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Assessment of an unstructured exponential scheme discrete ordinates radiation model for non-gray media

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Abstract

For radiative transfer in complex geometries, Sakami and his co-workers have developed a discrete ordinates method (DOM) exponential scheme for unstructured meshes which was mainly applied to gray media. The present study investigates the application of the unstructured exponential scheme to a wider range of non-gray scenarios found in fire and combustion applications, with the goal to implement it in an in-house Computational Fluid Dynamics (CFD) code for fire simulations. The original unstructured gray exponential scheme is adapted to non-gray applications by employing a statistical narrow-band/correlated-k (SNB-CK) gas model and meshes generated using the authors’ own mesh generator. Different non-gray scenarios involving spectral gas absorption by H2O and CO2 are investigated and a comparative analysis is carried out between heat flux and radiative source terms predicted and literature data based on ray-tracing and Monte Carlo methods. The maximum discrepancies for total radiative heat flux do not typically exceed 5%.

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Abbreviations

C :

Gas concentration

f(k):

k-distribution function in correlated-k (CK) gas model

g j :

jth quadrature point in CK gas model

g(k):

Cumulative distribution function in CK gas model

I :

Radiation intensity (W m−2 sr−1)

\( I_{km}^{i} \) :

Mean lateral intensity of side k bounding cell i for direction Ωm (W m−2 sr−1)

\( I_{m}^{i} \) :

Mean lateral intensity of cell i for direction Ωm (W m−2 sr−1)

k :

Absorption coefficient (m−1)

L :

Thickness of the medium (m)

\( l_{k}^{i} \) :

Length of side k bounding cell i (m)

M:

Number of discrete directions Ωm

m :

Relative to a discrete direction Ωm

n :

Unit normal vector of a surface

q :

Radiative heat flux (W m−2)

\( S_{k}^{i} \) :

Area of the side k bounding cell i (m2)

s :

Distance on the path of a ray, si and sf for the beginning and end of the path, or coordinate along direction Ω (m)

T :

Temperature (K)

t :

s i  − s f (m)

w :

Weight

X:

Molar fraction

x :

Coordinate (m)

y :

Coordinate (m)

z :

Coordinate (m)

α, γ:

Angle relative to an elementary cell, defined by Fig. 1 (rad)

β :

Extinction coefficient (m−1)

∆ν:

Wavenumber range (cm−1)

ε :

Wall emissivity

μ, ξ, η :

Direction cosines

v :

Wavenumber (cm−1)

ρ :

Medium density (kg m−3)

σ :

Scattering coefficient (m−1)

τ :

Optical thickness

Φ:

Scattering phase function

\( \varphi_{km}^{i} \) :

\( \Upomega_{m} \cdot n_{k}^{i} \)

Ψ:

Element surface (m2)

Ω:

Solid angle (sr)

ω:

Scattering albedo

b :

Black body

g :

Gas or medium

i :

Cell i

inco :

Incoming

k :

Related to side k of cell

leav :

Leaving

m :

Outgoing ordinate direction

m’:

Incoming ordinate direction

s :

Surface

x :

Coordinate direction

y :

Coordinate direction

z :

Coordinate direction

λ :

Spectral

’:

Incoming direction

i :

Relative to cell i

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Authors and Affiliations

  1. Faculty of Engineering, Kingston University, Roehampton Vale Campus, Friars Avenue, London, SW15 3DW, UK

    S. Dembele, K. L. M. Lima & J. X. Wen

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  1. S. Dembele
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  2. K. L. M. Lima
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  3. J. X. Wen
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Correspondence to S. Dembele.

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Dembele, S., Lima, K.L.M. & Wen, J.X. Assessment of an unstructured exponential scheme discrete ordinates radiation model for non-gray media. Heat Mass Transfer 47, 1349–1362 (2011). https://doi.org/10.1007/s00231-011-0805-9

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