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ABSTRACT
In this study, a radiation code based on the method of lines solution of the discrete ordinates method for the prediction of radiative heat transfer in nongray gaseous media is developed by incorporation of two different spectral gas radiative property models, banded spectral line-based weighted sum of gray gases (banded SLW) and gray wide band (GWB) approximation in the presence of nongray absorbing–emitting–scattering particles. The aim is to introduce an accurate and CPU efficient spectral gas radiation model, which is compatible with spectral fuel/ash particle property models. Input data required for the radiation code and its validation are provided from two combustion tests previously performed in a 300 kWt atmospheric bubbling fluidized bed combustor test rig burning low calorific value Turkish lignite with high volatile matter/fixed carbon (VM/FC) ratio in its own ash. The agreement between wall heat fluxes and source term predictions obtained by global and banded SLW models reveal that global SLW model can be converted to an accurate wide band gas model (banded SLW) which can directly be coupled with spectral particle radiation. Furthermore, assessment of GWB approximation by benchmarking its predictions against banded SLW model shows that GWB gives reasonable agreement with a higher CPU efficiency when the particle absorption coefficient is at least in the same order of magnitude with the gas absorption coefficient.
Nomenclature
| a | = | gray gas weight |
| B | = | particle load (kg m−3) |
| c | = | speed of light (m s−1) |
| Cabs | = | absorption cross section (m2 mol−1) |
| = | supplemental absorption cross section | |
| dp | = | particle diameter (m) |
| F | = | conventional blackbody fractional function |
| g | = | asymmetry factor |
| gc | = | k-distribution function for CO2 |
| gw | = | k-distribution function for H2O |
| I | = | radiative intensity (W m−2 sr−1) |
| Ib | = | blackbody intensity (W m−2 sr−1) |
| k | = | imaginary part of complex refractive index |
| kt | = | time constant with dimension (m−1 s−1) |
| Lm | = | mean beam length (m) |
| m | = | complex refractive index |
| n | = | real part of complex refractive index |
| N | = | molar density (mol m−3) |
| NB | = | number of wide bands in the solution of RTE |
| NGC | = | number of gray gases for CO2 |
| NGW | = | number of gray gases for H2O |
| Nt | = | number of discrete sizes in particle size measurement |
| P | = | pressure (Pa) |
| Qabs | = | absorption efficiency |
| Qscat | = | scattering efficiency |
| r | = | position vector |
| T | = | temperature (K) |
| t | = | time |
| w | = | quadrature weight |
| wi | = | differential weight of particle size i |
| α | = | absorptivity |
| γ | = | angular differencing coefficient |
| ε | = | emissivity |
| = | relative error tolerance | |
| η | = | direction cosine |
| Θ | = | scattering angle (rad) |
| κ | = | absorption coefficient (m−1) |
| λ | = | wavelength (µm) |
| µ | = | direction cosine |
| ξ | = | direction cosine |
| ρp | = | particle density (kg m−3) |
| σ | = | scattering coefficient (m−1) |
| τ | = | transmissivity |
| ϕ | = | azimuthal angle (rad) |
| Φ | = | scattering phase function (sr−1) |
| Ω | = | direction of radiation intensity |
| Subscripts | ||
| b | = | blackbody |
| g | = | gas |
| i | = | quadrature point |
| j | = | spectral wide band number |
| k | = | gray gas number index for H2O |
| l | = | gray gas number index for CO2 |
| ℓ | = | index for a discrete direction |
| = | incoming discrete direction | |
| m | = | ordinate index |
| m′ | = | incoming ordinate |
| p | = | particle |
| tr | = | transport approximation |
| w | = | wall |
| λ | = | wavelength (µm) |
| ν | = | wavenumber (cm−1) |
| Superscripts | ||
| m | = | ordinate index |
| m′ | = | incoming ordinate |
| ℓ | = | index for a discrete direction |
| ℓ′ | = | incoming discrete direction |
Nomenclature
| a | = | gray gas weight |
| B | = | particle load (kg m−3) |
| c | = | speed of light (m s−1) |
| Cabs | = | absorption cross section (m2 mol−1) |
| = | supplemental absorption cross section | |
| dp | = | particle diameter (m) |
| F | = | conventional blackbody fractional function |
| g | = | asymmetry factor |
| gc | = | k-distribution function for CO2 |
| gw | = | k-distribution function for H2O |
| I | = | radiative intensity (W m−2 sr−1) |
| Ib | = | blackbody intensity (W m−2 sr−1) |
| k | = | imaginary part of complex refractive index |
| kt | = | time constant with dimension (m−1 s−1) |
| Lm | = | mean beam length (m) |
| m | = | complex refractive index |
| n | = | real part of complex refractive index |
| N | = | molar density (mol m−3) |
| NB | = | number of wide bands in the solution of RTE |
| NGC | = | number of gray gases for CO2 |
| NGW | = | number of gray gases for H2O |
| Nt | = | number of discrete sizes in particle size measurement |
| P | = | pressure (Pa) |
| Qabs | = | absorption efficiency |
| Qscat | = | scattering efficiency |
| r | = | position vector |
| T | = | temperature (K) |
| t | = | time |
| w | = | quadrature weight |
| wi | = | differential weight of particle size i |
| α | = | absorptivity |
| γ | = | angular differencing coefficient |
| ε | = | emissivity |
| = | relative error tolerance | |
| η | = | direction cosine |
| Θ | = | scattering angle (rad) |
| κ | = | absorption coefficient (m−1) |
| λ | = | wavelength (µm) |
| µ | = | direction cosine |
| ξ | = | direction cosine |
| ρp | = | particle density (kg m−3) |
| σ | = | scattering coefficient (m−1) |
| τ | = | transmissivity |
| ϕ | = | azimuthal angle (rad) |
| Φ | = | scattering phase function (sr−1) |
| Ω | = | direction of radiation intensity |
| Subscripts | ||
| b | = | blackbody |
| g | = | gas |
| i | = | quadrature point |
| j | = | spectral wide band number |
| k | = | gray gas number index for H2O |
| l | = | gray gas number index for CO2 |
| ℓ | = | index for a discrete direction |
| = | incoming discrete direction | |
| m | = | ordinate index |
| m′ | = | incoming ordinate |
| p | = | particle |
| tr | = | transport approximation |
| w | = | wall |
| λ | = | wavelength (µm) |
| ν | = | wavenumber (cm−1) |
| Superscripts | ||
| m | = | ordinate index |
| m′ | = | incoming ordinate |
| ℓ | = | index for a discrete direction |
| ℓ′ | = | incoming discrete direction |
Notes
1With the PSD given in , these particle loads lead to a total geometric cross sectional area (S = πr2) of 0.77 and 5.46 m2/m3 for the particle cloud.
2Spectral bands used in banded SLW models covers the same spectral ranges as in 8 Banded GWB shown in .