CHE312: Problem set #1
1. (1.371)
The energy consumption associated with a home water heater has two
components: (i) the energy that must be supplied to bring the temperature of ground water to
the heater storage temperature, as it is introduced to replace hot water that has been used, and
(ii) the energy needed to compensate for heat losses incurred while the water is stored at the
prescribed temperature. In this problem, we will evaluate the first of these components for a
family of four, whose daily hot water consumption is approximately 100 gallons. If ground
water is available at 15oC, what is the annual energy consumption associated with heating the
water to a storage temperature of 55oC? For a unit electrical power cost of $0.08/kWh, what
is the annual cost associated with supplying hot water by means of (a) electric resistance
heating or (b) a heat pump having a COP of 3 and a compressor efficiency of 85%
(conversion from electrical energy to mechanical work).
Water density = 993 kg/m3 at 0.5(15 + 55) = 35oC = 308 K, Cpw = 4.187 kJ/kgK, 100 gal =
0.3785 m3
(a) Annual cost associated with supplying hot water by means of electric resistance heating
Resistance heating cost = (6369 kWh)($0.08/kWh) = $510
(b) Annual cost associated with supplying hot water by means of a heat pump
Heat pump cost = (2498 kWh)($0.08/kWh) = $200
Operating costs are lower for a heat pump, capital costs are higher.
1
Fundamentals of Heat Transfer by Incropera and DeWitt
2. (1.491)
Liquid oxygen, which has a boiling point of 90oK and a latent heat of
vaporization of 214 kJ/kg, is stored in a spherical container whose outer surface is of 500mm diameter and at a temperature of 10oC. The container is housed in a laboratory whose
air and walls are at 25oC.
(a)
If the surface emissivity is 0.20 and the heat transfer coefficient associated with free
convection at the outer surface of the container is 10 W/m2K, what is the rate, in kg/s, at
which oxygen vapor must be vented from the system?
(b)
Moisture in the ambient air will result in frost formation on the container, causing the
surface emissivity to increase. Assuming the surface temperature and convection coefficient
to remain at 10oC and 10 W/m2K, respectively, compute the oxygen evaporation rate (kg/s)
as a function of surface emissivity over the range 0.2    0.94.
(a)
mevap = 1.4110-3 kg/s
(b)
Compute the oxygen evaporation rate (kg/s) as a function of surface emissivity over
the range 0.2    0.94.
3. (1.551)
The roof of a car in a parking lot absorbs a solar radiant flux of 800 W/m2,
while the underside is perfectly insulated. The convection coefficient between the roof and
the ambient air is 12 W/m2K.
(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.
(c)
The convection coefficient depends on airflow conditions over the roof, increasing
with increasing air speed. Compute and plot the roof temperature as a function of h
for 2  h  200 W/m2K.
Solution
(a)
86.7oC
(b)
47oC
(c)
Compute and plot the roof temperature as a function of h for 2  h  200 W/m2K.
4. (1.591)
A surface whose temperature is maintained at 400oC is separated from an air
flow by a layer of insulation 25 mm thick for which the thermal conductivity is 0.1 W/mK.
If the air temperature is 35oC and the convection coefficient between the air and the outer
surface of the insulation is 500 W/m2K, what is the temperature of this outer surface?
T2 = 37.9oC
5. (1.611)
An experiment to determine the convection coefficient associated with airflow
over the surface of a thick steel casting involves insertion of thermocouples in the casting at
distances of 10 and 20 mm from the surface along a hypothetical line normal to the surface.
The steel has a thermal conductivity of 15 W/mK. If the thermocouples measure
temperatures of 50 and 40oC in the steel when the air temperature is 100oC, what is the
convection coefficient?
h = 15,000/(100  60) = 375 W/m2K
6. (1.651)
Plate glass at 600oC is cooled by passing air over its surface such that the
convection heat transfer coefficient is h = 5 W/m2K. 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 that the temperature of the air equals that of the surroundings.
T = 618 K = 345oC
7.2
A cylinder of radius ro, length L, and thermal conductivity k is immersed in a fluid of
convection coefficient h and unknown temperature T. At a certain instant the temperature
distribution in the cylinder is T(r) = a + br2, where a and b are constants. Obtain expressions
for the heat transfer rate at ro and the fluid temperature.
q|ro =  2roLk
T
r
=  2roLk(2bro) = 2roLh(T|ro  T)
ro
(2bkro) = h(T|ro  T)  T = a + bro2 
2
2bkro
h
Fundamentals of Heat Transfer by Incropera and DeWitt
8.2
In the two-dimensional body illustrated, the gradient at surface A is found to be T/y
= 30 K/m. What are T/y and T/x at surface B?
Solution
T
x
= 0,
A
T
y
=0
B
Energy balance over the body
qA = qB  qy,A = qx,B
 k(1)(wA)
T
x
=
B
T
T
=  k(1)(wB)
y A
x
w A T
wB y
B
= (2)(30) = 60 K/m
A