Problem 4: Numerical Methods (Shooting Method) (25 Points)
Working for a company called Fin Design Inc., you are asked to design a cylindrical fin
to increase the heat transfer rate from a pipe as shown in the figure below.
Temperature of
ambient air = Ta
The relevant values in the figure are:
R₁ = 0.8 in
R₁= 1.0 in
T,= 100 F
Tv = 300 F
R₁
Temperature T = T
at r = Ro/nThe equation governing the temperature variation within the fin is given by:
d²T 1 dr -4T=0
dr² r dr
You may assume the tip of the fin is at the temperature of the surrounding air.
Your goal is to determine the rate of heat transfer through this fin but for this you need to
obtain the temperature distribution within the fin first.
a. Dividing your fin into 4 equally spaced segments, use the shooting method to get the
temperature profile within the fin.
Fill in your values in the table below (if you used Goal seek/solver and obtained the
correct value of w=dT/dr in your first try...that is OK!) (10 points)
Estimate of
slope (w =
dT/dx)
r (inches)
0.8
0.85
0.9
0.95
1.0
Fill in 1
estimate of
"(w=dT/dr)"
here
T(r) - 1"
estimate
Fill in 2nd
estimate of
"(w=dT/dr)"
here
T(r) -2nd
estimate
Fill in 3rd
estimate of
"(w=dT/dr)"
here
T(r)-3rd
estimate
Fill in 4th
estimate of
"(w=dT/dr)"
here
T(r)-4th
estimate
b. Show a sample step-by-step calculation for any one of the columns that you report in
the tables above (i.e., choosing any "r" location and any T (r) of your choosing, show
how you used the shooting method to predict the T at the next spatial location - i.e.,
the number in the row below it). Please show the calculations clearly to enable me
to replicate your results (10 points)
Temperature at the first spatial location:
Spatial-step size:
Calculation of the slopes (dT/dr and dw/dr) at that spatial location:/n(Please make space here)
Temperature at the next spatial location:
(Please make space here)
c. How did you ensure (or why do you feel) that you have arrived at the numerically
correct answer? (5 points)
Fig: 1
Fig: 2
Fig: 3
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