4-21
4-31 The temperature of a gas stream is to be measured by a thermocouple. The time it takes to register 99 percent of the
initial
T is to be determined.
Assumptions 1 The junction is spherical in shape with a diameter of D = 0.0012 m. 2 The thermal properties of the junction
are constant. 3 The heat transfer coefficient is constant and 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).
Properties The properties of the junction are given to be
C W/m.35 =k
,
3
kg/m 8500=
, and
CJ/kg. 320 =
p
c
.
Analysis The characteristic length of the junction and the Biot number are
1.0 000629.0
C W/m.35
)m 0002.0)(C. W/m110(
m 0002.0
6
m 0012.0
6
6/
2
2
3
surface
=
==
=====
k
hL
Bi
D
D
D
A
L
c
c
V
Gas
4-22
4-32 The temperature of a gas stream 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 junction diameter 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 constant and uniform over the junction 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).
Properties The properties of the junction are given to be k = 35 W/mK, ρ = 8500 kg/m3, and cp = 320 J/kgK.
Analysis The characteristic length of the thermocouple junction is
6
6/
2
3D
D
D
A
L
s
c===
V
The time period for the thermocouple to read 99% of the initial temperature difference is determined from
h
h
hA
s1
K) W/m250(6
6
2
4-23
4-24
4-34 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
particle 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).
Properties The properties of coal are k = 0.26 W/mK, ρ = 1350 kg/m3, and cp = 1260 J/kgK (from Table A-8).
Analysis The characteristic length and the Biot number of the coal particle are
mm 5.0
3
V
4-25
4-26
4-36 Alumina particles are injected into a plasma jet. The time it would take for the particles to reach their melting point is to
be determined.
Assumptions 1 The thermal properties of alumina are constant. 2 The heat transfer coefficient is uniform over the entire
particle 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).
Properties The properties of alumina are given as k = 30 W/mK, ρ = 3970 kg/m3, and cp = 800 J/kgK.
Analysis The characteristic length and the Biot number of the alumina particle are
μm 60
6/
3
D
V
4-27
4-38 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
lumped system analysis is applicable (this assumption will be verified).
Properties The thermal conductivity, density, and specific heat of the balls are given to be k = 54 W/m.C, = 7833 kg/m3,
and cp = 0.465 kJ/kg.C.
Analysis The characteristic length of the balls and the Biot number are
m 0013.0
6
m 008.0
6
6/
2
3
=====
D
D
D
A
L
s
c
V
Furnace
4-28
4-29
4-40E A number of brass balls are to be quenched in a water bath at a specified rate. The temperature of the balls after
quenching and the rate at which heat needs to be removed from the water in order to keep its temperature constant are to be
determined.
Assumptions 1 The balls are spherical in shape with a radius of ro = 1 in. 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 lumped
system analysis is applicable (this assumption will be verified).
Properties The thermal conductivity, density, and specific heat of the brass balls are given to be k = 64.1 Btu/h.ft.F, = 532
lbm/ft3, and cp = 0.092 Btu/lbm.F.
Analysis (a) The characteristic length and the Biot
number for the brass balls are
ft 02778.0
6
ft 12/2
6
6/
2
3
=====
D
D
D
A
L
s
c
V
Brass balls, 250F
Water bath, 120F
4-30
4-31
4-42 An electronic device is on for 5 minutes, and off for several hours. The temperature of the device at the end of the 5-min
operating period is to be determined for the cases of operation with and without a heat sink.
Assumptions 1 The device and the heat sink are isothermal. 2 The thermal properties of the device and of the sink are
constant. 3 The heat transfer coefficient is constant and uniform over the entire surface.
Properties The specific heat of the device is given to be cp = 850 J/kg.C. The specific heat of the aluminum sink is 903
J/kg.C (Table A-3), but can be taken to be 850 J/kg.C for simplicity in analysis.
Analysis
(a) Approximate solution
This problem can be solved approximately by using an average temperature for the device
when evaluating the heat loss. An energy balance on the device can be expressed as
devicegenerationoutdevicegenerationoutin TmctEtQEEEE p=+−⎯→⎯=+−
−=
−
+
TT
Note that the temperature of the electronic device drops considerably as a result of attaching it to a heat sink.
(b) Exact solution
This problem can be solved exactly by obtaining the differential equation from an energy balance on the device for a
differential time interval dt. We will get
s
E
hA
TTd generation
)(
−
Electronic
device, 18 W
4-32
Transient Heat Conduction in Large Plane Walls, Long Cylinders, and Spheres with Spatial Effects
4-44C A cylinder whose diameter is small relative to its length can be treated as an infinitely long cylinder. When the
4-45C The Fourier number is a measure of heat conducted through a body relative to the heat stored. Thus a large value of
4-33
4-50 The temperature at the center plane of a brass plate after 3 minutes of cooling by impinging air jet is to be determined.
Assumptions 1 Heat conduction is one-dimensional. 2 Thermal properties are constant. 3 Convection heat transfer coefficient
is uniform. 4 Heat transfer by radiation is negligible.
Properties The properties of the brass plate are given as
= 8530 kg/m3, cp = 380 J/kg ∙ K, k = 110 W/m ∙ K, and
= 33.9
10−6 m2/s.
Analysis This geometry can be considered to be a large plane wall with a thickness of 2L = 20 cm subjected to convection at
both sides. The surface with insulation becomes the center surface of the wall. Then the Biot number for this process is
)m 10.0)(K W/m220(2
hL
4-34
4-35
4-52 ASTM A240 410S stainless steel plate is exposed to hot fluid at T∞ = 700°C and h = 323 W/m2·K. The plate has an
initial temperature of 20°C. How long can the plate expose to the hot fluid before reaching its maximum use temperature?
Assumptions1 Heat conduction is transient and one dimensional. 2 Thermal properties are constant. 3 Radiation effects are
negligible. 4 The convection heat transfer coefficient is constant. 5 The Fourier number is > 0.2 so that the one term-term
approximate solutions areapplicable.
Properties The thermal properties given are cp = 460 J/kg·K, k = 26.9
W/m·K, and ρ = 7730 kg/m3.
Analysis The plate is considered as a large plane wall with a thickness
of 2L = 5 cm subjected to convection on both sides. The Biot number
for this process is
4-36
4-53 The temperature at the center plane of an aluminum plate with
TTs
, after 15 seconds of heating, is to be determined.
Assumptions 1 Heat conduction is one-dimensional. 2 Thermal properties are constant. 3 Heat transfer by radiation is
negligible.
Properties The thermal diffusivity erties of the aluminum plate is given as
= 97.1 10−6 m2/s.
Analysis For
TTs
, it implies that
→h
. Thus, the Biot number is
hL
4-37
4-54 Large brass plates are heated in an oven. The surface temperature of the plates leaving the oven is to be determined.
Assumptions 1 Heat conduction in the plate is one-dimensional since the plate is large relative to its thickness and there is
thermal symmetry about the center plane. 2 The thermal properties of the plate are constant. 3 The heat transfer coefficient is
constant and uniform over the entire surface. 4 The Fourier number is > 0.2 so that the one-term approximate solutions are
applicable (this assumption will be verified).
Properties The properties of brass at room temperature are given to be k = 110 W/mC,
= 33.910-6 m2/s
Analysis The Biot number for this process is
)m 015.0)(C. W/m80(2
hL
4-38
4-55 Prob. 4-54 is reconsidered. The effects of the temperature of the oven and the heating time on the final surface
temperature of the plates are to be investigated.
Analysis The problem is solved using EES, and the solution is given below.
“GIVEN”
L=(0.03/2) [m]
T_i=25 [C]
T_infinity=700 [C]
time=10 [min]
h=80 [W/m^2-C]
“PROPERTIES”
k=110 [W/m-C]
alpha=33.9E-6 [m^2/s]
“ANALYSIS”
Bi=(h*L)/k
“From Table 4-2, corresponding to this Bi number, we read”
lambda_1=0.1039
A_1=1.0018
tau=(alpha*time*Convert(min, s))/L^2
(T_L-T_infinity)/(T_i-T_infinity)=A_1*exp(-lambda_1^2*tau)*Cos(lambda_1*L/L)
T
[C]
TL
[C]
500
321.6
525
337.2
550
352.9
575
368.5
600
384.1
625
399.7
650
415.3
675
430.9
700
446.5
725
462.1
750
477.8
775
493.4
800
509
825
524.6
850
540.2
875
555.8
900
571.4
300
350
400
450
500
550
600
TL [C]
4-39
time
[min]
TL
[C]
2
146.7
4
244.8
6
325.5
8
391.9
10
446.5
12
491.5
14
528.5
16
558.9
18
583.9
20
604.5
22
621.4
24
635.4
26
646.8
28
656.2
30
664
0 5 10 15 20 25 30
100
200
300
400
500
600
700
time [min]
TL [C]
4-40