On Numerical Methods for Radiative Heat Transfer · · 2017-09-13On Numerical Methods for...

On Numerical Methods for Radiative Heat Transfer

by

Professor Bengt Sundén,

Department of Energy Sciences, Lund University

Background

A principle sketch of the heat transfer mechanisms for a combustor liner for a gas turbine engine

Luminous radiation from soot particles Non-luminous radiation from gases

• Inhomogeneous medium• High temperature• High heat flux

Another Combustor

Sketch of grate furnace for biofuels combustion

Combustion zones in a grate furnace

Thermal Radiation – Radiative Heat Transfer

• Radiation without participating media» Surface radiation> Surface phenomena> No change in radiation intensity> Basic radiation quantity – emissive power> Analysis simple

Thermal Radiation – Radiative Heat Transfer

• Radiation with participating media» Gas and particle radiation> Volumetric phenomena> Change in radiation intensity> Basic radiation quantity – intensity of radiation> Analysis complex

Participating MediaCharacterization of the participating medium

• Absorption: attenuation of intensity→ absorption coefficient κ

• Emission: augmentation of intensity → absorption coefficient κ

• Scattering → sca ering coefficient σs, scattering albedo ω and scattering phase function ϕ- in-scattering: augmentation of intensity- out-scattering: attenuation of intensity

Radiation Parameters

• Extinction coefficient: β = κ + σsabsorption coefficient plus scattering coefficient

• Optical thickness:

• Scattering albedo: ω =

Radiative Heat Transfer Related to Combustion

What we need: radiation properties of combustion gases, coal and fly ash particles, small soot particlesSoot particles are produced in fuel-rich flames as a result of incomplete combustion of hydrocarbon fuels.Electron microscope studies show that soot particles are generally small and spherical of sizes 50 – 800 Å (0.005-0.08 μm). Volume fractions (fv) of soot in diffusion flames of hydrocarbons: 10-4 – 10-6

%

General considerations - 1

• Surface-to-surface radiation- no participating media → net exchange formulation based on radiosities, configuration factors (view factors) and surface properties: the general theory outlined in course MMV031 Heat Transfer – textbook ”B. Sunden, Introduction to Heat Transfer, WIT Press, UK, 2012”


• Radiation in participating media →Total Radiation Transfer Equation (RTE) –coupling with general energy equation for a fluid flow

RTE and how to solve it will be presented and exemplified


• Radiation in participating media →Simplified methods available for idealized cases – MMV031 Heat Transfer – Textbook ”B. Sunden, Introduction to Heat Transfer, WIT Press, UK, 2012”

Basic Features of Thermal Radiation Heat Transfer

Thermal radiation

4B TE

Stefan-Boltzmann’s law

81067.5 W/m2K4

Thermal Radiation

4TEg

gray body

Thermal radiation - Intensity

d

Ir

dA

dA'

d

Thermal radiation-solid angle

2rAdd

Solid angle dω

Thermal radiation-Intensity

2coscos)(

rAddAIddAIdAEd

The radiation intensity is defined as the radiant energyper unit area projected perpendicular to a given directionand per solid angle unit viewed from the radiating surface.

Thermal radiation-Intensity

)( dAEdEdA

ddd sin

dddAIdAEd sincos)(

2/

0

2

0sincos

dAIdddAIdAE

IE

Thermal Radiation – View Factors

A2

A1dA1

dA2

ndA1ndA2

r

1111

cos

dAIdd

222

1cosr

dAd


,

1 2

21 21221

1coscos

A AAA dAdA

rI

11111AIAEA

121 12 AAA F

1 2

21221

112

coscos1

A A

dAdArA

F


1 2

21221

221

coscos1

A A

dAdArA

F

212121 FAFA

1........211

niii

nj

jji FFFF

Reciprocal Identity

Enclosures

Thermal Radiation- Exchange Black Surfaces

Gi

Ji

i

k

iiii GJAQ

ii EJ ,B

k

ikkik

ikkkii FJAFJAGA

Thermal Radiation- Exchange Black Surfaces

)( k

kikiii JFJAQ

)()( ,B,B k

kikiik

kikiii EEFAJJFAQ

Thermal Radiation, Exchange Non-Black Surfaces

Gi

Ji

i

k

iiii GJAQ

iiiii GEJ ,B

ii 1


)(1 ,B ii

i

iii JEAQ

k

kikiii JJFAQ )(


Surface to Surface Radiative Heat Exchange ends up in a system of equations to be solved as radiosities are not known a priori, i.e., the equations on previous slide

Participating media

Elementary gases like H2, O2 and N2 emit almost nothermal radiation and are almost transparent ( = 1) forradiative transfer.

In engineering applications CO2 and H2O-steam are most important as these are good emitters and usually are present in high concentrations.CO, SO2 and CH4 are also good emitters but are usually present in small concentrations only.

Particles like soot may also contribute in the radiative exchange

Gas radiation - absorptance for CO2

102 3 4 5 6 7 8 200102030405060708090

100

%

4

4

4

431

2

[m]

(1) 5 cm gas layer(2) 3 cm gas layer(3) 6.3 cm gas layer(4) 100 cm gas layer

Gas radiation - absorption in a gas layer

x dx

I

I

dxIadI

Gas Radiation - absorption in a gas layer

a depends on pressure and temperature

If pressure and temperature are uniform, i.e., constant in the gas layer one has

,0a xI I e

Beer's law

Beer’s law

Consider a layer of thickness L

,⁄

If << 1 optically thin medium ⇒simpli ication

If >> 1 optically thick medium⇒exchangeonlybetweenneighboringvolumeelements

If ~ 1 ⇒integro-differential equations to be solved

Gas radiation - absorption in a gas layer

xae

0Commonly one as

which means

xae

1

Gas Radiation; Mean Beam Length Equivalent Beam Length

AVL 6.3

Beam lengths are different, average needed

Gas Radiation, emittance for CO2

,

0.01

0.1

500 1000 1500 2000 2500

0.3

0.05

0.003

0.005

0.03

pCO2L = 105 Pa m6x104

3x104

10 4

6x103

3x10 3

1000

600

300100

60

30

Temperature [ K ]

2COε

Gas Radiation - emittance for CO2, pressure correction

pCO2L=8x104 Pa m

3x104

1.5x104

8x103

4x103

1.5x103

600

0.05 0.08 0.1 0.2 0.3 0.5 0.8 1.0 2.0 3.0 5.0

2.0

1.5

1.0

0.8

0.5

0.6

0.4

0.3

Total pressure atm

2COC

Gas Radiation; Heat Exchange between a Gas and a Chamber with Black Surfaces

emitted gas radiation – absorbed wall radiation

AQ

Radiative Exchange between a Gas and a Chamber with Black Surfaces

4wwg

4ggg )()( TTTT

AQ

OHCOwg 22)(T

)( gg T is determined at Tg

depends on both Tg and Tw

Detailed Analysis of Radiative Exchange in Participating Media

• Overall aim is to enable accurate heat load calculations

• Key is the so-called radiative transfer equation (RTE), plus

• Determine absorption coefficients for gases

• Determine scattering coefficients of particles

Radiation in Gases

•Gases, depending on the molecular structure,absorb and emit photons in some Vibration and/or Rotation bands. Each band consists of some absorption lines.

Absorption spectra

H2O

0

100

A

bsor

ptio

n (%

)

CO2100

0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

CO100

0x-axis in μm

Importance and specifications

The products in combustion consist of gases such as carbon dioxide, water vapor, carbon monoxide etc., and in some cases consist of particles like soot, droplets, fly ash,…

The Importance of gas radiation in industry was recognized in the 1920,

The accurate prediction of radiative transfer problem not only depends on the solution method of the Radiative Transfer Equation (RTE), but also depends on models for determining the radiative properties!

General statements

• Empirical (analytical) models, preliminary design (50 - 100 K, deviation)

• Numerical models (CFD), detailed design

• Empirical models could give better results if parameters are tuned properly (by using, e.g., CFD)

• Many numerical models do not include soot

• Pressure and fuel type are important (soot formation)

• Empirical models use a luminosity factor to take care of the influence of soot

S1 0 0

Uj

h

Y

k

j

j j jt xS

U

x x

2 23 3

j kt ij ij

i j i k

U UP kx x x x

,R j

j

qPt x

kP

1 1 2 2kf C P f Ck k

2 /t f C k

t

/ /t tPr Pr

/ /t tSc Sc

/t k

/t

Governing equations - CFD

RTE-radiative transfer equation

•Will provide source term in energy equation

• Radiation may be important depending on pressure and fuel used in the combustion chamber

• Good predictions of the RTE will give better predictions of the temperature and flow fields and accordingly heat load calculations

Pencil of rays for radiative energy balance

RTE-derivation• Consider a gas layer with thickness ds;

absorption(dIλ)abs = - κλ•Iλ•ds

• Scattering(dIλ)sca = - σsλ•Iλ•ds

• Emission(dIλ)em = κλ•Ibλ•ds

cross section dAds

s

SidΩ

dΩi

Scattered Radiation from a Volume Element

Radiation within the solid angle dΩi impinging onthe surface element dA:

•

Scattered part:

• •

Only a small fraction of the radiative energy flow within the solid angle dΩi is scattered into the solid angle dΩ.A scattering function Φλ is introduced to handle this.Φλ gives the probability that a ray from the direction will be scattered into a certain direction .

• 4

4ˆˆ ˆ ˆ( ) = ds ( ) ( ,s)

4s

i sca i i idI s I s s d

By integrating the previous equation overall incident directions, the total in-scattered radiative energy into a solid angle dΩ becomes

For isotropic scattering, Φλ = 1

Pencil of rays for radiative energy balanceEnergy Balance for thermal radiation propagating in a direction s

Out - In = emission-absorption-scattering (away) + all in scattering

• Energy Balance for thermal radiation propagating in a direction s

4ˆˆ ˆ ˆ ˆ ˆ ˆ( , , ) I ( , , ) ( , ) ( , , ) ( , , ) ( ) ( ,s)

4s

b s b i i iI s ds s t dt s s t I s t ds I s s t ds I s s t ds I s s d ds

By introducing c = ds/dt one can find

4

1 1 ˆ

ˆ ˆ ˆ( ) ( , )4

b

si i i

I I I I I Ic t s c t

I d

s

s s s

Radiative Transfer Equation, RTE (omitted transient term)

4

ˆ ˆ ˆ ˆ( ) ( , )4

sb i i i

dI I I I I dds

s s s s

βλ = κλ + σsλ

Divergence of the Radiative Heat Flux

yx zqq q dxdydz dV

x y z

q

( , , )x y zq q qq

Net heat flux into volume element

Divergence of radiative heat flux

4 4 4 4

ˆ ˆ ˆ ˆ ˆ4 ( ) ( ) ( , )4

sb i i iI d I I d I d d

s s s s s

4

ˆI d

q s

4

ˆ ˆ( , ) 4i d

s s

Divergence of radiative heat flux

s

4

4 4b bI I d I G

q

with

where Gλ is the irradians

For a gray medium κλ = κ

4 2 4

4

4 4T Id n T G

q

The Complete Energy Equation

pT T T Tc u v wt x y z

q

conduction radiation radiationk T q q q q

radiationq according to previous slide

Solution methods for the radiative transfer equation,

RTE1) Spherical Harmonics Method - also called the

PN – approximation method2) Discrete Ordinates Method (DOM)3) Discrete Transfer Method (DTM)4) Finite Volume Method (FVM)5) Zonal Method6) Monte Carlo Method (MC)

Radiation models available in ANSYS FLUENT

Discrete Ordinates Model (DOM)Discrete Transfer Radiation Model

(DTRM)P1 Radiation ModelRosseland ModelSurface-to-Surface (S2S)

Radiation models – general advices for selection

• Low computational effort with reasonable accuracy: P1

• Accuracy: DTRM and DOM• Optical thin media: DTRM and DOM• Optical thick media: P1

• Scattering: P1 and DOM• Localized heat sources/sinks: DTRM,

DOM with large number of rays or ordinates

RTE coupling with Energy Equation

Energy equation

RTE

NS-equations

2 44R RS divq n T G

Species equations

Radiative Transfer Equation (RTE)

iiis

sb dsssIIIIsI

tI

c

)ˆ,ˆ()ˆ(4

1

4

Solution of RTE requires the absorption coefficient ()

and scattering coefficient s

Brief descriptions of methods to solve the RTE

Rosseland ModelIf the participating medium is optically thick, it is possible to simplify the numerical simulations by means of the so-called

Rosseland approximation.Instead of solving an equation for G (the irradiance), it is possible

to assume a theoretical value prescribed by the black body expression and then to group the consequent radiation terms in

some modified transport coefficients for the energy eqution:

16

Differential Approximation (PN)• The basic idea is that the intensity in a participating

medium can be represented as a rapidly converging series whose terms are based on orthogonal spherical harmonics. Usually a small number is adequate (P1, P3, ...). For the P1 approximation only zeroth and first order moments of the intensity are considered.

• Essentially the deviation of the local intensity from its local mean value is expressed in terms of local gradients. This allows derivation of a diffusion expression for the radiation flux and consequently the derivation of an advection-diffusion equation for the local mean intensity.

The P1 radiation modelThe directional dependence in the RTE is integrated outRTE becomes easy to solve with limited CPU demandEffects of scattering by particles, droplets and soot can

be includedWorks well where the optical thickness is large (e.g.,

combustion)The limitations are: All surfaces are assumed to be

diffuse, less accurate for small optical thickness, may overpredict the radiative fluxes from localized heat sources and sinks.

The P1 radiation model• The P1, model solves an advection-diffusion

equation for the mean local incident radiation (irradiance) G

• Consequently the gradient of the radiation flux can be directly substituted into the energy equation to account for heat sources or sinks due to radiation

4

• 4 0

Hottel’s zonal method• The equations for emitting and absorbing media

can be easily solved and find the conditions leading to a uniform temperature distribution within a medium being bounded by infinite parallel black walls.

• Then it is always possible to subdivide the generic domain into isothermal zones of appropriate shape such that the constant stepwise profile approximates the actual solution. The analysis, i.e., Hottel’s method, allows to recover the consistent thermal fluxes at the boundaries of the arbitrary zones.

Discrete Ordinates Model (DO or DOM)• The RTE is solved for a discrete number of finite

solid angles• The method is conservative which means that

heat balance is achieved even for coarse discretization

• Accuracy is increased by using a finer discretization

• Accounts for scattering, semi-transparent media, specular surfaces and wavelength-dependent transmission using banded-gray option

• CPU intensive

Angular discretization - DOM

Discrete Transfer Radiation Model (DTRM or DTM)

• It assumes that radiation leaving a surface element within a specified range of solid angles can be approximated by a single ray.

• It uses a ray-tracing technique to integrate radiant intensity along each ray.

• Relatively simple model• Accuracy can be increased by increasing the

number of rays• Can be applied to a wide range of optical

thicknesses• It assumes that all surfaces are diffuse• Effect of scattering not included• CPU intensive if a large number of rays are used

DTRM-Ray arriving at an surface element Q

DTRM-Analytical solution for a ray

1 – e-αs) + I0e-αs

DTRM

• The method as it was originally proposed consists of determining the intensity for each of N (Nφ x Nθdiscretized angles) rays arriving at each surface element Q in an enclosure as shown in previous Fig.

• The ray and its associated solid angles are equally distributed over the surface of a hemisphere centred over the receiving element, rather than being chosen and weighted through a quadrature technique as for the DOM.

• Radiative Properties of Gases

Gas radiative properties

H2O24 atm, 1.704 m path

78


H2O, CO2

24 atm, 1.704 m path

79

H2O, CO2, CO

Line overlapping and broadening


4,3 µm

4,7 µm

6,3 µm

80

H2O, CO2, CO



81

Importance of CO

Position c, 24 atm


82

On importance of CO

•CO radiation can make substantial contribution to radiation in combustion environments with high CO concentration

•The CO contribution to the total radiative heat flux increases with decreasing pressure

•The line overlapping between radiative species decreases as pressure decreases

83

Models for radiative properties: importance and specifications

Modelling of radiative properties in the infrared region with sufficient accuracy,

Compatibility with selected RTE solver,

Capability to handle non-homogeneous cases,

Small database,

Low computational efforts.

Radiative Properties of Gases

• Line-By-Line (LBL)• Gray Gas (GG) models• Spectral models; Statistical Narrow

Band Models (SNB) (with a correlated k-distribution method (CK)), Wide Band models (WB), Weighted-Sum-of-Gray-Gases (WSGG), Spectral-Line-Weighted-sum-of-gray-gases (SLW), Wide Band Correlated k-model (WBCK)

Prediction of Radiation in Gases

• Empirical - Based on measurements of Total Emissivity • Less sophisticated, more engineering approaches

• Line by line calculations (Spectroscopic databases)

• Numerical Band models (Narrow bands, Wide bands, ..)

• Correlation models (Weighted Sum of Grey Gases, …)

EWBM (exponential wide band model)

Advantages: Experimental base and wide range of application,

Small database and fairly less computational efforts,

Possibility for utilization in nonhomogeneous mixtures.

Deficiencies: Calculates band absorptance and band transmissivity,

Can not be used in reflective walls and scattering media,

Low accuracy.

The Weighted-Sum-of-Gray-Gases Model

• The weighted-sum-of-gray-gases model (WSGGM) is a reasonable compromise between the oversimplified gray gas model and a complete model which takes into account particular absorption bands. The basic assumption of the WSGGM is that the total emissivity over the distance s can be presented as

• where aε,i is the emissivity weighting factor for the ithfictitious gray gas, the bracketed quantity is the ithfictitious gray gas emissivity, κi is the absorption coefficient of the ith gray gas, p is the sum of the partial pressures of all absorbing gases, and s is the path length.

Spectral-Line-Weighted-sum-of-gray-gases-model (SLW)

• The SLW might be regarded as an improved WSGG model, obtained from line-by-line spectra of the absorbing species.

• SLW-x where x is the number of gray gases considered.

Wide Band Correlated-k model (WBCK)

• WB models yield transmittances and cannot be easily coupled to differential solutions methods of the RTE.

• WBCK gives an absorption coefficient for each spectral interval. The weight of each spectral interval is calculated from the blackbody fraction of the interval.

RTE, radiative properties:

Gas properties

Particle properties (soot)

Rayleigh

Mie

RDG - PFA

LBLNBMEWBMGlobal models

Thermal Properties Radiation

Spheres Cylinder

Particles Surfaces

Properties(absorption coefficient)

Gases and particles contribute to (total = partikel + gas)

There are various geometric forms of the particles

Type of Particles

Analysis of ash samples from cyclones and Electro-Static Precipitator (ESP) in a grate biomass boiler, fueled by bark and wood chips, shows:

• The fly-ash and char are the main particles, and

• The ratio of fly-ash and char to total amount of ash is 0.72 and 0.28, respectively.

Particle Compositions

• Sub-micron particles (< 1 m) are mainly composed of C (in form of soot), S, Cl and heavy metals

• Super-micron particles:

Fly-ash particles are concentrated mostly in the range of some microns and formed mainly by SiO2, CaO, Al2O3, MgO, K2O and Fe2O3

Char particles are formed mainly by C

Study of effects of particles on radiative heat transfer in biomass boilers

• Determining types and amounts of particles.

• Prediction of non-gray radiative properties and accurate phase functions, and

• Evaluation of particle effects on radiative heat transfer by designated test cases.

Interaction of an electromagnetic wave with a

particle

Introduce

Interaction of an electromagnetic wave with a particle

X << 1 Rayleigh scatteringX ~ 1 Mie scatteríngX >> 1 geometric opticsapplicable

98/21

• Scattering of soot

Radiation properties of soot

μm

Main difference between particle and gas: Scattering

Mie theory

Rayleigh theory

Suitable for optical large particles

Suitable for optical small particles

Selected Methods

• Using measured data and statistical methods

• Investigation of composition of particles,• Applying the Mie scattering theory,

• Summarizing the data from the Mie calculations by using statistical distributions.

Small soot particles

• Scattering is negligible

= va

C f

C and a are empirical constants

Results

height, m

heat

sour

ce,

kW/m

3

0 1 2 3 4-300

-250

-200

-150

-100

-50

0

12L2H3L3H

Calculated Heat Source in the Test Case A, (gas+soot=0.01 m-1)

1: Without Particle L: Low Particle Concentration

2: With Particles, Absorption Effect H: High Particle Concentration3: With Particles, Absorption and Scattering Effects

On Solution of the RTE

• Different radiation models will give different results

• Some radiation models are better suited for different situations, see next table

• Good if solution method of RTE is compatible with that of other equations

• Be able to handle scattering (due to soot agglomerates)

• Be able to handle spectral calculations

• Be able to handle complex geometries

Comparison of Methods for Solving the RTE

Angular resolution

Spatialresolution

Spectral resolution

Scattering medium, 2D

Scattering medium, 3D

DTM 2 4 4 1 1DOM 3 4 4 3 3Spherical harmonics 2 3 4 3 3

Zonalmethod 2 3 2 2 2MonteCarlo 4 3 3 4 4Finite

element/volume

techniques

2 4 3 2 2

4 = very good, 3 = good, 2 = acceptable, 1 = not good

Applications of Radiative Heat Transfer with Participating Media

• Furnace model

Numerical Simulations in a Small Furnace

μm

Radiation heat transfer and radiative properties are based on temperature and species fields

Combustor

0.25 m

AirInlet

SwirlerKerosene ejector

0.048 m

Outlet

Swirler

• Volume flow rate of air 31.6 m3/h • Volume flow rate of kerosene 6.6 litre/h• DOM (the so-called S8 model was used)• SNB for CO2 and H2O

Results

9/16

Flow field - Fluent results

The flow in the furnace is very non-homogeneous and complex as it passes the swirler

a) Temperature b) Pressure (+101325Pa)

c) Mole fraction of H2O

d) Mole fraction of CO2

Numerical Simulation in a Small Furnace

a) With scattering b) Without scattering

• Soot radiation contribution(a)

W/m

(b)

W/m

Participating media are gases H2O, CO2, CO and soot

Numerical Simulation in a Small Furnace

a) Radiative Heat Flux b) Radiation Contribution

• Soot radiation contribution

0.00 0.05 0.10 0.15 0.20 0.25 0.3005

10152025303540

z (m)

Rad

iatio

n he

at fl

ux (k

w/m

2 )

Two gases Three gases Soot-With scattering Soot-Without scattering

(a)

0.00 0.05 0.10 0.15 0.20 0.25 0.300

5

10

15

20

25

30

35

Rad

iatio

n C

ontri

butio

n (%

)

z (m)

Soot-With scattering Soot-Without scattering

(b)

Soot has a big influence on the radiative heat fluxScattering has no influence on the radiative heat flux

Test facility atB & W VölundEsbjerg, Denmark

Pellets in the tests

Control room in the operation

Gas temperature distribution central plane

Furnace wall irrradiation

On Numerical Methods for Radiative Heat Transfer · · 2017-09-13On Numerical Methods for...

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