20–41
20–46 A cylindrical resistance heater is placed horizontally in a fluid. The outer surface temperature of the resistance wire is
to be determined for two different fluids.
1–
25
K 001294.0
K)273500(
11
,6986.0Pr
/sm 10804.7
C W/m.05572.0
f
T
k
1–
26
K 000377.0
32.4Pr
/sm 106582.0/
C W/m.631.0
k
repeat the calculations if necessary. The characteristic length in this case is the outer diameter of the wire,
m. 005.0 DLc
Then,
7.214)6986.0(
)/sm 10804.7(
)m 005.0(C)201200)(K 001294.0)(m/s 81.9(
Pr
)(
225
3-12
2
3
DTTg
Ra s
919.1
6986.0/559.01
)7.214(387.0
6.0
Pr/559.01
387.0
6.0
2
27/8
16/9
6/1
2
27/8
16/9
6/1
Ra
Nu
2
2
m 01178.0)m 75.0)(m 005.0(
C. W/m38.21)919.1(
m 005.0
C W/m.05572.0
DLA
Nu
D
k
h
s
197,92)32.4(
)/sm 106582.0(
)m 005.0)(K 2040)(K 000377.0)(m/s 81.9(
Pr
)(
226
3-12
2
3
DTTg
Ra s
986.8
32.4/559.01
)197,92(387.0
6.0
Pr/559.01
387.0
6.0
2
27/8
16/9
6/1
2
27/8
16/9
6/1
Ra
Nu
C W/m.631.0 2
k
Air
T = 20C
Resistance
heater, Ts
300 W
L = 0.75 m
D = 0.5 cm
20–44
20–49 An insulated electric wire is exposed to calm air. The temperature at the interface of the wire and the plastic insulation
is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The local atmospheric
pressure is 1 atm.
3.339)7255.0(
)/sm 10702.1(
)m 006.0)(K 3050)(K 003195.0)(m/s 81.9(
Pr
)(
225
3-12
2
3
DTTg
Ra s
101.2
7255.0/559.01
)3.339(387.0
6.0
Pr/559.01
387.0
6.0
2
27/8
16/9
6/1
2
27/8
16/9
6/1
Ra
Nu
2
2
m 2262.0m) m)(12 006.0(
C. W/m327.9)101.2(
C W/m.02662.0
DLA
Nu
k
h
s
The rate of heat generation, and thus the rate of heat transfer is
20–45
20–50 A steam pipe extended from one end of a plant to the other with no insulation on it. The rate of heat loss from the
steam pipe and the annual cost of those heat losses are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The local atmospheric
pressure is 1 atm.
1–
25
K 002717.0
K)27395(
11
7122.0Pr
/sm 10254.2
C W/m.0306.0
f
T
k
Analysis The characteristic length in this case is the outer diameter of the pipe,
m 0.0603 DLc
. Then,
6
3-12
3
)m 0603.0)(K 20170)(K 002717.0)(m/s 81.9(
)(
DTTg
Steam
L = 60 m
D =6.03 cm
= 0.7
Air
20–46
20–51 Prob. 20–50 is reconsidered. The effect of the surface temperature of the steam pipe on the rate of heat loss from
the pipe and the annual cost of this heat loss is to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
D=0.0603 [m]
T_s=170 [C]
T_infinity=20 [C]
epsilon=0.7
g=9.807 [m/s^2] “gravitational acceleration”
“ANALYSIS”
delta=D
Ra=(g*beta*(T_s-T_infinity)*delta^3)/nu^2*Pr
Nusselt=(0.6+(0.387*Ra^(1/6))/(1+(0.559/Pr)^(9/16))^(8/27))^2
145
21168
8923
20000
Cost [$]
20–50
20–55 Hot engine flows in a horizontal pipe with a known inner surface temperature. The pipe outer surface is covered with a
layer of insulation. The outer surface temperature is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Surface temperatures are constant. 3 Thermal conductivities of pipe and
insulation are constant. 4 Contact resistance is negligible. 5 Radiation heat transfer is negligible. 6 Local atmospheric
pressure is 1 atm.
38.17
7228.0
559.0
1
)108633.1(387.0
6.0
Pr
559.0
1
Ra387.0
6.0Nu
2
27/8
16/9
6/16
2
27/8
16/9
6/1
K W/m02735.0
k
20–53
20–58 A hot liquid flowing inside a horizontal pipe with a known mass flow rate and temperature difference of the pipe inlet
and outlet. The pipe outer surface temperature is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Surface temperatures are constant. 3 Local atmospheric pressure is 1 atm.
4 The film temperature is 35°C (this assumption will be verified). 5 The Tsurr is the same as the air temperature.
Properties We first assume the film temperature is Tf = 35°C. Then, the properties of air at Tf = 35°C are k = 0.02625 W/m∙K,
21.13
7268.0
559.0
1
)065,703(387.0
6.0
Pr
559.0
1
Ra387.0
6.0Nu
2
27/8
16/9
6/1
2
27/8
16/9
6/1
K W/m02625.0
k
20–54
20–59E The average surface temperature of a human head is to be determined when it is not covered.
Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The local atmospheric
pressure is 1 atm. 4 The head can be approximated as a 12-in.-diameter sphere.
Properties The solution of this problem requires a trial-and-error approach since the determination of the Rayleigh number
1–
24
R 001852.0
R)46080(
11
7290.0Pr
/sft 10697.1
FBtu/h.ft. 01481.0
f
T
k
Air
T = 70F
Head
Q
= ¼ 240 Btu/h
D = 12 in
= 0.9
20–55
20–60 The equilibrium temperature of a light glass bulb in a room is to be determined.
Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The local atmospheric
pressure is 1 atm. 4 The light bulb is approximated as an 8–cm-diameter sphere.
Properties The solution of this problem requires a trial-and-error approach since the determination of the Rayleigh number
and thus the Nusselt number depends on the surface temperature which is unknown. We start the solution process by
“guessing” the surface temperature to be 170C for the evaluation of the properties and h. We will check the accuracy of this
guess later and repeat the calculations if necessary. The properties of air at 1 atm and the anticipated film temperature of
20–56
20-61 Water in a tank is to be heated by a spherical heater. The heating time is to be determined.
57.5C are (Table A-15)
26
/sm 10493.0
C W/m.6515.0
k
Resistance
heater
PROPRIETARY MATERIAL. © 2017 McGraw-Hill Education. Limited distribution permitted only to teachers and educators for course preparation. If
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currents will develop in the enclosure since the lighter (hot) fluid will always be on top of the heavier (cold) fluid.
The thermal resistance of air space will be zero only when the convection coefficient approaches infinity, which is never the
case. However, when the air space is eliminated, so is its thermal resistance.
20–68 Conduction thermal resistance of a medium is expressed as
)/(kALR
. Thermal resistance of a rectangular enclosure
can be expressed by replacing L with characteristic length of enclosure Lc, and thermal conductivity k with effective thermal
conductivity
eff
k
to give
Lc
Q
20–60
20–69 A rectangular enclosure consists of two surfaces separated by an air gap, and the ratio of the heat transfer rate for the
horizontal orientation (with hotter surface at the bottom) to that of vertical orientation is to be determined.