Examples and Problems

1
Examples and Problems
2
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.
• 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.

3
Problem 1.31 (Contd.)

1. 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?

4
Problem 1.44

• Radioactive wastes are packed in a long,
  thin-walled cylindrical container. The wastes
  generate thermal energy non-uniformly according
  to the relation , where is
  the local rate of energy generation per unit
  volume, 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.

5
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 mm.
• 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.
• 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.

6
Problem 2.22

• Uniform internal heat generation 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, ? 1100
  kg/m3, and cp 800 J/kg.K.

7
Problem 2.22 (contd.)

1. What is the rate of heat transfer per unit length
   of the rod at r 0 (the centerline) and r 25
   mm (the surface)?
2. If the reactor power level is suddenly increased
   to W/m3, what is the initial time rate of
   temperature change at r 0 and r 25 mm?

8
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.

9
Problem 2.26 (contd.)

1. Applying an overall energy balance to the wall,
   calculate the internal energy generation rate
2. 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.

10
Problem 3.2 (a)

• The rear window of an automobile is defogged by
  passing warm air over its inner surface.
• 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?

11
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.
• Derive an expression for the temperature
  distribution T(x) in symbolic form, assuming 1-D
  conditions. Sketch the temperature distribution.
• Calculate the heat rate qx.

12
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.

13
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 T8 around the tube. Is there an
   optimum thickness associated with application of
   insulation to the tube?

14
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 , while
  there is no generation in walls A and C. The
  temperatures at the interfaces are T1 261oC and
  T2 211oC.
• a) Assuming a negligible contact resistance at
  the interfaces, determine the volumetric heat
  generation and the thermal conductivity kB.

15
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.

16
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 .
• Derive the differential equation that determines
  the steady-state temperature distribution T(x) in
  the plate.
• 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.

17
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 T8, with a convection
  coefficient h.
• Derive the differential equation that determines
  the steady-state temperature distribution T(x) in
  the plate.

18
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.

19
Problem 3.134 (contd.)
20
Problem 3.134 (contd.)

1. Sketch the equivalent thermal circuit for the
   pin-chip-board assembly, assuming
   one-dimensional, steady-state conditions and
   negligible contact resistance between the pins
   and the chip. In variable form, label appropriate
   resistances, temperatures, and heat rates.
2. 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.

21
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?

22
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.

23
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 T25oC and h30
  W/m2K. At time t 0, the plate is 200?C.
• Find the Biot number and check the validity of
  lumped analysis
• Find the plate temperature at t10sec.
• Find the time rate of change of the plate
  temperature at t0.
• (Hint Neglect radiation)
• Properties of Aluminum k200W/m.K, c900
  J/kg.K, ?2700kg/m3.

24
2nd Major Exam (062)
25
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.

26
Problem 5.111 (modified, contd.)

1. Consider the 5-node network shown 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 ?t 1 s.
2. 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.

27
Solution of Problem 5.111
28
Explicit method

• Node 1
• Node 2
• Node 3
• Node 4
• Node 5

29

• Stability condition

30
Implicit method

• Node 1
• Node 2
• Node 3
• Node 4
• Node 5

31
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 form
• Estimate the surface temperature of the chip if
  it is dissipating 30 mW.

32
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 , 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 ,
  where (W/m3) is a constant. A uniform
  convection coefficient h may be assumed to exist
  between the surface of the rod and the water.

33
Problem 8.13 (Contd.)

1. Obtain expressions for the local heat flux q(x)
   and the total heat transfer q from the fuel rod
   to the water.
2. Obtain an expression for the variation of the
   mean temperature Tm(x) of the water with distance
   x along the tube.
3. 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 maximized.

34
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.

35
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?

36
Problem 7.88

• A tube bank uses an aligned arrangement of
  30-mm-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.

37
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.
• If cooling water from a large lake is pumped
  through the condenser tubes at
  , 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.

38
Problem 11.7(contd.)

1. 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?
2. 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.

39
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.

40
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, respectively.
• 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.

41
Problem 11.25 (contd.)

1. Determine the outlet temperature of the water.
2. 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?

42
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 steady-state temperature of 2900 K.
• What is the total hemispherical emissivity when
  the filament temperature is 2900 K?

43
Problem 12.29 (contd.)

1. Assuming the surroundings are at 300 K, what is
   the initial rate of cooling of the filament when
   the current is switched off?

44
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.