Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
June 8 23, 2015
Chapter 2: Lecture 3a:
Method of Separation of Variables
Brian G. Higgins
Department of Chemical Engineering and
Materials Sci
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
June 8 23 2015
Chapter 2: Lecture 3b:
Linear Operators
Brian G. Higgins
Department of Chemical Engineering and
Materials Science
University
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
June 8 23, 2015
Chapter 1: Lecture 1:
Overview
Brian G. Higgins
Department of Chemical Engineering and
Materials Science
University of Cal
Problem 1.4.1(f)
The differential equation and BCs at steady state are given by
d u
Q
du
+ , u H0L = T , HLL = 0
Ko
dx
dx2
2
where Q K0 = x2 . The general solution (after integrating twice) is
x
u HxL =  + c1 x + c2
12
4
Applying the BCs gives
c2 = T
a
HANOI UNIVERSITY OF MINING and GEOLOGY
Advanced Program in Chemical Engineering
ECH140: Mathematical Methods for Chemical Engineering
Final Exam Solution
September 17, 2013 (9:0011:00 )
(Closed book and notes no cell phones; no questions asked or answere
ECH140 Practice Final
September 2013
No solutions will be provided, as at least one of the problems below will be on the Final Exam!
Problem 1
You are required to analyze the temperature distribution in a spherical shell with a heat source. The
mathematic
Department of Chemical Engineering & Materials Science
University of California, Davis
ECH 140 Practice Exam
November 2006
(Closed book, closed notes, no calculators, no cell phones)
Problem 1 (15 points)
Determine the general solution for the following O
Hanoi University of Mining and Geology
Chemical Engineering Program
ECH 140 Mathematical Methods
September 2013
Homework Assignment #3: Solutions
Problem 2.3.1(a)
The PDE is given by
u
k
u
= Jr N
t
r r
r
We let u Hr, tL = G HtL f HrL and on subsitution
Hanoi University of Mining and Geology
Chemical Engineering Program
ECH 140 Mathematical Methods
September 2013
Homework Assignment #1: Solution
Problem_1.3
Find the general solution of the following ODE
1
x
2
x
Ix2 yM + y = 0
Solution
First we rearrange
One one is faced with solving a nonhomogeneous PDE by separation of variables, often the approach
one must take is to first solve the steady state problem and then using the linearity properties of the
PDE is to writte the solution as
( ) = ( ) + ()
suc
Undergraduate Lectures
ECH140: Mathema8cal Methods for Chemical Engineers
March 27 April 7, 2017
Lecture 9:
Introduction to Laplace Transforms
Brian G. Higgins
Department of Chemical Engineering and
Materials Science
University of California, Da
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
March 27 April 7, 2017
Chapter 2: Lecture 3b:
Linear Operators
Brian G. Higgins
Department of Chemical Engineering and
Materials Science
University of California, Davis
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
June 8 23, 2015
Chapter 1: Lecture 2c:
Cooling of a Sphere
Brian G. Higgins
Department of Chemical Engineering and
Materials Science
Univers
There are several ways to program Mathematica to implement a finite difference method for solving
ODEs and PDEs. In these notes we illustrate a method that makes use of
to solve a linear boundary value problem.
With this approach the finite difference
Consider steady state heat conduction in a rod of length , and radius . The mathematical formulation of the problem is as follows:
+
=
< < < <
( ) =
( ) =
( ) =
where is a constant.
Use the method of separation of variables to find the ste
ECH140: Mathematical Methods for Chemical Engineering
Please note that at least one or perhaps two of the practice exam problems will be on the First exam
Determine the general solution for the following ODEs:
()
()
()
+
Find the general solution of the following ODE
+ =
Find the general solution to the following ODE
+ =
The growth of a biological population P(t) is given by the Verhulst Pearl equation
= 
where P is the limiting size of the population, beyond which t
Find the general solution of the following ODE
+ =
First we rearrange the equation as follows
= 
Then we introduce the new variable p = 2 y. Our ODE becomes
= 
Integrating once gives
() = 
Finally we make the substitution p = 2 y to get
() = 

Use the method of Laplace Transforms to solve the following ODEs :
+
+ = () () =
() =
() = (  )  (  ) + (  )
Use the method of Laplace Transforms to solve the following ODEs :
Hint:
 = () () =
() =
You will nee
In this homework assignment you are required to determine the steady state temperature ina 1D rod.
The governing equation is
+ =
Solving gives
() = 
+ +
The constants C1 and C2 are determined by the BCs
= () =
() =
Thus
() = 
+ +
() = =
HANOI UNIVERSITY OF MINING and GEOLOGY
Advanced Program in Chemical Engineering
ECH140: Mathematical Methods for Chemical Engineering
First Exam Solution
September 9, 2013 (9:0011:00 )
(Closed book and notes no cell phones; no questions asked or answered
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
June 8 23, 2015
Chapter 1: Lecture 2a:
Derivation of Heat Conduction Equation
Brian G. Higgins
Department of Chemical Engineering and
Materials
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
June 823, 2015
Chapter 1: Lecture 4C:
Steady State Conduction in a Sphere with Source
Brian G. Higgins
Department of Chemical Engineering and
Ma
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
June 8 23, 2015
Chapter 1: Lecture 2b:
Derivation of 1D Heat Conduction Equation
with Variable CrossSection
Brian G. Higgins
Department of Chemical
Undergraduate Lectures
ECH140: Mathema8cal Methods for Chemical Engineers
June 8 June 23, 2015
Chapter 3: Lecture 4d:
Fourier Series
Brian G. Higgins
Department of Chemical Engineering and
Materials Science
Universi
Use partial fractions to determine the coefficients A< B< C, in the following expression
1
s
(s + 1)2
=
A
s
+
B
(s + 1)
+
C
(s + 1)2
Consider the following ODE :
d2 y
dt2
+2
dy
dt
+ y = f (t), y (0) = 0,
dy
dt
(0) = 0
where
f (t) = (t  1)  2 (t  2) +
Undergraduate Lectures on
ECH140: Mathema9cal Methods for Chemical Engineers
March 27 April 7, 2017
Chapter 2: Lecture 4b:
Transient Heat Conduction:
Flux Boundary Conditions
Brian G. Higgins
Department of Chemical Engineering and
Materials Scien
Undergraduate Lectures
ECH140: Mathema8cal Methods for Chemical Engineers
March 27 April 7, 2017
Chapter 5: Lecture 6a:
SturmLiouville Eigenvalue Problem
Brian G. Higgins
Department of Chemical Engineering and
Materials Science
University of Ca
ECH140: Mathematical Methods for Chemical Engineering
Total Mass Balance for fixed Control volume
t
V + v n A = 0
Species Mass Balance
t
A V + A vA n A = rA V, A = 1, 2, , N
Table of Laplace Transforms
Consider the following PD
ECH140: Mathematical Methods for Chemical Engineering
Consider the following problems:
u
(a)
t
= k ( x)
BC1 :
BC2 :
u
t
+ q(x),
0<x<L
=
x
K ( x, u)
u
x
+ 2 u = 0,
0 < x < e,
u(x, 0) = f (x)
IC :
(c)
x2
u(x, 0) = f (x)
u
(0, t ) = G1
x