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Examples and
Problem 1.31
Chips of width L = 15 mm on a side are mounted
to a substrate that is installed in an enclosure
whose walls and air are maintained at a
temperature of Tsur = T = 25oC. The chips have
an emissivity of  = 0.60 and a maximum allowable
temperature of Ts = 85oC.
a) If heat is rejected from the chips by radiation and
natural convection, what is the maximum operating
power of each chip? The convection coefficient
depends on the chip-to-air temperature difference
and may be approximated as h = C(Ts – T)1/4,
where C = 4.2 W/m2.K5/4.
Problem 1.31 (Contd.)
b) If a fan is used to maintain air flow
through the enclosure and heat transfer
is by forced convection, with h = 250
W/m2.K, what is the maximum
operating power?
Problem 1.44
Radioactive wastes are packed in a long, thinwalled cylindrical container. The wastes
according to the relation q  qo 1 r ro 2  , where q is
the local rate of energy generation per unit
volume, q o is a constant, and ro is the radius of
the container. Steady-state conditions are
maintained by submerging the container in a
liquid that is at T and provides a uniform
convection coefficient h.
Obtain an expression for the total rate at which
energy is generated in a unit length of the
container. Use this result to obtain an expression
for the temperature Ts of the container wall.
Problem 1.63
A rectangular forced air heating duct is suspended
from the ceiling of a basement whose air and walls
are at a temperature of T = Tsur = 5oC. The duct is
15 m long, and its cross-section is 350 mm × 200
a) For an uninsulated duct whose average surface
temperature is 50oC, estimate the rate of heat loss
from the duct. The surface emissivity and
convection coefficient are approximately 0.5 and 4
W/m2.K, respectively.
b) If heated air enters the duct at 58oC and a velocity
of 4 m/s and the heat loss corresponds to the result
of part (a), what is the outlet temperature? The
density and specific heat of the air may be
assumed to be r = 1.10 kg/m3 and cp = 1008
J/kg.K, respectively.
Problem 2.22
Uniform internal heat generation q  5107 at W/m3
is occurring in a cylindrical nuclear reactor fuel
rod of 50-mm diameter, and under steady-state
conditions the temperature distribution is of the
form T(r) = a + br2, where T is in degrees
Celsius and r is in meters, while a = 800oC and b
= -4.167 × 105 oC/m2. The fuel rod properties are
k = 30 W/m.k, r = 1100 kg/m3, and cp = 800
Problem 2.22 (contd.)
a) What is the rate of heat transfer per unit length
of the rod at r = 0 (the centerline) and r = 25
mm (the surface)?
b) If the reactor power level is suddenly increased
to q 2  108 W/m3, what is the initial time rate of
temperature change at r = 0 and r = 25 mm?
Problem 2.26 (a) & (b)
One-dimensional, steady-state conduction with
uniform internal energy generation occurs in a
plane wall with a thickness of 50 mm and a
constant thermal conductivity of 5 W/m.K. For
these conditions, the temperature distribution
has the form, T(x) = a + bx + cx2. The surface at
x = 0 has a temperature of T(0) = To = 120oC
and experiences convection with a fluid for
which T = 20oC and h = 500 W/m2.K. The
surface at x = L is well insulated.
Problem 2.26 (contd.)
a) Applying an overall energy balance to
the wall, calculate the internal energy
generation rate q .
b) Determine the coefficients a, b, and c by
applying the boundary conditions to the
prescribed temperature distribution. Use
the results to calculate and plot the
temperature distribution.
Problem 3.2 (a)
The rear window of an automobile is
defogged by passing warm air over its
inner surface.
a) If the warm air is at T,i = 40oC and the
corresponding convection coefficient is hi =
30 W/m2.K, what are the inner and outer
surface temperatures of 4-mm-thick window
glass, if the outside ambient air temperature
is T,o = -10oC and the associated
convection coefficient is ho = 65 W/m2.K?
Problem 3.29
The diagram shows a conical section fabricated
from pure aluminum. It is of circular cross
section having diameter D = ax1/2, where a =
0.5 m1/2. The small end is located at x1 = 25
mm and the large end at x2 = 125 mm. The end
temperatures are T1 = 600 K and T2 = 400 K,
while the lateral surface is well insulated.
a) Derive an expression for the temperature
distribution T(x) in symbolic form, assuming 1-D
conditions. Sketch the temperature distribution.
b) Calculate the heat rate qx.
Example 3.5
The possible existence of an optimum insulation
thickness for radial systems is suggested by the
presence of competing effects associated with an
increase in this thickness. In particular, although the
conduction resistance increases with the addition of
insulation, the convection resistance decreases due
to increasing outer surface area. Hence there may
exist an insulation thickness that minimizes heat
loss by maximizing the total resistance to heat
transfer. Resolve this issue by considering the
following system.
Example 3.5 (Contd.)
1. A thin-walled copper tube of radius ri is used to
transport a low-temperature refrigerant and is
at a temperature Ti that is less than that of the
ambient air at T∞ around the tube. Is there an
optimum thickness associated with application
of insulation to the tube?
Problem 3.73
Consider 1-D conduction in a plane composite
wall. The outer surfaces are exposed to a fluid at
25oC and a convection heat transfer coefficient of
1000 W/m2.K. The middle wall B experiences
uniform heat generation q B , while there is no
generation in walls A and C. The temperatures at
the interfaces are T1 = 261oC and T2 = 211oC.
Assuming a negligible contact resistance at the
interfaces, determine the volumetric heat
generation q B and the thermal conductivity kB.
Problem 3.73 (Contd.)
b) Plot the temperature distribution, showing its
important features.
c) Consider conditions corresponding to a loss of
coolant at the exposed surface of material A ( h =
0). Determine T1 and T2 and plot the temperature
distribution throughout the system.
Problem 3.101
A thin flat plate of length L, thickness t, and width
W »L is thermally joined to two large heat sinks
that are maintained at a temperature To. The
bottom of the plate is well insulated, while the net
heat flux to the top surface of the plate is known to
have a uniform value of qo'' .
a) Derive the differential equation that determines the
steady-state temperature distribution T(x) in the
b) Solve the foregoing equation for the temperature
distribution, and obtain an expression for the rate
of heat transfer from the plate to the heat sinks.
Problem 3.102
Consider the flat plate of problem 3.101, but
with the heat sinks at different temperatures,
T(0) = To and T(L) = TL, and with the bottom
surface no longer insulated. Convection heat
transfer is now allowed to occur between this
surface and a fluid at T∞, with a convection
coefficient h.
a) Derive the differential equation that determines
the steady-state temperature distribution T(x) in
the plate.
Problem 3.134
As more and more components are placed
on a single integrated circuit (chip), the
amount of heat that is dissipated continues to
increase. However, this increase is limited by
the maximum allowable chip operating
temperature, which is approximately 75oC. To
maximize heat dissipation, it is proposed that
a 4 × 4 array of copper pin fins be
metallurgically joined to the outer surface of a
square chip that is 12.7 mm on a side.
Problem 3.134 (contd.)
Problem 3.134 (contd.)
a) Sketch the equivalent thermal circuit for the
pin-chip-board assembly, assuming onedimensional, steady-state conditions and
negligible contact resistance between the pins
and the chip. In variable form, label
appropriate resistances, temperatures, and
heat rates.
b) For the conditions prescribed in Problem 3.27,
what is the maximum rate at which heat can be
dissipated in the chip when the pins are in
place? That is, what is the value of qc for Tc =
75oC? The pin diameter and length are Dp =
1.5 mm and Lp = 15 mm.
Problem 4.10
A pipeline, used for transport of crude oil, is
buried in the earth such that its centerline is a
distance of 1.5 m below the surface. The pipe
has an outer diameter of 0.5 m and is insulated
with a layer of cellular glass 100 mm thick. What
is the heat loss per unit length of pipe under
conditions for which heated oil at 120oC flows
through the pipe and the surface of the earth is
at temperature of 0oC?
Problem 4.23
A hole of diameter D = 0.25 m is drilled through
the center of a solid block of square cross
section with w = 1 m on a side. The hole is
drilled along the length, l = 2 m, of the block,
which has a thermal conductivity of k = 150
W/m.K. The outer surfaces are exposed to
ambient air, with T,2 = 25oC and h2 = 4 W/m2.K,
while hot oil flowing through the hole is
characterized by T,1 = 300oC and h1 = 50
W/m2.K. Determine the corresponding heat rate
and surface temperatures.
2nd Major Exam (062)
An aluminum heated plate is being cooled by
air flowing over both sides and parallel to the
plate as shown in the figure below with T=25oC
and h=30 W/m2K. At time t = 0, the plate is
Find the Biot number and check the validity of
lumped analysis
ii. Find the plate temperature at t=10sec.
iii. Find the time rate of change of the plate
temperature at t=0.
(Hint: Neglect radiation)
Properties of Aluminum: k=200W/m.K, c=900 J/kg.K,
2nd Major Exam (062)
Aluminum Plate
Problem 5.111 (modified)
A plane wall of thickness 20 mm is insulated on
the left face and subjected to convection
condition on the right face, as shown below.
Problem 5.111 (modified, contd.)
a) Consider
schematically. Write the implicit form of the
finite-difference equations for the network and
determine temperature distributions for t = 50,
100, and 500 s using a time increment of Δt = 1
b) Use the one-term approximation given in
section 5.5 to obtain the temperature at the
same location and times as in (a). Compare the
two results.
Solution of Problem 5.111
 x  L / 4  0.020 / 4  0.005m
h x
500 0.005
Bi 
 16
Explicit method
p 1
 2 FoT
 1  2 FoT
Node 1:
Node 2:
p 1
 p
 Fo T  T   1  2 FoT
3 
 1
Node 3:
p 1
 p
 Fo T  T   1  2 FoT
4 
 2
Node 4:
p 1
 p
 Fo T  T 5   1  2 FoT
 3
Node 5:
p 1
 2 FoT
 1  2 Fo  2 Bi FoT 5  2 Bi FoT
Stability condition:
1  2 Fo  0
1  2 Fo  2 Bi Fo  0
 Fo 
2 1  Bi
  t  2.45
Implicit method
Node 1: 1  2 FoT 1p 1
 2 FoT
Node 2: 1  2 FoT 2p 1
p 1 
 p 1
 Fo  T
Node 3: 1  2 FoT 3p 1
p 1 
 p 1
 Fo  T
p 1
Node 4:
p 1 
 p 1
 Fo  T
T 5
p 1
Node 5:
p 1
 T
 2 FoT
p 1
 T 5  2 Bi FoT
Problem 6.26
Forced air at T = 25oC and V = 10 m/s is used to
cool electronic elements on a circuit board. One
such element is a chip, 4 mm by 4 mm, located 120
mm from the leading edge of the board.
Experiments have revealed that flow over the board
is disturbed by the elements and that convection
heat transfer is correlated by an expression of the
0.85 1 3
Nu x  0.04 Re x
Estimate the surface temperature of the chip if it is
dissipating 30 mW.
Problem 8.13
Consider a cylindrical nuclear fuel rod of length L
and diameter D that is encased in a concentric tube.
Pressurized water flows through the annular region
between the rod and the tube at a rate m , and the
outer surface of the tube is well insulated. Heat
generation occurs within the fuel rod, and the
volumetric generation rate is known to vary
sinusoidally with distance along the rod. That is
q,x  q o sin  x L, where q o (W/m3) is a constant. A
uniform convection coefficient h may be assumed to
exist between the surface of the rod and the water.
Problem 8.13 (Contd.)
a) Obtain expressions for the local heat flux q’’(x)
and the total heat transfer q from the fuel rod to
the water.
b) Obtain an expression for the variation of the
mean temperature Tm(x) of the water with
distance x along the tube.
c) Obtain an expression for the variation of the
rod surface temperature Ts(x) with distance x
along the tube. Develop an expression for the x
location at which this temperature is
Problem 8.31
To cool a summer home without using a vapor
compression refrigeration cycle, air is routed
through a plastic pipe (k = 0.15 W/m.K, Di = 0.15
m, Do = 0.17 m) that is submerged in an
adjoining body of water. The water temperature
is nominally at T = 17oC, and a convection
coefficient of ho = 1500 W/m2.K is maintained at
the outer surface of the pipe.
Problem 8.31 (contd.)
If air from the home enters the pipe at a
temperature of Tm,i = 29oC and volumetric flow
rate of Vi = 0.025 m3/s, what pipe length L is
needed to provide a discharge temperature of
Tm,o = 21oC? What is the fan power required to
move the air through this length of pipe if its
inner surface is smooth?
Problem 7.88
A tube bank uses an aligned arrangement of 30mm-diameter tubes with ST = SL = 60 mm and a
tube length of 1 m. There are 10 tube rows in the
flow direction (NL = 10) and 7 tubes per row (NT
= 7). Air with upstream conditions of T = 27oC
and V = 15 m/s is in cross flow over the tubes,
while a tube wall temperature of 100oC is
maintained by steam condensation inside the
tubes. Determine the temperature of air leaving
the tube bank, the pressure drop across the
bank, and the fan power requirement.
Problem 11.7
The condenser of a steam power plant contains N =
1000 brass tubes (kt = 110 W/m.K), each of inner
and outer diameters, Di = 25 mm and Do = 28 mm,
respectively. Steam condensation on the outer
surfaces of the tubes is characterized by a
convection coefficient of ho = 10,000 W/m2.K.
a) If cooling water from a large lake is pumped
through the condenser tubes at m c  400kg / s , what
is the overall heat transfer coefficient Uo based on
the outer surface area of a tube? Properties of the
water may be approximated as μ = 9.60 × 10-4
N.s/m2, k = 0.60 W/m.K, and Pr = 6.6.
Problem 11.7(contd.)
b) If, after extended operation, fouling provides a
resistance of R''f,i = 10-4 m2.K/W, at the inner
surface, what is the value of Uo?
c) If water is extracted from the lake at 15oC and
10 kg/s of steam at 0.0622 bars are to be
condensed, what is the corresponding
temperature of the water leaving the
condenser? The specific heat of the water is
4180 J/kg.K.
Problem 11.44
A shell-and-tube heat exchanger is to heat
10,000 kg/h of water from 16 to 84oC by hot
engine oil flowing through the shell. The oil
makes a single shell pass, entering at 160oC
and leaving at 94oC, with an average heat
transfer coefficient of 400 W/m2.K. The water
flows through 11 brass tubes of 22.9-mm inside
diameter and 25.5-mm outside diameter, with
each tube making four passes through the shell.
(a) Assuming fully developed flow for the water,
determine the required tube length per pass.
Problem 11.25
In a diary operation, milk at a flow rate of 250
liter/hour and a cow-body temperature of 38.6oC
must be chilled to a safe-to-store temperature of
13oC or less. Ground water at 10oC is available at a
flow rate of 0.72 m3/h. The density and specific heat
of milk are 10300 kg/m3 and 3860 J/kg.K,
a) Determine the UA product of a counterflow heat
exchanger required for the chilling process.
Determine the length of the exchanger if the inner
pipe has a 50-mm diameter and the overall heat
transfer coefficient is U = 1000 W/m2.K.
Problem 11.25 (contd.)
b) Determine the outlet temperature of the water.
c) Using the value of UA found in part (a),
determine the milk outlet temperature if the
water flow rate is doubled. What is the outlet
temperature if the flow rate is halved?
Problem 12.29
The spectral, hemispherical emissivity of
tungsten may be approximated by the
distribution depicted below. Consider a
cylindrical tungsten filament that is of diameter
D = 0.8 mm and length L = 20 mm. The
filament is enclosed in an evacuated bulb and
is heated by an electrical current to a steadystate temperature of 2900 K.
a) What is the total hemispherical emissivity
when the filament temperature is 2900 K?
Problem 12.29 (contd.)
b) Assuming the surroundings are at 300 K,
what is the initial rate of cooling of the
filament when the current is switched off?
Problem 13.50
A very long electrical conductor 10 mm in
diameter is concentric with a cooled cylindrical
tube 50 mm in diameter whose surface is diffuse
and gray with an emissivity of 0.9 and
temperature of 27oC. The electrical conductor
has a diffuse, gray surface with an emissivity of
0.6 and is dissipating 6.0 W per meter of length.
Assuming that the space between the two
surfaces is evacuated, calculate the surface
temperature of the conductor.

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