Soal Bab 2
Soal Bab 2
Soal Bab 2
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CHAPTER 1
3. F. P. Incropera and D. P. DeWitt. Introduction to Heat 5. M. N. Ozisik. Heat TransferA Basic Approach.
Transfer. 4th ed. New York: John Wiley & Sons, 2002. New York: McGraw-Hill, 1985.
4. S. S. Kutateladze. Fundamentals of Heat Transfer. 6. F. M. White. Heat and Mass Transfer. Reading, MA:
New York: Academic Press, 1963. Addison-Wesley, 1988.
PROBLEMS*
114
HEAT TRANSFER
steady or transient heat transfer problem? Also, would you con- ity and heat generation in its simplest form, and indicate what
sider the heat transfer to be one-dimensional or multidimen- each variable represents.
sional? Explain. 220 Write down the one-dimensional transient heat conduc-
214E The resistance wire of a 1000-W iron is 15 in. long tion equation for a long cylinder with constant thermal con-
and has a diameter of D 0.08 in. Determine the rate of heat ductivity and heat generation, and indicate what each variable
generation in the wire per unit volume, in Btu/h ft3, and the represents.
heat flux on the outer surface of the wire, in Btu/h ft2, as a re- 221 Starting with an energy balance on a rectangular vol-
sult of this heat generation. ume element, derive the one-dimensional transient heat con-
q duction equation for a plane wall with constant thermal
conductivity and no heat generation.
D g 222 Starting with an energy balance on a cylindrical shell
volume element, derive the steady one-dimensional heat con-
duction equation for a long cylinder with constant thermal con-
FIGURE P214E ductivity in which heat is generated at a rate of g.
L
Solar
x pond
r + r
FIGURE P217 0 r R r
115
CHAPTER 1
116
HEAT TRANSFER
Boundary and Initial Conditions; Express the radiation boundary condition on the outer surface
Formulation of Heat Conduction Problems of the shell.
234C What is a boundary condition? How many boundary 244 A container consists of two spherical layers, A and B,
conditions do we need to specify for a two-dimensional heat that are in perfect contact. If the radius of the interface is r0,
transfer problem? express the boundary conditions at the interface.
235C What is an initial condition? How many initial condi- 245 Consider a steel pan used to boil water on top of an
tions do we need to specify for a two-dimensional heat transfer electric range. The bottom section of the pan is L 0.5 cm
problem? thick and has a diameter of D 20 cm. The electric heating
236C What is a thermal symmetry boundary condition? unit on the range top consumes 1000 W of power during cook-
How is it expressed mathematically? ing, and 85 percent of the heat generated in the heating element
is transferred uniformly to the pan. Heat transfer from the top
237C How is the boundary condition on an insulated sur- surface of the bottom section to the water is by convection with
face expressed mathematically? a heat transfer coefficient of h. Assuming constant thermal
238C It is claimed that the temperature profile in a medium conductivity and one-dimensional heat transfer, express the
must be perpendicular to an insulated surface. Is this a valid mathematical formulation (the differential equation and the
claim? Explain. boundary conditions) of this heat conduction problem during
steady operation. Do not solve.
239C Why do we try to avoid the radiation boundary con-
ditions in heat transfer analysis?
Steel pan
240 Consider a spherical container of inner radius r1, outer
radius r2, and thermal conductivity k. Express the boundary
condition on the inner surface of the container for steady one-
dimensional conduction for the following cases: (a) specified Water
temperature of 50C, (b) specified heat flux of 30 W/m2 toward x
the center, (c) convection to a medium at T
with a heat trans- L
fer coefficient of h.
0
Spherical container
FIGURE P245
117
CHAPTER 1
118
HEAT TRANSFER
119
CHAPTER 1
r
Electric heater
FIGURE P264
r2
265 Repeat Problem 264 for a heat flux of 950 W/m2 and
a surface temperature of 85C at the left surface at x 0. r1
120
HEAT TRANSFER
the range of r r1 to r r2, and discuss the results. Use the 276C Consider uniform heat generation in a cylinder and a
EES (or other) software. sphere of equal radius made of the same material in the same
environment. Which geometry will have a higher temperature
270 In a food processing facility, a spherical container of at its center? Why?
inner radius r1 40 cm, outer radius r2 41 cm, and thermal
conductivity k 1.5 W/m C is used to store hot water and to 277 A 2-kW resistance heater wire with thermal conductiv-
keep it at 100C at all times. To accomplish this, the outer sur- ity of k 20 W/m C, a diameter of D 5 mm, and a length
face of the container is wrapped with a 500-W electric strip of L 0.7 m is used to boil water. If the outer surface temper-
heater and then insulated. The temperature of the inner surface ature of the resistance wire is Ts 110C, determine the tem-
of the container is observed to be nearly 100C at all times. As- perature at the center of the wire.
suming 10 percent of the heat generated in the heater is lost
through the insulation, (a) express the differential equation and
the boundary conditions for steady one-dimensional heat con-
duction through the container, (b) obtain a relation for the vari- 110C
ation of temperature in the container material by solving the
differential equation, and (c) evaluate the outer surface tem-
perature of the container. Also determine how much water at 0
100C this tank can supply steadily if the cold water enters r
at 20C. D
Insulation Resistance
heater
Electric
heater
Hot FIGURE P277
water
0 r1 r2 r 278 Consider a long solid cylinder of radius r0 4 cm and
thermal conductivity k 25 W/m C. Heat is generated in the
100C
cylinder uniformly at a rate of g0 35 W/cm3. The side surface
of the cylinder is maintained at a constant temperature of Ts
80C. The variation of temperature in the cylinder is given by
g r02
T
Spherical 2
r
container T(r) 1 r s
k 0
FIGURE P270
Based on this relation, determine (a) if the heat conduction is
271 Reconsider Problem 270. Using the relation ob- steady or transient, (b) if it is one-, two-, or three-dimensional,
tained for the variation of temperature in the con- and (c) the value of heat flux on the side surface of the cylinder
tainer material, plot the temperature as a function of the radius at r r0.
r in the range of r r1 to r r2, and discuss the results. Use
the EES (or other) software. 279 Reconsider Problem 278. Using the relation
obtained for the variation of temperature in the
Heat Generation in a Solid cylinder, plot the temperature as a function of the radius r in
the range of r 0 to r r0, and discuss the results. Use the
272C Does heat generation in a solid violate the first law of EES (or other) software.
thermodynamics, which states that energy cannot be created or
destroyed? Explain. 280E A long homogeneous resistance wire of radius r0
0.25 in. and thermal conductivity k 8.6 Btu/h ft F is being
273C What is heat generation? Give some examples. used to boil water at atmospheric pressure by the passage of
274C An iron is left unattended and its base temperature
r
rises as a result of resistance heating inside. When will the rate T
of heat generation inside the iron be equal to the rate of heat Water r0 h
loss from the iron?
275C Consider the uniform heating of a plate in an envi- 0
ronment at a constant temperature. Is it possible for part of the Resistance
heater
heat generated in the left half of the plate to leave the plate
through the right surface? Explain. FIGURE P280E
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electric current. Heat is generated in the wire uniformly as a coefficient of 44 W/m2 C. Explain where in the plate the
result of resistance heating at a rate of g 1800 Btu/h in3. highest and the lowest temperatures will occur, and determine
The heat generated is transferred to water at 212F by con- their values.
vection with an average heat transfer coefficient of h 820
285 Reconsider Problem 284. Using EES (or other)
Btu/h ft2 F. Assuming steady one-dimensional heat transfer,
software, investigate the effect of the heat trans-
(a) express the differential equation and the boundary condi-
fer coefficient on the highest and lowest temperatures in the
tions for heat conduction through the wire, (b) obtain a relation
plate. Let the heat transfer coefficient vary from 20 W/m2 C
for the variation of temperature in the wire by solving the dif-
to 100 W/m2 C. Plot the highest and lowest temperatures as
ferential equation, and (c) determine the temperature at the
a function of the heat transfer coefficient, and discuss the
centerline of the wire. Answer: (c) 290.8F
results.
281E Reconsider Problem 280E. Using the relation
286 A 6-m-long 2-kW electrical resistance wire is made of
obtained for the variation of temperature in the
0.2-cm-diameter stainless steel (k 15.1 W/m C). The re-
wire, plot the temperature at the centerline of the wire as a
sistance wire operates in an environment at 30C with a heat
function of the heat generation g in the range of 400 Btu/h in3
transfer coefficient of 140 W/m2 C at the outer surface. De-
to 2400 Btu/h in3, and discuss the results. Use the EES (or
termine the surface temperature of the wire (a) by using the ap-
other) software.
plicable relation and (b) by setting up the proper differential
282 In a nuclear reactor, 1-cm-diameter cylindrical uranium equation and solving it. Answers: (a) 409C, (b) 409C
rods cooled by water from outside serve as the fuel. Heat is
287E Heat is generated uniformly at a rate of 3 kW per ft
generated uniformly in the rods (k 29.5 W/m C) at a rate
length in a 0.08-in.-diameter electric resistance wire made of
of 7 107 W/m3. If the outer surface temperature of rods is
nickel steel (k 5.8 Btu/h ft F). Determine the temperature
175C, determine the temperature at their center.
difference between the centerline and the surface of the wire.
175C
288E Repeat Problem 287E for a manganese wire (k
4.5 Btu/h ft F).
g Uranium rod 289 Consider a homogeneous spherical piece of radioactive
material of radius r0 0.04 m that is generating heat at a con-
stant rate of g 4 107 W/m3. The heat generated is dissi-
FIGURE P282 pated to the environment steadily. The outer surface of the
sphere is maintained at a uniform temperature of 80C and
283 Consider a large 3-cm-thick stainless steel plate (k the thermal conductivity of the sphere is k 15 W/m C. As-
15.1 W/m C) in which heat is generated uniformly at a rate suming steady one-dimensional heat transfer, (a) express the
of 5 105 W/m3. Both sides of the plate are exposed to an en- differential equation and the boundary conditions for heat con-
vironment at 30C with a heat transfer coefficient of 60 W/m2 duction through the sphere, (b) obtain a relation for the varia-
C. Explain where in the plate the highest and the lowest tem- tion of temperature in the sphere by solving the differential
peratures will occur, and determine their values. equation, and (c) determine the temperature at the center of the
sphere.
284 Consider a large 5-cm-thick brass plate (k 111
W/m C) in which heat is generated uniformly at a rate of 80C
2 105 W/m3. One side of the plate is insulated while the other
side is exposed to an environment at 25C with a heat transfer g
0 r0 r
Brass
plate
g
h
FIGURE P289
T
290 Reconsider Problem 289. Using the relation ob-
tained for the variation of temperature in the
sphere, plot the temperature as a function of the radius r in the
range of r 0 to r r0. Also, plot the center temperature of
0 the sphere as a function of the thermal conductivity in the
L x
range of 10 W/m C to 400 W/m C. Discuss the results. Use
FIGURE P284 the EES (or other) software.
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HEAT TRANSFER
291 A long homogeneous resistance wire of radius r0 always equivalent to the conductivity value at the average tem-
5 mm is being used to heat the air in a room by the passage of perature?
electric current. Heat is generated in the wire uniformly at a 299 Consider a plane wall of thickness L whose thermal
rate of g 5 107 W/m3 as a result of resistance heating. If conductivity varies in a specified temperature range as k(T)
the temperature of the outer surface of the wire remains at k0(1 T2) where k0 and are two specified constants. The
180C, determine the temperature at r 2 mm after steady op- wall surface at x 0 is maintained at a constant temperature of
eration conditions are reached. Take the thermal conductivity T1, while the surface at x L is maintained at T2. Assuming
of the wire to be k 8 W/m C. Answer: 212.8C steady one-dimensional heat transfer, obtain a relation for the
heat transfer rate through the wall.
r
2100 Consider a cylindrical shell of length L, inner radius
180C r1, and outer radius r2 whose thermal conductivity varies
r0
linearly in a specified temperature range as k(T) k0(1 T)
g where k0 and are two specified constants. The inner surface
0
of the shell is maintained at a constant temperature of T1, while
FIGURE P291 the outer surface is maintained at T2. Assuming steady one-
dimensional heat transfer, obtain a relation for (a) the heat
transfer rate through the wall and (b) the temperature distribu-
292 Consider a large plane wall of thickness L 0.05 m.
tion T(r) in the shell.
The wall surface at x 0 is insulated, while the surface at x
L is maintained at a temperature of 30C. The thermal conduc-
tivity of the wall is k 30 W/m C, and heat is generated in
the wall at a rate of g g0e0.5x/L W/m3 where g0 8 106
W/m3. Assuming steady one-dimensional heat transfer, (a) ex-
press the differential equation and the boundary conditions for Cylindrical
shell
heat conduction through the wall, (b) obtain a relation for the
variation of temperature in the wall by solving the differential T2
equation, and (c) determine the temperature of the insulated
surface of the wall. Answer: (c) 314C T1
h k(T)
293 Reconsider Problem 292. Using the relation 0 r1
given for the heat generation in the wall, plot the r2
heat generation as a function of the distance x in the range of r
x 0 to x L, and discuss the results. Use the EES (or other)
software. FIGURE P2100
123
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124
HEAT TRANSFER
Plane Tsurr
wall
hi
Ti h 45C
T0
ho
0 r1
r2 h
r T
FIGURE P2125
0 x
L
2126 The boiling temperature of nitrogen at atmospheric
pressure at sea level (1 atm pressure) is 196C. Therefore, ni-
trogen is commonly used in low temperature scientific studies FIGURE P2128
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CHAPTER 1
2129 A 1000-W iron is left on the iron board with its base surface temperature of the roof is T1 62F, determine the
exposed to ambient air at 20C. The base plate of the iron has outer surface temperature of the roof and the rate of heat loss
a thickness of L 0.5 cm, base area of A 150 cm2, and ther- through the roof when steady operating conditions are reached.
mal conductivity of k 18 W/m C. The inner surface of the
base plate is subjected to uniform heat flux generated by the re- 2132 Consider a long resistance wire of radius r1 0.3 cm
sistance heaters inside. The outer surface of the base plate and thermal conductivity kwire 18 W/m C in which heat is
whose emissivity is 0.7, loses heat by convection to ambi- generated uniformly at a constant rate of g 1.5 W/cm3 as a
ent air at T
22 C with an average heat transfer coefficient result of resistance heating. The wire is embedded in a 0.4-cm-
of h 30 W/m2 C as well as by radiation to the surrounding thick layer of plastic whose thermal conductivity is kplastic 1.8
surfaces at an average temperature of Tsurr 290 K. Dis- W/m C. The outer surface of the plastic cover loses heat by
regarding any heat loss through the upper part of the iron, convection to the ambient air at T
25C with an average
(a) express the differential equation and the boundary con- combined heat transfer coefficient of h 14 W/m2 C. As-
ditions for steady one-dimensional heat conduction through suming one-dimensional heat transfer, determine the tempera-
the plate, (b) obtain a relation for the temperature of the outer tures at the center of the resistance wire and the wire-plastic
surface of the plate by solving the differential equation, and layer interface under steady conditions.
Answers: 97.1C, 97.3C
(c) evaluate the outer surface temperature.
T
Iron Tsurr h
base
plate
Wire
h g r2
r1
T
r
Plastic cover
0 FIGURE P2132
L x
Tsky Ts
D Fuel rod g
y
T
h FIGURE P2134
L
126
HEAT TRANSFER
radiation, (a) express the differential equations and the bound- 2139 Write an interactive computer program to calcu-
ary conditions for steady one-dimensional heat conduction late the heat transfer rate and the value of tem-
through the wall, (b) obtain a relation for the variation of tem- perature anywhere in the medium for steady one-dimensional
perature in the wall by solving the differential equation, and heat conduction in a long cylindrical shell for any combination
(c) evaluate the temperatures at the inner and outer surfaces of of specified temperature, specified heat flux, and convection
the wall. boundary conditions. Run the program for five different sets of
2136 Consider a water pipe of length L 12 m, inner ra- specified boundary conditions.
dius r1 15 cm, outer radius r2 20 cm, and thermal conduc- 2140 Write an interactive computer program to calculate the
tivity k 20 W/m C. Heat is generated in the pipe material heat transfer rate and the value of temperature anywhere in
uniformly by a 25-kW electric resistance heater. The inner and the medium for steady one-dimensional heat conduction in
outer surfaces of the pipe are at T1 60C and T2 80C, re- a spherical shell for any combination of specified tempera-
spectively. Obtain a general relation for temperature distribu- ture, specified heat flux, and convection boundary conditions.
tion inside the pipe under steady conditions and determine the Run the program for five different sets of specified boundary
temperature at the center plane of the pipe. conditions.
2137 Heat is generated uniformly at a rate of 2.6 106 2141 Write an interactive computer program to calculate the
W/m3 in a spherical ball (k 45 W/m C) of diameter 30 cm. heat transfer rate and the value of temperature anywhere in the
The ball is exposed to iced-water at 0C with a heat transfer co- medium for steady one-dimensional heat conduction in a plane
efficient of 1200 W/m2 C. Determine the temperatures at the wall whose thermal conductivity varies linearly as k(T)
center and the surface of the ball. k0(1 T) where the constants k0 and are specified by the
user for specified temperature boundary conditions.
Computer, Design, and Essay Problems
2138 Write an essay on heat generation in nuclear fuel rods.
Obtain information on the ranges of heat generation, the varia-
tion of heat generation with position in the rods, and the ab-
sorption of emitted radiation by the cooling medium.