Question

A microbial cell may be modeled as a well-mixed reactor in which nutrients (“corn steep liquor”) diffuse into the cell and are converted to products (including penicillin) which diffuse out of the cell. Write unsteady-state material balances on the nutrients (A) and on the products (B) in terms of the rate of consumption of A (rA), the yield of B (YB/A) defined as the mass of B produced per mass of A consumed, a mass-transfer coefficient (kA) for transport of nutrients into the cell, a mass-transfer coefficient (kP) for transport of product out of the cell, the surface area of the cell (A), the volume of the cell (V), the concentrations of nutrients outside and inside the cell (CAo, CA), and the concentrations of products outside and inside the cell (CPo, CP).

Answer 1

Defining the variables and parameters :

C_A = Concentration of A at any time 't' , mol/m³

C_P = Concentration of P at any time 't' , mol/m³

Rate of consumption of A = r_Amol/m³.sec

Yield of B = Y_BA moles of B / Mole of A consumed

Mass transfer coefficient of Nutrients into the cell = k_A , m/sec

Mass transfer coefficient of Products out of the cell = k_P , m/sec

Surface area of the cell = A, m²

Volume of the cell = V, m³

Concentration of nutrient outside = C_A0, mol/m³

Concentration of products outside the cell = C_P0, mol/m³

Assumptions :

1) The whole cell is considered as a single reactor so there at each point the concentration is same.

2) The Mass transfer occurs due to the gradient of concentration between the inside and outside of the the cell.

3) The reaction occurs inside the cell with the products leaing the cell due to concentration gradient of product.

4) All the working conditions such as temperature, pressure are assumed to be constant.

5) The reaction is complete that is no reactant leaves in the product side.

Rate of change of concentration of A inside the cell at any time t = Rate of mass transfer insie the cell - Rate of consumption of A

VdC_A/dt =  (k_AA)( C_A0 - C_A ) - Vr_A.

Rate of change of concentration of P inside the cell at any time t  = Rate of production of B - Rate of Transfer of Products from Cell to outside

VdC_p/dt = Y_BA * (V*r_A) - (k_PA)(C_P0 - C_P)

If the reaction isn't complete.then A will also be coming out of the cell. Let the mass transfer coefficient of A leaving the cell be k_A' and let C_A' be the steady state concentration of A exiting the cell

Rate of change of concentration of A inside the cell at any time t = Rate of mass transfer inside the cell - Rate of consumption of A - Rate of transfer of A exiting the cell

VdC_A/dt =  (k_AA)( C_A0 - C_A ) - Vr_A- (k_A'A)( C_A - C_A' )