CEE 291 Problem Solving Using Computer Tools
Homework 5: Matlab
Assigned: 2/15/2010
Due: 2/22/2010 5:00pm in the ECOW2 drop box and in class on Monday.
1.) This problem assumes you are familiar with the background information given in hmwk 2 #1.
Given the two governing equations for steady, one dimensional concentration of waste (L) and
Oxygen Deficit (D) are given below. In this problem, we would like to write a function that
numerically integrates these equations and then use this function in a program that optimizes
the location of treatment plants given an initial set of conditions.
Governing Equations
L
k
x
L
U
r

=
∂
∂
)
(
L
k
D
k
x
D
U
d
a


=
∂
∂
Analytical Solution
U
x
k
r
e
L
L

=
0


+
=



U
x
k
U
x
k
r
a
d
U
x
k
a
r
a
e
e
k
k
L
k
e
D
D
0
0
Numerical Technique
We will use the Euler explicit forward difference method to numerically integrate our
governing equations. Forward difference means we use dependant variable values at the
previous step to approximate the value at the current stop. In our case, we will use L and D at
the previous x location to calculate L and D at the next x location. This procedure is commonly
written as:
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1


+
∆

=
→

=
∂
∂
i
i
r
i
r
L
x
U
L
k
L
U
L
k
x
L
1
1
1



+

∆

=
i
i
d
i
a
i
D
U
L
k
D
k
x
D
NOTE:
*The subscript “i” represents the current x location and “i1” is the x location
previous.
* i was chosen here because the most common looping variable in Matlab is i. Numerical
solutions are often written with other variables.
*Everything is constant except L and D, which have indices
*In order to start, we need initial conditions
*Use constants
U = 1,500 m/day,
k
a
= 0.6 days
1
, k
r
= 1.2 days
1
, k
d
= 1.0 days
1
a.)
Write a function in matlab that numerically integrates the governing equations. Your top line
should look like:
function [L,D,x] = BOD_num_sol(deltaX, final_x, L_init, D_init, U, ka, kd, kr)
The function should take as input :
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 Spring '10
 Hoopes
 matlab, Numerical Analysis, Analytical Solution, Numerical Solution, numerical solution function

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