Heat Transfer Handbook part 63 docx

8.68 may be split into two sets ofN equations each, one set for each spectral range, and with different radiative properties for each set.. In a typical combustion process this interacti

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Figure 8.21 Arrangement of parallel or concentric radiation shields

where

j−1 A j−1 +

 1

j − 1

 1

The analysis of radiation shields is one of the few applications where analysis of spec-ularly reflecting surfaces is relatively simple and may lead to substantially different answers for concentric shields with strongly varying radii For a specularly reflecting shieldA j (withA j−1 being specular or diffuse), the radiative resistance becomes

R j−1,j =

 1

j−1 + 1

j − 1

 1

A j−1 (A j specular) (8.73) Note that it is desirable to make shields highly reflective (low), and this tends

to make them specularly reflecting (also desirable, because it also increases the resistance)

Further simplifications arise if all shields are of identical material (2 = 3 =

· · · = N−1); on the other hand, eqs (8.71) through (8.73) remain valid for shields

with different emittances on both of its sides (different values forj inR j−1,j and

R j,j+1).

While the network analogy can (and has been) applied to configurations with more than two surfaces seeing each other, this leads to very complicated circuits (because there is only one resistance between any two surfaces) For such problems the network analogy is not recommended, and the net radiation method, eq (8.68), should be employed

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8.3.5 Radiative Exchange between Diffuse Nongray Surfaces

In a number of important engineering problems the assumption of gray surface prop-erties may not provide adequate accuracy (when propprop-erties exhibit strong spectral variations across the important range of the spectrum) To deal with such effects, two

simple models, known as the semigray approximation and the band approximation,

will be described

Semigray Approximation Method This method employs the principle of su-perposition: The radiative flux at any given point is the sum of the contributions from the various emitters in the enclosure, each one acting independently In some applica-tions there is a natural division of the radiative energy within an enclosure into two or more distinct spectral regions For example, in a solar collector the incoming energy comes from a high-temperature source with most of its energy below 3µm, whereas radiation losses for typical collector temperatures are at wavelengths above 3µm

In the case of laser heating and processing, the incoming energy is monochromatic (at the laser wavelength); reradiation takes place over the entire near- to midinfrared (depending on the workpiece temperature) In such a situation, eq (8.68) may be split into two sets ofN equations each, one set for each spectral range, and with

different radiative properties for each set For example, consider an enclosure subject

to external irradiation, which is confined to a certain spectral range (1) The surfaces

in the enclosure, owing to their temperature, emit over spectral range (2).*Then from

eq (8.68),

1

(1) i

Q (1) i

A j −

N

j=1

1

j (1) − 1 F i−j

Q (1) i

A j + H oi= 0 (8.74a)

1

(2) i

Q (2) i

A j −

N

j=1

1

j (2)− 1 F i−j

Q j (2)

A j = E bi−

N

j=1

F i−j E bj (8.74b)

Q i

A i = Q (1) i

A i +Q (2) i

A i i = 1, 2, , N (8.74c) wherej (1) is the average emittance for surfacej over spectral interval (1), and so

on The semigray approximation is not limited to two distinct spectral regions Each surface of the enclosure may be given a set of absorptances and reflectances, one value for each different emission temperature (with its different emission spectra) How-ever, while simple and straightforward, the method can never become exact no matter how many different values of absorptance and reflectance are chosen for each surface

Band Approximation Method Anothercommonly used method to deal with

nongray surfaces is the band approximation method This method employs the fact

* Note that spectral ranges (1) and (2) do not need to cover the entire spectrum, and indeed, they may overlap.

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that even for nongray materials, eq (8.68) remains valid on a spectral basis (replacing emissive powerE bby spectral emissive powerE bλ, related to the total emissive power

byE b=0∞ E bλ dλ) In this method the spectrum is broken up into M bands, over which the radiative properties of all surfaces in the enclosure are constant Therefore,

1

(m) i

Q (m) i

A i −N

j=1

1

(m) i − 1 F i−j

Q j (m)

A j + H oi (m) = E (m) bi −N

j=1

F i−j E bj (m)

i = 1, 2, , N, m = 1, 2, , M (8.75a)

E bj =

M

m=1

E bj (m) Q A j

M

m=1

Q j (m)

A j H oi=

M

m=1

H oi (m) (8.75b)

Here,E b (m)is the fractional emissive power contained in bandm and so on Equations

(8.75) are, of course, nothing but a simple numerical integration of the spectral version

of eq (8.68), using the trapezoidal rule with varying steps This method has the advantage that the widths of the bands can be tailored to the spectral variation of properties, resulting in good accuracy with relatively few bands For very few bands the accuracy of this method is similar to that of the semigray approximation but is a

little more cumbersome to apply On the other hand, the band approximation method

can achieve any desired accuracy by using many bands

8.4 RADIATIVE PROPERTIESOF PARTICIPATING MEDIA

In many high-temperature applications, when radiative heat transfer is important, the medium between surfaces is not transparent but is “participating”; that is, it absorbs, emits, and (possibly) scatters radiation In a typical combustion process this interaction results in (1) continuum radiation due to tiny, burning soot particles (of dimension < 1 µm) and also due to larger suspended particles, such as coal

particles, oil droplets, and fly ash; (2) banded radiation in the infrared due to emission and absorption by moleculargaseous combustion products, mostly watervaporand carbon dioxide; and (3) chemiluminescence due to the combustion reaction itself

While chemiluminescence may normally be neglected, particulates as well as gas radiation generally must be accounted for

8.4.1 Molecular Gases

When a photon (or an electromagnetic wave) interacts with a gas molecule, it may

be absorbed, raising the energy level of the molecule Conversely, a gas molecule may spontaneously lower its energy level by the emission of an appropriate photon

This leads to large numbers of narrow spectral lines, which partially overlap and together form vibration–rotation bands As such, gases tend to be transparent over most of the spectrum but may be almost opaque over the spectral range of a band

The absorption coefficient κλ is defined as a measure of how strongly radiation is

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␩ (cm )⫺1

0 100 200 300 400

␬␳␩

T= 300 K, = 1 bar,p pCO2= 0 bar

Figure 8.22 Absorption coefficient spectrum for the CO24.3-µm band

absorbed or emitted along a path in a participating medium Figure 8.22 shows the absorption coefficient of the important 4.3-µm vibration–rotation band of CO2(per partial pressure of CO2), forsmall amounts of CO2 contained in nitrogen, with a temperature of 300 K and a mixture pressure of 1 bar, generated from the HITRAN database (Rothman et al., 1998) The figure shows that the band consists of a large number of strong spectral lines, and a number of weak lines can also be observed In reality, there are many more spectral lines than appear in the figure However, at the relatively high total pressure of 1 bar, the lines strongly overlap, giving a relatively smooth appearance Lowering the pressure would decrease line overlap, and more and more of the≈ 12,500 lines contained in the HITRAN database forthis band would become distinguishable Similarly, with increasing temperature, lines become narrower (less overlap), and many additional “hot lines” must be considered, which are negligible at room temperature The new HITEMP database (Rothman et al., 2000), which is designed for temperatures up to 1000 K, includes≈ 185,000 lines forthe 4.3-µm CO2band alone!

Fortunately, for many engineering problems, for simple heat transfer calculations,

it is sufficient to determine the total emissivity for an isothermal, homogeneous path

of lengthL,

 = E1

b

∞ 0

(1 − e−κ λL )E bλ (T g ) dλ (8.76) Fora mixture of gases the total emissivity is a function of path lengthL, gas

tem-perature T g, partial pressure(s) of the absorbing gas(es) p a, and total pressure p.

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Especially important in combustion application, the total emissivity in mixtures of nitrogen with watervaporand/orcarbon dioxide may be calculated from Leckner (1972) First, the individual emissivities forwatervaporand carbon dioxide, respec-tively, are calculated separately from

(p a L,p,T g ) = 0(p a L,T g )



0(p a L,p,T g ) =



1−(a − 1)(1 − P a + b − 1 + P E )

e exp



−c

 log10(p a L) m

p a L

2

(8.77b)

0(p a L,T g ) = exp



M

i=0

N

j=0

c ji

T

g

T0

j log10 p a L (p a L)0

i

Here0is the total emissivity at a reference state, which isp = 1 bar total pressure and

p a → 0 (but p a L > 0) The correlation constants a, b, c, c ji , P E , (p a L)0, (p a L) m, andT0are given in Table 8.4 forwatervaporand carbon dioxide (forconvenience, plots of0are given in Fig 8.23 for CO2and Fig 8.24 forH2O) The total emissivity

of a mixture of nitrogen with both water vapor and carbon dioxide is calculated from

CO2 +H 2 O= CO2+ H2 O− ∆ (8.78)

∆ =

 ζ

10.7 + 101ζ − 0.0089ζ10.4

  log10(pH 2 O+ pCO2)L

(p a L)0

2.76 (8.79)

TABLE 8.4 Correlation Constants for the Determination of the Total Emissivity for Water Vapor and Carbon Dioxide

. −0.85667 −0.93048 −0.14391 −1.2710 −1.1090 −1.0195 −0.21897

0.225t2, t > 0.7

1.888 − 2.053 log10t, t > 0.75

Source: Leckner(1972).

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Figure 8.23 Total emissivity of carbon dioxide at a total pressure of 1 bar and zero partial pressure (From Leckner, 1972.)

Figure 8.24 Total emissivity of water vapor at a total pressure of 1 bar and zero partial pressure (From Leckner, 1972.)

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where

ζ = pH 2 O

pH 2 O+ pCO2 and where the∆ compensates foroverlap effects between H2O and CO2bands, and theCO2andH2 Oare calculated from eq (8.77)

If radiation emitted externally to the gas (e.g., by emission from an adjacent wall

at temperatureT w) travels through the gas, the total amount absorbed by the gas is of interest This leads to the absorptivity of a gas path atT g with a source atT w:

α(p a L, p, T g , T w ) = E 1

b (T w )

∞ 0 [1− e−κ λ(T ) g L]E bλ (T w ) dλ (8.80) which forwatervapororcarbon dioxide may be estimated from

α(p a L, p, T g , T w ) =



T g

T w

1/2





p a L T w

T g , p, T w



(8.81)

where is the emissivity calculated from eq (8.77) evaluated at the temperature of the surface,T w, and using an adjusted pressure path length,p a LT w /T g Formixtures

of watervaporand carbon dioxide, band overlap is again accounted forby taking

αCO2 +H 2 O= αCO2+ αH2 O− ∆ (8.82) with∆ evaluated for a pressure path length of p a LT w /T g.

8.4.2 Particle Clouds

Nearly all flames are visible to the human eye and are therefore called luminous

(sending out light) Apparently, there is some radiative emission from within the flame

at wavelengths where there are no vibration–rotation bands for any combustion gases

This luminous emission is today known to come from tiny char (almost pure carbon) particles, called soot, which are generated during the combustion process The dirtier

the flame, the higherthe soot content and the more luminous the flame

Soot Soot particles are produced in fuel-rich flames, or fuel-rich parts of flames,

as a result of incomplete combustion of hydrocarbon fuels As shown by electron microscopy, soot particles are generally small and spherical, ranging in size between approximately 5 and 80 nm and up to about 300 nm in extreme cases Although mostly spherical in shape, soot particles may also appear in agglomerated chunks and even as long agglomerated filaments It has been determined experimentally in typical diffusion flames of hydrocarbon fuels that the volume percentage of soot generally lies in the range 10−4to 10−6%

Because soot particles are very small, they are generally at the same temperature as the flame and therefore strongly emit thermal radiation in a continuous spectrum over

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the infrared region Experiments have shown that soot emission often is considerably stronger than the emission from the combustion gases

Fora simplified heat transferanalysis it is desirable to use suitably defined mean absorption coefficients and emissivities If the soot volume fractionf vis known as well as an appropriate spectral average of the complex index of refraction of the soot,

m = n − ık(ı = √−1), one may approximate the spectral absorption coefficient

from Felske and Tien (1977) as

κλ= C0 fλv C0 =(n2− k236πnk+ 2)2+ 4n2k2 (8.83) and a total or spectral-average value may be taken as

κm=3.72f v C0T

C2

(8.84)

whereC2 = 1.4388 cm · K is the second Planck function constant Substituting eq.

(8.84) into eq (8.76) gives a total soot cloud emissivity of

(f v TL) = 1 − e−κm L = 1 − e −3.72C0f v T L/C2 (8.85)

Pulverized Coal and Fly Ash Dispersions To calculate the radiative proper-ties of arbitrary size distributions of coal and ash particles, one must have knowledge

of their complex index of refraction as a function of wavelength and temperature

Data for carbon and different types of coal indicate that its real part,n, varies little

over the infrared and is relatively insensitive to the type of coal (anthracite, lignite, bituminous), while the absorptive index,k, may vary strongly across the spectrum

and from coal to coal If the number and sizes of particles are known and if a suitable average value for the complex index of refraction can be found, the spectral absorption coefficient of the dispersion may be estimated by a correlation given by Buckius and Hwang (1980) They observed spectral behavior to be weak (similar to that of small soot particles) and that spectrally averaged properties do not depend appreciably on the optical properties of the coal However, due to their larger size, coal particles tend

to scatter radiation as well as absorb and emit radiation, leading to the definition of

the scattering coefficientσs and extinction coefficientβ = κ + σs Interpolating the

data of Buckius and Hwang, crude approximations for spectrally averaged absorption and extinction coefficients may be determined from

κm

f A =







0.0032



1+

 φ 425

1.8−6/5

+



10.99

φ0.02

−6/5



−5/6

(8.86)

βm

f A =







0.0032



1+

 φ 650

2.0−5/4

+



13.75

φ0.13

−5/4



−4/5

(8.87)

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wheref Ais the total projected area of particles per unit volume (e.g.,f A = πa2N for

uniform spheres of radiusa and a particle density of N particles/unit volume), and φ

is a size parameter defined as

φ = ¯aT ¯a = 3f v

where ¯a is an average particle size This leads to total coal cloud emissivity and

absorptivity:

α(φ) = (φ) = 1 − e−κm L (8.89)

On the other hand, if one is interested in transmitted radiation (i.e., radiation not absorbed or scattered away), the cloud transmissivity becomes

If both soot as well as larger particles are present in the dispersion, the absorption coefficients of all constituents must be added before applying eqs (8.89) and (8.90)

Mixtures of Molecular Gases and Particulates To determine the total emis-sivity of a mixture, it is generally necessary to find the spectral absorption coefficient

κλof the mixture (the sum of the absorption coefficient of all contributors), followed

by numerical integration of eqs (8.89) and (8.90) However, because molecular gases tend to absorb only over a small part of the spectrum, to some degree of accuracy

mix gas+ particulates (8.91) Equation (8.91) gives an upper estimate because overlap effects result in lower emis-sivity [compare eq (8.78) for gas mixtures]

8.5 RADIATIVE EXCHANGE WITHIN PARTICIPATING MEDIA

To calculate the radiative heat transfer rates within—and to the bounding wall of—a

participating medium, it is necessary to solve the radiative transfer equation (RTE),

dIλ

ds = ˆs · ∇Iλ= κλI bλ− βλIλ+

σsλ 4π

4 π Iλ(ˆs i )Φλ(ˆs i , ˆs)dΩ i (8.92)

to some degree of accuracy, followed by integration over all directions and all wave-lengths, to obtain the radiative heat flux desired Hereκλis the medium’s absorption

coefficient,σsλ its scattering coefficient,βλ = κλ + σsλ is known at the extinction coefficient, andΦλis the scattering phase function.

As demonstrated in Fig 8.25, this equation states that spectral radiative intensity

Iλ along a paths in the direction of ˆs is augmented by emission along the path,

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Figure 8.25 Coordinates for the formal solution to the radiative transfer equation

diminished by extinction or absorption and outscattering (scattering of radiation away

from ˆs), and augmented by in-scattering (scattering from all other directions ˆsi into

direction ˆs);κλ gives a measure of how much radiation is absorbed and/or emitted,

σsλgives a measure of how much is scattered, andΦλis the probability that radiation

is scattered from direction ˆsiinto direction ˆs.

Finding a solution to eq (8.92), which is an integrodifferential equation in five independent variables (three space coordinates and two direction coordinates), is a truly daunting task for all but the most trivial situations, even at the spectral level

Integration over all wavelengths, due to the complicated nature of radiative properties, tends to add another dimension to the level of difficulty Consequently, the literature abounds with solutions to very simplistic scenarios as well as with approximate solution methods A few of these simplified cases and methods are outlined in this section

In most engineering applications scattering can be neglected, and eq (8.92) can

be formally integrated along a straight path froms = 0 at a bounding wall to a point

s = s inside the medium, to yield

Iλ(s) = Iλ(0)e−κ λs+

s 0

I bλ (s )e−κ λ(s−s )κλds (8.93)

where it was also assumed that κλ is constant along the path If the medium is isothermal along the path, eq (8.93) can be reduced further to

Iλ(s) = Iλ(0)e−κ λs + I bλ (1 − e−κ λs ) (8.94) or

Iλ(s) = Iλ(0)τλ(s) + I bλλ(s) (8.95)