ChE 101 2012
Problem Set 1
1.
Computational problem of the week:
Every week, we will assign one problem designed
to increase your exposure to numerical methods, improve your algorithmic thinking, and get
you familiar with Matlab (you’ll thank us once you get to 103C, 126, and 163). This week
we address perhaps the most important topic in the study of chemical kinetics: the solution
of nonlinear ordinary differential equations. Stripped of its physical richness, all of kinetics
can be reduced to solving the most general initial value problem of the form
y
0
(
t
) =
f
(
t,
y
)
,
y
(
t
0
) =
y
0
(1)
Here,
y
is a vector of coupled variables
{
y
1
, y
2
, ..., y
n
}
, which can include species concen
trations, reactor temperature (for nonisothermal problems), etc.
Unfortunately, Equation
1 cannot be solved analytically except for very special cases in which the equation is either
separable or can be made separable with an integrating factor, so we would like to develop
numerical methods to obtain approximate solutions with arbitrary accuracy.
(a)
Euler’s method:
The simplest of all numerical integrators, and probably a method
you’ve seen in high school calculus.
If we wish to integrate Equation 1 from
t
0
up
to
t
f
, then we divide up the interval [
t
0
, t
f
] into
n
segments with length
h
≡
Δ
t/n
,
where Δ
t
=
t
f

t
0
.
Along each segment, we approximate the function
y
(
t
) by a
straight line, with slope given by
f
(
t,
y
) evaluated at the
preceding
mesh point.
As
such, Euler’s method can be viewed as a linear interpolation scheme for ODEs, with the
simple updating rule
y
i
+
1
=
y
i
+
h
·
f
(
t
i
,
y
i
)
i. Implement Euler’s method as a Matlab function that takes in a function handle for
the right hand side of Equation 1, the initial and final time points
t
0
and
t
f
, the
number of steps
n
, and the vector
y
0
of initial conditions.
The function should
return a matrix containing the values of
y
(
t
) at every mesh point in the simulation.
ii. To see how this method performs on a real problem, consider the firstorder, re
versible isomerization
A
k
f
GGGGGGB
FGGGGGG
k
b
B
A. Write out the system of coupled differential equations governing the kinetics of
this reaction in a batch reactor.
B. Solve the system analytically with the initial conditions
c
A
(0) =
c
A,
0
and
c
B
(0) =
c
B,
0
.
Confirm that your solution is consistent with thermodynamic
equilibrium for the two species. Show all your work!
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '11
 ARNOLD
 Numerical Analysis, Chemical reaction, Reaction rate constant

Click to edit the document details