W
Ceramic
Steel plate
insulation
Heating
?
kst
Wall
kc
q1
?,k
q2
T(x)
T2
T3
Liquid
h1, T?,1 W
111
Air
h2, T?,2
Ti
Liquid
h1, T?,1
111
L1
3-D View
T4
Air
h2, T?,2
x
x=0
x=L1
Side View
Problem #2:
Now just consider the heat-generating wall in Problem #1. We can set up the coordinate system as shown
in the figure. Then the energy equation for the temperature function, T(x), within the heating wall
is governed by the following second-order, ordinary differential equation,
$\frac{d^2T}{dx^2} + \frac{\dot{q}}{k} = 0$
You are asked to do the following:
(1)
1. Find the general solution of Eq. (1), T(x), with two integral constants C? and C?.
2. Write down two boundary conditions for Eq. (1) at x = 0 and x = L?, respectively. (You can
use your temperature results, T? and T?, obtained from Problem #1.)
3. Derive the mathematical expressions for two integral constants C? and C?, based on your
two boundary conditions. (Detailed derivation processes must be given here.)
4. Calculate two integral constants C? and C? based on given information.
5. Determine the maximum temperature inside the heating wall, Tmax, and its location xmax.
(Hint: You can take first derivative of your temperature function and let it be zero to locate
the location where the temperature is maximum.)