Thus a large value of the fourier number indicates

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Unformatted text preview: ong cylinder. After all, nothing in this world is infinite. A plate whose thickness is small relative to the other dimensions can be modeled as an infinitely large plate, except very near the outer edges. But the edge effects on large bodies are usually negligible, and thus a large plane wall such as the wall of a house can be modeled as an infinitely large wall for heat transfer purposes. Similarly, a long cylinder whose diameter is small relative to its length can be analyzed as an infinitely long cylinder. The use of the transient temperature charts and the one-term solutions is illustrated in the following examples. T h→ Ts Ts T h→ Ts = T (b) Infinite convection coefficient FIGURE 4–16 The specified surface temperature corresponds to the case of convection to an environment at T with a convection coefficient h that is infinite. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 224 224 HEAT TRANSFER . Qmax t=0 EXAMPLE 4–3 T = Ti T= T m, Cp h T (a) Maximum heat transfer (t → ) . An ordinary egg can be approximated as a 5-cm-diameter sphere (Fig. 4–19). The egg is initially at a uniform temperature of 5°C and is dropped into boiling water at 95°C. Taking the convection heat transfer coefficient to be h 1200 W/m2 · °C, determine how long it will take for the center of the egg to reach 70°C. SOLUTION An egg is cooked in boiling water. The cooking time of the egg is to be determined. Q t=0 T = Ti T = T (r, t) m, Cp h T Bi = . . . h2α t —— = Bi2τ = . . . k2 Q —— = . . . Qmax (Gröber chart) (b) Actual heat transfer for time t FIGURE 4–17 The fraction of total heat transfer Q/Qmax up to a specified time t is determined using the Gröber charts. L L Boiling Eggs L Assumptions 1 The egg is spherical in shape with a radius of r0 2.5 cm. 2 Heat conduction in the egg is one-dimensional because of thermal symmetry about the midpoint. 3 The thermal properties of the egg and the heat transfer coefficient are constant. 4 The Fourier number is 0.2 so that the one-term approximate solutions are applicable. Properties The water content of eggs is about 74 percent, and thus the thermal conductivity and diffusivity of eggs can be approximated by those of water at the average temperature of (5 70)/2 37.5°C; k 0.627 W/m · °C and k/ Cp 0.151 10 6 m2/s (Table A-9). Analysis The temperature within the egg varies with radial distance as well as time, and the temperature at a specified location at a given time can be determined from the Heisler charts or the one-term solutions. Here we will use the latter to demonstrate their use. The Biot number for this problem is 47.8 which is much greater than 0.1, and thus the lumped system analysis is not applicable. The coefficients 1 and A1 for a sphere corresponding to this Bi are, from Table 4–1, · Qconducted · Q (1200 W/m2 · °C)(0.025 m) 0.627 W/m · °C hr0 k Bi 1 3.0753, A1 1.9958 Substituting these and other values into Eq. 4–15 and solving for · Qstored · Qconducted αt Fourier number: τ = —– = ———— · 2 L Qstored FIGURE 4–18 Fourier number at time t can be viewed as the ratio of the rate of heat conducted to the rate of heat stored at that time. To Ti T T A1e → 2 1 70 5 95 95 1.9958e (3.0753)2 → gives 0.209 which is greater than 0.2, and thus the one-term solution is applicable with an error of less than 2 percent. Then the cooking time is determined from the definition of the Fourier number to be t ro2 (0.209)(0.025 m)2 0.151 10 6 m2/s 865 s 14.4 min Therefore, it will take about 15 min for the center of the egg to be heated from 5°C to 70°C. Discussion Note that the Biot number in lumped system analysis was defined differently as Bi hLc /k h(r /3)/k. However, either definition can be used in determining the applicability of the lumped system analysis unless Bi 0.1. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 225 225 CHAPTER 4 EXAMPLE 4–4 Heating of Large Brass Plates in an Oven In a production facility, large brass plates of 4 cm thickness that are initially at a uniform temperature of 20°C are heated by passing them through an oven that is maintained at 500°C (Fig. 4–20). The plates remain in the oven for a period of 7 min. Taking the combined convection and radiation heat transfer coefficient to be h 120 W/m2 · °C, determine the surface temperature of the plates when they come out of the oven. SOLUTION 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 and the heat transfer coefficient are constant. 3 The Fourier number is 0.2 so that the one-term approximate solutions are applicable. Properties The properties of brass at room temperature are k 110 W/m · °C, 8530 kg/m3, Cp 380 J/kg · °C, and 33.9 10 6 m2/s (Table A-3). More accurate results are obtained by using properties at average temperature. Analysis The tempe...
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