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MAE 105D, Fall 2009Prof. A. LavineMidterm, Open Book and
NotesPROBLEM AVERAGES: 11.0/20, 11.6/20, 15.5/20 = total 38.1
1.
Consider a slab of thickness 2 L = 6 cm. The thermal
conductivity, density, andspecific heat are 10 W/m-K, 1000 kg/m 3,
and 2000 J/kg-K, respectively. At timezero the slab temperature is
110 C and it is exposed to a convection environmenton both sides
with heat transfer coefficient of 50 W/m 2-K and temperature of
10C. Dont forget that heat loss occurs through two sides.
Fill in the table below (or copy to your own paper), by using
two differenttransient methods (state what they are) to find, at a
time of 3.2 s:
a) the surface temperature, and b) the surface heat flux. (Note
that once you have the surface temperature, it
is trivial to find the surface heat flux.)Then:c) For each
method, use the appropriate criterion to determine if the
method
is valid. Which method do you believe is more accurate?
T s q s
Method 1
Method 2
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2. Metal hydrides can be used to store hydrogen to be usedas a
fuel. When heated, a reaction occurs that releases thehydrogen
atoms from the metal hydride, thereby
producing hydrogen gas. Consider a slab of metal hydrideof
thickness 2 L, in which a reaction is occurring that
produces hydrogen gas (species A) at a uniform,
constantvolumetric generation rate An (kg/m3-s). The gas
diffuses
to the outer surfaces of the slab and out into theenvironment.
The binary mass diffusion coefficient forhydrogen gas diffusing
through the solid metal hydride is
D AB, a constant. This process maintains a density ofhydrogen
gas just inside the surface of the slab of As.Under steady-state*
conditions, and assuming theconcentration of hydrogen gas to be
small, derive anexpression for the maximum density of hydrogen gas
in the slab, using onlysymbols provided in the problem
statement.
*Extra information not needed to solve problem: The conditions
arent trulysteady because the amount of hydrogen contained in solid
form in the metalhydride is decreasing with time. But the density
of hydrogen gas can reach asteady-state.
3. Consider a conical fin of length L on a base with temperature
T b. The radius ofthe fin varies linearly with distance along the
fin, x:
R = R b (1 x/ L)
The fin is exposed to convection and radiation to large
surroundings. Derive thedifferential equation for the temperature
of the fin. Introduce any neededsymbols. You do not need to solve,
nor do you need to state the boundaryconditions.
x
2 L
An
As
Rb
L
