On Numerical Methods for Radiative Heat Transfer. by Professor Bengt Sundén, Department of Energy Sciences, Lund University

On Numerical Methods for Radiative Heat Transfer by Professor Bengt Sundén, Department of Energy Sciences, Lund University

Background

Luminous radiation from soot particles Non-luminous radiation from gases A principle sketch of the heat transfer mechanisms for a combustor liner for a gas turbine engine

Another Combustor Inhomogeneous medium High temperature High heat flux

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 Media Characterization of the participating medium Absorption: attenuation of intensity absorption coefficient κ Emission: augmentation of intensity absorption coefficient κ Scattering scattering coefficient σ s, scattering albedo ω and scattering phase function ϕ - in-scattering: augmentation of intensity - out-scattering: attenuation of intensity

Radiation Parameters Extinction coefficient: β = κ + σ s absorption coefficient plus scattering coefficient Optical thickness: s s τ = න൫κ + σ s )ds = න β ds 0 0 Scattering albedo: ω =

Radiative Heat Transfer Related to Combustion What we need: radiation properties of combustion gases, coal and fly ash particles, small soot particles Soot 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 (f v ) 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

General considerations - 2 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

General considerations - 3 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 E B T 4 8 W/m 2 K 4 5.67 10 Stefan-Boltzmann s law

Thermal Radiation gray body E T g 4

Thermal radiation - Intensity d da' r I da d

Thermal radiation-solid angle Solid angle dω d da r 2

Thermal radiation-intensity The radiation intensity is defined as the radiant energy per unit area projected perpendicular to a given direction and per solid angle unit viewed from the radiating surface. d( EdA) I dacos d da I dacos 2 r

Thermal radiation-intensity EdA d( EdA) d sin d d d( EdA) I dacos sin d d EdA 2 0 / 2 I da d cos sin d 0 I da E I

Thermal Radiation View Factors da 2 A 2 n da1 r n da2 da 1 A 1 d d 1 d 1 I 1 da da 2 1 2 cos cos r 2 1

Thermal Radiation View Factors A A 1 2 2 A A 1 2 I 1 cos cos 1 r 2 2 da 1 da A E 1 1A1 I1A1 A1 A F12 A 2 1 F 12 1 A 1, A A 1 2 cos cos 1 r 2 2 da 1 da 2

Thermal Radiation View Factors F 21 1 A 2 A A 1 2 cos cos 1 r 2 2 da 1 da 2 A F A 1 12 2F21 Reciprocal Identity j n j1 F i j F i1 F i2... F in 1 Enclosures

Thermal Radiation- Exchange Black Surfaces G i J i i k i i i i G J A Q i i E J, B k ik k i k k i k k i i F J A F J A G A

Thermal Radiation- Exchange Black Surfaces Q i A i ( J i F k ik J k ) Q i A i k F ik ( J i J k ) Ai Fik ( EB, i EB, k k )

Thermal Radiation, Exchange Non- Black Surfaces k i J i G i Q i A i J i G i J i ieb, i G i i 1 i i

Thermal Radiation, Exchange Non- Black Surfaces Q i A i i 1 i ( E B, i J i ) Q A F ( J J ) i i k ik i k

Thermal Radiation, Exchange Non- Black Surfaces 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 H 2, O 2 and N 2 emit almost no thermal radiation and are almost transparent ( = 1) for radiative transfer. In engineering applications CO 2 and H 2 O-steam are most important as these are good emitters and usually are present in high concentrations. CO, SO 2 and CH 4 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 CO 2 % 100 90 80 70 60 50 40 30 20 10 0 4 4 1 2 4 2 3 4 5 6 7 8 10 20 4 3 [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 I I x dx di a I dx

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 I I e,0 a x Beer's law

Beer s law Consider a layer of thickness L I λ = I λ,0 e a λlxτl If a λ L << 1 optically thin medium simplification If a λ L >> 1 optically thick medium exchange only between neighboring volume elements If a λ L ~ 1 integro-differential equations to be solved

Gas radiation - absorption in a gas layer a e x Commonly one has which means 0 1 a e x

Gas Radiation; Mean Beam Length Equivalent Beam Length Beam lengths are different, average needed L 3. 6V A

Gas Radiation, emittance for CO 2 0.3 p CO2 L = 10 5 Pa m 0.1 6x10 4 ε CO 2 3x10 4 0.05 10 4 6x10 3 0.03 3x10 3 1000 600 0.01 100 300 60 0.005 30 0.003, 500 1000 1500 2000 2500 Temperature [ K ]

Gas Radiation - emittance for CO 2, pressure correction 2.0 1.5 C CO2 1.0 0.8 0.6 0.5 0.4 p CO2 L=8x10 4 Pa m 3x10 4 1.5x10 4 8x10 3 4x10 3 1.5x10 3 600 0.3 0.05 0.08 0.1 0.2 0.3 0.5 0.8 1.0 2.0 3.0 5.0 Total pressure atm

Gas Radiation; Heat Exchange between a Gas and a Chamber with Black Surfaces Q A emitted gas radiation absorbed wall radiation

Radiative Exchange between a Gas and a Chamber with Black Surfaces Q A ( T ) T ( T T 4 g g g g w ) 4 w g ( T g ) is determined at T g (T ) g w CO2 H2O depends on both T g and T w

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 100 H 2 O Absorption (%) 0 100 0 100 CO 2 CO 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 x-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

Governing equations - CFD U j t x j x j x j S S 1 0 0 U j h t / Pr / Pr t t P U j 2 U k 2 t ij k ij xi x j xi 3 xk 3 P qr, j t x j Y k / Sc / Sc / t t k / t t P k f C P f C k 1 1 k 2 2 t f C k 2 / k

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 (di λ ) abs = - κ λ I λ ds Scattering Emission (di λ ) sca = - σ sλ I λ ds (di λ ) em = κ λ I bλ ds

Scattered Radiation from a Volume Element S i dω i ds cross section da dω s

Radiation within the solid angle dω i impinging on the surface element da: I λ ( sƹ i )(dasƹ i s)dω Ƹ i dλ Scattered part: σ sλ (I λ ( sƹ i )(dasƹ i s)dω Ƹ i dλ) ds sƹ i sƹ Only a small fraction of the radiative energy flow within the solid angle dω i is scattered into the solid angle dω. A scattering function Φ λ is introduced to handle this. Φ λ gives the probability that a ray from the direction sƹ i will be scattered into a certain direction s. Ƹ σ sλ I λ ( sƹ i )dadω i dλ)ds Φ λsƹ i sƹ dω 4π

By integrating the previous equation overall incident directions, the total in-scattered radiative energy into a solid angle dω becomes di ( sˆ ) = ds I ( sˆ ) ( sˆ,s) ˆ d 4 s i sca 4 i i i For isotropic scattering, Φ λ = 1

Pencil of rays for radiative energy balance Energy Balance for thermal radiation propagating in a direction ŝ Out - In = emission-absorptionscattering (away) + all in scattering

Energy Balance for thermal radiation propagating in a direction ŝ s I (, ˆ, ) I (, ˆ, ) (, ) (, ˆ, ) (, ˆ, ) ( ˆ ) ( ˆ,s) ˆ s ds s t dt s s t Ib s t ds I s s t ds sib s s t ds I s 4 4 i si d ids By introducing c = ds/dt one can find 1 I I 1 I sˆ I Ib I c t s c t s I ( ˆ ) ( ˆ, ˆ i i ) di 4 s s s 4

Radiative Transfer Equation, RTE (omitted transient term) di ds sˆ I I I I ( ˆ ) ( ˆ, ˆ) d s s s s b i i i 4 4 β λ = κ λ + σ sλ

Divergence of the Radiative Heat Flux Net heat flux into volume element q q q x y z x y z dxdydz qdv q ( q, q, q ) x y z

Divergence of radiative heat flux I sˆ d 4 I I ( sˆ ) d I ( ˆ ) ( ˆ, ˆ) d d s s s s b i i i 4 4 4 4 4 q 4 I sˆ d 4 ( sˆ, sˆ) d 4 i

Divergence of radiative heat flux with s q 4I I d 4I G b b 4 where G λ is the irradians For a gray medium κ λ = κ q 4 4 4 4 2 4 T Id n T G

The Complete Energy Equation T T T T cp u v w q t x y z q q q kt q conduction radiation radiation qradiation according to previous slide

Repeat and continuation

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

Governing equations in CFD approaches U j t x j x j x j S S 1 0 0 U j h t / Pr / Pr t t P U j 2 U k 2 t ij k ij xi x j xi 3 xk 3 P qr, j t x j Y k / Sc / Sc / t t k / t t P k f C P f C k 1 1 k 2 2 t f C k 2 / k

Radiative Transfer Equation, RTE (omitted transient term) di ds sˆ I I I I ( ˆ ) ( ˆ, ˆ) d s s s s b i i i 4 4 β λ = κ λ + σ sλ

The Complete Energy Equation T T T T cp u v w q t x y z q q q kt q conduction radiation radiation qradiation according to previous slide

Divergence of radiative heat flux with s q 4I I d 4I G b b 4 where G λ is the irradians For a gray medium κ λ = κ q 4 4 4 4 2 4 T Id n T G

Solution methods for the radiative transfer equation, RTE 1) Spherical Harmonics Method - also called the P N approximation method 2) Discrete Ordinates Method (DOM); S N 3) Discrete Transfer Method (DTM, DTRM) 4) Finite Volume Method (FVM) 5) Zonal Method 6) Monte Carlo Method (MC)

Radiation models available in ANSYS FLUENT Discrete Ordinates Model (DOM) Discrete Transfer Radiation Model (DTRM) P 1 Radiation Model Rosseland Model Surface-to-Surface (S2S)

Radiation models general advices for selection Low computational effort with reasonable accuracy: P 1 Accuracy: DTRM and DOM Optical thin media: DTRM and DOM Optical thick media: P 1 Scattering: P 1 and DOM Localized heat sources/sinks: DTRM, DOM with large number of rays or ordinates

Optical Thicknesses Consider a layer of thickness L I λ = I λ,0 e a λlxτl If a λ L << 1 optically thin medium simplification If a λ L >> 1 optically thick medium exchange only between neighboring volume elements If a λ L ~ 1 integro-differential equations to be solved

RTE coupling with Energy Equation RTE 2 4 SR divqr 4n T G Energy equation Species equations NS-equations

Radiative Transfer Equation (RTE) i i i s s b d s s s I I I I s I t I 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 Model If 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 equation: q = k R dt dz k R = 16n2 σt 3 3β R

Differential Approximation (P N ) 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 (P 1, P 3,...). For the P 1 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 P 1 radiation model The directional dependence in the RTE is integrated out RTE becomes easy to solve with limited CPU demand Effects of scattering by particles, droplets and soot can be included Works 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 P 1 radiation model The P 1, 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 1 3β A 1 σ s G = κ(4n 2 σt 4 G)

The P 1 radiation model Boundary conditions for this model are tricky to handle. Commonly Mark s or Marshak s methods are employed.

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 di ds + αi = ασt4 π I s = σt4 π (1 e-αs ) + I 0 e -αs

DTRM The method as it was originally proposed consists of determining the intensity for each of N (N φ x N θ discretized 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 24 atm, 1.704 m path H 2 O 86

Gas radiative properties 24 atm, 1.704 m path H 2 O, CO 2 87

Line overlapping and broadening 24 atm, 1.704 m path H 2 O, CO 2, CO 4,3 µm 4,7 µm 6,3 µm 88

Gas radiative properties 24 atm, 1.704 m path H 2 O, CO 2, CO 89

24 atm, 1.704 m path Importance of CO Position c, 24 atm 90

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 91

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-Sumof-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.

where a ε,i is the emissivity weighting factor for the ith fictitious gray gas, the bracketed quantity is the ith fictitious gray gas emissivity, κ i 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. 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

Spectral-Line-Weighted-sum-ofgray-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.

Correlated k-distribution The rapid oscillating absorption coefficient κ attains the same value many times (at different wavelengths) but resulting in the same intensity and radiative heat flux. In the k-distribution the absorption coefficient is re-ordered resulting in a smooth dependence κ versus λ

Line overlapping and broadening 24 atm, 1.704 m path H 2 O, CO 2, CO 4,3 µm 4,7 µm 6,3 µm 100

RTE, radiative properties: Gas properties LBL NBM EWBM Global models Rayleigh Particle properties (soot) Mie RDG - PFA

RDG-PFA Rayleigh-Debye-Gans Polydisperse Fractal Aggregate approximation

Thermal Properties Radiation Properties (absorption coefficient) Particles Surfaces Spheres Cylinder 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 SiO 2, CaO, Al 2 O 3, MgO, K 2 O and Fe 2 O 3 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 scattering X ~ 1 Mie scatteríng X >> 1 geometric optics applicable

μm Radiation properties of soot Scattering of soot Main difference between particle and gas: Scattering Mie theory Suitable for optical large particles Rayleigh theory Suitable for optical small particles 109/21

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 C f v = a C and a are empirical constants

Results 0 heat source, kw/m 3-50 -100-150 -200-250 1 2L 2H 3L 3H -300 0 1 2 3 4 height, m 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 Concentration 3: 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 Spatial resolution Spectral resolution Scattering medium, 2D Scattering medium, 3D DTM 2 4 4 1 1 DOM 3 4 4 3 3 Spherical harmonics 2 3 4 3 3 Zonal method 2 3 2 2 2 Monte Carlo 4 3 3 4 4 Finite 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

μm Numerical Simulations in a Small Furnace model Furnace Kerosene ejector Swirler Air Inlet Combustor 0.048 m Outlet 0.25 m Radiation heat transfer and radiative properties are based on temperature and species fields Swirler

Volume flow rate of air 31.6 m 3 /h Volume flow rate of kerosene 6.6 litre/h DOM (the so-called S 8 model was used) SNB for CO 2 and H 2 O

Results Flow field - Fluent results a) Temperature b) Pressure (+101325Pa) c) Mole fraction of d) Mole fraction of H 2 O CO 2 The flow in the furnace is very non-homogeneous and complex as it passes the swirler 9/16

Numerical Simulation in a Small Furnace Soot radiation contribution (a) (b) W/m W/m a) With scattering b) Without scattering Participating media are gases H2O, CO2, CO and soot

(a) Radiation heat flux (kw/m 2 ) Numerical Simulation in a Small a) Radiative Heat Flux Furnace Soot radiation contribution 40 35 30 25 20 15 10 5 Two gases Three gases Soot-With scattering Soot-Without scattering 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 z (m) (b) Radiation Contribution (%) 35 30 25 20 15 10 5 Soot-With scattering Soot-Without scattering 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 z (m) b) Radiation Contribution Soot has a big influence on the radiative heat flux Scattering has no influence on the radiative heat flux

Test facility at B & W Völund Esbjerg, Denmark

Pellets in the tests

Control room in the operation

Gas temperature distribution central plane

Furnace wall irrradiation

Case Study for Evaluation of a New Concept of EFGT A cylindrical combustor with the following conditions: Duct side: 3 atm, 923 K and 0.1 kg/s Combustor: 1 atm, 710 K and 0.1 kg/s 1.5 mm 100 mm L= 300 mm 150 mm Insulation Bed, combustion of wood chips Duct Combustor

Porous Duct Results In the similar case with the empty duct, by keeping Dd-Dc = 0.050 m constant, d p = 0.0075 m, ε = 0.426. The pressure drop calculated by Forchheimer extension of the Darcy model which is function of particle diameter and porosity of the porous media. Temperature (K) 1400 To Tw 1300 DP/Pin 1200 1100 1000 900 0.08 0.12 0.16 0.2 D c (m) 0.2 0.16 0.12 0.08 0.04 0 DP/Pin (-) 25 Heat transfer by radiation and convection from the combustor to the porous duct increased up to 16 kw. Heat Transfer (kw) 20 15 10 5 Qrc Qcc Qcd 0 0.08 0.12 0.16 0.2 D c (m)