ChE 384 Numerical Methods for Chemical Engineers, Fall 2018

Exam Preparation Assignment No.1 (due by 11:59 pm on Wednesday, August 29)

Please note the following:

• No late problem sets will be accepted; do not leave the problem set for the last minute.

• Your homework solution must show your individual effort! Do not forget to provide

a signed ChE 461 Student Academic Integrity statement to the instructor. You can

find it in the Course Information section of the myASU website.

• Please submit a clear and legible scan of any handwritten responses to facilitate grading

of your assignment; as noted below,

Problem 1. Consider the falling parachutist problem described in lecture, and represented

by the differential equation

m

dν

dt = mg − cν (1)

a) Substitute the backward difference approximation for a derivative in (1), and rearrange

the equation to obtain an expression in the form

ν(ti+1) = f(ν(ti), h)

where h is the step size. Hint: first obtain an expression in terms of ν(t

0

i

) = f(ν(t

0

i−1

), h).

Then perform a change of variables (i.e., t

0

i = ti+1, t

0

i−1 = ti) to get the desired form.

b) Use MATLAB to compare the numerical solution developed in a) with the numerical

solution based on forward differences and the analytical solution. A nearly complete

M-file to assist you in this comparison has been provided with this homework set.

Augment your program to plot the responses with a thicker linewidth (than default) and

also to include a legend. Submit a plot showing the responses of all three techniques

for a final time of 12 sec and an step size h of 2 seconds. Please answer the following

questions:

1. At this step size, are the approximations “accurate” and/or “precise” (explain)?

2. Re-execute your program using a step size of 0.2 seconds; include a plot of the

resulting response. What has occurred to the precision/accuracy of the numerical

approximations?

Please use Publish to both document and evaluate your m-file; submit PDFs of your

results, as well as your source .m file. Please note that you may have to comment the

input statements (and replace these with fixed values) in order to be able to evaluate

the .m file with Publish. Submit your source file as XYZfirst.m (where XYZ are your

initials).

c) (Extra credit): Augment your program (call this XYZfirst2.m) to include default values

in the input statements (as illustrated by the instructor). Submit the .m file, as well

as proof that you have a working program.

Problem 2. Chapra Problem 4.11.

% ChE 384 Fall 2018

% OUR FIRST MATLAB M-FILE: The Falling Parachutist Problem

g = 9.8; % Acceleration due to gravity, m/sec^2

c = 12.5; % drag coefficient, kg/sec

m = 68.1; % parachutist mass, kg

% This initializes the vectors needed for the computation

nu = [0];

nu2 = [0];

nu3 = [0];

% Input information

% tf = 12;

% delt = 2;

tf = input(’ Enter final time: ’);

delt = input(’ Enter step size: ’);

incr = round(tf/delt); % Round to nearest integer

for ti = 1:incr

% Numerical Solution – Forward Difference

nu(ti+1) = nu(ti) + (g – c*nu(ti)/m)*delt;

% Analytical Solution

nu2(ti+1) = (m*g/c)*(1 – exp(-c*ti*delt/m));

% Numerical Solution – Backward difference

nu3(ti+1) = ** answer from part a) implemented here **

end

time = delt*[0:incr];

plot(time,nu,’–’,time,nu2,’-’,time,nu3,’:’)

xlabel(’ Time (sec) ’)

ylabel(’ Velocity (m/sec) ’)

title([’YOUR NAME HERE; Solid: analytical; dashed: forward; dotted: backward; step size is ’, num2str(delt)])

# Substitute the backward difference approximation for a derivative in (1), and rearrange the equation to obtain an expression in the form

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