APPLICATION TO NONGRAY MEDIA
Following from: Discrete ordinates and finite volume methods
The application of the discrete ordinates method (DOM) and the finite volume method (FVM) to nongray gaseous media, with or without particles, is described in this article. The formulation depends on the model used to calculate the radiative properties of the gaseous medium. The radiative properties of molecular gases are a strong function of wave number, while the radiative properties of particles, although also being dependent on the wave number, are much smoother. As a consequence, the particles in gas-particle media are often treated as gray within a band, in contrast to the gases. Then, the absorption coefficient of the particles is added to that of the gaseous mixture.
Line-by-Line method
The most accurate method to calculate the radiative properties of gases is the line-by-line method (Taine, 1983). However, this method requires the solution of the RTE for several hundred thousand wave numbers, followed by integration over the spectrum (Modest, 2003a). Since the spectrum of common gases comprises hundreds of thousand lines, and the number of lines that need to be taken into account greatly increases with the temperature, line-by-line calculations are too computationally demanding for practical applications. Accordingly, this method is mostly used in 1D cases to provide benchmark solutions against which the accuracy of other approximate methods may be evaluated. An example of line-by-line benchmark results for 1D problems obtained using the DOM is found in Denison and Webb (1995a). Radiative calculations of a 2D, axisymmetric, thermal plasma were carried out by Menart (2000) using the DOM and the line-by-line method.
Narrow band Model
Narrow band models (NBMs) (Ludwig et al., 1973) have traditionally been considered as the most accurate ones suitable for practical applications (the more recent k-distribution methods, when applied to narrow bands, provide results of similar accuracy), but they are difficult to apply using the DOM or the FVM. In fact, an NBM gives the spectral transmissivity averaged over a narrow band for a homogeneous and isothermal path length. Various NBMs are available, e.g., the model of Elsasser (1943) and the statistical models of Goody (1952) and Malkmus (1967). The parameters required to compute the mean transmissivity over a narrow band are derived theoretically and/or obtained from spectroscopic databases, e.g., Soufiani and Taine (1997).
An NBM cannot easily be coupled to differential solution methods of the RTE, such as the DOM and the FVM, where knowledge of the spectral absorption coefficient or its average over an interval is required. The absorption coefficient is easily related to the transmissivity on a spectral basis or in the case of a gray medium, but not for a band of finite width, since Beer’s law, given by Eq. (3) below, does not hold for such a band.
Integration of the RTE for an emitting-absorbing medium over a narrow band yields
(1) |
where the subscript ν denotes the wave number, and the overbar represents an average value over a band. It has been assumed that the width of the band is small enough such that the Planck function, which is smooth over the spectrum, can be assumed as constant within the band, i.e., κ_{ν}I_{bν} ≈ κ_{ν}I_{bν}. A similar argument cannot be used to simplify the absorption term of the RTE, since the absorption coefficient, and consequently the radiation intensity, is strongly dependent on the wave number, even over a narrow band of the width typically employed in the NBM. In fact, a few hundred of narrow bands are generally considered in these models, and each one of them may comprise hundreds of spectral lines with significantly different strength. In order to solve the RTE using an NBM along with the DOM or the FVM, it is necessary to write Eq. (1) in terms of the average transmissivity over a narrow band. This is accomplished following the procedure described in Kim et al. (1991), which yields
(2) |
where the spectral transmissivity τ_{ν} is defined as
(3) |
and the subscript w denotes the wall. In the derivation of Eq. (2), besides assuming that the medium does not scatter, it is further assumed that the emissivity of the wall is high enough so that only emission from the wall is considered, and the spectral radiation intensity leaving the boundary is uncorrelated with the spectral transmissivity. The formulation is more involved if this assumption does not hold, as described in Menart et al. (1993).
Equation (2) accounts for the correlation between the spectral absorption coefficient and the radiation intensity when the RTE is averaged over a wave number interval, and is referred to as a correlated formulation. Kim et al. (1991) applied the DOM to solve Eq. (2) in 1D media. The following discretized equation is obtained for the ith control volume, whose grid node is i + 1/2, and cell face indices are i and i + 1:
(4) |
where τ ^{m}_{ν, k → i} denotes the spectral transmissivity averaged over the band under consideration, for direction m, and along the path between cell faces k and i, i.e., between s_{k} = x_{k} / |ξ^{m}| and s_{i} = x_{i} / |ξ^{m}|, where ξ^{m} is the direction cosine of the mth direction.
The extension of this procedure to multidimensional problems is not feasible, unless a noncorrelated formulation is used. The noncorrelated formulation (de Miranda and Sacadura, 1996) relies on the following approximation:
(5) |
This allows Eq. (4) to be written, after some algebra, in the following form:
(6) |
The noncorrelated formulation, although satisfactory for some situations (de Miranda and Sacadura, 1996; Liu et al., 1998b), does not maintain good accuracy in general applications (Zhang et al., 1988; Marakis, 2001).
Kim and Song (1996) applied the weighted-sum-of-gray-gases model, which is addressed in the last section of the present article, to narrow bands of a width typical of an NBM. For a narrow band, the mean spectral emissivity is represented by a weighted sum of gray gas emissivities expressed in terms of absorption coefficients. This allows the solution of the differential form of the RTE for multidimensional problems by methods such as the DOM and the FVM. Although accurate solutions can be obtained using this method, it has not received much attention, since the correlated k-distribution method, also addressed below, has similar advantages and a sounder theoretical basis.
Calculations based on the assumption that the medium is gray within a narrow band have also been reported. The standard form of the RTE (see article “Mathematical formulation”), and therefore also the DOM and FVM, may be applied to every band when this assumption is employed, since it allows the evaluation of a mean absorption coefficient, which is the radiative property that appears in the RTE. The mean absorption coefficient within a narrow band is estimated from Beer’s law based on the global mean beam length (Kim et al., 1991; Liu et al. 1998c) or on the mean beam length for the local control volume under consideration (Liu et al., 1998b,c). It should be realized that the mean absorption coefficient obtained in this way has generally no physical meaning (Taine et al., 1998), as pointed out above. Nevertheless, the results reported by Liu et al. (1998b,c) based on a local mean beam length are comparable to those based on the noncorrelated formulation, but are only in fair agreement with the benchmark results calculated using the correlated formulation.
Wideband Models
Wideband models (WBMs) (Edwards, 1976), rely on the fact that spectral lines of molecular gases are concentrated in a few bands of the spectrum, outside which the gases are transparent to radiation. A WBM yields the total band absorptance for every band. Hence, this model cannot be easily coupled to differential solution methods of the RTE, as also observed above for the NBM. In order to solve the RTE using a WBM along with the DOM or the FVM, the procedure used to obtain Eq. (2) from Eq. (1) is followed, but the integration is performed over the whole spectrum yielding
(7) |
This equation is valid for nonscattering media and black walls. The band absorptance of an isolated band, which is calculated by the WBM, is defined as
(8) |
where ν_{l} and ν_{u} denote the lower and upper limits of the band, respectively. Equation (7) may be rewritten in terms of the band absorptances as follows:
(9) |
where the summation extends over all the bands, and the subscript j stands for the band index. It was assumed that I_{bν} and I_{wν} are essentially constant over each band. Band overlapping may be handled as described in Edwards and Balakrishnan (1973) or Ströhle and Coelho (2002). The discrete form of the DOM equations for 1D media may be written as (Kim et al, 1991)
(10) |
where A^{m}_{j, k → i} is the absorptance of the jth band for the mth direction and along the path between cell faces k and i, i.e., between s_{k} = x_{k} / |ξ^{m}| and s_{i} = x_{i} / |ξ^{m}|. The width of the bands may be relatively large, so that it is more accurate to evaluate the band absorptance at the band center or at the upper or lower limit of the band, depending on how the position of the band head is prescribed (see Edwards, 1976), and to compute the blackbody fraction of radiative energy emitted within the band. As in the case of the NBM, the extension of this procedure to multidimensional problems is not feasible, unless a noncorrelated formulation is used.
The WBM is reasonably accurate and more economical than the NBM if applied to the evaluation of the radiative properties of an isothermal and homogeneous medium. However, the width of the bands in the WBM is a function of the temperature and chemical composition of the medium, and so changes along an optical path in the case of nonisothermal and/or nonhomogeneous media. The computational requirements significantly increase if this variation is taken into account. Moreover, the calculation of the band parameters is more computationally intensive in the WBM than in the NBM, although curve fits may be used to speed up the calculations, as described in Lallemant and Weber (1996).
Band models, including the NBM and the WBM, are difficult to apply to scattering media and/or problems with reflecting walls, and require an additional approximation, e.g., the Curtis-Godson approximation (Young, 1977), in the case of nonisothermal and/or nonhomogeneous media. This approximation is not needed for the noncorrelated formulation, which only requires knowledge of the transmissivity or the band absorptance over a control volume.
Correlated k-Distribution Method
In the correlated k-distribution (CK) method (Goody et al., 1989; Lacis and Oinas, 1991), the spectrum is divided into bands sufficiently narrow such that the Planck function may be assumed constant within every band, as in the NBM. The spectral absorption coefficient κ_{ν} within a band is reordered into a smooth monotonically increasing function. Then, the integration over the wave number of any function dependent on the spectral absorption coefficient is replaced by the integration over the absorption coefficient. The spectral transmissivity of a single absorbing species over a band is averaged as
(11) |
where
is the k-distribution function. The summation extends over all points such that κ_{ν} = k (see, e.g., Modest, 2003a, for details). The cumulative k-distribution function, g(k), is given by
(12) |
The distribution function f(k) is either determined from high-resolution spectroscopic databases or from the inverse Laplace transformation of the mean transmissivity over a narrow band given by a statistical NBM. In the latter case, it is referred to as the statistical narrow band correlated-k (SNBCK) method.
The integral over the cumulative k-distribution function in Eq. (11) is evaluated using a Gaussian type quadrature, yielding
(13) |
where N_{Q} is the number of quadrature points, ω_{j} is the jth quadrature weight, and L is the path length. The absorption coefficient k_{j} corresponding to the jth quadrature point g_{j} is obtained from inversion of the cumulative k-distribution function, i.e., from Eq. (12). This inversion is carried out using a numerical method, e.g., the Newton-Raphson method.
When the CK method is used, the RTE for quadrature point g_{j} in a band of width Δν_{i} is given by
(14) |
where I_{b, Δνi} is the spectral blackbody intensity averaged over the width of the band. The discretization of this equation using the DOM or the FVM closely follows the procedure described in the article “Mathematical formulation,” and the solution algorithm is similar to that described in the article “Solution algorithm.” In the case of the DOM, the total radiation intensity is given by
(15) |
where N_{b} is the total number of bands over which the spectrum is divided, each one with width Δν_{i}. The incident heat flux on a surface whose normal is n H and the incident radiation, G, are determined as follows
(16) |
(17) |
where M is the total number of directions along which the RTE is solved. The FVM yields equations similar to these.
In the case of a mixture of absorbing species, the k-distributions are assumed to be statistically uncorrelated at overlapping bands (Taine and Soufiani, 1999), and Eq. (11) is written as
(18) |
The integrals in these equations are evaluated using a Gaussian-type quadrature. In a mixture of two absorbing species, typically H_{2}O and CO_{2} in reactive flows, Eq. (13) is written as
(19) |
Other treatments of overlapping bands are discussed in Liu et al. (2001) and Shi et al. (2009). The generalization of Eqs. (14)–(17) to media containing a mixture of absorbing species is straightforward. In every band, the RTE needs to be solved for all i and j quadrature points, i.e., for all k_{i} and k_{j} absorption coefficients in Eq. (19). Equations (15)–(17) involve a double summation over the quadrature points. Scattering may also be included (Tang and Brewster, 1994).
In the case of a nonhomogeneous medium, an additional assumption is needed, as in the NBM and WBM. The correlated k-distribution method assumes that the maximum absorption coefficients across the spectrum always occur at the same wave number, and similarly for all intermediate values. In problems with large temperature gradients, the correlated assumption is less accurate, and the the correlated-k fictitious gas model (Rivière et al., 1992) may be used. The previous equations remain valid for the CK method in nonhomogeneous media.
The CK method provides accuracy comparable to that of the NBM when applied to narrow bands, but it is easily coupled with any solution method of the RTE, including the DOM and FVM, in contrast with the NBM. Unfortunately, the CK method is also time consuming, even though lumping bands (Liu et al., 2000b) and optimizing the order of quadrature (Liu et al., 2000a) may significantly reduce the computational requirements with a marginal impact on the solution accuracy.
Most applications of the CK method have been restricted to 1D problems, due to the computational cost. However, some applications to multidimensional problems using the DOM have been reported, e.g., Goutiere et al. (2000, 2002), Coelho (2002), and Joseph et al. (2004, 2009). Radiation from both participating gases and soot is considered in Coelho et al. (2003a), Perez et al. (2005), and Demarco et al. (2011).
The CK method has also been applied to wide bands (WBM-CK), and different approaches have been reported. Marin and Buckius (1996, 1997) calculated cumulative k-distribution functions for each wide band and assumed the blackbody radiation intensity as constant within every band. Simple correlations to calculate the absorption coefficient directly from the cumulative distribution function are given in Marin and Buckius (1998a,b). Lee et al. (1996) and Parthasarathy et al. (1996) developed reordered wave number distribution functions of the absorption coefficient based on continuous correlations for the wideband absorption. Denison and Fiveland (1997) presented a correlation in closed form for the reordered wave number that closely approximates the four-region expression for the wideband absorption. This correlation has been revisited by Runstedtler and Hollands (2002), who developed a new model for the smooth absorption coefficient.
The WBM-CK formulation of Dension and Fiveland (1997) was compared with various computational implementations of the WBM in Ströhle and Coelho (2002). One- and two-dimensional problems were solved using ray tracing for the correlated formulation of the WBM, and using the DOM for the noncorrelated formulation of the WBM and for the WBM-CK. The latter implementation was found to be a good compromise between computational requirements and accuracy for multidimensional problems. Ströhle (2008) provides additional insight into the performance of the reordered wideband model along with the DOM by examining the performance of different formulations, averaging procedures, and scaling methods. Further assessment of the WBM-CK formulation of Dension and Fiveland (1997) is reported in Çayan and Selçuk (2007), who compared its performance against that of a global model, a gray gas approach, and line-line benchmark results, using the method of lines solution of the DOM (see article “Alternative formulations”). Despite the computational savings enabled by the wideband implementation of the CK method in comparison to the CK method, most engineering calculations are currently performed using global models, which are addressed below.
Global Models
Global models treat the entire spectrum directly and may significantly reduce the computational time in radiative transfer calculations, although in most cases their application is limited to problems with gray walls and/or particles. The weighted-sum-of-gray-gases (WSGG) model (Hottel and Sarofim, 1967; Lallemant et al., 1996) is probably the most widely used global model for the calculation of gas radiative properties in combustion systems. The WSGG model expresses the total emissivity of a participating gas as a weighted sum of N_{g} gray gas emissivities as
(20) |
where a_{j} and κ_{j} are the emission weighting factor and the absorption coefficient for the jth gray gas component, respectively. The coefficients a_{j} and κ_{j} are obtained from a fit to total emissivity with the constraint that the sum of coefficients a_{j} is equal to unity (e.g., Taylor and Foster, 1974, Truelove, 1976, and Smith et al., 1982). The transparent regions of the spectrum are accounted for by the term j = 0. The extension to mixtures of gray gases is straightforward. The model can also be applied to mixtures of gray gases and soot (Taylor and Foster, 1975; Truelove, 1976).
It is worth pointing out that the WSGG model has often been used assuming that the medium is gray. In such a case, the absorption coefficient of the medium is calculated from the total gas emissivity using Beer’s law. To compute the local absorption coefficient at a control volume of a computational grid, some authors take the mean effective path length of the control volume under consideration (e.g., Gosman and Lockwood, 1973), and others take the mean beam length for the whole enclosure (e.g., Liu et al., 1998a). Although neither of these options can be considered a satisfactory solution, the first one has the additional problem of yielding grid-dependent results. However, this procedure has no sound theoretical foundation, regardless of the choice of the path length (Lallemant et al., 1996), since Beer’s law only holds for gray media or for a particular wavelength. This arbitrariness does not exist if the WSGG is applied as a nongray gas model, solving the RTE for the N_{g} gray gases plus one clear gas, as outlined below. As a consequence of this, the errors resulting from using the WSGG model along with the assumption of gray medium may be quite large (Liu et al., 1998a; Coelho, 2002), even though they are attenuated for increasing soot load (Bressloff, 1999).
The spectral line–based weighted-sum-of-gray-gases (SLW) model (Denison and Webb, 1993a) is an improved version of the WSGG model. The total emissivity is still given by Eq. (20), but the blackbody weights a_{j} are determined from an absorption-line blackbody distribution function, which represents the fraction of blackbody energy in the portion of the spectrum where the high-resolution spectral absorption cross section of the gas is less than a prescribed value. This function is obtained from the line-by-line spectra of the absorbing species. Simple approximating correlations for CO_{2} and H_{2}O have been developed by Denison and Webb (1993b, 1995a). The absorbing cross-sectional domain is discretized into N_{g} intervals, each one being represented by a single gray gas absorption coefficient. The scaling approximation is used in the case of nonisothermal and/or nonhomogeneous media (Denison and Webb, 1995b), while several approximations, such as double integration, have been proposed for CO_{2}-H_{2}O gas mixtures (Denison and Webb, 1995c). The extension of the SLW model to multicomponent gas mixtures is addressed in Solovjov and Webb (2000), while gas mixtures with soot are considered in Solovjov and Webb (2001).
The absorption distribution function (ADF) model (Rivière et al., 1996) is rather similar to the SLW method, the only difference being in the calculation of the blackbody weights a_{j}. An extension to handle strongly nonisothermal media, referred to as absorption distribution function with fictitious gases (ADFFG) is described in Pierrot et al. (1999b).
When the WSGG, SLW, or ADF methods are used, the RTE for the jth gray gas component may be expressed as (Modest, 1991)
(21) |
The discretization of this equation using the DOM or the FVM is similar to that described for gray media in the article “Mathematical formulation.” The discrete equations are solved for every gray gas and for the clear gas component. The total radiation intensity is calculated from
(22) |
The incident heat flux on a surface whose normal is n and the incident radiation are determined from
(23) |
(24) |
where M is the total number of directions along which the RTE is solved. The FVM yields equations similar to these.
The accuracy of several global models (WSGG, SLW, ADF, and ADFFG) and band models (SNB, CK, and CKFG) has been investigated by Pierrot et al. (1999a) in 1D planar media. Line-by-line results were taken as a reference. It was concluded that the SLW and ADF predictions are typically within 10–20% of the reference line-by-line solution, but care must be taken in the choice of the reference temperature for nonisothermal media. However, only the fictitious gas–based models lead to accurate results for long-range sensing of hot gases. The WSGG and/or the SLW have been widely employed along with the DOM and FVM in multidimensional problems, both for academic and industrial configurations (see, e.g., Liu et al., 1998a; Goutiere et al., 2000, 2002; Coelho, 2002; Coelho et al., 2003b; Çayan and Selçuk, 2007; Selçuk and Doner, 2009). Some of these applications include radiation from nongray gases with soot and other gray particles (Yu et al., 2000, 2001; Baek et al., 2002; Trivic, 2004; Borjini et al., 2007; Byun and Baek, 2007).
The cumulative wave number (CW) model (Solovjov and Webb, 2002, 2010) is closely related to the SLW model. It introduces the cumulative wave number distribution function, which is convenient for the spectral integration and for the treatment of gas mixtures with nongray particles and boundaries. The CW model handles nonisothermal media by means of a local correction factor that is calculated under the assumption of local spectrum correlation. An application of this model to 2D enclosures in the framework of the DOM is presented in Ismail and Salinas (2005).
The full-spectrum k-distribution (FSK) method (Modest and Zhang, 2002; Mazumder and Modest, 2002) is a more general global model, such that the WSGG, SLW, and ADF methods may be regarded as crude implementations of the FSK method. In the FSK method, as in the CK method, the spectral absorption coefficient is reordered into a monotonically increasing function. However, while in the CK method this reordering is performed over a narrow band, within which the Planck function remains approximately constant, in the FSK method, the reordering is performed across the entire spectrum. This implies that the dependence of the Planck function on the wave number needs to be taken into account, which is accomplished by defining a Planck function weighted k-distribution. The full-spectrum cumulative k-distributions are evaluated from line-by-line spectroscopic databases. The FSK method is exact for homogeneous media, apart from the numerical errors in the numerical Gaussian quadrature. In the case of nonhomogenous media, the method requires a correlated or scaled approximation (Modest, 2003b). It has been shown that the scaled approximation performs better for full-spectrum calculations. Several improvements and extensions of the FSK method have been reported, providing accurate solutions for strongly nongray media, and allowing for nongray particles and nongray walls (Zhang and Modest, 2002, 2003; Modest and Riazzi, 2005).
An application of the FSK method to multidimensional enclosures is presented in Porter et al. (2010). A detailed comparison of the WSGG, SLW, and FSK global models, as well as the SNBCK and two crude implementations of the NBM and WBM based on the assumption of gray bands, is reported in Demarco et al. (2011). Both 1D and 2D geometries containing a mixture of H_{2}O, CO_{2}, and soot were considered, and the RTE was solved using the FVM. The FSCK and the SLW methods were found to be the best compromise in terms of accuracy and computational requirements.
The application of the DOM and FVM to nongray media modeled using a global model is similar, regardless of the selected model. Differences lie on how the coefficients in Eq. (20) are evaluated, and on how gaseous mixtures are treated.
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