Excert from the roceedings of the COMSOL Conference 8 Boston A Finite Element Analysis on the Modeling of Heat Release Rate, as Assessed by a Cone Calorimeter, of Forming olycarbonate David L. Statler Jr. * and Rakesh K. uta Mid-Atlantic Technology, Research and Innovation Center, South leston, WV Deartment of Chemical Engineering, West Virginia University, Morgantown, WV *Corresonding author: Building 74, 3 Kanawha Turnike, South leston, WV, 533, david.statler@matricresearch.com Abstract: During the yrolysis and combustion of s, heat is released and is tyically measured with a cone calorimeter to better assess the s flammability. Key data from a cone calorimeter include the rate of heat release, eak heat release rate, and time to ignition. Modeling heat release rate, as assessed by cone calorimetry, has not been extensively studied for -forming s, such as, olycarbonate. Here we determine the heat release rate with the hel of a one-dimensional transient finiteelement model using commercially available COMSOL Multihysics software. This model considers the heat and mass transort henomena taking lace throughout the thickness of the and the growing that is forming during the yrolysis and combustion reaction. This model accurately redicts the heat release rate curve for forming olycarbonate. This model should also be alicable to other systems that contain flame retardants where only the yrolysis kinetics are different. Keywords: Modeling Heat Release Rate, Forming olycarbonate.. Introduction The cone calorimeter is a small-scale testing aaratus that can effectively redict real world fire behavior. It can determine fire acteristics, such as, ignition time, weight loss, heat and smoke release rates, heat of combustion, and the average secific extinction area. A samle laced on a load cell is heated with a conical heater, sarked with an igniter and combusted in atmosheric air. The exhaust gasses are analyzed to determine heat and smoke release rates. The heat release rate is found by knowing the ercentage of oxygen consumed during combustion. g of oxygen consumed in burning of most organic materials leads to the release of about 3kJ of heat.. Descrition of Heat and Mass Transfer rocesses The modeling of heat and mass transfer during the heating, yrolysis, and formation of a ic material to redict the heat release rate, as assessed by the Cone Calorimeter, has not been examined. One aer by Fredlund 3 looked at the modeling of wood structures under fire but is quite comlex and does not aly the results to heat release rates. To describe the combustion rocess in detail, a ste by ste analysis follows. First, a flat late of material (3mm x mm x mm) is laced in a holding device that is held u by a thick layer of insulating material. The cone calorimeter uses a circular radiant cone heater to deliver heat to the samle. This cone heater is reheated and equilibrated before any exeriment is erformed. Once the cone is at equilibrium, the samle (at ambient temerature) is laced under the cone heater. The samle is first heated by radiation from the cone heater. As the samle heats, the begins to break down into smaller gas molecules through a rocess called yrolysis. The yrolysis gases then diffuse out of the samle and into the atmoshere just above the samle where air is resent. These gases then react with oxygen in the air, and thus, combustion takes lace. A small amount of the heat of combustion is then fed back to the roviding more heat to the samle, however, most of the heat is lost to the atmoshere in heating u the surrounding air. During the rocess, some s form a layer (intumescent, carbon, ceramic like), which has a lower thermal conductivity and diffusivity than the host. These roerties hel in lowering the eak heat release rate and sread the heat release rate over a much broader time frame. Ultimately, the will be consumed during the reaction, residual will be left behind, and combustion will cease.
olycarbonate is a forming. Tyical results from cone calorimeter data for olycarbonate are shown in Figure. The data were taken at a frequency of Hz and the radiant heat flux was 5 kw/m. This heat flux corresonds to a cone temerature of 975 K 4. Examining the heat release rate curve, for forming s, after the eak heat release rate, a lateau region is observed before returning to zero. This is due to formation. During the combustion, formation occurs during yrolysis. As the grows thicker, the hysics of the roblem balance and give rise to a lateau and second eak. Thus, heat is released at a lower rate and over a broad time range, which is a major advantage for forming s. HRR (W/m ) 6.E+5 5.E+5 4.E+5 3.E+5.E+5.E+5.E+ 5 5 5 3 35 4 Time (sec) Figure. Heat Release Rate data from Cone Calorimetry for olycarbonate. 5kW/m Radiant Heat Flux. To model the heat release rate observed in a cone calorimeter, artial differential equations, of mass and heat transfer that accurately deict the yrolysis reaction and the temerature rofile, must be solved. Due to the nature of this roblem, the artial differential mass and heat transfer equations are couled together, and an analytical solution does not exist. Therefore, a numerical solution must be sought. However, this three dimensional transient roblem can be simlified to a one dimensional transient roblem, lowering the comutational ower required to solve this system of equations. Figure shows the hysical situation involved, where x -x (3mm) is the zone, and x - x (nm initially, grows to ~ cm) is the formation zone, where a very thin layer is introduced and grows at a velocity corresonding to the rate of yrolysis. The numerical technique used is a finite element model. The commercial software ackage COMSOL Multihyics was urchased and used for this modeling. COMSOL Multihysics is a user friendly, efficient, finite element method rogram, where many equations can be couled together at one time and solved. Insulation x y z olymer Zone x Cone Heater olymer late (3mm thick) x x, x Formation Zone (very thin, then grows) 3 Dimensional Transient Dimensional Transient Combustion (% Heat of Rxn fed back to Boundary) Radiant Heat Convective Heat Lost Figure. Schematic and geometry of hysical model. The artial differential equations that describe this system are discussed next. 3. Mass Transfer Uon heating, the yrolyzes into gas and. To model the roblem, two distinct regions have been selected to account for the hysics involved. The zone accounts for the yrolysis reaction; is consumed and gas is roduced. The formation zone accounts for the growth and acts only as a barrier to heat and mass transfer. 3. yrolysis Reaction inside the olymer Zone [x -x ] The must first react in the zone. Reaction mechanisms have been greatly simlified to bring out the essential hysics: goes to gas lus. The reaction scheme is shown: k olymer as + or k α + ( α ) C During this reaction, mass of is consumed and roduces a fraction, α, of gas and
the remaining. The first order reaction rate for mass consumtion is: k m where m Mass of [kg], t Time [sec], k Rate constant for yrolysis reaction [/sec]. Further, the rate constant for the yrolysis reaction, k, is a function of temerature and is better described by the Arrhenius relationshi: E k A ex R T where A re-exonential factor of yrolysis reaction [/sec], E Activation energy of A yrolysis reaction [kj/mol], R as constant [J/mol/K], T Temerature [K]. The re-exonential factor and activation energy can be found by thermogravimetric analysis. Considering the hysics of the roblem, in reality, the boundary, x, would move to the left as is consumed. During the numerical modeling, it is assumed that zone, x -x, is constant, thus, the volume is constant, and instead concentration (or density) changes with time. By dividing each side by a constant volume gives a concentration equation, which is easier to work with in COMSOL Multihysics. Re-writing the equation: k r A c where r Rate of consumtion during yrolysis [kg/m 3 /sec], c Concentration of [kg/m 3 ]. When the is consumed during the reaction, gas is roduced. The roduction of gas in the zone [x -x ] must also diffuse through this zone to reach the formation zone [x -x ]. Writing the mass balance on the gas secies that is roduced: c D α k r c where r Rate of gas evolution during yrolysis [kg/m 3 /sec], c Concentration of yrolysis gases [kg/m 3 ], D Diffusion coefficient of yrolysis gases through [m /sec], x Length in the x-direction [m], α Mass fraction of gas that is roduced. 3. Mass Transfer in Formation Zone [x -x ] When the gas secies that is roduced during yrolysis in the zone [x -x ] reaches the formation zone [x -x ], it must diffuse through to reach the outside boundary that is exosed to atmosheric oxygen. Writing the mass balance in the formation zone: D c where D Diffusion coefficient of yrolysis gases through [m /sec]. 4. Heat Transfer Before the cone calorimeter exeriment takes lace, the thin (3mm) laque of is laced in a samle holder and susended by a thick layer of insulating material. The starts off at ambient conditions. When the is laced under the cone heater, the outside surface is heated by radiation. Heat is transferred throughout the by conduction. The yrolysis reaction may also give off or consume heat. If there are other additives or fillers in the, they too may react in an exo- or endothermic fashion. When the gas is roduced and it diffuses out of the layer, it reacts with oxygen to roduce more heat. art of the heat of combustion of this reaction is fed back to the surface, whereas the rest leaves to the atmoshere. 4. Heat Transfer in olymer Zone [x -x ] In the zone [x -x ], heat is transferred by conduction. Heat of volatilization from yrolysis and heat from other chemical rocesses (for examle, endothermic reaction of aluminum hydroxide) are also shown:
T T C k H k c + H e Constant [J/m /K 4 /sec], cone heater [K]. T Temerature of Cone where Density of [kg/m 3 ], C Heat caacity of [J/kg/K], k Thermal conductivity of [J/m/K/sec], H Heat of volatilization of [J/kg], H Heat released or absorbed e in other chemical rocesses (e.g. decomosition of additives) in, negligible [J/kg]. 4. Heat Transfer in Formation Zone [x -x ] In the formation zone [x -x ], heat is transferred through the by conduction. The energy balance leads to: C T k T where Density of [kg/m 3 ], C Heat caacity of [J/kg/K], k Thermal conductivity of [J/m/K/sec]. 4.3 Combustion on the Surface Boundary At the surface of the, the mechanism of heat transfer is very comlex. Heat is being introduced to the surface by radiation. Some of this heat may be reflected back. Combustion is taking lace on the surface and a fraction of this heat of reaction is fed back to the boundary. Heat is also leaving the surface by convection to the outside air that is assing the samle and being swet away through the exhaust tube. At this boundary, the heat flux entering the surface is described as: Q Surface φ H ( ) D h atm σ cone T Surface 4 4 ( T T ) + ε ( T ) where Q Heat flux at surface Surface [J/m /sec], φ ercent heat transferred by heat of combustion, H Heat of combustion [J/mol], h Heat transfer coefficient of gas at surface [J/m /K/sec], T Temerature of atmoshere atm [K], ε Emissivity, σ Stefan-Boltzmann For modeling convenience, the convection heat loss term can be combined into the heat of combustion term, therefore, eliminating one arameter that is not exactly known. Now, it is easy to adjust one arameter, φ, which sets the ercentage of heat that is transferred back to the surface. The heat of combustion term, H, can be found from the cone calorimeter exerimental data by integrating the heat release rate versus time curve or by simle means of disassociation of bond energies. 4.4 Heat Release Rate The heat release rate versus time is measured by heat of combustion multilied by the flux of gas diffusing out of the surface: HRR H ( ) Surface D where HRR Heat Release Rate [W/m ]. 5. Moving Mesh during Formation For the model geometry, the formation zone [x -x ] is initially set to nm, as to try to minimize the influence of this barrier at the beginning of the comutations. As time roceeds, the heats, and as the temerature rises, gas is roduced, but ignition does not start until some minimum level of gas surface concentration is reached (or minimum level of loss). When this minimum level of gas surface concentration is reached, ignition begins along with moving mesh to simulate formation. Inside the model, each subdomain zone is set to free dislacement, meaning that, the mesh is not fixed in each subdomain and can move. For x and x the mesh dislacement was set to zero, meaning that it was fixed in sace. For x, the mesh growth velocity must be set so that the grows at the correct rate. To arrive at the mesh growth velocity term, it is imortant to recognize the governing henomena that control this comlex system.
First, rate of mass growth of is equal to the same yrolysis kinetics which can be equated as: C ( α) ( α) where m C mass of [kg]. k m Dividing by the left hand side becomes C mc V Ax A where V is the volume and A is the area and x is the dislacement of the. The comlete balance is ( ) m ( ) k A α α Now by dividing both sides by A and multilying the right hand side to and bottom by a constant thickness, χ, which is the distance between x and x, 3mm. ( α) χ A χ ( α) k m χ m A χ By recognizing that (A. χ) is a constant volume, m /(A. χ) c and m /(A. χ) c. Therefore, the mesh velocity, mvel, can be defined as: mvel ( α) χ ( α) k χ c This relationshi directly shows how the hysical henomena are related to one another during growth. For modeling uroses, the c time derivative is used and must be secified at a articular boundary, because c is at boundary x where growth is taking lace. Therefore, the growth mesh velocity becomes: ( α ) χ mvel ( ) x where mvel /mesh formation velocity [m/s], χ Thickness constant, x -x [m]. This function should only become active when c reaches the minimum level of gas surface concentration for ignition and growth to take lace (or a minimum level of loss). If it were active at all times, then growth would be taking lace in the very beginning. Even though this value might be quite small, it would affect the heat transfer of the roblem enough to have drastic effects on the end results. In transient numerical modeling, a basic ste function (on/off) can not be used due to convergence issues. Rather, a smoothed ste function is used that lets the transition take lace over some given range. The ste function used in COMSOL Multihysics is called flchs. Now mvel can be written as: mvel flchs ( c x ( α) χ min, scale) ( ) where flc hs Smoothing ste function in COMSOL Multihysics, min Minimum level of concentration reduction for ignition and growth to begin [kg/m 3 ], scale Range over which transition takes lace [kg/m 3 ]. For the solver to use ( ) x and c x, they must be defined in COMSOL Multihysics in the Extrusion Couling Variables / Boundary Variables section. Here, names are created, ct3 and c3, that will be valid on surface x and will equal the negative of the c time derivative evaluated at x and the concentration evaluated at x, resectively. 6. Initial and Boundary Conditions Initial conditions for each zone of the roblem are shown in Table. Table. Initial Conditions, t. Initial Condition (x,) olymer [x -x ] Zone Formation [x -x ] c c c c c c T T T initial T T initial x
where T Initial Temerature [K]. initial Boundary conditions were set at the x and x boundaries. At x all boundary conditions are assumed to be constant interfacial boundary conditions where temerature and concentration are held constant at the interface. Boundary conditions are shown in Table. Boundary Condition Table. Boundary conditions at x and x. Boundary (x ;x 3,t) x x 3 c Flux Flux c Flux c Flux T Flux 4 φ H ( ) D + ε σ Tcone T 4 ( ) 7. Constants & Variables; COMSOL Conversion t The names of the variables and constants described above must be changed to use in COMSOL Multihysics. Conversion ts for the variables, constants, and arameters are shown in Table 3-5. Also included in the tables are the values for the constants and arameters used to best fit the exerimental data. All arameters in Table 5 is referenced at the bottom of the table. Some of the arameters come from TA data and came from the hysical roerties of olycarbonate. The arameters that had to be adjusted to fit the data are D,, C, k, φ, ε, scale, and c. These arameters min are within a realistic range. Table 3. Variable conversion t for COMSOL. Variables COMSOL Name Unit c c_ mol/m 3 c c_ mol/m 3 T T K t t sec x x m Table 4. Constants conversion t for COMSOL. Constants COMSOL Value Unit Name R Rconst 8.34 J/(mol. K) T Cone Theater 975 K T initial Tinitial 3 K σ sigma 5.673e-8 J/(m. K 4. sec) χ mconst.3 m Table 5. arameters conversion t for COMSOL. arameter COMSOL Name Value Unit A kconst 454 a /s a E Eact 66.59 A k k kconst*ex(- Eact/Rconst/T) b D D r c kj/mol /s m /s kg/m 3 C C c J/(kg. K) k k b J/(m. K. s) D D.8* -6 d m /sec r 35 d kg/m 3 C C d J/(kg. K) k k.5 d J/(m. K. s) e H DH 3655 J/kg a α alha.8 φ HRfrac d ε Emissivity.3 d scale scale d kg/m 3 c min 5 d kg/m 3 min mvel mvel Flchs(r- c3- min,scale)*(ct3)/rch ar*mconst*(-alha) m/s a ~ TA exerimentally determined data. b ~ olymer zone assumed to diffuse and heat equally. c ~ Tyical olycarbonate data. d ~ Reasonably adjusted arameters to fit exerimental data. e ~ H found by integrating HRR curve of C (see Figure ) and dividing by χ. With everything mentioned, the model is comlete. Adjustments to the arameters can then be made to fine tune the system. Results of the comutations are shown next. 8. Numerical Modeling Results 8. olycarbonate with No Flame Retardant Fine tuning of the model arameters must be made initially to fit the time of ignition data. This is done by setting the velocity to zero and adjusting the emissivity, ε, value. One other iece of data to check is the Total Heat Released, H, which is the area under the curve of the heat release rate versus time lot.
Now that the emissivity term has been set and the total area under the curves are the same, growth can now be imlemented. Turning the formation velocity term on, the values min and scale can be set. These values come from looking at a lot of c versus time and observing when formation should occur. The value that works best is a /4 loss of concentration for formation to begin. Therefore, min 5 and scale, so that formation begins at (5-)/ /4 loss of concentration. Figure 3 shows the modeled COMSOL data comared to the cone data. arameters used are those listed in Table 5. HRR (W/m ) 6.E+5 5.E+5 4.E+5 3.E+5.E+5.E+5.E+ Cone Data COMSOL Results 5 5 5 3 35 4 45 5 55 6 Time (sec) Figure 3. Heat Release Rate versus Time of C No FR; COMSOL Model Results with rowth comared to Cone Data. It is instructive to exlain in more detail the hysical henomena taking lace. These exlainations can be found in Statler s disseration 5 and will be resented at the COMSOL 8 Boston Conference. 8. The Influence of Flame Retardant Additive The use of two different flame retardants have be exaimed and numerical data fit well with the exerimental data. These data can be found in Statler s disseration 5 and will be resented at the COMSOL 8 Boston Conference. 9. Conclusions A mathematical model has been develoed to redict accurately the heat release rate of a forming ic material during combustion described under the cone calorimeter. From the model, time to ignition and eak heat release rate can be redicted for forming materials using comutational methods. The HRR after the eak (lateau region) can also be modeled with minor error. For this model to redict the heat release rate curves of a ic material some hysical roerties must be known, such as, : thermal conductivity, density, heat caacity, and : thermal conductivity, density, heat caacity, and diffusivity. These are quantities that can be measured by other instruments. Kinetic rate constants and amount of formed can be found by thermogravimetric analysis. Heat of combustion of a can be calculated by simle bond energies. From these data, heat release rate curves can be redicted and the use of a cone calorimeter is not necessary. However, a cone calorimeter will revail until confidence has been built in the use of the model for different ic materials.. References. C. Huggett, Estimation of Rate of Heat Release by Means of Oxygen Consumtion Measurements, Fire and Materials, 4, 6-65, 98. V. Babrauskas, Ten Years with Heat Release Research with the Cone Calorimeter, Fire Science and Technology Inc., htt://www.doctorfire.com/cone.html, 5 3. B.Fredlund, Modeling of Heat and Mass Transfer in Wood Structures During Fire, Fire Safety Journal,, 39-69, 993 4. J. Rychly and L. Costa, Modeling of olymer Ignition and Burning Adoted for Cone Calorimeter Measurements: The Correlation between the Rate of Heat Release and Oxygen Index, Fire and Materials, 9, 5-, 995 5. D. Statler, A Mechanistic and Modeling Study of Virgin and Recycled Flame Retarded olycarbonate, Dissertation, West Virginia University, 8.. Acknowledgements eorge and Carolyn Berry for the Berry Fellowshi. The Cone calorimeter data included here were obtained courtesy of rofessor les Wilkie of Marquette University.