800 ENGINEERING THERMODYNAMICS

dharm
\M-therm\Th15-2.pm5

Let r 1 , r 2 = Inner and outer radii ;
t 1 , t 2 = Temperature of inner and outer surfaces, and
k = Constant thermal conductivity within the given temperature range.
Consider an element at radius ‘r’ and thickness ‘dr’ for a length of the hollow cylinder
through which heat is transmitted. Let dt be the temperature drop over the element.

Hollow cylinder

(Length = L)

Q (Heat flows

radially outwards)

t> t 12

Element

No heat flows

in the axial

r (^2) r dr direction
r 1
t 1 dt
dr
t 2
Q t 1 t 2 Q
R=th 2kLπ^1 ln (r / r ) 21
Fig. 15.17
Area through which heat is transmitted. A = 2π r. L.
Path length = dr (over which the temperature fall is dt)
∴ Q = – kA.
dt
dr
F
HG
I
KJ = – k. 2πr. L^
dt
dr
per unit time or Q.
dr
r = – k. 2πL.dt
Integrating both sides, we get
Q
dr
r r
t
1
2
z = – k.2πL^ t dt
t
1
2
z or Q^ ln ( )r r
L r
N
M
O
Q
P
1
2
= k. 2 πL t
t
L t
N
M
O
Q
P
1
2
or Q.ln(r 2 /r 1 ) = k.2πL(t 2 – t 1 ) = k.2πL(t 1 – t 2 )
∴ Q =
kLt t
rr
.( )
ln ( / )
(^212)
21
π−

()
ln( / )
tt
rr
kL
12
21
2
−
L
N
M
O
Q
π P
...(15.33)
15.2.7.2. Heat conduction through a composite cylinder
Consider flow of heat through a composite cylinder as shown in Fig. 15.18.
Let thf = The temperature of the hot fluid flowing inside the cylinder,
tcf = The temperature of the cold fluid (atmospheric air),