Post on 20-Dec-2015
02 October 2007
KKKQ 3013KKKQ 3013PENGIRAAN BERANGKAPENGIRAAN BERANGKA
Week 13 – Partial Differential Equations02 October 2007
8.00 am – 9.00 am
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Topics
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Tutorial Example 1 (adapted courtesy of ref. [1])
[1] KQ3013 Lecture Notes & Excercise
A boiler is supported by four 6cm tall steel legs. Initially, the temperatures on the legs are 20oC. The boiler is then filled with boiling 100oC water, while at the base of its legs, the temperature is constantly cooled to 20oC. If the coefficient of thermal diffusivity k = 0.835 cm2/s, using the Crank-Nicolson method, determine the temperatures at T1 and T2 from t = 1s to t = 5s. Plot the graph of temperature vs. time. (Use a timestep t = 1s)
What happens as t ∞ (i.e. steady state) ?
Boiler filled with 100oC water Boiler
base
02 October 2007 Week 13 Page 4
Tutorial Example 1 (adapted courtesy of ref. [1])
[1] KQ3013 Lecture Notes & Excercise
What we have here, is a case of transient 1-D heat conduction problem :
t
T
x
Tk
2
2
where : 0 , 60 , txtxT
with boundary and initial condition :
CxTCtTCtT ooo 20)0,( ,20),6( ,100),0(
At x = 0 and all time t, T = 100oC
At x = 6 and all time t, T = 20oC
At all x and at time t = 0 (initially), T = 20oC
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Tutorial Example 1
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Tutorial Example 1
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Tutorial Example 1
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Tutorial Example 1
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Tutorial Example 1
02 October 2007 Week 13 Page 10
Tutorial Example 1
0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60 70
time (s)
Tem
per
atu
re (
deg
. C
)
T2
T1
Plot of variation in temperature at point 1 and 2 with time (using Crank-Nicolson method)