The Trapezoidal Integration Method for Calculating Fo Values in Fishery Products Canning

A mathematical method in which the time-temperature data are used to measure changes I in lethality during heating and cooling. By using standard time intervals the lethal value: is computed in stages and the cumulative L value for the process is found without the need for graphical representation of the heating and cooling curves.

The Fo value for the process is calculated by summing all the L values and multiplying this value by the standard time interval between readings.

The trapezoidal method also allows simple calculation of the contribution to total process lethality of the heating and cooling portions of the process.

In Table 2 are shown L values and in Table 3 is shown a worked example in which temperature was recorded at 5 minute intervals for a process of 60 minutes at 121.1 °C.

To calculate Fo for the process: Summing the L values gives 2.925 which when multiplied by 5 (the time interval between readings) gives an Fo value of 14.6 min.

To calculate Fo for the heating phase: The sum of L values at time 25 and 60 min (0 and 0.776) is divided by 2 and this value (0.388) is added to the sum of L values from time 30 to 55 min. This gives 1.730 which when multiplied by 5 yields on Fo of 8.6 min for the process lethality at the stage when the steam was turned off.

The Improved General Method which relies on a temperature-time plot. for the entire process is the most accurate of all methods for calculating Fo value and for this reason is frequently quoted as the "reference method". Like the Trapezoidal Method there are no assumptions made regarding product heating and cooling characteristics, however the benefits of accuracy have to be balanced against the lack of versatility. Data from one set of trials cannot easily be used to calculate Fo values when product temperature and/or retort temperature are (is) altered. This means that once process conditions are altered new temperature-time data must be collected under the new experimental conditions.

Table 2. Values of L for temperature ranging 
from 90 ºC to 130.9 ºC in 0.1 ºC intervals

oC	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
90	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001
91	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001
92	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.001	0.002
93	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.002
94	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.002	0.002
95	0.002	0.003	0.003	0.003	0.003	0.003	0.003	0.003	0.003	0.003
96	0.003	0.003	0.003	0.003	0.003	0.003	0.004	0.004	0.004	0.004
97	0.004	0.004	0.004	0.004	0.004	0.004	0.004	0.005	0.005	0.005
98	.005	0.005	0.005	.005	0.005	0.005	0.006	0.006	0.006	0.006
99	0.006	0.006	0.006	0.007	0.007	0.007	0.007	0.007	0.007	0.008
100	0.008	0.008	0.008	0.008	0.009	0.009	0.009	0.009	0.009	0.010
101	0.010	0.010	0.010	0.010	0.011	0.011	0.011	0.011	0.012	0.012
102	0.012	0.013	0.013	0.013	0.013	0.014	0.014	0.014	0.015	0.015
103	0.015	0.016	0.016	0.017	0.017	0.017	0.018	0.018	0.019	0.019
104	0.019	0.020	0.020	0.021	0.021	0.022	0.022	0.023	0.023	0.024
105	0.025	0.025	0.026	0.026	0.027	0.028	0.028	0.029	0.030	0.030
106	0.031	0.032	0.032	0.033	0.034	0.035	0.035	0.036	0.037	0.038
107	0.039	0.040	0.041	0.042	0.043	0.044	0.045	0.046	0.047	0.048
108	0.049	0.050	0.051	0.052	0.054	0.055	0.056	0.058	0.059	0.060
109	0.062	0.063	0.065	0.066	0.068	0.069	0.071	0.072	0.074	0.076
110	0.078	0.079	0.081	0.083	0.085	0.087	0.089	0.091	0.093	0.095
111	0.098	0.100	0.102	0.105	0.107	0.110	0.112	0.115	0.117	0.120
112	0.123	0.126	0.129	0.132	0.135	0.138	0.141	0.145	0.148	0.151
113	0.155	0.158	0.162	0.166	0.170	0.174	0.178	0.182	0.186	0.191
114	0.195	0.200	0.204	0.209	0.214	0.219	0.224	0.229	0.234	0.240
115	0.245	0.251	0.257	0.263	0.269	0.275	0.282	0.288	0.295	0.302
116	0.309	0.316	0.324	0.331	0.339	0.347	0.355	0.363	0.372	0.380
117	0.389	0.398	0.407	0.417	0.427	0.437	0.447	0.457	0.468	0.479
118	0.490	0.501	0.513	0.525	0.537	0.550	0.562	0.575	0.589	0.603
119	0.617	0.631	0.646	0.661	0.676	0.692	0.708	0.724	0.741	0.759
120	0.776	0.794	0.813	0.832	0.851	0.871	0.891	0.912	0.933	0.955
121	0.977	1.000	1.023	1.047	1.072	1.096	1.122	1.148	1.175	1.202
122	1.230	1.259	1.288	1.318	1.349	1.380	1.413	1.445	1.479	1.514
123	1.549	1.585	1.622	1.660	1.698	1.738	1.778	1.820	1.862	1.905
124	1.950	1.995	2.042	2.089	2.138	2.188	2.239	2.291	2.344	2.399
125	2.455	2.512	2.570	2.630	2.692	2.754	2.818	2.884	2.951	3.020
126	3.090	3.162	3.236	3.311	3.388	3.467	3.548	3.631	3.715	3.802
127	3.890	3.981	4.074	4.169	4.266	4.365	4.467	4.571	4.677	4.786
128	4.898	5.012	5.129	5.248	5.370	5.495	5.623	5.754	5.888	6.026
129	6.166	6.310	6.457	6.607	6.761	6.918	7.079	7.244	7.413	7.586
130	7.762	7.943	8.128	8.318	8.511	8.710	8.913	9.120	9.333	9.550

Note:

z = 10 ºC
T = product temperature

Table 3. Trapezoidal method for integration of 

lethal rate data to calculate Fo value

Time (min)	Temperatur (oC)	L	L/t	Fo (min)
0	24	0
5	24.5	0
10	34	0
15	54	0
20	72.5	0
25	87	0
30	98	0.005
35	105	0.025
40	110.5	0.087
45	114.5	0.219
50	117	0.389
55	119	0.617
60	120	0.776	1.730	8.6
* STEAM OFF
65	120	0.776
70	106	0.031
75	88	0	2.925	14.6

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