Principles of Thermal

Processing of
Packaged Foods

Ricardo Simpson
Universidad Técnica Federico Santa María, Chile

Helena Nuñez
Universidad Técnica Federico Santa María, Chile

Cristian Ramírez
Universidad Técnica Federico Santa María, Chile
https://​doi​.org/​10​.21061/​IntroBiosystemsEngineering/​Food​_Thermal​_Processing

How to cite this chapter:

Simpson, R., Nuñez, H., & Ramírez, C. (2020). Principles of Thermal Processing of Packaged Foods.
In Holden, N. M., Wolfe, M. L., Ogejo, J. A., & Cummins, E. J. (Ed.), Introduction to Biosystems Engineering.
https://​doi​.org/​10​.21061/​IntroBiosystemsEngineering/​Food​_Thermal​_Processing

This chapter is part of Introduction to Biosystems Engineering

International Standard Book Number (ISBN) (PDF): 978-­1-­949373-­97-­4
International Standard Book Number (ISBN) (Print): 978-­1-­949373-­93-­6
https://​doi​.org/​10​.21061/​IntroBiosystemsEngineering

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Principles of Thermal
Processing of
Packaged Foods
Ricardo Simpson Cristian Ramírez
Departamento de Ingeniería Química y Ambiental, Departamento de Ingeniería Química y Ambiental,
Universidad Técnica Federico Santa María, Valparaíso, Chile Universidad Técnica Federico Santa María, Valparaíso, Chile
Centro Regional de Estudios en Alimentos y Salud (CREAS) Centro Regional de Estudios en Alimentos y Salud (CREAS)
Conicyt-­Regional GORE Valparaíso Project R17A10001, Conicyt-­Regional GORE Valparaíso Project R17A10001,
Curauma, Valparaíso, Chile Curauma, Valparaíso, Chile

Helena Nuñez
Departamento de Ingeniería Química y Ambiental,
Universidad Técnica Federico Santa Maria, Valparaíso, Chile

KEY TERMS

Heat transfer Bacterial inactivation Food sterilization

Microorganism heat resistance Decimal reduction time Commercial sterilization

Variables
α = thermal diffusivity
ρ = density
Cp = specific heat
CUT = time required to come up to retort temperature
D = decimal reduction time
F0 = cumulative lethality of the process from time 0 to the end of the
process
I = inactivation
k = rate constant
Kt = thermal conductivity
N = number
t = time

1
 T = temperature
Tref = reference temperature
TRT = retort temperature
z = temperature change

Introduction
Thermal processing of foods, like cooking, involves heat and food. However, thermal
processing is applied to ensure food safety and not necessarily to cook the food.
Thermal processing as a means of preservation of uncooked food was invented in
France in 1795 by Nicholas Appert, a chef who was determined to win the prize
of 12,000 francs offered by Napoleon for a way to prevent military food supplies
from spoiling. Appert worked with Peter Durand to preserve meats and vegetables
encased in jars or tin cans under vacuum and sealed with pitch and, by 1804, opened
his first vacuum-­packing plant. This French military secret soon leaked out, but
it took more than 50 years for Louis Pasteur to provide the explanation for the
effectiveness of Appert’s method, when Pasteur was able to demonstrate that
the growth of microorganisms was the cause of food spoilage.
The preservation for storage by thermal treatment and removal of atmo-
sphere is known generically as canning, regardless of what container is used
to store the food. The basic principles of canning have not changed dramati-
cally since Appert and Durand developed the process: apply enough heat to
food to destroy or inactivate microorganisms, then pack the food into sealed or
“airtight” containers, ideally under vacuum. Canned foods have a shelf life
of one to four years at ordinary temperatures, making them convenient, afford-
able, and easy to transport.

Outcomes
After reading this chapter, you should be able to:

• Identify the role of heat transfer concepts in thermal processing of packaged foods

• Describe the principles of commercial sterilization of foods

• Describe the inactivation conditions needed for some example microorganisms important for food safety

• Define some sterilization criteria for specific foods

• Apply, in simple form, the main thermal food processing evaluation techniques

Concepts
The main concepts used in thermal processing of foods include: (a) heat transfer;
(b) heat resistance of microorganisms of concern; and (c) bacterial inactivation.

2 • Principles of Thermal Processing of Packaged Foods

Heat Transfer
The main heat transfer mechanisms involved in the thermal processing of pack-
aged foods are convection and conduction. Heat transfer by convection occurs
due to the motion and mixing of flows. The term natural convection refers to
the case when motion and mixing of flow is caused by density differences in
different locations due to temperature gradients. The term forced convection
refers to the case when motion and mixing of flow is produced by an outside
force, e.g., a fan. Heat transfer by conduction occurs when atoms and molecules
collide, transferring kinetic energy. Conceptually, atoms are bonded to their
neighbors, and if energy is supplied to one part of the solid, atoms will vibrate
and transfer their energy to their neighbors and so on.
The main heat transfer mechanisms involved in the thermal processing of
packaged foods are shown in figure 1. Although the figure shows a cylindrical can
(a cylinder of finite diameter and height), a similar situation will arise when pro-
cessing other types of packaging such as glass containers, retortable pouches,
and rigid and semi-­rigid plastic containers. In general, independent of shape,
food package sizes range from 0.1 L to 5 L (Holdsworth and Simpson, 2016).
The main mechanism of heat transfer from the heating medium (e.g., steam
or hot water) to the container or packaging is convection. Then heat transfers
by conduction through the wall of the container or package. Once inside the
container, heat transfer through the covering liquid occurs by convection, and
in solid foods mainly by conduction. In case of liquid foods, the main mecha-
nism is convection.
The rate of heat transfer in packaged foods depends on process factors,
product factors, and package types. Process factors include retort tempera-
ture profile, process time, heat transfer medium, and container agitation.

Figure 1. Main heat transfer mechanisms involved in the thermal processing of packaged foods.

Principles of Thermal Processing of Packaged Foods • 3

 Product factors include food composition, consistency, initial temperature,
initial spore load, thermal diffusivity, and pH. Factors related to package
type are container material, because the rate of heat transfer depends on
thermal conductivity and thickness of the material, and container shape,
because the surface area per unit volume plays a role in the heat penetra-
tion rate.
For liquid foods, the heating rate is determined not only by the thermal
diffusivity α, but also by the viscosity. The thermal diffusivity is a material
property that represents how fast the heat moves through the food and is
determined as:

α = Kt/(ρ Cp) (1)

where α = thermal diffusivity (m2/s)

Kt = thermal conductivity (W/m-­K)
ρ = density (kg/m3)
Cp = specific heat (W/s-­kg-­K)

It is extremely difficult to develop a theoretical model for the prediction of

a time-­temperature history within the packaging material. Therefore, from a
practical point of view, a satisfactory thermal process (i.e., time-­temperature
relationship) is usually determined using the slowest heating point, the cold
spot, inside the container.

Heat Resistance of Microorganisms of Concern

The main objective in the design of a sterilization process for foods is the inac-
tivation of the microorganisms that cause food poisoning and spoilage. In order
to design a safe sterilization process, the appropriate operating conditions
(time and temperature) must be determined to meet the pre-­established
sterilization criterion. To establish this
Table 1. Some typical microorganisms heat resistance data criterion, it is necessary to know the heat
(Holdsworth and Simpson, 2016). resistance of the microorganisms (some
examples are given in table 1), the ther-
Organism Conditions for Inactivation
mal properties of the food and packag-
Vegetative cells 10 min at 80°C ing, and the shape and dimensions of the
Yeast ascospores 5 min at 60°C packaged food. From these, it is possible
Fungi 30–­60 min at 88°C to determine the retort temperature and
Thermophilic organisms: holding time (that is, the conditions for
Bacillus stearothermophilus 4 min at 121.1°C inactivation), how long it will take to reach
Clostridium thermosaccharolyticum 3–­4 min at 121.1°C that temperature (the come-­up time), and
Mesophilic organisms: how long it will take to cool to about 40°C
(the cooling time) (Holdsworth and Simp-
Clostridium botulinum spores 3 min at 121.1°C
son, 2016).
Clostridium botulinum toxins Types A & B 0.1–­1 min at 121.1°C
The pH of the food is extremely relevant
Clostridium sporogenes 1.5 min at 121.1°C
to the selection of the sterilization process
Bacillus subtilis 0.6 min at 121.1°C parameters, i.e., retort temperature and

4 • Principles of Thermal Processing of Packaged Foods

holding time, because microorganisms grow better in a less acid environ-
ment. That is why the standard commercial sterilization process is based on
the most resistant microorganism (Clostridium botulinum) at the worst-­case
scenario conditions (higher pH) (Teixeira et al., 2006). The microorganism heat
resistance is greater in low-­acid products (pH ≥ 4.5–­4.6). On the other hand,
medium-­acid to acidic foods require a much gentler heat treatment (lower
temperature) to meet the sterilization criterion. Based on that, foods are clas-
sified into three groups:

• low-­acid products: pH > 4.5–­4.6 (e.g., seafood, meat, vegetables, dairy

products);
• medium-­acid products: 3.7 < pH < 4.6 (e.g., tomato paste);
• acidic products: pH < 3.7 (e.g., most fruits).

Bacterial Inactivation
Abundant scientific literature supports the application of first-­order kinetics to
quantify bacterial (spores) inactivation as (Esty and Meyer, 1922; Ball and Olson,
1957; Stumbo, 1973, Holdsworth and Simpson, 2016):

 dN 
   kN (2)
 dt  I
where N = viable bacterial (microbial) concentration (microorganisms/g) after
process time t
t = time
I = inactivation
k = bacterial inactivation rate constant (1/time)

Instead of k, food technologists have utilized the concept of decimal

reduction time, D, defined as the time to reduce bacterial concentration by
ten times. In other words, D is the required time at a specified tempera-
ture to inactivate 90% of the microorganism’s population. A mathematical
expression that relates the rate constant, k, from equation 2 to D is devel-
oped by separating variables and integrating the bacterial concentration
from the initial concentration, N0, to N0/10 and from time 0 to D, therefore
obtaining:

ln 10 2.303 (3)
k 
D D

or

ln 10 2.303 (4)
D 
k k

where k = reaction rate constant (1/min)

D = decimal reduction time (min)

Principles of Thermal Processing of Packaged Foods • 5

 A plot of the log of the survivors (log N) against D is called
a survivor curve (figure 2). The slope of the line through one
log cycle (decimal reduction) is −1/D and

t
log N  log N 0  (5)
D
where N = number of survivors
N0 = N at time zero, the start of the process

Temperature Dependence of the Decimal

Reduction Time, D
Every thermal process of a food product is a function of the
thermal resistance of the microorganism in question. When
the logarithm of the decimal reduction time, D, is plotted
against temperature, a straight line results. This plot is called
the thermal death time (TDT) curve (figure 3). From such a
plot, the thermal sensitivity of a microorganism, z, can be
determined as the temperature change necessary to vary
Figure 2. Semilogarithmic survivor curve. TDT by one log cycle.
Bigelow and co-­workers (Bigelow and Esty, 1920; Bigelow,
1921) were the first to coin the term thermal death rate to
relate the temperature dependence of D. Mathematically,
the following expression has been used:

T  Tref (6)
log D  log Dref 
z

or
Tref T
(7)
D  Dref 10 z

where D = decimal reduction time at temperature T (min)

Dref = decimal reduction time at reference temperature Tref (min)
   z = temperature change necessary to vary TDT by one log
cycle (°C), e.g., normally z = 10°C for Clostridium
botulinum
Figure 3. Thermal death time (TDT) curve.
T = temperature (°C)
Tref = reference temperature (normally 121.1°C for sterilization)

The D value is directly related to the thermal resistance of a given micro-

organism. The more resistant the microorganism to the heat treatment, the
higher the D value. On the other hand, the z value represents the temperature
dependency but has no relation to the thermal resistance of the target micro-
organism. Then, the larger the z value the less sensitive the given microorgan-
ism is to temperature changes. D values are expressed as DT. For example, D140
means the time required to reduce the microbial population by one log cycle
when the food is heated at 140°C.

6 • Principles of Thermal Processing of Packaged Foods

Food Sterilization Criterion and Calculation
Sterilization means the complete destruction or inactivation of microorgan-
isms. The food science and engineering community has accepted the utiliza-
tion of a first-­order kinetic for Clostridium botulinum inactivation (equation 2).
Again, this pathogen is the target microorganism in processes that use heat to
sterilize foods. Theoretically, the inactivation time needed to fully inactivate
Clostridium botulinum is infinite. According to equations 2 and 3 and assuming
a constant process temperature and that k is constant, the following expres-
sion is obtained:
ln10
 t
N f  N 0 e-kt  N 0 e D (8)

This equation shows that the final concentration of Clostridium botulinum

(Nf ) tends to zero when time (t) tends to infinity; therefore, it is not possible
to reach a final concentration equal to zero for the target microorganism.
Thus, it is necessary to define a sterilization criterion (or commercial steril-
ization criterion) to design a process that guarantees a safe product within
a finite time.
The level of microbial inactivation, defined by the microbial lethality value or
cumulative lethality, is the way in which the sterilization process is quantified.
Specifically, the sterilizing value, denoted by F0, is the required time at 121.1°C
to achieve 12 decimal reductions (12D). In other words, F0 is the time required to
reduce the initial microorganism concentration from N0 to N0/1012 at the pro- F can be calculated for
cess temperature of 121.1°C. a process temperature
The 12D sterilization criterion is an extreme process (i.e., overkill) designed other than 121.1°C.
to ensure no cells of C. botulinum remain in the food and, therefore, prevent
illness or death. According to the FDA (1972), the minimum thermal treat-
ment for a low-­acid food should reach a minimum F0 value of 3 min (that
is larger than 12D; D for C. botulinum at 121.1°C is 0.21 min, then 12 × 0.21
= 2.52 min, which is lower than 3 min). Thus, a thermal process for com-
mercial sterilization of a food product should have an F0 value greater than
3 minutes.
The F0 attained for a food can be calculated easily when the temperature at
the center of the food during the thermal processing is known by:
t T Tref

F0   10 z
dt (9)
0

where F0 = cumulative lethality of the process from time 0 to the end of the process (t)
T = temperature measured at the food cold spot, which is the place in the food that
heats last
Tref = temperature of microorganism reference; for sterilization of low-­acid foods,
Tref = 121.1°C for C. botulinum
z = temperature change necessary to reduce D value by ten times; in the case of
sterilization of low-­acid foods, z = 10°C for C. botulinum
t = process time to reach F0

Principles of Thermal Processing of Packaged Foods • 7

 Equation 9 can be calculated according to the general method proposed
by Bigelow and co-­workers 100 years ago (Bigelow et al., 1920; Simpson et al.,
2003).
If the food is heated instanta-
neously to 121.1°C and maintained
at this temperature for 3 min, then
the F0 value for this process will be
3 min. From equation 9,
t 121.1121.1 t 3

F0 = 10 10
dt = 100 dt= 1 dt
0 0 0

Since the time interval is between

0 to 3 min, then the integral solution
3

is 3 min or 1 dt = 3 –­0 = 3. However,

0

in practice, due to the resistance

of the food to the transfer of heat,

the thermal sterilization process
requires a longer time in order to
get a F0 ≥ 3 min, because a signifi-
cant part of the processing time is
needed to raise the cold-­spot tem-
perature of the food and later to
cool the food.

Applications
Commercial Sterilization
Process
A general, simplified flow diagram
for a typical commercial canning
factory is presented in figure 4.
Stage 1: Selecting and prepar-
ing the food as cleanly, rapidly, and
precisely as possible. Foods that
maintain their desirable color, fla-
vor, and texture through commer-
cial sterilization include broccoli,
corn, spinach, peas, green beans,
peaches, cherries, berries, sauces,
purees, jams and jellies, fruit and
vegetable juices, and some meats
(Featherstone, 2015). The prepara-
tion must be performed with great
Figure 4. Stages of a typical commercial food canning process. care and with the least amount of

8 • Principles of Thermal Processing of Packaged Foods

damage and loss to minimize the monetary cost of the operation. If foods
are not properly handled, the effectiveness of the sterilization treatment is
compromised.
Stage 2: Packing the product in hermetically sealable contain-
ers ( jars, cans, or pouches) and sealing under a vacuum to elimi-
nate residual air. A less common approach is to sterilize the food first
and then aseptically package it (aseptic processing and packaging of
foods).
Stage 3: Stabilizing the food by sterilizing through rigorous thermal pro-
cessing (i.e., high temperature to achieve the correct degree of sterilization or
the target destruction of the microorganisms present in the food), followed by
cooling of the product to a low temperature (about 40°C), at which enzymatic
and chemical reactions begin to slow down.
Stage 4: Storing at a temperature below 35°C, the temperature below which
food-­spoilage organisms cannot grow.
Stage 5: Labeling, secondary packaging, distribution, marketing, and con-
sumption. Although not part of the thermal process per se, this stage addresses
the steps required for commercialization of the treated foods.
Stage 3, thermal processing, is the focus of this chapter. The aim of the
thermal process is to inactivate, by the
effect of heat, spores and microorganisms
present in the unprocessed product. The
thermal process is performed in vessels
known as retorts or autoclaves to achieve
the required high temperatures (usually
above 100°C).
As depicted in figure 5, a typical ster-
ilization process has three main steps:
come-­up time, operator process time,
and cooling. The first step, the come-
­up time (CUT ), is the time required to
reach the specified retort temperature
( TRT ), i.e., the target temperature in
the retort. The second step is the hold-
ing time (Pt), also called operator pro-
cess time, which is the amount of time
that the retort temperature must be main-
Figure 5. Temperature profiles for a typical thermal process, where CUT is
tained to ensure the desired degree of
come-­up time and Pt is operator time.
lethality. This depends on the target
microorganism or the expected micro-
biological contamination. The final step is the cooling, when the temperature
of the product is decreased by introducing cold water into the retort. The
purpose of cooling the food is to minimize the excessive (heat) processing
of the food, and avoid the risk of thermophilic microorganism development. Thermophilic organisms
During the cooling cycle, it may be necessary to inject sterile air into the thrive on heat.
food packaging to avoid sudden internal pressure drops and prevent pack-
age deformation.

Principles of Thermal Processing of Packaged Foods • 9

 The concepts described in this chapter describe the key principles for
applying a thermal process to packaged food to achieve the required lethality
for food safety. These concepts can be used to design a thermal process to
ensure adequate processing time and food safety while avoiding over pro-
cessing the packaged food. This should ensure safe, tasty, and nutritious
packaged foods.

Examples
Example 1: Calculation of microbial count after a given
thermal process
Problem:
The D120°C value for a microorganism is 3 minutes. If the initial microbial con-
tamination is 1012 cells per gram of product, how many microorganisms will
remain in the sample after heat treatment at 120°C for 18 minutes?

Solution:
Calculate the number of remaining cells using equation 5 with N0 = 1012 cells/g,
t = 18 minutes, and D120°C = 3 minutes.
From equation 5,

t
log N (t) = log N 0 
D

cells 18 min
log N 18  log 1012 
g 3 min
Solving for N(18) yields:

N (18)  106 cells/g

Discussion:
Starting with a known microbial concentration (N0), the final concentration of
a specific microorganism for a given thermal process at constant temperature
can be calculated if the thermal resistance of the microorganism at a given
temperature is known. In this case, D120°C = 3 min.

Example 2: Calculation of z value for a particular

microorganism
Problem:
D of a given bacterium in milk at 65°C is 15 minutes. When a food sample that
has 1010 cells of the bacterium per gram of food is heated for 10 minutes at 75°C,
the number of survivors is 2.15 × 103 cells. Calculate z for this bacterium.

10 • Principles of Thermal Processing of Packaged Foods

Solution:
First, calculate D at the process temperature of 75°C, D75°C, using equation 5.
Then calculate z using equation 6 with D65°C = 15 minutes, N0 = 1010 cells/g, and
t = 10 minutes at T = 75°C.

t (5)
log N (t) = log N 0 
D

10 min
log 2.15  103 cells/g  log 1010 cells/g 
D 75C
and D75°C = 1.5 min.

To calculate z, recall equation 6:

T  Tref
log D = log Dref 
z
Solving for z, equation 6 can be expressed as:

T
z
D
log  1 

 D2 
with ∆T = (75 –­65)°C, D1 = D65°C and D2 = D75°C,

75  65
z  10°C
15
log
1.5

Discussion:
As previously explained, the z value represents the change in process tempera-
ture required to reduce the D value of the target microorganism by ten times. In
this case, the z value is 10°C and accordingly the D value was reduced 10 times,
from 15 minutes to 1.5 minutes.

Example 3: Lethality of thermal processing of a can of

tuna fish
Problem:
Table 2 presents the values of temperature measured in the retort (TRT) and
the temperature measured at the cold spot of a can of tuna fish (Tcold spot) during
a thermal process. The total process time was 63 min until the product was
cold enough to be withdrawn from the retort.

(a) Determine CUT (the time required to come up to TRT), operator process
time Pt, and cooling time.
(b) Determine the lethality value (F0) attained for the can of tuna fish.

Principles of Thermal Processing of Packaged Foods • 11

 Table 2. Retort temperature (TRT ) and cold spot (Tcold spot) during thermal
processing of tuna fish in a can.

Time (min) TRT (°C) Tcold spot (°C)

0.97 29.7 45.0
1.97 39.7 45.0
2.97 49.7 45.0
3.97 59.7 45.0
4.97 69.7 44.9
5.97 79.7 44.9
6.97 89.7 44.8
7.97 99.7 44.7
8.97 109.7 44.7
9.97 119.7 44.8
10.97 120.0 45.0
11.97 120.0 45.4
12.97 120.0 46.0
13.97 120.0 46.9
14.97 120.0 48.0
15.97 120.0 49.3
16.97 120.0 50.8
17.97 120.0 52.6
18.97 120.0 54.4
19.97 120.0 56.4
20.97 120.0 58.5
21.97 120.0 60.6
22.97 120.0 62.8
23.97 120.0 65.0
24.97 120.0 67.1
25.97 120.0 69.3
26.97 120.0 71.4
27.97 120.0 73.5
28.97 120.0 75.5
29.97 120.0 77.5
30.97 120.0 79.4
31.97 120.0 81.2
32.97 120.0 83.0
33.97 120.0 84.7
34.97 120.0 86.3
35.97 120.0 87.9
36.97 120.0 89.4
37.97 120.0 90.8
(continued)

12 • Principles of Thermal Processing of Packaged Foods

Time (min) TRT (°C) Tcold spot (°C)
38.97 120.0 92.2
39.97 120.0 93.6
40.97 120.0 94.8
41.97 120.0 96.0
42.97 120.0 97.2
43.97 120.0 98.3
44.97 120.0 99.3
45.97 120.0 100.3
46.97 120.0 101.3
47.97 120.0 102.2
48.97 120.0 103.0
49.97 120.0 103.9
50.97 120.0 104.7
51.97 120.0 105.4
52.97 120.0 106.1
53.97 120.0 106.8
54.97 120.0 107.4
55.97 120.0 108.0
56.97 120.0 108.6
57.97 120.0 109.2
58.97 120.0 109.7
59.97 120.0 110.2
60.97 120.0 110.7
61.97 120.0 111.1
62.97 120.0 111.6
63.97 120.0 112.0
64.97 120.0 112.4
65.97 120.0 112.8
66.97 120.0 113.1
67.97 120.0 113.4
68.97 120.0 113.8
69.97 120.0 114.1
70.97 120.0 114.4
71.97 120.0 114.6
72.97 120.0 114.9
73.97 120.0 115.2
74.97 120.0 115.4
76 25.0 115.6
77 25.0 115.8
78 25.0 116.0
79 25.0 116.2
(continued)

Principles of Thermal Processing of Packaged Foods • 13

 Table 2. Retort temperature (TRT ) and cold spot (Tcold spot) during thermal
processing of tuna fish in a can. (continued )
Time (min) TRT (°C) Tcold spot (°C)
80 25.0 116.2
81 25.0 116.0
82 25.0 115.5
83 25.0 114.6
84 25.0 113.4
85 25.0 111.8
86 25.0 110.0
87 25.0 107.9
88 25.0 105.6
89 25.0 103.1
90 25.0 100.6
91 25.0 97.9
92 25.0 95.3
93 25.0 92.6
94 25.0 89.9
95 25.0 87.3
96 25.0 84.7
97 25.0 82.1
98 25.0 79.6
99 25.0 77.2
100 25.0 74.9
101 25.0 72.6
102 25.0 70.5
103 25.0 68.4
104 25.0 66.3
105 25.0 64.4
106 25.0 62.6
107 25.0 60.8
108 25 59.08
109 25 57.46
110 25 55.91
111 25 54.43
112 25 53.01
113 25 51.67
114 25 50.38
115 25 49.15
116 25 47.99
117 25 46.87
118 25 45.81
(continued)

14 • Principles of Thermal Processing of Packaged Foods

 Time (min) TRT (°C) Tcold spot (°C)
119 25 44.8
120 25 43.84
121 25 42.92
122 25 42.05
123 25 41.22
124 25 40.43

Solution:
(a) To determine CUT and Pt, plot TRT and
Tcold spot against time, which produces the
thermal profiles in figure 6.
Figure 6 shows that the CUT is approxi-
mately 10 min and Pt, during which process
temperature is maintained constant at
120°C, is approximately 64 min.
(b) The lethality value, F0, can be obtained
through numerical integration of equation 9
using the trapezoidal rule (Patashnik, 1953).
The calculations can be completed as
follows or using software such as Excel.
As presented in table 3, for each time, we
can evaluate equation 9: Figure 6. Temperature profile of thermal processing data in table 2.
t T 121.1
F0   10 10
dt (9)
0

  where T = T
 cold spot and Tref and z-value for
Clostridium botulinum are 121.1°C
and 10°C, respectively.

Given that F0 corresponds to the integral

of 10[(Tcold spot −­Tref)/z], this can be solved
numerically by the trapezoidal rule method,
i.e., by determining the area under the
curve by dividing the area into trapezoids,
computing the area of each trapezoid, and
summing all trapezoidal areas to yield F0.
(More details about the trapezoidal rule are
included in the appendix.) The calculations
are summarized in table 3. In this particular
case, F0 was about 6.07 min. The change of
F0 along the thermal process is shown as Figure 7. Thermal process temperature profiles including the
the blue line in figure 7. cumulative lethality value (F at any time t ).

Principles of Thermal Processing of Packaged Foods • 15

 Table 3. Numerical integration of equation 9 for the estimation of F0.
Time TRT Tcold spot (Tcold spot 10 [(Tcold Trapezoi- Sum of
(min) (°C) (°C) − Tref )/z spot −Tref )/z]
dal Area Areas
0.97 29.67 45 −­7.6 0.000 0.000 0.000
1.97 39.67 45 −­7.6 0.000 0.000 0.000
2.97 49.67 44.99 −­7.6 0.000 0.000 0.000
3.97 59.67 44.97 −­7.6 0.000 0.000 0.000
4.97 69.67 44.93 −­7.6 0.000 0.000 0.000
5.97 79.67 44.85 −­7.6 0.000 0.000 0.000
6.97 89.67 44.76 −­7.6 0.000 0.000 0.000
7.97 99.67 44.69 −­7.6 0.000 0.000 0.000
8.97 109.67 44.68 −­7.6 0.000 0.000 0.000
9.97 119.67 44.77 −­7.6 0.000 0.000 0.000
10.97 120 45 −­7.6 0.000 0.000 0.000
11.97 120 45.41 −­7.6 0.000 0.000 0.000
12.97 120 46.03 −­7.5 0.000 0.000 0.000
13.97 120 46.88 −­7.4 0.000 0.000 0.000
14.97 120 47.97 −­7.3 0.000 0.000 0.000
15.97 120 49.29 −­7.2 0.000 0.000 0.000
16.97 120 50.83 −­7.0 0.000 0.000 0.000
17.97 120 52.55 −­6.9 0.000 0.000 0.000
18.97 120 54.42 −­6.7 0.000 0.000 0.000
19.97 120 56.41 −­6.5 0.000 0.000 0.000
20.97 120 58.49 −­6.3 0.000 0.000 0.000
21.97 120 60.63 −­6.0 0.000 0.000 0.000
22.97 120 62.79 −­5.8 0.000 0.000 0.000
23.97 120 64.97 −­5.6 0.000 0.000 0.000
24.97 120 67.14 −­5.4 0.000 0.000 0.000
25.97 120 69.29 −­5.2 0.000 0.000 0.000
26.97 120 71.41 −­5.0 0.000 0.000 0.000
27.97 120 73.48 −­4.8 0.000 0.000 0.000
28.97 120 75.5 −­4.6 0.000 0.000 0.000
29.97 120 77.46 −­4.4 0.000 0.000 0.000
30.97 120 79.36 −­4.2 0.000 0.000 0.000
31.97 120 81.2 −­4.0 0.000 0.000 0.000
32.97 120 82.97 −­3.8 0.000 0.000 0.001
33.97 120 84.67 −­3.6 0.000 0.000 0.001
34.97 120 86.31 −­3.5 0.000 0.000 0.001
35.97 120 87.89 −­3.3 0.000 0.001 0.002
36.97 120 89.4 −­3.2 0.001 0.001 0.003
37.97 120 90.84 −­3.0 0.001 0.001 0.004
38.97 120 92.22 −­2.9 0.001 0.002 0.005
39.97 120 93.55 −­2.8 0.002 0.002 0.007
(continued)

16 • Principles of Thermal Processing of Packaged Foods

Time TRT Tcold spot (Tcold spot 10 [(Tcold Trapezoi- Sum of
(min) (°C) (°C) − Tref )/z spot −Tref )/z]
dal Area Areas
40.97 120 94.81 −­2.6 0.002 0.003 0.010
41.97 120 96.02 −­2.5 0.003 0.004 0.014
42.97 120 97.17 −­2.4 0.004 0.005 0.018
43.97 120 98.27 −­2.3 0.005 0.006 0.024
44.97 120 99.31 −­2.2 0.007 0.007 0.032
45.97 120 100.31 −­2.1 0.008 0.009 0.041
46.97 120 101.27 −­2.0 0.010 0.012 0.053
47.97 120 102.17 −­1.9 0.013 0.014 0.067
48.97 120 103.04 −­1.8 0.016 0.017 0.084
49.97 120 103.86 −­1.7 0.019 0.021 0.105
50.97 120 104.65 −­1.6 0.023 0.025 0.130
51.97 120 105.39 −­1.6 0.027 0.029 0.159
52.97 120 106.1 −­1.5 0.032 0.034 0.193
53.97 120 106.78 −­1.4 0.037 0.040 0.233
54.97 120 107.42 −­1.4 0.043 0.046 0.279
55.97 120 108.04 −­1.3 0.049 0.053 0.332
56.97 120 108.62 −­1.2 0.056 0.060 0.393
57.97 120 109.18 −­1.2 0.064 0.068 0.461
58.97 120 109.7 −­1.1 0.072 0.077 0.538
59.97 120 110.21 −­1.1 0.081 0.086 0.624
60.97 120 110.69 −­1.0 0.091 0.096 0.720
61.97 120 111.14 −­1.0 0.101 0.106 0.826
62.97 120 111.57 −­1.0 0.111 0.117 0.943
63.97 120 111.99 −­0.9 0.123 0.129 1.072
64.97 120 112.38 −­0.9 0.134 0.140 1.212
65.97 120 112.75 −­0.8 0.146 0.153 1.365
66.97 120 113.11 −­0.8 0.159 0.165 1.530
67.97 120 113.44 −­0.8 0.171 0.178 1.708
68.97 120 113.76 −­0.7 0.185 0.191 1.899
69.97 120 114.07 −­0.7 0.198 0.205 2.104
70.97 120 114.36 −­0.7 0.212 0.219 2.323
71.97 120 114.63 −­0.6 0.225 0.233 2.555
72.97 120 114.9 −­0.6 0.240 0.247 2.802
73.97 120 115.15 −­0.6 0.254 0.261 3.063
74.97 120 115.38 −­0.6 0.268 0.275 3.338
76 25 115.61 −­0.5 0.282 0.290 3.628
77 25 115.83 −­0.5 0.297 0.304 3.932
78 25 116.02 −­0.5 0.310 0.316 4.248
79 25 116.17 −­0.5 0.321 0.322 4.570
80 25 116.19 −­0.5 0.323 0.315 4.885
81 25 115.97 −­0.5 0.307 0.290 5.175
(continued)

Principles of Thermal Processing of Packaged Foods • 17

 Table 3. Numerical integration of equation 9 for the estimation of F0. (continued )

Time TRT Tcold spot (Tcold spot 10 [(Tcold Trapezoi- Sum of

(min) (°C) (°C) − Tref )/z spot −Tref )/z]
dal Area Areas
82 25 115.45 −­0.6 0.272 0.248 5.422
83 25 114.58 −­0.7 0.223 0.196 5.618
84 25 113.36 −­0.8 0.168 0.143 5.761
85 25 111.81 −­0.9 0.118 0.097 5.858
86 25 109.96 −­1.1 0.077 0.062 5.920
87 25 107.87 −­1.3 0.048 0.038 5.958
88 25 105.57 −­1.6 0.028 0.022 5.980
89 25 103.12 −­1.8 0.016 0.012 5.992
90 25 100.57 −­2.1 0.009 0.007 5.999
91 25 97.94 −­2.3 0.005 0.004 6.003
92 25 95.26 −­2.6 0.003 0.002 6.005
93 25 92.58 −­2.9 0.001 0.001 6.006
94 25 89.91 −­3.1 0.001 0.001 6.007
95 25 87.26 −­3.4 0.000 0.000 6.007
96 25 84.66 −­3.6 0.000 0.000 6.007
97 25 82.11 −­3.9 0.000 0.000 6.007
98 25 79.63 −­4.1 0.000 0.000 6.007
99 25 77.22 −­4.4 0.000 0.000 6.007
100 25 74.88 −­4.6 0.000 0.000 6.007
101 25 72.62 −­4.8 0.000 0.000 6.007
102 25 70.45 −­5.1 0.000 0.000 6.007
103 25 68.35 −­5.3 0.000 0.000 6.007
104 25 66.34 −­5.5 0.000 0.000 6.007
105 25 64.41 −­5.7 0.000 0.000 6.007
106 25 62.55 −­5.9 0.000 0.000 6.007
107 25 60.78 −­6.0 0.000 0.000 6.007
108 25 59.08 −­6.2 0.000 0.000 6.007
109 25 57.46 −­6.4 0.000 0.000 6.007
110 25 55.91 −­6.5 0.000 0.000 6.007
111 25 54.43 −­6.7 0.000 0.000 6.007
112 25 53.01 −­6.8 0.000 0.000 6.007
113 25 51.67 −­6.9 0.000 0.000 6.007
114 25 50.38 −­7.1 0.000 0.000 6.007
115 25 49.15 −­7.2 0.000 0.000 6.007
116 25 47.99 −­7.3 0.000 0.000 6.007
117 25 46.87 −­7.4 0.000 0.000 6.007
118 25 45.81 −­7.5 0.000 0.000 6.007
119 25 44.8 −­7.6 0.000 0.000 6.007
120 25 43.84 −­7.7 0.000 0.000 6.007
(continued)

18 • Principles of Thermal Processing of Packaged Foods

 Time TRT Tcold spot (Tcold spot 10 [(Tcold Trapezoi- Sum of
(min) (°C) (°C) − Tref )/z spot −Tref )/z]
dal Area Areas
121 25 42.92 −­7.8 0.000 0.000 6.007
122 25 42.05 −­7.9 0.000 0.000 6.007
123 25 41.22 −­8.0 0.000 0.000 6.007
124 25 40.43 −­8.1 0.000 0.000 6.007

Discussion:
The cumulative lethality, F0, was about 6.01 min, meaning that the process is
safe according to FDA requirements, i.e., F0 ≥ 3 min (see the Food Sterilization
Criterion and Calculation section above).

Example 4: Lethality of thermal processing of a can of mussels

Problem:
Temperatures measured in the retort and the temperature measured at the
cold spot of a can of mussels during a thermal process performed at 120°C were
recorded. The total process time was 113 min until the product was cold enough
to be withdrawn from the retort. The measured
thermal profiles (TRT and Tcold spot) were plotted, as
was done in Example 3. The resulting plot (figure 8)
shows that CUT was approximately 10 min and
Pt was approximately 53 min. The lethality value,
F0, was obtained through numerical integration
of equation 9. In this case, F0 attained in the
mussels can with a processing temperature of
120 °C was 2.508 min. The evolution of F0 along
the thermal process is shown in figure 9 as the
blue line.
Figure 8. Temperature profile of thermal processing data for a can
Discussion: of mussels.
The cumulative lethality, F0, attained along
the thermal process was 2.5 min, meaning that the
process is not safe according to FDA requirements
(F0 ≥ 3 min). Thus, the thermal processing time
of canning process of mussels must be extended
in order to reach the safety value recommended
by the FDA.

Example 5: Processing time at

different retort temperatures
Problem:
Determine the required processing time to get Figure 9. Thermal process temperature profiles including the
a lethality of 6 min (F0 = 6 min) when the retort cumulative lethality value (F at any time t ).

Principles of Thermal Processing of Packaged Foods • 19

 temperature is (a) 120°C and considered equal to the cold spot temperature,
(b) 110°C, and (c) 130°C.

Solution:
The F0 is typically set for the 12D value to give a 12 log reduction of heat-­resistant
species of mesophilic spores (typically taken as C. botulinum). The Tref = 121.1°C
and z = 10°C. Therefore, equation 9 can be used directly by replacing T by the
retort temperature, given that cold spot temperature can be assumed equal
to retort temperature:
t T 121.1
F0  10 z
dt (9)
0

t 120 121.1
6   10 10
dt
0

6
t 120 121.1
10 10

Solving the integral yields a processing time, t, of 7.7 min.

(b) When the temperature of the retort is reduced to 110°C, the lethality must
be maintained at 6 min. Solving equation 9:
t 110 121.1
6  10 10
dt
0

gives the required processing time t of 77.2 min.

(c) When the temperature of the retort is increased to 130°C, and maintaining
the F0 = 6 min, the processing time is reduced to 0.77 min

t 130 121.1
6   10 10
dt
0

Discussion:
The results showed that as the temperature in the food increased in 10°C incre-
ments, the processing time was reduced by one decimal reduction. This variation
is due to a z value of 10°C.

Appendix: The Trapezoidal Rule

A trapezoid is a four-­sided region with two opposite sides parallel (figure 10).
The area of a trapezoid is the average length of the two parallel sides multiplied
by the distance between the two sides. In figure 11, the area (A) under function
f(x) between points x0 and xn is given by:

b
A   f ( x)dx (10)
a

Figure 10. Example of a An approximation of the area A is the sum of the areas of the individual trap-
trapezoid. ezoids (T), where T can be calculated using equation 11:

20 • Principles of Thermal Processing of Packaged Foods

 1 1 1
T x  f (xo )  f (x1 )   x2  f (x1 )  f (x2 )   ...  xn  f (xn 1 )  f (x n )  (11)
2 1 2 2

where xi  xi  xi 1, for i = 1, 2, 3, . . . , n

In the particular case where ∆x1 = ∆x2 = ∆x3 = . . . = ∆xn = ∆x,

equation 11 can be expressed as:

 f (x0 ) f (xn )  (12)

T  x   f (x1 )  f (x2 )  f (x3 )  .. 
 2 2 

or, in the following reduced form: Figure 11. Curve divided into n equal parts
each of length ΔX.
 f (x0 ) n 1 f (xn ) 
(13)
T  x    f (xi ) 
 2 i 1 2 

Finally, to estimate area A under the trapezoidal rule,

xn 1 1 1
A =  f (x)dx  x 1  f (xo )  f (x1 )   x2  f (x1 )  f (x2 )   ...  xn  f (xn 1 )  f (x n )  (14)
x 0 2 2 2

When all intervals are of the same size (∆x1 = ∆x2 = ∆x3 = . . . = ∆xn = ∆x), the
following expression can be applied:

xn f ( x0 ) n 1 f ( xn ) 1 n 1
A= f ( x)dx x(   f ( xi )  )  x( f ( x0 )  2 f ( xi )  f ( xn )) (15)
x0 2 i 1 2 2 i 1

Example
Problem:
Using the heat penetration data at the cold spot of Table 4. Heat penetration data at the slowest heating
a canned food in table 4, calculate the cumulative point.
lethality, F0, in the range of 23 to 27 min using the
Time (min) Temperature (C)
trapezoidal rule.
... ...
Solution: 23 118.5
From equation 9, 24 118.7
25 118.9
T 121.1
27
26 119.1
Fo   10 10
dt
23
27 119.3
Applying the trapezoidal rule and considering ... ...
that all time steps are equal (∆t = 1 min), calculate
F0 using equation 15,
T 121.1
27 1
Fo   10 10
dt   f (23)  2 f (24)  2 f (25)  2 f (26)  f (27)
23 2

where ∆t = 1 (1 min interval), and:

118.5121.1
f (23)  10 10
 0.549541

Principles of Thermal Processing of Packaged Foods • 21

 118.7 121.1
f (24)  10 10
 0.57544
118.9 121.1
f (25)  10 10
 0.6025596
119.1121.1
f (26)  10 10
 0.63095734
119.3121.1
f (27)  10 10
 0.66069345

Replacing into equation (15):

T 121.1
27 1
Fo   10 10
dt  (0.549541  2  0.57544  2  0.6025596  2  0.63095734  0.66069345)
23 2

Therefore, F0 ~ 2.41407394 ~ 2.41 min.

Discussion:
The applied process to sterilize the target food is not safe since F0 < 3 minutes.

Image Credits
Figure 1. Simpson, R. (CC By 4.0). (2020). Main heat transfer mechanisms involved in the thermal
processing of packaged foods. Retrieved from https://​onlinelibrary​.wiley​.com
Figure 2. Holdsworth, S. Donald-­Simpson, R. (CC By 4.0). (2020). Semilogarithmic survivor
curve. Retrieved from https://​www​.springer​.com/​la/​book/​9783319249025
Figure 3. Holdsworth, S. Donald-­Simpson, R. (CC By 4.0). (2020). Thermal death time (TDT)
curve. Retrieved from https://​www​.springer​.com/​la/​book/​9783319249025.
Figure 4. Simpson, R. (CC By 4.0). (2020). Stages of a typical food commercial canning factory.
Figure 5. Ramírez, C. (CC By 4.0). (2020). Temperature profiles for a typical thermal process,
where CUT is come-­up time and Pt is operator time.
Figure 6. Ramírez, C. (CC By 4.0). (2020). Temperature profile of thermal processing data in
table 2.
Figure 7. Ramírez, C. (CC By 4.0). (2020). Thermal process temperature profiles including the
cumulative lethality value (F at any time t ).
Figure 8. Ramírez, C. (CC By 4.0). (2020). Temperature profile of thermal processing data
(Table 4).
Figure 9. Ramírez, C. (CC By 4.0). (2020). Thermal process temperature profiles including the
cumulative lethality value (F at any time t ).
Figure 10. Simpson, R. (CC By 4.0). (2020). Example of a trapezoid.
Figure 11. Simpson, R. (CC By 4.0). (2020). Curve divided into n equal parts each of length ΔX.

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Principles of Thermal Processing of Packaged Foods • 23