The design of depth filters

Equation (5.15), the log penetration relationship, is the same form as equation (5.3) in the derivation of thermal-death kinetics. In the case of heat sterilization the theory predicts that an infinite time is required to reduce the population to zero, whereas the theory of filtration predicts that a filter of infinite length is required to remove all organisms from an air stream. Thus, it is not surprising that the same approach is adopted in the design of filters and heat-sterilization cycles, in that an acceptable probability of contamination is determined. The probability of one fermentation in a thousand being contaminated is frequently used in filter design, as it is in the design of heat-sterilization cycles. Having arrived at an acceptable probability of contamination and determined the filtration characteristics (i.e. the value of K) of the material to be used, a filter may be designed to filter a certain volume of air containing a certain number of organisms; the following example illustrates the design calculation approach used by Richards (1967):

It is required to provide a 20-m3 fermenter with air at a rate of 10 m3 min-1 for a fermentation lasting 100 hours. From an investigation of the filter material to be used, the optimum linear air velocity was shown to be 0.15 m sec^1, at which the value of K was 1.535 cm-1. The dimensions of the filter may be calculated as follows:

The log penetration relationship states that:

The air in the fermentation plant contained approximately 200 micro-organisms m~3. Therefore,

N0 = total amount of air provided X 200,

= 12 X 106 organisms.

The acceptable degree of contamination is one in a thousand, therefore N = 10 \

In {10~3/(12 x 106)} = -Kx, In 8.33 X 10~n = -Kx, In 8.33 X 10-11 = -1.535*, x = -23.21/- 1.535 = 15.12 cm.

Therefore, the filter to be used should be 15.12 cm long.

The cross-sectional area of the filter is given by the volumetric air flow rate divided by the linear air velocity:

where r is the radius of the filter r = 0.59 m.

Thus the filter to be employed should be 15.12 cm long and 0.59 m radius.

However, as Humphrey (1960) pointed out, the efficient operation of the filter is dependent on the supply of air at the optimum linear velocity. If the air velocity increases or decreases the value of K will decrease, resulting in a loss of filtration efficiency. Considering the example calculation, if the linear air velocity were to drop to 0.03 m sec™1, then the value of K would decline to 0.2 cm-1. The number of organisms which would enter the fermentation in 1 minute at this reduced air-flow rate would be as calculated below:

At a linear air velocity of 0.03 m sec™1, in 1 minute 0.03 X 60 X the cross-sectional area of the filter m™3 of air would enter the filter, i.e. 1.98 m3. At a microbial contamination level of 200 organisms m™3 this means that 396 organisms would enter the filter in 1 minute. Thus:

Therefore, 19.24 organisms would have entered the fermenter in 1 minute at the decreased air-flow rate. If the filter had been designed to meet this contingency, then the length would have been:

Thus a filter length of 64.4 cm would have been required to have maintained the same probability of contamination over the 1 minute of reduced air flow.

This example illustrates the hazards of attempting very precise design and the necessity to consider the reliability of ancillary equipment in any design calculation.

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