1492 01413 01339 01267 01198 01132 erfc 114 116 118

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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 one-dimensional heat conduction problems associated with a large plane wall, a long cylinder, a sphere, and a semi-infinite 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 semi-infinite cylinder or plate, and even three-dimensional problems associated with geometries such as a rectangular prism or a semi-infinite 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 (two-dimensional) 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 one-dimensional geometries whose intersection is the multidimensi...
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