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

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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)