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Unformatted text preview: Table 4–1 to be
1 0.970, A1 1.122 Substituting these values into Eq. 4–14 gives
0 To
Ti T
T A1e 2
1 1.122e (0.970)2(1.07) 0.41 cen58933_ch04.qxd 9/10/2002 9:12 AM Page 228 228
HEAT TRANSFER and thus To T 0.41(Ti T) 200 0.41(600 200) 364°C The value of J1( 1) for 1 0.970 is determined from Table 4–2 to be 0.430.
Then the fractional heat transfer is determined from Eq. 4–18 to be Q
Qmax 1 2 J1( 1)
0 1 2 1 0.41 0.430
0.970 0.636 and thus Q 0.636Qmax 0.636 (47,354 kJ) 30,120 kJ Discussion The slight difference between the two results is due to the reading
error of the charts. 4–3 Plane
surface
T
h 0 x FIGURE 4–22
Schematic of a semiinfinite body. I TRANSIENT HEAT CONDUCTION
IN SEMIINFINITE SOLIDS A semiinfinite solid is an idealized body that has a single plane surface and
extends to infinity in all directions, as shown in Fig. 4–22. This idealized body
is used to indicate that the temperature change in the part of the body in which
we are interested (the region close to the surface) is due to the thermal conditions on a single surface. The earth, for example, can be considered to be a
semiinfinite medium in determining the variation of temperature near its surface. Also, a thick wall can be modeled as a semiinfinite medium if all we are
interested in is the variation of temperature in the region near one of the surfaces, and the other surface is too far to have any impact on the region of interest during the time of observation.
Consider a semiinfinite solid that is at a uniform temperature Ti. At time
t 0, the surface of the solid at x 0 is exposed to convection by a fluid at a
constant temperature T , with a heat transfer coefficient h. This problem can
be formulated as a partial differential equation, which can be solved analytically for the transient temperature distribution T(x, t). The solution obtained is
presented in Fig. 4–23 graphically for the nondimensionalized temperature
defined as
1 (x, t) 1 T(x, t ) T
Ti T T(x, t ) Ti
T
Ti (421) against the dimensionless variable x/(2
t) for various values of the parameter h
t/k.
Note that the values on the vertical axis correspond to x 0, and thus rept/k
resent the surface temperature. The curve h
corresponds to h → ,
which corresponds to the case of specified temperature T at the surface at
x 0. That is, the case in which the surface of the semiinfinite body is suddenly brought to temperature T at t 0 and kept at T at all times can be handled by setting h to infinity. The specified surface temperature case is closely cen58933_ch04.qxd 9/10/2002 9:12 AM Page 229 229
CHAPTER 4
1.0 Ambient
T(x, t)
T ,h 0.5
0.4
T(x, t) – T
1 – ————— = 1 – θ(x, t)
Ti – T 0.3 3
2 0.2 x 1
0.5
0.4
0.3
0.2 0.1 0.1 0.05
0.04 hα
t
k=
0.0
5 0.03
0.02 0.01
0 0.25 0.5 0.75 1.0 1.25 1.5 ξ = —x
—–
2 αt FIGURE 4–23
Variation of temperature with position and time in a semiinfinite solid initially at Ti subjected to convection to an
environment at T with a convection heat transfer coefficient of h (from P. J. Schneider, Ref. 10). approximated in practice when condensation or boiling takes place on the
surface. For a finite heat transfer coefficient h, the surface temperature
approaches the fluid temperature T as the time t approaches infinity.
The exact solution of the transient onedimensional heat conduction problem in a semiinfinite medium that is initially at a uniform temperature of Ti
and is suddenly subjected to convection at time t 0 has been obtained, and
is expressed as
T(x, t) Ti
T
Ti erfc x
2 t hx
k exp x
h2 t
erfc
k2
2
t h t
k (422) where the quantity erfc ( ) is the complementary error function, defined as
erfc ( ) 1 2 e u2 du (423) 0 Despite its simple appearance, the integral that appears in the above relation
cannot be performed analytically. Therefore, it is evaluated numerically for
different values of , and the results are listed in Table 4–3. For the special
case of h → , the surface temperature Ts becomes equal to the fluid temperature T , and Eq. 4–22 reduces to
T(x, t) Ti
Ts Ti erfc x
2 t (424) cen58933_ch04.qxd 9/10/2002 9:12 AM Page 230 230
HEAT TRANSFER TABLE 4–3
The complementary error function
erfc ( )
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36 erfc ( ) 1.00000
0.9774
0.9549
0.9324
0.9099
0.8875
0.8652
0.8431
0.8210
0.7991
0.7773
0.7557
0.7343
0.7131
0.6921
0.6714
0.6509
0.6306
0.6107 0.38
0.40
0.42
0.44
0.46
0.48
0.50
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
0.72
0.74 0.5910
0.5716
0.5525
0.5338
0.5153
0.4973
0.4795
0.4621
0.4451
0.4284
0.4121
0.3961
0.3806
0.3654
0.3506
0.3362
0.3222
0.3086
0.2953 erfc ( )
0.76
0.78
0.80
0.82
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
1.06
1.08
1.10
1.12 0.2825
0.2700
0.2579
0.2462
0.2349
0.2239
0.2133
0.2031
0.1932
0.1837
0.1746
0.1658
0.1573
0.1492
0.1413
0.1339
0.1267
0.1198
0.1132 erfc ( )
1.14
1.16
1.18
1.20
1.22
1.24
1.26
1.28
1.30
1.32
1.34
1.36
1.38
1.40
1.42
1.44
1.46
1.48
1.50 0.10...
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