## Solving CSTR design equations

Posted February 18, 2013 at 09:00 AM | categories: nonlinear algebra | tags: reaction engineering | View Comments

Updated March 06, 2013 at 04:29 PM

Given a continuously stirred tank reactor with a volume of 66,000 dm^3 where the reaction \(A \rightarrow B\) occurs, at a rate of \(-r_A = k C_A^2\) (\(k=3\) L/mol/h), with an entering molar flow of F_{A0} = 5 mol/h and a volumetric flowrate of 10 L/h, what is the exit concentration of A?

From a mole balance we know that at steady state \(0 = F_{A0} - F_A + V r_A\). That equation simply states the sum of the molar flow of A in in minus the molar flow of A out plus the molar rate A is generated is equal to zero at steady state. This is directly the equation we need to solve. We need the following relationship:

- \(F_A = v0 C_A\)

from scipy.optimize import fsolve Fa0 = 5.0 v0 = 10. V = 66000.0 # reactor volume L^3 k = 3.0 # rate constant L/mol/h def func(Ca): "Mole balance for a CSTR. Solve this equation for func(Ca)=0" Fa = v0 * Ca # exit molar flow of A ra = -k * Ca**2 # rate of reaction of A L/mol/h return Fa0 - Fa + V * ra # CA guess that that 90 % is reacted away CA_guess = 0.1 * Fa0 / v0 CA_sol, = fsolve(func, CA_guess) print 'The exit concentration is {0} mol/L'.format(CA_sol)

The exit concentration is 0.005 mol/L

It is a little confusing why it is necessary to put a comma after the CA_sol in the fsolve command. If you do not put it there, you get brackets around the answer.

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