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Nu6 = 0.42Ra<P Pr°012(S/#)03
for 10 < H/d < 40, 1 < Pr < 2 X 104, and 104 < Ras < 107.
43.3.4 The Log Mean Temperature Difference
The simplest and most common type of heat exchanger is the doublepipe heat exchanger, illustrated
in Fig. 43.15. For this type of heat exchanger, the heat transfer between the two fluids can be found
by assuming a constant overall heat transfer coefficient found from Table 43.8 and a constant fluid
specific heat. For this type, the heat transfer is given by
q=UA &Tm
where
A72  A7\
=
2
i_
m
ln(Ar2/A7\)
In this expression, the temperature difference, A7m, is referred to as the logmean temperature dif
ference (LMTD); AT^ represents the temperature difference between the two fluids at one end and
A72 at the other end. For the case where the ratio A^/AT^ is less than two, the arithmetic mean
temperature difference (AT2 + A7\)/2 may be used to calculate the heattransfer rate without intro
ducing any significant error. As shown in Fig. 43.15,
A7\ = ThJ  rc,
AT2  Thf0  Tc,0
for parallel flow
AT; = Thti  Tc^0
A72 = Th^0  Tci
for counterflow
CrossFlow Coefficient
In other types of heat exchangers, where the values of the overall heat transfer coefficient, [/, may
vary over the area of the surface, the LMTD may not be representative of the actual average tem
perature difference. In these cases, it is necessary to utilize a correction factor such that the heat
transfer, q, can be determined by
q = UAF AT;
Here the value of Arm is computed assuming counterflow conditions, A7\ = Thti — TCti and A72 =
Th,0 ~ TCt0. Figures 43.16 and 43.17 illustrate some examples of the correction factor, F, for various
multiplepass heat exchangers.
43.4 RADIATION HEAT TRANSFER
Heat transfer can occur in the absence of a participating medium through the transmission of energy
by electromagnetic waves, characterized by a wavelength, A, and frequency, v, which are related by
c = Xv. The parameter c represents the velocity of light, which in a vacuum is c0 = 2.9979 X 108
m/sec. Energy transmitted in this fashion is referred to as radiant energy and the heat transfer process
that occurs is called radiation heat transfer or simply radiation. In this mode of heat transfer, the
energy is transferred through electromagnetic waves or through photons, with the energy of a photon
being given by hv, where h represents Planck's constant.
In nature, every substance has a characteristic wave velocity that is smaller than that occurring
in a vacuum. These velocities can be related to c0 by c = c0/n, where n indicates the refractive index.
The value of the refractive index n for air is approximately equal to 1. The wavelength of the energy
given or for the radiation that comes from a surface depends on the nature of the source and various
wavelengths sensed in different ways. For example, as shown in Fig. 43.18 the electromagnetic
spectrum consists of a number of different types of radiation. Radiation in the visible spectrum occurs
in the range A = 0.40.74 /mi, while radiation in the wavelength range 0.1100 /mi is classified as
thermal radiation and is sensed as heat. For radiant energy in this range, the amount of energy given
off is governed by the temperature of the emitting body.
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 Spring '10
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 Mechanical Engineering

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