Bioreactions are often carried out in batch reactors. The fed-batch bioreactor model in Section 2.4.9 is also applicable to batch reactors if the feed flow rate F is set equal to zero. Using the available information shown below, determine how much time is required to achieve a 90% conversion of the substrate. Assume that the volume V of the reactor contents is constant. Available information:

(i) Initial conditions:

X(0) = 0.05 g/L, 5(0) = 10 g/L, P(0) = 0 g/L.

(ii) Parameter values:

V = 1 L, μm = 0.20 hr–1, Ks = 1.0 g/L,

YX/S = 0.5 g/g, YP/S = 0.1 g/g, YP/X 0.2g/g.