Suppose that the fed-batch bioreactor in Fig. 2.11 is converted to a continuous, stirred-tank bioreactor (also called a chemostat) by adding an exit stream. Assume that the inlet and exit streams have the same mass flow rate F and thus the volume of liquid V in the chemostat is constant. (a) Derive a dynamic model for this chemostat by modifying the fed-batch reactor model in Section 2.4.9. (b) Derive the steady-state relationship between growth rate f.L in Eq. 2-93 and dilution rateD where by definition, D = FIV. Suggest a simple control strategy for controlling the growth rate based on this result. (c) An undesirable situation called washout occurs when all of the cells are washed out of the bioreactor and thus cell mass X becomes zero. Determine the values of D that result in washout. (Hint: Washout occurs if dX/dt is negative for an extended period of time, until X= 0.) (d) For the numerical values given below, plot the steadystate cell production rate DX as a function of dilution rate D. Discuss the relationship between the values of D that result in washout and the value that provides the maximum production rate. The parameter values are: f.Lm = 0.20 h - 1; Ks = 1.0 g/1,

Problem 2.15Suppose that the fed-batch bioreactor in Fig. 2.11 is converted to a continuous, stirred-tankbioreactor (also called a chemostat) by adding an exit stream. Assume that the inlet and exitstreams have the same mass flow rate F and thus the volume of liquid V in the chemostat isconstant.(a) Derive a dynamic model for this chemostat by modifying the fed-batch reactor model inSection 2.4.9.(b) Derive the steady-state relationship between growth rate f.L in Eq. 2-93 and dilution rateDwhere by definition, D = F/V. Suggest a simple control strategy for controlling the growth ratebased on this result.(c) An undesirable situation called washout occurs when all of the cells are washed out of thebioreactor and thus cell mass X becomes zero. Determine the values of D that result inwashout. (Hint: Washout occurs if dX/dt is negative for an extended period of time, until X= 0.)(d) For the numerical values given below, plot the steady state cell production rate DX as afunction of dilution rate D. Discuss the relationship between the values of D that result inwashout and the value that provides the maximum production rate. The parameter values are: m 0.20 h ; Ks= 1.0 g/1, x/s= 0.5 g/g. The steady state condition is D =0.1 hX= 2.25 g/L ,S = 1.0 g/L, and Sf= 10 g/L. Step-by-step solution Step 1 of 7 ^An exit stream added to the bioreactor is converted to a continuous stirred-tank bioreactorcalled chemostat. The inlet and exit streams have flow rate, F. The volume of the liquid V in thechemostat is constant.(a)The dynamic model for the chemostat is obtained as follows:Write the differential equation for cells. V dX = V r FX dt gSimplify further to obtain the following expression. dX = rg ( ) X dt VThe differential equation for the product can be expressed as follows: dP V dt = V r p FPSimplify to obtain an expression as follows:\n dP dt = r p ( ) P V