HEAT AND MASS TRANSFER

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H e a t a

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M a s s T

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Question 3: Transient Heating of a Composite Wall (3 marks) The composite wall of Q.1 consisting of two materials A and B with thermal conductivities kA =

5 W/m⋅K and kB = 1 W/m⋅K has warm air (40 °C) on one side and cold air (10 °C) on the other.

The thermal diffusivities of the materials A and B are 7.0×10 -6

m 2 /s and 2.0×10

-6 m

2 /s,

respectively. The heat transfer coefficients are 500 W/m 2 K and 450 W/m

2 K for the warm and

cold sides respectively. The thicknesses are as given in the figure below. The composite has

a uniform initial temperature of 15 °°°°C. Neglecting contact resistance between the layers and using the method of finite differences, calculate the wall surface temperature on the warm side as a function of time during the first 60 seconds. Show your solution as a graph on the spreadsheet. (Make sure you include axis labels and units on the graph). Hint: The calculation will probably give you less trouble if you use a fully implicit time scheme rather than a time marching scheme. Note the governing differential equations for this problem are:

2

2 1

x

T

t

T AA

A ∂

∂ =

∂

∂

α ( )mx 002.00 <≤ (1)

2

2 1

x

T

t

T BB

B ∂

∂ =

∂

∂

α ( )mxm 01.0002.0 ≤< (2)

The relations at the interface:

x

T k

x

T k

B

B

A

A

∂

∂ =

∂

∂ ( )mx 002.0= (3)

TTT BA

== ( )mx 002.0= (i.e. no contact resistance) (4) The boundary conditions:

( ) Awarmwarm

A

A TTh

x

T k −=

∂

∂ − ( )0=x (5)

( ) coldBcold

B

B TTh

x

T k −=

∂

∂ − ( )mx 01.0= (6)

The initial condition:

CT °= 15 ( )mxt 01.00;0 ≤≤= (7)

Tsurface = ? x

Twarm = 40 °C

hwarm = 500 W/m 2 K hcold = 450 W/m

2 K

Tcold = 10 °C

A B

8 mm 2 mm