TITLE.PM5

806 ENGINEERING THERMODYNAMICS

dharm

\M-therm\Th15-2.pm5

15.2.8.2. Heat conduction through a composite sphere
Considering Fig. 15.23 as cross-section of a composite sphere, the heat flow equation can be
written as follows :

Hot

fluid thf

hhf

A

B

Q

Cold fluid (air)

tcf

hcf

t tcf

t 1 t 2 3

r 1

r 2

r 3

Fig. 15.23. Steady state conduction through a composite sphere.

Q = hhf. 4πr 12 (thf – t 1 ) =^44 12 1^2

21

23 1 3

32

ππkrr t t

rr

krr t t

rr

AB()

()

()

()

−

−

=

−

−

= hcf. 4πr 32 (t 3 – tcf)

By rearranging the above equation, we have

thf – t 1 =

Q

hrhf.4π 12

...(i)

t 1 – t 2 = Qr r

krrA

()

.

21

(^412)
−
π
...(ii)
t 2 – t 3 = Qr r
krrB
()
.
32
(^423)
−
π
...(iii)
t 3 – tcf =
Q
hrcf.4π 32
...(iv)
Adding (i), (ii), (iii) and (iv), we get
Q
hr
rr
krr
rr
(^4) hf ABkrrhrcf
11
1
2
21
12
32
π. 23 32
()
.
()

..

• − + − +

L

N

M

M

O

Q

P

P

= thf – tcf

∴ Q =

4

11

1

2

21

12

32

23 32

π()

.

()

.

()

..

tt

hr

rr

krr

rr

krrhr

hf cf

hf ABcf

−

+ − + − +

L

N

M

M

O

Q

P

P

...(15.38)