1. If you bale hay without allowing it to dry sufficiently then the hay bales will contain a lot
of water. Besides making the bales heavy and therefore difficult to put in the barn, the water
in the hay bails causes an exothermic chemical reaction to occur within the bale (i.e., the hay
is rotting). The chemical reaction proceeds at a rate that is related to temperature and the
bales may be thermally isolated (they are placed in a barn and surrounded by other hay
bales); as a result, the hay can become very hot and even start a barn fire. Figure 1 illustrates
a cross-section of a barn wall with hay stacked against it.
The air within the barn is maintained at Ta,in = 20°C and the heat transfer coefficient between
the air and the inner surface of the hay is a,in h = 15 W/m2-K. The outside air is at Ta,out = -5°C
with ha,out = 45 W/m2-K. Neglect radiation from the surfaces in this problem. The barn wall is
composed of wood (kw = 0.11 W/m-K) and is thw = 1 cm thick. The hay has been stacked L =
5 m thick against the wall. Hay is a composite structure composed of plant fiber and air.
However, hay can be modeled as a single material with an effective conductivity kh = 0.05
W/m- K. The volumetric generation of the hay due to the chemical reaction is given by:
where T is temperature in K.
a.) Develop a numerical model that can predict the temperature distribution within the hay.
b.) Prepare a plot that shows the temperature distribution as a function of position in the hay.
c.) Prepare a plot that shows that you are using a sufficient number of nodes in your
numerical solution.
d.) Verify that your solution is correct by comparing it with an analytical solution in an
appropriate limit. Prepare a plot that overlays your numerical solution and the analytical
solution in this limit.
e.) What is the maximum allowable thickness of hay (Lmax) based on keeping the maximum
temperature below Tfire?
f.) If L = Lmax from (e) then how much of the hay will remain usable (what percent of the hay
is lost to heat degradation)?
2. A current of 100 amps passes through a bare stainless-steel wire of D = 1.0 mm diameter.
The thermal conductivity and electrical resistance per unit length of the wire are k = 15 W/mK and 𝑅𝑒′ = 0.14 ohm/m, respectively. The wire is submerged in an oil that is maintained at
T∞ = 30°C. The steady-state temperature at the center of the wire is measured to be 180°C,
independent of axial position within the oil bath.
a) What is the temperature at the outer surface of the wire?
b) Estimate the convection coefficient between the submerged wire and the oil.
c) A plastic material (kp = 0.05 W/m-K) can be applied to the outer surface of the wire. Can
the insulation result in a reduction of the center temperature? If so, what insulation thickness
should be applied?
3. Figure 2 illustrates a design for a superconducting heat switch.
The heat switch is made by embedding eight square superconducting strands in a polymer
matrix. The width of switch is W = 10 mm (into the page). The size of the strands are a= 5
mm and the width of polymer that surrounds each strand is b = 1 mm. The conductivity of the
polymer is kp = 2.5 W/m-K. The heat switch is surrounded by a magnet. When the heat switch
is on (i.e., the thermal resistance through the switch in the x-direction is low, allowing heat
flow from TH to TC), the magnet is on. Therefore, the magnetic field tends to drive the
superconductors to their normal state where they have a high thermal conductivity, km,normal =
50 W/m-K. To turn the heat switch off (i.e., to make the thermal resistance through the switch
high, preventing heat transfer from TH to TC), the magnet is deactivated. The superconductors
return to their superconducting state, where they have a low thermal conductivity, km,sc = 0.2
W/m-K. The edges of the switch are insulated.
a.) Develop a model using a 1-D resistance network that provides a lower bound on the
resistance of the switch when it is in its off state (i.e., km = km,sc).
b.) Develop a model using a 1-D resistance network that provides an upper bound on the
resistance of the switch when it is in its off state (i.e., km = km,sc).
c.) Plot your answers from parts (a) and (b) as a function of km for km,sc < km < km,normal.
d.) The performance of a heat switch is provided by the resistance ratio; the ratio of the
resistance of the switch in its off state to its resistance in the on state. Use your model
to provide an upper and lower bound on the resistance ratio of the switch.
e.) Plot the ratio of your answer from part (b) to your answer from part (a) as a function of km
for km,sc < km < km,normal. Explain the shape of your plot.
Hint: Use resistance approximations for conduction problems