where G is the substrate feed rate (g dm-3 h~') and Y is the yield factor.
Hut d.i/dt = fxx, thus substituting for dx/dt in equation (2.34) gives:
l'unidcd that GY/x does not exceed /u,max then the limning substrate will be consumed as soon as it enters the fermenter and ds/dt ~ 0. However, dx/dt may not he equated to zero, as in the case of variable \okiine fed-batch, because the biomass concentration, as well as the total amount of biomass in the fer-mciik-i. will increase with time. Biomass concentration 1«. mwn by the equation:
when. r, is the biomass after operating in fed batch for t hours
•nul cs is the biomass concentration at the onset of fed-batch culture. As biomass increases then the specific growth rate will decline according to equation (2.35). The behaviour of a fixed volume fed-batch culture is illustrated in Fig. 2.13b from which it may be seen that fi declines 'aiYDuling to equation (2.35)), the limiting substrate concentration remains virtually constant, biomass increases and the concentration of the non-limiting nutrients declines.
Pirt (1979) described the product balance in a fixed volume fed-batch system as:
The life of a variable volume fed-batch fermentation may be extended beyond the time it takes to fill the fermenter by withdrawing a portion of the culture and using the residual culture as the starting point for a further fed-batch process. The decrease in volume results in a significant increase in the dilution rate (assuming that the flow rate remains constant) and thus, eventually, in an increase in the specific growth rate. The increase in ¡x is then followed by its gradual decrease as the quasi steady state is re-established. Such a cycle may be repeated several times resulting in a series of fed-batch fermentations. Thus, the organism would experience a periodic shift-up in growth rate followed by a gradual shift-down. This periodicity in growth rate may be achieved in fixed volume fed-batch systems by diluting the culture when the biomass reaches a concentration which cannot be maintained under aerobic conditions. Dilution results in a decline in x and, thus, according to equation (2.35) an increase in fx. Subsequently, as feeding continues, the growth rate will decline gradually as biomass increases and approaches the maximum sustainable in the vessel once more, at which point the culture may be diluted again. Dilution would be achieved by withdrawing culture and refilling to the original level with sterile water or medium not containing the feed substrate.
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