A Novel Fast Wavenumber Selection Scheme for Line-by-Line Monte Carlo Simulations of High-Temperature Gases

Alinejad, Farid; Bordbar, Hadi; Hostikka, Simo

doi:10.1115/1.4066172

Abstract

Implementing line-by-line absorption spectra in the Monte Carlo method provides benchmark solutions for radiative heat transfer in participating media. It is, however, a computationally demanding calculation, and therefore its application is limited to small and medium-scale cases. In gas mixtures, one of the most time-consuming parts is the wavenumber selection for each emission bundle (also known as photon). Due to interdependency of spectral emission of individual species, trial-and-error is needed to obtain the emission wavenumber of species. Doing trial-and-error with tight convergence threshold through large set of data increases the simulation time. This paper presents a novel scheme to select the wavenumber by adding a new dataset which circumvents the need for trial-and-error. The performance of the new scheme is exhibited in four different test cases containing CO₂, H₂O, and CO. An excellent agreement was observed between the results of the new and old schemes. Compared to the previously published hybrid selection scheme, the wavenumber selection is around seven times faster using the new scheme.

1 Introduction

Utilizing a substantial number of photons and achieving fine spectral resolution, line-by-line Monte Carlo (LBL-MC) calculations have emerged as a powerful tool for accurately solving thermal radiation transfer in nonhomogeneous participating media. LBL-MC results are commonly employed as a benchmark against which the performance of other simplified thermal radiation models is measured. With advancements in thermal radiation research and the continuous growth in computational power, the application of LBL-MC calculations in engineering simulations is becoming increasingly feasible. Ren et al. [1] employed LBL-MC to simulate radiation heat transfer within high-pressure industrial gas turbines. Wu et al. [2] utilized LBL-MC to provide detailed modeling of small-scale heptane pool fires, Wu and Zhao [3] applied LBL-MC in simulating laminar nonpremixed methane/air flames, and Paul et al. [4] provided benchmark radiation modeling data for nonsooting turbulent methanol flame and sooting ethylene flame. These examples underscore the growing application of LBL-MC in addressing engineering challenges.

Utilizing the LBL-MC method for radiation heat transfer applications entails the sampling of various random numbers to determine the emission location, direction, absorption, scattering, and reflection of photon bundles, as detailed in Ref. [5]. Within the LBL-MC framework, one of the most time-intensive processes is the selection of the emission wavenumber. This challenge arises due to the pronounced fluctuations in the spectral absorption coefficients of gas components like H₂O, CO₂, and CO within narrow wavenumber intervals. Consequently, a precise approach is essential when choosing the emission wavenumber. To be accurate in selecting the emission wavenumber in LBL-MC method, it is crucial to maintain a fine resolution when creating a database that correlates random numbers (⁠ $R_{η}$ ⁠) with wavenumbers (⁠ $η$ ⁠).

The LBL-MC method was initially formulated by Farmer and Howell [6,7] for a scattering media with a single gas species (i.e., CO₂). They established the $R_{η} - η$ relation which provides $R_{η}$ values corresponding to different $η$ at a given state. They created a tabulated dataset for $R_{η} (η)$ ⁠, which was then inverted to obtain $η (R_{η})$ ⁠. This inversion process was straightforward for a single gas species. However, when dealing with gas mixtures encompassing numerous potential states, generating tabulated data for $η (R_{η})$ became a laborious task. To address this challenge, Wang and Modest [8] introduced an equation that establishes the connection between $R_{η}$ (random number for a gas mixture) and $R_{η, i}$ (random number for gas species $i$ ⁠). This innovative approach only required the creation of tabulated data for $R_{η, i} (η)$ for each individual species. Subsequently, through a trial-and-error procedure, the appropriate value of $η$ corresponding to the sampled $R_{η}$ was determined. When dealing with a medium containing three gas species, this trial-and-error process needed to be conducted among six distinct databases [8], leading to an inefficient computational time.

Ozawa et al. [9], in their study on hypersonic nonequilibrium radiation, made a crucial observation that allowed for the consideration of emissions from different gas species separately within a mixture. Following this discovery, the methodology involved first selecting the emitting species and then determining the emission wavenumber using tabulated data of $η (R_{η, i})$ ⁠. In their approach, wavenumber determination was achieved through a straightforward interpolation, resulting in improved computational efficiency. This approach was further refined by Feldick and Modest [10], who introduced an enhanced databasing scheme designed to expedite the wavenumber determination process.

Ren and Modest [11] found that utilizing interpolation based on $R_{η, i}$ does not yield an accurate solution for gas mixtures, necessitating the use of a trial-and-error procedure for determining $η$ ⁠. To address this, they adopted a hybrid method, referred to here as the “old scheme,” which combined the species emission separation technique proposed by Ozawa et al. [9] and the highly accurate $η$ determination method developed by Wang and Modest [8]. The old scheme not only significantly reduced computational time compared to Wang and Modest's approach [8] but also restricted the trial-and-error process to a single species, preserving accuracy. In the context of selecting wavenumbers for 10⁶ photon bundles, the hybrid scheme boasted a speed improvement of approximately tenfold compared to the methods presented by Wang and Modest [8]. To the best of our knowledge, the hybrid scheme represents the most up-to-date wavenumber selection strategy, widely adopted in various Monte Carlo simulations [2,12].

The primary objective of this study is to enhance the computational efficiency of LBL-MC by introducing a novel wavenumber selection scheme. While the hybrid scheme has already improved wavenumber selection for LBL-MC, it relies on a trial-and-error approach for selecting the emission wavenumber, which can be time-consuming. In this research, we propose a new scheme by adding $η - R_{η}$ dataset for wavenumber selection in LBL-MC, eliminating the need for time-consuming trial-and-error. To test the accuracy of the new scheme, its results will be compared to the old scheme. To confirm the correct LBL-MC method implementation, we included the results of analytical (for constant temperature media) and LBL methods. To calculate the LBL values, we employed a finite volume method utilizing eight segments for polar angles and 16 segments for azimuthal angles within a hemisphere.

2 Wavenumber Selection Scheme

In the absence of scattering media, the radiative transfer equation can be expressed as

\frac{d I_{η}}{d s} = κ_{η} I_{b η} - κ_{η} I_{η}

(1)

where $I_{η}$ ⁠, $s$ ⁠, $κ_{η}$ ⁠, and $I_{b η}$ are radiation intensity, distance along the path, absorption coefficient of the gas mixture, and blackbody radiation intensity, respectively. The absorption coefficient of a gas mixture is determined by aggregating the absorption coefficients of its individual components $κ_{η} = \sum_{i = 1}^{n_{s}} κ_{η, i}$ ⁠. The determination of $κ_{η}$ in Eq. (1) necessitates the selection of a wavenumber for the photon bundles. In this section, we will first review the old scheme described in Ref. [11] and then present an improved approach that builds upon the foundations of the old scheme.

2.1 The Old Scheme.

The old scheme by Ren and Modest [11] relies on assuming the independence of emissions from different species. In this method, the emitting species is initially chosen, followed by the determination of the wavenumber using only two datasets. The emission of each species

i

⁠,

E_{i}

⁠, is

E_{i} = \int_{0}^{\infty} κ_{η, i} I_{b η} d η

(2)

A concise overview of this scheme is provided below. In accordance with this approach, the relationship between

R_{η}

and

η

is as follows:

R_{η} = \frac{\sum_{i = 1}^{j - 1} E_{i}}{E_{tot}} + \frac{\int_{0}^{η} κ_{η, s} I_{b η} d η}{E_{tot}}

(3)

where

j

and

E_{tot}

are the number index of the emitting species and total emission of all species. The total emission is

E_{tot} = \sum_{i = 1}^{n_{s}} E_{i}

⁠, where

n_{s}

is the number of species in gas mixture. By partitioning the

R_{η}

interval (i.e., 0 to 1) using the emission share of each species at a given state, the emitting species is picked using

\frac{\sum_{i = 1}^{j - 1} E_{i}}{E_{tot}} \leq R_{η} \leq \frac{\sum_{i = 1}^{j} E_{i}}{E_{tot}}

(4)

which returns the

j th

component as the emitting species. Next step is the rescaling of

R_{η}

as

0 \leq R_{η}^{*} = \frac{R_{η} E_{tot} - \sum_{i = 1}^{j - 1} E_{i}}{E_{j}} \leq 1

(5)

where $R_{η}^{*}$ is the rescaled random number for the species $j$ ⁠.

To determine the emission wavenumber, an initial wavenumber guess is made, and the corresponding $R_{η}^{i}$ is computed, where superscript “ $i$ ” represents the iteration number. If the difference between $R_{η}^{*}$ and $R_{η}^{i}$ is smaller than the specified convergence threshold, the initial wavenumber guess is selected as the emission wavenumber. Otherwise, the process is repeated until the convergence threshold of 10⁻⁶, as defined in this article, is reached.

2.2 The New Scheme.

In the new approach, we aim to eliminate the need for trial-and-error in the determination of emission wavenumbers. The new approach is a “geometrical” approach which is based on $η - R_{η}$ and $R_{η} - η$ diagrams. Since high resolution of $η - R_{η}$ and $R_{η} - η$ databases are being used in LBL Monte Carlo calculations, the selected points for interpolations are assumed to be very close together on the diagrams. An exception to this claim is the jump sections of $η - R_{η}$ diagram. Such a jump in $η - R_{η}$ marks an interval of $η$ which has a negligible effect in the calculation of heat source due to very small amounts of corresponding $κ_{η}$ values.

The approach aligns with the hybrid approach outlined in Eqs. (4) and (5) for identifying emitter species. The selection of emission wavenumbers involves the following steps: first, we sample $R_{η}$ and then rescale it according to the specific emitter species, i.e., $R_{η}^{*}$ ⁠. Subsequently, we refer to Table 1 for a guided selection process from the available databases. A visual representation of the various steps within the new scheme is shown in Fig. 1.

Fig. 1

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Schematic of emission wavenumber determination using the new scheme

Table 1

Implementation of the new scheme

Case	Description
Step 1	Determine $η_{1}$ using the dataset $S_{2}$ (dataset $η - R_{η})$ for $R_{η}^{*}$
Step 2	Determine $R_{η, 1}$ using $η_{1}$ and the dataset $S_{1}$ (dataset $R_{η} - η$ ⁠)
Step 3	Determine $Δ R_{η, 1}$ using the given state, $R_{η}^{}$ ⁠, and $R_{η, 1}$ by linear interpolation through temperature: $Δ R_{η, 1} = \frac{T_{1} - T^{}}{T_{2} - T_{1}} (R_{η}^{*} - R_{η, 1})$
Step 4	Determine $η_{2}$ using the dataset of $S_{1}$ (dataset $η - R_{η}$ ⁠)
Step 5	Determine $R_{η, 2}$ using $η_{2}$ and the dataset $S_{2}$ (dataset $R_{η} - η$ ⁠)
Step 6	Determine $Δ R_{η, 2}$ using the given state, $R_{η}^{}$ ⁠, and $R_{η, 2}$ by linear interpolation through temperature: $Δ R_{η, 2} = \frac{T_{1} - T^{}}{T_{2} - T_{1}} (R_{η, 2} - R_{η}^{*})$
Step 7	Adding $Δ R_{η}^{} = (\frac{T^{} - T_{1}}{T_{2} - T_{1}}) Δ R_{η, 1} + (\frac{T_{2} - T^{}}{T_{2} - T_{1}}) Δ R_{η, 2}$ to $R_{η}^{}$ and determining $η^{*}$ from $S_{1}$

Case	Description
Step 1	Determine $η_{1}$ using the dataset $S_{2}$ (dataset $η - R_{η})$ for $R_{η}^{*}$
Step 2	Determine $R_{η, 1}$ using $η_{1}$ and the dataset $S_{1}$ (dataset $R_{η} - η$ ⁠)
Step 3	Determine $Δ R_{η, 1}$ using the given state, $R_{η}^{}$ ⁠, and $R_{η, 1}$ by linear interpolation through temperature: $Δ R_{η, 1} = \frac{T_{1} - T^{}}{T_{2} - T_{1}} (R_{η}^{*} - R_{η, 1})$
Step 4	Determine $η_{2}$ using the dataset of $S_{1}$ (dataset $η - R_{η}$ ⁠)
Step 5	Determine $R_{η, 2}$ using $η_{2}$ and the dataset $S_{2}$ (dataset $R_{η} - η$ ⁠)
Step 6	Determine $Δ R_{η, 2}$ using the given state, $R_{η}^{}$ ⁠, and $R_{η, 2}$ by linear interpolation through temperature: $Δ R_{η, 2} = \frac{T_{1} - T^{}}{T_{2} - T_{1}} (R_{η, 2} - R_{η}^{*})$
Step 7	Adding $Δ R_{η}^{} = (\frac{T^{} - T_{1}}{T_{2} - T_{1}}) Δ R_{η, 1} + (\frac{T_{2} - T^{}}{T_{2} - T_{1}}) Δ R_{η, 2}$ to $R_{η}^{}$ and determining $η^{*}$ from $S_{1}$

As shown in Fig. 1, to calculate the location of

η^{*}

on

η - R_{η}

diagram at temperature T₁, two distances (⁠

Δ η_{1}

and

Δ η_{2}

⁠) from the imaginary diagram of

η - R_{η}

at temperature T* are calculated. Then,

Δ R_{η}^{*}

is calculated from

Δ R_{η, 1}

and

Δ R_{η, 2}

⁠. The effect of these two parameters on

Δ R_{η}^{*}

depends on T*. As shown in Fig. 2, when T* gets closer to T₂,

Δ R_{η}^{*}

is getting closer to

Δ R_{η, 1}

⁠. While, as shown in Fig. 1, when T* is in the middle of [T₁, T₂] interval, an equal effect of

Δ R_{η, 1}

and

Δ R_{η, 2}

on

Δ R_{η}^{*}

is expected. Here, we use a linear interpolation with respect to temperature to determine

Δ R_{η}^{*}

Δ R_{η}^{*} = (\frac{T^{*} - T_{1}}{T_{2} - T_{1}}) Δ R_{η, 1} + (\frac{T_{2} - T^{*}}{T_{2} - T_{1}}) Δ R_{η, 2}

(6)

Fig. 2

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The effect of temperature (T*) on determining $Δ R_{η}^{*}$ from $Δ R_{η, 1}$ and $Δ R_{η, 2}$

To implement the new scheme, two distinct types of datasets are required. The $η - R_{η}$ database is utilized to calculate $η$ during steps 1 and 4, while the $R_{η} - η$ database is employed to determine $R_{η}$ in steps 2 and 5. Consequently, the new scheme incorporates an additional database compared to the old scheme. Adding an extra database can affect the accuracy of the wavenumber selection. A small change in the selected wavenumber would impact the corresponding absorption coefficient due to the highly fluctuating nature of the gas absorption spectra. To minimize this effect, a high resolution (i.e., 10⁻⁶ for the values of tabulated $R_{η}$ ⁠) in preparing the new database was applied.

Fig. 3

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The considered one-dimensional slab for the simulations

3 Results and Discussion

To evaluate the performance difference between the new and old schemes, we conducted three separate one-dimensional test cases. The test medium, spanning a length of 2 m, comprises a mixture of CO₂, H₂O, and CO gases, enclosed by two cold walls as depicted in Fig. 3.

Fig. 4

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The considered temperature profiles inside the slab for different test cases

Figure 4 and Table 2 provides temperature profiles within the medium for different test cases, as shown in Fig. 4. The $H$ function in test case 2 is the Heaviside step function.

Table 2

The considered temperature profiles for the test cases

Test case	Temperature profile (K)
1	950
2	$450 H (x) + 1200 H (x - 1)$
3	$300 + 1500 sin (\frac{π}{2} x)$

Ren and Modest [11] found that utilizing 10⁶ random numbers enabled the reconstruction of a major part of the absorption coefficient spectra, revealing fluctuations around small values of the absorption coefficient. Increasing the number of random points by tenfold resolved the aforementioned fluctuation issue. Here, assuming negligible error caused by small values of the absorption coefficient, we incorporated 10⁶ random numbers in all of our simulations. We utilized the spectral database of absorption coefficients created by Bordbar et al. [13], where the temperature intervals were set at 100 K. In each test case, we conducted 50 simulations, and the results are presented for the arithmetic mean and standard deviation of the radiative heat source as ${\dot{q}}_{rad}^{″'} \pm σ$ ⁠. To assess the accuracy of the wavenumber selection using our new and old schemes, we separately compared them to the exact $R_{η} - η$ relation. This evaluation was carried out for various species at a medium temperature of 1750 K, as depicted in Fig. 5. Similar comparisons were performed for other temperatures, revealing a difference close to 0% between the exact $R_{η} - η$ diagram and the reconstructed $R_{η} - η$ diagrams using our new and old schemes.

Fig. 5

The calculated Rη−η diagram for 0.1CO2, 0.2H2O, and 0.05CO at temperature of 1750 K

View large Download slide

The calculated $R_{η} - η$ diagram for 0.1CO₂, 0.2H₂O, and 0.05CO at temperature of 1750 K

One of the occasions on which the new method does not show high accuracy is demonstrated in Fig. 6. The main weakness of the new scheme is the vertical section of the $η - R_{η}$ diagram, which corresponds to very small values of $κ_{η}$ ⁠. In Fig. 6, the nearly vertical line represents a portion of 0.1CO₂ absorption spectra that lies between two peaks and is supposed to have a negligible effect on the calculation of the heat source. For the rest of the $η - R_{η}$ diagram, the new scheme shows great agreement with the old scheme.

Fig. 6

The calculated η−Rη diagram for a gas mixture of 0.1CO2, 0.1H2O, and 0.1CO at temperature of 680 K (top) and η−Rη diagram for [0.04, 0.09] interval of Rη together with the absorption spectra of 0.1CO2

View large Download slide

The calculated $η - R_{η}$ diagram for a gas mixture of 0.1CO₂, 0.1H₂O, and 0.1CO at temperature of 680 K (top) and $η - R_{η}$ diagram for [0.04, 0.09] interval of $R_{η}$ together with the absorption spectra of 0.1CO₂

The simulation results for test case 1 are presented in Fig. 7. Since the medium maintains a constant temperature throughout, we have provided the analytical solution as a benchmark for test case 1. Using the given equation in Ref. [14], the averaged relative difference between the analytical and LBL-MC results is less than 1% reaffirming the simulations' high accuracy.

Fig. 7

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The radiative heat source for Test case 1

In Fig. 8, we present the simulation results for the second test case. The relative difference between the new and old schemes is 1.29%. We have included the LBL results in Fig. 8 solely for the purpose of comparison, with the understanding that they are not expected to be precisely congruent.

Fig. 8

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The radiative heat source for Test case 2

In Fig. 9, we can see the simulation results for Test case 3, finally. Here, the averaged relative difference between the results obtained from the new and old schemes is less than 1%. We have included the LBL simulation results in Fig. 9 purely for the sake of comparison.

Fig. 9

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The radiative heat source for Test case 3

Figures 7–9 demonstrate that the new scheme produces results that closely match those obtained using the old scheme. The key advantage of adopting the new scheme becomes evident in the calculation time required for wavenumber selection. With the new scheme, selecting 10⁶ wavenumbers takes just approximately 0.21 s, whereas the old scheme demands around 1.44 s. Consequently, the new scheme significantly enhances computational efficiency while maintaining results for ${\dot{q}}_{rad}^{″'}$ at a comparable level, albeit at the expense of utilizing an additional database.

To further demonstrate the performance of the new scheme, a simulation was conducted for the test case presented in Ref. [11]. This test case involves a one-dimensional slab of a mixture of 0.1CO₂, 0.1H₂O, and 0.1CO gases at a temperature of 650 K. The walls of the slab were assumed to be black and cold. Similar to Test cases 1–3, a good agreement between the results of the new scheme and the old scheme is observed in Fig. 10.

Fig. 10

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The divergence of the heat flux for presented test case in Ref. [11]

4 Conclusions

With the current computational resources and advancements in thermal radiation modeling, the application of LBL-MC simulations has become feasible not only for research studies but also for engineering simulations. In order to enhance the computational efficiency of LBL-MC, this research introduced a novel wavenumber selection scheme that is approximately seven times faster than the old scheme. The old scheme relies on the $η - R_{η}$ database and involves a trial-and-error process for wavenumber selection. In contrast, the new scheme streamlines this process by incorporating an additional database of $R_{η} - η$ ⁠. To assess the accuracy of the proposed scheme, simulations were conducted for a gas mixture composed of H₂O, CO₂, and CO within a one-dimensional slab, assuming three different temperature profiles. The results obtained using both the new and old schemes demonstrated a difference around 1%.

Acknowledgment

The authors greatly acknowledge the support of the Academy of Finland under Grant No. 314487 and the Finnish Fire Protection Fund (Palosuojelurahasto) under Grant No. SM1922147.

Funding Data

Academy of Finland (Grant No. 314487; Funder ID: 10.13039/501100002341).
Finnish Fire Protection Fund (Palosuojelurahasto) (Grant No. SM1922147; Funder ID: 10.13039/100009541).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

E_i =: emission of species $i$
E_tot =: total emission of components
$I_{b η}$ =: blackbody radiative intensity
$n_{s}$ =: number of emitting species
${\dot{q}}_{rad}^{″'}$ =: radiative heat source
$R_{η}$ =: random number
T =: temperature
x =: location

$η$ =: wavenumber
$κ_{η}$ =: absorption coefficient

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A Novel Fast Wavenumber Selection Scheme for Line-by-Line Monte Carlo Simulations of High-Temperature Gases

Abstract

1 Introduction

2 Wavenumber Selection Scheme

2.1 The Old Scheme.

2.2 The New Scheme.

3 Results and Discussion

4 Conclusions

Acknowledgment

Funding Data

Data Availability Statement

Nomenclature

References

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A Novel Fast Wavenumber Selection Scheme for Line-by-Line Monte Carlo Simulations of High-Temperature Gases

Abstract

1 Introduction

2 Wavenumber Selection Scheme

2.1 The Old Scheme.

2.2 The New Scheme.

3 Results and Discussion

4 Conclusions

Acknowledgment

Funding Data

Data Availability Statement

Nomenclature

References

Get Email Alerts

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