 
Homework assignments posted here are subject to correction in class or
through other means. Problems as assigned here are for your convenience
but are not a substitute for obtaining assignments in class.
Assignments as issued in class supersede these assignments unless otherwise
noted.
Homework Assignment: 1 2 3
4 5 6 7
8 9 10 11
12 13 14
Modeling and Computing
Problems for Class:
1. You will likely use three primary computer based tools during this
course. One will be a programming language, such as FORTRAN or Visual Basic. The
second will be a CAS, such as Mathcad or Maple. The last will be a spreadsheet.
For the program (or language) you expect to use in each of these three
categories, you should determine the number of significant digits in the
mantissa for a floating point variable and an integer variable. You may express
this as a range of base 10 values. Note that in some cases this will involve
consulting a manual or help file, but in others may require another approach.
2. Explain why the machine epsilon is often used as a convergence
criterion in an iterative calculation, instead of continuing to iterate until
the difference in values between successive iterations is zero.
3. Give one example of an engineering problem where each of the following
classes of numerical methods can come in handy. If possible, draw from your
experience in class and in readings or from any professional experience you have
had to date.
 Roots of equations 
 Linear algebraic equations 
 Curve fitting; regression and interpolation 
 Optimization 
 Integration 
 Ordinary differential equations 
 Partial differential equations 
Problems for Submission:
4. Examine your engineering textbooks and find four examples where
mathematical models are used to describe the behavior of physical systems. List
the independent and dependent variables as the parameters and forcing functions.
5. Problem 3.8 in your text
6P (based on 3.3 in your text) Determine the machine epsilon for the
computers in the computer lab. You may program in any language that requires you
to manually write the code to determine the machine epsilon (you may not use the
MATLAB system variable eps, for example).
You may use one of the templates available on the course web site for
programming in VB or FORTRAN. Download the ZIP file and expand it to a directory
called epsilon on your Z: drive.
FORTRAN: In your Z:\epsilon\ folder there are several files. Start Compaq
Visual Fortran and open the workspace epsilon.dsw in the aforementioned folder.
The only source file you need to edit is epsilon.f90. All of the other files,
however, are required to generate the working windows program. The comments in
epsilon.f90 contain information on the variables already defined for you that
you must use for the program to function as intended. You may need to declare
other variables in your code.
VB: In your Z:\epsilon\ folder is the MachineEpsilon project file. You will
edit the Epsilon module. The form and its associated code, including the
interface between the form and the global variables is already created. Note the
comments in the module file.
In either language, you can compile the code without editing anything. The
results will be meaningless, but you will be able to see how the finished
program will operate. There is very little error checking, so take care to use
appropriate input, most importantly keep the stepping factor between 0 and 1.
In both cases, submit a printout of your code and a screenshot of the result
you obtain. Does the result change if your inputs change?
Note that the "proper" way is to
start with a value of one with a stepping factor of 0.5.
Assignment Learning Objectives:
 Establish the role of modeling in chemical engineering (3, 4) 
 Analyze models in terms of applicable numerical methods (3) 
 Show the role of the machine epsilon and related truncation errors in
computing (1, 2, 5, 6) 
Reading Assignments:

Thursday (1/14): Chapters 13: Computers and Numerical Methods 

Tuesday (1/19): Chapter 4: Taylor Series;
Error Propogation and Uncertainty 


Taylor Series, RootFinding
Problems for Class:
1. Examine the Taylor series about x
=0 for e^{x} when x=4. The fourth term in the series, x^{3}/3!
is larger than the third term, x^{2}/2! Does this mean the series is
divergent? Explain the apparent anomaly.
2. 4.12 a, b from your text
3. 5.6 from your text
Problems for Submission:
4. 4.8 from your text
5. In a coffeepot, water flows by gravity from a reservoir into a heated
tube. The water is heated from room temperature (20 °C)
to 100 °C. Some of
the water is vaporized, and the expansion of steam from the volume occupied by
the equivalent mass of water to the volume occupied by the steam at atmospheric
pressure provides the work to move the water from the heated tube at the base of
the coffeepot to the top of the coffeepot. The water then flows through the
basket of coffee grounds via gravity. Assuming no energy is lost to the
surroundings, what is the minimum amount of energy that must be supplied by the
electric heater (which heats the water in the tube) to make 750 mL of coffee?
The base of the reservoir is 5 cm above the heated tube. The top of the basket
into which the coffee flows is 20 cm above the heated tube. CLEARLY STATE THE
MODEL EQUATIONS REQUIRED TO SOLVE THIS PROBLEM.
6. Consider two functions g(x) and h(x), related in such a way that g’(x)=h(x)
and h’(x)=g(x) and that g(0)=0 and h(0)=1. Find the Taylor series expansions
for g(x) and h(x) about x=0 using only this information
Assignment Learning Objectives:
 Use the Taylor series to approximate functions (1, 6) 
 Estimate the error associated with a series of calculations (4) 
 Determine the conditioning of functions (2) 
 Model systems using chemical engineering principles (5) 
 Apply numerical methods for finding roots of algebraic equations (3) 
Reading Assignments:
 Thursday (1/21): pp. 99130 Bracketing Methods for Root Finding 
 Tuesday (1/26): Chapter 6: Open Methods for Root Finding 
Open and Closed Case for RootFinding
Problems for Class:
1. A cylindrical tank in a
horizontal configuration (lying on its side) has a radius of 2 m and a
length of 5 m. If the volume of liquid in the tank is 8 m^{3}
, find the maximum depth of the liquid.
2. 6.2 (by hand) (no 4th edition
equivalent)
3. Cutlip 2.2 Note that
solutions to 2.1 are available on the book website.
Problems for Submission:
4. 6.14 (by spreadsheet) (no
4th edition equivalent)
5. 8.4
6. 8.10
7. Cutlip 9.3
Assignment Learning Objectives:
 Develop model equations for geometric systems (1) 
 Apply rootfinding techniques to chemical engineering models (3, 5, 6, 7) 
 Examine the importance of functional form in fixedpoint iteration (6) 
 Apply numerical methods for finding roots of algebraic equations (16) 
Reading Assignments:
 Thursday (1/28): pp. 176184; 212215
Computer applications of rootfinding; epilogue
pp. 217230; Chapter 9 Introduction to Linear Systems; Gauss
Elimination 
 Tuesday (2/02): Chapter 11
GaussSiedel (HW3 Due) 
Beat the System
Problem for Class:
1.
Chapra 9.11
2.
Cutlip 8.8a, b
3.
The following reactions are proposed to be descriptive of a chemical
process:
1)
CH_{4}
+ CO_{2}
> 2CO
+ 2H_{2}
2)
CO
+ H_{2}O
> CO_{2}
+ H_{2}
3)
CH_{4}
+ H_{2}O
> CO
+ 3H_{2}
4)
CH_{4}
+ 2H_{2}O
> CO_{2}
+ 4H_{2}
Determine how many of these reactions are independent, and give the equations
for two of the possible independent reactions.
Problems for Submission:
4.
Chapra 12.9 (no 4^{th} edition equivalent)
5.
Cutlip 7.14
6.
Cutlip 2.3b,c
7.
Consider heat conduction through the walls of a pipe when the inside wall
temperature is 200^{o}C and the outside wall temperature is 80^{o}C
and the thickness of the wall is 0.05 m.
The inside radius (r_{0}) is 0.05m and the outside radius
is 0.1m.
The differential equation
that describes the temperature distribution is
Applying a finite difference representation of the derivatives of
T with
respect to r gives a set of equations of the following form:
where
and
T_{1}=200 °C and T_{11}=80 °C
Solve for the temperature distribution using a Gauss Siedel program you develop
using the Excel/VBA template on the course website. Check your solution by using the matrix inversion
approach in Excel.
Remember that r is not constant.
Assignment Learning Objectives:
 Solve systems of linear equations using Gaussian
Elimination and other methods
(1) 
 Develop systems of equations describing chemical
engineering systems (25) 
 Apply numerical methods for finding roots of systems of algebraic
equations (26) 
 Solve systems of linear equations using GaussSiedel Iteration (7) 
 Apply concepts of rank and independence to chemical systems (3) 
Reading Assignments:
Thursday (2/04):
Chapter 12: Applications, Epilogue: Part III
Tuesday (2/09):
Chapter 17 CurveFitting. HW4 Due, HW5 AssignedThursday (2/11):
Exam 1 [Modeling, Taylor Series, Rootfinding, Linear Systems]
An assignment you’ll always regress
Problems for Class:
1. 18.1
2. 20.2
3. 20.8
4. Estimate the enthalpy of superheated steam at 741
^{o}F and 400
psia using both linear and quadratic interpolation.
Problems for Submission:
5. Using the following finite difference expression for the second
derivative:
It was found that the error of the approximation was 10^{1}
using a step size Dx of 10^{1}.
How small should Dx be so that the
error of the approximation equals 10^{4}?
6. A heterogeneous reaction is known to occur at a rate described by the
following LangmuirHinshelwood expression:
The following data was obtained at 400K:
P_{A} (atm) 
1 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
P_{R} (atm) 
0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
r 
3.4 x 10^{5} 
3.6 x 10^{5} 
3.7 x 10^{5} 
3.9 x 10^{5} 
4.0 x 10^{5} 
4.1 x 10^{5} 
4.2 x 10^{5} 
A) Using multiple linear regression, try to estimate the values of
k_{1},
K_{A} and
K_{R
}B) Try part A) again, this time using nonlinear regression.
C) The rate constant k_{1} is later determined from initial rate
measurements as 0.015 mol/(s·g catalyst·atm). Reestimate K_{A}
and K_{R} using this information and linear regression.
Make sure you report appropriate units with all calculated values!
7. Consider the following data on the index of refraction of aqueous
solutions of sucrose at 20 ^{o}C:
Percent Sucrose 
Index of Refraction 
10 
1.3479 
15 
1.3557 
20 
1.3639 
25 
1.3723 
30 
1.3811 
35 
1.3902 
Estimate the composition of a water/sucrose solution which has an index of
refraction of 1.3606 by using interpolation and by using regression.
Assignment Learning Objectives:
 Interpolate amongst known data points using first and second order methods
(1, 3, 4, 7) 
 Apply leastsquares regression methods to linear, multivariable linear,
and nonlinear systems (2, 6, 7) 
 Analyze the error associated with numerical differentiation (5) 
Reading Assignments:

Tuesday (2/16):
Exam 1 [Modeling, Taylor Series, Rootfinding, Linear Systems]


Thursday (2/18):
Chapter 20: Epilogue: Applications of Curve Fitting


Tuesday (2/23):
Chapter 23: Numerical Differentiation


Thursday (2/25):
Chapter 21: Numerical Integration


Tuesday (3/1):
Part 4 Prologue, Chapter 13: Optimization 
The integral of the derivative results in this assignment
Problems for Class:
1. Using steam table
data, estimate the value of the following integral at 700^{o}F for
superheated steam using both the trapezoidal method and Simpson’s rule:
2. 23.1
3. 23.14 (using Mathcad or Matlab)
4. Consider the integration of a known function for a fixed interval. If the
step size is cut in half, what would happen to the total truncation error for:
A. the trapezoidal rule?
B. Simpson’s rule?
Problems for Submission:
5. 24.6
6. 24.8
You should review the section on Gauss Quadrature for
guidance
7. 24.9
8. 24.11
Assignment Learning Objectives:
 Apply methods of numerical differentiation to models (2, 3, 5, 7) 
 Apply methods of numerical integration to model equations (1, 6, 8) 
 Analyze the error associated with numerical integration (4) 
Reading Assignments:
 Thursday (2/25): Chapter 21: Numerical Integration 
 Tuesday (3/2): Part 4 Prologue, Chapter 13: 1D Optimization (HW 6
Due, HW 7 Assigned) 
 Thursday (3/4): Chapter 14: Gradient Methods of Optimization. 
 Tuesday (3/9): Chapter 15: Linear Programming (HW 7 Due, HW 8
Assigned) 
 Thursday (3/11): Exam 2 
 Tuesday (3/16): Spring Break 
 Thursday (3/18): Spring Break 
 Tuesday (3/23): Felder Rousseau Chapter 11 
Optimize your time as you prepare for the exam
Problems for Class:
1.
15.11
2.
16.7
3.
16.12
4.
Find a
journal article referring to the use of a heuristic (random search) optimization
technique in a field related to chemical engineering.
Some algorithms to choose from include: simulated annealing, tabu
search, genetic algorithms, and
neural networks.
Each member of the class should choose a different method.
I suggest you limit yourself to journals obtainable online.
In class, you will be expected to briefly explain the method and its
application in a presentation to last no longer than 5 minutes.
Any images you wish to use should be submitted to me electronically at
least 1 hour before class.
Problems for Submission:
5.
13.11
6.
16.6
7.
16.10
8.
You are required to produce an alloy that has at least 30% Pb and at
least 30% Zn by mixing a number of available PbZnSn alloys.
Find the cheapest blend using the method of Lagrange Multipliers.
Available Alloys 
Analysis (%) 
Cost ($/lb) 

Pb 
Zn 
Sn 

1 
20 
20 
60 
6.0 
2 
10 
40 
50 
6.3 
3 
40 
50 
10 
7.5 
4 
50 
30 
20 
8.0 
9.
Submit the complete text of the journal
article (from problem 4), along with a summary (<1 page) indicating how the
technique was used.
You should also
include comments on why this method was chosen by the author and provide a brief
description of the method.
Assignment Learning Objectives:
 Apply methods of unidimensional optimization (5) 
 Find optimal input values for models using computer software (1,
2, 3, 6, 7) 
 Formulate optimization problems (1, 2, 3, 6, 7, 8)

 Develop optimization models and find optimal parameters for those models
using Lagrange multipliers (8) 
 Develop skills required for lifelong learning (4, 9)

 Describe heuristic methods of optimization (4, 9)

Reading Assignments:
 Thursday (3/11): Exam 2 
 Tuesday (3/16): Spring Break 
 Thursday (3/18): Spring Break 
 Tuesday (3/23): Felder Rousseau Chapter 11 
Returning to Where the Fun Began
Problems for Class:
1. Problem 11.4 from Felder/Rousseau (FR)
2. Problem 11.25 (FR)
Problems for Submission:
3. Problem 11.16 (FR)
4. Problem 11.26 (FR)
Assignment Learning Objectives:
 Write and solve separable differential equations representing material and
energy balances 
Reading Assignments:
 Tuesday (3/23):
Felder Rousseau Chapter 11. HW 7 Due, HW 8 Assigned 

Thursday (3/25):
Chapter 25: Numerical Approaches to solving ODEs, HW8 Due, HW9 Assigned 
Don’t be overanalytical
In addition to the instructions for each problem, you should also:
 Completely classify the equation you are solving (which determines the
analytical
method you use to solve) 
 Plot your results. 
Problems for Class:
1. Desalinization plants are used to
purify sea water so it is suitable for drinking. Sea water containing 8 g
salt/kg solution is pumped into a wellmixed tank at a rate of 10 kg/min. Assume
that the balance of the solution is pure water. Because of faulty design work,
water is evaporating from the tank at a rate of 0.5 kg/min. The salt solution
leaves the tank at a rate of 10 kg/min.
A) If the tank is filled initially with 1000 kg of the inlet solution, how
long after the outlet pump is turned on will the tank run dry?
B) Determine the salt concentration in the tank as a function of time.
Problems for Submission:
2. A spherical ice cube (an "ice
sphere") that is 5 cm in diameter is removed from a 0
°C
freezer and placed on a mesh screen at room temperature T_{a}=20
°C.
What will be the diameter of the ice cube as a function of time out of the
freezer (assuming that all the water that has melted immediately drips through
the screen)?
3. Compound
A diffuses through a 4cmlong tube and reacts as it
diffuses. The equation governing diffusion with reaction is
At
one end of the tube, there is a large source of A at a concentration of
0.1 M. At the other end of the tube there is an adsorbent material that
quickly adsorbs any A, making the concentration 0M. If
D = 1 × 10^{6}
cm^{2}/s and
k = 4 × 10^{6} s^{1}, what is the
concentration of A as a function of distance in the tube?
Assignment Learning Objectives:
 Write and solve separable differential equations representing material and
energy balances 
 Solve first and second order differential equations using analytical
methods 
Reading Assignments:
 Thursday (3/25): Chapter 25: Numerical Approaches
to solving ODEs. HW 8 Due, HW 9 Assigned 
 Tuesday (3/30): RungeKutta Methods

 Thursday (4/1): Chapter 27: BoundaryValue
Problems. HW 9 Due, HW 10 Assigned 
 Tuesday (4/6): Review shell balances

Everything Changes With Time
Problems for Class:
1. 25.1
2. 25.8 (Analytically and numerically with RungeKutta)
3. 28.3 (your choice of method)
4. 28.12
Problems for Submission:
5. 10.2 from Cutlip
6. Consider the semibatch reactor discussed in class, now
with the following reaction scheme:
where concentrations are given in mol/L and the reaction rates in mol/Lsec.
Determine the reaction time necessary to attain the maximum concentration of the
intermediate B.
Assignment Learning Objectives:
 Write and solve separable differential equations representing material and
energy balances 
 Solve first and second order differential equations using analytical and
numerical methods 
Reading Assignments:
 Tuesday (4/8):
Chapter 25: Numerical Approaches to solving ODEs. HW 9 Due, HW 10
Assigned 
 Thursday (4/10):
RungeKutta Methods, Review shell balances 
 Tuesday (4/15):
Chapter 27: BoundaryValue Problems. HW 10 Due, HW 11 Assigned 
Oodles of ODEs
Problems for Submission:
Problem for Class:
1. Chapra Problem 28.10 solved using a computer program of your choice.
Problems for Submission:2. Consider the cooling
fin pictured below.
A shell balance results in the following energy balance
expression
where T =T_{w} at
z = 0 and
at z =
L. When
h and
k are constant, an analytical
solution of this BVP can be obtained. But in general, the heat transfer
coefficient h will be a function of
z and the thermal conductivity
of the fin material k will very with temperature. Assume the following
forms for h and
k:
A) Consider the temperature dependent of
k and show that
the governing equation for this system is
B) Solve this problem to three significant figures in
T
using the SOR method and the following values:
h_{0} = 40 Btu/hrft^{2}^{o}F k_{0}
= 60 Btu/hrft^{o}F
a
= 0.02 ^{o}F^{1}
T_{w} = 450
^{o}F T_{0} = 77
^{o}F T_{a}
= 90 ^{o}F
L = 1.5 in B = 0.02 in
3.
Cutlip
Problem
8.3
Assignment Learning Objectives:
 Find solutions to BVP numerically (2,3) 
 Use engineering software to solve systems of ordinary differential
equations (13) 
Reading Assignments:
 Thursday (4/15): Modeling and Simulation

 Tuesday (4/20): HW 11 Due, HW 12 Assigned

 Thursday (4/22): Exam 3

 Tuesday (4/27): HW 12 Due

 Thursday (4/29): Review

 Tuesday (5/4): Comprehensive Final Exam
(10:451:15) 
You Can Simulate This If Needed
Problems for Submission:
1. Problem 10.13 from FelderRousseau
(FR).
2. Develop a simulation in Aspen for the same scenario in 10.13a. Compare
results with your spreadsheet and hand calculations. Why the differences?
Correct for those differences in your Aspen simulation wherever possible.
Assignment Learning Objectives:
 Use engineering software to simulate model processes 
Reading Assignments:
 Thursday (4/22): Exam 3

 Tuesday (4/27): HW 12 Due

 Thursday (4/29): Review

 Tuesday (5/4): Comprehensive Final Exam
(10:451:15) 
Assignment Learning Objectives:
Reading Assignments:
Assignment Learning Objectives:
Reading Assignments:
