ME 410/1&2 – Heat
Transfer
Fall 2003
Solutions of HW # (4a)
& (4b)
(Chapter 4: TRANSIENT
HEAT CONDUCTION)
HW # (4a)
Lumped System Analysis
4-2C The
lumped system analysis is more likely to be applicable for the body cooled
naturally since the Biot number is proportional to the convection heat transfer
coefficient, which is proportional to the air velocity. Therefore, the Biot
number is more likely to be less than 0.1 for the case of natural convection.
4-6C Biot
number represents the ratio of conduction resistance within the body to
convection resistance at the surface of the body. The Biot number is more
likely to be larger for poorly conducting solids since such bodies have larger
resistances against heat conduction.
4-8C The
cylinder will cool faster than the sphere since heat transfer rate is
proportional to the surface area, and the sphere has the smallest area for a
given volume.
4-14 The
temperature of a gas stream is to be measured by a thermocouple. The time it
takes to register 99 percent of the initial DT 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 
, 
, and 
.
Analysis The characteristic length of the junction and the
Biot number are
Since 
, the lumped system analysis is applicable. Then the time
period for the thermocouple to read 99% of the initial temperature difference
is determined from
4-23 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 r0 = 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, r = 7833 kg/m3, and Cp
= 0.465 kJ/kg.°C.
Analysis The
characteristic length of the balls and the Biot number are
Therefore,
the lumped system analysis is applicable. Then the time for the annealing
process is determined to be
The amount
of heat transfer from a single ball is
Then the
total rate of heat transfer from the balls to the ambient air becomes
4-24
"!PROBLEM
4-24"
"GIVEN"
D=0.008 "[m]"
"T_i=900
[C], parameter to be varied"
T_f=100 "[C]"
T_infinity=35
"[C]"
h=75 "[W/m^2-C]"
n_dot_ball=2500
"[1/h]"
"PROPERTIES"
rho=7833 "[kg/m^3]"
k=54 "[W/m-C]"
C_p=465 "[J/kg-C]"
alpha=1.474E-6
"[m^2/s]"
"ANALYSIS"
A=pi*D^2
V=pi*D^3/6
L_c=V/A
Bi=(h*L_c)/k
"if Bi < 0.1, the lumped sytem
analysis is applicable"
b=(h*A)/(rho*C_p*V)
(T_f-T_infinity)/(T_i-T_infinity)=exp(-b*time)
m=rho*V
Q=m*C_p*(T_i-T_f)
Q_dot=n_dot_ball*Q*Convert(J/h,
W)
|
Ti [C] |
time [s] |
Q [W] |
|
500 |
127.4 |
271.2 |
|
550 |
134 |
305.1 |
|
600 |
140 |
339 |
|
650 |
145.5 |
372.9 |
|
700 |
150.6 |
406.9 |
|
750 |
155.3 |
440.8 |
|
800 |
159.6 |
474.7 |
|
850 |
163.7 |
508.6 |
|
900 |
167.6 |
542.5 |
|
950 |
171.2 |
576.4 |
|
1000 |
174.7 |
610.3 |
4-29C The Fourier number is a measure of heat conducted through a body relative to the heat stored. Thus a large value of Fourier number indicates faster propagation of heat through body. Since Fourier number is proportional to time, doubling the time will also double the Fourier number.
4-30C
This case can be handled by setting the heat transfer coefficient h to infinity 
since the temperature
of the surrounding medium in this case becomes equivalent to the surface
temperature.
4-36 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. 3 The thermal properties of the plate are constant. 4 The heat transfer coefficient is constant and uniform over the entire surface. 5 The Fourier number is t > 0.2 so that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption will be verified).
Furnace, 700°C
Properties The properties of brass at room temperature are
given to be k = 110 W/m.°C, a =
33.9´10-6
m2/s
Analysis
The Biot number for this process is
The constants 
corresponding to this Biot number are, from Table 4-1,
The Fourier number is
Therefore, the one-term approximate solution (or the transient temperature charts) is applicable. Then the temperature at the surface of the plates becomes
Discussion This problem can
be solved easily using the lumped system analysis since Bi < 0.1, and thus
the lumped system analysis is applicable. It gives
which is almost identical to the result obtained above.
4-37 "!PROBLEM 4-37"
"GIVEN"
L=0.03/2 "[m]"
T_i=25 "[C]"
T_infinity=700 "[C],
parameter to be varied"
time=10 "[min],
parameter to be varied"
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 4-1,
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 |
|
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 |
4-40E Long cylindrical steel rods are heat-treated in an oven. Their centerline temperature when they leave the oven is to be determined.
Assumptions 1 Heat conduction in the rods is one-dimensional since the rods are long and they have thermal symmetry about the center line. 2 The thermal properties of the rod are constant. 3 The heat transfer coefficient is constant and uniform over the entire surface. 4 The Fourier number is t > 0.2 so that the one-term approximate solutions (or the transient temperature charts) are applicable (this assumption will be verified).
Properties The properties of AISI stainless steel rods are given to be k = 7.74 Btu/h.ft.°F, a = 0.135 ft2/h.
Analysis The time the steel rods stays in the oven can be determined from