Application of the full-spectrum k-distribution method to photon Monte Carlo solvers

Abstract

Accurate prediction of radiative heat transfer is key in most high temperature applications, such as combustion devices and fires. Among the various solution methods for the radiative transfer equation (RTE), the photon Monte Carlo (PMC) method is potentially the most accurate and the most versatile. The implementation of a PMC method in multidimensional inhomogeneous problems, however, can be limited by its demand for large computer storage space and its CPU time consumption. This is particularly true if the spectral absorption coefficient is to be accurately represented, due to its irregular behavior. On the other hand, the recently developed full-spectrum k-distribution (FSK) method reorders the irregular absorption coefficient into smooth k-distributions and, therefore, provides an efficient and accurate scheme for the spectral integration of radiative quantities of interest. In this paper the accuracy of the PMC method in solving the RTE and the efficiency and storage advantage provided by the FSK method are combined. The advantages of the proposed PMC/FSK method is described in detail. The accuracy and the efficiency of the method are demonstrated by sample calculations that consider inhomogeneous problems.

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Journal of Quantitative Spectroscopy &
Radiative Transfer 104 (2007) 297–304
Application of the full-spectrum k-distribution method to
photon Monte Carlo solvers
L. Wang, J. Yang, M.F. Modest?, D.C. Haworth
The Pennsylvania State University, Department of Mechanical and Nuclear Engineering, University Park, PA 16802, USA
Received 10 July 2006; accepted 28 July 2006
Abstract
Accurate prediction of radiative heat transfer is key in most high temperature applications, such as combustion devices
and fires. Among the various solution methods for the radiative transfer equation (RTE), the photon Monte Carlo (PMC)
method is potentially the most accurate and the most versatile. The implementation of a PMC method in multidimensional
inhomogeneous problems, however, can be limited by its demand for large computer storage space and its CPU time
consumption. This is particularly true if the spectral absorption coefficient is to be accurately represented, due to its
irregular behavior. On the other hand, the recently developed full-spectrum k-distribution (FSK) method reorders the
irregular absorption coefficient into smooth k-distributions and, therefore, provides an efficient and accurate scheme for
the spectral integration of radiative quantities of interest. In this paper the accuracy of the PMC method in solving the
RTE and the efficiency and storage advantage provided by the FSK method are combined. The advantages of the
proposed PMC/FSK method is described in detail. The accuracy and the efficiency of the method are demonstrated by
sample calculations that consider inhomogeneous problems.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Radiative heat transfer; Inhomogeneous problems; Full-spectrum k-distribution; Combustion; Fire modeling
1. Introduction
Thermal radiation plays an important role in fires and combustion applications due to its usually fourth
power dependence on temperature. In fires, fire growth and spread and containment are strongly influenced by
radiative heat transfer. In combustion applications, flame structure and pollutant emission, such as NOxand
soot, are intimately coupled with radiative heat transfer. The determination of thermal radiation is thus of
great importance in fire and pollutant emission control.
Computational fluid dynamics (CFD) has been commonly used in fire and combustion simulations.
Accurate and efficient modeling of thermal radiation, however, has been an issue in these simulations [1,2].
One of the major difficulties is the solution to the radiative transfer equation (RTE). Various solution methods
[3–5] have been employed and commonly used methods include the spherical harmonics (P1) method (e.g. [6]),
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0022-4073/$-see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jqsrt.2006.07.024
?Corresponding author. Fax: +18148634848.
E-mail address: MFModest@psu.edu (M.F. Modest).

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parameterized narrow-band CK model using more than 1000 pseudo-spectral points. They employed their
PMC methods to model radiation heat transfer as a post-process in their study of a turbulent sooty flame. In
the coupling of their PMC methods with the spectral models, a discrete representation of the spectral variation
was used.
Here we propose a PMC method that is coupled with the state-of-the-art FSK method for spectral
integration of radiative quantities in CFD simulations of fires and flames. The FSK method provides a smooth
and continuous representation of spectral variations. It not only preserves the accuracy of the LBL model but
the discrete ordinates (SN) method (e.g. [7]), and the discrete transfer method (e.g. [8]). These methods
transform the RTE into a set of simultaneous partial differential equations that can be readily solved in a CFD
program. Each of these methods, however, has drawbacks, for example, the discrete ordinates method suffers
from the so-called ‘‘ray effect’’ and ‘‘false scattering’’ [9], and the P1method is not suitable for quantitative
radiation modeling in jet flame configurations [10]. In cases where accurate determination of temperature is
important, for example, in the prediction of NOxemission, the use of these method is not sufficient. In other
cases where turbulence radiation interactions become important, the use of these methods inevitably invokes
the employment of the ‘‘optically thin eddy’’ approximation [3,6,7].
The photon Monte Carlo (PMC), or Monte Carlo ray tracing method, is well suited for radiative transfer
problems since radiative energy travels in discrete photons over relatively long distance along a straight path
before interaction with participating media. It also can be applied to problems of arbitrary difficulty with
relative ease [3]. PMC calculations yield answers with statistical fluctuations about the ‘‘exact’’ answer and, in
general, the results are reliable if a sufficient number of photon bundles are utilized. In the case of turbulent
combustion, PMC methods facilitate the treatment of turbulence-radiation interactions without any
approximation [11]. The disadvantages of the method are that it is computationally expensive and memory
demanding, as well as subject to statistical noise. The computational cost and the memory requirements derive
partly from the fact that the absorption coefficient of participating media (e.g., water and carbon dioxide)
varies irregularly across the spectrum and this variation must be accurately represented in a PMC method for
spectral integration. In addition, tracing a large number of photon bundles in a CFD mesh is also time
consuming. For these reasons the PMC method has seldom been employed in CFD simulations of fires and
flames.
With continuous improvements in numerical methods and computer capabilities, photon tracing in complex
three-dimensional unstructured CFD meshes is no longer a concern. For instance, parallel computing and
multi-grids scheme [12] have greatly reduced the cost of photon tracing. The spectral integration of radiative
properties, on the other hand, is still a time-consuming task. Simplified spectral models, e.g., a gray model,
may be employed but may lead to large errors.
Several methods have been proposed to improve the efficiency of the spectral integration of radiative
quantities. These include the weighted-sum-of-gray-gases (WSGG) model [3], the spectral-line-based
weighted-sum-of-gray-gases (SLW) model [13,14], the absorption distribution function (ADF) method
[15,16], and the recent full-spectrum k-distribution (FSK) method [17]. Whereas the SLW and ADF methods
are weighted-sum-of-gray-gases approaches (i.e., the—assumed to be correlated—absorption coefficient is
reduced to a few discrete values) the FSK method distinguishes itself in that it is an exact method for a
correlated absorption coefficient, utilizing a continuous k-distribution over the whole spectrum, allowing a
quadrature scheme of arbitrary order of accuracy to be employed. Compared to the SLW/ADF/WSGG
models, the FSK method is exact for homogeneous problems. The FSK method achieves line-by-line (LBL)
accuracy for homogeneous media with only a tiny fraction of LBLs computational cost. Since its introduction,
the FSK method has undergone several major developments [18–23], and has become one of the most
promising spectral model for radiative transfer in participating media.
Up to date nobody has attempted a LBL PMC implementation because of the accompanying difficulties
that will be discussed in the following. It appears that the first nongray PMC simulation for molecular
gas radiation with spectral line structure was done by Modest [24] in 1992, who employed a statistical
narrow-band model in the simulation. Recently, the PMC method has been coupled with other spectral
models and used in a couple of CFD simulations of combustion. In his study of buoyant turbulent
nonpremixed flames, Snegirev [25] incorporated a PMC method into a CFD code using the WSGG model for
spectral integrations. Tesse ´ et al. [26,27] proposed two Monte Carlo methods that are coupled with a
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also facilitates the implementation of a PMC method in a CFD code by increasing the efficiency of the PMC
method and greatly decreasing the memory requirements.
2. Photon Monte Carlo method for nongray gases
2.1. Line-by-line considerations
The only difference between the PMC method applied to gray and to nongray media is the need to
determine wavenumbers for every emitted photon bundle according to a representative random number
relationship for the nongray case. The random number relationship to find the spectral location of an emitted
photon is written as [3]
ZZ
Here the spectral variable is wavenumber Z, s is the Stefan–Boltzmann constant, kp is the Planck-mean
absorption coefficient, kZis the spectral absorption coefficient, IbZis the Planck function or spectral blackbody
intensity, and T is the temperature at the point of emission. The quantity RZis a random number and the
upper limit of the integral Z is the wavenumber being sought.
For combustion gases this random number relationship is rather cumbersome to use because of the strongly
irregular behavior of the absorption coefficient kZ. As an example, Fig. 1 shows the spectral variation of the
absorption coefficient kZ for a mixture containing 10% CO2 and 20% H2O at 1500K and 1atm. The
absorption coefficient is obtained from the HITEMP [28] (for H2O) and CDSD [29] (for CO2) spectral
databases. This seemingly random variation raises several issues in implementing the spectral PMC method in
a CFD program considering that the inversion of the random number relation must be carried out many times
for every computational cell/grid and for every time/iteration step during a CFD run.
The first issue is the computer memory requirement. It is impractical to store the absorption coefficient kZ
for every computational cell and it is inefficient to invert the random number relation using the local
absorption coefficient for every cell and every time/iteration step. The issue of storing absorption coefficients
may be managed by pretabulating pressure-based absorption coefficients at specified temperatures and species
concentrations. Linear interpolation may then be used to obtain absorption coefficients at arbitrary
RZ¼
p
kpsT4
0
kZIbZdZ.(1)
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Absorption Coefficient (1/cm)
Wavenumber (1/cm)
36503660 3670 36803690 3700
10-2
10-1
100
10-1
10-2
10-3
10-4
10-5
102
101
100
06000 40002000 800010000
Fig. 1. Absorption coefficient of a mixture of 20% H2O and 20% CO2at 1500K.
L. Wang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 297–304
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10-4
conditions. If pressure-based absorption coefficients are pretabulated for 25 temperatures from 300 to 2700K,
3mol fractions from 0 to 0:5, and 3 species including CO2, H2O, and CO, then somewhat more than 1GB
memory is needed for single precision storage.
For the inversion of the random number relation a table may also be established for the Z vs. RZrelation as
a function of temperature, species, and species mole fraction. Given a random number, linear interpolation
may then be used to obtain the wavenumber according to the local condition. If one total pressure is
considered, then one more gigabyte of memory is required for the pretabulation of the random number
relationship. The total requirement of more than two gigabyte memory is high, since radiation calculations
usually constitute only a small part of an overall CFD simulation of fires and flames.
Another issue of implementing the LBL/PMC method is efficiency. First of all, the data arrays for the
pretabulated absorption coefficients and random number relation are very large and, therefore, the multiple
interpolations needed to find local absorption coefficients and the wavenumbers of emitted photons are
relatively slow. Secondly, the minimum number of photons emitted during one PMC test/trial must be on the
order of several million to adequately resolve the irregular variation of absorption coefficient. Thus, the
computational overhead resulting from multiple interpolations and tracing photons is substantial.
2.2. The PMC method coupled with the FSK method
The various issues that arise in a spectral PMC method can be addressed effectively by the FSK method
without much compromise in accuracy even for inhomogeneous problems. The essence of the FSK method is
a reordering process. It has been observed that the oscillatory absorption coefficient attains the same value k
many times at different wavenumbers even across a very small portion of the spectrum (narrow band), each
time resulting in identical radiative intensity if the RTE is solved for a homogeneous medium. This
observation has led to the reordering of the absorption coefficient across such small wavenumber intervals into
a monotonic absorption coefficient distribution against normalized artificial wavenumber. This is known as
the narrow band k-distribution approach [30,31]. The continuous k-distribution can also be obtained for the
entire spectrum [18], leading to the FSK method and, therefore, facilitating the application of the k-
distribution approach to practical engineering problems.
Fig. 2 shows the full-spectrum k-distribution generated by reordering the absorption coefficient shown in
Fig. 1. Unlike the irregular absorption coefficient in wavenumber space, the k-distribution in g-space, which is
the cumulative distribution function of the absorption coefficient and serves as the Planck-function-weighted
normalized spectral variable, is smooth and monotonically increasing. The tedious spectral integration over
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g
k(1/cm)
a
00.20.40.6 0.81
101
10-1
100
10-2
10-3
10-5
0
1
2
3
4
5
6
7
8
9
10
k-g distribution
Stretching factor a
Fig. 2. k–g distribution and ‘‘a’’ function of the mixture of 20% H2O and 20% CO2at 1500K.
L. Wang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 297–304
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