Heat Transfer Handbook part 60 pdf

It may be shown through another variation of Kirchhoff ’s law that, at least on a spectral, directional basis, This is also true for hemispherical values if either the directional emitta

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the enclosure Thus, in the case of opaque walls it is customary to speak of absorption

by and emission from a “surface,” although a thin surface layer is implied

If radiation impinging on a solid or liquid layer is considered, a fraction of the

energy will be reflected (reflectance ρ, often also referred to as reflectivity), another

fraction will be absorbed (absorptance α, often also referred to as absorptivity), and

if the layer is thin enough, a fraction may be transmitted (transmittanceτ, often also

referred to as transmissivity) Because all radiation must be either reﬂected, absorbed,

ortransmitted,

If the medium is sufﬁciently thick to be opaque, thenτ = 0 and

All surfaces also emit thermal radiation (or, rather, radiative energy is emitted

within the medium, some of which escapes from the surface) The emittance is

deﬁned as the ratio of energy emitted by a surface as compared to that of a black surface at the same temperature (the theoretical maximum)

All of these four properties may vary in magnitude between the values 0 and 1; for a black surface, which absorbs all incoming radiation and emits the maximum possible,

α = = 1 and ρ = τ = 0 They may also be functions of temperature as well

as wavelength and direction (incoming and/or outgoing) One distinguishes between spectral and total properties (an average value over the spectrum) and also between directional and hemispherical properties (an average value over all directions)

It may be shown (through another variation of Kirchhoff ’s law) that, at least on a

spectral, directional basis,

This is also true for hemispherical values if either the directional emittance or the

incoming radiation are diffuse (they do not depend on direction) It is also true for

total values if eitherthe spectral emittance does not depend on wavelength orif the spectral behavior of the incoming radiation is similar to blackbody radiation at the same temperature

Typical directional behavior is shown in Fig 8.4a (fornonmetals) and b (metals).

In these ﬁgures the total, directional emittance, a value averaged over all wavelengths,

is shown For nonmetals the directional emittance varies little over a large range of polar angles but decreases rapidly at grazing angles until a value of zero is reached at

θ = π/2 Similartrends hold formetals, except that at grazing angles, the emittance

ﬁrst increases sharply before dropping back to zero (not shown) Note that emittance levels are considerably higher for nonmetals

A surface whose emittance is the same for all directions is called a diffuse emitter, ora Lambert surface No real surface can be a diffuse emitter because

electromag-netic wave theory predicts a zero emittance at θ = π/2 forall materials However,

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a

b

Figure 8.4 Directional variation of surface emittances: (a) forseveral nonmetals; (b) for

several metals (From Schmidt and Eckert, 1935.)

little energy is emitted into grazing directions, as seen from eq (8.16), so that the assumption of diffuse emission is often a good one

Typical spectral behavior of surface emittances is shown in Fig 8.5 for a few materials, as collected by White (1984) Shown are values for directional emittances

in the direction normal to the surface However, the spectral behavior is the same

for hemispherical emittances In general, nonmetals have relatively high emittances,

which may vary erratically across the spectrum, and metals behave similarly for short wavelengths but tend to have lower emittances with more regular spectral dependence

in the infrared

Mathematically, the spectral, hemispherical emittance is deﬁned in terms of

emis-sive poweras

λ(T,λ) ≡ Eλ(T,λ)

E bλ (T,λ) (8.27)

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Copper Carbon

Gold

Aluminum

Tungsten

Nickel

Aluminum oxide Magnesium oxide

White enamel

Wavelength ( m)␭ ␮

Aluminum

Silicon carbide

,nλ

Figure 8.5 Normal, spectral emittances for selected materials (From White, 1984.)

This property may be extracted from the spectral, directional emittanceλby inte-grating over all directions,

λ(T,λ) = π1

λ(T,λ,θ,ψ) cos θ dΩ

= π1

2π 0

π/2 0

λ(T,λ,θ,ψ) cos θ sin θ dθ dψ (8.28)

and ﬁnally, the total, hemispherical emittance may be related to the spectral

hemi-spherical emittance through

(T ) = E E(T )

b (T ) =

∞

0 Eλ(T,λ) dλ

E b (T ) =

1

n2σT4

∞ 0

λ(T,λ)E bλ (T,λ) dλ (8.29)

Here a prime and subscriptλ have been added temporarily to distinguish directional

from hemispherical properties, and spectral from total (spectrally averaged) values

If the spectral emittance is the same for all wavelengths, eq (8.29) reduces to

Such surfaces are termed gray, and forthe very special case of a gray, diffuse surface,

this implies that

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Although no real surface is truly gray, it often happens thatλis relatively constant over that part of the spectrum where E bλ is substantial, making the simplifying assumption of a gray surface warranted

Wavelength Dependence Electromagnetic theory states that the radiative prop-erties of interfaces are strong functions of the material’s electrical conductivity Met-als are generally excellent electrical conductors because of an abundance of free elec-trons For materials with large electrical conductivity, both the real and imaginary parts of the complex index of refraction,m = n − ık(ı =√−1), become large and

approximately equal for long wavelengths, sayλ > 1 µm, leading to an approximate

relation for the normal, spectral emittance of the metal, known as the Hagen–Rubens

relation (Modest, 2003),

nλ 2n √ 2

0.003λσdc

= 1 − ρnλ λ in µm, σdcinΩ−1· cm−1 (8.32)

whereσdc is the dc conductivity of the material Equation (8.32) indicates that for clean, polished metallic surfaces the normal emittance can be expected to be small, and the reﬂectance large (using typical values for conductivity,σdc), with a 1/√λ

wavelength dependence Comparison with experiment has shown that for sufﬁciently long wavelengths, the Hagen–Rubens relationship describes the radiative properties

of polished (not entirely smooth) metals rather well, in contrast to the older, more

sophisticated Drude theory (Modest, 2003) However, for optically smooth metallic

surfaces (such as vapor-deposited layers on glass), radiative properties closely obey electromagnetic wave theory, and it is the Drude theory that gives excellent results

Directional Dependence The spectral, directional reﬂectance for an optically

smooth interface is given by Fresnel’s relations (Modest, 2003) As noted before,

in the infrared,n and k are generally fairly large for metals, and Fresnel’s relations

simplify to

ρ= (n cos θ − 1)2+ (k cos θ)2

(n cos θ + 1)2+ (k cos θ)2 (8.33a)

ρ⊥= (n − cos θ)2+ k2

(n + cos θ)2+ k2 (8.33b) Hereρ is the spectral reﬂectance for parallel-polarized radiation, which refers to electromagnetic waves whose oscillations take place in a plane formed by the surface normal and the direction of incidence Similarly,ρ⊥ is the spectral reﬂectance for perpendicular-polarized radiation, which refers to waves oscillating in a plane normal

to the direction of incidence In all engineering applications (except lasers), radiation consists of many randomly oriented waves (randomly polarized or unpolarized), and the spectral, directional emittance and reﬂectance can be evaluated from

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Figure 8.6 Spectral, directional reﬂectance of platinum atλ = 2µm

λ= 1 − ρλ= 1 −1

2

The directional behaviorforthe reﬂectance of polished platinum atλ = 2 µm

is shown in Fig 8.6 and is also compared with experiment (Brandenberg, 1963;

Brandenberg and Clausen, 1965; Price, 1947) As already seen from Fig 8.4, re-ﬂectance is large for near-normal incidence, and the unpolarized rere-ﬂectance remains fairly constant with increasingθ However, near grazing angles of θ 80–85°, the

parallel-polarized component undergoes a sharp dip before going toρ = ρ⊥ = 1

atθ = 90° This behavioris responsible forthe lobe of strong emittance

neargraz-ing angles commonly observed for metals Fortunately, these near-grazneargraz-ing angles are fairly unimportant in the evaluation of radiative ﬂuxes, due to the cosθ in eq (8.21);

that is, even metals can usually be treated as “diffuse emitters” with good accuracy It needs to be emphasized that the foregoing discussion is valid only for relatively long wavelengths (infrared) For shorter wavelengths, particularly the visible, the assump-tion of large values forn and k generally breaks down, and the directional behavior

of metals resembles that of nonconductors (discussed in the next section)

Hemispherical Properties Equation (8.33) may be integrated analytically over all directions to obtain the spectral, hemispherical emittance Figure 8.7 is a plot of the ratio of the hemispherical and normal emittances,λ/ nλ Forthe case ofk/n = 1 the

dashed line represents results from integrating equation (8.33), while the solid lines were obtained by numerically integrating the exact form of Fresnel’s relations For most metalsk > n > 3, so that, as shown in Fig 8.7, the hemispherical emittance is

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Refractive index, n

k n/ = 4 2

1

⑀␭

⑀n␭

Figure 8.7 Ratio of hemispherical and normal spectral emittance for electrical conductors as

a function ofn and k (From Dunkle, 1965.)

larger than the normal value, due to the strong emission lobe at near grazing angles

Again, this statement holds only forrelatively long wavelengths

Total Properties Equation (8.32) may be integrated over the spectrum, using eq

(8.29), and applying the correction given in Fig 8.7 to convert normal emittance

to hemispherical emittance This leads to an approximate expression for the total,

hemispherical emittance of a metal,

(T ) = 0.766

T

σdc

1/2

−

0.309 − 0.0889 ln T

σdc

T

whereT is in K and σdcis inΩ−1· cm−1. Because eq (8.35) is based on the Hagen–Rubens relation, this expression is valid only for relatively low temperatures (where most of the blackbody emissive power lies in the long wavelengths; see Fig 8.1) Figure 8.8 shows that eq (8.35) does an excellent job predicting the total hemispherical emittances of polished metals as com-pared with experiment (Parker and Abbott, 1965), and that emittance is essentially linearly proportional to(T /σdc)1/2.

Surface Temperature Effects The Hagen–Rubens relation, eq (8.32), predicts that the spectral, normal emittance of a metal should be proportional to 1/√σdc Because the electrical conductivity of metals is approximately inversely proportional

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Theoretical (Hagen-Rubens) Tungsten

Tantalum Niobium Molybdenum Platinum

Gold Silver Copper Zinc Tin Lead

公 ␴T/ dc ( cm K)⍀ 1/2

Figure 8.8 Total, hemispherical emittance of various polished metals as a function of tem-perature (From Parker and Abbott, 1965.)

to temperature, the spectral emittance should therefore be proportional to the square root of absolute temperature for long enough wavelengths This trend should also hold for the spectral, hemispherical emittance Experiments have shown that this is indeed true for many metals A typical example is given in Fig 8.9, which shows the spectral dependence of the hemispherical emittance fortungsten fora numberof temperatures

Note that the emittance for tungsten tends to increase with temperature beyond a

crossover wavelength of approximately 1.3µm, while the temperature dependence

is reversed for shorter wavelengths Similar trends of a single crossover wavelength have been observed for many metals Because the crossover wavelength is fairly short for many metals, the Hagen–Ruben temperature relation often holds for surprisingly high temperatures

Electrical nonconductors have few free electrons and thus do not display high re-ﬂectance/opacity behavior across the infrared as do metals

Wavelength Dependence Reﬂection of light by insulators and semiconductors tends to be a strong, sometimes erratic function of wavelength Crystalline solids generally have strong absorption–reﬂection bands (largek) in the infrared commonly

known as Reststrahlen bands, which are due to transitions of intermolecular

vibra-tions These materials also have strong bands at short wavelengths (visible to ultravi-olet), due to electronic energy transitions In between these two spectral regions there

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Tungsten

Wavelength

T = 1600 K

T = 2000 K

T = 2400 K

T = 2800 K

⑀␭

␭ ␮ ( m)

Figure 8.9 Temperature dependence of the spectral, hemispherical emittance of tungsten

(From Weast, 1988.)

generally is a region of fairly high transparency (and low reflectance), where absorp-tion is dominated by impurities and imperfecabsorp-tions in the crystal lattice As such, these spectral regions often show irregular and erratic behavior Defects and impurities may vary appreciably from specimen to specimen and even between different points on the same sample As an example, the spectral, normal reflectance of silicon at room temperature is shown in Fig 8.10 The strong influence of different types and levels of impurities is clearly evident Therefore, looking up properties for a given material in published tables is problematical unless a detailed description of surface and material preparation is given

In spectral regions outside Reststrahlen and electronic transition bands the absorp-tive index of a nonconductoris very small; typically,k < 10−6fora pure substance.

While impurities and lattice defects can increase the value ofk, one is very unlikely to

find values ofk > 10−2fora nonconductoroutside the Reststrahlen bands This im-plies that Fresnel’s relations can be simplified significantly, and the spectral, normal reflectance may be evaluated as

ρnλ=

n − 1

n + 1

2

(8.36)

Therefore, for optically smooth nonconductors the radiative properties may be cal-culated from refractive index data Refractive indices for a number of semitransparent materials at room temperature are displayed in Fig 8.11 as a function of wavelength

All of these crystalline materials show similar spectral behavior: the refractive index drops rapidly in the visible region, then is nearly constant (declining very gradually) until the midinfrared, wheren again starts to drop rapidly This behavior is explained

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Figure 8.10 Spectral, normal reﬂectance of silicon at room temperature (Redrawn from the data of Touloukian and DeWitt, 1972.)

by the fact that crystalline solids tend to have an absorption band due to electronic transitions near the visible and a Reststrahlen band in the infrared: The ﬁrst drop in

n is due to the tail end of the electronic band; the second drop in the midinfrared is

due to the beginning of a Reststrahlen band

Directional Dependence Foroptically smooth nonconductors, forthe spectral region between absorption–reflection bands, experiment has been found to closely follow Fresnel’s equations of electromagnetic wave theory Figure 8.12 shows a comparison between theory and experiment for the directional reflectance of glass (blackened on one side to avoid multiple reflections) for polarized, monochromatic irradiation Becausek2 n2, the absorptive index may be eliminated from Fresnel’s relations, and the relations for a perfect dielectric become valid For unpolarized light incident from vacuum (or a gas), this leads to

λ= 1 − 1

2

ρ+ ρ⊥= 1 −1

2





n2cosθ −n2− sin2θ

n2cosθ +n2− sin2θ

2

+

cosθ −n2− sin2θ

cosθ +n2− sin2θ

2

Comparison with experiment agrees well with elecromagnetic wave theory for a large numberof nonconductors

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Figure 8.11 Refractive indices for various semitransparent materials (From American Insti-tute of Physics, 1972.)

Temperature Dependence The temperature dependence of the radiative prop-erties of nonconductors is considerably more difﬁcult to quantify than for metals

Infrared absorption bands in ionic solids due to excitation of lattice vibrations (Rest-strahlen bands) generally increase in width and decrease in strength with tempera-ture, and the wavelength of peak reﬂection–absorption shifts toward higher values

The reﬂectance for shorter wavelengths depends largely on the material’s impurities

Often, the behavioris similarto that of metals, that is, the emittance increases with temperature for the near infrared while it decreases with shorter wavelengths On the

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