A Fifth Set of Problems demonstrating some of the capabilities of Maple:
Laboratory Problems
These problems may be completed with help from any of your fellow students (as well as the instructor). You may not copy anyone else's work, but you can get other users to give you suggestions and point out mistakes that should be corrected.
Problems should be completed using the tools suggested and the results stored in a file that the instructor can look at.
1) Example problem 15.3 from Systematic Methods of Chemical Process Design considers the following problem:
The total feed to two reactors (x0 kmol/hr) of compound A is split so that x1 kmol/hr goes to the first reactor and the rest (x2) to the second reactor. The reaction A->B occurs in each reactor. The conversion is 80% in the first reactor and 66.7% in the second. The total production of B from the reactors must be 10 kmol/hr. Compound A costs $5/kmol. The first reactor costs 5.5(x1)0.6 $/hr to operate. The second reactor costs 4.0(x2)0.6 $/hr to operate. Use Maple (together with mass balances) to determine the total cost of the operation. Check your result with the formula given in Lab Problem 3 of assignment 2.
2) Use Maple to differentiate the result of problem 1 with respect to the flow to reactor 1. Does this agree with the result given in test problem 3 in assignment 2?
3) Suppose we have a binary solution of two compounds: A and B. The molecular weights of the compounds are: MA and MB. In one cubic meter of the solution, there is rA kg of A and rB kg of B. The total mass density is r and is the sum of rA and rB. The mass fraction of A is wA and is rA/r.
Analogous to the mass quantities: molar quantities may be defined: cA and cB for molar concentrations of the species and c for the total. Thus cA = rA/MA and c = cA + cB. Define xA and xB as the mol fractions in the solution.
The molecular weight of the mixture is defined as M=r/c.
From these basic definitions, use Maple to show that:
Problem (4) was particular to the Rice University course and has been omitted.
Test Problems
You may work these problems with help ONLY from the course instructor.
1) For the same problem described in Lab problem 3:
2) The set of autocatalytic reactions
A + B -> 2 B, with rate =
k1[A]2[B]
A + C -> 2 B, with rate =
k2[A][C]
B + C -> 2 C, with rate =
k3[B][C]
is carried out in an isothermal, perfectly mixed,
stirred tank reactor.. The rate constants are k1=1
liter2 mole-2 s-1, k2=1
liter mole-1 s-1, and k1=1
liter mole-1 s-1. The feed to the reactor has
none of species B or C, and has a concentration of A given by [A]o.
The reactor residence time is t.
From the reaction stoichiometry, we can see that [C] = [A]o
-[A] - [B]. The reactor mole balances may therefore be written as
t([A]o-[A])
- k1[A]2[B] - k2[A]([A]o
-[A] - [B]) = 0, and
t[B]
+ k1[A]2[B] + k2[A]([A]o
-[A] - [B]) - k3[B]([A]o -[A] - [B]) = 0
Use maple to solve these equations for [A] and
[B]. Examine all of the possible solutions and determine which can
be physically realistic (all concentrations greater than or equal to zero)
and for what values of [A]o and t
they are physically realistic.
The stability of a given steady state operating
condition may be determined by examining the eigenvalues of the Jacobian
of the mole balance equations. Use Maple to find the Jacobian of
the equations (the matrix whose elements are the derivatives of the equations
with respect to the solution variables ([A] and [B]). Use Maple to
construct the Jacobian of the above two equations and find its eigenvalues.
Evaluate these eigenvalues for each of the physically realistic steady-state
solutions. Put in numbers for [A]o and t
and
determine the steady state solutions and evaluate their stability.
Repeat this for a few values of [A]o and t.