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18-2
Lumped System Analysis
18-1C Biot number represents the ratio of conduction resistance within the body to convection resistance at the surface of the
18-2C In heat transfer analysis, some bodies are observed to behave like a "lump" whose entire body temperature remains
essentially uniform at all times during a heat transfer process. The temperature of such bodies can be taken to be a function of
18-4C The lumped system analysis is more likely to be applicable for a golden apple than for an actual apple since the
thermal conductivity is much larger and thus the Biot number is much smaller for gold.
18-6C The lumped system analysis is more likely to be applicable for the body cooled naturally since the Biot number is
18-7C The lumped system analysis is more likely to be applicable for the body allowed to cool in the air since the Biot
18-8C The temperature drop of the potato during the second minute will be less than 4C since the temperature of a body
18-9C The temperature rise of the potato during the second minute will be less than 5C since the temperature of a body
18-3
18-10C The heat transfer is proportional to the surface area. Two half pieces of the roast have a much larger surface area than
18-12 Relations are to be obtained for the characteristic lengths of a large plane wall of thickness 2L, a very long cylinder of
radius ro and a sphere of radius ro.
Analysis Relations for the characteristic lengths of a large plane
wall of thickness 2L, a very long cylinder of radius ro and a
18-4
18-14 The time required to cool a brick from 1100°C to a temperature difference of 5°C from the ambient air temperature is
to be determined.
Assumptions 1 Thermal properties are constant. 2 Convection heat transfer coefficient is uniform. 3 Heat transfer by
radiation is negligible.
Properties The properties of the brick are given as
= 1920 kg/m3, cp = 790 J/kg ∙ K, and k = 0.90 W/m ∙ K.
difference of 5°C from the ambient air temperature is
14
3
2
s 10128.2
)m 01549.0)(KJ/kg 790)(kg/m 1920(
K W/m5 -
cpp
s
Lρc
h
c
hA
b
V
bt
TtT
)(
18-6
18-16 Prob. 18-15 is reconsidered. The effects of the heat transfer coefficient and the final plate temperature on the
time it will take for the plate to reach this temperature are to be investigated.
A=0.03 [m^2]
T_infinity=22 [C]
T_i=T_infinity
h=12 [W/m^2-C]
h
[W/m2.C]
time
[s]
5
7
9
11
13
15
17
19
21
23
25
51
51.22
51.43
51.65
51.88
52.1
52.32
52.55
52.78
53.01
53.24
51
51.5
52
52.5
53
53.5
h [W/m2-C]
time [s]
18-7
Tf
[C]
time
[s]
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
3.428
7.728
12.05
16.39
20.74
25.12
29.51
33.92
38.35
42.8
47.28
51.76
56.27
60.8
65.35
69.92
74.51
79.12
20 40 60 80 100 120 140 160 180 200
0
20
40
60
80
Tf [C]
time [s]
18-8
4–17 Metal plates are heated in an oven. The temperature of the plates exiting the oven is to be determined.
Assumptions 1 The thermal properties are constant. 2 The heat transfer coefficient is uniform over the entire surface of all the
metal plates. 3 Radiation effects are negligible. 4 The Biot number is Bi < 0.1 so that the lumped system analysis is
applicable (this assumption will be verified).
Properties The properties of the metal plates are given as k = 180 W/mK, ρ = 2800 kg/m3, and cp = 880 J/kgK.
18-10
4–19 Stainless steel plates are heat treated as they move through a furnace. The effect of the plate velocity on the plate
temperature at the furnace exit is to be determined.
Assumptions 1 The thermal properties are constant. 2 The heat transfer coefficient is uniform over the entire surface. 3
Radiation effects are negligible. 4 The Biot number is Bi < 0.1 so that the lumped system analysis is applicable (this
assumption will be verified).
Discussion As the plate velocity increases, the duration of the plates being heated in the furnace decreases. Thus, the plate
temperature at the furnace exit decreases with increasing plate velocity.
18-12
18-22 Steel rods are quenched in a hardening process. The average temperature of rods when they are taken out of oven is to
be determined.
Assumptions 1 Thermal properties are constant. 2 Convection heat transfer coefficient is uniform. 3 Heat transfer by
4
4
DL
A
s
1.0102.0
K W/m9.63
)m 01.0)(K W/m650( 2
k
hL
Bi c
Water, 50C
h = 650 W/m2C
18-13
18-23 Milk in a thin-walled glass container is to be warmed up by placing it into a large pan filled with hot water. The
warming time of the milk is to be determined.
Assumptions 1 The glass container is cylindrical in shape with a radius of
than 0.1). However, the lumped system analysis is still applicable since
Milk
18-16
4–26 The temperature of an air flow is to be measured by a thermocouple. The time it takes to register 99 percent of the
initial T should be within 5 s, and the air flow velocity (given as a function of h) is to be determined.
Assumptions 1 The junction is spherical in shape. 2 The thermal properties of the junction are constant. 3 The heat transfer
coefficient is uniform over the entire surface. 4 Radiation effects are negligible. 5 The Biot number is Bi < 0.1 so that the
lumped system analysis is applicable (this assumption will be verified).
i
TT
Dc
p
)s 5(
m) 0005.0K)(J/kg 320)(kg/m 8500(
6
exp01.0 3
h
Thus,
m/s 4.50V
Since this analysis was carried out under the assumption that it is a lumped system, and for this assumption to be applicable,
the condition Bi < 0.1 needs to be satisfied
)m 0005.0(K) W/m77.208(
2
hD
hLc
18-17
18-27 Coal particles suspended in hot air flow. The time it takes for the particles to reach 2/3 of the initial temperature
difference is to be determined.
Assumptions 1 The thermal properties of coal particles are constant. 2 The heat transfer coefficient is uniform over the entire
18-19
18-29 A number of carbon steel balls are to be annealed by heating them first and then allowing them to cool slowly in
ambient air at a specified rate. The time of annealing and the total rate of heat transfer from the balls to the ambient air are to
be determined.
Assumptions 1 The balls are spherical in shape with a radius of ro = 4 mm. 2 The thermal properties of the balls are constant.
3 The heat transfer coefficient is constant and uniform over the entire surface. 4 The Biot number is Bi < 0.1 so that the
