30079_43b.pdf

Nu6 = 0.42Ra<P Pr°-012(S/#)0-3

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 double-pipe 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 log-mean 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 heat-transfer 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

Cross-Flow 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

multiple-pass 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 = 0.4-0.74 /mi, while radiation in the wavelength range 0.1-100 /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.

43.4.1 Black-Body Radiation

All objects in space are continuously being bombarded by radiant energy of one form or another and

all of this energy is either absorbed, reflected, or transmitted. An ideal body that absorbs all the

radiant energy falling upon it, regardless of the wavelength and direction, is referred to as a black

body. Such a body emits the maximum energy for a prescribed temperature and wavelength. Radiation

from a black body is independent of direction and is referred to as a diffuse emitter.

Parallel flow Counterflow

Fig. 43.15 Temperature profiles for parallel flow and counterflow in double-pipe heat exchanger.

Fig. 43.16 Correction factor for a shell-and-tube heat exchanger with one shell and any

multiple of two tube passes (two, four, etc., tube passes).

The Stefan-Boltzmann Law

The Stefan-Boltzmann law describes the rate at which energy is radiated from a black body and

states that this radiation is proportional to the fourth power of the absolute temperature of the body

eb = crT4

where eb is the total emissive power and a is the Stefan-Boltzmann constant, which has the value

5.729 X 10-8W/m2-K4 (0.173 X ICT8 Btu/hr -ft2-°R4).

Planck's Distribution Law

The temperature dependent amount of energy leaving a black body is described as the spectral

emissive power e8b and is a function of wavelength. This function, which was derived from quantum

theory by Planck, is

exb = 27rC1/A5[exp(C2/Ar) - 1]

where e^ has a unit W/m2 • pun (Btu/hr • ft2 • jum).

Values of the constants Cl and C2 are 0.59544 X lO'16 W • m2 (0.18892 X 108 Btu • Mm4/hr ft2)

and 14,388 /.cm • K (25,898 ^m • °R), respectively. The distribution of the spectral emissive power

from a black body at various temperatures is shown in Fig. 43.19, where, as shown, the energy

emitted at all wavelengths increases as the temperature increases. The maximum or peak values of

the constant temperature curves illustrated in Fig. 43.20 shift to the left for shorter wavelengths as

the temperatures increase.

The fraction of the emissive power of a black body at a given temperature and in the wavelength

interval between Xl and A2 can be described by

I /pi fA2 \

^A,r-A2r = -^A e^dX - exbd\ I = F0_XlT - F0_X2T

crl \Jo

Jo I

Fig. 43.17 Correction factor for a shell-and-tube heat exchanger with two shell passes and

any multiple of four tubes passes (four, eight, etc., tube passes).

where the function F0_AT = (1/oT4) /£ exbd\ is given in Table 43.16. This function is useful for the

evaluation of total properties involving integration on the wavelength in which the spectral properties

are piecewise constant.

Wien's Displacement Law

The relationship between these peak or maximum temperatures can be described by Wien's displace-

ment law,

Fig. 43.18 Electromagnetic radiation spectrum.

Fig. 43.19 Hemispherical spectral emissive power of a black-body for various temperatures.

Amaxr= 2897.8 jim-K

Amaxr= 5216.0 Mm-0R

43.4.2 Radiation Properties

While, to some degree, all surfaces follow the general trends described by the Stefan-Boltzmann and

Planck laws, the behavior of real surfaces deviates somewhat from these. In fact, because black

bodies are ideal, all real surfaces emit and absorb less radiant energy than a black body. The amount

of energy a body emits can be described in terms of the emissivity and is, in general, a function of

the type of material, the temperature, and the surface conditions, such as roughness, oxide layer

thickness, and chemical contamination. The emissivity is in fact a measure of how well a real body

radiates energy as compared with a black body of the same temperature. The radiant energy emitted

into the entire hemispherical space above a real surface element, including all wavelengths, is given

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