Ethyl acetate (A) undergoes a reaction with sodium hydroxide (B) to form sodium acetate and ethyl alcohol: 

The reaction is carried out at steady state in a series of stirred-tank reactors. The output from the ith reactor is the input to the (i + 1)st reactor. The volumetric flow rate between the reactors is constant at V̇(L/min), and the volume of each tank is V(L). 

The concentrations of A and B in the feed to the first tank are C_A0 and C_B0 (mol/L). The tanks are stirred sufficiently for their contents to be uniform throughout, so that C_A and C_B in a tank equal C_A and C_B in the stream leaving that tank. The rate of reaction is given by the expression 

where k[L/(mol
min)] is the reaction rate constant. 

(a) Write a material balance on A in the ith tank, and show that it yields 

where τ = V/V̇ is the mean residence time in each tank. Then write a balance on B in the ith tank and subtract the two balances, using the result to prove that

(b) Use the equations derived in Part (a) to prove that 

and from this relation derive an equation of the form 

where α, β, and γ are functions of k;C_A0;C_B0; C_A;i-1, and τ. Then write the solution of this equation for C_Ai . 

(c) Write a spreadsheet or computer program to calculate N, the number of tanks needed to achieve a fractional conversion x_AN ≥ x_Af at the outlet of the final reactor. Your program should implement the following procedure: 

(i) Take as input values of k, V̇, V, C_A0 (mol/L), C_B0(mol/L), and xAf. 

(ii) Use the equation for C_Ai derived in Part (b) to calculate C_A1; then calculate the corresponding fractional conversion x_A1. 

(iii) Repeat the procedure to calculate C_A2 and x_A2, then C_A3 and x_A3, continuing until x_Ai ≥ x_Af.

Test the program supposing that the reaction is to be carried out at a temperature at which k = 36.2 L/(mol•min), and that the other process variables have the following values:

Use the program to calculate the required number of tanks and the final fractional conversion for the following values of the desired minimum final fractional conversion, x_Af: 0.50, 0.80, 0.90, 0.95, 0.99, 0.999. Briefly describe the nature of the relationship between N and x_Af and what probably happens to the process cost as the required final fractional conversion approaches 1.0. Hint: If you write a spreadsheet, it might appear in part as follows: 

(d) Suppose a 95% conversion is desired. Use your program to determine how the required number of tanks varies as you increase (i) the rate constant, k; (ii) the throughput, V̇; and (iii) the individual reactor volume, V. Then briefly explain why the results you obtain make sense physically.

Transcribed Image Text:

CH3COOC,H5 + NaOH CH3COONA + C,H5OH (А) (В)