Handin 2: Numerical Solution of the CSTR Problem

What follows is a formulation of the equations for a reactioncarried our in a continuous stirred tank reactor. It is somewhat different from that found in most textbooks since the intention here is that the equations should be solved numerically.

The reaction which occurs is:

A -> B

This reaction is unimolecular, but not necessarily assumed to be first order. The reactor is fed with pure A at a rate F m³/hr. The molar density of A (and B) is d kmol/m³. The following equations apply:

Rate equation:
R = - V k C_aⁿ
Species A balance:
F d = F C_a - R
Species B balance:
0 = F C_b + R

the flows of A and B out of the reactor may be calculated by noting that these are equal to the product of the respective concentration and the feed rate F since the total molar flowrates in and out of the reactor must be equal. Other symbols:

R is the extensive rate (extent) of the reaction in kmol/hr
V is reactor volume in m³
k is the rate constant on a per hour basis; its units depend on reaction order
n is reaction order and may be nonintegral
C_x is the concentration of species x in kmol/m³

These equations have an analytical solution only for integral values of reaction order n, and convenient solutions only for zero and first order, when all equations are linear. for the first order case the well known analytical solution can be obtained:

C_a = d / (1 + kV/F)

Exercise

Before starting it is suggested you read these sections of the online notes.

A reactor of 4m³ is fed with 10m³/hr of A whose molar density is 10kmol/m³. Determine, by solving the above equations numerically, the concentration of A and B in the reactor and its outlet and the production rate of B in kmol/hr, p_b, given by:

p_b = F C_b

Set up the equations with n as a parameter so that you may change its value, which will not necessarily be one.

Case 1: the reaction may be assumed to be first order (i.e. n=1) with rate constant 2h^-1. Compare analytical and numerical solutions for the reactor A concentration.
Case 2: the reaction is believed to have effective order 1.47 with a rate constant 1.7 units/hour.

In preparing your submission should do the following:

Identify and list the variables and parameters in this problem.
Write down the relevant equations and confirm that there are enough to solve the problem. (Note that you may have to define additional variables and/or equations.)
Produce an ordered incidence table for the system.
Write and solve the models.
Record your answers. To set up and solve the model you should use the online ordered equation solver in the Modelling Lab.