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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 – 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 results’ 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 we’ll 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:
𝜕𝐼(𝑟⃗,𝑠,𝑡)
𝜕𝑡
+
𝜕𝐼(𝑟⃗,𝑠,𝑡)
𝜕𝑠
= −𝛽(𝑟⃗)𝐼(𝑟⃗, 𝑠, 𝑡) + 𝜅(𝑟⃗)𝐼𝑏 (𝑟⃗, 𝑡) +
𝜎(𝑟⃗)
4𝜋
∫4𝜋 𝐼(𝑟⃗, 𝑠 ′ , 𝑡)𝜙(𝑠 ′ , 𝑠)𝑑Ω′
(1)
Where I is radiation intensity, 𝑟⃗ spatial position, s angular direction, 𝛽 the extinction coefficient,
𝜅 the absorption coefficient, the underscript b refers to the blackbody reference value, 𝜎 is the StefanBoltzman constant, 𝜙 is the scattering function and 𝑑Ω′ represents the angular variation.
The heat flux equation is written as:
(2)
𝑞𝑖 (𝑟⃗, 𝑡) = ∫2𝜋 𝐼(𝑟⃗, 𝑠, 𝑡)(𝑠 . 𝑖)𝑑Ω
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.
𝜕
On the P-N model, the direction vector 𝜕𝑠 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:
𝑞𝑟 = −
1
3(𝑎+ 𝜎𝑠 )
(3)
− 𝐶𝜎𝑠 ∇𝐺
Where a is the absorption coefficient, 𝜎𝑠 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:
𝑑𝐼 𝑙
𝜇 ′ 𝑑𝑧 = − 𝛽𝐼 𝑙 + 𝜅𝐼𝑏 +
𝜎
4𝜋
′
′
′
∑𝐿𝑙′ =1 𝐼 𝑙 𝜙̅ 𝑙 𝑙 𝑤 𝑙
(4)
Where 𝜇 ′ is the directional cosine of the angle, superscripts l and l’ 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 – from Kim & Ryou (2003)
The present work makes use of the data available in Kim & Ryou’s (2003) paper, regarding one
of the methanol pool fires configuration. Table 1 summarizes the experiment’s 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 room’s 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, it’s possible to see that the Discrete-Ordinates model is able to predict
a temperature behavior that better agrees with what’s 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, it’s possible to observe that the Discrete-Ordinates model performed better in
predicting the temperature behavior in our recreation of Kim & Ryou’s (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.