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)