Sample Heat Transfer Problems with Solutions
References are from:
from: Incropera, F.P. and De Witt,
Witt, D.P., Introduction to Heat Transfer, John
nd
Wiley and Sons,
Sons, 2 Edition, New York, 1996.
S.1 The heat flux through a wood slab 50 mm thick, whose inner and outer surface temperatures
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are 40 and 20°C, respectively, has been determined to be 40 W/m . What is
is the thermal
conductivity of the wood?
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S.2 The inner and outer
o uter surface temperatures of a glass window 5 mm thick are 15 and 5°C,
respectively. What is the heat
heat loss through a window
window that is 1 m by 3 m on a side? The thermal
conductivity of glass is 1.4 W/mK.
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S.3 The convection heat transfer coefficient between a surface at 40°C and ambient air at 20°C is
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20 W/m K. Calculate the heat flux leaving the surface by convection.
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S.4 Air at 300°C flows over a flat plate of dimensions 0.50 m by 0.25 m. If the convection heat
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transfer coefficient is 250 W/m K, determine the heat transfer rate from the air to one side of the
plate when the plate is maintained at 40°C.
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2
S.5 A surface of area 0.5 m , emissivity 0.8, and t emperature 150°C is placed in a large,
evacuated chamber whose walls
wa lls are maintained at 25°C. What is the rate at which radiation is
emitted by the surface? What is the net rate at which radiation is exchanged between the surface
and the chamber walls?
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S.6 A hot water pipe with outside radius r1 has a temperature
t emperature T1. A thick insulation applied to
reduce the heat loss has an outer radius r2 and temperature T2. On T±r coordinates, sketch the
temperature distribution in the insulation for one-dimensional, steady-state heat transfer with
constant properties. Give a brief explanation, justifying
justifying the shape o f the curve shown.
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S.7 Sections of the Alaskan pipeline run above the ground and are supported by vertical steel
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shafts (k = 25 W/mK) which are 1 m long and have a cross-sectional area of 0.005 m . Under
normal operating conditions, the temperature variation along the length of a shaft is known to be
2
governed by an expression
e xpression of the form: T = 100 ± 150 x + 10 x
where T and x have units of °C and meters, respectively. Temperature variations are negligible
negligible
over the shaft cross section. Evaluate the
t he temperature and conduction heat rate at the
shaft/pipeline joint (x = 0) and at the shaft/ground interface (x = 1 m). Explain the difference
d ifference in
heat rates.
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S.8 The rear window of
o f an automobile is defogged by passing war m air at 40°C over its inner
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surface, and the associated co nvection coefficient is 30 W/m K. Under conditions for which the
outside ambient air temperature is -10°C and the a ssociated convection coefficient is 65 W/m2K,
what are the inner and o uter surface temperatures of the window? The window glass is 4 mm
thick.
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S.9 The rear window of
o f an automobile is defogged by attaching
att aching a thin, transparent, film-type
heating element to its inner
inner surface. By electrically heating this
this element, a uniform heat flux
flux
may be established at the inner surface. What is the electrical power that must be pro vided per
unit window area to maintain
maintain an
a n inner surface temperature of 15°C when the interior air
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temperature and convection coefficient are 25°C and 10 W/m K and the exterior (ambient) air
2
temperature and convection coefficient are -10°C and 65 W/m K? The window glass is 4 mm
thick.
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S.10 For laminar flow over a flat plate, the local heat transfer coefficient, hx, is known to vary as
-1/2
x , where x is the distance from the leading edge (x=0)
x=0) of the plate. What is the ratio of the
average coefficient between the leading edge and some location x on the plate to the local
coefficient at x?
x?
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S.11 A fan that can provide air speeds up to 50 m/s is to be used in a low speed wind tunnel with
atmospheric air at 25°C. If one wishes to use the wind tunnel to study
stud y flat plate boundary layer
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behavior up to Reynolds numbers of 10 , what is the minimum plate length? At what distance
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from the leading edge wou ld transition occur, if the critical Reynolds number is 5×10 ?
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S.12 Experimental results for heat transfer over a flat plate with an extremely rough surface were
found to be correlated by an expression of the form
Nu x 0.04 Re 0x.9 Pr 1 / 3
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where Nux is the local value of the Nusselt number at a position x measured from the leading
edge of the plate. Obtain an expression for the ratio of the average heat transfer coefficient
between the leading edge and a location x to the local heat transfer
tr ansfer coefficient at x.
x.
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S.13 Consider steady, parallel flow of
o f atmospheric air over a flat plate. The air has a temperature
and free stream velocity of 300 K and 25 m/s.
a) Evaluate the boundary layer thickness at distances of x = 1, 10, and 100 mm from the
leading edge. If a second plate were installed parallel to and at a distance of 3 mm from
the first plate, what would be the distance from the leading edge at which the boundary
layer merger would occur?
b) Evaluate the surface shear stress and the
t he y-velocity component at the outer edge
edg e of the
boundary layer for the single plate at x = 1, 10, and 100 mm.
c) Comment on the validity of
o f the boundary layer appro ximations.
ximations.
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S.14 Consider water at 27°C in parallel flow over an isothermal, 1 m long flat plate with a
velocity of 2 m/s. Plot the variation
variat ion of the local heat transfer coefficient with distance along t he
plate. What is the value of the average coefficient?
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S.15 A thin, flat plate o f length L = 1 m separates two airstreams that are in parallel flow over
opposite surfaces of the plate. One a irstream has a temperature of 200°C and a velocity o f 60
m/s, while the other has a temperature
t emperature of 25°C and a velocity of 10 m/s. What is the heat flux
between the two streams at the
t he midpoint of the plate?
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S.16 Consider a rectangular fin that is used to cool a motorcycle engine. The
T he fan is 0.15 m long
and at a temperature of 250°C, while the motorcycle is moving at 80 km/hr in air at 27°C. The
air is in parallel flow over both surfaces of
o f the fin, and turbulent flow conditions may be assumed
to exist throughout. What is the rate
rat e of heat removal per unit width of fin?
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S.17 One-hundred electrical components, each dissipating 25 W, are attached to one surface of a
0.2 m square copper plate, and all of the dissipated
d issipated energy is transferred to water in parallel flow
over the opposite surface. A protuberance
protu berance at the leading edge of
o f the plate acts to trip the
boundary layer, and the p late itself may be assumed to be isothermal
isot hermal.. The water
wat er velocity and
temperature are 2 m/s and 17°C.
17 °C.
a) What is the temperature of the copper plate?
2
b) If each component has a plate contact surface area of 1 cm and the corresponding area-4
2
specific contact resistance is 2×10 m K/W, what is the component temperature?
t emperature?
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S.18 A square aluminum plate 5 mm thick and 200 mm on a side is cooled while vertically
suspended in quiescent air
a ir at 40°C. Determine the average heat transfer coefficient for the plate
when its temperature is 15°C.
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S.19 An aluminum plate, heated to a uniform temperature of 227°C, is allowed
allowed to cool while
vertically suspended in a room
roo m where the ambient air and surroundings are at 27°C. The plate is
0.3 m square with a thickness
t hickness of 15 mm and an e missivity
missivity of
o f 0.25.
a) Develop an expression for the
t he time rate of change of the p late temperature, assuming the
temperature to be uniform at any t ime.
b) Determine the initial rate of cooling (K/s) when t he plate temperature is 227°C.
c) Justify the uniform plate temperature assumption.
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S.20 The plate described
descr ibed in the previous problem has been used in an experiment to determine
the free convection heat transfer
t ransfer coefficient. At an instant of time when the p late temperature
was 127°C, the time rate of
o f change of this temperature was observed to be -0.0465 K/s. What is
the corresponding free convection
convect ion heat transfer coefficient? Compare this result with an estimate
based on a standard empirical correlation.
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S.21 A household oven door of 0.5 m height and 0.7 m width reaches an average surface
temperature of 32°C during operation. Estimate the heat loss to the room with ambient air at
22°C. If the door has an emissivi
e missivity
ty of 1.0 and the surroundings are a lso at 22°C, comment on the
heat loss by free convection
convect ion relative to that by radiation.
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S.22 An enclosure has an
a n inside area of 100 m2, a nd its inside surface is black and is maintained
at a constant temperature. A small opening in the enclosure has an area of 0.02 m2. The radiant
power emitted from this opening is 70 W. What is the temperature of the interior enclosure wall?
If the interior surface is maintained at this temperature, but is now polished, what will be the
value of the radiant power emitted from the opening?
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S.23 Under steady state operation, a 50 W incandescent light bulb has a surface temperature of
135°C when the room air is at a temperature of 25°C. If the bulb may be approximated as a 60
mm diameter sphere with a diffuse, gray surface o f emissivity
emissivity 0.8, what is the radiant heat
transfer from the bulb surface to its surroundings?