- Email: [email protected]
Modelling of transport processes in a developing char
Polymer Degradation and Stability 93 (2008) 1205–1213
Contents lists available at ScienceDirect
Polymer Degradation and Stability journal homepage: www.elsevier.com/locate/polydegstab
Modelling of transport processes in a developing char } i b, Gy. Marosi b E. Farkas a, *, Z.G. Meszena a, A. Toldy b, S. Matko´ b, B.B. Marosfo a b
Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, Szt. Gelle´rt te´r 4., 1111 Budapest, Hungary ¨ egyetem rkp.3., P.O. Box 1521, 1111 Budapest, Hungary Department of Organic Chemistry and Technology, Budapest University of Technology and Economics, Mu
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 April 2007 Received in revised form 9 January 2008 Accepted 20 February 2008 Available online 8 March 2008
A mathematical model, simulation results and laboratory experiments are reported to describe the degradation of fire retardant polymeric materials. The model describes the heat and mass balances of a polymer layer with finite thickness. The degradation is initiated by a constant heat flux at the top of the layer. It is assumed that the polymer degrades to a fixed mass of char and volatile gas in an instantaneous step, at the moment when the temperature reaches a critical value. The most important heat transport mechanism is conduction, which dominates the temperature profile. The mass transport of gas is described by Darcy’s law, with a simplifying condition that the overall solid volume is constant during degradation. The transport processes have been modelled in one spatial dimension. Calculations and experiments have been carried out to establish the effects of critical parameters such as layer thickness, heat flux and material properties. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Char forming polymer Mathematical model Moving boundary Mass loss calorimeter Epoxy resin
1. Introduction Our everyday environment is full of synthetic polymers and plastics. It is important to be aware of the risks associated with the use of these materials. In addition to endurance problems and health risks, important issues have to be considered regarding flammability and the release of volatile gases. Modelling and computer simulation is a valuable tool to accompany or, sometimes, substitute expensive or often infeasible experiments. Simulations became more and more important tools to predict material parameters, however the complexity of these systems makes it difficult to mimic reality. The problem is not new – numerous articles and reviews were published in this research field [1–4]. There are two main types of polymer degradation: charring and decomposition without char formation. The moving boundaries of the char layer and the heat and mass transport in this continuously growing zone require special mathematical handling. To decrease the complexity of the problem simplifying conditions are frequently applied in these models. Some models do not include the material balance of the char; the pressure changes or the accumulation of gasses are ignored [5] and they focus only on the heat effect of reaction and the heat transfer of char layer [6]. Plastic materials are organic in nature and therefore very combustible. In the current state-of-the-art, it is not possible to model the manner in which they decompose and burn from the
* Corresponding author. Tel.: þ36 1 463 3174; fax: þ36 1 463 3953. E-mail address: [email protected] (E. Farkas). 0141-3910/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymdegradstab.2008.02.010
basic chemical reactions. In the numerical models the approximation of constant reaction heat or an Arrhenius expression is often applied [4]. A widely used simplification is that the polymer degrades into a fixed fraction of char and gas at the critical temperature. The reason for this simple description is the shortage of knowledge about the mechanism of gas transport in porous media containing a molten polymer. Model restrictions of course draw barriers to represent in detail the real systems, but can give basis for further investigations and model development. The phenomenon which has been modelled in this work is a polymer degradation process indicated by a constant heat flux according to mass loss calorimeter measurements. The algorithm is based on Staggs’s published model [7] but has been modified to fit the circumstances of a mass loss calorimetric measurement. The model describes both heat and mass changes and gives the whole temperature and pressure profiles of the system. In this present work we intend to describe the different behaviour of flame retarded epoxy resins and reference sample and determine the model parameters from the measurement data.
2. Experimental 2.1. Mass loss calorimeter measurement The fire resistance of the polymer has been characterized by mass loss calorimeter (according to ISO 13927, Fire Testing Technology, heat flux of 50 kW/m2) (Fig. 1).
1206
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
m
Nomenclature apressure athermal d 3 s m r s 4 f B c H1 H2 Hc K keff l L
volatile diffusivity of char (m2/s) thermal diffusivity of polymer (m2/s) relative location of char/polymer boundary emissivity of exposed surface Stephan–Boltzmann constant (W/m K4) dynamic viscosity of gas (Pa s) density (kg/m3) dimensionless time rate of re-radiated combustion heat char porosity permeability of porous media (m2) heat capacity (J/kg K) convection heat transfer coefficient at the exposed surface (W/K m2) convection heat transfer coefficient at the unexposed surface (W/K m2) combustion heat (J/kg) heat transport coefficient (W/m K) heat transport coefficient in char–gas system (W/m K) overall thickness (m) latent heat of gasification (J/kg)
The mass loss calorimeter measures mass loss rate, time to ignition and heat release rate during the burning of material or product specimens. At the core of the instrument is a radiant electrical heater in the shape of a truncated cone. This heating element irradiates a flat horizontal sample of 100 mm 100 mm and up to 50 mm thickness, placed beneath it, at a preset heating flux of up to 100 kW/m2 (in our case 50 kW/m2). The sample is placed on a load cell for continuous monitoring of its mass as it burns. Ignition can be optionally forced by an intermittent spark igniter located above the sample. Heat release is then determined as a product of the effective heat of combustion and the mass loss rate. Once the effective heats of combustion are known for a material over a range of conditions the heat release rates can be calculated from mass loss data.
P b P q_ 000 qg r s T b T t Ta Tcone Tc v x y
mass flux of the gas on the upper surface per unit area (kg/s m2) pressure (Pa) absolute pressure external heat flux of cone heater (W/m2) gas flux per unit area (m/s m2) char rate char/polymer interface (m) temperature (K) absolute temperature time (s) ambient temperature (room temperature) (K) temperature over the sample (K) critical temperature (K) pore velocity (m/s) transformed spatial coordinate spatial coordinate (m)
Subscripts a ambient c char g gas p polymer
2.2. Materials The polymer matrix was epoxy resin (ER) type Eporezit AH-16 (non-modified, resin like reacive diluent, epoxy equivalent: 160– 175; density at 258 C: 1.24 g/cm3) applied with Eporezit T-58 curing agent (amine number: 460–470 mg KOH/g; density at 208 C: 0.944 g/cm3) supplied by PþM Polimer Ke´mia Kft., Hungary. A synthesized phosphorus-containing reactive amine, TEDAP [8] (amine number: 510–530 mg KOH/g; N-content: 37.5 mass%, P-content: 13.84 mass%) was used as flame retardant. Pangel S9 (product of Tolsa Ltd.) type sepiolite (SEP) was applied as mineral clay additive. This way ER samples containing 1%, 2%, 5% SEP were prepared. The sample thickness was l ¼ 4 mm.
3. Model description
Fig. 1. Cone calorimeter.
As in the mass loss calorimeter measurements, the upper surface of polymer layer is exposed to constant level of heat irradiance ðq_ 000 Þ. At the onset of preheating, the sample is at ambient temperature (Ta) and its environment under the cone is approximately at 390–400 C (Tcone), based on the temperature measurement of sample space. After the preheating session, when the polymer surface reaches the critical temperature (Tc), it is assumed to degrade to a constant mass fraction of gas and char instantaneously. Over the sample the evolved gases are mixed with oxygen and ignite. In the mass loss calorimeter measurements the launch of degradation and the ignition are not at the same moment. The weight loss and slow gas emission are measured before seconds of the ignition. The model doesn’t handle this intermediate stage, only one critical temperature has been assumed for both phenomena. The beginning of the degradation is denoted as the moment of ignition. The ignition time is a characteristic value of the fire resistance of materials. During the degradation the heat sources are the flame and the cone heater. The absorbed heat increases the temperature of the sample and gives the required heat of gasification. The char is the porous solid phase, which remains after the decomposition and builds an increasing insulating layer over the virgin polymer.
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
The thickness of the degradation zone is assumed to be infinitely small and there is a sharp interface between the char and virgin polymer, therefore the gas and the molten polymer cannot coexist in the same volume of the model (Fig. 2). Each zone consists of a single material (virgin polymer or char) and has constant thermal parameters during the degradation process. In real systems between the char and the virgin polymer is a reaction zone, which is a mix of gas and molten polymer. This layer has a very complex mass transport, which is dependent on the viscosity change of molten polymer and the building and moving of gas bubbles. This liquid zone has a major importance if the sample is placed vertical, because the char layer can slip on this surface and therefore can lead to the deformation or break of the insulating board. In our work the samples are horizontally placed, therefore the detailed model of this layer has been neglected at this stage to simplify the model, it handles only the gas–char system. Most of the produced gas leaves the sample, the pressure gradient in the char depends on the gas diffusivity of the char. The reaction rate depends on the thermodynamic properties of the polymer and char. The char builds an insulating layer between the degradation zone and cone heater and hinders the heat transport to the virgin polymer. If the char has low thermal diffusivity, the virgin polymer layer attached to the thicker and thicker char reaches the critical temperature slowly and the present degradation process slows down. The main parameter of the calculated thermal conductivity is the gas–char ratio, which is considered to be constant during the degradation. In real systems it depends also on the structure of the char which can be different in the depth of the sample, but the model has been assumed to be a homogeneous char layer. As a further simplification it is assumed that the overall thickness of the sample does not change in the degradation process. The heat and mass transport are modelled in one spatial dimension. 3.1. Mass transport The released gas leaves the system through the char. The geometry of this porous medium in the mathematical model is simplified to a regular array of tubes. The modelling of gas flow in the char is based on the Navier–Stokes equation and with the simplified geometry, where it has been assumed that the produced gas flows through regular tubes in the char. A special form of Navier Stokes
1207
equation, the Darcy expression can be applied, which averages the flow over a representative volume giving a correlation between the gas pressure gradient and the average velocity of flow: B qg [ v,f [ L VP m
(1)
qg is the gas flux (discharge per unit area, with units of length per time, m/s), v is the pore velocity. The flux is divided by porosity to account for the fact that only a fraction of the total formation volume is available for flow. Darcy’s law can be applied only at low Reynolds numbers. In the cases studied the gas velocity is low, so it is reasonable to assume that the Reynolds number is still in the allowed range. It is combined with the expression of conservation of mass vr DVðrvÞ [ 0 vt
(2)
leading to the mass transport equation applied: v vt
b b BPa P P b V VP [ b b fm T T
(3)
The mass loss of the sample is the difference of the gas produced by decomposition and the fraction trapped in the char. The mass flux of the gas on the upper surface per unit area:
dm ds [ Lð1LrÞrp L dt dt
Zl f
drg dy dt
(4)
sðtÞ
3.2. Heat transport In the heat transport model three transport mechanisms have been built in: bulk conduction, convection by gas releasing and radiation across the pores. Heat transport has been separated into two parts. Each part has a different heat transport system. The first part is preheating, where the temperature change of the virgin polymer is modelled and this preheating phase is followed by degradation with a more complex polymer–char–gas system. During the preheating step the only heat transport mechanism assumed in the bulk polymer is conduction rp cp
b b vT v2 T [ kp 2 vt vy
(5)
and that on the surfaces is convection (Eqs. (6) and (7)). On the upper surface the effect of radiation under the ignition temperature can be neglected at lower temperatures (see in Section 5 Fig. 7.). Lkp
b vT b [ h2 1L T vy
at the lower surface of sample ðy [ 0Þ
(6)
b 3q_ 00 vT b b cone L T [ 0 Dh1 T vy Ta at the upper surface of sample ðy [ 1Þ without radiation
ð7Þ
b 3q_ 00 vT b 4 LT b D3sT 3 T b4 b cone L T [ 0 Dh1 T a cone vy Ta at the upper surface of sample ðy [ 1Þ assuming radiation
ð8Þ
kp
kp
Fig. 2. Schematic of sample.
The surface reaches the critical temperature at time tc and the upper side of polymer begins to degrade. The reaction layer is at the top, s(t) ¼ l at the beginning of the decomposition. This zone goes through the whole sample and the degradation ends if it reaches the lower side s(t) ¼ 0.
1208
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
The heat transport in the char takes place by conduction in the mixture of char and gas and by convection of the flowing gas in the char. The radiation within the char has been neglected. The gas and the surrounding char are in local thermal equilibrium all the time.
frg cg Dð1LfÞrc cc
b vT keff vy
y¼sD
rp cp
b b vT v vT Drg cg q [ keff vy vy vy vy
v T b
b vT Lkp vy
b b vT v2 T [ kp 2 vt vy
y¼sL
rp L ds [L Ta dt
in the char
(9)
b 3q_ 00 vT b D3sT 3 T b 4 D43Hc dm b cone L T b 4 LT [ 0 Dh1 T a cone vy Ta Ta dt
b vT b [ h2 1L T vy
v v 1Lx dd v 0 D vt vt d dt vx
(17)
v 1 v 0 ; vy lð1LdÞ vx
(11)
Dimensionless time has been applied, the time scale being the heat diffusivity of virgin polymer:
(12)
bt [
v v 1Dx dd v 0 D vt vt 1Ld dt vx
(18)
kp t rp cp l2
(19)
After carrying out all transformations the pressure equation may be written as d2
b b b b cg dd v P apressure v P v P vP Dð1LrÞ d [ b b b vx cp dbt vx T athermal vx T vbt T
(20)
(13)
where the factor apressure ¼ BPa =fm is the volatile diffusivity of char and athermal ¼ kp =rp cp is the thermal diffusivity of polymer. If the ratio of the pressure conductivity and thermal conductivity is large, the pressure can be given [7] in the form of
(14)
0 11=2 Z1 2rp athermal dd b b @ Pz 1D T dxA d rg;a apressure d bt
At the bottom of the sample ( y ¼ 0): Lkp
Coordinate values in the remaining polymer are in the range 1 x < 0. The degradation zone is fixed at x ¼ 0. The transformation of the derivatives in the char–gas system is
(10)
In the degradation zone y ¼ s(t): b [ Tc T Ta
(16)
and that in the polymer is
Boundary conditions on the upper surface: conduction, radiation and the heat source of combustion and cone heater. At the bottom: the same heat loss as in the preheating phase. On the top ( y ¼ 1): keff
yLsðtÞ sðtÞ
v 1 v 0 ; vy ld vx
in the degradation zone
in the virgin polymer
x[
(21)
x
Consequently, the heat balance equation of char is 4. Numerical solution The preheating and the degradation sessions were solved separately. The preheating equations build a set of partial differential equation system, which was solved without any numerical problem (i) with a 4th order Runge–Kutta method and (ii) with the Euler method. Over the time of resolution of 0.1 s there is no difference between the two methods. Because of the easier numerical handling the Euler method was applied in further calculations. The initial temperature profile for the degradation simulation is the result of the preheating calculations at the moment of ignition. One of the challenges in the model is the solution of the moving grid problem also known as the non-linear Stephan problem. The reaction zone separates the sample into two parts with different materials. The local coordinate of the char/polymer interface as well as the mass of char is changing during degradation. Theuns et al. [9] solved the space discretization with a uniform grid in the char and a non-uniform grid in the virgin polymer. The global solution of this set of equations is obtained with an iterative method. The numerical solution of the model equations with detailed heat and mass balance terms demands an iteration circle alone because of the interdependence of temperature and pressure equations. A second iteration circle leads to convergence problems and numerical failures. Another attempt to solve the Stephan problem is the Landau transformation [10], which results in a fixed interface of the char and virgin polymer, so the degradation zone and the surfaces have always constant local coordinates [5,7]. In Landau’s transformation the spatial coordinate x is defined as x[
yLsðtÞ lLsðtÞ
Coordinate values in the char are in the range of 0 < x 1.
(15)
ð1LrÞ
[
rg;a cg b r g Dr rp cc
b keff v2 T kp vx2
!
d2
b b
b cg dd v T vT dd v T Dð1LrÞ d Dð1LxÞd cp dbt vx vbt dbt vx ð22Þ
It has been assumed that the densities of char and virgin polymer are the same. The spatial dimension of the partial differential equations of the model has been discretised with the method of finite differences. The resulting set of ordinary differential equations has been attempted to be solved (i) with a 4th order Runge–Kutta method and (ii) with an implicit Euler method. Convergence of the numerical solution was only possible with the implicit Euler method. With Runge–Kutta numerical method, fluctuations in the temperature profiles appeared which are independent of the resolution of time and spatial coordinates. The equation set for preheating is solved in the original time and spatial dimension. The resulting temperature profile has been used as the initial condition of the degradation. In the degradation calculation loop first the relative location of the reaction zone is calculated. After that the pressure equation has been solved, followed by the calculation of new temperature profiles in the char and in the virgin polymer with the pressure profile. The process has been repeated until the difference in the temperature profiles was smaller than 0.01 C. The numerical stability of the simulation has been tested with various time and spatial coordinates. The numerical method is based on coordinate-transformation, in the early stage of degradation when the thickness of char is near zero, the temperature derivates are imprecise. Furthermore, the location of the degradation zone fluctuates and may result in negative gas flux values. The fluctuation and negative values decrease with higher time
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
1209
Fig. 3. Numerical stability at various time resolutions.
resolution. The shape and the location of the peak on the mass flux curves show no significant deviation, the bimodal character of the curve (Fig. 3) is not due to numerical failure. The model gives stable results at relatively low resolution of spatial coordinates, numerical instability appears under 20 local points. Simulation parameters applied based on Staggs’s paper [7]: kp [ 0:3 W=m K; kg [ 0:01 W=m K; rp [ 900 kg=m3 ; rg;a [ 1:16 kg=m3 ; cp [ 2500 J=kg K; cg [ 1000 J=kg K; l [ 104 ; Tc [ 600 K; Ta [ 293 K; Tcone [ 293 K; h1 [ 10 W=m2 ; r [ 0:5; q_ 000 [ 50 kW=m2 ; l [ 0:02 m; 3 [ 0:9; B=m [ 6:66310L9 m2 =Pa s; 4 [ 0; L [ 1000 kJ=kg
The heat transport coefficient in char–gas system has been derived from the Maxwell expression [7].
5. Results and discussion Simulations were carried out for characterizing the heat transport of the reference sample (AH-16 T-58 epoxy resin) and the fire resistant samples and identifying the differences. The preheating phase and the degradation phase were investigated in two steps. The degradation simulations were compared only with the reference sample AH-16 T-58, because the simplifying condition of constant sample thickness is valid approximately only in this case.
Fig. 4. Predicted and measured temperatures of surface in the preheating phase at various heat conductivity and heat capacity combinations, the measured ignition is at 40 s and at 432 C.
1210
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
Fig. 5. Predicted and measured ignition temperatures of sample at various heat conductivity and heat capacity combinations at 40 s.
In the preheating process the key parameters are heat capacity and heat conductivity of the virgin polymer. Published values of heat capacity of epoxy resins are in a wide range between 1000 and 2100 J/kg K [4,11–13]. Heat conductivity varies between 0.2 [11,13–15] and 0.8 W/m K [16]. Because of the significant differences in the published data, several parallel calculations have been carried out with various heat capacity and conductivity values to find the best fit to measured temperature profile. In three of the cases constant heat capacity and heat conductivity were assumed, in one of the cases only the heat capacity was increased with the temperature [11], and in two cases the heat capacity was changed linearly, while the heat conductivity was a function of square-root of temperature [13]. Temperature measurements were carried out to characterize the heat parameters of the AH-16 T-58 epoxy resin (Figs. 4 and 5).
Thermocouples were placed on the surface of the sample and the temperature was measured for the whole preheating process. Ignition was observed at 41 s, being in good agreement with the published data [8]. The surface temperature was 432 C at ignition. Fig. 6. shows the predicted local temperature of the sample at the end of preheating. Table 1 contains the predicted ignition times needed to reach the measured critical temperature. The predictions reproduce the tendency observed experimentally. The higher the heat conductivity of the epoxy resin, the higher is the preheating time: the absorbed heat reaches the lower region of the sample faster and is distributed more uniformly in the sample, therefore the difference between the temperatures of the top and the bottom surfaces is smaller. The heat capacity of polymer does not have any effect on the temperature profile of the preheating process, it determines the preheating time (Fig. 6). Higher heat capacity requires more
Fig. 6. Temperature profiles of sample at various heat conductivity and heat capacity combinations.
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213 Table 1
1211
Table 2
cp (J/kg K)
kp ¼ 0.2 W/m K
kp ¼ 0.6 W/m K
kp ¼ 0.8 W/m K
Material
Time to ignition (s)
1000 1500 2000 1110 at 0 C 1520 at 100 C 2110 at 200 C (A)
14.44 21.66 28.88 30.23
30.62 45.94 61.25 66.24
33.05 49.57 66.01 73.06
AH-16 T-58 AH-16 TEDAP AH-16 TEDAP þ Pangel S9 (1%, 2%, 5%)
40 96 186, 184, 125
42.21
significant (Fig. 7). With these conditions the predicted surface temperatures of the sample are over 500 C. Samples with additive Pangel S9 show higher preheating time. With the same heat conductivity values the simulated surface temperature is over 600 C (Fig. 7). After ignition the degradation depends on the latent heat of gasification, the char fraction and the heat conductivity of char. The measured char fraction for the AH-16 T-58 sample was r ¼ 0.055 [8]. Most of the published modelling works do not take into account combustion heat absorption. The recharged heat dramatically changes the shape of the simulated mass flux curves. It can be assumed that only a little part of the combustion heat returns back into the sample, the bigger part leaves the system absorbed by the surrounding medium and with the gas flow of cone calorimeter. Simulations were carried out for 5% and 10% adsorbed combustion heat, in the 10% case the gas emission peak appears nearly 100 s earlier and the maximum value increases by 30%. The peak is more asymmetric because of the faster decomposition rate at early stages of the reaction. The time needed until the degradation zone reaches the lower surface decreased from 1058 s to 894 s (Fig. 8.). The emitted gas burns continuously over the sample surface and higher heat emission leads to higher gas emission, hence the degradation runs faster. The higher amount of evolved gas builds up a higher pressure in the char. If the pressure reaches a critical value, the char layer cracks and loses its heat insulating property. With the values of char permeability applied, the pressure increases in the char by 0.005–0.01 bar, which can be critical to epoxy resin samples (Fig. 9). After the formation of a thicker insulating char layer, the reaction slows down and the gas emission decreases also. This tendency can be seen in the measured gas emissivity of AH-16 T-58 too, but the degradation time is nearly one order of magnitude less than and the peak value of the curve is 800% of the predicted value. It can be assumed that the sample
(B) [13]
cp ¼ 1100 J/kg K, kp ¼ 0.19 W/m K at 20 C cp ¼ 2670 J/kg K, kp ¼ 0.3 W/m K at 432 C cp ¼ 1700 J/kg K, kp ¼ 0.19 W/m K at 20 C cp ¼ 4000 J/kg K, kp ¼ 0.3 W/m K at 432 C
53.24
absorbed heat, therefore it takes more time to reach the critical temperature (Fig. 4). Comparing experiments to simulation results, the kp values must be lower than 0.6 W/m K at constant heat capacity. The best fit to measured data was found with constant values of cp ¼ 1400 J/kg K and kp ¼ 0.5 W/m K, however it is unlikely that these parameters do not change with temperature. If temperature dependent heat conductivity and heat capacity values were used in the model, the measured ignition time is best reproduced in case A (Table 1), but the predicted temperature curve is higher than the measured surface temperature (Figs. 4 and 5). With higher initial heat capacity the measured ignition time is far bellow the calculated one. It can be concluded that the heat capacity increase is not linear with temperature. To get better input parameters for the model, DSC and m-TA measurements are planned. Better parameter fitting could be reached by performing more detailed temperature measurements in different depths of the sample, however, placing thermocouples into the melting polymer at defined depth is hardly feasible. As another reference curve the temperature of bottom surface is planned to be measured. In the degradation simulations constant values of cp ¼ 2670 J/kg K, kp ¼ 0.3 W/m K are used, similar to most published values [11,13–15]. The fire resistant samples (where T-58 is substituted with TEDAP) reach higher ignition times (Table 2). Assuming that the thermal properties of the samples do not change, TEDAP increases the thermal stability of polymer [8] and the surface can bear higher temperature. Over 450 C the heat loss by radiation became
Fig. 7. Simulated preheating time using thermal properties A (Table 1) and assuming radiation.
1212
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
Fig. 8. Predicted gas flux curves at different combustion heat adsorption levels and the measured gas flux of AH-16 T-58 sample.
absorbs more heat during the measurement than that predicted in the simulation. More work is needed for a better fit of the predicted values to the experimental results. It can be concluded that the amount of re-radiated heat influences the degradation process significantly and, therefore, cannot be neglected in the model. Applied simulation parameters for epoxy simulations based on published data [7,8,13]: kp [ 0:19 W=m K ðat 293 KÞ; kp [ 0:3 W=m K ðat 705 KÞ; kg [ 0:01 W=m K; rp [ 1200 kg=m3 ; rg;a [ 1:16 kg=m3 ; cg [ 1000 J=kg K; cp [ 2670 J=kg K ðin degradationÞ; Tc [ 705 K; Ta [ 293 K; Tcone [ 653 K; h1 [ 10 W=m2 ; r [ 0:055; q_ 000 [ 50 kW=m2 ; l [ 0:004 m; 3 [ 0:9; B=m [ 3:26310L10 m2 =Pa s; 4 [ 0; 0:05; 0:1; Hc [ 20; 400 kJ=kg; L [ 1600 kJ=kg
The heat transport coefficient in char–gas system was given by the Maxwell expression.
6. Topics of further model development In the next stage of model and simulation development input parameters should be improved and measured to expand the applicability of the model for all epoxy resin samples. It has been shown that data taken from the literature are not enough, measured values are needed especially for the novel fire retardant epoxy resins in case of which no published data are available. Heat capacity and heat conductivity investigations will be carried out based on differential scanning calorimetry and m-TA for the virgin polymer samples and for the char as well. The degradation model of fire retardant epoxy resins has to handle the volume change at charring; to solve this problem in the mass and heat equations it is planned to include a function of the changing char-gas fraction, which is able to describe also the
Fig. 9. Predicted evolution of pressure on the char/gas interface (x ¼ 0) at different combustion heat adsorptions.
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
gradient structure of char. For the investigation of intumescence laser pyrolysis treatment is applicable. Another possible way of development is the kinetic modelling of degradation. At the same time in our department the investigation of decomposition mechanism is carried out [17]. 7. Conclusion Laboratory experiments together with a mathematical model and simulation results are reported to describe the degradation of pure and flame retarded epoxy resins initiated by a constant heat flux. The applied model is based on Staggs’s work and describes both the heat and mass changes and predicts the whole temperature and pressure profiles of the system. Experiments in a mass loss calorimeter and computer simulations have been carried out to establish the effects of critical parameters such as layer thickness, heat flux and material properties. The predicted ignition times and critical temperatures were in good agreement with the experimental data. Furthermore it could be concluded that the heat capacity of polymer does not have any effect on the temperature profile of the preheating process, it determines the preheating time instead. The effect of re-radiated combustion heat was established and it has been found that the amount of absorbed pyrolysis heat is an important factor in the degradation model. The simplification applied in this work can be accepted only as a first attempt to describe the behaviour of the reference material. In order to achieve better correlation with the experimental results of fire retarded sample improvement of the model is planned. Acknowledgement Authors acknowledge the financial supports received through the EU-6 Framework Programs (NMP3-CT-2004-505637 and IP 026685-2) and OTKA (project nos. 046460 and T49121).
1213
References [1] Di Blasi C. Modeling and simulation of combustion processes of charring and non-charring solid fuels. Progress in Energy and Combustion Science 1993;19: 71–104. [2] Staggs JEJ. A simple model of polymer pyrolysis including transport of volatiles. Fire Safety Journal 2000;34:69–80. [3] Hopkins Jr D, Quintiere JG. Material fire properties and predictions for thermoplastics. Fire Safety Journal 1996;26:241–68. [4] Davies JM, Wang YC, Wong PMH. Polymer composites in fire. Composites: Part A 2006;37:1131–41. [5] Lin WS. Quasi-steady solutions for the ablation of charring materials. Heat and Mass Transfer 2007. [6] Fei You, Lin Zhang, Jianjun Zhou, Chow TT. Direct numerical simulation on the pyrolysis of materials with volatile residue layer. Applied Mathematical Modelling 2006. [7] Staggs JEJ. Heat and mass transport in developing chars. Polymer Degradation and Stability 2003;82:297–307. [8] Toldy A, To´th N, Anna P, Keglevich Gy, Kiss K, Marosi Gy. Flame retardancy of epoxy resin with phosphorus containing reactive amine and clay minerals. Polymers for Advanced Technologies 2006;17:1–4. [9] Theuns E, Vierendeels J, Vandevelde P. Validation of the integral model for the pyrolysis of charring materials with a moving grid. Journal of Computational and Applied Mathematics 2004;168:471–9. [10] Landau HG. Heat conduction in a melting solid. Quarterly of Applied Mathematics 1950;8(1):81–94. [11] Asher J, editor. Kaye and Laby, tables of physical and chemical constants. UK: Longman. 16th ed. (web edition), http://www.kayelaby.npl.co.uk/chemistry/ 3_11/3_11_1.html; 1995, http://www.kayelaby.npl.co.uk/general_physics/2_3/ 2_3_6.html; 1995. [12] Swier S, van Assche G, Van Hemelrijck A, Rahier H, Verdonck E, Van Mele B. Characterization of reacting polymer systems by temperature-modulated differential scanning calorimetry. Journal of Thermal Analysis 1998;54: 585–604. [13] Lyon RE, Janssens ML. Polymer flammability (report). Southwest Research Institute; 2005. [14] 15th ed.. In: Dean JA, editor. Lange’s handbook of chemistry. New York: McGraw-Hill; 1999 [15] Harper CA, editor. Modern plastics handbook. New York: McGraw-Hill; 2000. [16] Domininghaus H, editor. Kunststoffe. Du¨sseldorf: VDI-Verl; 1969. [17] Toldy A, Anna P, Csontos I, Szabo´ A, Marosi Gy. Intrinsically flame retardant epoxy resin – fire performance and background – part 1. Polymer Degradation and Stability 2007. available on-line.
Contents lists available at ScienceDirect
Polymer Degradation and Stability journal homepage: www.elsevier.com/locate/polydegstab
Modelling of transport processes in a developing char } i b, Gy. Marosi b E. Farkas a, *, Z.G. Meszena a, A. Toldy b, S. Matko´ b, B.B. Marosfo a b
Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, Szt. Gelle´rt te´r 4., 1111 Budapest, Hungary ¨ egyetem rkp.3., P.O. Box 1521, 1111 Budapest, Hungary Department of Organic Chemistry and Technology, Budapest University of Technology and Economics, Mu
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 April 2007 Received in revised form 9 January 2008 Accepted 20 February 2008 Available online 8 March 2008
A mathematical model, simulation results and laboratory experiments are reported to describe the degradation of fire retardant polymeric materials. The model describes the heat and mass balances of a polymer layer with finite thickness. The degradation is initiated by a constant heat flux at the top of the layer. It is assumed that the polymer degrades to a fixed mass of char and volatile gas in an instantaneous step, at the moment when the temperature reaches a critical value. The most important heat transport mechanism is conduction, which dominates the temperature profile. The mass transport of gas is described by Darcy’s law, with a simplifying condition that the overall solid volume is constant during degradation. The transport processes have been modelled in one spatial dimension. Calculations and experiments have been carried out to establish the effects of critical parameters such as layer thickness, heat flux and material properties. Ó 2008 Elsevier Ltd. All rights reserved.
Keywords: Char forming polymer Mathematical model Moving boundary Mass loss calorimeter Epoxy resin
1. Introduction Our everyday environment is full of synthetic polymers and plastics. It is important to be aware of the risks associated with the use of these materials. In addition to endurance problems and health risks, important issues have to be considered regarding flammability and the release of volatile gases. Modelling and computer simulation is a valuable tool to accompany or, sometimes, substitute expensive or often infeasible experiments. Simulations became more and more important tools to predict material parameters, however the complexity of these systems makes it difficult to mimic reality. The problem is not new – numerous articles and reviews were published in this research field [1–4]. There are two main types of polymer degradation: charring and decomposition without char formation. The moving boundaries of the char layer and the heat and mass transport in this continuously growing zone require special mathematical handling. To decrease the complexity of the problem simplifying conditions are frequently applied in these models. Some models do not include the material balance of the char; the pressure changes or the accumulation of gasses are ignored [5] and they focus only on the heat effect of reaction and the heat transfer of char layer [6]. Plastic materials are organic in nature and therefore very combustible. In the current state-of-the-art, it is not possible to model the manner in which they decompose and burn from the
* Corresponding author. Tel.: þ36 1 463 3174; fax: þ36 1 463 3953. E-mail address: [email protected] (E. Farkas). 0141-3910/$ – see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymdegradstab.2008.02.010
basic chemical reactions. In the numerical models the approximation of constant reaction heat or an Arrhenius expression is often applied [4]. A widely used simplification is that the polymer degrades into a fixed fraction of char and gas at the critical temperature. The reason for this simple description is the shortage of knowledge about the mechanism of gas transport in porous media containing a molten polymer. Model restrictions of course draw barriers to represent in detail the real systems, but can give basis for further investigations and model development. The phenomenon which has been modelled in this work is a polymer degradation process indicated by a constant heat flux according to mass loss calorimeter measurements. The algorithm is based on Staggs’s published model [7] but has been modified to fit the circumstances of a mass loss calorimetric measurement. The model describes both heat and mass changes and gives the whole temperature and pressure profiles of the system. In this present work we intend to describe the different behaviour of flame retarded epoxy resins and reference sample and determine the model parameters from the measurement data.
2. Experimental 2.1. Mass loss calorimeter measurement The fire resistance of the polymer has been characterized by mass loss calorimeter (according to ISO 13927, Fire Testing Technology, heat flux of 50 kW/m2) (Fig. 1).
1206
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
m
Nomenclature apressure athermal d 3 s m r s 4 f B c H1 H2 Hc K keff l L
volatile diffusivity of char (m2/s) thermal diffusivity of polymer (m2/s) relative location of char/polymer boundary emissivity of exposed surface Stephan–Boltzmann constant (W/m K4) dynamic viscosity of gas (Pa s) density (kg/m3) dimensionless time rate of re-radiated combustion heat char porosity permeability of porous media (m2) heat capacity (J/kg K) convection heat transfer coefficient at the exposed surface (W/K m2) convection heat transfer coefficient at the unexposed surface (W/K m2) combustion heat (J/kg) heat transport coefficient (W/m K) heat transport coefficient in char–gas system (W/m K) overall thickness (m) latent heat of gasification (J/kg)
The mass loss calorimeter measures mass loss rate, time to ignition and heat release rate during the burning of material or product specimens. At the core of the instrument is a radiant electrical heater in the shape of a truncated cone. This heating element irradiates a flat horizontal sample of 100 mm 100 mm and up to 50 mm thickness, placed beneath it, at a preset heating flux of up to 100 kW/m2 (in our case 50 kW/m2). The sample is placed on a load cell for continuous monitoring of its mass as it burns. Ignition can be optionally forced by an intermittent spark igniter located above the sample. Heat release is then determined as a product of the effective heat of combustion and the mass loss rate. Once the effective heats of combustion are known for a material over a range of conditions the heat release rates can be calculated from mass loss data.
P b P q_ 000 qg r s T b T t Ta Tcone Tc v x y
mass flux of the gas on the upper surface per unit area (kg/s m2) pressure (Pa) absolute pressure external heat flux of cone heater (W/m2) gas flux per unit area (m/s m2) char rate char/polymer interface (m) temperature (K) absolute temperature time (s) ambient temperature (room temperature) (K) temperature over the sample (K) critical temperature (K) pore velocity (m/s) transformed spatial coordinate spatial coordinate (m)
Subscripts a ambient c char g gas p polymer
2.2. Materials The polymer matrix was epoxy resin (ER) type Eporezit AH-16 (non-modified, resin like reacive diluent, epoxy equivalent: 160– 175; density at 258 C: 1.24 g/cm3) applied with Eporezit T-58 curing agent (amine number: 460–470 mg KOH/g; density at 208 C: 0.944 g/cm3) supplied by PþM Polimer Ke´mia Kft., Hungary. A synthesized phosphorus-containing reactive amine, TEDAP [8] (amine number: 510–530 mg KOH/g; N-content: 37.5 mass%, P-content: 13.84 mass%) was used as flame retardant. Pangel S9 (product of Tolsa Ltd.) type sepiolite (SEP) was applied as mineral clay additive. This way ER samples containing 1%, 2%, 5% SEP were prepared. The sample thickness was l ¼ 4 mm.
3. Model description
Fig. 1. Cone calorimeter.
As in the mass loss calorimeter measurements, the upper surface of polymer layer is exposed to constant level of heat irradiance ðq_ 000 Þ. At the onset of preheating, the sample is at ambient temperature (Ta) and its environment under the cone is approximately at 390–400 C (Tcone), based on the temperature measurement of sample space. After the preheating session, when the polymer surface reaches the critical temperature (Tc), it is assumed to degrade to a constant mass fraction of gas and char instantaneously. Over the sample the evolved gases are mixed with oxygen and ignite. In the mass loss calorimeter measurements the launch of degradation and the ignition are not at the same moment. The weight loss and slow gas emission are measured before seconds of the ignition. The model doesn’t handle this intermediate stage, only one critical temperature has been assumed for both phenomena. The beginning of the degradation is denoted as the moment of ignition. The ignition time is a characteristic value of the fire resistance of materials. During the degradation the heat sources are the flame and the cone heater. The absorbed heat increases the temperature of the sample and gives the required heat of gasification. The char is the porous solid phase, which remains after the decomposition and builds an increasing insulating layer over the virgin polymer.
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
The thickness of the degradation zone is assumed to be infinitely small and there is a sharp interface between the char and virgin polymer, therefore the gas and the molten polymer cannot coexist in the same volume of the model (Fig. 2). Each zone consists of a single material (virgin polymer or char) and has constant thermal parameters during the degradation process. In real systems between the char and the virgin polymer is a reaction zone, which is a mix of gas and molten polymer. This layer has a very complex mass transport, which is dependent on the viscosity change of molten polymer and the building and moving of gas bubbles. This liquid zone has a major importance if the sample is placed vertical, because the char layer can slip on this surface and therefore can lead to the deformation or break of the insulating board. In our work the samples are horizontally placed, therefore the detailed model of this layer has been neglected at this stage to simplify the model, it handles only the gas–char system. Most of the produced gas leaves the sample, the pressure gradient in the char depends on the gas diffusivity of the char. The reaction rate depends on the thermodynamic properties of the polymer and char. The char builds an insulating layer between the degradation zone and cone heater and hinders the heat transport to the virgin polymer. If the char has low thermal diffusivity, the virgin polymer layer attached to the thicker and thicker char reaches the critical temperature slowly and the present degradation process slows down. The main parameter of the calculated thermal conductivity is the gas–char ratio, which is considered to be constant during the degradation. In real systems it depends also on the structure of the char which can be different in the depth of the sample, but the model has been assumed to be a homogeneous char layer. As a further simplification it is assumed that the overall thickness of the sample does not change in the degradation process. The heat and mass transport are modelled in one spatial dimension. 3.1. Mass transport The released gas leaves the system through the char. The geometry of this porous medium in the mathematical model is simplified to a regular array of tubes. The modelling of gas flow in the char is based on the Navier–Stokes equation and with the simplified geometry, where it has been assumed that the produced gas flows through regular tubes in the char. A special form of Navier Stokes
1207
equation, the Darcy expression can be applied, which averages the flow over a representative volume giving a correlation between the gas pressure gradient and the average velocity of flow: B qg [ v,f [ L VP m
(1)
qg is the gas flux (discharge per unit area, with units of length per time, m/s), v is the pore velocity. The flux is divided by porosity to account for the fact that only a fraction of the total formation volume is available for flow. Darcy’s law can be applied only at low Reynolds numbers. In the cases studied the gas velocity is low, so it is reasonable to assume that the Reynolds number is still in the allowed range. It is combined with the expression of conservation of mass vr DVðrvÞ [ 0 vt
(2)
leading to the mass transport equation applied: v vt
b b BPa P P b V VP [ b b fm T T
(3)
The mass loss of the sample is the difference of the gas produced by decomposition and the fraction trapped in the char. The mass flux of the gas on the upper surface per unit area:
dm ds [ Lð1LrÞrp L dt dt
Zl f
drg dy dt
(4)
sðtÞ
3.2. Heat transport In the heat transport model three transport mechanisms have been built in: bulk conduction, convection by gas releasing and radiation across the pores. Heat transport has been separated into two parts. Each part has a different heat transport system. The first part is preheating, where the temperature change of the virgin polymer is modelled and this preheating phase is followed by degradation with a more complex polymer–char–gas system. During the preheating step the only heat transport mechanism assumed in the bulk polymer is conduction rp cp
b b vT v2 T [ kp 2 vt vy
(5)
and that on the surfaces is convection (Eqs. (6) and (7)). On the upper surface the effect of radiation under the ignition temperature can be neglected at lower temperatures (see in Section 5 Fig. 7.). Lkp
b vT b [ h2 1L T vy
at the lower surface of sample ðy [ 0Þ
(6)
b 3q_ 00 vT b b cone L T [ 0 Dh1 T vy Ta at the upper surface of sample ðy [ 1Þ without radiation
ð7Þ
b 3q_ 00 vT b 4 LT b D3sT 3 T b4 b cone L T [ 0 Dh1 T a cone vy Ta at the upper surface of sample ðy [ 1Þ assuming radiation
ð8Þ
kp
kp
Fig. 2. Schematic of sample.
The surface reaches the critical temperature at time tc and the upper side of polymer begins to degrade. The reaction layer is at the top, s(t) ¼ l at the beginning of the decomposition. This zone goes through the whole sample and the degradation ends if it reaches the lower side s(t) ¼ 0.
1208
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
The heat transport in the char takes place by conduction in the mixture of char and gas and by convection of the flowing gas in the char. The radiation within the char has been neglected. The gas and the surrounding char are in local thermal equilibrium all the time.
frg cg Dð1LfÞrc cc
b vT keff vy
y¼sD
rp cp
b b vT v vT Drg cg q [ keff vy vy vy vy
v T b
b vT Lkp vy
b b vT v2 T [ kp 2 vt vy
y¼sL
rp L ds [L Ta dt
in the char
(9)
b 3q_ 00 vT b D3sT 3 T b 4 D43Hc dm b cone L T b 4 LT [ 0 Dh1 T a cone vy Ta Ta dt
b vT b [ h2 1L T vy
v v 1Lx dd v 0 D vt vt d dt vx
(17)
v 1 v 0 ; vy lð1LdÞ vx
(11)
Dimensionless time has been applied, the time scale being the heat diffusivity of virgin polymer:
(12)
bt [
v v 1Dx dd v 0 D vt vt 1Ld dt vx
(18)
kp t rp cp l2
(19)
After carrying out all transformations the pressure equation may be written as d2
b b b b cg dd v P apressure v P v P vP Dð1LrÞ d [ b b b vx cp dbt vx T athermal vx T vbt T
(20)
(13)
where the factor apressure ¼ BPa =fm is the volatile diffusivity of char and athermal ¼ kp =rp cp is the thermal diffusivity of polymer. If the ratio of the pressure conductivity and thermal conductivity is large, the pressure can be given [7] in the form of
(14)
0 11=2 Z1 2rp athermal dd b b @ Pz 1D T dxA d rg;a apressure d bt
At the bottom of the sample ( y ¼ 0): Lkp
Coordinate values in the remaining polymer are in the range 1 x < 0. The degradation zone is fixed at x ¼ 0. The transformation of the derivatives in the char–gas system is
(10)
In the degradation zone y ¼ s(t): b [ Tc T Ta
(16)
and that in the polymer is
Boundary conditions on the upper surface: conduction, radiation and the heat source of combustion and cone heater. At the bottom: the same heat loss as in the preheating phase. On the top ( y ¼ 1): keff
yLsðtÞ sðtÞ
v 1 v 0 ; vy ld vx
in the degradation zone
in the virgin polymer
x[
(21)
x
Consequently, the heat balance equation of char is 4. Numerical solution The preheating and the degradation sessions were solved separately. The preheating equations build a set of partial differential equation system, which was solved without any numerical problem (i) with a 4th order Runge–Kutta method and (ii) with the Euler method. Over the time of resolution of 0.1 s there is no difference between the two methods. Because of the easier numerical handling the Euler method was applied in further calculations. The initial temperature profile for the degradation simulation is the result of the preheating calculations at the moment of ignition. One of the challenges in the model is the solution of the moving grid problem also known as the non-linear Stephan problem. The reaction zone separates the sample into two parts with different materials. The local coordinate of the char/polymer interface as well as the mass of char is changing during degradation. Theuns et al. [9] solved the space discretization with a uniform grid in the char and a non-uniform grid in the virgin polymer. The global solution of this set of equations is obtained with an iterative method. The numerical solution of the model equations with detailed heat and mass balance terms demands an iteration circle alone because of the interdependence of temperature and pressure equations. A second iteration circle leads to convergence problems and numerical failures. Another attempt to solve the Stephan problem is the Landau transformation [10], which results in a fixed interface of the char and virgin polymer, so the degradation zone and the surfaces have always constant local coordinates [5,7]. In Landau’s transformation the spatial coordinate x is defined as x[
yLsðtÞ lLsðtÞ
Coordinate values in the char are in the range of 0 < x 1.
(15)
ð1LrÞ
[
rg;a cg b r g Dr rp cc
b keff v2 T kp vx2
!
d2
b b
b cg dd v T vT dd v T Dð1LrÞ d Dð1LxÞd cp dbt vx vbt dbt vx ð22Þ
It has been assumed that the densities of char and virgin polymer are the same. The spatial dimension of the partial differential equations of the model has been discretised with the method of finite differences. The resulting set of ordinary differential equations has been attempted to be solved (i) with a 4th order Runge–Kutta method and (ii) with an implicit Euler method. Convergence of the numerical solution was only possible with the implicit Euler method. With Runge–Kutta numerical method, fluctuations in the temperature profiles appeared which are independent of the resolution of time and spatial coordinates. The equation set for preheating is solved in the original time and spatial dimension. The resulting temperature profile has been used as the initial condition of the degradation. In the degradation calculation loop first the relative location of the reaction zone is calculated. After that the pressure equation has been solved, followed by the calculation of new temperature profiles in the char and in the virgin polymer with the pressure profile. The process has been repeated until the difference in the temperature profiles was smaller than 0.01 C. The numerical stability of the simulation has been tested with various time and spatial coordinates. The numerical method is based on coordinate-transformation, in the early stage of degradation when the thickness of char is near zero, the temperature derivates are imprecise. Furthermore, the location of the degradation zone fluctuates and may result in negative gas flux values. The fluctuation and negative values decrease with higher time
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
1209
Fig. 3. Numerical stability at various time resolutions.
resolution. The shape and the location of the peak on the mass flux curves show no significant deviation, the bimodal character of the curve (Fig. 3) is not due to numerical failure. The model gives stable results at relatively low resolution of spatial coordinates, numerical instability appears under 20 local points. Simulation parameters applied based on Staggs’s paper [7]: kp [ 0:3 W=m K; kg [ 0:01 W=m K; rp [ 900 kg=m3 ; rg;a [ 1:16 kg=m3 ; cp [ 2500 J=kg K; cg [ 1000 J=kg K; l [ 104 ; Tc [ 600 K; Ta [ 293 K; Tcone [ 293 K; h1 [ 10 W=m2 ; r [ 0:5; q_ 000 [ 50 kW=m2 ; l [ 0:02 m; 3 [ 0:9; B=m [ 6:66310L9 m2 =Pa s; 4 [ 0; L [ 1000 kJ=kg
The heat transport coefficient in char–gas system has been derived from the Maxwell expression [7].
5. Results and discussion Simulations were carried out for characterizing the heat transport of the reference sample (AH-16 T-58 epoxy resin) and the fire resistant samples and identifying the differences. The preheating phase and the degradation phase were investigated in two steps. The degradation simulations were compared only with the reference sample AH-16 T-58, because the simplifying condition of constant sample thickness is valid approximately only in this case.
Fig. 4. Predicted and measured temperatures of surface in the preheating phase at various heat conductivity and heat capacity combinations, the measured ignition is at 40 s and at 432 C.
1210
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
Fig. 5. Predicted and measured ignition temperatures of sample at various heat conductivity and heat capacity combinations at 40 s.
In the preheating process the key parameters are heat capacity and heat conductivity of the virgin polymer. Published values of heat capacity of epoxy resins are in a wide range between 1000 and 2100 J/kg K [4,11–13]. Heat conductivity varies between 0.2 [11,13–15] and 0.8 W/m K [16]. Because of the significant differences in the published data, several parallel calculations have been carried out with various heat capacity and conductivity values to find the best fit to measured temperature profile. In three of the cases constant heat capacity and heat conductivity were assumed, in one of the cases only the heat capacity was increased with the temperature [11], and in two cases the heat capacity was changed linearly, while the heat conductivity was a function of square-root of temperature [13]. Temperature measurements were carried out to characterize the heat parameters of the AH-16 T-58 epoxy resin (Figs. 4 and 5).
Thermocouples were placed on the surface of the sample and the temperature was measured for the whole preheating process. Ignition was observed at 41 s, being in good agreement with the published data [8]. The surface temperature was 432 C at ignition. Fig. 6. shows the predicted local temperature of the sample at the end of preheating. Table 1 contains the predicted ignition times needed to reach the measured critical temperature. The predictions reproduce the tendency observed experimentally. The higher the heat conductivity of the epoxy resin, the higher is the preheating time: the absorbed heat reaches the lower region of the sample faster and is distributed more uniformly in the sample, therefore the difference between the temperatures of the top and the bottom surfaces is smaller. The heat capacity of polymer does not have any effect on the temperature profile of the preheating process, it determines the preheating time (Fig. 6). Higher heat capacity requires more
Fig. 6. Temperature profiles of sample at various heat conductivity and heat capacity combinations.
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213 Table 1
1211
Table 2
cp (J/kg K)
kp ¼ 0.2 W/m K
kp ¼ 0.6 W/m K
kp ¼ 0.8 W/m K
Material
Time to ignition (s)
1000 1500 2000 1110 at 0 C 1520 at 100 C 2110 at 200 C (A)
14.44 21.66 28.88 30.23
30.62 45.94 61.25 66.24
33.05 49.57 66.01 73.06
AH-16 T-58 AH-16 TEDAP AH-16 TEDAP þ Pangel S9 (1%, 2%, 5%)
40 96 186, 184, 125
42.21
significant (Fig. 7). With these conditions the predicted surface temperatures of the sample are over 500 C. Samples with additive Pangel S9 show higher preheating time. With the same heat conductivity values the simulated surface temperature is over 600 C (Fig. 7). After ignition the degradation depends on the latent heat of gasification, the char fraction and the heat conductivity of char. The measured char fraction for the AH-16 T-58 sample was r ¼ 0.055 [8]. Most of the published modelling works do not take into account combustion heat absorption. The recharged heat dramatically changes the shape of the simulated mass flux curves. It can be assumed that only a little part of the combustion heat returns back into the sample, the bigger part leaves the system absorbed by the surrounding medium and with the gas flow of cone calorimeter. Simulations were carried out for 5% and 10% adsorbed combustion heat, in the 10% case the gas emission peak appears nearly 100 s earlier and the maximum value increases by 30%. The peak is more asymmetric because of the faster decomposition rate at early stages of the reaction. The time needed until the degradation zone reaches the lower surface decreased from 1058 s to 894 s (Fig. 8.). The emitted gas burns continuously over the sample surface and higher heat emission leads to higher gas emission, hence the degradation runs faster. The higher amount of evolved gas builds up a higher pressure in the char. If the pressure reaches a critical value, the char layer cracks and loses its heat insulating property. With the values of char permeability applied, the pressure increases in the char by 0.005–0.01 bar, which can be critical to epoxy resin samples (Fig. 9). After the formation of a thicker insulating char layer, the reaction slows down and the gas emission decreases also. This tendency can be seen in the measured gas emissivity of AH-16 T-58 too, but the degradation time is nearly one order of magnitude less than and the peak value of the curve is 800% of the predicted value. It can be assumed that the sample
(B) [13]
cp ¼ 1100 J/kg K, kp ¼ 0.19 W/m K at 20 C cp ¼ 2670 J/kg K, kp ¼ 0.3 W/m K at 432 C cp ¼ 1700 J/kg K, kp ¼ 0.19 W/m K at 20 C cp ¼ 4000 J/kg K, kp ¼ 0.3 W/m K at 432 C
53.24
absorbed heat, therefore it takes more time to reach the critical temperature (Fig. 4). Comparing experiments to simulation results, the kp values must be lower than 0.6 W/m K at constant heat capacity. The best fit to measured data was found with constant values of cp ¼ 1400 J/kg K and kp ¼ 0.5 W/m K, however it is unlikely that these parameters do not change with temperature. If temperature dependent heat conductivity and heat capacity values were used in the model, the measured ignition time is best reproduced in case A (Table 1), but the predicted temperature curve is higher than the measured surface temperature (Figs. 4 and 5). With higher initial heat capacity the measured ignition time is far bellow the calculated one. It can be concluded that the heat capacity increase is not linear with temperature. To get better input parameters for the model, DSC and m-TA measurements are planned. Better parameter fitting could be reached by performing more detailed temperature measurements in different depths of the sample, however, placing thermocouples into the melting polymer at defined depth is hardly feasible. As another reference curve the temperature of bottom surface is planned to be measured. In the degradation simulations constant values of cp ¼ 2670 J/kg K, kp ¼ 0.3 W/m K are used, similar to most published values [11,13–15]. The fire resistant samples (where T-58 is substituted with TEDAP) reach higher ignition times (Table 2). Assuming that the thermal properties of the samples do not change, TEDAP increases the thermal stability of polymer [8] and the surface can bear higher temperature. Over 450 C the heat loss by radiation became
Fig. 7. Simulated preheating time using thermal properties A (Table 1) and assuming radiation.
1212
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
Fig. 8. Predicted gas flux curves at different combustion heat adsorption levels and the measured gas flux of AH-16 T-58 sample.
absorbs more heat during the measurement than that predicted in the simulation. More work is needed for a better fit of the predicted values to the experimental results. It can be concluded that the amount of re-radiated heat influences the degradation process significantly and, therefore, cannot be neglected in the model. Applied simulation parameters for epoxy simulations based on published data [7,8,13]: kp [ 0:19 W=m K ðat 293 KÞ; kp [ 0:3 W=m K ðat 705 KÞ; kg [ 0:01 W=m K; rp [ 1200 kg=m3 ; rg;a [ 1:16 kg=m3 ; cg [ 1000 J=kg K; cp [ 2670 J=kg K ðin degradationÞ; Tc [ 705 K; Ta [ 293 K; Tcone [ 653 K; h1 [ 10 W=m2 ; r [ 0:055; q_ 000 [ 50 kW=m2 ; l [ 0:004 m; 3 [ 0:9; B=m [ 3:26310L10 m2 =Pa s; 4 [ 0; 0:05; 0:1; Hc [ 20; 400 kJ=kg; L [ 1600 kJ=kg
The heat transport coefficient in char–gas system was given by the Maxwell expression.
6. Topics of further model development In the next stage of model and simulation development input parameters should be improved and measured to expand the applicability of the model for all epoxy resin samples. It has been shown that data taken from the literature are not enough, measured values are needed especially for the novel fire retardant epoxy resins in case of which no published data are available. Heat capacity and heat conductivity investigations will be carried out based on differential scanning calorimetry and m-TA for the virgin polymer samples and for the char as well. The degradation model of fire retardant epoxy resins has to handle the volume change at charring; to solve this problem in the mass and heat equations it is planned to include a function of the changing char-gas fraction, which is able to describe also the
Fig. 9. Predicted evolution of pressure on the char/gas interface (x ¼ 0) at different combustion heat adsorptions.
E. Farkas et al. / Polymer Degradation and Stability 93 (2008) 1205–1213
gradient structure of char. For the investigation of intumescence laser pyrolysis treatment is applicable. Another possible way of development is the kinetic modelling of degradation. At the same time in our department the investigation of decomposition mechanism is carried out [17]. 7. Conclusion Laboratory experiments together with a mathematical model and simulation results are reported to describe the degradation of pure and flame retarded epoxy resins initiated by a constant heat flux. The applied model is based on Staggs’s work and describes both the heat and mass changes and predicts the whole temperature and pressure profiles of the system. Experiments in a mass loss calorimeter and computer simulations have been carried out to establish the effects of critical parameters such as layer thickness, heat flux and material properties. The predicted ignition times and critical temperatures were in good agreement with the experimental data. Furthermore it could be concluded that the heat capacity of polymer does not have any effect on the temperature profile of the preheating process, it determines the preheating time instead. The effect of re-radiated combustion heat was established and it has been found that the amount of absorbed pyrolysis heat is an important factor in the degradation model. The simplification applied in this work can be accepted only as a first attempt to describe the behaviour of the reference material. In order to achieve better correlation with the experimental results of fire retarded sample improvement of the model is planned. Acknowledgement Authors acknowledge the financial supports received through the EU-6 Framework Programs (NMP3-CT-2004-505637 and IP 026685-2) and OTKA (project nos. 046460 and T49121).
1213
References [1] Di Blasi C. Modeling and simulation of combustion processes of charring and non-charring solid fuels. Progress in Energy and Combustion Science 1993;19: 71–104. [2] Staggs JEJ. A simple model of polymer pyrolysis including transport of volatiles. Fire Safety Journal 2000;34:69–80. [3] Hopkins Jr D, Quintiere JG. Material fire properties and predictions for thermoplastics. Fire Safety Journal 1996;26:241–68. [4] Davies JM, Wang YC, Wong PMH. Polymer composites in fire. Composites: Part A 2006;37:1131–41. [5] Lin WS. Quasi-steady solutions for the ablation of charring materials. Heat and Mass Transfer 2007. [6] Fei You, Lin Zhang, Jianjun Zhou, Chow TT. Direct numerical simulation on the pyrolysis of materials with volatile residue layer. Applied Mathematical Modelling 2006. [7] Staggs JEJ. Heat and mass transport in developing chars. Polymer Degradation and Stability 2003;82:297–307. [8] Toldy A, To´th N, Anna P, Keglevich Gy, Kiss K, Marosi Gy. Flame retardancy of epoxy resin with phosphorus containing reactive amine and clay minerals. Polymers for Advanced Technologies 2006;17:1–4. [9] Theuns E, Vierendeels J, Vandevelde P. Validation of the integral model for the pyrolysis of charring materials with a moving grid. Journal of Computational and Applied Mathematics 2004;168:471–9. [10] Landau HG. Heat conduction in a melting solid. Quarterly of Applied Mathematics 1950;8(1):81–94. [11] Asher J, editor. Kaye and Laby, tables of physical and chemical constants. UK: Longman. 16th ed. (web edition), http://www.kayelaby.npl.co.uk/chemistry/ 3_11/3_11_1.html; 1995, http://www.kayelaby.npl.co.uk/general_physics/2_3/ 2_3_6.html; 1995. [12] Swier S, van Assche G, Van Hemelrijck A, Rahier H, Verdonck E, Van Mele B. Characterization of reacting polymer systems by temperature-modulated differential scanning calorimetry. Journal of Thermal Analysis 1998;54: 585–604. [13] Lyon RE, Janssens ML. Polymer flammability (report). Southwest Research Institute; 2005. [14] 15th ed.. In: Dean JA, editor. Lange’s handbook of chemistry. New York: McGraw-Hill; 1999 [15] Harper CA, editor. Modern plastics handbook. New York: McGraw-Hill; 2000. [16] Domininghaus H, editor. Kunststoffe. Du¨sseldorf: VDI-Verl; 1969. [17] Toldy A, Anna P, Csontos I, Szabo´ A, Marosi Gy. Intrinsically flame retardant epoxy resin – fire performance and background – part 1. Polymer Degradation and Stability 2007. available on-line.