The food is deemed sterilized when Cl. Botulinum spores are reduced by a factor of
10-12
𝑁
𝑁0
= 10−12
(1)
In order to calculate the time required to reach this level of sterilization, we must first
determine the decimal reduction time, D, given by equation 2, where k represents the reaction
velocity constant. (Geankoplis 2003)
2.303
𝑘
𝐷=
(2)
This represents the time required to reduce the original number of microbes by 1/10.
As our sterilization requirement as already been determined to be 10^-12, we must multiply
this value by 12 in order to determine our thermal process time, given by equation 3
(Geankoplis 2003).
𝑡 = 12 𝐷
(3)
As we are dealing with Cl. Botulinum, we can determine our thermal processing time
from experimental data of thermal death rates given in literature. The decimal reduction time
for Cl botulinum spores at 250oF is given as well as the experimental value z. Z represents
the increase in temperature in oF that will increase the death rate by a factor of ten.
D250 = 3 min
Z = 18 oF
These values can be used to find the decimal reduction time for any other
temperature, given by equation 4 (Geankoplis 2003).
𝐷𝑇 = 𝐷250 𝑥10
250−𝑇
𝑧
(4)
Our processing temperature is 190oF. Plugging these values into Equation 4 will yield
the decimal reduction time for 190 degrees. This value can then be used with equation 3 to
yield the thermal death time of 7.756 x 10^4 seconds for 190
𝐷190 = 𝐷250 𝑥10
250−190
𝑧
o
= 6.643𝑥103
F.
𝑡 = 12𝐷190 = 7.756𝑥104
We use the t value to find the processing time using equation 5 as thermal death
time does not necessarily equal processing time.
𝐹𝑜 = 𝑡 ∗ 10
(𝑇𝑜 𝐹−250)
𝑧
(5
𝐹𝑜 =∗ 10
(190−250)
𝑧
= 36 min
Our minimum processing time to guarantee elimination of the Cl. Botulinum is therefore
36 minutes.
We also need to calculate the heat transfer through the cooking chamber. To do so,
we must use the specified parameters as well as the Choi Okos Equation to calculate the
specific heat capacity of the salsa, given in equations 6-11 in J/kg*K.
Parameters:
𝑇1 = 355.37 𝐾
𝑇2 = 366.48 𝐾
𝛿=
𝑇2 + 𝑇1
= 360.925
2
𝑇 = 377.594 𝐾
𝐶𝑝𝑓𝑖 = 1.8459.103 + 1.8306𝛿 − 4.6509.10−3 𝛿 2 = 1.901.103
𝐽
𝑘𝑔 𝐾
𝐶𝑝𝑤 = 4.1289.103 − 9.0864.10−2 𝛿 + 5.4732. 10−3 𝛿 2 = 4.809.103
𝐽
𝑘𝑔 𝐾
𝐶𝑝𝑝 = 2.0082.103 + 1.2089𝛿 − 1.3129.10−3 𝛿 2 = 2.273.103
𝐽
𝑘𝑔 𝐾
𝐶𝑓𝑎𝑡 = 1.9842.103 + 1.4733𝛿 − 4.8008.10−3 𝛿 2 = 1.891.103
𝐽
𝑘𝑔 𝐾
𝐶𝑐𝑎𝑟𝑏 = 1.5488.103 + 1.9625𝛿 − 5.9399.10−3 𝛿 2 = 1.483.103
𝐽
𝑘𝑔 𝐾
𝐶𝑎𝑣𝑔 = 0.0267𝐶𝑝𝑝 + 0.0059𝐶𝑓𝑎𝑡 + 0.1328𝐶𝑐𝑎𝑟𝑏 + 0.0306𝐶𝑝𝑓𝑖 + 0.8133𝐶𝑝𝑤 = 4.238.103
𝐽
𝑘𝑔 𝐾
In addition to the heat capacity of the salsa, we must also calculate the thermal
conductivity. The Choi-Okos equation can be used to calculate this value, given in equations
12-16. Equation 17 is the average of all the calculated thermal conductivities and represents
the overall thermal conductivity for the system in W/m*K.
𝐾𝑝𝑟𝑜 = 1.7881 ∗ 10−1 + 1.1958 ∗ 10−3 ∗ 𝛿 − 2.7178 ∗ 10−6 ∗ 𝛿 2 = 0.256
(12)
𝐾𝑓𝑎𝑡 = 1.8071 ∗ 10−1 + 2.7604 ∗ 10−4 ∗ 𝛿 − 1.7749 ∗ 10−7 ∗ 𝛿 2 = 0.058
(13)
𝐾𝑐𝑎𝑟𝑏 = 2.0141 ∗ 10−1 + 1.3874 ∗ 10−3 ∗ 𝛿 − 4.3312 ∗ 10−6 ∗ 𝛿 2 = 0.138
(14)
𝐾𝑓𝑖𝑏 = 1.8331 ∗ 10−1 + 1.2497 ∗ 10−3 ∗ 𝛿 − 3.1683 ∗ 10−6 ∗ 𝛿 2 = 0.222
(15)
(𝐾𝑤𝑎𝑡 = 0.57109 + 1.7625 ∗ 10−3 ∗ 𝛿 − 6.7306 ∗ 10−6 ∗ 𝛿 2 = 0.256)
(16)
𝐾𝑎𝑣𝑔 = 0.0267 ∗ 𝐾𝑝𝑟𝑜 + 0.0059 ∗ 𝐾𝑓𝑎𝑡 + 0.1328 ∗ 𝐾𝑐𝑎𝑟𝑏 + 0.0306 ∗ 𝐾𝑓𝑖𝑏 + 0.8133 ∗
𝐾𝑤𝑎𝑡 = .301
𝑊
𝑚∗𝐾
(17)
The Choi-Okos equation is once again used in calculating the densities of each
component of the product, given in equations 18-22. Equation 23 represents the average
density of the overall product.
𝜌𝑝 = 1.399 ∗ 103 − 5.1840 ∗ 10−1 ∗ (𝛿 − 273)
(18)
𝜌𝑓𝑎𝑡 = 9.2559 ∗ 102 − 4.1757 ∗ 10−1 ∗ (𝛿 − 273)
(19)
𝜌𝑐 = 1.5991 ∗ 103 − 3.1046 ∗ 10−1 ∗ (𝛿 − 273)
(20)
𝜌𝑓𝑖𝑏 = 1.3115 ∗ 103 − 2.8063 ∗ 10−1 ∗ (𝛿 − 273)
(21)
𝜌𝑤𝑎𝑡 = 9.9718 ∗ 102 + 3.1439 ∗ 10−3 ∗ (𝛿 − 273) − 3.7574 ∗ 10−3 ∗ (𝛿 − 273)
(22)
𝜌𝑎𝑣𝑔 = 0.0267 ∗ 𝜌𝑝 + 0.0059 ∗ 𝜌𝑓𝑎𝑡 + 0.1328 ∗ 𝜌𝑐 + 0.03066 ∗ 𝜌𝑓𝑖𝑏 + 0.8133 ∗ 𝜌𝑤𝑎𝑡 = 1.075 ∗
103
(23)
In order to calculate the processing time, we must also know the parameters of the
jar itself, such as height, diameter, and heat capacity given below.
H = 0.12065 m
height of the glass jar
d = 0.08255 m
diameter of the glass jar
h = 6000 W/m2K
heat capacity
These parameters can be used in conjunction with the Heisler charts to find total
processing time. We can calculate total radius, x1, from the given parameters.
𝑥1 =
𝑑
= 0.041
2
𝑥=0
As we consider the center of the can to be a radius of 0, or x = 0, we can use
Equations 24-27 and our values for the average density, thermal conductivity, and heat
capacity of the salsa to to determine the n, m,α, and Y values used on the Heisler charts
(Geankoplis 2003).
𝑛=
𝑥
=0
𝑥1
𝑚=
𝐾𝑎𝑣𝑔
ℎ∗𝑥1
10−3
= 1.216 ∗
(25)
𝛼=
𝐾𝑎𝑣𝑔
𝜌𝑎𝑣𝑔 ∗𝐶𝑎𝑣𝑔
= 6.606.10
−8
(26)
𝑇−𝑇2
𝑌 = 𝑇−𝑇1 = 0.5
(27)
After applying these values to the Heisler charts, we arrive at a value of 0.25 for X.
We can then use equations 28 and 29 to calculate our processing time of 107.457 minutes.
Equation 29 is the time given in minutes, Equation 28 in seconds.
Calculated value X = 25
𝑡𝑖𝑚𝑒 =
𝑋∗(𝑥1)2
𝛼
= 6.447 ∗ 103 𝑠
(28)
𝑇=
𝑡𝑖𝑚𝑒
60
= 107.457 𝑚𝑖𝑛
(29)
As our final processing time of 107.457 minutes is greater than our thermal processing
time, the salsa will be adequately sterilized to prevent foodborne illness. Along with that, the
high temperature will melt the rubber in the lid and seal the jar. The canning process will
ensure a longer shelf life for our low acid food product.