Mechanical Engineer´s Handbook P61 docx

The Stefan-Boltzmann Law The Stefan-Boltzmann law describes the rate at which energy is radiated from a black body andstates that this radiation is proportional to the fourth power of th

Nu6 = 0.42Ra<P Pr°-012(S/#)0-3for 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 fluidspecific heat For this type, the heat transfer is given by

q = U A &Tmwhere

dif-A7\ = ThJ - rc, AT2 - Thf0 - Tc,0 for parallel flowAT; = Thti - Tc^0 A72 = Th^0 - Tci for counterflowCross-Flow Coefficient

In other types of heat exchangers, where the values of the overall heat transfer coefficient, [/, mayvary 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 heattransfer, 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 variousmultiple-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 108m/sec Energy transmitted in this fashion is referred to as radiant energy and the heat transfer processthat occurs is called radiation heat transfer or simply radiation In this mode of heat transfer, theenergy is transferred through electromagnetic waves or through photons, with the energy of a photonbeing 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 energygiven or for the radiation that comes from a surface depends on the nature of the source and variouswavelengths sensed in different ways For example, as shown in Fig 43.18 the electromagneticspectrum 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 asthermal radiation and is sensed as heat For radiant energy in this range, the amount of energy givenoff 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 andall of this energy is either absorbed, reflected, or transmitted An ideal body that absorbs all theradiant energy falling upon it, regardless of the wavelength and direction, is referred to as a blackbody Such a body emits the maximum energy for a prescribed temperature and wavelength Radiationfrom 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 andstates that this radiation is proportional to the fourth power of the absolute temperature of the body

eb = crT4where eb is the total emissive power and a is the Stefan-Boltzmann constant, which has the value5.729 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 spectralemissive power e8b and is a function of wavelength This function, which was derived from quantumtheory 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 powerfrom a black body at various temperatures is shown in Fig 43.19, where, as shown, the energyemitted at all wavelengths increases as the temperature increases The maximum or peak values ofthe constant temperature curves illustrated in Fig 43.20 shift to the left for shorter wavelengths asthe temperatures increase

The fraction of the emissive power of a black body at a given temperature and in the wavelengthinterval between Xl and A2 can be described by

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

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 theevaluation of total properties involving integration on the wavelength in which the spectral propertiesare piecewise constant

Wien's Displacement Law

The relationship between these peak or maximum temperatures can be described by Wien's ment law,

displace-Fig 43.18 Electromagnetic radiation spectrum

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

Amaxr= 2897.8 jim-Kor

Amaxr= 5216.0 Mm-0R43.4.2 Radiation Properties

While, to some degree, all surfaces follow the general trends described by the Stefan-Boltzmann andPlanck laws, the behavior of real surfaces deviates somewhat from these In fact, because blackbodies 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 ofthe type of material, the temperature, and the surface conditions, such as roughness, oxide layerthickness, and chemical contamination The emissivity is in fact a measure of how well a real bodyradiates energy as compared with a black body of the same temperature The radiant energy emittedinto the entire hemispherical space above a real surface element, including all wavelengths, is given

Fig 43.20 Configuration factor for radiation exchange between surfaces of area dA, and dAj.

by e — eoT4, where e is less than 1.0, and is called the hemispherical emissivity (or total spherical emissivity to indicate integration over the total wavelength spectrum) For a given wave-length, the spectral hemispherical emissivity eA of a real surface is defined as

hemi-£x = ^lexbwhere ex is the hemispherical emissive power of the real surface and exb is that of a black body atthe same temperature

Spectral irradiation GA (W/m2 • /x,m) is defined as the rate at which radiation is incident upon asurface per unit area of the surface, per unit wavelength about the wavelength A, and encompassesthe incident radiation from all directions

Spectral hemispherical reflectivity pl is defined as the radiant energy reflected per unit time, perunit area of the surface, per unit wavelength/GA

Spectral hemispherical absorptivity ctK, is defined as the radiant energy absorbed per unit area ofthe surface, per unit wavelength about the wavelength/GA

Spectral hemispherical transmissivity is defined as the radiant energy transmitted per unit area ofthe surface, per unit wavelength about the wavelength/GA

For any surface, the sum of the reflectivity, absorptivity and transmissivity must equal unity, thatis,

«A + PATA = 1When these values are integrated over the entire wavelength from A = 0 to <*> they are referred

to as total values Hence, the total hemispherical reflectivity, total hemispherical absorptivity, andtotal hemispherical transmissivity can be written as

p = I pAGAJA/GJo

a = «AGAdA/GJo

and

r = I r,G,dX/GJo

of the incident radiation

The terms reflectance, absorptance, and transmittance are used by some authors for the realsurfaces and the terms reflectivity, absorptivity, and transmissivity are reserved for the properties ofthe ideal surfaces (i.e., those optically smooth and pure substances perfectly uncontaminated) Sur-

51 f s 33 1 3 2 a o 3 f

AT

//,m • K urn • °R FO-AT

XTjurn-K /xm^R

340035003600370038003900400041004200430044004500460047004800490050005100520053005400550056005700580059006000610062006300

612063006480666068407020720073807560774079208100828084608640882090009180936095409720990010,08010,26010,44010,62010,80010,98011,16011,340

0.36170.38290.40360.42380.44340.46240.48090.49870.51600.53270.54880.56430.57930.59370.60750.62090.63370.64610.65790.66940.68030.69090.70100.71080.72010.72910.73780.74610.75410.7618

64006500660068007000720074007600780080008200840086008800900010,00011,00012,00013,00014,00015,00020,00025,00030,00035,00040,00045,00050,00055,00060,000

11,52011,70011,88012,24012,60012,96013,32013,68014,04014,40014,76015,12015,48015,84016,20018,00019,80021,60023,40025,20027,00036,00045,00054,00063,00072,00081,00090,00099,000108,000

0.76920.77630.78320.79610.80810.81920.82950.83910.84800.85620.86400.87120.87790.88410.89000.91420.93180.94510.95510.96280.96890.98560.99220.99530.99700.99790.99850.99890.99920.9994

faces that allow no radiation to pass through are referred to as opaque, that is, TA = 0, and all of theincident energy will be either reflected or absorbed For such a surface,

«A + Px = land

a + p = 1Light rays reflected from a surface can be reflected in such a manner that the incident and reflectedrays are symmetric with respect to the surface normal at the point of incidence This type of radiation

is referred to as specular The radiation is referred to as diffuse if the intensity of the reflected radiation

is uniform over all angles of reflection and is independent of the incident direction, and the surface

is called a diffuse surface if the radiation properties are independent of the direction If they areindependent of the wavelength, the surface is called a gray surface, and a diffuse-gray surface absorbs

a fixed fraction of incident radiation from any direction and at any wavelength, and «A =

sx = a = s

Kirchhoff's Law of Radiation

The directional characteristics can be specified by the addition of a ' to the value For example thespectral emissivity for radiation in a particular direction would be denoted by «A For radiation in aparticular direction, the spectral emissivity is equal to the directional spectral absorptivity for thesurface irradiated by a black body at the same temperature The most general form of this expressionstates that a[ = s'x- If the incident radiation is independent of angle or if the surface is diffuse, then

ax = SA for the hemispherical properties This relationship can have various conditions imposed,depending on whether the spectral, total, directional, or hemispherical quantities are beingconsidered.19

Emissivity of Metallic Surfaces

The properties of pure smooth metallic surfaces are often characterized by low emissivity and sorptivity values and high values of reflectivity The spectral emissivity of metals tends to increasewith decreasing wavelength and exhibits a peak near the visible region At wavelengths A > ~5 ^m,the spectral emissivity increases with increasing temperature; however, this trend reverses at shorterwavelengths (A < —1.27 /^m) Surface roughness has a pronounced effect on both the hemisphericalemissivity and absorptivity, and large optical roughnesses, defined as the mean square roughness ofthe surface divided by the wavelength, will increase the hemispherical emissivity For cases wherethe optical roughness is small, the directional properties will approach the values obtained for smoothsurfaces The presence of impurities, such as oxides or other nonmetallic contaminants, will changethe properties significantly and increase the emissivity of an otherwise pure metallic body A summary

ab-of the normal total emissivities for metals is given in Table 43.17 It should be noted that thehemispherical emissivity for metals is typically 10-30% higher than the values typically encounteredfor normal emissivity

Emissivity of Nonmetallic Materials

Large values of total hemispherical emissivity and absorptivity are typical for nonmetallic surfaces

at moderate temperatures and, as shown in Table 43.18, which lists the normal total emissivity ofsome nonmetals, the temperature dependence is small

Absorptivity for Solar Incident Radiation

The spectral distribution of solar radiation can be approximated by black-body radiation at a perature of approximately 5800 K (10,000°R) and yields an average solar irradiation at the outer limit

tem-of the atmosphere tem-of approximately 1353 W/m2 (429 Btu/ft2 -hr) This solar irradiation is called thesolar constant and is greater than the solar irradiation received at the surface of the earth, due to theradiation scattering by air molecules, water vapor, and dust, and the absorption by O3, H2O, and CO2

in the atmosphere The absorptivity of a substance depends not only on the surface properties butalso on the sources of incident radiation Since solar radiation is concentrated at a shorter wavelength,due to the high source temperature, the absorptivity for certain materials when exposed to solarradiation may be quite different from that for low-temperature radiation, where the radiation is con-centrated in the longer-wavelength range A comparison of absorptivities for a number of differentmaterials is given in Table 43.19 for both solar and low-temperature radiation

43.4.3 Configuration Factor

The magnitude of the radiant energy exchanged between any two given surfaces is a function of theemisssivity, absorptivity, and transmissivity In addition, the energy exchange is a strong function of

how one surface is viewed from the other This aspect can be defined in terms of the configurationfactor (sometimes called the radiation shape factor, view factor, angle factor, or interception factor).

As shown in Fig 43.20, the configuration factor Fz_7 is defined as that fraction of the radiation leaving

a black surface i that is intercepted by a black or gray surface j, and is based upon the relativegeometry, position, and shape of the two surfaces The configuration factor can also be expressed interms of the differential fraction of the energy or dFt_dj, which indicates the differential fraction ofenergy from a finite area Af that is intercepted by an infinitesimal area dAj Expressions for a number

of different cases are given below for several common geometries

Infinitesimal area dAt to infinitesimal area dAj

COS0 COS0,dF^-—^^

Infinitesimal area dAt to finite area Aj

Wrought iron, polished

Cast iron, rough, strongly oxidized

Mild steel, polished

Sheet with rough oxide layer

Tin, polished sheet

480-870373370-810350310-1370310310310370-870310-530700-760310-530310-530310-530310280-370310-3030310-530293920-1530530-810310-81090-420530-920295310310-810310-810295

NormalTotalEmissivity

0.038-0.060.0950.20-0.330.340.08-0.400.020.150.780.018-0.0350.05-0.070.14-0.380.280.950.06-0.080.430.09-0.120.05-0.290.04-0.060.110.59-0.860.06-0.100.01-0.030.07-0.140.27-0.310.810.050.03-0.080.02-0.050.23-0.28Table 43.17 Normal Total Emissivity of Metals9

Aluminum, highly polished

Copper, highly polished

Flat black lacquer

White paints, various types of pigments

ForSolarRadiation0.150.180.650.940.370.460.900.750.290.960.12-0.16

For TemperatureRadiation(-300 K)0.040.030.750.210.600.950.900.930.850.950.90-0.95

Low-Table 43.19 Comparison of Absorptivities of Various Surfaces to Solarand Low-Temperature Thermal Radiation9

Oil, all colors

Lacquer, flat black

Normal TotalEmissivity0.960.290.930.950.940.9660.69-0.550.92-0.960.96-0.980.950.910.920.920.83-0.900.83-0.960.820.960.75Table 43.18 Normal Total Emissivity of Nonmetals*

"Adapted from Ref 20 after J P Holman, Heat Transfer, McGraw-Hill, New York,1981

f cos 9, cosfl,F^-l—^dA,Finite area Af to finite area Aj

1 f f cos a cos 0,Ft-i = T —^ dAidAi' J A- JAt JAj 7TR2 l JAnalytical expressions of other configuration factors have been found for a wide variety of simplegeometries A number of these are presented in Figs 43.21-43.24 for surfaces that emit and reflectdiffusely

Reciprocity Relations

The configuration factors can be combined and manipulated using algebraic rules referred to asconfiguration factor geometry These expressions take several forms, one of which is the reciprocalproperties between different configuration factors that allow one configuration factor to be determinedfrom knowledge of the others:

dA.dF^ = d^d¥dj_dldA.dF^ - AjdFj_aAfFf.j = AjFj_iThese relationships can be combined with other basic rules to allow the determination of the config-uration of an infinite number of complex shapes and geometries form a few select, known geometries.These are summarized in the following sections

The Additive Property

For a surface Ai subdivided into N parts (A, , At, , , At ) and a surface Aj subdivided into M parts(AA,AA, ,AJ,

N MV^SEAJWRelation in an Enclosure

When a surface is completely enclosed, the surface can be subdivided into N parts having areas At,A2, , AN, respectively, and

NEF,_,= 17=1Black-Body Radiation Exchange

For black surfaces Ai and A7 at temperatures Tt and Tj, respectively, the net radiative exchange qtj can

be expressed as

<7ff = Afr-fW - 77)and for a surface completely enclosed and subdivided into N surfaces maintained at temperatures

Tl, T2, , TN, the net radiative heat transfer qf to surface area Ai is

9, = E AF,-Xn - 7?> = 2 qti7=1 7=143.4.4 Radiative Exchange among Diffuse-Gray Surfaces in an Enclosure

One method for solving for the radiation exchange between a number of surfaces or bodies is throughthe use of the radiosity /, defined as the total radiation that leaves a surface per unit time and perunit area For an opaque surface, this term is defined as

/ = eo-r4 + (i - c)GFor an enclosure consisting of N surfaces, the irradiation on a given surface / can be expressedas

Area dAi of differential widthand any length, to infinitely longstrip dA 2 of differential width andwith parallel generating line to dA \:COS <f

dFdi-d2 = — — d v = V4cf(sin<p)

Two infinitely long plates ofunequal widths h and w, havingone common edge, and at anangle of 90° to each other:

* = *wFi-a = V*(l + H - VI +H2)Two infinitely long, directlyopposed parallel plates of thesame finite width:

H = *WFi-2 = F2-i = VI +H2 ~ HInfinitely long enclosure formed

by three plane areas:

r2F-2-2 == 1 T~

r2Concentric spheres:

F!-2=l'-fe)1'"-fe)'Differential or finite areas on theinside of a spherical cavity:

Fig 43.22 Configuration factor for coaxial parallel circular disks.

Fig 43.23 Configuration factor for aligned parallel rectangles

Fig 43.24 Configuration factor for rectangles with common edge.

G, = S 7,F,,,7=1and the net radiative heat-transfer rate at given surface i is

qt = AM - G.) = T^-L (oT* - JJ

®/

For every surface in the enclosure, a uniform temperature or a constant heat transfer rate can bespecified If the surface temperature is given, the heat transfer rate can be determined for that surfaceand vice versa Shown below are several specific cases that are commonly encountered

Case I: Temperatures Tt (i = 1,2, , N) are known for each of the N surfaces and the values

of the radiosity Jt are solved from the expression

S {fy - (1 - *,)*•,_,}/, = e^Tf I s i s N7=1

The net heat-transfer rate to surface i can then be determined from the fundamental relationship

4t = A T-1— (oTt - 7Z) 1 < / < N

1 — Sjwhere 8tj = 0 for i ± j and 8ij = 1 for / = j

Case II: The heat transfer rates qt(i = 1, 2, , N) are known for each of the N surfaces andthe values of the radiosity Ji are determined from

2 {fy-F^J./^fc/A, l < / < t f7=1

The surface temperature can then be determined from

fi/i-e^ M1/4T;= \-( ~ + 4 1 < / < 7 VLO-\ ez Af /J

Case III: The temperatures Tt (i = 1, , NJ for Nf surfaces and heat-transfer rates q{ (i = A^+ 1, , N) for (N-Nf) surfaces are known and the radiosities are determined by

E {60 - (1 - ejFt-JJj = e^Ti I < i < N,7=1

2 {fy - F,-,}J, = 7 tf, + 1 ss i s AT7=1 A'

The net heat-transfer rates and temperatures can be found as

9, = A, 7T- (CT7? -4) i s«'s ^i1 e,fi/1-^ M1/4

T; = - T + Ji)\ Nt + l < i < Nl<r\ e{ A{ J]

Two Diffuse-Gray Surfaces Forming an Enclosure

The net radiative exchange ql2 for two diffuse-gray surfaces forming an enclosure is shown in Table43.20 for several simple geometries

Radiation Shields

Often, in practice, it is desirable to reduce the radiation heat transfer between two surfaces This can

be accomplished by placing a highly reflective surface between the two surfaces For this ration, the ratio of the net radiative exchange with the shield to that without the shield can beexpressed by the relationship

configu-#12 with shield _ 1

#12 without shield * ~*~ XValues for this ratio x f°r shields between parallel plates, concentric cylinders, and concentric spheres

Table 43.20 Net Radiative Exchange between Two Surfaces Forming an EnclosureLarge (Infinite) Parallel Planes

A l = A2 = A _ Ao{T\ - T$

?'2= 1 + 1_!

el B2Long (Infinite) Concentric Cylinders

AI r\ oAiVj ~ 7*)A2 r2 q» 1 ( 1 - «2 /rA

Cl £2 V2/

Concentric Sphere

Ai = r? oA^T* - nA2 r\ qi2 1 t 1 - «2 /rA2

*1 £2 \r2/Small Convex Object in a Large Cavity

A, ^12 = oAlCl(rt-rj)A2~*