MECH 3705
Assignment # 1
Due: Thursday, October 4th, 2018, 8:35
1. The thermal conductivity of a sheet of rigid, extruded insulation is reported to be k = 0.029
W/m·K. The measured temperature difference across the 20 mm thick sheet of the material is
T1 - T2 = 10ºC.
a) What is the heat flux through a 2 m × 2 m sheet of the insulation?
b) What is the rate of heat transfer through the sheet of insulation?
2. A freezer compartment consists of a cubical cavity that is 2 m on a side. Assume the bottom
to be perfectly insulated. What is the minimum thickness of Styrofoam insulation (k = 0.030
W/m·K) that must be applied to the top and side walls to ensure a heat load of less than
500 W, when the inner and outer surfaces are -10ºC and 35 ºC?
3. You've experienced convection cooling if you've ever extended your hand out the window of
a moving vehicle or into a flowing water stream. With the surface of your hand at a
temperature of 30ºC, determine the convection heat flux for:
a) a vehicle speed of 35 km/h in air at -5ºC with a convection coefficient of 40
W/m2K;
b) a velocity of 0.2 m/s in a water stream at 10ºC with a convection coefficient of
900 W/m2K.
c) Which condition would feel colder? Contrast there results with a heat loss of
approximately 30 W/m2 under normal room conditions.
4. An electric resistance heater is embedded in a long cylinder of diameter 30 mm. When water
with a temperature of 25ºC and velocity of 1 m/s flows crosswise over the cylinder, the power
per unit length required to maintain the surface at a uniform temperature of 90ºC is 28 kW/m.
When air, also at 25ºC, but with a velocity of 10 m/s is flowing, the power per unit length
required to maintain the same surface temperature is 400 W/m. Calculate and compare the
convection coefficients for the flows of water and air.
5. The temperature controller for a
clothes dryer consists of a
bimetallic switch mounted on an
electrical heater attached to a wallmounted insulation pad.
The
switch is set to open at 70ºC, the
maximum dryer air temperature.
To operate the dryer at a lower air
temperature, sufficient power is
supplied to the heater such that the
switch reaches 70ºC (Tset) when the air temperature T is less than Tset. If the convection heat
transfer coefficient between the air and the exposed switch surface of 30 mm2 is 25 W/m2K,
how much heater power Pe is required when the desired dryer temperature is T∞ =50 ºC?
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6. A spherical interplanetary probe of 0.5 m diameter contains electronics that dissipate 150 W.
If the probe surface has an emissivity of 0.8 and the probe does not receive radiation from any
other surfaces (the sun for example), what is its surface temperature?
7. A 50 mm × 45 mm × 20 mm cell phone charger has a surface temperature of Ts = 33ºC when
plugged into an electrical wall outlet but not in use. The surface of the charger is of
emissivity ε = 0.92 and is subject to a free convection heat transfer coefficient of h = 4.5
W/m2K. The room air and wall temperatures are T∞ = 22ºC and Tsur = 20ºC, respectively. If
electricity costs C = $0.18/kWh, determine the daily cost of leaving the charger plugged in
when not in use.
8. Electronic power devices are mounted to a heat sink having an
exposed surface area of 0.045m2 and an emissivity of 0.8. When
the devices dissipate a total of 20 W and the air and
surroundings are at 27ºC, the average sink temperature is 42ºC.
What average temperature will the heat sink reach when the
devices dissipate 30 W for the same environmental condition?
9. The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m2, and the underside
is perfectly insulated. The convection coefficient between the roof and the ambient air is 12
W/m2K.
a) Neglecting radiation exchange with the surroundings, calculate the temperature
of the roof under steady-state conditions if the ambient air temperature is 20ºC;
b) For the same ambient air temperature, calculate the temperature of the roof if its
surface emissivity is 0.8.
10. The emissivity of galvanized steel sheet, a common roofing material, is ε = 0.13 at
temperatures around 300 K, while its absorptivity for solar radiation is αs = 0.65. Would the
neighborhood cat be comfortable walking on a roof constructed of this material on day when
Gs = 750 W/m2, T∞ = 16ºC, and h = 7 W/m2K? Assume the bottom surface of the steel is
insulated.
11. During its manufacture, plate glass at 600ºC is cooled by passing air over its surface such that
the convection heat transfer coefficient is h = 5 W/m2K. To prevent cracking, it is known that
the temperature gradient must not exceed 15ºC/mm at any point in the glass during the
cooling process. If the thermal conductivity of the glass is 1.4 W/m·K and its surface
emissivity is 0.8, what is the lowest temperature of the air that can initially be used for the
cooling? Assume the temperature of the air equals that of the surroundings.
12. A dormitory at a large university, built 50 years ago (sounds familiar!), has exterior walls
constructed of Ls = 25 mm thick sheathing with thermal conductivity of ks = 0.1 W/m·K. To
reduce heat losses in the winter, the university decides to encapsulate the entire dormitory by
applying an Li = 25 mm thick layer of extruded insulation characterized by ki = 0.029 W/m·K
to the exterior of the original sheathing. The extruded insulation is, in turn, covered with a
Lg = 5 mm thick architectural glass with kg = 1.4 W/m·K. Determine the heat flux through the
original and retrofitted walls when the interior and exterior air temperatures are T∞,i = 22ºC
and T∞,o = -20ºC respectively. The inner and outer convection coefficients are hi = 5 W/m2K
and ho = 25 W/m2K, respectively.
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13. The wind chill, which is experienced on a cold, windy day, is related to increased heat
transfer from exposed human skin to the surrounding atmosphere. Consider a layer of fatty
tissue that is 3 mm thick and whose interior surface is maintained at a temperature of 36ºC.
On a calm day, the convection coefficient at the outer surface is 25 W/m2K, but with 30 km/h
winds, it reaches 65 W/m2K. In both cases the ambient air temperature is -15ºC:
a) What is the ratio of the heat loss per unit area from the skin for the calm day to
that for the windy day?
b) What will be the skin outer surface temperature for the calm day? For the windy
day?
c) What temperature would the air have to assume on the calm day to produce the
same heat loss occurring with the air temperature of -15ºC on the windy day?
14. Two stainless steel plates 10 mm thick are subjected to a contact pressure of 1 bar under
vacuum conditions for which there is an overall temperature drop of 100ºC across the plates.
What is the heat flux through the plates? What is the temperature drop across the contact
plane?
15. Electric current flows through a long rod
generating thermal energy at a uniform
volumetric rate of q''' = 2 × 106 W/m3.
The rod is concentric with a hollow
ceramic cylinder, creating an enclosure
that is filled with air. The thermal
resistance per unit length due to radiation
between the enclosure surfaces is R'rad =
0.30 m·K/W, and the coefficient
associated with free convection in the
enclosure is h = 20 W/m2K.
a) Construct a thermal circuit that can be used to calculate the surface temperature
of the rod Tr. Label all temperatures, heat rates, and thermal resistances, and
evaluate each thermal resistance.
b) Calculate the surface temperature of the rod for the prescribed conditions.
16. The wall of a spherical tank of 1-m diameter contains an exothermic chemical reaction and is
at 200ºC when the ambient air temperature is 25ºC. What thickness of urethane foam is
required to reduce the exterior temperature to 40ºC, assuming the convection coefficient is
20W/m2K for both situations? What is the percentage reduction in heat rate achieved by
using the insulation?
17. The cross section of a long cylindrical fuel element in a nuclear
reaction is shown on the following figure. Energy generation
occurs uniformly in the thorium fuel rod, which is of diameter
D = 25 mm and is wrapped in a thin aluminum cladding.
It is proposed that under steady-state condition, the system operates
with a generation rate q'''= 7 × 108 W/m3 and cooling system
characteristics of T∞ = 95ºC and h = 7000 W/m2K. Is this proposal
satisfactory?
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