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

Submit your assignment

Click here to order this assignment @Essaywriting.us.No Plagiarism.Written from scratch by professional writers.

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

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)])

Click here to order this assignment @Essaywriting.us.No Plagiarism.Written from scratch by professional writers.

Leave a Reply

Your email address will not be published. Required fields are marked *