CHEMICAL ENGINEERING

126 CHEMICAL ENGINEERING VOLUME 1 SOLUTIONS

PROBLEM 9.2

Calculate the time for the distant face to reach 470 K under the same conditions as
Problem 9.1, except that the distant face is not perfectly lagged but a very large thickness
of material of the same thermal properties as the brickwork is stacked against it.

Solution

This problem involves the conduction of heat in an infinite medium where it is required
to determine the time at which a point 0.45 m from the heated face reaches 470 K.
The boundary conditions are therefore:

D 0 ,tD0; D^0 ,t > 0 for all values ofx

D 

870  290 D580 deg K,xD 0 ,t > 0

D 0 ,xD1,t> 0

D 0 ,xD 0 ,tD 0

∂

∂t

DDH

(

∂^2 

∂x^2

C

∂^2 

∂y^2

C

∂^2 

∂z^2

)

DDH

∂^2 

∂x^2

(for unidirectional heat transfer) (equation 9.29)

The Laplace transform of: DND

∫ 1

0

eptdt (i)

and hence:

d^2 N

dx^2

D

p

DH

N

NtD 0

DH

(ii)

Integrating equation (ii):NDB 1 ex

p

p/DH (^) CB 2 ex
p

p/DH (^) CtD 0 /p (iii)
and:
dN
dx

DB 1

p



p/DH ex

p

p/DH (^) B
2
p

p/DH ex
p

p/DH (^) (iv)
In this case, Nt> 0
xD 0

D

∫ 1

0

^0 eptdtD^0 /p

and:

(

∂

∂t

)

t> 0

xD 0

D

∫ 1

0

(

∂

∂t

)

eptdtD 0

Substituting the boundary conditions in equations (iii) and (iv):

Nt> 0

xD 0

D^0 t> 0

xD 0

/pDB 1 CB 2 CtD 0 /p or B 1 CB 2 D^0 t> 0

xD 0

/p

and:

(

∂N

∂t

)

t> 0

xD 0

D 0 DB 1

p



p/DH e^1 B 2

p



p/DH e1

∴ B 1

p



p/DH D0andB 1 D 0 ,B 2 D^0 t> 0

xD 0

/p