8/19/2019 2016 Che 151 Homework 5
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ChE 151: Process Simulation and Mathematical Techniques for
Chemical EngineersProf. Davis
Spring 2016 Homework 5: 30 points totalDue Thursday, March
31st (at the beginning of class)
1. (9 points) Answer the following in 2-3 sentences (3
points each):
A. What is a stiff differential equation and why is it
bad? (See section 5.8 of the book.)
B. Explain the primary difference between single step and
multistep methods for solvingIVPs numerically.
C. How can predictor-corrector methods be better than
single-step methods for solving
IVPs? In what way are they worse?
2.
(21 points) You’re working as a process engineer for a
pharmaceutical company and you’rein charge of a batch reactor. The
reactor, which is supposed to operate isothermally at 320 K,makes a
certain isomer of a drug from a racemic mixture:
k
A B→
One day you notice there’s a slow periodic disturbance in the
temperature in the reactor. Youcheck your controller data and the
reactor temperature (T) seems to be following a regular
pattern. You plot the data and you were able to fit the
curve with a periodic function; the
equation T = λ 1cos(λ 2t) + 320, where λ 1 =
60 K and λ 2 = π/120 s-1
seemed to fit it okay.Assuming the reaction is first order
and elementary, the design equation for the batch reactor is:
3 1 0 3
0 0, 4 10 , 5000 , 3.06 /act
E
act A RT
A A A
E dC kC k e C k s K C mol m
dt R
−−
= − = − = × = =
A. (6 points) Write your own algorithm in Matlab which
implements Euler’s method tosolve numerically for CA(t). I want a
code that solves THIS PROBLEM, not a general
problem; you should be able to run the algorithm in one
*.m file without declaring an
anonymous function or making/using any subroutines. State the
step size you picked(and state why you picked it) on your hard copy
submission of your code. The batch runs
for 30 minutes; the output of the code should be the
concentration of B (not A) at 30
minutes, since you care about making B. Code with not enough
comments is bad code.B. (6 points) Modify your Euler code
from part A. so that it solves the problem for CA(t)
using the Runge-Kutta order 4 method as well. Call the vector of
guesses “RK” and not
“w” (the guesses from the old code should be called “Euler”). No
need to write a
separate file; calculate this answer in the same loop as the
Euler answer. I want everyoneto write their own *.m file named
“LastnameFirstnameEulerRK.m” which solves the
problem using both methods and submit it electronically to
Moodle. If your code does
8/19/2019 2016 Che 151 Homework 5
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not run on my machine as a single *.m file, you will not get
full credit. Don’t forget to
submit a screen shot of your command window with the output,
CB(30 min).
C. (3 points) Note in your submission (on your paper, not
in the comments of your code) the primary differences between
RK4 and Euler.
D. (3 points) Run your algorithm for step sizes of 1
second, 10 seconds, and 1 minute. Make
a table of the results at t = 30 minutes for Euler and RK for
each of the three cases. Also plot the concentration of A for
each step size and submit a copy of your plot.
E. (3 points) Comment on the differences between your
plots in part D. and what they imply
for the accuracy of the two methods in 2-3 sentences, i.e. how
much error is there in each
approximation method for different h values?
3. (0 points) List the number of hours you spent on work
for this class for this week, not
counting time spent in class. Approximate to the nearest half
hour. Put it on the very top ofyour homework, under your name and
the names of your other group members like this:
Name 1
Name 2 Name 3 (if applicable)
Hours spent on 151 this week: 5.5 hours
Objectives addressed on this assignment:
• Identify the general type of numerical method needed to
solve a problem (IVP, system of
NLEs, etc.)
• Explain what Euler’s method is and how it relates to
solving ODEs
• Give bounds on the error in the solution to a problem
using a numerical method
• Write an algorithm to solve a chemical reactor design
IVP using a Runge-Kutta method
•
Effectively implement algorithms in Matlab to solve various
problems from chemicalengineering numerically
