Assessment study of different radiation models in the simulation of 1 m methanol pool fire

By comparing the improved models, the WCassol model and Cassol model have the best performance


Introduction
Pool fires often occur when liquid fuel is accidentally spilled and ignited during fuel storage and transportation, and they can also induce a series of events that may amplify the hazards of the accident, leading to the well-known domino effect.The study has shown that 43% of domino accidents result initially from fire, and 80% of them are pool fires [1] and are dominated by radiation heat transfer [2,3] .For liquid pool fires, the thermal feedback from flame radiation can, on the one hand, intensify fuel evaporation and thus increase the flame spreading area; on the other hand, it poses a significant threat to nearby fuel tanks and can potentially accelerate the development of fire [4] .Therefore, it is essential to establish reasonable models to account for the radiation heat transfer in pool fires.
However, a reliable calculation of gas radiation remains a challenging task, not only because of the difficulty in solving the radiation transport equation (RTE) but also because of the highly temperature-dependent radiation properties of the participating species (e.g., water, carbon dioxide, and soot) that emit and absorb thermal energy in the infrared part of the radiation spectrum, thus performing heat transfer.The approach in most studies [5−7] commonly adopts the assumption of a gray gas medium where the gaseous radiation properties are assumed to be constant at all wavelengths.This is also known as the gray gas model (GG model).Compared to the complex LBL (line-by-line) method, the GG model avoids the integration of radiation variations across hundreds of thousands of spectral lines and significantly reduces the computational cost.These advantages make the GG model favorable, especially in fire science, where radiation has to be solved along with large-scale fluid flow and turbulent combustion.Therefore, further improvement of the GG model's accuracy for various fire scenarios is the most desirable.
Hostikka et al. [8] evaluated the default GG model in the Fire Dynamics Simulator (FDS) code, which is widely used in fire research, and found that the model can only qualitatively predict the pool size dependence of the burning rate.A similar observation was indicated by Ref. [9] in the simulation of a small-scale methane flame, where the results of default FDS's GG model were very close to those obtained by nongray calculations.Recently, Fernandes et al. [5] studied the modeling options for the evaluation of medium emittance in GG models by modeling small-to medium-scale pool fires.They found that the default formulation [10] in FDS can still provide a good prediction in the weekly sooting flames, although the evaluation method for medium emittance matters in determining the prediction accuracy.However, their work lacks an in-depth analysis of the difference between different modeling options, which is unbeneficial for the further improvement of GG models.Meanwhile, previous studies were limited to small pool fires, while large-scale fires are known to be more radiation-dominated.Therefore, in this work, a large-eddy simulation of a 1 m large-scale methanol pool fire is carried out employing different modeling strategies for evaluating medium absorption properties.Compared to Ref. [5] , weighted-sum-of-gray-gases (WSGG)-based models adopting different spectroscopic databases (i.e., HITEMP) and methods to evaluate mixture emittance are also investigated, along with a discussion of additional models for Planck-mean absorption coefficients.In total, five radiation models are considered, which are detailed in the next section.The purpose of this paper is to extend the FDS source code, and the program associated with all different radiation models is integrated into the source code to ensure that the fluid flow and combustion are solved with the same solvers.For a comprehensive parameter analysis, the inhouse code is also developed to extract the various fire data from the FDS computations.

Numerical modeling
In this work, the modeling strategy for flame radiation is thoroughly evaluated in the large-eddy simulation of a 1 m diameter methanol pool fire [11] .Methanol fuel is chosen because the soot generation of methanol combustion is low, which can mitigate the coupling effect on radiation exchange due to soot and is beneficial to evaluate the accuracy of different models in predicting the more complex spectral dependence of the radiative coefficients of gas-phase species.This would eventually help to establish a theoretical basis for selecting a suitable radiation model to simulate various complex pool fires.The detailed data records in this experiment will help to validate the model.

Radiation models
The radiative transport equation (RTE) must be solved to obtain the radiation field.Since the heat dissipation albedo of submicron soot particles produced by hydrocarbon flames is very low, we generally ignore scattering in solving the RTE equation.Therefore, the RTE equation for a nonscattering gas under the GG model assumption with its spectrally integrated form can be written as Ref. [10] ： where is the radiation intensity, is the blackbody radiation intensity, and is the absorption coefficient of the graygas medium.In this paper, in addition to the use of the FDS default method to calculate , the other four GG models are κ m employed by modification of the FDS source code.Generally, these five radiation models adopt different assumptions to determine the value that should have a highly complex spectral dependence for gaseous species.Below is a description of each model used.
Barlow's model [6] .This model calculates the gray gas absorption coefficient of each species by curve fitting the Planck-mean absorption coefficient from the narrow-band RADCAL database [12] , and then the absorption coefficient for each species is given as： where j represents the different species (e.g., H 2 O, CO 2 , etc. ), is the partial pressure, T is the temperature, and is the polynomial fit coefficient.For the CO 2 − H 2 O − soot mixture, its spectral absorption coefficient can be expressed as = + + .Here, the spectral absorption coefficient of soot is determined by Eq. ( 3), in which =1864.32/(mK atm), indicates the volume concentration of soot.
Cassol's model [7] .It fits the Planck-averaged absorption coefficient curves of water and carbon dioxide by using the latest spectral database HITEMP2010 [13] , and the species coefficient is therefore given by Eq. ( 4), where NP denotes the number of coefficients.The fitting coefficients in Eq. ( 2) and ( 4) are shown in Table 1.
Smith's WSGG-based model [14] .Instead of curve-fitting the Planck average absorption coefficient, this method obtains the gray gas through the total emissivity ( ) of the gas mixture that is calculated by the WSGG-based model and Bouguer's formula, i.e., Eq. ( 5), where S denotes the optical path length.The emissivity fitted by Smith's model is derived from the data provided by the exponential broad-band model and can be written as: Barlow [6] Cassol [7] CO J u s t A c c e p t e d where is the pressure of the mixture, and , as shown in Eq. ( 6), is the temperature-dependent coefficient of jth gray gas (the number of gray gas is 3, i.e., ) and represents the proportion of blackbody emission in the corresponding spectral range of each gray gas: The Smith model considers three gray gases and assumes a constant molar ratio of 2:1 between H 2 O and CO 2 in the mixture.The model parameters are listed in Table 2.However, this brings the first limitation of this model since in the combustion field, the local partial pressure of H 2 O and CO 2 would vary from one point to another.Furthermore, this model relies on an old spectral database to generate the coefficients, and the study has shown that this could lead to lower accuracy compared to the new versions of HITEMP2010 [13] .
Cassol's WSGG-based model [15] : In contrast to Smith's model, Cassol et al. [15] can account for the variations in the molar ratios between H 2 O and CO 2 throughout the practical combustion field based on the newly updated spectral data of HITEMP2010.Since the WSGG correlations for soot are provided in this model, the total emissivity is calculated as follows: For a mixture of H 2 O-CO 2 -Soot, the temperature coefficient of each species ( ) and the mixture absorption coefficient ( ) are determined following a probabilistic argument [16] as: The number of gray gases considered for water, carbon dioxide, and soot in this model is 4. A previous study by [17] showed that marginal effects on the accuracy were observed when the number of gray gases exceeded four.The coefficients used in this model are given in Table 3.
These five models are labeled FDS, Barlow, Cassol, WSmith, and WCassol, respectively.Overall, the former three models rely on the Planck-mean absorption coefficient to determine the medium emittance, using different databases for the curve fittings; the latter two resort to the WSGG-based correlations, and these two models are differentiated mainly by the spectral database and assumption for H 2 O and CO 2 mixture molar ratio.Meanwhile, the latter four models are integrated into the FDS by the newly coded program.Their performance in the prediction of large-scale pool fires is thoroughly evaluated in a later section.

Computational setup
The effective filtered form of the transport equations for species mass fraction, momentum and energy for a three-dimensional, transient, low Mach number compressible flow was solved in a Cartesian coordinate system [18] .The fluid is assumed to be an ideal gas, and an additional Poisson equation is introduced for the pressure-velocity coupling [19] .

J u s t A c c e p t e d
acy Resolution Method) in the second-order accuracy TVD (total variation diminishing) format.A low-pass box filter with width proportional to the grid cell size is used for LES spatial filtering, and turbulence closure is achieved using a Deardorff eddy viscosity model for the Navier-Stokes equations and standard gradient assumptions for species and energy transport, assuming a constant Pr and Sc number of 0.5.
In addition, for the stability of the calculations, the time step is limited by the use of the CFL number and the Von Neumann Constraint, which range from 0.8 to 1. Combustion is modeled based on mixing-controlled, single-step, and infinitely fast chemical reactions [10] .A fixed proportion of the consumed mass of fuel determines soot production.

ṁ'' ∆h c ∆h g
A simple pyrolysis model is used to simulate fuel evaporation by specifying a constant mass loss rate per unit area.According to the experiments, the pool has a diameter of 1 m and a depth of 0.15 m.The distance between the liquid surface and the pool's edge is maintained at 1 cm.It was found [20] that the mass loss rate of methanol fuel ( ) is related to the ratio of the heat of combustion ( ) and the heat of gasification ( ), i.e., as shown in Eq. ( 10): where the smoke production rate θ is set to 0.001, so the resulting is 0.019.In addition, the grid cell size dx has a significant impact on the simulation calculation.In general, the numerical error decreases with increasing D*/dx, while the flame characteristic diameter D* is a function of the total heat release rate ( ): Here, denotes the ambient density, denotes the specific heat of air, denotes the ambient temperature, and g is the gravitational acceleration.It is shown that the simulation results are in good agreement with the experiments when the grid size is taken between [D*/16, D*/4] [10] .Following this criterion, in this work, unless otherwise specified, the calculations are performed using a 1 cm grid with a computational domain of 3.2 m * 1.2 m * 2.4 m, as schematically shown in Fig. 1.The total calculation time is 30 s, and since the results vary with the time series, all the results are averaged over 10 s ̶ 20 s, during which the combustion reaches a stable stage, and the combustion rate reaches a predetermined value.

Verification of the models
It is necessary to verify the correctness of the new code before the calculation of the complex turbulent fire combustion, so this subsection first presents the results of the radiation computation for the domain between two infinitely long flat plates (i.e., the plate height is 12.5 times the plate spacing) [21] .This configuration can be assumed to be a quasi-one-dimensional, steady-state heat transfer problem, ignoring the effect of complex flow and thus providing reliable data for comparing and validating the model [22−24] .A grid of 200 cells is used along the x-direction (i.e., the direction of plate spacing), and the y-direction (i.e., the plate height direction) is discretized into 63 grid cells.Forty-eight control angles are used for the angular discretization in the RTE equation.In the x-direction, the initial molar concentrations of components H 2 O and CO 2 , the volume concentration of soot, and the temperature are given by Eq. ( 12) and ( 13): T (x) where the value for H 2 O is 0.2 and the of CO 2 is 0.1.The value for carbon fume is .Fig. 2 shows the radiation heat source Sr predicted from the four newly coded models (i.e., Barlow, Cassol, WSmith, WCassol) in comparison to the one calculated in previous Ref. [25].The local error associated with the FDS computations is defined as .It can be seen that these four models coupled into FDS agree well with the literature results, along with a maximum error of no more than 3.5%.These results verify the correctness of the models and imply that the modified FDS code can provide reliable predictions for later discussion.

Centerline gas temperature
The gas temperature along the centerline of the pool fire is shown in Fig. 3.The measured maximum temperature is located approximately 0.3 m above the burner exit.This position of peak temperature was better predicted by four other models except for the Barlow model, which peaks the temperature at approximately 0.2 m.In comparison to the experimental value of 1370 K, The peak temperature predicted by Barlow and FDS is 1 301.39K and 1 316.67 K, with errors of 5% and 3.8%, respectively.The errors are higher than those calculated by the other three models, with an error of almost 3%.The experiments show that the temperature profile changes steeply near the fuel surface, and all five radiation models reproduce this trend well.The gas temperature at 0.05 m above the burner is approximately 1 144 K ± 424 K.In general, compared with the default FDS and the Barlow models, the other three mod- J u s t A c c e p t e d els are in better agreement with the experiments, especially in the range of Z<0.6.Moreover, WCassol performs best given that the underestimation of temperature in the downstream region is observed for the WSmith model.This result indicates that when the WSGG-based model is adopted, the variation in the concentration ratio of the local components (H 2 O and CO 2 ) in the medium emittance calculation should have an important impact on the accuracy of radiation prediction.Meanwhile, since the Cassol model, relying on the Planck-mean absorption coefficient, also shows superiority to the default FDS and Barlow models, the spectral database may play an important role in radiation calculation.Among all the radiation models, the Cassol and WCassol models appear to be the best candidates for present flame computation.This can be attributed to the same database of HITEMP2010 that both models used.The high temperature during combustion causes gas particles to acquire higher energy, and therefore, the spectral lines resulting from jumps between higher energy levels, commonly known as hot lines, appear in the spectrum.The coefficients of the Barlow model are derived from the narrow band model of the database RADCAL, and the database of the WSmith model is an out-of-date version of HITEMP, both of which do not contain information on the spectral properties of hot lines in the high-temperature range.

Pulsation frequency
Pool fires always exhibit periodic pulsation behavior.The flame pulsation frequency is one of the fundamental and essential parameters of pool fires, which can help to understand the buoyancy effect and its interaction with the structure of combustion and annular vortex [26] .The interaction between

J u s t A c c e p t e d
the vertical structure of combustion and the annular vortex dominates the flow field [27] .In previous studies, the pulsation frequency of pool fires has been measured based on the fluctuating frequency of flame thermal radiation, the frequency of smoke buoyancy or vortex shedding, and the variation in flame height [28] .This work calculates the pulsation phenomenon through the dynamics of vertical velocity.The instantaneous vertical velocity is extracted at the positions of X = 0 m and Y = 0.505 m by referring to the experimental work of Tieszen et al. [29] using PIV to measure the flow field.The Fourier fast transform (FFT) was used to convert the time series of vertical velocity data into a function of frequency and amplitude.Since the time series started from 0 s, a highpass FFT was performed to eliminate the right-angle offset before the FFT.The pulsation periods obtained for the different radiation models are shown in Fig. 4. The pulsation period of flame height measured in the NIST experiment using the camera is 1.37 Hz, which is comparable to the model prediction of a value of approximately 1.37 ± 0.04 Hz.Generally, good agreement is observed.The result of the Cassol model is closer to the experiment.

Instantaneous distribution of the flame characteristics κ m
Snapshots of the velocity vector, temperature, radiation heat flux (Q r ), absorption coefficient , and species mole fraction (X H2O , X CO2 ) calculated using different radiation models are shown in Fig. 5.The radiation heat flux is determined by： The instantaneous velocity vector field is also colored by the local fuel methanol concentration.The velocity field is highly consistent with the gradient of the temperature field.
Close to the burner (Z<0.5 m), the flow is more turbulent with strong mixing, and the temperature rises quickly to the peak value (see Fig. 3).When moving downstream, turbulence is weak, and the temperature and density gradients become more considerable, causing a larger-scale vortex structure.Comparatively, the differences in the temperatures predicted by different models are less pronounced than those shown for the radiative heat flux (Q r ).Meanwhile, the intense radiation region seems less correlated to the high-temperature domain, especially downstream.The distribution of combustion products of X H2O and X CO2 could explain this.According to Ref. [30] , the radiation intensity is generally a function of the fourth power of temperature and the medium emittance.Therefore, the concentration of radiative species that determines the mixture absorption coefficient also plays a critical role in the distribution of Q r .Furthermore, by comparing X H2O , X CO2 , and Q r , the species of water appears to be more dominant in the correlations with Q r in the present flame.
In addition, theoretically or in cases of laminar conditions, the high-temperature region usually overlaps with the area featuring a high concentration of primary products, i.e., H 2 O and CO 2 .However, in turbulent pool fires, the displacement between temperature and concentration maxima would be induced because of buoyancy and air-entrainment effects, and the region of elevated product concentration tends to present near the pool surface.Thus, the present results also indicate that for pool fires, the evaluation of radiation characteristics should consider both the distribution of temperature and the species concentration.
The normalized absorption coefficient of is also shown in the figure . is the maximum absorption coefficient predicted by different models and is shown in Table 4.For the models based on the Planck average absorption coef- J u s t A c c e p t e d ficient, Cassol's maximum is two orders of magnitude smaller than that of Barlow and FDS.In contrast, the WSmith and WCassol models do not differ much.

Radiation heat flux
The magnitude of radiation heat flux determines the hazard posed by the fire and directly affects the rate of fire spread.Figure 6  For these five GG radiation models, gray gas is assumed, and the radiative properties of the participating gases are a constant value relative to the wavenumber spectrum.This simplification avoids the direct coupling of property variations over hundreds of thousands of spectral lines, but it undoubtedly leads to a decrease in accuracy.It is well known that H 2 O, CO 2 , and soot are the leading absorbing-emitting energy media in fire scenarios.In contrast to the two diatomic gases of H 2 O and CO 2 , the dependence of soot on wavenumber can often be approximately linear.Due to the fuel properties of methanol, the soot production rate is marginal.In contrast, H 2 O and CO 2 become the dominant absorption medium, with a much higher spectral dependence than soot.This constitutes the main difficulty for accurately modeling the radiation in weekly soot methanol pool fires based on the GG model.Therefore, all GG radiation models used in this work underestimate the radiation heat flux.It is worth noting that the errors in measurement could also contribute partly to the discrepancy.

Computational efficiency
All LES computations are carried out using the same scenario file and a server computing system of Centos7 with two Intel® Xeon® Platinum8375C@2.99GHz central processors with a total of 128 cores and 256 G RAM.The information on the computational elapsed time, maximum errors in temperature and radiant heat flux Q r , and pulsation periods for the different radiation models are presented in Table 5.The original FDS model has the highest computational efficiency, while the computational time of the WCassol model is significantly higher than that of the other radiation models.This is because the WCassol model is equivalent to the usage of 125 gray gases and thus requires additional computational power to calculate different temperature-dependent coefficients and absorption coefficients.It is worth mentioning that the calculation efficiency of the WSmith model does not differ much from that of the Barlow, FDS, and Cassol models because only the fixed component ratios and three gray gases are considered in the WSmith model.There is almost no difference in the calculation efficiency of the Barlow and Cassol models compared to the FDS model because both of them assume one gray gas.Overall, the GG model based on the Planck-averaged absorption coefficient is faster than the GG model using the WSGG-based correlations.Referring to the previous discussion, the predictions from the Cassol model and WCassol model have the best agreement with the experiments.Furthermore, if the CPU computational time is considered, the Cassol model seems to be the best choice that compromises between accuracy and efficiency.However, this conclusion should be further studied in the simulations of other complex pool fires, including strong sooting flames, which will be the focus of future work.

Conclusion
The prediction performance and accuracy of different radiation models were thoroughly evaluated in an LES of 1 m methanol pool fires.This is achieved by the modifications of the FDS source code that will then couple different gray gas radiation models, i.e., the traditional ones based on the Planck average absorption coefficient and the one relying on the WSGG-based correlations for medium emittance.The main conclusions are as follows: (Ⅰ) The accuracy of the radiation models (WSmith, WCassol) based on the WSGG correlation is better in the temperature calculation.Although the Cassol model does not use the WSGG method to obtain the absorption coefficients,  J u s t A c c e p t e d its predictions are close to those of the WSmith and WCassol models because it employs the latest spectral database HITEMP2010.In addition, both the FDS model and the Barlow model underestimate the maximum temperature value.The pulsation period is well predicted by all radiation models.(Ⅱ) The Cassol and WCassol models have the best accuracy among all radiation models, but the latter is computationally time-consuming.The Cassol model is concluded to be a good choice that nicely balances computational efficiency and accuracy.
This work presents a newly updated assessment of radiation modeling for large-scale pool fires and reveals the importance of both temperature and concentration distribution in accurately evaluating the radiation flux and their special correlations in pool fires.To the author's best knowledge, there is limited research on this topic.The default GG model in FDS usually underestimated the radiant heat flux when calculating low-sooty pool fires, which can affect fire hazard assessments.The new coupled model improves the accuracy of the radiation calculation, and the results also provide ideas for future study on the establishment of a suitable radiation model for other complex liquid-fueled pool fires under various fuel-type and flame-sooting conditions.

Fig. 2 .
Fig. 2. Validation of the results of 1D RTE calculations for the newly coupled radiation models.

Fig. 3 .
Fig. 3. Temperature distribution along the central axis of the pool fire

Fig. 6 .
Vertical distribution of radiation flux at 2.8 m from the center of the pool.Assessment of different radiation models in the simulation of 1 m methanol pool fire Zhou et al. -8 DOI: 10.52396/JUSTC-2023-0021 JUSTC, 2023, 53(X):

Table 1 .
Polynomial fit coefficients used in Barlow and Cassol's model.

Table 2 .
Smith's WSGG correlation coefficient of the H 2 O -CO 2

Table 3 .
The coefficients for Cassol's WSGG-based model compares the radiation heat flux calculated by the five radiation models with those obtained from the NIST measurements at 2.8 m away from the fire centerline.The Barlow model deviates from the observed values with a maximum error of 50%, followed by the WSmith model with 36%.Consistent with the temperature prediction, the WCassol model reaches the best agreement with only a 15% deviation.The default FDS model and the Cassol model share a similar prediction with the WCassol model.

Table 5 .
Comparison of the computational performance of different radiation property models.