A) Write a pair of differential equations which describe the concentrations of b

Question

A) Write a pair of differential equations which describe the concentrations of biomass and substrate in the reactor. For simplicity, assume the volume, V is one. B)question on image 1. Consider a bioreactor containing cells (biomass) and nutrients (substrate). Let ri and t2 be the concentrations (mass/volume) of the biomass and substrate. Let ciy and z2f be the correspond- ing concentrations in the feed stream, which flows in at rate u (volume/time). The well-mixed solution in the reactor flows out at rate u, so that the volume V is constant. Let ri denote the rate of biomass generation (mass/volume/time) and r2 the rate of substrate consumption (mass/volume/time). C1 C1 C2 (a) Write a pair of differential equations which describe the concentrations of biomass and substrate in the reactor. For simplicity, assume the volume, V, is one. (b) Suppose the generation rate is given by and that the yield, Y = is constantly equal to and that the volumne of the tank is one unit. Taking zif = 0, 2a,-5, and u find all steady states of the system. (Steady state means there is no change in the state over time.) (c) Find the matrices (A, B) which describe the linearization of the system about the positive (that is, nonzero biomass) equilibrium point found in part (b). Note: define the state and input variables for which this equilibrium lies to be at the origin.

Explanation / Answer

The differential equations would be

dX1 / dt = - K x1

dX2 / dt = - Kx2

Here k = rate constant

x1 and x2 are the concetration of biomass and substrate. These are ist order differential equations describing the concentration changing with respect to time of both Biomass and substrate.

c) The value of x1 and x2 at orignin would be found by using solution of these two differnetial equation. The solution would be

x1 = X1o e-kt

x2 = X2o e-kt

At t = o

x1= x1o

x2= x20

The equilibrium lies on the origin when t = 0 ......

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