Problem 2
A large beaker with a shallow pool of liquid Heptane of depth
l
, is under an array
of mercury lamps which provide an energy flux of
q
00
s
, as shown in the figure.
From experiments it was determined that the energy flux inside the heptane pool
decays as
q
00
s
=
q
00
0
e

γx
, where
q
00
0
and
γ
are positive constants. The temperature
of the room is
T
∞
, the heat transfer coefficient at the liquid surface is
h
and the
thermal conductivity of liquid heptane is
k
. The bottom of the beaker is made
of a transparent material that is perfectly insulating. At steady state conditions,
assuming that the atmosphere is perfectly transparent to radiation and that there is
no mass loss to the atmosphere:
a. Show that the differential equation for variation of temperature in the hep
tane pool is
d
2
T
d
x
2
+
γq
00
0
k
e

γx
= 0
b. Setup the boundary conditions at the upper and lower boundaries.
c. Solve the equation to get the temperature profile in the heptane pool. (Hint:
Integrate the equation twice and apply the two boundary conditions)
1
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 Spring '09
 LaRue
 Thermodynamics, Heat, dx dx dx, heptane pool

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