Before any analysis, what do you believe the final steady-state temperature distribution in the bar will be?

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Problem 2: Heat Equation Problem

A bar of aluminum is fixed in place with one end held at a temperature of zero degrees and the other end held at a higher temperature To.

u(x,0)

To 

Metal bar with end at x = 0 held at zero degrees and end at x = L held at temperature To

L - d L x

The distribution of temperature in the bar as a function of position and time, u(x,t), obeys the one- dimensional heat equation

𝜕𝜕2𝑢 1   𝜕𝜕𝑢 

𝜕𝜕𝑥2 − 𝑐2 

= 0

𝜕𝜕𝑡 

where 𝑐2 = 𝐾

𝜎𝜌 

, with K = thermal conductivity,  = specific heat, and  = mass density of the aluminum.

a) Before any analysis, what do you believe the final steady-state temperature distribution in the bar will be?

b) Determine the general solution for u(x,t) for this situation.

c) Determine the specific solution for u(x,t) with the initial condition u(x,0) as shown in the figure above. 

d) Repeat part (b) assuming 𝑢(𝑥, 0) = 𝑇𝑜

2 

3𝜋

1 − cos( 𝑥)

𝐿 

e) In Matlab, implement your solution from part (c), assuming that To = 10°K and the following parameters for the aluminum bar

L = 80cm; d = 3cm; K = 205 𝑤𝑤𝑎𝑡𝑡;  = 900 𝑗𝑗𝑜𝑢𝑙𝑒;  = 2700 𝑘𝑘𝑘 

𝑚 °𝐾 

𝑘𝑘𝑘 °𝐾 

𝑚3 

plot the solution for several times between t = 0 and when the temperature distribution reaches its steady state.

f) In Matlab, repeat part (e) assuming the initial condition from part (d).

Attachments:

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