Influence of detailed in-depth radiation modeling on the computational predictions of liquid pool burning rates

Since thermal radiation substantially contributes to the flame heat feedback, a significant impact is anticipated for the in-depth radiation absorption in pool fires. This research focuses on investigating the influence of a detailed in-depth radiation absorption model on the accurate prediction of the flammability characteristics ( 𝑖.𝑒. , ignition time and burning rate) of liquids in using computational fluid dynamics. First, we determined the radiative properties of hydrogenated tetrapropylene (TPH) (i.e., fuel used in the OECD PRISME project) using UV–Vis–NIR and FTIR spectroscopy and derived an expression for the depth-dependent absorption coefficient and average surface reflectivity. Then, a new approach was introduced for the modeling of in-depth radiation absorption. Subsequently, simulations were conducted for three test cases: heptane evaporation, pool fire, and spill fire scenarios involving both heptane and TPH. Results indicated mild effects of in-depth absorption modeling on heptane evaporation, negligible changes in heptane pool fires, and significant alterations in spill fires compared to the predictions with gray calculations. For all the studied cases except heptane pool fire, the ignition times decreased significantly. Sensitivity analysis revealed that the assumed source temperature had negligible influence on the mass loss rate predictions. The importance of the liquid’s Nusselt number ( Nu l ) calculation varied between the studied scenarios.


Introduction
Over the past few decades, the widespread utilization of liquid fuels in industries and the occurrence of numerous incidents involving pool fires have captured the attention of researchers [1].Researchers have devoted significant efforts to understanding the relevant heat and mass transfer processes and underlying physical processes, such as liquid evaporation [2,3], the feedback of flame heat to the liquid [4], heat transfer within the pool [5], flame structure [6,7], interaction between flame and ventilation [8], and the impact of air pressure on flame behavior [9], among numerous others.
One of the pivotal mechanisms involves radiative heat transfer, which stems from the presence of hot gases and soot particles within the flame and buoyant plume.This transfer of thermal radiation significantly contributes to the heat feedback, with estimated share between 68 and 96% [10,11] for various fuels using a pool diameter of 30 cm.The share of thermal radiation in the flame heat feedback rises with increasing the pool diameter [12].The radiative component of heat feedback penetrates into the liquid and is absorbed in-depth.Due to its high importance, the interaction between radiative heat transfer and turbulence has attracted attentions in pool fire research [13,14].However, a detailed investigation of radiation absorption within liquids has rarely been done in the past.
Besides in-depth radiation, heat is transferred inside the liquid pool by buoyancy-driven convective motions, thermocapillary forces, arising from the variations in convective and radiative heat transfer distributions, and heat conduction through pool wall.The contact between bottom of the flame with burner edge makes the burner wall hotter than the adjacent liquid.The transfer of heat from hot wall to the adjacent liquid creates a vortex which transfers the hot liquid towards the center of the pool (buoyancy-driven convection) [15].Fukumoto et al. [16] have concluded that convective motions driven by buoyancy within the liquid play an important role in the prediction of the small methanol pool fire, surpassing other mechanisms such as thermocapillary force and wall thermal conduction.A similar conclusion about the influence of convective motions was drawn in the study by Xu and Wen [17].The in-depth radiation may create unstable temperature distributions and participate in the formation of convective motions.Convective motions transport the absorbed energy to deeper layers of the liquid, while https://doi.org/10.1016/j.firesaf.2024 The transfer of thermal radiation within liquids is often simulated using a gray model with a constant absorption coefficient.This approach is chosen for simplicity or due to limited spectral data for fuels.Sikanen and Hostikka [18] used gray model to simulate pool fires of various liquid fuels in Fire Dynamics Simulator (FDS) [19].They devised two methods to calculate the mean absorption coefficient: one for accurate heat flux estimation at the bottom of the liquid layer (total energy conservation) and another for minimizing the mean error in heat flux over liquid layer thickness.In another study, Sikanen and Hostikka [20] applied this model to predict heat release from liquid TPH fires using FDS, determining the mean absorption coefficient using spectral absorption data from DTE medium oil.Beji et al. [21] examined radiation absorption in liquid heptane and methanol, finding pool mass loss rate (MLR) sensitivity to absorption coefficients below 1000 m −1 .A more detailed approach to gray modeling for liquid fuels was presented by Alinejad et al. [22].This approach takes into consideration several key factors such as layer thickness and source temperature, which influence the absorption coefficient values.
Advancements in modeling spectral in-depth thermal radiation absorption by liquids have been documented in literature.Isojärvi et al. [23] pioneered non-gray modeling of radiation penetration into liquid fuels by developing a k-distribution method for liquid heptane.Building upon this work, Alinejad et al. [24,25] introduced the full-spectrum k-distribution (FSK) and separated full-spectrum k-distribution (SFSK) methods for liquid fuels, which demonstrated enhancements over the classical gray method.
In addition to considering the spectral characteristics of thermal radiation, the directional distribution of this flame radiation feedback within condensed materials plays a crucial role.The disparity in refractive indexes between gases and liquids leads to a redistribution of thermal radiation that penetrates through the liquid medium [26].To address this phenomenon, Dombrovsky et al. [27] made modifications to the two-flux method for solving the radiative transfer equation (RTE) across Fresnel interfaces.Liou and Wu [28] introduced the composite discrete ordinate method, which employs distinct sets of ordinates on either side of the interface.For the finite volume method (FVM), Murthy and Mathur [29] employed finer angular divisions to achieve a precise distribution of thermal radiation on the second side.However, this approach comes with a significant computational burden and is not feasible for practical implementation in engineering CFD codes such as FDS.To overcome this computational challenge, Alinejad et al. [30] introduced the concept of ordinate weighting within the FVM framework, providing a computationally efficient alternative.These same researchers subsequently proposed a simplified approach to account for the impact of refractive index differences at the interface, making the assumption of a diffuse interface condition [31].
Implementing all the detailed spectral and directional radiation submodels in simulating mass/heat release from condensed materials in fires is neither feasible nor necessary.Implementing all such models into a commonly used software, such as FDS, may be excessive in terms of usability, computational effort and required maintenance.Addressing this concern, Alinejad et al. [31] explored varying levels of thermal radiation intricacy in their pyrolysis model for black PMMA.Their findings distinguished an optimal configuration, involving a combination of the two-flux method as the RTE solution, utilization of the gray method using depth and source temperature-dependent absorption coefficients, and incorporation of diffuse boundaries with equivalent reflectivities.
Incorporating the outlined details of the thermal radiation into pool fire simulations necessitates access to the optical constants of the liquid fuels.However, for many liquids, these optical constants remain elusive.Consequently, a presumed value for radiative properties is employed within the simulations.Within the scope of our pool fire simulations, the chosen liquid fuels include heptane and TPH.The optical constants for liquid heptane have been detailed within a wavelength range of 0.65-14.92μm in Refs.[32][33][34].However, the optical constants for liquid TPH have not yet been established.To bridge this gap, current research employs spectroscopy techniques to ascertain the spectral optical constants pertaining to liquid TPH.This has been accomplished through the utilization of a newly derived equation for calculating the absorption coefficient.In current research, we take the recommended models by Alinejad et al. [31] and initially implement them into FDS.Subsequently, we apply this refined approach to simulate pool fires encompassing both liquid heptane and TPH.In the present research, we try to figure out how the detailed modeling of in-depth thermal radiation transfer affects predicting ignition time and MLR (or evaporation rate) of pool and spill fires.

Heat transfer inside the liquid
The liquid component is modeled as a one-dimensional layer material, and the heat transfer equation is formulated as follows: ,   ,  , , , , and q′′′ rad are density, specific heat capacity, temperature, time, distance from the liquid surface, thermal conductivity, and radiative heat source, respectively.Given the omission of the momentum equation for liquids, the convective term in the heat transfer equation is intentionally disregarded.Following the methodology pioneered by Abramzon and Sirignano [35], Sikanen and Hostikka [18] introduced an adapted thermal conductivity value in their simulations to compensate for the enhanced heat transfer caused by convective motions.By utilizing the Nusselt number (representing the ratio of convective heat flux to conductive heat flux), the resulting effective or modified thermal conductivity value ( ef f ) can be expressed as follows [36]: Sikanen and Hostikka [37] employed  ef f for the entire layer in their original work.However, Beji [38] emphasized that  ef f should be exclusively applied within the mixing (or boiling) layer.The thickness of this mixing layer exhibits non-uniformity across the pool's surface and is influenced by diverse parameters, such as the temperature of the bottom wall [16,39], the material of the vessel [40], the thickness of the pool [41], and other undisclosed factors.The complexity of convective motions within the pool is encapsulated within  ef f , rendering the derivation of an equation for calculating liquid Nusselt number (Nu l ) challenging and perhaps even impractical.Considering the main focus of the present research, which is to quantify the importance of indepth radiation absorption in pool fire simulations, Nu l was specified as a calibration parameter.The boundary condition for Eq.(1) can be described as follows: where ℎ  and ṁ′′ are heat of vaporization and evaporation rate, respectively, and q′′  is the convective heat flux: ,   ,   , , and  are an empirical coefficient for natural convection (1.52 for a horizontal plate), gas temperature in the center of the gas cell adjacent to the liquid surface, liquid surface temperature, characteristic length (the default value is 1 m), and normal grid size, respectively.The Nusselt number of gas side (Nu g ) is calculated as: where Re, Pr, ||, and  are Reynolds number, Prandtl number, gas velocity at the center of the adjacent cell to the liquid surface, and dynamic viscosity of the gas, respectively.There are limitations inherent to Eq. ( 5) which may not be accurate to apply for pool fires simulations as discussed by Hong et al. [3].The given Nu g correlation in Eq. ( 5) was developed for forced convection with turbulent flow over a flat surface where the temperature of the surface is higher than environment.The default length scale in calculating Re is considered 1 m which is not representing the geometry of the arbitrary pools.

Liquid evaporation
For a single component liquid, the evaporation rate is given by [19]: where ℎ  and  f ilm are mass transfer coefficient and density of fuel vapor and gas mixture, respectively and  is the Spalding mass transfer number: where   and   are the mass fractions of fuel vapor and gas within the adjacent grid to the liquid surface.  is related to the fuel vapor volume fraction (  ) by: where   and  air are the molecular weights of the fuel vapor and air, respectively.Assuming an equilibrium condition at the liquid surface,   is a function of   and boiling temperature (  ) according to the Clausius-Clapeyron equation: where ℎ  and  are heat of vaporization and gas constant, respectively.The mass transfer coefficient is Sh and  are Sherwood number and gas diffusivity, respectively.Sherwood, Schmidt, and Reynolds numbers are The density of the film is calculated by 0 is the atmospheric pressure and  f ilm is the temperature of a film consists of the fuel vapor and air within the cell adjacent to the liquid surface.In Eq. ( 11), the Sherwood number applies to turbulent flow over flat plates.It may not accurately capture liquid evaporation from pool fires without wind or ventilation.A more precise approach could involve a natural convection-based correlation, as suggested by Hong et al. [3].
In the absence of solving the momentum equation for the liquid phase, the primary consideration for evaporation is directed towards the liquid surface.This situation can potentially lead to the internal temperature (  ) of the liquid surpassing its designated boiling point.To counter this, a supplementary amount of liquid evaporation is introduced in conjunction with surface evaporation.This additional evaporation serves the purpose of preventing the temperature from surpassing the boiling point [19]: where  is the layer thickness.

Common approach for the in-depth radiation absorption
This approach is based on the previous implementation of the thermal radiation transfer inside the condensed material in FDS.The two-flux method [42] is used as the RTE solution method: where  + ,  ef f , and   are the forward radiation intensity, effective (or gray) absorption coefficient, and total blackbody intensity, respectively.Replacing the   =  4 ()∕ in Eq. ( 14) and multiplying the both sides of the equation by  gives: where ( q′′ ) + is the forward radiative heat flux.The boundary condition for Eq. ( 15) is: where , ρ, and ( q′′ ) − are emissivity, reflectivity, and backward radiative heat flux, respectively.Analogous formulations of the RTE and boundary conditions are employed for the backward radiative heat flux.In accordance with Eq. ( 16), the reflectivity on either side of the interface is denoted as 1 − .Empirical observations by Hamins et al. [10] revealed liquid surface reflectivity ranging from 5% to 8% for heptane, toluene, and methanol.For the purpose of simulating pool fires, it is a common practice to assume a 5% surface reflectivity [2,16].However, due to the minor influence of this reflective parameter, some studies have omitted it from their simulations [18,20].In the present investigation, we adhere to the prevailing approach of considering a 5% radiation reflection from both sides of the interface.
The second approach of Sikanen and Hostikka [18] is applied to calculate  ef f : where ( q′′ ex ) + () is the exact integrated radiative heat flux at depth  calculated using the line by line integration as: where  , ,  S ,   , and  =  are spectral blackbody emissive power, source (flame) temperature, exponential integral function of order , and optical thickness, respectively.Obtaining ( q′′ ) + and ( q′′ ) − through solving the RTEs, the radiative heat source is calculated as:

New approach for the in-depth radiation absorption
The thermal radiation model proposed by Alinejad et al. [31] is employed in the present research to simulate pool fires.This model utilizes the two-flux method as the solution method for the RTE and considers the refractive index of the liquid ( n) in determining the emission term: Furthermore, we apply the approach outlined by Alinejad et al. [22] to compute  ef f : where  tot and   are total transmissivity and spectral absorption coefficient, respectively.Eqs. ( 21) and ( 22) yield  ef f which takes into consideration both the source temperature and the depth below the liquid surface, thereby incorporating the spectral characteristics of the absorption coefficient.Employing these equations, we construct a comprehensive look-up table encompassing diverse values of  s and .The depth parameter was introduced in calculating  ef f since Linteris et al. [43] observed a nonlinear dependence between the measured broadband (here effective) absorption coefficient and applied thicknesses of the materials.
To prevent the integration of extensive datasets into the computational code, a correlation was employed to model the data within each individual look-up table, outlined as follows: The function  ( s ) exhibits variation based on the source temperature and is characterized by the following form: and the function () depicts the depth-dependence and is structured as follows: The coefficients  1 through  7 are optimized to minimize the difference between the provided  ef f value from Eq. ( 22) and the value derived from Eq. ( 23).The boundary condition for RTE is: where ρ01 and ρ10 are the average surface reflectivity at gas and liquid sides of the interface, respectively.The variability in reflectivity is dependent on direction and is determined using the Fresnel relations, as referenced in [26].In the two-flux method, a single ordinate is applied to each side of the interface, necessitating the use of an averaged surface reflectivity value.The specific methodology for calculating this averaged reflectivity is detailed in Appendix B. When addressing scenarios involving pool fires, it is important to note that the incident radiation from the flames, along with the radiation directed backward beneath the interface, spans the entire range of solid angles (hemispheres).As a result, the calculations for ρ01 and ρ10 are as follows: F. Alinejad et al.
where  0 and  01 are polar angle and reflectivity at gas side and  1 and  10 are the polar angle and reflectivity at the liquid side of the interface. 01 and  10 are computed using the Fresnel relations.With  0 and  1 being interconnected through Snell's law, it is important to note that ρ01 and ρ10 cannot be treated as independent.If we introduce the concept of a bulk emissivity, denoted as  and defined by  = 1 − ρ01 , ρ10 is calculated using the following correlation: where n is the averaged refractive index and is calculated from We derived Eqs. ( 29) and ( 30) through a process of curve fitting.By establishing bulk emissivity values for each material, we can subsequently compute the reflectivities at the interface ( ρ01 and ρ10 ) using the aforementioned correlations.

Spectroscopy and calculation of optical constants
To determine the optical constants of a liquid through spectroscopy, the material's spectral transmissivity and reflectivity within the desired wavelength range need to be measured.The liquid under investigation, in this case TPH, is introduced into the confined region between two KBr windows, as depicted in Fig. 1.
The comprehensive derivations of the transmissivity equation for the illustrated configuration in Fig. 1 are provided in Appendix A. Simplifying the aforementioned equation through the omission of terms involving reflectivities multiplication, due to their expected negligible values compared to 1, we derive the following expression for the spectral transmissivity of liquid TPH ( ,TPH ): where   ,  ,1 ,  ,2 ,  1 , and  2 are reflectivities at different sides of the interfaces, absorption coefficient of KBr window, absorption coefficient of TPH, beam path length inside the KBr window, and beam path length inside the TPH, respectively.Eq. ( 31) reveals that the observed spectral transmissivity of TPH using the current setup is influenced by the absorption characteristics of the KBr windows.To nullify the impact of these window absorptions, it is essential to reiterate the transmissivity measurements using air as the intermediate layer.Given the negligible absorption of radiation by air at exceptionally thin thicknesses, the resultant spectral transmissivity can be articulated as: Calculating the logarithm of the ratio between  ,TPH and  ,Air yields the following equation: Conducting measurements for various liquid thicknesses allows us to compute  ,2 through the utilization of Eq. ( 33) in combination with linear regression.Another essential factor in radiation treatment at the interface is the refractive index (  ).The determination of this parameter needs the calculation of the absorptive index (  ) by [44] Then, the refractive index is calculated using the Kramers-Kronig relations [44]: where  is wavenumber.The two primary challenges with the utilization of Eq. ( 35) are the unavailability of measured value of   at high and low wavenumbers and the absence of knowledge regarding the refractive index value at an infinite wavenumber ( ∞ ).In order to tackle the former issue, a substitution was made with the absorption index at a wavenumber of 49000 cm −1 .Addressing the latter problem involved connecting the   -spectrum to zero at 0 cm −1 using a linear interpolation method, as suggested by Bertie [44], to account for very low wavenumbers.The resolution for the second concern necessitated the calculation of  ∞ by imposing a known refractive index value at a specific wavenumber [37,45].
In pursuit of calculating the radiative properties, inspired by the methodology of [18,45], the initial step involves an examination of the liquid TPH's transmissivity.This evaluation is conducted across varying thicknesses: 25, 50, and 100 μm, utilizing a FTIR spectrometer; and and 3000 μm, employing a UV-Vis-NIR spectrometer.The UV-Vis-NIR and FTIR spectrometers encompass wavelengths of 0.2 to 2.5 μm and 2.5 to 25 μm, respectively which is adequately broad for fire-related applications.Concurrently, the refractive index of the liquid TPH is measured at a wavelength of 0.589 μm using a refractometer.

Heptane pool evaporation
The experimental data presented by Beji et al. [21] serves as a validation benchmark for heptane evaporation rates within a controlled atmosphere cone calorimeter experiment.In this study, a square steel pan with dimensions of 10 cm on each side and a height of 4 cm was employed as the liquid container.The initial liquid depth within the container was 2.5 cm, leaving an initial lip height of 1.5 cm.To insulate the container, both its sides and bottom were covered with 18 mmthick and 57 mm-thick layers of calcium silicate, respectively.During the experiments, two distinct heat fluxes of 25 and 50 kW∕m 2 were applied.The liquid's weight variation was continuously monitored using a precision weighing system with a resolution of 0.01 g.Temperature measurements of the liquid were taken at depths of 6 mm and 19 mm from the center, as well as 1 cm from the side.The initial temperature range of the liquid was maintained between 15 • C and 19 • C.

VTT room tests for heptane pool fire
In the present research, we employ the experimental data presented by Hostikka et al. [46] for the validation of our study, focusing on heptane pool fires.The experiments took place in a room with dimensions of 10 × 7 × 5 m 3 , which included an opening of dimensions 2.4 × 3.0 m 2 .The room's construction materials included lightweight concrete for the walls and ceiling, both with varying thicknesses of 0.3 and 0.25 m, respectively.The floor, on the other hand, was constructed from standard concrete.The experiments involved heptane pools of different sizes, ranging from 0.4 to 1.07 m 2 , positioned at three distinct locations within the room as indicated in Fig. 2. Heptane was burned on top of a water layer to prevent heating and deformations of the steel pool.The distance between the water surface and the pool's edge was consistently upheld at 0.13 cm.Details are summarized in Table 2.

OECD PRISME tests for TPH pool fire
In the context of a TPH pool fire scenario, we utilize experimental data provided by Prétrel et al. [47] and Audouin et al. [48] from the Institute de Radioprotection et de Sûreté Nucléaire (IRSN).These experiments specifically pertain to situations within mechanically ventilated compartments.The experiment involved a circular TPH pool positioned at the center of a room with dimensions of 5 m by 6 m and a height of 4 m.The room's construction featured concrete walls insulated with stone wool, and an additional layer of insulation was applied beneath the pool itself.At the initiation of the experiments, an initial lip height of the pool was maintained at approximately 5 cm.To enable ventilation, two rectangular inlet and exhaust ducts were integrated into the ceiling, each measuring 0.3 m by 0.6 m.These ducts played a crucial role in exploring the various effects of different ventilation strategies.Throughout the experimental series, adjustments were made to the positioning of the inlet duct and the nominal air renewal rates, offering an extensive investigation into how ventilation influences the dynamics of the fire scenarios.Detailed specifics of the selected experiments chosen for analysis are documented in Table 3.

Spectroscopy measurements and optical constants
The spectral transmissivity of liquid TPH was determined through two different spectrometers: UV-Vis-NIR for the wavelength range of 0.2 to 2.5 μm and FTIR for the wavelength range of 2.5 to 28 μm.The corresponding transmissivity data is illustrated in Fig. 3. Due to TPH's strong radiation absorption in the FTIR range, spectroscopic measurements were done using thinner liquid layers compared to the UV-Vis-NIR range.
Referring to Eq. ( 33), determining the absorption coefficient of the TPH relies on knowing the transmissivity of the utilized KBr windows.The spectral transmissivity of the used two KBr windows is depicted in Fig. 4, covering wavelengths from 0.2 to 25 μm.Both KBr windows   were used concurrently during measurements to account for potential variations in micro cracks that might affect each window differently.Now, with the spectral transmissivities of the TPH and KBr windows at hand, we can proceed to calculate the absorption coefficient of the TPH.This calculation utilizes Eq. ( 33) within the wavelength range of 0.2 to 25 μm, as illustrated in Fig. 5.For the determination of the optical constants, we first compute the absorptive index directly using Eq.(34).By accessing this absorptive index, we subsequently derive the refractive index through the application of Eq. ( 35).The measured refractive index of the TPH at a wavelength of 0.589 μm yields a value of 1.4224.Further examination involves extrapolating the refractive index value to the wavenumber of infinity, resulting in  ∞ = 1.4225.The spectrum of the refractive index shown in Fig. 5 is then determined by employing Maclaurin's formula, as outlined in [49].
To simulate the pool fires in current research, for the common approach, we utilized fixed values of 333 m −1 for heptane and 300 m −1 for TPH as outlined by Sikanen and Hostikka [18] and Sikanen and Hostikka [20], respectively.In contrast, our new approach introduces a variable absorption coefficient depending on the liquid surface's depth.The variation in  ef f between both methodologies is depicted in Fig. 6. Specific values of calculated  ef f coefficients in our new approach for liquid heptane, TPH, and water can be found in Table 4.For an expanded list of materials, refer to Appendix C for the corresponding computed coefficients.

Mesh independence study
The simulations were done using FDS [50], which is an open source computational fluid dynamics solver for fire-driven flows.The described new radiation model was implemented in a dedicated branch of the open source repository. 1 The version of FDS 6.7.9 was used for the implementation and simulations.The absorption of thermal radiation by released fuel vapors is considered in the FDS simulations.Table 5 reports the material properties were applied in the simulations.
Prior to conducting the primary simulations, a study on mesh independence was performed for VTT heptane pool fires using Test 1, and for PRISME TPH pool fires using Test 6.During the mesh generation phase, the entire domain was segregated into two zones: the flame zone and the outer zone.In order to capture the intricacies of flame physics accurately, finer grids were employed in the flame zone compared to the outer zone.The specific meshes utilized for our mesh independence study are outlined in Table 6.However, due to the substantial computational time involved, opting for even finer meshes than those presented would have proven excessively costly.
The results of mass loss rate predictions utilizing different mesh configurations are presented in Fig. 7. Across all simulations, a consistent    Nu l value of 10 was maintained.The initial segment of the mass loss rate diagram lacks result convergence while the intermediate section reveals a close alignment between Mesh 4 results and those from finer meshes (Mesh 5 and Mesh 6).Consequently, Mesh 4 was chosen for the primary simulations.In the context of heptane evaporation simulation, the scale of the geometry renders the use of highly refined grids practical, making a mesh independence study unnecessary.

Results for heptane evaporation
The simulation results for heptane evaporation using both common (Gray model with constant , ρ/ ρ) and new (Gray model with varying (,   ), ρ01 / ρ10 ) approaches are depicted in Fig. 8.These simulations were conducted within a controlled environment cone calorimeter experiment, considering two initial distinct heat flux levels.Due to using 3D geometry of the cone in the simulations, the level of heat flux at the surface varies with the pool height.By changing the value of Nu l , several simulations were done using the new approach.The value of Nu l that yielded the closest results to the experimental data was selected as  the calibrated value of Nu l .For the heptane evaporation simulation, the calibrated value of Nu l was determined to be 15.
This calibrated Nu l value was subsequently applied to simulations using the common approach, enabling a comparative analysis of the predictive capabilities of both models.The results highlight that with appropriate calibration of the Nu l number, the common approach can achieve comparable accuracy to the new approach.However, it is worth noting that in the displayed outcomes, the common approach tends to slightly underestimate the initial phase, which could bear significance in predicting ignition events.Additionally, using the common approach the peak evaporation rate occurs slightly earlier than the new approach.
The temperature profiles of the liquid (T l ) at two distinct depths are depicted in Fig. 9.The Nu l number was fine-tuned for a heat flux of 50 kW∕m 2 and the same parameter was subsequently employed for the lower heat flux conditions.This strategy accounts for the improved predictive capabilities exhibited by both methodologies for T l under the 50 kW∕m 2 heat flux.Employing the same Nu l value reveals a tendency in the common approach to overestimate T l at both depths.This may stand in direct contradiction to expectations, especially when considering the common approach's utilization of lower effective absorption coefficient values ( ef f ) compared to the new approach.The nonsmooth profiles of the liquid temperature can be related to the cell size.
For all the simulations, the CELL_SIZE_FACTOR of 0.1 was utilized.This choice makes the grid sizes inside the liquid 10 times finer than the default grid size.While the heights of temperature measurements points from the bottom of the pool remains constant during the simulation, the location of the grid which contains the measurement point can slightly change which causes a fluctuation in temperature readings.
In our study of radiative energy distribution in liquids, we computed the radiative heat source in a 1D heptane slab using a MATLAB code used by Alinejad et al. [24].The resulting heat source distributions depicted in Fig. 10 highlight a noteworthy observation: the new approach concentrates a considerable amount of radiative energy within an exceedingly thin layer near the surface.Conversely, the conventional approach maintains a higher heat source across the remaining thickness.The mentioned difference between two approaches justifies the given temperature profiles in Fig. 9.This distinction becomes particularly evident when considering potential temperature measurements within the ultra-thin surface layer.Such measurements would provide a clearer understanding of the elevated temperatures associated with the new approach.
To demonstrate the effect of high thermal radiation absorption using new method, the surface temperature prediction for both new and common approaches at a time of 10 s and the heat flux of 25 kW∕m 2  are shown in Fig. 11.As expected, the new approach yields a higher surface temperature compared to the common approach.
Numerous parameters affect the predictions of MLR in pool fires.This investigation narrows its focus to two specific parameters: Nu l Fig. 12.The effect of Nu l variation on the evaporation rate of the liquid heptane at heat flux of 50 kW∕m 2 .number (which accounts for convective motions) and source temperature (which directly influences the determination of the  ef f coefficients).
To analyze the role of Nu l in predicting MLR effects, we conducted simulations for heptane's evaporation.Variations in Nu l impact primarily the initial and middle section of the evaporation rate curve in opposing ways, while the evaporation rate peak and the final portion remain relatively insensitive to Nu l values as seen in Fig. 12.
In the new approach, the  ef f coefficients are determined as a function of the depth from the liquid surface and the source temperature [22].Unlike the depth from the liquid surface, due to the ambiguity regarding the average (or equivalent) temperature of incident radiation from hot gases, the choice of the source temperature introduces an uncertainty into the simulations.To address this uncertainty, we conducted a series of simulations for heptane evaporation within a controlled environment cone calorimeter test.These simulations were carried out under a heat flux of 50 kW∕m 2 , while considering a range of source temperatures spanning from 500 to 2000 o C. The results of these simulations and applied profiles of  ef f , as depicted in Fig. 13, highlight that the predictions of MLR exhibit a remarkable insensitivity to the specific choice of heat source temperature.

Results for heptane pools
Five distinct simulations were conducted for VTT heptane pool fires, and the outcomes are depicted in Fig. 14.Fine tuning of Nu l was executed for Test 1 (pool size: 0.4 m 2 ), Test 2 (pool size: 0.61 m 2 ), and Test 4 (pool size: 1.07 m 2 ), yielding Nu l values of 26, 130, and 40, respectively.Subsequently, the same Nu l value was applied to each respective pool size in other locations.These Nu l values were then applied to the same pool sizes in other locations, leading to consistent outcomes between the approaches.The FDS default settings were utilized for the ignition and extinction of the fire.The default value for auto-ignition temperature in FDS is −273 • C, causing ignition whenever a sufficient concentration of fuel and oxidizer mix.Therefore, there is no need to add an igniter to simulate piloted ignition.The critical flame temperature for heptane, used in the extinction models, is 1497 • C. For predicting the amount of produced fuel vapor, the default evaporation model of FDS was used (Eqs.( 6) to ( 13)).The simulations reveal a dynamic interplay between in-depth radiation absorption and convective motions within the liquid.The former mechanism concentrates energy within an exceedingly narrow layer near the surface, while the latter mechanism seeks to distribute the energy deeper into the liquid.Notably, the disparity between the outcomes of the two approaches is more pronounced for Test 1 compared to the other tests, owing to the lower Nu l value assigned to Test 1.For Test 2 and Test 3, same Nu l values were used within the simulations.Nonetheless, the simulation outcomes for Test 3 diverge significantly from the measured mass loss rate (MLR), indicating intricate underlying dynamics that demands further investigation.The present approach to liquid phase modeling has limitations that hinder the representation of liquid flow within the container, a lack of consideration for wall heat flux, an inadequate representation of the possible non-ideal pool geometry during testing, and reliance on an equilibrium assumption for evaporation modeling.The inaccuracy of this assumptions especially at the experiment's outset, hinders the precise capture of both the initial and final segments of the measured mass loss rate.
The temperatures of the upper layer (T up ) for Test 1 to Test 5 can be found in Fig. 15.The methodology for calculating T up is elaborated in [46].The outcomes reveal a notable concurrence between the projected temperature distribution in the gas phase and the actual measurements.In the case of Test 1, the distinction in predicted T up between the two approaches highlights the impact of a lower Nu l value.Temperature distribution inside the liquid layer at time 20 s is shown in Fig. 16.Within the very thin layer close to the pool surface, using the new approach yields a higher liquid temperature compared to the common approach.The reason for the observed temperature profile can be explained by the difference in the thermal radiation absorption between the two approaches, as given in Fig. 10.
To show the effect of Nu l variation in predicting MLR of pool fires, we conducted simulations for Test 1 as presented in Fig. 17.The most significant Nu l -driven effects are observed in pool fires compared to heptane evaporation and spill fire, attributed to the thicker liquid layer.
However, beyond a certain Nu l threshold, the MLR curve exhibits less sensitivity to Nu l . .A visualization of the VTT heptane pool fire simulations are shown in Fig. 18.The domain could be extended more to get the flow field more accurate.However, due to heavy computational load and getting relatively good accuracy in prediction of upper layer temperature (.., Fig. 15) which can be directly related to the smoke flow outside the opening, we used lowest possible extension for the domain (here 1 m).

Results for TPH pools
The simulation results concerning PRISME TPH pool fires are shown in Fig. 19.Manual adjustment of the Nu l number was only carried out for Test 6, yielding a value of 13.5.The way of doing Nu l adjustment was explained in Section 4.3.The tuned Nu l value was subsequently employed in the other two simulations.The critical flame temperature for TPH is 1427 • C. The default settings mentioned for the heptane pool simulations were also used for TPH pool fire simulations.Unlike heptane, the TPH pool did not ignite, possibly due to its higher boiling point.In order to ignite the liquid TPH, a 20 × 20 × 20 cm 3 block at 1300 • C was positioned approximately 20 cm above the pool's surface.This setup was maintained for 37 s at the simulation's commencement using the new approach.The same ignition source utilized within the common approach failed to elicit ignition.Achieving similar results with the common approach necessitated an extension of the ignition source activation time to 90 s.These findings highlight a distinct divergence introduced by new approach, particularly pronounced when   dealing with liquids possessing higher boiling points resulting in discernible variations in ignition behavior.It is crucial to acknowledge that simulation results are also significantly responsive to adjustments in ignition source settings.Furthermore, in addition to the already mentioned limitations contributing to inaccuracies in simulating heptane pool fires, it is crucial to recognize that uncertainties stemming from the ignition source could equally contribute to the observed deviations in the context of Test 7 and Test 8. Sikanen and Hostikka [20] investigated the reason of erroneous oscillations for Test 8. Based on their observations, the oscillations arises from the interplay between the combustion model, evaporation model, and extinction model.The oscillations partially arises from the simplification of the combustion model which assumes combustion happens whenever fuel vapor and oxygen mix.This causes the reignition of the fuel vapor inside the compartment when there is enough oxygen.This reignition can be prevented by setting the auto-ignition temperature higher than 373 K. Another reason for the large difference between the predictions and experiments could be the interaction between ventilation conditions and the flame, which may affect the convective motions inside the pool (.., the value of Nu l ).Therefore, applying different values of Nu l can reduce the prediction errors in Tests 7 and 8.
The gas temperature (T g ) predictions at two distinct heights and the variations in gas pressure within the fire compartment for Test 6 are presented in Fig. 20.These predictions exhibit a favorable alignment with the measured data.

Spill fire
To comprehensively characterize the impacts of in-depth radiation absorption, simulations were conducted for spill fires involving liquid heptane and TPH.An imaginary case consisted of a square container containing the respective liquid fuels each with dimensions of 2 m by 2 m placed on a 30 cm thick concrete floor in an open atmosphere.At the simulation's commencement, the initial liquid thickness was 5 mm.For liquid TPH, the ignition source with dimensions of 80 cm by 80 cm by 20 cm and temperature of 1300 • C was employed for a duration of 20 s, using both approaches.Similar to the heptane pool fire simulations, no igniter was applied for heptane spill fire.The simulation domain was established at 3 m by 3 m by 5 m, employing the grid size of 2 × 2 × 2 cm 3 .The open boundary condition was applied for sides and upper face of the domain.Fig. 21 shows the applied geometry in the simulations.The default settings of FDS were applied for the extinction and ignition of fire, as well as for fuel evaporation, as explained in previous sections.
Fig. 22 presents the MLR predictions from both approaches.Considering the significant impact of the wall effect (and the minimal influence of convective motion) within thin liquid layers, we adopted a Nu l value of 5 for the simulation of both liquid scenarios.Unlike for the large-scale pool fires, noticeable differences can be observed in the MLR predictions obtained using the common and new approaches of internal radiation.This observation happens since a thin layer of the liquid reaches boiling point very soon, showing an insensible effect of convective motions compared to the in-depth radiation absorption.Subsequent simulations, incorporating a range of Nu l values, consistently yielded similar observations.In the event of a heptane spill, both ignition and MLR peak are notably influenced by the in-depth absorption of radiation.In the context of a TPH spill, the ignition source appears to possess considerable strength, thus leading to a less conspicuous distinction in ignition behavior compared to a TPH pool exposed to the same ignition source.Nonetheless, the results highlight a noteworthy distinction in the magnitudes of MLR peaks.
The effect of Nu l variation in predicting MLR for the spill fires was numerically investigated for the heptane spill fire as shown in Fig. 23.Smallest Nu l -induced changes are noticed in spill fires compared to the heptane evaporation and pool fires, where MLR curve variance is minimal with Nu l values exceeding 1.This emphasizes the limited role of convective motion in spill fires, shedding light on the prominence of indepth radiation absorption.Our findings underscore liquid height as a critical factor determining the prevalence of convective motions versus in-depth radiation absorption as dominant heat transfer mechanisms within the liquid.

Conclusion and remarks
The heat feedback to the liquid fuels from liquid pool fires is largely attributed to thermal radiation.This research investigated the sensitivity of the evaporation and burning rate predictions of small to medium-scale pool fires to the treatment of phase interface effects and in-depth radiation.We introduced a new approach for in-depth radiation transfer modeling.This approach employs a gray method incorporating absorption coefficients that vary based on depth and source temperature.At the interface, a diffuse approximation is employed, utilizing equivalent reflectivity values derived from the Fresnel relations.In comparison to the commonly used approach, this approach employs varying absorption coefficient values inside the liquid and higher reflectivity at the liquid side of the interface.The new approach provides a well-defined, physical-based method with one less calibration parameter (.., absorption coefficient) compared to commonly used gray method, which uses a calibrated absorption coefficient.Simulations were done for liquid heptane and TPH.The radiative properties of the liquid TPH were measured using spectroscopy technique for the wavelength region of 0.25 to 25 μm.The impact of using the new approach was investigated concerning the heptane evaporation rates in controlled environment cone calorimeter tests, pool fires, and spill fires of liquid heptane and TPH.In terms of MLR predictions, the impact was small for heptane evaporation rates, minimal for pool fires, and substantial for spill fires.Within liquids, heat transfer is governed by the interplay between convective motions and in-depth radiation absorption.Since the convective motions exert a dominant effect in thick liquid layers (all test cases except spill fires), utilizing a detailed model for in-depth radiation absorption becomes less important.The new approach had a significant effect on ignition times especially for TPH pool fire and spill fires.Finally, the heptane evaporation rates were shown to be insensitive to the selected value of the radiation source temperature used in the new approach for estimating gray absorption coefficient.This brings huge advanatage to the method since the general and global value of source temperature can be very difficult to define.

Fig. 1 .
Fig. 1.Multilayer geometry of liquid TPH and KBr windows inside the spectrometer.

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Fig. 4 .
Fig. 4. The measured transmissivity of the applied two KBr windows in measurements using UV-Vis-NIR and FTIR spectrometers.

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Fig. 6 .
Fig.6.The calculated  ef f using new at source temperature of 800 o C and common approaches for (left) heptane and (right) TPH.

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Fig. 7 .Fig. 8 .
Fig. 7.The mesh independence study for a 0.4 m 2 pool of (left) heptane in open atmosphere and (right) TPH in ventilated environment with nominal air renewal rate of 4.7 hr −1 .

Fig. 9 .Fig. 10 .
Fig. 9.The predicted temperature inside the heptane pool at two different depths by common and new approaches for heat fluxes of (left) 25 kW/m 2 and (right) 50 kW/m 2 .

Fig. 11 .
Fig. 11.The surface temperature contours of liquid heptane at heat flux of 25 kW∕m 2 and time 10 s: (left) new approach and (right) common approach.

Fig. 13 .
Fig.13.The heptane evaporation rate prediction in controlled environment cone calorimeter experiment at the heat flux of 50 kW∕m 2 (left) and applied  ef f (right) using different source temperatures.

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Fig. 15 .
Fig. 15.The upper layer temperature prediction of the two introduced approaches for Test 1 to Test 5.

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Fig. 16 .
Fig. 16.The liquid temperature with respect to depth from the surface of the pool at time of 20 s: (left) Test 1 and (right) Test 4.

Fig. 17 .
Fig. 17.The effect of Nu l variation on the mass loss rate prediction of Test 1.

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Fig. 19 .
Fig. 19.The prediction of mass loss rate for 0.4 m 2 TPH pool fires using new and common approaches.

Fig. 20 .
Fig.20.Gas temperature at two distinct heights (left) and variation of gas pressure inside the fire compartment predicted by two approaches for Test 6.

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.104128 Received 10 December 2023; Received in revised form 9 February 2024; Accepted 27 February 2024 + Forward radiative heat flux (W∕m 2 ) ( q′′ exp ) + Exact integrated radiative heat flux (W∕m 2 ) ( q′′ ) − Backward radiative heat flux (W∕m 2 ) q′′  Convective heat flux (W∕m 2 ) q′′ in Incident radiative heat flux (W∕m 2 )the in-depth radiation absorption concentrates the energy to the layers very close to the liquid surface.The present study aims to explore the comparative impact of convective motions and precisely modeled indepth radiation absorption on liquid evaporation, pool fires, and spill fires.
Table gives the details of experiments.

Table 1
The characteristics of the measurements devices (spectrometers and refractometer) used for measuring transmissivity and refractive index together with the sample thicknesses.

Table 2
The location and geometry of the pool and the size of the opening in VTT experiments for liquid heptane.

Table 3
Geometry of the pool and ventilation characteristics in IRSN experiments for liquid TPH.
Fig. 2. Room dimensions and pool locations in VTT room tests for heptane pool fire.

Table 4
The emissivity and calculated absorption coefficient parameters of the liquid heptane, TPH, and water.

Table 5
The material properties applied in the simulations.

Table 6
Applied grid sizes in the mesh independence study for 0.4 m 2 heptane pool fire simulation.

Table C . 7
The emissivity and calculated absorption coefficient parameters of different fuels (Part a).The emissivity and calculated absorption coefficient parameters of different fuels (Part b).