36

CHAPTER 3

SOLUTION METHODS AND APPLICATIONS

OF

RADIATIVE TRANSFER EQUATION

3.1. BACKGROUND

The basic equations of radiative heat transfer including the radiative transfer

equation are discussed. Spectro-radiometric curves and their use in deciding the accuracy

and precision of reflectometers and imagery systems are presented. Finally a brief review of

different solution techniques of RTE is reviewed and selected Monte Carlo (MC) method as

the method of solution for the modeling of incident radiative heat flux from the exhaust

plume of a solid rocket.

3.2. SPECTRAL EMISSIVE POWER

Radiation is emitted by bodies by virtue of their temperature. Its importance in

thermal calculations is limited to the wavelength band 0.1 to 100 µm. The total quantity of

radiation emitted by a body per unit area and time is called the total emissive power and the

amount of radiation emitted at wave length λ is referred to monochromatic emissive power

denoted by Eλ. The monochromatic emissive power of a black body, denoted as Ebλ is

given by the Planck’s law as a function of wave length λ and absolute temperature T as

follows.

)1(

2

2

5

1

−

=T

cb

e

cE

λ

λ

λ

π (3.1)

Where C1 = 0.59552197 E-16 W. m2 / sr , C2 = 0.01438769 m.K.

37

Equation (3.1) shows that monochromatic emissive power is a function of both

wavelength and temperature. The area under the spectro-radiometric curve is the total

amount of radiation emitted over all the wavelengths and Stefan-Boltzmann law expresses a

quantitative relationship between the temperatures.

0

5000

10000

15000

5 10 15

Temperature - 1000KTemperature - 900K

Wavelength, µµµµm

He

mis

ph

eri

ca

l s

pe

ctr

al e

mis

siv

e p

ow

er(

W/(

m2. µµ µµ

m))

Figure 3.1: Hemispherical spectral emissive power up to 1000 K

0

1x105

2x105

3x105

4x105

5x105

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00

Temperature = 2000 KTemperature =1800 KTemperature =1700 KTemperature =1600 KTemperature =1500 KTemperature = 1000 K

Wavelength, µµµµmHem

isp

heri

cal s

pe

ctr

al em

issiv

e p

ow

er,

W/(

m2. µµ µµ

m)

Figure 3.2: Hemispherical spectral emissive power -1000 K to 2000 K

38

and the total emissive power of a blackbody in its most simple and popular form as

4TEb σ= (3.2)

The curves of monochromatic emissive power for different temperatures are known

as spectro-radiometric curves. Figure 3.1 shows spectral radiometric curves for two typical

temperatures 900 K and 1000 K, while Figure 3.2 shows the same for temperatures ranging

from 1000 K to 2000 K. These figures reveal the drastic variation of hemispherical spectral

emissive power as a function of temperature. The spectro-radiometric curves are widely

used in the engineering applications to find the bandwidth of wavelengths where minimum

and maximum energy is emitted. The following section gives a small description of the

engineering applications of the spectro-radiometric curves.

3.3. APPLICATIONS OF SPECTRO-RADIOMETRIC CURVES

The spectro-radiometric curves are used to calculate the content of incident energy

in a given spectral band width. This data finds extensive use in thermal and optical

applications and also in configuring sophisticated equipments such as

Reflectometer/Emissometer, selecting advanced thermal imagery systems and pyrometers.

Reflectometer works on the principle of Integrating Sphere to measure the total

reflectance spectrum of surfaces. There are separate instruments to measure spectral

reflectance in Ultra Violet (UV) to Near Infra Red (NIR) in the spectral range 250 to 2500 nm

and another in the Mid IR ranges of 2500 to 20,000 nm.

For opaque surfaces, absorptivity can be derived from spectral reflectance and

absorptivity. Absorptivity is an important parameter indicating the level of absorbed radiant

energy in the thermal simulation studies. Also absorptivity is the scaling factor of the

convective heat flux in laboratory tests for simulation of re-entry aerodynamic heating, using

radiative sources.

Transmittance of materials can also be determined using the Reflectometer. This

parameter assumes importance in thermal diagnostics when the object under investigation

is inside a chamber and the pyrometer (or thermal imagery) is probing the object through an

optical window. Primarily, the emissive power of the object has to be determined using

spectro-radiometric curves and from the transmittance bandwidth of the window material

39

(usually hardened glass or fused silica) the energy emanating through the window can be

determined.

3.4. THE UNIFIED SPECTRO-RADIOMETRIC CURVE

It is seen from figures (3.1) and (3.2) that to calculate the spectral radiant energy in

a specific band, one needs to evaluate the spectrum corresponding to the temperature of

the source. Thus it becomes cumbersome, if one needs to work with variable source

temperature. A transformation of the parameter, wavelength to wavelength-temperature

product, transforms equation (3.1) to an equation with a single variable ζ as given below.

)1(

22

5

1

5

−

=ς

λ

ς

πc

b

e

c

T

E

(3.3)

Where Tλς = , the wavelength- temperature product.

Equation (3.3) unifies all the spectro-radiometric curves into a single curve and Fig.

(3.3) depicts the same.

0

0.5x10-11

1.0x10-11

1.5x10-11

0 4000 8000 12000 16000

Ebλλλλ

/T5

λT

Em

issiv

e p

ow

er/

T5,

W

/(m

2.K

5.µ

m)

\

Figure 3.3: Spectral distribution of blackbody hemispherical power as a function of wavelength temperature product.

Equation (3.3) can be integrated numerically as a function of ζ and generate the

fractional emissive power from 0 to λT, denoted as F0-λT, to estimate conveniently the

intercepted radiant energy on a target in a wavelength band as discussed below.

40

0

0.2

0.4

0.6

0.8

1.0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Wavelength-temperature product, λλλλT (µµµµm.K)

Fra

cti

on

al

em

iss

ive

po

we

r (F

0- λλ λλ

T)

Figure 3.4 : Fractional emissive power up to 92%

0

0.2

0.4

0.6

0.8

1.0

0 2300 4600 6900 9200 11500 13800 16100 18400 20700 23000

Wavelength-temperature product λλλλT (µµµµm.K)

Fra

cti

on

al

em

iss

ive

po

we

r (F

0- λλ λλ

T)

Figure 3.5: Fractional emissive power up to 99.00%

The numerical integration of equation (3.3) in discrete steps of unit width of λT is

carried out for achieving better accuracy. Figures 3.4 and 3.5 show the fractional emissive

power up to 92 % and 99.8% respectively. These figures indicate that to capture 7.0 % of

the energy in the terminal region of the emissive power, the wavelength-temperature

product has to run from 10000 µm.K to 23000 µm.K . Modern thermal imagery systems are

fast acquiring the status as reliable non-intrusive diagnostic tools for obtaining the whole

field temperature distribution. The Infra Red (IR) thermal imagery systems are usually very

41

expensive and proper selection of the equipment is essential for obtaining measurement

accuracy in the specific application. Similarly caution has to be exercised while employing

an existing imagery system for any given application. The first check is to ensure that the

emission spectrum of the application is covered by the input spectrum of the equipment. If

not, based on the energy content of the emitted source energy (Fig.3.4 and Fig.3.5), the

system has to be calibrated for measuring the actual surface temperature. Further, the

media could be participating and hence the spectral band where the media is dormant has

to be selected for operation. Hence, the energy content in the bandwidth where the media is

dormant has to be determined for that bandwidth where the source is operating and based

on this, the calibration factor has to be applied for the thermal imagery system. Thus to

achieve a reasonably good accuracy, integration is carried out in discrete steps of unit width

of λT for obtaining fractional emissive power up to 99.90 %. A slight improvement in

accuracy of the values of λT corresponding to the emissive power beyond 75 % is noticed in

the present work, when compared to the values provided by Robert Siegel and Howell,

[2002]-page:24, as shown in Table 3.1.

Table 3.1: Comparison of values of fractional emissive power of radiant energy corresponding to wavelength temperature product

λT, (µm.K)

[Siegel & Howell ]

λT, (µm.K)

Present Study F0-λT

1448 1448 0.01

2898 2897 0.25

4107 4107 0.50

6148 6147 0.75

22890 22811 0.99

Thus, for an efficient use of the values of fractional emissive power, Table 3.2 is

provided to infer the wavelengths corresponding to any temperature to obtain the required

accuracy of the emissive power. For example, the values of the fractional emissive power,

F0-λT, provided in Table 3.2 can be used to distinguish the wavelength corresponding to

the distribution of radiant energy level up to

(1) 33.33% (corresponding to λT=3267) (2) 50% (corresponding to λT=4107)

(3) 66.6% (corresponding to λT=5269) (4) 75% (corresponding to λT=6147)

(5) 90% (corresponding to λT=9373) (6) 98% (corresponding to λT=17694)

42

The quantum of energy in between two wavelengths λ1 and λ2, denoted by, 21 λλ −F,

can be obtained from the Table 3.2, by using the following equation

TTFF 2121 λλλλ −− = (3.4)

TTTT FFF 102021 λλλλ −−− −= (3.5)

Table 3.2 : Fractional emissive power of radiant energy

λτ

( mKµ )

F0-λτ

(ND)

λτ

( mKµ )

F0- λτ

(ND)

2195 0.100 9373 0.900

2447 0.150 9800 0.910

2676 0.200 10293 0.920

2897 0.250 10875 0.930

3119 0.300 11577 0.940

3267 0.333 12451 0.950

3582 0.400 13591 0.960

4107 0.500 15185 0.970

5269 0.666 17694 0.980

5589 0.700 22811 0.990

6147 0.750 29183 0.995

6862 0.800 34830 0.997

7846 0.850 39950 0.998

3.5. RADIATION INTENSITY AND EMISSIVE POWER

The rate of radiative heat transfer, dq1-2, from an elemental emitting area dA1 to

another elemental area A2, which is centered at dA1 is proportional to the

(1) emitted surface area dA1

(2) solid angle from dA1 extending in the direction of dA2, namely dΩ

(3) cosine of the zenith angle, η (ranging from 00 to 900)

In mathematical form, the radiative heat transfer from dA1 to dA2 is

ηcos121 Ω∝− ddAdq .

Upon putting the proportionality constant, I , named radiation intensity, the above

proportionality gives rise to the following equation

43

ηcos121 Ω=− ddAIdq (3.6)

The solid angle dΩ can be defined in terms of the zenith angle η and the azimuthal

angle φ as given below.

ϕηη ddd sin=Ω (3.7)

where the range of angles are as 00 3600&900 ≤≤≤≤ ϕη

The radiative heat transfer from unit elemental area of the emitting surface, dE, is

obtained by dividing eqn. (3.6) with dA1. The resulting equation when integrated over the

entire hemisphere, the emissive power, E, is obtained. Thus the intensity and the emissive

power is related as

∫ ∫∫ ==Ω=Ω

π ππηηηϕη

2

0

2/

0sincoscos IdIddIE (3.8)

In general, the intensity is a function of direction and wavelength and is denoted as

‘I’ to convey the spectral dependence.

3.6. HEAT TRANSFER EQUATION IN A PARTICIPATING MEDIUM

The basic equation employed in the radiative analysis of heat transfer by radiation,

conduction and convection in a medium with absorption and scattering characteristics is

vhrp qqTkD

Dp

D

DTC φ

ττρ ++−∇∇+= ).( (3.9)

Where ∇+

∂

∂= .v

tD

D

τ is the total derivative.

The symbol zk

yj

xi

∂

∂+

∂

∂+

∂

∂=∇

where kandji , are the unit vectors in the

x,y,z directions respectively. Left hand side of equation (3.9) represents the local

acceleration and convective terms, and the right side consists of the pressure work,

conduction, radiation chemical heat generation and viscous heat dissipation respectively.

The term qh is chemical heat generation rate while the term qr is the radiative heat flux

vector. The term vφis viscous heat dissipation rate. The radiative heat flux vector which

44

represents the total radiant heat energy transferred across a certain cross section is

expressed as

rzryrxr qkqjqiq ++= (3.10)

∫∫∫∏∏∏

Ω∂′+Ω∂′+Ω∂′=4

0

4

0

4

0

coscoscos γβα IkIjIi

where ∂Ω is the differential solid angle in the range 0 to 4π. qrx, qry and qrz are the

net radiative heat flux along the unit vectors kandji , respectively. The angles α, β and γ

are the angles between the directions of the I’ and the x,y and z axes respectively.

The term rq.∇ is the divergence of radiative heat flux which is net radiant energy

emitted from a unit volume. It is a scalar quantity and a function of position consisting of

terms of self radiation, the attenuation of the incident radiation by absorption and scattering

and the emission resulting from scattering from nearby regions. Mathematically, the

divergence of radiative heat flux is described by the equation

∫∫∞∞

++−′=∇00

)]()](),([4)(),(4. λλσλσλπλλλπ λλλλ dITCdITCq bspapabsbabsr

λλλλσπ π

λλ dddI iiis ΩΩΩΩΦΩ′+ ∫ ∫∫∞

),,(),()(

4

0

4

00

(3.11)

First term in the right side of equation (3.11) represents the self radiation where

Cabs(λ,T) is the monochromatic absorption coefficient of gas. The second term represents

the attenuation of the incident radiation by absorption and scattering and the terms σapλ and

σspλ are the monochromatic absorption and scattering cross sections of particles contained

in the medium. The third term represents the emission resulting from all the incoming

scattered radiant energy from the nearby regions. The function ),,( iΩΩΦ λis the

dimensionless scattering phase function, which determines the fraction of intensity of

scattered energy in any direction. The term I appearing in eqn. (3.6) stands for the

averaged value of all radiant energy incident on a point and is defined as

∫ Ω′=π

λ λπ

λ4

0

)(4

1)( dII (3.12)

45

3.7. SOURCE FUNCTION OF RADIATION INTENSITY

The source function is defined as an increase in the radiation intensity per unit

thickness in the Ω direction at the location on the abscissa at an optical length of λκ and is

denoted as ),( Ωλλ κi. It is similar to the radiation intensity, a function of the location,

direction and wavelength and is expressed as

iiib dIIi ΩΩΩΦΩ′+′−=Ω′

∫ ),,(),(4

)()1(),(

4

0

λκπ

ωκωκ

π

λλλ

λλλλλ (3.13)

Where ωλ is the scattering albedo which is the ratio of scattering to attenuation of

radiant energy λλ

λλ

σσ

σω

sa

s

+=

. The first term on the RHS denotes the local self-

radiation at the optical length λκ and the second term is the directional component of a

change in the radiant energy in the Ω direction due to scattering.

3.8. RADIATIVE TRANSPORT EQUATION

The intensity of radiant energy incident on a point on the abscissa, at optical length

of λκ and in the Ω direction is expressed in the integral form as

λκκ

λ

κ

λκ

λλλλλ

λ

λ κκ *)(*

0

*

),()0(),( dkeieII−− −Ω′+′=Ω′ ∫

(3.14)

The first term on the RHS is the portion of the incident radiation from the boundary

that is not attenuated but instead reaches the gas element. The second term represents the

portion of radiant energy emitted by all other gas elements, expressed by the source

function, ),( Ω′ λλ κi, that reaches the element.

3.8.1. Solution Procedure

The conventional radiation heat transfer analysis calls for a direct solution of the

basic heat balance equation in an integro-differential form. The four independent variables

at each point in the domain are the following:

46

(1) Temperature, T, which determines the self emission from each point

into the surrounding area.

(2) The source function ),( Ω′ λλ κiwhich includes the self emission and

the scattering from each point into the surrounding area.

(3) Radiation intensity λI ′,defined as the radiant energy passing through

each point in each direction.

(4) The radiant heat flux rq which is the total radiant energy passing

through a cross section integrated in all directions around the point.

The source function and the radiation intensity are coupled by the equations (3.13)

and (3.14). The radiant heat flux is determined from equation (3.11) using the two variables

tained from equations (3.13) and (3.14) and is related to convection and conduction via eq

(3.9). Therefore, to obtain the solution for a heat transfer problem with a combination of

conduction, convection and radiation mechanisms, these four coupled equations are to be

solved simultaneously. The source function and radiation intensity at each point are

expressed as the integrations of the effects at all other points in the system. They are

functions of not only the locations but also the direction and thus the analysis becomes

extremely difficult.

3.8.2. Approximations to the Solutions

Analytical or numerical solutions to the radiation heat transfer analysis resort to

certain assumptions to simplify the problem and the most common simplifications are the

following:

(1) Only isotropic scattering is considered in order to avoid the difficulty due to

the directional dependence of these variables.

(2) Three dimensional analysis is generally skipped

(3) Only constant physical properties are considered in the analysis.

(4) Severance of coupling of heat transfer modes is diluted by neglecting the

effects of conduction and convection.

47

3.9. EXISTING METHODS OF SOLUTION

A number of solution methods are available in literature with their variance lying on

the domain of computation and end use of the solution. The use of geometric mean beam

lengths in the solution of RTE is explained by Andersen FMB [1997]. However its use is

restricted to a constant temperature and absorption coefficient in the medium. Chih-Yang

Wu (1989) used integral equations for the solution. The solution of integro-differential

equation of RTE is a function of both position and direction; whereas the solution of integral

equations in terms of moments of the radiation intensity reduced the dual dependence of

position and direction to the dependence of position only. However its scope is limited to

simple geometries. The spherical harmonics method, (e.g., Menguc and Viskanta 1986) is

not accurate for low order approximations, except in optically thick media. Both the two-flux

and zone methods have been used extensively for the solution of radiative transport

equations. The discrete ordinates method is an extension of two-flux method where the

general transfer relations are represented by a set of equations for an intensity that is

angularly averaged over each of a finite number of ordinate directions. Discrete transfer

method illustrated by Coelho PJ and Carvalho MG [1997], is a variant of discrete ordinance

method which was originally formulated to provide faster algorithm for incorporating

radiative transfer into codes for combustion in complex geometries. Ye Wang and Yildiz

[2002] worked with discrete ordinate method with wavelets to evaluate the spectral radiative

intensity. Double rays method due to Hong-Shun Li and Gilles Flamant [2002] is a variant of

discrete ordinates method which tries to rectify the discontinuous nature of radiation

intensity. Subash C. Mishra etal., [2005] attempted a modified collapsed dimension method

which is very similar to the discrete ordinates method. However solutions of these methods

often suffer from discontinuities in temperature and radiative flux, known as ray effect. Later,

Zhao and Liu [2008], worked with discontinuous spectral element approach for solving RTE

by making use of Crank-Nicolsan scheme for discretization of time domain and the spatial

discretization using discontinuous spectral element approach.

A comparison of accuracy of the two-flux, spherical harmonics and discrete

ordinates method for predicting radiative transfer in a planar, highly forward scattering and

absorbing medium is made by Menguc MP and Viskanta R [1983]. Radiation element

method by ray emission model was described by Maruyama and Aihara [1997] as a tool for

48

solving RTE which proposes good accuracy with a small number of rays simulated from

each control volume. However not much work is reported on this method.

The following three sub sections, 3.8.1, 3.8.2 and 3.8.3 give a little more details

about three well known methods of solution to facilitate the right choice of the computational

tool for the present study.

3.9.1. Two Flux Method

The two-flux method is basically a one dimensional approach wherein the radiation

intensities in the positive and negative directions of the coordinate axis are computed as I+

and I-.The two flux method, as explained by Brewster and Tien [1982], approximates that

angular distributions of both I+ and I- are uniform and constant, but with different values.

However, the assumption of one dimensionality makes it possible to solve the complete

equation of transfer with relative simplicity. In dealing with the radiant energy that travels

through a gas layer of infinitesimal thickness and perpendicular to the coordinate axis, both

assume that for one-dimensional energy transfer, the intensity in the positive and negative

directions is isotropic with different values. Two-flux method gives good agreement with the

exact solution, but slightly under predicted in the case of higher wall emissivities and thicker

optical thickness. However, the method is known to deviate from the exact solution in the

case of a two or three dimensional system or in the presence of scattering, especially strong

non-isotropic scattering. The two and six flux methods, [Ludwig, Malkmus etal, [1982], follow

lines of sight through the plume and approximate the scattering process by accounting for

either two or six components of scattering at each point.

3.9.2. Zone Method

Wen-Jei Yang, Hiroshi and Kudo [1995] give an excellent description of the zone

method. In the zone method, the domain of computation is divided into many gas and wall

elements and the temperature is assumed to be constant within each element. Then total

exchange areas are calculated from direct exchange areas, which represent the radiative

energy exchange between each element when all the wall elements in the system are

assumed to be black. Then the temperature and the wall heat flux distributions in the

system can be obtained by solving the set of energy equations of the wall and gas

elements.

49

Since the direct exchange area is defined for the radiative energy emitted or

reflected diffusely from the wall element, a specular reflecting wall cannot be treated by the

zone method.

3.9.3. Monte Carlo Method

The terms of source function and radiation intensity are present in the radiative

transport equation only as a means of expressing the macroscopic behavior of a radiative

process. Physically energy transport by radiation is a combination of scattering and

absorption of a large number of independent photons, which are emitted from either gas

species or solid wall containing the gas species. This is simulated in Monte Carlo method by

tracing and collecting the scattering and absorption behavior of a large number of

independent radiative energy particles named photons which are emitted from each control

volume in the system. Monte-Carlo methods use random numbers to define paths of energy

bundles through the plume beginning with an emission and followed by either

absorption/reemission or scattering events until the energy bundle exits the plume. The ratio

of paths leaving the plume to the number of emission and absorption/re-emission provides

information on the portion of total plume radiant emission, which escapes. Thus the real

physical phenomenon is simulated to estimate the portion of energy reaching any nearby

surface and thus avoids the requirement of directly evaluating the view factor. Algorithms for

designed three dimensional particle trajectories incorporating absorption, emission and

scattering phenomenon and for determining the incident radiant heat from the system to the

desired locations are required in the MC method. These algorithms necessarily incorporate

the probability density functions, which translate the mechanism of absorption, emission and

scattering into mathematical expressions and appropriate random numbers. This facilitates

the evaluation of incident radiation that is absorbed by each differential volume defined in

the system and hence various postulations and restrictions are eliminated for evaluating the

direction-dependent source function and the radiation intensity. Thus the radiative heat

transfer analysis is greatly simplified through the use of MC method and thus this method is

selected as the tool for analysis of the RTE in the present study.

Tracing of all photons requires an enormous amount of computational time and this

difficulty is overcome to a great extent by the advent of high-speed computers and the

50

introduction of some efficient methods like the radiant energy absorption distribution method

in the transient analysis. However, one has to ensure the size of minimum sample space for

the convergence of solution.

MC method is very popular in dealing with the problems consisting of data

uncertainties. In these types of problems each datum is assigned with different degree of

uncertainties following a statistical distribution and solutions are sought with each set of

data. Thus from a large number of solution set, one can determine the most probable

solution with its statistical parameters like range mean and median. Chacko, Mani and

Shukla [1987] and Chacko and Shukla [1988] demonstrated the procedure for this type of

MC method in thermal management of electronic equipments for space applications.

However, MC method as a solution of RTE problems is not well explained in the current

literature. A large portion of the details of MC method is available in papers followed by

technical reports and text books. This study makes an attempt to bring out a comprehensive

description of the MC method as a solution technique of RTE, in chapter-8.

3.10. STRATEGY OF THIS STUDY

In view of the objectives outlined, it was necessary to model the plume radiosity

using an appropriate strategy. Basically the strategy consists of the following:

Use Monte-Carlo method to model the plume radiosity, since it is more

amenable to incorporate the basic phenomena like scattering, absorption

and emission present in the Radiative Transfer Equation.

Use Mie theory to model the scattering of radiant energy emanating from

the inner region of the plume to incorporate the scattering efficiency more

accurately as a function of particle radius and wavelength of the radiant

energy.

Use Rosin-Rammler distribution to model the particle spectrum in the

exhaust plume, which defines a more accurate domain for applying the Mie

theory of scattering.

Use Computational fluid Dynamics as a tool to generate the particle

trajectories to model the computational domain consisting of gas-particle

mixture with a varying cross section of particle sizes.

51

Finally compare the predicted plume radiosity with flight data of a single

rocket motor to avoid the uncertainties arising due to the interference of

adjacent motors.

3.11. CONCLUSIONS

The basic equations in the filed of Radiative Transport Equation in a participating

medium are presented. Planck’s law for the spectral emissive power of a black body,

radiation intensity and emissive power are introduced as a prerequisite for the solution of

RTE. The concept of radiative source function is introduced along with the RTE. A brief

review of certain popular methods like, two flux method and zone method and exist in the

field of solution of RTE presented with their merits and demerits. The basis of selection of

MC method in the current study is explained. A brief description of the use of spectral

distribution of emissive power in choosing reflectometers and thermal imagery systems is

also provided.