PROBLEM 2.31
KNOWN: Wall thickness, thermal conductivity, temperature distribution, and fluid temperature.
FIND: (a) Surface heat rates and rate of change of wall energy storage per unit area, and (b)
Convection coefficient.
SCHEMATIC:
ASSUMPTIONS: (1) One-dimensional conduction in x, (2) Constant k.
ANALYSIS: (a) From Fourier’s law,
()
x T
q k 200 60x k
x
∂
∂
′′ =− = − ⋅
′′ =′′ =×
⋅=qq C
m
W
mK W/m
in x=0 2
200 1 200
o
<
()
2
out x=L
q q 200 60 0.3 C/m 1 W/m K=182 W/m .
′′ ′′
==−× × ⋅
o <
Applying an energy balance to a control volume about the wall, Eq. 1.12c,
&& &
′′ −′′ =′′
EE E
in out s
t
&.
′′ =′′ −′′ =Eqq W/m
st in out 2
18 <
(b) Applying a surface energy balance at x = L,
()
out
qhTLT
∞
′′ ⎡⎤
=−
⎣⎦
() ()
2
out
q182 W/m
h= TL T 142.7-100 C
∞
′′ =
−o
h = 4.3 W/ m K.
2⋅ <
COMMENTS: (1) From the heat equation,
(∂T/∂t) = (k/ρcp) ∂2T/∂x2 = 60(k/ρcp),
it follows that the temperature is increasing with time at every point in the wall.
(2) The value of h is small and is typical of free convection in a gas.