MATHEMATICAL FORMULATION
Following from: Discrete ordinates and finite volume methods
The radiative transfer equation (RTE) for an emitting-absorbing-scattering medium may be written as follows (Modest, 2003):
(1) |
where subscript ν denotes the wave number. In this equation, I(s) is the radiation intensity in direction s, r is the position vector, I_{b} is the blackbody radiation intensity, κ, β, and σ_{s} are the absorption, extinction, and scattering coefficients of the medium, respectively, and Φ(s′, s) is the scattering phase function. Although the discrete ordinates method (DOM) and the finite volume method (FVM) may be applied to nongray media (see article “Application to nongray media”), the present analysis is restricted to gray media, so that the subscript ν will be dropped from now on. The direction of propagation s does not depend on spatial coordinates. Therefore, Eq. (1) may also be written as
(2) |
The boundary condition for a gray surface that emits and reflects diffusely is given by (Modest, 2003)
(3) |
where I(r_{w}, s) is the radiation intensity leaving the boundary surface at position r_{w}, I(r_{w}, s′) is the radiation intensity arriving at the same position of that surface in the s′ direction, I_{b}(r_{w}) = I_{bw} is the blackbody radiation intensity at the temperature of the boundary surface, ε is the surface emissivity, ρ is the surface reflectivity, and n is the outward (pointing into the medium) unit vector normal to the surface (see Fig. 1). More details about boundary conditions are given in the article “Boundary conditions.”
Figure 1. Radiation leaving a boundary surface.
Equation (2) is independent of the coordinate system. The spatial discretization may be carried out using the FVM, and is valid for both the DOM and the FVM, even though several other spatial discretization methods have been used in the case of the DOM (see article “Alternative formulations”). Hence, integrating both the left-hand-side and the right-hand-side terms over an arbitrary control volume V centered at grid node P (a few typical control volumes are shown in Fig. 2), and assuming that the values on the right side of Eq. (2) remain constant within the control volume, yields
Figure 2. Typical control volumes.
(4) |
Applying the Gauss divergence theorem to the term on the left-hand side, the following equation is obtained:
(5) |
The cell faces are assumed to be flat, so that the outer unit vector normal to a cell face n does not change along the face. Accordingly, Eq. (5) may be written as
(6) |
where subscript f denotes a cell face and F is the total number of cell faces of the control volume under consideration. The mean radiation intensity at cell face f along direction s is denoted by I_{f}(s). The symbol n_{f} represents the outer unit vector normal to cell face f, whose area is A_{f}, and V is the volume of the control volume under consideration. This is a common approach in the FVM, which approximates curved grid lines by piecewise linear segments.
The mean radiation intensity at cell face f needs to be expressed in terms of radiation intensities at grid nodes, so that the number of unknowns is equal to the number of equations, which is equal to the number of control volumes. In this article, for simplicity, we will use only the step scheme. This scheme was often used in the past by the computational fluid dynamics (CFD) community, where it is referred to as the upwind scheme. The step scheme is first-order accurate, and it is well known that it may yield large errors. In particular, it produces the so-called false diffusion (see article “Ray effects and false scattering”). More accurate schemes are available, and should be preferred (see article “Spatial discretization schemes”). In the step scheme, the mean radiation intensity at cell face f is assumed to be equal to the radiation intensity at the grid node in the center of the upstream control volume I_{U,f}, yielding (see Fig. 3)
Figure 3. Cell face intensity calculated according to the step scheme: (a) s · n_{f} > 0; (b) s · n_{f} < 0.
(7a) |
(7b) |
In the case of a cell face coincident with a boundary of the domain, the radiation intensity leaving the boundary may be readily available, as in the case of a black boundary with prescribed temperature, or is determined iteratively from the boundary condition [e.g., Eq. (3)], as addressed in the article “Solution algorithm.”
It is important to realize that spatial discretization schemes employed in the DOM may also be used in the FVM, and vice versa, provided that the FVM is employed for spatial discretization, as in the present article. Moreover, in general, schemes developed for the discretization of the convective term of a transport equation in the CFD community may be used for the discretization of the term on the left-hand side of Eqs. (1) and (2).
Inserting Eq. (7) into Eq. (6) yields
(8) |
The previous equations are valid for both the DOM and the FVM. Now, the angular discretization, which differs in the two methods, will be carried out.
Discrete Ordinates Method
In the DOM, Eq. (8) is replaced by a discrete set of M coupled differential equations that describe the radiation intensity field along M directions. Integrals over solid angles are replaced by a quadrature of order M yielding
(9) |
where superscript m (1 ≤ m ≤ M) denotes the mth direction and w_{l} is the quadrature weight for the lth direction. While the choice of the directions may be arbitrary, a few rules are generally followed. This subject is addressed in the article “Angular discretization methods.” The in-scattering term [last term of Eq. (9)] may be split into two parts, i.e., one that accounts for the contribution of the mth direction, which is treated implicitly, and the other one accounting for all the other directions, which is treated explicitly. This yields the following discretized set of M equations for control volume P:
(10) |
This splitting of the in-scattering term, originally proposed by Chai et al. (1994b) is not strictly needed, but it improves the convergence rate whenever there is a strong contribution of forward scattering.
The boundary condition given by Eq. (3) is approximated as follows:
(11) |
Notice that all previous equations do not depend on the coordinate system employed, and may be readily applied to unstructured grids. Further details on application of the DOM to general body-fitted and unstructured grids may be found in Vaillon et al. (1996), Liu et al. (1997, 2000), Sakami et al. (1998), and Koo et al. (2003). The solution algorithm is described in the article “Solution algorithm.”
The radiative heat flux vector is determined as follows (Modest, 2003):
(12) |
so that the incident heat flux (hemispherical irradiation) on a surface whose outer normal is n is computed as
(13) |
The incident radiation is evaluated as (Modest, 2003)
(14) |
If a Cartesian coordinate system is employed, then the first term on the left side of Eq. (10) may be expressed as
(15) |
where ξ^{m}, η^{m}, and μ^{m} are the direction cosines of the s^{m} direction in the x, y, and z directions, respectively, and A_{x}, A_{y}, and A_{z} are the areas of the faces of the control volume normal to the x, y, and z directions, respectively. Notice that A_{x} is the area of the east face of the control volume if ξ^{m} > 0 (n_{f} = i), and the area of the west face if ξ^{m} < 0 (n_{f} = −i), and similarly for the other directions. In addition, the first term on the right side of Eq. (10) may be written as
(16) |
The subscripts of the cell face intensities represent the direction (x, y, or z) and the upstream (in) face. If ξ^{m} > 0, then I^{m}_{x,in} is the radiation intensity at west cell face, along the mth direction. Conversely, if ξ^{m} < 0, then I^{m}_{x,in} is the radiation intensity at east cell face, along the mth direction. The direction cosines may be expressed in terms of the polar angle θ and the azimuthal angle φ employed in a standard spherical coordinate system (see Fig. 4),
Figure 4. Polar and azimuthal angles.
(17a) |
(17b) |
(17c) |
Inserting Eqs. (15) and (16) into Eq. (10) yields the final form of the discretized DOM equations for Cartesian coordinates using the step scheme (Carlson and Lathrop, 1968; Fiveland, 1984, 1987, 1988; Jamaluddin and Smith, 1988a; Truelove, 1987, 1988),
(18) |
See the article “Solution algorithm” for information on the solution algorithm procedure.
In the case of 2D axisymmetric problems, the discretized DOM equations using the step scheme may be written as follows (Carlson and Lathrop, 1968; Fiveland, 1982; Jamaluddin and Smith, 1988b):
(19) |
where ξ^{m} and μ^{m} are the direction cosines of the axial (z) and radial (r) directions (see Fig. 5, where e_{z} and e_{r} are the unit vectors along the axial direction and the local radial direction, respectively). The indices in (out) denote a cell face where radiation propagates into (out of) the control volume. The directions m ± 1/2 define the edges of the azimuthal angle Δψ associated with the mth direction, and the superscript m − 1 denotes the direction defined by θ^{m−1} = θ^{m} and ψ^{m−1} = ψ^{m} − Δψ. The geometrical coefficients α satisfy the following relation, drawn on the basis of isotropic radiation (Carlson and Lathrop, 1968; Fiveland, 1982; Jamaluddin and Smith, 1988b):
Figure 5. Polar and azimuthal angles for cylindrical coordinates.
(20) |
The coefficient α_{1/2} is equal to zero, and the coefficients α_{m+1/2} for the other directions are determined recursively from Eq. (20). The direction cosines are related to the polar angle θ measured between the s^{m} direction and the axial direction, and the azimuthal angle ψ measured on a plane normal to the axial direction from the local radial direction (see Fig. 5), as follows:
(21a) |
(21b) |
(21c) |
Additional information on the application of the DOM to axisymmetric geometries may be found in Jendoubi et al. (1993), Baek and Kim (1997), Kim and Baek (1998), and Liu et al. (2000), while Jamaluddin and Smith (1992) applied the DOM to nonaxisymmetric cylindrical enclosures.
Finite Volume Method
In the FVM, Eq. (8) is integrated over a solid angle ΔΩ^{m}, also referred to as the control angle, resultant from the discretization of the 4π solid angle. It is assumed that the value of the radiation intensity remains constant within that control volume, as in the DOM. However, in contrast to the DOM, the direction of the radiation intensity is allowed to vary within a solid angle. This yields
(22) |
The last term of Eq. (22) may be evaluated as follows:
(23) |
Let us define the following quantities:
(24a) |
(24b) |
(24c) |
If the in-scattering term [last term of Eq. (22)] is split again into two parts, as in the derivation of the DOM equations, and if Eqs. (23) and (24) are inserted into Eq. (22), then we obtain the following discretized set of M equations for control volume P using the step scheme:
(25) |
The boundary condition given by Eq. (3) is approximated as follows:
(26) |
The previous equations for the FVM are independent of the coordinate system employed, and may be readily applied to unstructured grids. Further details on application of the FVM to general body-fitted and unstructured grids may be found in Raithby and Chui (1990), Chui and Raithby (1993), Chai et al. (1995), Baek et al. (1998), Murthy and Mathur (1998a), Moder et al. (2000), and Kim et al. (2008). The solution algorithms of the DOM and FVM are identical (see the article “Solution algorithm”).
The radiative heat flux vector is determined as follows (Modest, 2003):
(27) |
so that the incident heat flux (hemispherical irradiation) on a surface whose outer normal is n is computed as
(28) |
The incident radiation is evaluated as (Modest, 2003)
(29) |
If a Cartesian coordinate system is employed, then the first term on the left side of Eq. (25) may be expressed as
(30) |
where the integrals involved in the calculation of D^{m}_{cx}, D^{m}_{cy}, and D^{m}_{cz} may be determined analytically. The first term on the right-hand side of Eq. (25) may be written as
(31) |
Inserting Eqs. (30) and (31) into Eq. (25) yields the final form of the discretized FVM equations for Cartesian coordinates using the step scheme (Chai and Patankar, 2000; Chai et al., 1994a),
(32) |
In the case of 2D axisymmetric problems, the discretized FVM equations using the step scheme may be written as follows (Ben Salah et al., 2004):
(33) |
where D^{m}_{cz} and D^{m}_{cr} are defined according to Eq. (24a), taking the normal as the unit vector along the axial (z) and radial (r) directions, respectively. The indices in (out) denote a cell face where radiation propagates into (out of) the control volume. The directions m ± 1/2 define the edges of the azimuthal angle Δψ associated with the mth direction. The geometrical coefficients α_{m ± 1/2} satisfy the following relation, drawn on the basis of isotropic radiation (Ben Salah et al., 2004):
(34) |
The coefficient α_{1/2} is equal to zero, and the coefficients α_{m±1/2} for the other directions are determined recursively from Eq. (34). Equation (21), which relates the direction cosines to the polar angle θ, measured between the s^{m} direction and the axial direction, and the azimuthal angle ψ, measured on a plane normal to the axial direction from the local radial direction, remain valid.
Other applications of the FVM to axisymmetric geometries, which include different treatments of the angular distribution term of the RTE in cylindrical coordinates, may be found in Chui et al. (1992), Kim and Baek (1997, 2005), Murthy and Mathur (1998b), Tian and Chiu (2005), and Kim (2008), while Moder et al. (1996) applied the FVM to nonaxisymmetric cylindrical enclosures.
Comparison between the DOM and the FVM
The difference between the DOM and the FVM lies in the angular discretization procedure. The DOM approximates the angular dependence of the radiation intensity field by a set of radiation intensities along a finite prescribed set of directions, and assigns a quadrature weight to every direction. In contrast, the FVM performs a discretization of the solid angle of 4π corresponding to a sphere, and integrates the RTE over every discrete solid angle, assuming that the value of the radiation intensity is constant within a discrete solid angle, but allowing the direction to vary. Different spatial discretization methods and schemes for the evaluation of cell face radiation intensities may be used in the DOM and FVM, but they are not a characteristic of the methods.
A comparison between the discrete DOM and FVM equations, (10) and (25), shows that s^{m} · n_{f}, w^{m}, and Φ(s^{l}, s^{m}) in the DOM are replaced by D_{cf}^{m} ⁄ ΔΩ^{m}, ΔΩ^{m}, and Φ^{lm} in the FVM, respectively. All these changes are a consequence of the above-mentioned difference in the angular discretization procedure.
It is possible to use the same angular discretization in the DOM and in the FVM, e.g., dividing the angular space into discrete solid angles defined by lines of constant longitude and latitude, and taking the quadrature weights in the DOM as the values of the solid angles. If this choice is made, then w^{m} = ΔΩ^{m}. However, s^{m} · n_{f} and Φ(s^{l}, s^{m}) remain different from D_{cf}^{m} ⁄ ΔΩ^{m} and Φ^{lm}, respectively, because the direction s in Eq. (24a) and the directions s and s′ in Eq. (24c) depend on the polar and azimuthal angles.
As a consequence of the difference between the DOM and FVM highlighted above, the DOM may not strictly conserve energy in the case of anisotropic scattering and/or boundaries not aligned with the coordinate axes, while the FVM is fully conservative. If the phase function is analytically available, then the FVM ensures an exact evaluation of the average phase function Φ^{lm}. Even if the analytical evaluation of Φ^{lm} is not feasible, an accurate numerical integration may be carried out by subdividing the solid angles into smaller solid angles, as described in Chai and Patankar (2000). In contrast, the DOM does not guarantee energy conservation in the case of anisotropic scattering, i.e., there is no guarantee that the integral of the phase function over the sphere yields 4π, even though it is possible to use a correction factor of the in-scattering term to enforce conservation (Liu et al., 2002). In the case of boundaries not aligned with coordinate axes, the DOM does not guarantee exact evaluation of the incident/leaving heat flux in the case of isotropic radiation, while the FVM does. These issues are addressed in Raithby (1999), who presented an excellent discussion and comparison of the DOM and FVM.
Comparative calculations using the DOM and FVM are presented in Coelho et al. (1998), Kim and Huh (1999), and Boulet et al. (2007). Because of the similarity of the DOM and FVM methods, it is expected that they yield solutions of similar accuracy and with identical computational requirements for most problems, provided that the same spatial discretization method and schemes are employed along with a similar number of directions (DOM) and solid angles (FVM). However, in the case of strongly anisotropic media and/or complex geometries, the FVM has some advantages, as pointed out above.
REFERENCES
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Ben Salah, M., Askri, F., Slimi, K., and Ben Nasrallah, S., Numerical Solution of the Radiative Transfer Equation in a Cylindrical Enclosure with the Finite-Volume Method, Int. J. Heat Mass Transfer, vol. 47, pp. 2501−2509, 2004.
Boulet, P., Collin A., and Consalvi, J. L., On the Finite Volume Method and the Discrete Ordinates Method Regarding Radiative Heat Transfer in Acute Forward Anisotropic Scattering Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 104, no. 3, pp. 460−473, 2007.
Carlson, B. G. and Lathrop, K. D., Transport Theory–The Method of Discrete Ordinates, Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber and D. Okrent, Eds., New York: Gordon & Breach, 1968.
Chai, J. C., Lee, H. S., and Patankar, S. V., Finite Volume Method for Radiation Heat Transfer, J. Thermophys. Heat Transfer, vol. 8, no. 3, pp. 419−425, 1994a.
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Chai, J. C., Patankar, S. V., and Lee, H. S., Improved Treatment of Scattering Using the Discrete Ordinates Method, J. Heat Transfer, vol. 116, no. 1, pp. 260−263, 1994b.
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Chui, E. H., Raithby, G. D., and Hughes, P. M. J., Prediction of Radiative Transfer in Cylindrical Enclosures with the Finite Volume Method, J. Thermophys. Heat Transfer, vol. 6, no. 4, pp. 605−611, 1992.
Chui, E. H. and Raithby, G. D., Computation of Radiant Heat Transfer on a Nonorthogomal Mesh Using the Finite-Volume Method, Numer. Heat Transfer B, vol. 23, pp. 269−288, 1993.
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Использованная литература
- Baek, S. W. and Kim, M. Y., Modification of the Discrete-Ordinates Method in an Axisymmetric Cylindrical Geometry, Numer. Heat Transfer B, vol. 31, pp. 313−326, 1997.
- Baek, S. W., Kim, M. Y., and Kim, J. S., Nonorthogonal Finite-Volume Solutions of Radiative Heat Transfer in a Three-Dimensional Enclosure, Numer. Heat Transfer B, vol. 34, pp. 419−437, 1998.
- Ben Salah, M., Askri, F., Slimi, K., and Ben Nasrallah, S., Numerical Solution of the Radiative Transfer Equation in a Cylindrical Enclosure with the Finite-Volume Method, Int. J. Heat Mass Transfer, vol. 47, pp. 2501−2509, 2004.
- Boulet, P., Collin A., and Consalvi, J. L., On the Finite Volume Method and the Discrete Ordinates Method Regarding Radiative Heat Transfer in Acute Forward Anisotropic Scattering Media, J. Quant. Spectrosc. Radiat. Transfer, vol. 104, no. 3, pp. 460−473, 2007.
- Carlson, B. G. and Lathrop, K. D., Transport Theory–The Method of Discrete Ordinates, Computing Methods in Reactor Physics, H. Greenspan, C. N. Kelber and D. Okrent, Eds., New York: Gordon & Breach, 1968.
- Chai, J. C., Lee, H. S., and Patankar, S. V., Finite Volume Method for Radiation Heat Transfer, J. Thermophys. Heat Transfer, vol. 8, no. 3, pp. 419−425, 1994a.
- Chai, J. C., Parthasarathy, G., Lee, H. S., and Patankar, S. V., Finite Volume Radiative Heat Transfer Procedure for Irregular Geometries, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 410−415, 1995.
- Chai, J. C., Patankar, S. V., and Lee, H. S., Improved Treatment of Scattering Using the Discrete Ordinates Method, J. Heat Transfer, vol. 116, no. 1, pp. 260−263, 1994b.
- Chai, J. C. and Patankar, S. V., Finite-Volume Method for Radiation Heat Transfer, Advances in Numerical Heat Transfer, vol. 2, W. J. Minkowycz and E. M. Sparrow, Eds., New York: Taylor and Francis, pp. 109−141, 2000.
- Chui, E. H., Raithby, G. D., and Hughes, P. M. J., Prediction of Radiative Transfer in Cylindrical Enclosures with the Finite Volume Method, J. Thermophys. Heat Transfer, vol. 6, no. 4, pp. 605−611, 1992.
- Chui, E. H. and Raithby, G. D., Computation of Radiant Heat Transfer on a Nonorthogomal Mesh Using the Finite-Volume Method, Numer. Heat Transfer B, vol. 23, pp. 269−288, 1993.
- Coelho, P. J., GonÃ§alves, J. M., Carvalho, M. G., and Trivic, D., Modelling of Radiative Heat Transfer in Enclosures with Obstacles, Int. J. Heat Mass Transfer, vol. 41, nos. 4−5, pp. 745−156, 1998.
- Fiveland, W. A., A Discrete Ordinates Method for Predicting Radiative Heat Transfer in Axisymmetric Enclosures, ASME Paper No. 82−HT−20, 1982.
- Fiveland, W. A., Discrete-Ordinates Solutions of the Radiative Transport Equation for Rectangular Enclosures, J. Heat Transfer, vol. 106, pp. 699−706, 1984.
- Fiveland, W. A., Discrete Ordinate Methods for Radiative Heat Transfer in Isotropically and Anisotropically Scattering Media, J. Heat Transfer, vol. 109, pp. 809−812, 1987.
- Fiveland, W. A., Three-Dimensional Radiative Heat-Transfer Solutions by the Discrete-Ordinates Method, J. Thermophysics Heat Transfer, vol. 2, no. 4, pp. 309−316, 1988.
- Jamaluddin, A. S. and Smith, P. J., Predicting Radiative Transfer in Rectangular Enclosures Using the Discrete Ordinates Method, Combust. Sci. Technol., vol. 59, pp. 321−340, 1988a.
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