PGE 310 - HW6 -Solution - The University of Texas at Austin...

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The University of Texas at Austin
PGE 310: Formulation and Solution in Geosystems Engineering
Homework #6: Root Finding (Secant Method) and Solving Linear System of Equations
SOLUTION
By Hossein Roodi
Questions 1 is solved
for both
cases: using
global variables
and
not using global variables
Question #1 Solution: Not Using Global Variables:
1. Root Finding: Using Secant Method (MATLAB PROGRAMMING)
The Colebrook equation is used for friction factor in fully developed pipe flow:
! " # $ )
Re
51 . 2
7 . 3
/
log( 2
1
f
D
f
%
0 )
Re
51 . 2
7 . 3
/
log( 2
1
$ " " $
f
D
f
y
%
Where
f
is friction factor,
D / %
is the relative roughness of the pipe material, and
Re
is Reynolds number based
on pipe diameter
D
. You are supposed to solve this equation for
f
using secant method .
a) Write a MATLAB function called “ Colebrook.m” that when sent Darcy factor (
f
), Relative roughness (
eD
) and
Reynolds number (
Re
), it returns the value of function.
- Note: In MATLAB, log (x) is reserved for natural log of x . For logarithm in base of 10 you are supposed
to use log10(x).So in this problem if you write log instead of log10 , your root will be 0.0108 instead of
0.0572 and the number of iteration will change.
b) Write a MATLAB function called “ secant.m” that when sent a function (
func
), initial guesses (
0 x
&
1 x
), it
returns the root of equation (
root
) and the number of iterations. In your function file, choose a tolerance that you
think is appropriate .
Recall that secant method formulation is:
) ( ) (
) )( (
1
1
1
i i
i i i
i i
x f x f
x x x f
x x
#
#
# $
#
#
"
So, in each step two estimations are needed, like x
i
and x
i-1
and the next guess for root is obtained by the
above mentioned formulation. (x
i+1
is obtained based on x
i
and x
i-1
)
In the following function secant.m x0 and x1 are assigned as the two estimation values using in secant
method formulation. Aftereach iteration x0 and x1 are updated.

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c) Write a script file called “ frictionfac.m ” to calculate the value of friction factor for the following values
" Set
6
10 Re $
,
03 . 0 / $ D %
,
006 . 0
0
$ x
and
007 . 0
1
$ x
Results:
Print all your files and the results. Upload function files and script file on the Blackboard.
(35 %)

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Question #1 Solution Using Global Variables:
If global variables are used for eD and Re , then it is not required to declare them as input variables
neither in Colebrooks.m nor in secant.m . MATLAB assumes them global and recognizes them within
all functions.
To make them global, only thing you need to do is writing command“ global eD Re”
at the beginning of
your script file and your function files.
Also, since Colebrook.m will have only one input, you need to erase them from the end of feval functions in
secant.m .

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2. Developing System of Equations for a Real World Problem: (By HAND)
An oil company called OILHW6 wants to make decision about pumping from three oil wells in one of its reservoirs in
Texas. Unknowns are oil outputs from each well: Q1, Q2 and Q3
Q1 contains 30% Gasoline , 30% Diesel Fuel and 40% Kerosene
Q2 contains 60% Gasoline , 20% Diesel Fuel and 20% Kerosene
Q3 contains 50% Gasoline , 20% Diesel Fuel and 30% Kerosene
In order to separate these chemicals, OILHW6 transfers Q1, Q2