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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
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
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 = Ti
T = T (r, t) m, Cp
Bi = . . .
—— = Bi2τ = . . .
—— = . . .
(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 Boiling Eggs L Assumptions 1 The egg is spherical in shape with a radius of r0
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, ·
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
Fourier number: τ = —– = ————
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
T A1e → 2
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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