how much heat and
time are needed to kill bacteria
and keep food safe (1920,
1923).
The formulae for thermal death
time, which became an FDA
standard for calculating thermal
processes, is still in use today.
Food Quality, June/July, 2011
26. 26
Ball’s Formula Method
In the Foreword of the Ball and Olson (1957) classical book
written by L.V. Burton, he stated:
“… The ideas and the concepts, when originally proposed by the
authors, were far ahead of what their contemporary food
technologists were willing to accept. Skepticism, based on the
belief that biological phenomena were not susceptible to rigorous
mathematical treatment, lasted for about 10 years, but in the
ensuing decade there came about a slow acceptance of this
system of calculating and evaluating thermal processes. It was
because of this skepticism that Dr. Ball’s original publication in
1923 was a National Research Council Monograph instead of a
official bulleting from the industry that utilizes thermal sterilization
of foods. And even its second publication was by the University of
California Press for a similar reason.”
28. 28
Ball’s Formula Method
Heat Penetration Data
Ball used two empirical parameters, the fh και jh values, to
describe product temperature evolution during the heating
cycle of a thermal process.
29. 29
Heating curve
)
lo g(
))
(
lo g( T
T
T
T
j
f
t R T
IT
R T
h
h
119
118
117
116
115
114
113
112
111
110
100
90
80
70
60
50
40
30
20
-80
-180
-280
-380
-480
-580
-680
-780
-880
TIT I=TRT-TIT
TA jhI=TRT-TA
1
10
100
1000
0 20 40 60 80 100 120
Retort
-
Product
Temperature,
T
RT
-T
(°C)
Product
Temperature,
T
(°C)
Time (min)
Heating Curve
Straight line fitting (to the "linear"
portion of the experimental data)
Experimental Data
fh=60 min
89
.
1
100
189
IT
RT
A
RT
h
T
T
T
T
j
30. 30
t
T
α
x
T
1
2
2
p
C
ρ
k
α
c o n s ta n t
)
0
,
(
IT
T
t
x
T
0
0
x
x
T
*
2
)
cos(
cos
sin
sin
2
)
,
( *
1
*
*
t
β
m
m m
m
m
m
IT
RT
RT m
e
x
β
β
β
β
β
T
T
t
x
T
T
2
)
2
/
(
,
2
/
,
)
2
/
(
,
cot
of
root
:
L
t
t
L
x
x
k
L
h
Bi
Bi
n *
*
m
m
th
m
)
,
2
/
(
2
/
t
L
x
T
T
h
x
T
k RT
L
x
Heat transfer by conduction
37. 37
Cooling curve
For the straight line portion of
the cooling curve, use of fc
and jc parameters
c
c f
t
C W
g
c
C W T
T
j
T
T
/
)
(
1 0
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
110
120
220
320
420
520
620
720
820
920
1020
Tg
Tg-TCW
TB TB-TCW
1
10
100
1000
0 20 40 60 80 100 120
Product
-
Cooling
Water
Temperature,
T-T
CW
(°C)
Product
Temperature,
T
(°C)
Time (min)
Cooling Curve
Straight line fitting (to the "linear"
portion of the experimental data)
Experimental Data
fc=60 min
54
.
1
93
.
95
73
.
147
CW
g
CW
B
c
T
T
T
T
j
41. 41
du
u
e
x
x u
)
Ei( du
z
m
u
e
z
m
z
m
u
)
/
)
10
ln(
3
.
0
(
E
/
)
10
ln(
643
.
0
/
)
10
ln(
3
.
0
2
2
z
T
T RT
ref
e
F
U
/
)
)(
10
ln(
1
)
)
80
)(
10
ln(
Ei(
-
)
)
10
ln(
657
.
0
Ei(
E
)
10
ln(
5833
.
0
0.33172
)
)
10
ln(
Ei(
-
)
80
)
10
ln(
Ei(
)
10
ln(
1
/
)
10
ln(
/
)
10
ln(
300
.
0
/
)
10
ln(
343
.
0
/
)
10
ln(
/
)
10
ln(
z
g
m
z
m
e
e
m
z
e
z
g
-
z
-
e
e
U
f
z
m
z
m
z
m
z
g
z
g
h
Ball’s Formula Method
42. 42
Ball’s Formula Method
Assumptions:
• No temperature rise after steam-off
• jc = 1.41
• fc = fh (he proposed a correction)
• Constant retort temperature (he suggested to correct the
heating time, by shifting the zero heating time axis by
0.58CUT and apply his method, as for a process with
CUT = 0, based on this corrected ‘‘zero’’ axis)
• Start lethality accumulation when product temperature
reaches 80°F below TRT
• Straight-line heating curve (he proposed a correction for
“broken” heating curves)
43. 43
)
,
,
( g
m
z
g
f
U
fh
)
lo g(
))
(
lo g( g
T
T
j
f
B IT
R T
h
h
)
parametres
,
(B
f
F z
Tref
)
parametres
,
( z
Tref
F
f
B
Ball’s Formula Method
46. 46
Ball’s Formula Method
Ball’s method was questioned due to “discrepancies” between
published equations and tabulated values. However, the validity
of the method was reestablished (Larkin, 1990; Stoforos, 1991).
The fundamental equation was not the same in the 1923 and
1957 publications. In 1923 both z and fh values were based on
Napierian logarithms
Flambert et al. (1977) (logarithmic cooling was not
appropriately represented in the equation -exponential
integral definition)
Steele et al. (1979) (revised tables)
47. 47
Determine heating time for the following case:
fh=60 min
jc=2
TRT=250ºF
TIT=180ºF
TCW=70ºF
z=24ºF
F250°F,req=8.4 min
Example
48. 48
For
and m+g = TRT –TCW = 250 – 70 = 180ºF and
z=24ºF
we find log(g) =1 and substituting: Β= 68.8 min
Solution
)
lo g(
))
(
lo g( g
T
T
j
f
B IT
R T
h
h
50. 50
Use of Regression Equations
0.1
1
10
100
1000
-1 -0.5 0 0.5 1 1.5 2
f
h
/U
log(g) (g in °F)
z = 6°F
z = 10°F
z = 14°F
z = 18°F
z = 22°F
z = 26°F
0.1
1
10
100
1000
-2.5 -2 -1.5 -1 -0.5 0 0.5
f
h
/U
log(g/z)-z/zc
z = 6°F
z = 10°F
z = 14°F
z = 18°F
z = 22°F
z = 26°F
51. 51
du
u
e
x
x u
)
Ei( du
z
m
u
e
z
m
z
m
u
)
/
)
10
ln(
3
.
0
(
E
/
)
10
ln(
643
.
0
/
)
10
ln(
3
.
0
2
2
z
T
T RT
ref
e
F
U
/
)
)(
10
ln(
1
)
)
80
)(
10
ln(
Ei(
-
)
)
10
ln(
657
.
0
Ei(
E
)
10
ln(
5833
.
0
0.33172
)
)
10
ln(
Ei(
-
)
80
)
10
ln(
Ei(
)
10
ln(
1
/
)
10
ln(
/
)
10
ln(
300
.
0
/
)
10
ln(
343
.
0
/
)
10
ln(
/
)
10
ln(
z
g
m
z
m
e
e
m
z
e
z
g
-
z
-
e
e
U
f
z
m
z
m
z
m
z
g
z
g
h
Ball’s Formula Method
52. 52
Regression equations
7
5
4
2
1
6
3 1
1
a
e
a
a
e
a
a
y
x
a
x
a
(1)
Table 1. Values of the coefficients of Eq. (1) according to the definitions of the x and y variables.
m + g = 130°F m + g = 180°F
y log(g/z)-z/zc log(fh/U) log(g/z)-z/zc log(fh/U)
x log(fh/U) log(g/z)-z/zc log(fh/U) log(g/z)-z/zc
a1 -0.088335831 40.122199 -3.3545727 22.016510
a2 -0.96375429 38.533071 -0.34453049 21.598294
a3 0.028257272 2.3715954 0.42100067 2.4586869
a4 1.0711536 5.3058320 4.0057210 38.202986
a5 0.19518983 2.8885491 0.13211471 23.706331
a6 4.5699218 0.63534158 3.2971998 0.49435142
a7 - -0.63814873 - -0.74859566
zc (°F) 389.10600 405.49832 389.48491 468.11021
53. 53
0.1
1
10
100
1000
-1 -0.5 0 0.5 1 1.5 2
f
h
/U
log(g) (g in °F)
z = 6°F
z = 10°F
z = 14°F
z = 18°F
z = 22°F
z = 26°F
Comparison between
predicted, (lines) and
Ball’s tabulated fh/U vs
log(g) data (open cycles)
for m+g = 180°F.
54. 54
Comparison between predicted, (lines) and Ball’s tabulated
log(g) vs fh/U data (open cycles) for m+g = 130°F.
-1
-0.5
0
0.5
1
1.5
2
0.10 1.00 10.00 100.00 1000.00
log(g)
(g
in
°F)
fh/U
z = 26°F
z = 22°F
z = 18°F
z = 14°F
z = 10°F
z = 6°F
55. 55
Comparison between predicted
(lines) and Ball’s tabulated r vs
log(g) data (open symbols) for
m+g = 130°F και m+g = 180°F.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 1 10 100
r
log(g) (g in °F)
6°F 14°F 18°F 26°F
6°F 14°F 18°F 26°F
56. 56
Comparison between Ball’s predictions through the developed
regression equations and Stumbo’s data for large z values
0.1
1
10
100
-1 0 1 2
f
h
/U
log(g)
Comparison of Ball Predictions and
Stumbo Data for z = 54°F
Ball m+g = 130°F
Ball m+g = 180°F
Stumbo jc = 0.40
Stumbo jc = 1.00
Stumbo jc = 1.40
Stumbo jc = 2.00
0.1
1
10
100
-1 0 1 2
f
h
/U
log(g)
Comparison of Ball Predictions and
Stumbo Data for z = 90°F
Ball m+g = 130°F
Ball m+g = 180°F
Stumbo jc = 0.40
Stumbo jc = 1.00
Stumbo jc = 1.40
Stumbo jc = 2.00
0.1
1
10
100
-1 0 1 2 3
f
h
/U
log(g)
Comparison of Ball Predictions and
Stumbo Data for z = 200°F
Ball m+g = 130°F
Ball m+g = 180°F
Stumbo jc = 0.40
Stumbo jc = 1.00
Stumbo jc = 1.40
Stumbo jc = 2.00
57. 57
User friendly software (in EXCEL)
Using the algebraic equations, developed to replace Ball’s
graphs and tables, a freely accessible user friendly software
(in EXCEL) was developed to facilitate the use of Ball’s
original formula method
Part of Ms. Lydia Katsini master’s thesis
58. 58
General method is the reference method for F values
calculations
Ball’s method served food industry to establish both
process and required F values
Limitations of Ball’s method must be understood
Comparisons among different methods should be made
based on the underlined assumptions of each method,
rather than comparisons based on particular heat
penetration data
Conclusions
