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A COMPARATIVE STUDY BETWEEN THERMAL RADIATION MODELS P-1 AND DISCRETE ORDINATES USING CFD SOFTWARE OPENFOAM K. CID1, S. VIANNA² 1 Universidade Estadual de Campinas, Faculdade de Engenharia Química, Departamento de Sistemas Químicos ¹[email protected] ²[email protected] ABSTRACT β When it comes to Risk Analysis and Fire Safety Engineering, guaranteeing the reliability and the similarity between analyzed data and reality are extremely important. Computational Fluid Dynamics techniques (CFD) have been largely utilized to replicate fire accidents, enabling the analysis of temperature profiles and the quantification of thermal radiation incidence at specific points of the fire zone. The work here presented uses the open-source CFD toolbox named OpenFOAM, through the use of the fireFoam solver, present in OpenFOAM, we elaborate a comparative study between the thermal radiation models P-1, simpler and computationally less demanding, and Discrete Ordinates, more complex and robust, evaluating processing time and resultsβ confiability. Comparing the results to experimental data, we could observe the model DO predicted better results. A link between the number of angles, processing time and results is also shown. 1. INTRODUCTION In Fire Safety Engineering, the ability to estimate thermal radiation intensity and temperature profiles on accidents involving fires and explosions is essential. With a better understanding of fire behavior, it becomes possible to better prepare escape routes, properly design emergency systems, and to prevent serious injuries and eventual fatalities. Through the latest years there's been a rapid increase in the understanding of what governs the physical phenomena of fire (McCaffrey, 1979; Babrauskas, 1983; Drysdale, 1999; McAllister et al, 2011), thus giving scientists the capacity to mathematically model it. Such developments have enabled the use of CFD tools as a way of reliably simulating fire behavior. Fire modeling encompasses a large spectrum of different phenomena: flame height, soot emission, heat release rate, fuel consumption rate, thermal radiation intensity, heat fluxes, etc. Accurate and computationally optimized models for said phenomena translate into faster and more reliable fire simulations. The number of publications on heat transfer processes is considerably large, and thus weβll skip a comprehensive review of the literature regarding combustion modelling and heat transfer processes. The present work focuses on the evaluation of two different models for estimating heat radiation fluxes and intensity, namely the Discrete-Ordinates and the P-1 models. 2. EQUATIONS AND MODELS FireFOAM, part of the OpenFOAM CFD toolbox, is a solver for turbulent buoyancy-dominated fires. Using the Large Eddy model for turbulence, fireFoam solves Favre-filtered compressible NavierStokes equations. The energy equation is written in terms of total enthalpy. Mixture fraction is considered a conserved scalar, and is evaluated by solving a transport equation. Temperature comes from total enthalpy and species composition. The solver uses temperature-dependent specific heats, based on a NIST model (7-Nasa-Polynomial-Coefficients). The turbulent chemical reaction is modeled by the Eddy Dissipation model (which comes from the Infinitely-Fast Chemistry model), removing the need of evaluating chemical kinetics at time-steps. Sub-Grid species mass fractions are obtained through a probabilistic density function. The turbulent sub-grid scale is solved by the Smagorinsky oneequation model. (Chatterjee et al, 2010) According to Siegel & Howell (2010) the governing equation for heat transfer through thermal radiation is: ππΌ(πβ,π ,π‘) ππ‘ + ππΌ(πβ,π ,π‘) ππ = βπ½(πβ)πΌ(πβ, π , π‘) + π (πβ)πΌπ (πβ, π‘) + π(πβ) 4π β«4π πΌ(πβ, π β² , π‘)π(π β² , π )πΞ©β² (1) Where I is radiation intensity, πβ spatial position, s angular direction, π½ the extinction coefficient, π the absorption coefficient, the underscript b refers to the blackbody reference value, π is the StefanBoltzman constant, π is the scattering function and πΞ©β² represents the angular variation. The heat flux equation is written as: (2) ππ (πβ, π‘) = β«2π πΌ(πβ, π , π‘)(π . π)πΞ© Where the subscript i refers to the heat flux in direction i. 2.1. P-1 MODEL The P-1 model for thermal radiation is the simplest case of the P-N model, which is based on expanding the term of radiation Intensity through an orthogonal series of spherical harmonics. π On the P-N model, the direction vector ππ on Equation 1 is rewritten as a function of the directional cosines of the angles between the vectors and the coordination axes. The new equation then becomes a series of sums of orthogonal spherical harmonics. The P-1 model is the simplest version of this model, where only the first 4 terms of the series are taken into account. Using the P-1 model, Equation 2 becomes: ππ = β 1 3(π+ ππ ) (3) β πΆππ βπΊ Where a is the absorption coefficient, ππ the scattering coefficient and C a coefficient referring to how anisotropical the scattering function is. 2.2. DISCRETE-ORDINATES MODEL The Discrete-Ordinates model, developed by Chai & Patankar (2000) discretizes the thermal radiation field in a defined finite number of directions. In a one-dimensional problem, we can express the thermal radiation transfer equation in a discrete direction as: ππΌ π π β² ππ§ = β π½πΌ π + π πΌπ + π 4π β² β² β² βπΏπβ² =1 πΌ π πΜ π π π€ π (4) Where π β² is the directional cosine of the angle, superscripts l and lβ are angular directions, L is the total number of angles and w the weight of each angle. The DO model is computationally costlier, since it is necessary to solve Equation 4 for each of the discretized angles at every cell, while for model P-1 you only solve Equation 3 once for each cell. 3. CASE DESCRIPTION & SETUP The work of Kim & Ryou (2003) evaluated sprinklers as a suppression method for compartment pool fires of different configurations. In a closed room of dimensions 4.0m x 4.0m x 2.3m, controlled methanol and n-hexane pool fires were studied and the temperature in different points of the room was monitored with the use of sensors. Four distances from the center of the room were chosen (r = 0.5m, r = 1.0m, r = 1.5m and r = 1.8m) and, for each distance, sensors measured the temperature at four different positions, all at a height of 1.8m. The temperature was then averaged and evaluated over time for each of the pre-determined distances. Figure 1 illustrates the schematics for the experiment. Figure 1 - Experimental Room Schematics β from Kim & Ryou (2003) The present work makes use of the data available in Kim & Ryouβs (2003) paper, regarding one of the methanol pool fires configuration. Table 1 summarizes the experimentβs of interest configuration. Table 1 - Experiment Configuration Fuel Pool Diameter Fuel Consumption Rate Heat Release Rate Methanol 30 cm 0,0148 kg/m²s 26,64 kW We start by recreating the experimental roomβs configuration on OpenFOAM, as illustrated by Figure 2. We then select the appropriate boundary conditions and run initial tests to determine, through a sensitivity analysis, the correct grid size necessary to run the simulations properly. Determined the minimum grid size, we proceed with the simulations, switching between both models. We evaluate temperature predictions and computational time of the simulations using the thermal radiation models P-1, Discrete-Ordinates with 288 angles and a more refined Discrete-Ordinates, with 800 angles. Our simulations were ran on an Intel i7-2600 @ 3.4ghz octacore CPU, with 8 gb RAM. Running Ubuntu 12.04 64bits and OpenFOAM v. 2.4.0. Figure 2 - Computational Reconstruction of Experimental Room 4. RESULTS Figures 3 to 6 illustrate the temperature profiles obtained in the simulations. DO stands for Discrete-Ordinates with 288 angles and Refined DO for the model with 800 angles discretized. Table 2 summarizes the running time for each of the simulations. Table 2 - Running Time of Simulations Model P-1 Discrete-Ordinates Discrete-Ordinates Refined Running Time (s) 23964 24828 108736 Figure 3 - Temperature Profile for r = 0.5m Figure 4 - Temperature Profile for r = 1.0m Figure 5 - Temperature Profile for r = 1.5m Figure 6 - Temperature Profile for r = 1.8m 5. DISCUSSION AND CONCLUSION Analyzing Figures 3 to 6, itβs possible to see that the Discrete-Ordinates model is able to predict a temperature behavior that better agrees with whatβs observed experimentally. The P-1 model seems to overestimate temperatures, an effect that seemingly increases with time. Such differences can be explained by the difference in approach between both models. The P-1 model is far more simplified than the Discrete-Ordinates model. It is interesting to notice that a significant increase in the number of angles (from 288 to 800 angles) did not actually translate to a difference in temperature profiles. In regards to running time, we expected the P-1 model to be faster. However, as is shown by Table 2, there was no significant difference in computational time between running P-1 model and Discrete-Ordinate with 288 angles. We do observe a significant growth in running time when moving from 288 angles to 800 angles on the Discrete-Ordinates model. So, in conclusion, itβs possible to observe that the Discrete-Ordinates model performed better in predicting the temperature behavior in our recreation of Kim & Ryouβs (2003) experiments. It was also possible to observe that an increase in refinement of the model does not necessarily translates to better results. Comparing running times between models it is possible to conclude that P-1 offered no computational advantage when compared to our less refined Discrete-Ordinate model implementation. 6. REFERENCES MCCAFFREY, B.J. Purely Buyoant Diffusion Flames: Some Experimental Results. Washington: National Bureau Of Standards - Center For Fire Research, 1979 BABRAUSKAS, V. Estimating Large Pool Fire Burning Rates. Fire Technology , v. 19.4, p. 251261, 1983 DRYSDALE, D. An Introduction to Fire Dynamics. 2. ed. Chichester: Wiley, 1999. MCALLISTER, S.; CHEN, J.; FERNANDEZ-PELLO, A.C. Fundamentals of combustion processes. 1st edition, New York: Springer, 2011 CHATTERJEE, P.; WANG, Y.; DE RIS, J.L.. Large Eddy Simulation of Fire Plumes Proceedings of the Combustion Institute, v. 33, p. 2473-2480, 2010. KIM, S.C.; RYOU, H.S. An Experimental and Numerical Study on Fire Suppression Using a Water Mist in an Enclosure. Building And Environment. Hong Kong, p. 309-316. jun. 2003. SIEGEL, R.; HOWELL, J.R.. Thermal Radiation Heat Transfer. 5. ed. Boca Raton: Taylor & Francis, 2010. CHAI, J. C.; PATANKAR, S.V.. Discrete-Ordinates and Finite-Volume Methods for Radiative Heat Transfer. Handbook of Numerical Heat Transfer, 2000.