Heat Transfer - Sheet two conduction

1. The walls of a refrigerator are typically constructed by sandwiching a layer of insulation

between sheet metal panels. Consider a wall made from fiberglass insulation of thermal
conductivity k = 0.046 W/m.K and thickness 50 mm and steel panels each of thermal
conductivity k = 60 W/m.K and thickness 3 mm. If the wall separates refrigerated air at
4°C from ambient air at 25°C, what is the heat gain per unit surface area. Assume h i = ho =
5 W/m2.K.
2. A house has a composite wall of wood, fiber glass insulation and plaster board as shown in
figure. On a cold winter day the convection heat transfer coefficients are ho = 60 W/m2.K
and hi = 30 W/m2.K. The total wall surface area is 380 m2. Write the equation of the total
thermal resistance of the wall including inside and outside convection effects. Also
determine the total heat loss through the wall. If the wind were blowing violently raising
ho to 300 W/m2.K, determine the percentage increase in the heat loss. Discuss which
resistance is the controlling resistance that affects the amount of heat flow through the
wall.

3. The exterior walls of a building are composite consisting of 10mm thick plaster board,
50mm thick urethane foam and a 10mm soft wood. On a typical winter day the outside and
inside air temperatures are -15°C and 20°C respectively with inner and outer convection
coefficients of 15 W/m2.K and 5 W/m2.K respectively. What is the amount of heat
transferred through 1m2 of the composite wall. Calculate the amount of heat transferred if
the wall was replaced by a double pan glass window each 3mm thick where the glass
panes are separated by a 5mm thick stagnant air, kg = 1.4 W/m.K and kair = 0.024 W/m.K.
4. Concrete of normal strength loses about 75% of its strength at approximately 600°C and
explosive spalling can occur in high strength concrete between 350°C and 600°C. Both
effects can lead to structural collapse during fires. The walls of a room experiencing a fire
may receive radiant flux of 25kW/m2 from the fire. Also convection transfers heat to the
wall at fire side since the air temperature is 400°c and h = 200W/m2.K. Consider steady
state conditions for a 150mm thick concrete wall whose exterior temperature is 300°C.
Calculate the temperature at the fire side surface of the wall and comment on whether the
wall is likely to experience structural collapse under these conditions or not.

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5. The composite wall of an oven consists of three materials two of which are of known
thermal conductivity, kA = 20 W/m.K and kC = 50 W/m.K and their thicknesses are: LA =
0.3 m and LC = 0.15 m. The third material B which is sandwiched between materials A
and C is of known thickness, LB = 0.15 m but unknown thermal conductivity kB. Ts,o =
20°C and the inner surface temperature Ts,i = 600°C and the oven air temperature T∞ =
800°C. The inside convection coefficient h is 25 W/m2.K. What is the value of kB. Also
calculate the rate of heat transfer per unit area of that oven.

6. A 2m×1.5m section of wall of an industrial furnace is not insulated. The temperature at the
outer surface of this section is measured to be 110°C while the temperature of the furnace
room is 32°C. The combined convection and radiation heat transfer coefficient at the outer
surface of the furnace is 10W/m2.K. It is proposed to insulate this section of the furnace
wall with glass wool insulation (k = 0.038 W/m.K) in order to reduce the heat loss by
90%. Assuming the outer surface temperature of the metal section remains at about 110°C,
determine the thickness of insulation to be used. If the furnace operate continuously and
has an efficiency of 78%, also if the price of natural gas is 10P.T/therm (1 therm = 105500
kJ of energy content). If the installation of the insulation will cost 2500L.E. for materials
and labor, determine how long it will take for the insulation to pay for it self from the
energy it saves.
7. Some under graduate students at a university have rented a house in which the windows
are of single pane construction. One of the students is an engineer and like all good
engineers wishes to conserve energy and save money from fuel savings. She therefore
proposed that winter heat losses can be reduced by covering the windows with a
polystyrene insulation (kins = 0.027 W/m.K) during the evening hours. To estimate the
energy savings consider application of 25 mm thick insulation panels to 6mm thick
windows (kg = 1.4 W/m.K). The contact resistance between the glass and the insulation
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may be approximated as t,c = 0.002 m .K/W, while the convection coefficient at the
outer surface of the windows is ho = 20 W/m2.K. With the insulation at the inner surface hi
= 2 W/m2.K while without insulation hi = 5W/m2.K. Calculate the percentage reduction in
heat loss when using the insulation. If the total surface area of the windows in the house
was As = 12m2, what are the heat losses associated with the insulated and uninsulated
window for interior and exterior air temperatures of T ∞i = 20°C and T∞o = -12°C. If the
house is heated by a gas furnace operating at an efficiency of ηf = 0.8 and natural gas is
priced at 0.2 P.T. per MJ, what is the daily savings associated with covering the windows
for 12 hrs.
8. Consider a plane composite wall that is composed of two materials of thermal
conductivities kA =0.1 W/m.K and kB = 0.04 W/m.K. Their thicknesses LA = 10mm and LB
= 20mm. The contact resistance at the interface between the two materials is known to be
0.3 m2.K/W. Material A adjoins a fluid at 200°C for which h = 10 W/m2.K and material B
adjoins a fluid at 40°C for which h = 20 W/m2.K. Calculate the rate of heat transfer
through the wall if it is 2m high and 2.5m wide. Recalculate the amount of heat transferred
through the wall if no contact resistance exists and estimate the percentage reduction in
heat transfer due to presence of contact resistance.

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9. A composite wall separate combustion gases at 2600°C from a liquid coolant at 100°C
with gas and liquid side convection coefficients of 50 and a1000 W/m2.K. The wall is
composed of a 10mm thick layer of beryllium oxide (k= 272 W/m.K) on the gas side and a
20mm thick slab of stainless steel (AISI 304) on the liquid side (k= 14.9 W/m.K). The
contact resistance between the oxide and the steel is 0.05 m2.K/W. Calculate the heat lost
per unit surface area of the composite. Also sketch the temperature distribution from the
gas to the liquid. Draw the circuit of thermal resistances showing the value of each
resistance.
10. Two 5cm diameter, 15 cm long aluminum bars (k= 176 W/m.K) with ground surfaces are
pressed against each other with a pressure of 20 atm. The bars are enclosed in an insulation
sleeve and thus heat transfer from the lateral surfaces is negligible. If the top and bottom
surfaces of the two bar system are maintained at temperatures of 150°C and 20°C
respectively, determine the rate of heat transfer along the cylinders under steady conditions
and the temperature drop at the interface.
11. A composite cylindrical wall is composed of two materials of thermal conductivity kA and
kB which are separated by a very thin electric resistance heater for which interfacial contact
resistances are negligible. Liquid is pumped through the tube having a temperature T ∞i and
provides a convection coefficient hi at the inner surface of the composite. The outer
surface is exposed to ambient air which is at T∞o and provides a convection coefficient of
ho. Under steady state conditions a uniform heat flux ́ h is dissipated by the heater. Sketch
the equivalent thermal circuit of the system and express all resistances in terms of relative
variables. Also obtain an expression that can be used to determine the heater temperature.
Finally find an expression for the ratio of heat flows between the outer and inner fluids.

12. A stainless steel (AISI 304) tube (k = 14.9 W/m.K) is used to transport a chilled
pharmaceutical has an inner diameter of 36 mm and a wall thickness of 2mm. The
pharmaceutical and ambient air are at temperatures of 6°C and 23°C respectively, while
the corresponding inner and outer convection coefficients are 400 W/m2.k and 6W/m2.K
respectively. Calculate the heat gain per unit length of the tube. If a 10mm thick layer of
calcium cilicate insulation (kins = 0.05 W/m.K) is used, calculate the heat gain in this
case. Calculate also the critical radius of insulation for this material.

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13. Super heated steam at 575°C is routed from a boiler to the turbine of an electric power
plant through steel tubes (k=35 W/m.K) of 300 mm inner diameter and 30 mm wall
thickness. To reduce heat loss to the surroundings and to maintain a safe to touch outer
surface temperature, a layer of calcium silicate insulation (k = 0.1 W/m.K) is applied to the
tubes. The surrounding air in the power plant is at 27°C. Assuming that the inner surface
temperature of the steel tube corresponds to that of the steam. The outer convection
coefficient is 6 W/m2.K. Calculate the minimum insulation thickness required to make
sure that the outer surface of insulation material does not exceed 50°C. Also calculate the
amount of heat loss per meter length of the tube.
14. A 0.2m diameter thin walled steel pipe is used to transport saturated steam at a pressure of
20 bars. The room air temperature is 25°C and the convection heat transfer coefficient at
the outer surface of the pipe is 20 W/m2.K. Calculate the heat loss per unit length from the
bare pipe (no insulation). Calculate the heat loss per unit length if a 50mm thick layer of
insulation (magnesia, 85%) is added (kins = 0.055 W/m.K). The steam side convection
resistance may be neglected. The costs associated with generating the steam and installing
the insulation are known to be 4 P.T. per 109J and 100 L.E./m of pipe length respectively.
If the steam line is to operate 7500 hrs/yr, how many years are needed to pay back the
initial investment in insulation.
15. Steam flowing through a long thin walled pipe which maintains the pipe wall at a uniform
temperature of 500K. The pipe is covered with an insulation blanket comprised of two
different materials A and B. The interface between the two materials may be assumed to
have an infinite contact resistance and the entire outer surface is exposed to air for which
T∞ = 300K and h = 25 W/m2.K. Sketch the thermal circuit of the system. Calculate the
total heat loss from the pipe. Also determine the outer surface temperatures T s2A and Ts2B.

16. A nuclear reactor fuel element consists of a solid cylindrical rod of radius r1 and thermal
conductivity kf. The fuel rod is in good contact with a cladding material of outer radius r 2
and thermal conductivity kc. Consider steady state conditions for which uniform heat
generation occurs within the fuel at a volumetric rate ̇ and the outer surface of the
cladding is exposed to a coolant that is characterized by a temperature T∞ and a convection
coefficient h. Consider a Uranium oxide fuel rod for which k f = 2W/m.K and r1 = 6mm
and cladding for which kc = 25 W/m.K and r2 = 9mm. If ̇ = 2×108 W/m3, h =
2000W/m2.K and T∞ = 300K. Determine the maximum temperature in the fuel element.
17. A long cylindrical rod of diameter 200mm with thermal conductivity of 0.5 W/m.k
experiences uniform volumetric heat generation of 24000 W/m3. The rod is encapsulated
by a circular sleeve having an outer diameter of 400 mm and a thermal conductivity of 4
W/m.K. The outer surface of the sleeve is exposed to cross flow of air at 27°C with a
convection coefficient of 25 W/m2.K. Find the temperature at the interface between the rod
and sleeve, Also find the outer surface temperature. Calculate the temperature at the center
of the rod.