Benha University.

Benha faculty of engineering.

Mechanical Department. Heat and Mass Transfer(M1321).
3rd year (Power Section) - First semester 2021/2022.
Dr. Mahmoud Said.

Sheet 3
STEADY HEAT CONDUCTION

1. Consider a 1.2-m-high and 2-m-wide glass window whose thickness is 6 mm and thermal
conductivity is k = 0.78 W/m · °C. Determine the steady rate of heat transfer through this glass
window and the temperature of its inner surface for a day during which the room is maintained
at 24°C while the temperature of the outdoors is -5°C. Take the convection heat transfer
coefficients on the inner and outer surfaces of the window to be h1 = 10 W/m2 · °C and h2 =
25 W/m2 · °C and disregard any heat transfer by radiation.
2. Consider a 1.2-m-high and 2-m-wide double-pane window consisting of two 3-mm-thick
layers of glass (k = 0.78 W/m · °C) separated by a 12-mm-wide stagnant air space (k = 0.026
W/m · °C). Determine the steady rate of heat transfer through this double-pane window and
the temperature of its inner surface for a day during which the room is maintained at 24°C
while the temperature of the outdoors is -5°C. Take the convection heat transfer coefficients
on the inner and outer surfaces of the window to be h1 = 10 W/m2 · °C and h2 = 25 W/m2 · °C
and disregard any heat transfer by radiation.
3. Repeat Problem 2, assuming the space between the two glass layers is evacuated.
4. A 2-m X 1.5-m section of wall of an industrial furnace burning natural gas is not insulated, and
the temperature at the outer surface of this section is measured to be 80°C. The temperature of
the furnace room is 30°C, and the combined convection and radiation heat transfer coefficient
at the surface of the outer furnace is 10 W/m2 · °C. It is proposed to insulate this section of the
furnace wall with glass wool insulation (k = 0.038 W/m · °C) in order to reduce the heat loss
by 90 percent. Assuming the outer surface temperature of the metal section still remains at
about 80°C, determine the thickness of the insulation that needs to be used. The furnace
operates continuously and has an efficiency of 78 percent. The price of the natural gas is
$0.55/therm (1 therm = 105,500 kJ of energy content). If the installation of the insulation will
cost $250 for materials and labor, determine how long it will take for the insulation to pay for
itself from the energy it saves.

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Benha University.
Benha faculty of engineering.
Mechanical Department. Heat and Mass Transfer(M1321).
3rd year (Power Section) - First semester 2021/2022.
Dr. Mahmoud Said.

5. A1-mm-thick copper plate (k = 386 W/m · °C) is sandwiched between two 5-mm-thick epoxy
boards (k = 0.26 W/m · °C) that are 15 cm x 20 cm in size. If the thermal contact conductance
on both sides of the copper plate is estimated to be 6000 W/m · °C, determine the error involved
in the total thermal resistance of the plate if the thermal contact conductances are ignored.
6. A 4-m-high and 6-m-wide wall consists of a long 18-cm x 30-cm cross section of horizontal
bricks (k = 0.72 W/m · °C) separated by 3-cm-thick plaster layers (k = 0.22 W/m · °C). There
are also 2-cm-thick plaster layers on each side of the wall, and a 2-cm-thick rigid foam (k =
0.026 W/m · °C) on the inner side of the wall. The indoor and the outdoor temperatures are
22°C and -4°C, and the convection heat transfer coefficients on the inner and the outer sides
are h1 = 10 W/m2 · °C and h2 = 20 W/m2 · °C, respectively. Assuming one-dimensional heat
transfer and disregarding radiation, determine the rate of heat transfer through the wall.

7. Consider a 5-m-high, 8-m-long, and 0.22-m-thick wall whose representative cross section is
as given in the figure. The thermal conductivities of various materials used, in W/m · °C, are
kA = kF = 2, kB = 8, kC = 20, kD = 15, and kE = 35. The left and right surfaces of the wall are
maintained at uniform temperatures of 300°C and 100°C, respectively. Assuming heat transfer
through the wall to be one-dimensional, determine (a) the rate of heat transfer through the wall;
(b) the temperature at the point where the sections B, D, and E meet; and (c) the temperature
drop across the section F. Disregard any contact resistances at the interfaces.

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Benha University.
Benha faculty of engineering.
Mechanical Department. Heat and Mass Transfer(M1321).
3rd year (Power Section) - First semester 2021/2022.
Dr. Mahmoud Said.

8. A 5-m-internal-diameter spherical tank made of 1.5-cm-thick stainless steel (k = 15 W/m · °C)

is used to store iced water at 0°C. The tank is located in a room whose temperature is 30°C.
The walls of the room are also at 30°C. The outer surface of the tank is black (emissivity = 1),
and heat transfer between the outer surface of the tank and the surroundings is by natural
convection and radiation. The convection heat transfer coefficients at the inner and the outer
surfaces of the tank are 80 W/m2 · °C and 10 W/m2 · °C, respectively. Determine (a) the rate
of heat transfer to the iced water in the tank and (b) the amount of ice at 0°C that melts during
a 24-h period. The heat of fusion of water at atmospheric pressure is hif = 333.7 kJ/kg.
9. Steam at 320°C flows in a stainless steel pipe (k = 15 W/m · °C) whose inner and outer
diameters are 5 cm and 5.5 cm, respectively. The pipe is covered with 3-cm-thick glass wool
insulation (k = 0.038 W/m · °C). Heat is lost to the surroundings at 5°C by natural convection
and radiation, with a combined natural convection and radiation heat transfer coefficient of 15
W/m2 · °C. Taking the heat transfer coefficient inside the pipe to be 80 W/m2 · °C, determine
the rate of heat loss from the steam per unit length of the pipe. Also determine the temperature
drops across the pipe shell and the insulation.

3
Benha University.
Benha faculty of engineering.
Mechanical Department. Heat and Mass Transfer(M1321).
3rd year (Power Section) - First semester 2021/2022.
Dr. Mahmoud Said.

10. A50-m-long section of a steam pipe whose outer diameter is 10 cm passes through an open
space at 15°C. The average temperature of the outer surface of the pipe is measured to be
150°C. If the combined heat transfer coefficient on the outer surface of the pipe is 20W/m2 °C,
determine (a) the rate of heat loss from the steam pipe, (b) the annual cost of this energy lost
if steam is generated in a natural gas furnace that has an efficiency of 75 percent and the price
of natural gas is $0.52/therm (1 therm = 105,500 kJ), and (c) the thickness of fiberglass
insulation (k = 0.035 W/m · °C) needed in order to save 90 percent of the heat lost. Assume
the pipe temperature to remain constant at 150°C.
11. Consider a 2-m-high electric hot water heater that has a diameter of 40 cm and maintains the
hot water at 55°C. The tank is located in a small room whose average temperature is 27°C, and
the heat transfer coefficients on the inner and outer surfaces of the heater are 50 and 12 W/m2
· °C, respectively. The tank is placed in another 46-cm-diameter sheet metal tank of negligible
thickness, and the space between the two tanks is filled with foam insulation (k = 0.03 W/m ·
°C). The thermal resistances of the water tank and the outer thin sheet metal shell are very
small and can be neglected. The price of electricity is $0.08/kWh, and the home owner pays
$280 a year for water heating. Determine the fraction of the hot water energy cost of this
household that is due to the heat loss from the tank. Hot water tank insulation kits consisting
of 3-cm-thick fiberglass insulation (k = 0.035 W/m · °C) large enough to wrap the entire tank
are available in the market for about $30. If such an insulation is installed on this water tank
by the home owner himself, how long will it take for this additional insulation to pay for itself?
12. Hot water at an average temperature of 90°C is flowing through a 15-m section of a cast iron
pipe (k = 52 W/m · °C) whose inner and outer diameters are 4 cm and 4.6 cm, respectively.
The outer surface of the pipe, whose emissivity is 0.7, is exposed to the cold air at 10°C in the
basement, with a heat transfer coefficient of 15 W/m2 · °C. The heat transfer coefficient at the
inner surface of the pipe is 120 W/m2 · °C. Taking the walls of the basement to be at 10°C
also, determine the rate of heat loss from the hot water. Also, determine the average velocity
of the water in the pipe if the temperature of the water drops by 3°C as it passes through the
basement.

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Benha University.
Benha faculty of engineering.
Mechanical Department. Heat and Mass Transfer(M1321).
3rd year (Power Section) - First semester 2021/2022.
Dr. Mahmoud Said.

13. The boiling temperature of nitrogen at atmospheric pressure at sea level (1 atm pressure) is =
- 196°C. Therefore, nitrogen is commonly used in low-temperature scientific studies since the
temperature of liquid nitrogen in a tank open to the atmosphere will remain constant at -196°C
until it is depleted. Any heat transfer to the tank will result in the evaporation of some liquid
nitrogen, which has a heat of vaporization of 198 kJ/kg and a density of 810 kg/m3 at 1 atm.
Consider a 3-m-diameter spherical tank that is initially filled with liquid nitrogen at 1 atm and
-196°C. The tank is exposed to ambient air at 15°C, with a combined convection and radiation
heat transfer coefficient of 35 W/m2 · °C. The temperature of the thin-shelled spherical tank is
observed to be almost the same as the temperature of the nitrogen inside. Determine the rate
of evaporation of the liquid nitrogen in the tank as a result of the heat transfer from the ambient
air if the tank is (a) not insulated, (b) insulated with 5-cm-thick fiberglass insulation (k = 0.035
W/m · °C), and (c) insulated with 2-cm-thick superinsulation which has an effective thermal
conductivity of 0.00005 W/m · °C.
14. A2-mm-diameter and 10-m-long electric wire is tightly wrapped with a 1-mm-thick plastic
cover whose thermal conductivity is k = 0.15 W/m · °C. Electrical measurements indicate that
a current of 10 A passes through the wire and there is a voltage drop of 8 V along the wire. If
the insulated wire is exposed to a medium at T= 30°C with a heat transfer coefficient of h = 24
W/m2 · °C, determine the temperature at the interface of the wire and the plastic cover in steady
operation. Also determine if doubling the thickness of the plastic cover will increase or
decrease this interface temperature.
15. A5-mm-diameter spherical ball at 50°C is covered by a 1-mm-thick plastic insulation (k = 0.13
W/m · °C). The ball is exposed to a medium at 15°C, with a combined convection and radiation
heat transfer coefficient of 20 W/m2 · °C. Determine if the plastic insulation on the ball will
help or hurt heat transfer from the ball.

DR./ Mahmoud Said

Eng./ Osman Samir