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806 ENGINEERING THERMODYNAMICS

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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)
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