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Two lines are shown on the graph. One corresponds to the case in which the cooling water temperature is maintained at 38°C (£=0). In the other case (£=2) the temperature of the cooling water (Tw) is manipulated in response to the temperature measured at the top of the bed, halfway between the heat transfer plates (this temperature being denoted T*), according to the following equation:

where Topt is the optimum temperature for growth. This equation calculates the number of degrees by which the measured temperature exceeds the optimum for growth and then decreases the temperature of the cooling water by this temperature difference multiplied by a factor K. If possible, the value of K should be chosen so as not to require refrigeration of the cooling water to values below the temperature at which it is normally available. This will avoid the costs of building and operating a water refrigeration system. However, the ability to do this will depend on the optimum growth temperature of the organism in relation to the temperature of the available cooling water.

Whether or not it is advantageous to use this strategy to control the cooling water temperature depends on the spacing between the plates. If the spacing between the plates is large, of the order of 20 cm (1=10 cm), the cooling water has relatively little effect on much of the bed, and therefore the temperature control scheme brings little advantage (Fig. 24.11). If the spacing between the plates is small, of the order of 2 cm (1=1 cm), temperature control is reasonably efficient even without the temperature control scheme, so there is little advantage in having it. The temperature control scheme is most advantageous at intermediate plate spacings.

In fact, intermediate plate spacings are probably preferable. Although a 2 cm gap between plates gives near optimum performance (the minimum possible value of t90 for Xo=0.001 kg kg-1, Xm = 0.125 kg kg-1, and = 0.236 h-1 is 29.7 h), it is not a reasonable value, because a significant volume of the bioreactor will be occupied by the plates, leading to a low overall productivity and, additionally, the capital costs of the bioreactor will be much higher. On the other hand, the wider the space between the plates, the less effective they are in cooling the bed, and therefore the higher is the superficial air velocity that is needed to achieve the same cooling effect and, consequently, the higher are the operating costs. Essentially these will need to be balanced against each other. Mitchell et al. (2002b) use a model similar to the one presented here to explore these issues in more depth, identifying a gap of 6 cm (L=3 cm) as optimal in terms of productivity of the bio-reactor, calculated per m3 of overall bioreactor volume, for a microorganism with a specific growth rate of 0.324 h-1 and a superficial air velocity of 1 cm s-1. Obviously the optimal plate spacing will differ for different organisms and under different operating conditions. The mathematical model provides a tool that allows the optimum to be determined for any particular combination of growth kinetics, bioreactor design and operating conditions.

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