Unformatted text preview: 69
0.10090
0.09516
0.08969
0.08447
0.07950
0.07476
0.07027
0.06599
0.06194
0.05809
0.05444
0.05098
0.04772
0.04462
0.04170
0.03895
0.03635
0.03390 erfc ( )
1.52
1.54
1.56
1.58
1.60
1.62
1.64
1.66
1.68
1.70
1.72
1.74
1.76
1.78
1.80
1.82
1.84
1.86
1.88 0.03159
0.02941
0.02737
0.02545
0.02365
0.02196
0.02038
0.01890
0.01751
0.01612
0.01500
0.01387
0.01281
0.01183
0.01091
0.01006
0.00926
0.00853
0.00784 erfc ( )
1.90
1.92
1.94
1.96
1.98
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.20
3.40
3.60 0.00721
0.00662
0.00608
0.00557
0.00511
0.00468
0.00298
0.00186
0.00114
0.00069
0.00041
0.00024
0.00013
0.00008
0.00004
0.00002
0.00001
0.00000
0.00000 This solution corresponds to the case when the temperature of the exposed
surface of the medium is suddenly raised (or lowered) to Ts at t 0 and is
maintained at that value at all times. Although the graphical solution given in
Fig. 4–23 is a plot of the exact analytical solution given by Eq. 4–23, it is subject to reading errors, and thus is of limited accuracy.
EXAMPLE 4–6 Ts = –10°C
Soil x Water pipe
Ti = 15°C FIGURE 4–24
Schematic for Example 4–6. Minimum Burial Depth of Water Pipes to Avoid
Freezing In areas where the air temperature remains below 0°C for prolonged periods of
time, the freezing of water in underground pipes is a major concern. Fortunately, the soil remains relatively warm during those periods, and it takes weeks
for the subfreezing temperatures to reach the water mains in the ground. Thus,
the soil effectively serves as an insulation to protect the water from subfreezing
temperatures in winter.
The ground at a particular location is covered with snow pack at 10°C for a
continuous period of three months, and the average soil properties at that location are k 0.4 W/m · °C and
0.15 10 6 m2/s (Fig. 4–24). Assuming an
initial uniform temperature of 15°C for the ground, determine the minimum
burial depth to prevent the water pipes from freezing. SOLUTION The water pipes are buried in the ground to prevent freezing. The
minimum burial depth at a particular location is to be determined.
Assumptions 1 The temperature in the soil is affected by the thermal conditions at one surface only, and thus the soil can be considered to be a semiinfinite medium with a specified surface temperature of 10°C. 2 The thermal
properties of the soil are constant. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 231 231
CHAPTER 4 Properties The properties of the soil are as given in the problem statement.
Analysis The temperature of the soil surrounding the pipes will be 0°C after
three months in the case of minimum burial depth. Therefore, from Fig. 4–23,
we have h t (since h → ) k
1 x
T (x, t ) T
Ti T 0
15 1 ( 10)
( 10) 2 0.6 0.36 t We note that t 7.78 106 s m2/s)(7.78 106 s) (90 days)(24 h/day)(3600 s/h) and thus x 2 t 2 0.36 (0.15 10 6 0.77 m Therefore, the water pipes must be buried to a depth of at least 77 cm to avoid
freezing under the specified harsh winter conditions. ALTERNATIVE SOLUTION The solution of this problem could also be determined from Eq. 4–24: T (x, t ) Ti
Ts Ti erfc x
2 t 0 15
10 15 → erfc x
2 t 0.60 The argument that corresponds to this value of the complementary error function is determined from Table 4–3 to be
0.37. Therefore, x 2 t 2 0.37 (0.15 10 6 m2/s)(7.78 106 s) T
h T
h
T(r, t) 0.80 m Heat
transfer Again, the slight difference is due to the reading error of the chart. (a) Long cylinder 4–4 I TRANSIENT HEAT CONDUCTION IN
MULTIDIMENSIONAL SYSTEMS The transient temperature charts presented earlier can be used to determine the
temperature distribution and heat transfer in onedimensional heat conduction
problems associated with a large plane wall, a long cylinder, a sphere, and a
semiinfinite medium. Using a superposition approach called the product
solution, these charts can also be used to construct solutions for the twodimensional transient heat conduction problems encountered in geometries
such as a short cylinder, a long rectangular bar, or a semiinfinite cylinder or
plate, and even threedimensional problems associated with geometries such
as a rectangular prism or a semiinfinite rectangular bar, provided that all surfaces of the solid are subjected to convection to the same fluid at temperature T
h
T(r, x, t) Heat
transfer (b) Short cylinder (twodimensional) FIGURE 4–25
The temperature in a short
cylinder exposed to convection from
all surfaces varies in both the radial
and axial directions, and thus heat
is transferred in both directions. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 232 232
HEAT TRANSFER T
h Plane wall a
ro
Long
cylinder FIGURE 4–26
A short cylinder of radius ro and
height a is the intersection of a long
cylinder of radius ro and a plane wall
of thickness a.
Plane wall
T
h T , with the same heat transfer coefficient h, and the body involves no heat
generation (Fig. 4–25). The solution in such multidimensional geometries can
be expressed as the product of the solutions for the onedimensional geometries whose intersection is the multidimensi...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.
 Spring '10
 Ghaz

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