Combustion, Ezplosion, and Shock Waves, Voi. 33, No. 5, 1997
EFFECT OF VARIATION OF THE PHYSICOCHEMICAL
OF A P H E N O L I C C A R B O N P L A S T I C ON N O N S T A T I O N A R Y H E A T AND MASS EXCHANGE IN HIGH-TEMPERATURE
D E S T R U C T I O N IN A G A S F L O W
V. I. Zinchenko, V. V. Nesmelov,
A. S. Y a k i m o v , a n d G. F. K o s t i n
Unsteady heat and mass exchange characteristics of a phenolic carbon plastic in a flow of an inert gas have been studied experimentally and theoretically. Temperature fields and amounts of thermochemical destruction have been examined with variation of the density of the coke residue and the heating rate and thermal properties of the material. It is shown that, with increase in the convective heat fluxes and the gas deceleration temperature, it is necessary to take into account the dependence of the thermal properties of the material on the heating rate. A method of taking into account this dependence is proposed.
Because of high temperatures and high heating rates, typical of operating conditions of ablating thermal protective coatings (TPC), in mathematical modeling of heating and destruction, it is necessary to take into account the variation of the physicochemical properties of the material. The shift of physicochemical reaction zones to the high-temperature region with increase in the heating rate leads to ambiguity in choosing values of thermal conductivity As and specific heat cp,s at a given temperature Ts . The magnitude of this shift is determined by the kinetics of the destruction process and the external conditions of heat action. At the same time, the values of As and c~,s are primarily affected by the density of the material ps, which depends on the change in the structure and composition of the material at the heating temperature Ts. Empirical data for the functions As(Ts) and c~,8(Ts), obtained at various heating rates, are generalized as the universal dependences As(ps) and c~,s(ps) (Fig. 1) . This makes it possible to determine As and cp,s from the value of the current density ps, calculated by the equation of chemical kinetics
dPSdt = - k o ( p s - pc)n 9exp (-R-~sE).
Thus, one can take into account the effect of the heating rate on the thermal properties and, hence, on the thickness of the heated and coked layers and the TPC ablation. Here k0, E, and n are the preexponent, activation energy, and order of the thermal decomposition of the polymer binder, respectively, R is the universal gas constant, pc is the density of the coke residue, and t is time. Figure 1 also gives functions As(Ts) and cp,s(Ts) that do not take into account the effect of the heating rate . It is assumed, as a rule, that the coke density pc is constant and corresponds to the final state of the material at a linear heating rate to temperatures of ..~1000 K . It should be noted that pc is a function of the temperature Ts . This occurs because physicochemical transformations follow a multistep course and are of a relaxation character (the tendency of the process toward completion at a given temperature level). The number of steps and the character of their progress depend on many factors, the main ones of which are the chemical structure of the polymer and the conditions of thermal action . The heat effects of the
Institute of Applied Mathematics and Mechanics of Tomsk State University, Tomsk 634050. Translated from Fizika Goreniya i Vzryva, Vol. 33, No. 5, pp. 76-84, September-October, 1997. Original article submitted September 27, 1995; revision submitted May 6, 1997. 576
0010-5082/97/3205-0576 $18.00 (~) 1998 Plenum Publishing Corporation
~. , W/(rn- K) 1.8
~,s, W/(m. K)
cp,s, J/(kg. K)
cp, s, J/(kg.K)
// /" I 1500
t 1000 300 1050ps, kg/ma 1350
~ , K 750
Fig. 1. Thermal conductivity (a) and specific heat (b) versus the temperature and density of a carbon plastic: the solid curves refer to the experiment: a heating rate of ~1 (1) and 150 K/sec (2); the dashed curves are the commonly used functions As(Ts) and c~,s(Ts).
processes can superimpose completely or partially or cancel one another . The nature and mechanism of these processes and the temperature intervals within which they proceed are frequently unknown, but each of these processes contributes to the formation of the coke residue. Thus, to each value of the temperature Ts an eigenvalue of pc must correspond, which characterizes the limiting state of the material at the given temperature. Therefore, in describing the pyrolysis of polymers by one overall reaction with effective values of the thermokinetic constants E, k0, and n and the heat release Q, one should use the dependence pc(Ts). This makes it possible to take into account the contribution of individual steps to the ablation characteristics and thermal state of TPC in the temperature range studied. The goal of the present paper is to analyze the effects of variation of the thermal properties and density of the coke residue on the ablation, temperature profiles, and thickness of the coke layer under various conditions of thermal action on a composite consisting of carbon fiber and phenol-formaldehyde resin (mass content of the binder N42%). The thermogravimetric studies in  showed that, after thermal decomposition of a phenolic carbon plastic, ~-50% of the mass of the polymer binder and the carbon filler together form the coke residue of the material. Consequently, for the studied composition with an initial density of Pin = 1350 kg/m 3, the coke density for decomposition of the resin must be pc = 1067 kg/m 3. Special studies using the procedure of  were performed to verify this assumption and determine the dependence pc(Ts). Carbon plastic specimens were heated in an atmosphere of an inert gas (nitrogen) to a given temperature Ts and kept at this temperature to completion of ablation. The mass and linear dimensions of the specimens were then measured to calculate the density of the coke residue. In each subsequent experiment, the decomposition temperature was increased. It should be noted that the heating conditions affect only slightly the final value of pc up to the thermal stabilization temperature Ts. Figure 2a shows dependences of the current density of the material ~ps on the time ~- = t/theat (t is the current time and theat is the maximum time of heating) under various conditions of heating to a temperature Ts = 900 K with subsequent thermal stabilization. Evidently, irrespective of the heating rate, the value of ps tends to its limiting value pc ~ 1180 kg/m a, which is determined by the thermal-stabilization temperature. The dependence of the density of the coke residue of the carbon plastic on the heating temperature is given in Fig. 2b. As can be seen from the figure, the curve of pc(Ts) has two points of inflection - - at Ts N 1300 and 1850 K. This indicates that, at high temperatures, the value of pc is strongly affected by secondary processes due to thermal decomposition of the binder. In this case, in the temperature region Ts ~ 2500-2900 K, the value of pc is ~-720-750 kg/m a. This has a direct effect on the thermal diffusivity
Pc, kg/ma 14o0
: " ~ " ~-
.Pc. . . . ~
~ 600 500
Fig. 2. Variation in the material density under heating to Ts = 900 K followed by thermal stabilization of the specimen (a) and the density of the coke residue versus the heating temperature (b): dTs/dt = 50 (1) and 10 K/sec (2), theat = 80 sec.
as = As/ps~,s of the material and, hence, on the thickness of the heated and coked layers under interaction with a high-enthalpy gas flow. Thus, we obtain a complete set of data on the properties of the material to realize the proposed method, which takes into account the effect of the heating rate and the temperature on the parameters As, cp,s, and pc, and to compare the results with the traditional approach. Using generally adopted assumptions, we reduce the mathematical model to a system of equations of chemical kinetics (1) and conservation of energy for a body consisting of a T P C layer on an inert substrate (asbestos cement): cOTs cOTs cO (As cOTs'~ dps pscp,s -~ +c~,gG cOy -cOy _ "~'y ] + Q d t ' cO [
a~ k~'~'--#Uy )
with the initial and boundary conditions T~ ~=o = Ts.b ~=o = ~ '
P~ ,=o = Pin,
q~ - (pv)w(hw - h~ )) + Ae~fcr(T4 - T~) = -A~ coy ,=o Ij,w "~ (pV)wCj,w = (pv)~cj,~, (1)
= )~ub COT~u|, OT~.b
j = l, N, [
,=t~ = 0,
h~, = E hjcj,~,,
= ~j = alcp,
N m ; 1 = ~ cj,~,l~s,
7 = 0.2(me/m~) ~
j=l O~ (h. - hw), cp
q., = -
'] (\ dP" dt l) ~y,
A M , lO-Skg
5OO 3~ ~ / / 10
A M , l(~'Skg
0 t, sec
3OO ~ 0
~~ " ' 5
~ ~ ' f-
0 l, sec
Fig. 3. Experimental and calculated time dependences of the temperature at various distances from the heated surface (curves 1-3) and ablation of the carbon plastic (curves 4 and 5) in a nitrogen flow: (a) he,0 = 1.25.106 J/kg, (a/cp)o = 0.1 kg/(m 2sec), and y = 0 (1), 0.85 (2), and 2.75 mm (3); (b) he,0 = 1.2.106 J/kg, (c~/%)0 = 0.68 kg/(m 2. sec), and y = 0 (1), 1.9 (2), and 3.1 mm (3); the solid curves refer to the experiment, and the dashed curves refer to the calculation.
In Eqs. (2)-(8), the following notation is used: y is a spatial coordinate directed from the surface to the depth of the body, h is the enthalpy, G is the mass consumption of gaseous products from decomposition of the binder, (pv),~ is the ablation velocity in pyrolysis of the resin, qw is the convective heat flow from the gas phase,/3 is the mass-exchange coefficient, (a/c~)0 is the heat-exchange coefficient in the absence of blow-in in the boundary layer, cj, w is the mass concentration of the components at the interface, Q is the heat release due to resin pyrolysis, Aeff is the reduced emittance, a is the Stefan-Boltzmann constant, Iz and 12 are the thickness of the TPC and the substrate, and rnj is the molecular weight of the j t h component. The following subscripts are used: w for the values at the interface, in for the initial values of the quantities, s for the properties of the TPC, sub for the properties of the substrate material, e for the parameters on the outer surface of the boundary layer, and g for the gas parameters; the superscript (1) denotes the characteristics of the gas blown in from the side of the carbon plastic, and the subscripts 1 and 2 refer to the characteristics of the TPC and the substrate, respectively. In numerical integration of boundary-value problem (1)-(8) by the iteration-interpolation method of , we used the following data: E = 83,000 J/mole, k0 = 3.15 9 104 sec-z, Q = 3.35 9 10s J/kg, n = 1, Pin = 1350 kg/m 3, Psub = 1800 k g / m 3, Asub = 0.349 W / ( m . K), cp,sub = 837 J/(kg. K), R = 8.31 J/(mole. K), and Aetr = 0.9. The composition of the blown-in gases c~]) was identified with the blow-in of the components in ; E, k0, Q, and n are taken from [2, 4], and Asub, Psub, and c~,sub are taken from . To analyze the effect of variation of the physicochemical properties of the material on the temperature field and the ablation, we performed four calculation variants for the following four sets of parameters: No. 1. As = As(Ts),
cp,, = cp,s(Ts),
pc = 1067 k g / m 3 = const;
No. 2. As = As(ps),
cp,s = cp,s(ps),
Pc = 1067 k g / m 3 = const;
No. 3. As = As(T,),
cp,s = c~,,(T,),
pc = pc(Ts);
No. 4. As = As(ps),
cp,s = cp,s(ps),
pc = pc(Ts).
Results of numerical solution of the problem were compared with data of experiments in which, by the procedure of , we measured the temperature fields and ablation of thermal protective material specimens subjected to one-sided heating in a flow of an inert gas (nitrogen) under various thermal loads. Experimental and calculated time dependences of the temperature Ts at various distances y from the heated surface of the
Fig. 4. Temperature profiles (a) and density distribution (b) in the TPC cross section at various times: t = 4 (1) and 20 sec (2); the solid curves refer to the calculation for As = )~s(Ps), c~,s = c~,s(ps), and pc = pc(Ts), and the dashed curves refer to the calculation for As = A,(Ts), cp,s = cp,s(Ts), and pc = 1067 k g / m a.
specimen and the ablation A M = (/l~n - M), where M~n and M are the initial and current masses of the specimen, are shown in Fig. 3. It should be noted that, under the given conditions of thermal action, the difference in the dependences Ts(y, t) and AM(t) between calculation variant Nos. 1 and 2 was insignificant (not more than ,--1-3%). The same is true for variant Nos. 3 and 4. For this reason, in Fig. 3, we give only results for calculation variant Nos. 1 and 4. As can be seen in Fig. 3, ablation primarily depends on the form of the functional dependence pc(Ts). Specification of the constant value pc = 1067 kg/m 3, as might be expected, yields considerably overestimated ablation values (curves 4) compared with the results of the calculation (curves 5) using the dependence pc(Ts) given in Fig. 2. This is especially pronounced for low thermal loads (see Fig. 3a), for which the calculated values of A M differ by more than a factor of two after the twentieth second. However, because of the small mass rates of ablation, their effect on the temperature fields is insignificant (the difference between the temperature fields does not exceed the graphical error). At the same time, the method of specification of the thermal properties at heating temperatures Ts < 1100 K has little effect not only on the temperature profiles but also on the yield of gaseous decomposition products. Indeed, as follows from the data in Fig. 1, at Ts "~ 1100 K, the changes in the parameters As(ps) and ~,s(ps) are insignificant and, hence, cannot have a strong effect on the heating depth and the reaction-zone width. This is supported by calculations of density profiles of the material across the specimen thickness. During the experiment, the decrease in the carbon plastic density in the surface layer does not exceed ,~12% (,~1200 kg/m 3) and, as can be seen in Fig. 1, affects only slightly the parameters As and cp,s. From Fig. 1 it also follows that the variation of the thermal properties of the degrading TPC is of significance only with attainment of densities ps < 1060 k g / m 3, for which a sudden increase in the value of )~s is observed. To verify this assumption, we calculated the temperature fields, ablation, and density distribution across the TPC thickness for variants (9) for higher values of the incident gas flow parameters (a/cp)0 = 2.0 kg/ (m 2 9sec) and he = 3. l0 s J/kg. This made it possible to analyze the effect of the temperature dependence of )~s, Cp,s, and pc on the heat- and mass-exchange characteristics at higher temperatures. Calculation results are given in Figs. 4 and 5. Comparison of the data shows that, for the high-temperature pyrolysis regime, there is significant difference in the temperature and density profiles across the TPC thickness (see Fig. 4) and also in the ablation between calculation variant Nos. 1 and 4 of (9) (see Fig. 5). The use of dependences As(Ts) and Cp,s(Ts) that do not take into account the effect of the heating rate leads to an overestimated thermal 580
T8, K A M , IO-3g
t, sec 20
t, sec 20
Fig. 5. Temperature Ts at various depths y from the heated TPC surface (curves 1-3), ablation A M (curve 4), and the position of the boundary of the decomposition zone 3d (curve 5) versus time: the solid curves refer to the calculation for As = As(ps), c~,s = cp,s(ps), and pc = pc(Ts); the dashed curves refer to the calculation for As = As(Ts), cp,s = cp,s(Ts), and pc = 1067 kg/m3; y = 0 (1), 1.55 (2), 3.55 (3).
diffusivity as and, hence, to an increase in the calculated thickness of the reacting layer. Thus, for example, for Ts ~ 1800 K, according to the data of Fig. 1 (dashed curves), we have as,1 = 7.7- 10-7 m2/sec. To determine the value of as from the curves of As(ps) and cp,s(ps), we use the data in Figs. 4 and 1. It follows from Fig. 4 that, at Ts " 1800 K, the material density ps is ,~930 kg/m 3. Next, from Fig. 1 we find the thermal conductivity As ~ 0.73 W / ( m . K) and specific heat c,,s ~ 1900 J / ( k g . K) of the material layer that has reacted. This allows us to evaluate the thermal diffusivity of the material: as,2 ~ 4.13.10 -7 m:/sec. Thus, the two different methods of determining the thermal properties of the thermal protective material (ignoring and taking into account the effect of the heating rate) lead to significant differences in determining the calculated degrees of heating (Sh) , decomposition (~d), and coking (~c) of the TPC. Indeed, assuming that the thickness of the heated layer ~h "~ V~st , we obtain ~h,1/$h,2 ~ 1.36, i.e., the traditional method, which ignores the effect of the heating rate on As and c~,s, overestimates the calculated depth of the isotherm Ts "~ 1800 K by -.~36%. Note that the errors in determining *h, ~d, and ~c are not constant but depend on the conditions of external thermal action (the heating regime), the heating time of the material, and the temperatures of initiation 7]n,d and completion Tcompl of the pyrolysis reaction. Figure 5 gives values of ~d [which determines the position of the boundary of the decomposition zone, where (pin-Ps) = 0.1 k g / m 3] calculated by the above methods for a heating regime with the following incident gas flow parameters: (c~/c~)0 = 2.0 k g / m 2. sec and he = 3.106 3/kg. As can be seen from the figure, at initial times ($ ~ 3 sec), when the heating temperature is low and the thickness of the heated layer is small, the difference between the curves is insignificant. With time, the difference increases, and, at t ~ 24 sec, it reaches ,~28%. Comparisons between the calculation results of variant Nos. 1 and 3 and those of Nos. 2 and 4 of (9) show that, under significant thermal loads, the method of specifying the functional dependence pc(Ts) has little effect on the value of mass ablation. This allows one to use the limiting value pc = 1067 kg/m 3 = const to evaluate ablation, despite the qualitative difference between the density profiles in the TPC material. Thus, in predicting the service life of TPC in high-enthalpy gas flows, the accuracy of determining the thermal properties and density of the coke residue is important. In this case, for flow deceleration temperatures <~1000 K, the method of specifying thermal properties has little effect on temperature fields and ablation, although the latter depends strongly on the density of the coke residue. With increase in heat fluxes and 581
TPC temperatures (Ts > 1100 K), it is necessary to take into account the effect of the heati~ag rate on the thermal properties. In this case, it is convenient to use their expressions in terms of the current density of the material. At the same time, the method of specifying the density of the coke residue has a weaker effect on the destruction properties under such conditions. Results of comparison of experimental and calculated time dependences of ablation and temperature confirm the validity of the proposed method for studying and predicting the thermal state of ablating TPC in high-enthalpy gas flows. This work was supported by the Russian Foundation for Fundamental Research (Grant No. 96-0100964). REFERENCES 1.
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