- Email: [email protected]
On the analysis of linear pyrolysis experiments
Combustion Involving Solids Chairman: Dr. D. Altman
Vice Chairman: Dr. P. Nichols
(United Technology Corp.)
(A erojet-General Corp.)
ON THE ANALYSIS OF LINEAR PYROLYSIS EXPERIMENTS W. N A C t I B A R A N D F. A. W I L L I A M S
Linear pyrolysis experimental data have been used as surface boundary conditions in the analytical treatment of solid propellant combustion problems. The surface boundary conditions required for the analysis are functional relationships between normal regression velocity, gaseous reactant density, total pressure, temperature, and possibly derivatives of these quantities as well, evaluated at the interface between condensed and gaseous phases during combustion. Available experimental data indicates t h a t a relation based upon equilibrium conditions prevailing at the interface is not generally valid. There are available for many substances, however, hot-plate linear pyrolysis data which give an explicit relation between regression velocity of the conden~d phase and a measured hot-plate temperature. This hot-plate temperature has L~en assumed to be also the temperature at the interface. Ilowever, this assumption is thought in some cases to be questionable. In addition, the use of such a rate law in combustion analysis would imply t h a t the total pressure and the gaseous reactant partial pressures or densities, whieh are greatly different for the pyrolysis test as compared to the eombustion, have a negligible effect on the rate. This implication again is questionable. The pre~ent paper seeks to answer in part these questions on the validity of using linear pyrolysis data. A one-dimensional hot-plate linear pyrolysis cxperimenf, is proposed which uses a porous heated plate in place of the present impervious plate. The proposed experiment, having simpler fluid dynamics than present hot-plate experiments, is Capable of a reasonably precise analytical description. The analysis is carried out for a general case and is then specialized to simple and chainlike surface gasification processes. Conditions are developed to determine whether the surface process is either a rate process or is one of Imar-e.quilibrium. A description is presented of the possible use of data from the proposed pyrolysis experiments to compute the accommodation c(x~tticient and the vacuum sublimation r a ~ as functions of surf'~ce temperature. With this information, the surface boundary condition for a given material will then be specified. The pyrolysis rate of potassium chloride is calculated as an example of the use of the derived formulas. If fuid dynamic effects are not too i m p o r t a n t in the present experiments, and in some respects it is estimated t h a t they are not, interpretation from the results of this analysis serves present pyrolysis data as well. Interpretations are carried out for a number of compounds which have interest in propcllant combustion studies. From published data on ammonium perchloratc, for example, revised values are estimated for the heat of sublimation, for the activation energy in linear pyrolysis, and for the difference between the hot-l)l.~te temperature and the surface temperature in linear pyrolysis. 345
346
COMBUSTION INVOLVING SOLIDS
Introduction Schultz et al. t-6 have reported experiments in which hot-plate pyrolysis techniques are used to measure the linear regression rates of solids undergoing surface heating. These experiments have yielded pyrolysis rate data for many different materials at pressures and temperatures approaching (but not yet equaling) pressures and temperatures encountered in solid propellant rockets. These data are of practical importance in helping to predict, for example, the deflagration rate of pure ammonium perchlorate, ~-12 and thereby in promoting further understanding of the factors controlling the burning rate of composite solid propellants containing ammonium perchlorate. I t has also been proposed ta to use pyrolysis data for actual rocket propellants (if such data is available) to predict burning rates. The pressure is sufficiently high in these pyrolysis experiments so that the continuum equations of fluid mechanics may be applied to the gas adjacent to the solid. The geometry of the gas flow is so complicated, however, that unquestionable interpretations oi the significance of the measurements appear to be difficult to obtain. Consequently, it is of interest to investigate the possibility of modifying these experiments in order to yield data which can be interpreted more easily. In the present paper there is suggested a modified hot-plate linear pyrolysis experiment which should not be difficult to perform. A mathematical analysis of this experiment is caried out under simplifying assumptions which, in the main, do not appear to be unduly restrictive. However, the reader's attention is directed to the following assumptions made in this analysis. Except for the mathematically thin, plane surface in contact with the hot gas, the condensed phase is assumed here to be in a uniform phase and to be homogeneous. (We will refer in particular to the solid phase in our discussion.) Both gaseous and condensed phases are assumed to exhibit no chemical reaction. These assumptions may not be valid for certain materials under all conditions. There may occur in the solid energetic bulk chemical reactions, at temperatures less than the surface temperature, which transform one solid-phase state into another solid-phase state. 14 There may also occur, as in ammonium perchlorate, for example, a partial decomposition of the solid into a reactant gas plus a solid residue. The plane surface in this latter case is more exactly a layer of finite thickness in which the two phases exist simultaneously. These mechanisms may be important if the linear regression rate of the solid is sufficiently small. There appears to be no
essential difficulty, however, in extending the present analysis to the more complex cases where these assumptions are relaxed. The usefulness of such an extension in particular cases will depend upon whether the kinetics of the condensed-phase reactions and the degree of nonhomogeneity are known. Conditions under which the surface process is either a rate process or is one of near-equilibrium are determined for the modified experiment from our analysis, and interpretations of the significance of the measurements are given for these two limiting cases. The pyrolysis rate of potassium chloride is computed as an example. The mathematical simplicity of the system described here provides a strong incentive to construct such an apparatus for measuring regression rates. A degree of similarity of the present model to the previous hot-plate linear pyrolysis experiments also makes it possible to draw some conclusions concerning the previous measurements.
The Modified Experiment It is suggested that the impervious hot plate used in previous pyrolysis experiments be replaced by a porous hot plate. If suitable shielding can be provided to prevent gas from escaping without passing through the porous plate, then the flow would be approximately one-dimensional, and the arrangement would appear as illustrated in Fig. 1. The solid is assumed to be converted to gas at x equals 0, and the gas flows through the porous plate at x equals l, where l is the normal distance between the surface and the inner face of the porous plate. Steady-state conditions are assumed. For convenience in the analysis, the solid is considered to be semiinfinite. At x equals -- ~ , the solid has a density p, and a velocity v, directed towards the interface, which remains stationary. In analyzing this experiment, the gas-phase and the interface conditions will be considered in that order. The momentum conservation equation is considered in the Appendix. The solid sample is assumed to be homogeneous and nonreactive for x < 0.
~'$~P'I~E'~ 0 -co=
1
s
e SOT PLATEr + x p
co
FIo. 1. Schematic diagram of modified pyrolysis experiment.
ANALYSIS OF L I N E A R PYROLYSIS E X P E R I M E N T S
The one-dimensional, steady-state equations on 0 <: x < l for conservation of total mass, mass of species K (in the absence of gas-phase chemical reactions), and energy of a mixture of N species of ideal gases are, respectively15:
pv = m = p.v,~ = constant .dYE m dx
(
d dx oD d~x ]'
(1)
347
Denoting the boundary values of YK and H at x equals 1 as Y~ and H~, one finds the solutions to Eqs. (3), (5), and (6) to be H(~) ---- H ~ -
J[exp (~p) -- e~3
(7)
and Yt;(~) = YK,p exp(~--~p)
K =- 1, " " , N
(2)
+ e K [ 1 - - exp (~--~p)3,
K=
1,-..,Y
(8)
where
and
dH m dx = d-x pD -~x
-- ~it
9
(3)
Equation (2) implies that the binary diffusion coefficients of all pairs of chemical species are equal. Equation (3) relies upon this assumption, the approximation that the Lewis number is unity (k = pDhp), and the assumption that either the mass diffusivity equals the effective coefficient of viscosity (~ = pD) or the ~kinetic energy is negligible in comparison to the thermal plus chemical energy:
([~v~( Rff R3) ] ( ( h ) It is to be noted 5p, ),, and pD may depend upon both T and YK in this formulation. When the energy flux at x = ~ oo is equated to the energy flux at x = 0-t-, and when the conditions T = T~ and (dH/dx) = 0 are imposed at x = - - ~ , the equation
~, ~ m
s
[dx/oD 3
(9a)
[dx/oD 3
(9b)
While the experimenter often has direct control over the plate temperature Tp, the values of YK.p [which are needed along with Tp to determine Hp and Yp in Eqs. (7) and (8)-]* will basically be determined by a permeability Pt: of the porous plate of species K and by the partial pressure pK.~ of species K on the downstream side of the plate. A reasonable phenomenological expression is obtained by assuming that the mass flux ?beE for each species is proportional to the pressure difference across the plate for each species, viz.
PK(
rheK = --t- \-
~
)
PK.~o_ K =
rh [hT( T~) "-{- ( RffRa) 89
(lO)
.= ~h [K~ YK.,hK( T,) --}- (RI/R3) 89 -- [pD(dH/dx)~,
(4)
is obtained. This relation is a boundary condition for Eq. (3), and it is written as
oD ~
1, " " , N
, = ThJ
Here Pv is the total pressure at x = l, t is the thickness of the porous plate, and the first term inside the bracket is the partial pressure pK of the species K at x = 1. The mass-average molecular weight of the gases at x = l is Wv:
(5)
Conservation of mass across the phase interface requires that the net flux (i.e., mass per unit area per second) of species K leaving the surface to be equal to ~heg, which is the mass flux of species K produced in the gasification process. The eK'S are the mass flux fractions of species K in the decomposition of the solid, and they are determined by the over-all stoichiometry of the gasification process. With this boundary condition at x equals 0, the following integrals of Eq. (2) are obtained
~hl"K -- pD(dYK/dx) = ~he~:, K ~- 1, " ", N (6)
Equation (10) can be put into a convenient, nondimensional form by introducing a characteristic pressure p*, defined as
p* - t z.. (~:/PK) K=I
/fzdx, pD
/Jo
(11)
and a ratio ~, which is a measure of p~ and which is defined as
= p*/p~
(12)
* Strictly speaking, either vp must also be measured or the momentum equation (see Appendix) must be solved in order to determine completely the constants in Eq. (7). However, the kinetic energy will usually be negligibly small.
348
COMBUSTION INVOLVING SOLIDS
In view of these definitions and of Eq. (9b), Eq. (10) is written as -l/V, -
WK
y
o"
K,p ~
-( e e- / P K )
tp "4-
Pe,oo l
N
we and a e may be functions of Tz, ps, and all the Ye.,'s for complex surface processes. Equation (161) may be written in the nondimensional form
P$ |
F_, ~/P~
~pee = ilK(we -- YK,8),
J=l
K = 1, " " , M
(19)
[dx/(pD) ]
(20)
where K=
(13)
1,-..,N
Although only N -- 1 of the YK,p are linearly independent, the N relations in Eq. (13) are independent, since summing Eq. (13) over all N species gives the nontrivial equation
fie ~ o~e(pce), re
-
(21)
weE-e(pce).]-'
(14)
From Eq. (19) it follows that, in general, the surface process with respect to any evolved species K will be a rate process, similar to that
where p,~ is the total pressure maintained on the downstream side of the plate, v/z.
encountered in vacuum vaporization or sublimation experiments l~m if the following condition holds:
1 = a (#, -]- P,co ~-)
(Ye,./~)
N
(15)
P,oo = ~_, PK,o~ e~l
A general boundary condition at the solid surface is now obtained for each of the species evolved in the gasification of the solid. Let the species numbering be so ordered that, if M is an integer, 0 < M < N, then the first M species in the gas are evolved from the decomposition of the solid. Consequently, K = M -{- 1, ---, N denotes the chemically inert species in the pressurizing gas which are present at the surface through diffusion. If M = N, there are no inert species. It follows that eK ---- 0, by definition, for the inert species K -- M -[- 1, . . - , N. The general boundary condition for the evolved species is obtained by equating the net surface mass flux ~hee of each species to the difference between the vacuum sublimation rate WK, which is the mass flux of the species leaving the surface, and the mass flux of the species entering the surface, 16-~9 rheg = WE -- a e Ye,~(pce)~
K -=- 1, . ' . , M
(161) eK = 0
K=
M-[-1,
"",N
(16b) Here a/~ is a surface accommodation coefficient for species K, cK = [ ' ( R 3 T , ) / ( 2 7 r W K ) -]~
(17)
is the average velocity of molecules of species K (of the ideal gas mixture) in the - - x direction at the surface, and p = pI~/'/(R2T)
=
(~.~e/~e&)
<<
1 (22)
At the opposite extreme, near-equilibrium conditions will prevail for an evolved species K at the surface if (Ye,Jwe) ~ 1, that is, if
(~,eK/wefle) << 1
(23)
General Equations for Determination of Pyrolysis Rate The simple geometry of the modified pyrolysis experiment, and the simplifying assumptions which have been made in the analysis, allow the temperature and concentrations at each point of the gas to be expressed in an elementary form by Eqs. (7) and (8). The constants or parameters appearing in these equations must be determined uniquely with the use of the boundary conditions, however, in order to determine a unique solution. In order to do this, certain of the parameters must be known either from predetermined physical properties or from measurements taken during the pyrolysis experiment. Which of the parameters fall into this class is somewhat arbitrary. We will assume at first that the unknown parameters are:
{ Yg,p}, [ fg,.}, IeKI, Ts, r ~p
(24)
The notation {( )g} stands for the set ( )g, K = 1, - - . , N. Since there are 3N -{- 3 unknown parameters in group (24), then the same number of independent equations must exist to determine them. Evaluating Eqs. (7) and (8) at $ equals 0 gives N ~ 1 of these equations, viz. YK,, = YK,p exp (--~p) -{-eKF1 -- exp (--~p)l K--- 1 , . . . , N
(18)
according to the ideal gas law. In general, the
1 -
H, = H p -
JFexp ($p) -
1]
(25) (26)
ANALYSIS OF LINEAR
PYROLYSIS
Note that the identity
geometry of the gas flow. The modified pyrolysis experiment suggested in the present paper has certain advantages in this application, and this will bc dcmonstrated in the following sections.
N
~'-~eK=l is implied by the summation of Eqs. (25) from K= ltoK= N. Equations (13), (16b), (19), (25), (26) and the two identities N
N
Y~. YK,. = E YK,~, = 1
K~I
K~I
349
EXPERIMENTS
(27)
constitute a set of (3N + 3) transcendental equations to determine the unknown parameters in (24). For this problem it is assumed that { P K } , { W K } , and t are predetermined constants for an experiment, that l, Ps, {PK.oo}, Ti, and Tp are parameters measured in the experiment, and that ~p, pD, J, {ilK}, and {WE} are known functions of these parameters and of the parameters in group (24). If all of this information is given, then one could, for instance, compute the pyrolysis rate v~ expected in a pyrolysis test as a function of the plate temperature Tp and the downstream pressure p.r A numerical example of such a computation is given in one of the following sections. The application of the hot-plate, linearpyrolysis test results to practical problems which have been its chief motivation is not nearly so straightforward, however. An important practical problem is to obtain, for solid propellant constituents, the data which enter into the general surface conditions, Eqs. (16), so that these equations can be used to compute propellant deflagration rates from a combustion theory. If {WE} and {aK} are known functions, however, as was implied in the paragraph just preceding, then it would not be necessary, for application to this particular combustion problem at least, to perform the pyrolysis test. The necessary parameters for the general surface conditions, Eqs. (16), would be determined already. However, the central difficulty of this problem is that the data necessary to evaluate Eqs. (16) for propellant constituents are not usually available, and especially not for the range of surface temperatures obtained in deflagration. To apply Eqs. (16) to the propellant combustion problem, we should ideally like to be able to calculate from the linear pyrolysis data the vacuum sublimation rates {WK} and the surface accommodation coefficients {aK} as functions of Ts, Ps, and { YK,S}. Data from previous pyrolysis experiments 1-6 (i.e., v, as function of Tp for fixed P.oo) has not been applied to the calculation of these functions, however. Such application would appear to be generally quite difficult to perform in an unequivocal way because of the complicated
Simple, Chainlike Surface Processes The equations which are developed in the preceding two sections are now reduced to a much simpler set by the introduction of the following additional assumptions. a. The surface process is simple, that is, the fig and wK are functions only of T~ (i.e., independent of p~ and { YK.8}). b. The surface process is chainlike, meaning that parallel, competing surface reactions do not occur. A consequence of this assumption is that the e g are constants--independent of Ta, pa, and { Yg,8}--and each eg will be assumed to have a known value. c. The molecular weights of the gaseous species are all equal to a common molecular weight W. d. Changes in kinetic energy are neglected in comparison to changes in thermal energy to a first approximation (higher approximation can be computed through integration of the momentum equation). e. The isobaric heat capacities of all species are equal to an average value of the mixture heat capacity 5p between T, and Tp. Let the process indicated in (b) above be represented by M
M,(c) ~ E vg'MK(g)
(28)
K~I
The VK'S are the fixed number of moles of species K produced per mole of solid; as a consequence of (b) and (c) above, EK = VK!
PK!
Consider Eq. (28) in equilibrium at a fixed T~; let Pg,e be the equilibrium partial pressure of species K, and let pe be the total equilibrium pressure of all evolved species: M
Pe =
~_, PK,e K~I
As a consequence of the chainlike assumption (b),
PK.e = egpe
K = 1, " " , M,
(29)
the enthalpy difference AQ(T~) corresponding to Eq. (28) is, per unit mass of solid, M
AQ(T,) = E eghK( Ta) -- h,( T,) K=I
350
COMBUSTION
INVOLVING
Hence, the Van't Hoff equation for the equilibrium (28) is written as d [ i n K~I~I(PK,.)vK'] = W~. AQ dTJ(R1T~ 2) (30) The ehainlike assumption means dO'K') = O, K = l, ---, M. Since mass conservation requires M
SOLIDS
as a measure of To. The last equality in Eq. (36) can be derived from Eqs. (5), (25), and (26). For typical endothermic surface reactions, it is usually the case that r << 1. It is the case in many of the pyrolysis experiments reported previously. The first two of the following set of equations are obtained by summing Eqs. (13) and (25) from K = 1, . - ' , M; the third is obtained from Eq. (34) ; the fourth from Eq. (14) and the fifth from Eq. (26).
Yp = o-[~p-~ (p~,Jp*)-] then the following integral of Eq. (30) expresses p~ as a function of T. only:
p~( T~) = p.( T~) exp [ - fT~ W AQ dT] I_ .,To R~T ~
~(T.) = p-' ~ ~K P~ KL__I
(32)
(33)
(1-- Yp)exp(--$p)
~. = 811 - (Y,p./p~
(31)
Here T~ is an arbitrary reference temperature at which p~ is known. These results are now applied to Eqs. (19). The quantities/~ and w, which are functions only of T. [cf., Eqs. (17), (18), (20), (21), (31), and assumption (a)], are defined as
8( T.) = pa p-~ /~=1 (eg/flK)
1-- Y.=
(37a) (37b) (37c)
1 = a [-~p+ (p,~/p*)-]
(37d)
7- = exp (~p) -- 1
(37e)
In Eq. (37a), M P M , c o ~--
~ K~I
PK,~
Equations (37a, b, c, d) may be combined to eliminate Y, and Y.; the result is the equations ~, -~- p .
The summation of Eqs. (19) over K = 1, -. ", M then gives the equation
1--
-X exp (--~p)
pJp* = ~p + (p,~/p*)
(38a) (38b)
An approximate solution of the preceding equations is now obtained in order to discuss the physical aspects of these results. It is now assumed that v << 1, and so it follows from Eq. (37e) that
where Y, defined as M
Y--~_~YK K~I
is the total mass fraction of the evolved species. But Yaps in Eq. (34) is the partial pressure of the evolved species at the surface. For fixed T,, ~ (orTh) varies only with Y,p~, and as Y,p~-~ p~, the equilibrium requirement is Sp --~ 0. I t must therefore follow (under the assumptions already made) that w(T.) ~ 1 for all T,. Using this result, together with Eqs. (17), (18), (21), and (33), we find the following equation to hold:
~R2'T. j ~] ,~ WE(To) . ~ ( n ) = p~(n)
7" - -
-
-
g.
g
5,(Tp--
~p << 1
(35)
T~)
AQ(T~) -]- h,~(T~) -- h~-(T,) (36)
(39)
As a first approximation we set p. equal to p, (see Appendix). Then, neglecting terms which are of higher order than the first power of 7-, Eq. (38a) may be shown to take the following form; it is understood that p~ and/~ are evaluated at the plate temperature T, in this expression: P~-
P~
$P = t-P* -F P,o~ -- PM,~ -'~ Pe(ff-~ -'F A)']
Returning now to the development of a simplified set of equations, we find it convenient, as a consequence of assumptions (d) and (e), to define the nondimensional parameter % _ Yp-
7- ~
(40)
where
A --
5pR1Tp 2
eghg(Tp) -- h,~(T~)
For simplicity of presentation, we will neglect A as small compared to if-l, and we will also neglect PM,~ with respect to both pe and p,~. The limiting criteria for the surface process (cf. Eqs. (22) and (23)) are (~p//~) ~ 1 for a rate
ANALYSIS OF LINEAR PYROLYSIS EXPERIMENTS
process and (~p/~) (~ 1 for a near-equilibrium process. We may then write from Eq. (40) :
(~,/~) = i-1 + (p* + P,oo)(~/P~)]--'
(41)
The limiting criteria are therefore
(p* + p.~) (~/p~) << 1 for a rate process, and (p* ~ p,~) (fl/p~) >> 1 for a near-equilibrium process. An expression for (~/p~) follows from Eqs. (17), (18), (20), and (32); again for simplicity, we restrict attention to the case M = 1:
/ R3W1 fo Zdx
(42)
In most present pyrolysis experiments, ~-6 experimental conditions are approximately equivalent to setting p* ~- p,~ equal to p.~ equal to 1 atm. Surface temperatures are assumed to be in the range 300~ ~ T, ~ 900~ and for reasonable pD and WI(pD ~ 5 X 10-a g/em see, W~ = 75 g/mole) it follows from Eq. (42) that
p~l
351
series as a rate process; the other has the surface process as near-equilibrium. From Eq. (41) and the subsequent discussion, it follows that, in the rate process ~ ~ ~p, while in the equilibrium process p~ ~- (p* -t- p,~)~l. Hence, if ~h is measured as a function of Tp, then we determine p~(T,) in the surface-equilibrium test series, and we determine ~(T~) in the surface rate-process test series. Using this data we can determine a(T.) from Eq. (42) and w(T,) from Eq. (35). Pyrolysis R a t e of P o t a s s i u m Chloride As an example of the application of the preceding analysis we shall consider KCI, one of the few materials whose properties are sufficiently well known to permit an a priori calculation of the pyrolysis rate. Sublimation of KCI is a simple chainlike surface process (reference 23). In Fig. 2, the theoretical pyrolysis rate is plotted as a function of the plate temperature Tp for various values of 1. The computation was made from Eq. (40) with p* -[- p,r equal to 1 arm, PM,~ equal to 0, and T; equal to 300~ Here (references 23, 24,
= 5 X 103 (era-1)
10"3 - ~ - i
within a factor of 10. Since it is expected that one mean free path ~ 1 0 -5 cm, it is therefore estimated that 10-5 cm ~ l ~_ 10-2 cm, and it follows that
~
~---~
I0"4
0.05~ _~ (p* -~- P,~o) ~- ~ 50a P, Hence, for simple, chainlike surface processes with r << 1, the surface process will probably be near equilibrium for a equals 1 and a rate process for ~ < 0.01. We now discuss the determination of accommodation coefficients and vacuum sublimation rates, of substances for which these functions are not known, by means of the modified hot-plate, linear-pyrolysis test proposed in this paper. For this purpose we restrict attention to simple, ehainlike surface processes for which M = 1, and base this discussion on the simplified set of Eqs. (41) and (42). There are to be varied at our disposal in this experiment the dimensions and porosity of the hot plate, which are incorporated into the characteristic pressure p*, the pressure p.~ maintained on the downstream face of the hot plate, the hot-plate temperature Tp, and possibly also the sample spacing 1. It is proposed that suitable ranges of values for these parameters be found so that, on any one given material, two series of tests may be performed: one has the surface process for the test
~
10"5
:_~ - _ _____
V
~
e= 10"2cm
v
107 ......... 0.9
1,0
\
...... 1 \
1.1 1.2 1.3 Tp'l x10-3(OK-l)
-
1,4
Fro. 2. Linear pyrolysis rate of potassium chloride.
352
COMBUSTION INVOLVING SOLIDS
and 25) G = 0.117 cal/gm~ h,(T,) -h~-(Ti) = 0 . 1 6 6 ( T , - Ti) cal/gm, AQ = 693 eai/gm, pD ---- 5 X l0 -4 gin/era sec, a = 0.63, and p~ = 6.4 X l0 T exp (--26,070/T~) atm. The computations were completed only for T~ less than the melting point (1049~ The results imply that surface equilibrium exists (provided l > 10-5 cm). Since it was always found that Tp -- T~ ( 5~ Eq. (40) is actually valid here, and a pyrolysis rate measurement yields the equilibrium sublimation pressure.
V given by Eq. (36), relies upon none of the aspects of our preceding model except the heattransfer equation. The pyrolysis results, 2'5,6 are m/pT = 120 exp [--13,500/(RIT~)] cm/see (629~
Schultz and co-workers have reported linearpyrolysis rate measurements for NH4C1,~''~ NH4NO3, 5 and NH4CI04, 6 and have given consistent interpretations of their results on the basis of absolute reaction-rate theory. 1,5,~6 These interpretations rely upon the assumption that the surface process is an unopposed rate process (i.e., surface conditions are far from equilibrium). Our previous analysis of the surface process indicates that, under certain conditions, surface equilibrium may be approached in these experiments. In this section, we apply some of our preceding results to the interpretation of these pyrolysisrate measurements, and we suggest an alternative hypothesis, concerning the nature of the surface process, which leads to a simple explanation of the pyrolysis results whether the surface process is near or far from equilibrium. Since our preceding results were derived for a modified pyrolysis experiment, we will find it necessary in the following discussion to consider the extent to which the results that we use are applicable to the previous pyrolysis experiments. Difference between Plate Temperature and Surface Temperature. In the previous interpretations of the pyrolysis experiments it has been assumed that the plate and surface temperatures are approximately equal in order to compute activation energies for the surface process. It appears reasonable that in the pyrolysis experiments heat is transferred from the hot plate to the solid surface across the gas film by a conduction process which is essentially one-dimensional. If this is the case, then the difference between the plate temperature Tp and the surface temperature T, can easily be estimated in terms of the observed pyrolysis rate. The appropriate one-dimensional heat conduction equation has, in fact, been solved in Eq. (7), and the result is given by Eq. (37e), since the pyrolysis rates are so small that kinetic energy and viscous dissipation are negligible. It can be shown that Eq. (37e), with
(43a)
for NH4C1,
,i~/p~ = 12o e:,p [ - 7100/ ( R1T,) J em/seo (453~
Remarks on the Interpretation of Pyrolysis Experiments on Ammonia Salts
< T. ( 8 0 7 ~
< Tp < 573~
(43b)
for NH4NO3, and ~h//p~ = 4600 exp [--20,O00/(R1T~,)] cm/sec (753~
< Tp < 903~
(43c)
for NH4C104. Since p~ is 1.527 gm/cm 3 for NHaC1, 1.725 gm/cm a for NH4NOa, and 1.95 gm/cm 3 for NH4C104, Eqs. (43) imply that 7h never exceeds 0.35 gm/cm 2 sec. Utilizing the reasonable values that pD ~- 5 X 10-4 gm/cm sec and l ~- 10-4 cm, we find ~, -- ~h
Edx/(pD)] << 1,
whence Eq. (39) is valid. With the reasonable estimate 5p = 0.2 cal/gm~ the dimensional form of Eq. (39) becomes (Tp -- T,) --= rh [AQ + h,~(T,) -- h,~(T,)3 (44) where use has been made of Eq. (36). The quantity in the square brackets in Eq. (44) may be computed from the relations AQ = AH/W,~
(45)
and h~( T.) -- hT( T,) = cp,T,l( T t -
T~) + Aht
+ e.,.,2(T~-- Tt)
(46)
where AH is the enthalpy change in the gasification of one mole of the salt, Tt is a temperature at which a phase transition of the sample occurs, Aht is the heat of transition per unit mass, and c,.,~.~ and cp.~.2 are the heat capacities of the sample below and above the transition temperature, respectively. For NH4C1,~5,2~-29 W~=53.5 gm/molc, Tt = 457.6~ (a solid phase transition), AhtW,~ = 1000 cal/mole, cp.,~,lW~ -~ 20.1 cal/mole ~ and cp,~.~lV~ = 22.3 cal/mole ~ for NH4N03 (references 25, 29, 30), W~ = 80 gm/mole, Tt = 442,8~ (melting), AhtW~ = 1300 cal/mole, Cp,~-A]4'~w = 43.5 cal/mole ~ and cp.,~.2W~ = 38.5 cal/mole ~ for NH4C104,25 W~ = 117.5 gin/mole, Tt = 513~ (a solid phase
A N A L Y S I S OF L I N E A R P Y R O L Y S I S E X P E R I M E N T S
transition) but since heat of transition data and heat-capacity data appear to be absent, it will be assumed that Ah~ = 0, cp.~,l ~ cp,~2 = 0.5 cal/gm ~ and Ti = 300~ in all cases. Utilizing this data and Eqs. (43)-(46), one can solve for (Tp -- T,) as a function of T~ for all three salts, provided AH values are known. Using the values of AH given in Eqs. (48) (based on our alternative hypothesis concerning the surface process) we find that (Tp -- T,) varies from I~ at Tp equals 629~ to 14~ at Tp equals 807~ for NH4C1, from 9~ at Tp equals 453~ to 80~ at Tp equals 573~ for NH4NOa, and from 3~ at Tp equals 753~ to 40~ at Tp equals 903~ for NH4CIOa. Thus, the assumption Tp = T, is approximately valid for NH4C1, invalid for NH4N0a, and poor for NH4CIOa. It is interesting that the variation in (Tp -- T,) is such that the experimental checks employed indicate that the approximation T, ~ T, is valid; e.g., (Tp -- T,) is relatively small at the melting point of NH4N03, so that the break in the pyrolysis curve would occur at a value of Tp near the melting point. If the larger AH values in Eqs. (47) (which are usually assumed to hold) are used instead of those in Eqs. (48), then the calculated values of (Tp -- T,) are roughly doubled; it is clear that they must increase since the term #zAH/W~ enters on the rigbt-hand side of Eq. (44). The estimate of pDgp used in this calculation should not be wrong by more than a factor of 5 and therefore cannot substantially modify our conclusion. Varying l does not alter the conclusion; if 1 is increased the estimate of (Tp -- T~) increases [-see Eq. (37e)-], while when / i s decreased free molecule heat conduction sets in and the estimate of (T~, -- T~) changes very little. The result that ( Tp -- T.~) increases substantially as Tp increases for NH4N04 and NHaCIO4 appears to be inescapable. A theoretical analysis by Cantrell, 31 in which a thin-film lubrication theory was used to account for the non-onedimensional flow effects in pyrolysis experiments, also supports the contention that (Tp -- T,) is not small. The main effect of the difference ( Tp -- T~) is to modify the computed activation energy for the interpretation corresponding to Eqs. (47), and to modify the AH values for the interpretation corresponding to Eqs. (48). For NH4C1, NH4NO3, and NH4CI04, respectively, the corrected activation energies would be roughly 14 kcal/mole, 30 kcal/mole, and 30 kcal/mole and the corrected heats of sublimation would be very roughly 14 kcal/mole, 25 kcal/mole, and 25 kcal/mole.
Alternative Interpretation of Pyrolysis Experiments. Other investigators have assumed that in
353
the pyrolysis of ammonia salts the over-all surface process is one of dissociation, 2'5'6 viz., NH4CI(s) ~ NH~(g) ~- HCI(g)
All = 40,3 kcal/mole
(47a)
NH4NOa(1) --~ NHa(g) -t- HNO3(g) AH = 39.8 kcal/mole
(47b)
and NH,CIO4(s) --~ NH3(g) + HCIOt(g) AH = 56 kcal/mole
(47c)
This hypothesis is supported by some evidence for dissociation in vacuum sublimation experiments. 1,5,~4,26-2s,~ The AH values listed here are obtained mainly from equilibrium vapor-pressure measurementsy -3~ except in the case of NH4C10, for which AH must be calculated from the heats of formation since no vapor-pressure measurements exist. In the pressure and temperature range of the pyrolysis experiments, it may be true that either (1) the equilibrium composition is one with a negligible amount of dissociation, or (2) dissociation occurs in a gaseous process, the rate of which is small compared with the rate of escape of the gas from the film between the plate and the sample. In either of these cases, the over-all surface process would be sublimation to the associated vapor. If the surface processes are simple chainlike processes such as that encountered in KC1 sublimation, then our previous analysis suggests that the temperature dependence of the pyrolysis rate would determine the heat of sublimation regardless of the importance of the fluid-transfer processes. The restriction to simple chainlike surface processes can be removed if surface equilibrium prevails in the pyrolysis experiments. When the correction for Tp ;~ T, is neglected, Eqs. (43) would then imply that NH4el(s) --~ NH4Cl(g) AH --- 13.5 kcal/mole
(48a)
NH4NO3(I) --~ NH4NO:~(g) AH = 7.1 kcal/mole
(48b)
NH4CIOt(s) --+ NH4CIO4(g) AH = 20 kcal/mole
(48c)
and
respectively, in the temperature ranges indicated in Eqs. (43). Although these conclusions seem reasonable because of the similarity between our proposed modified pyrolysis experiment and the experiments of Schultz and co-workers, the present conclusions lack the rigor of those in the previous subsection. Finally, we shall briefly discuss each material
354
COMBUSTION INVOLVING SOLIDS
separately, pointing out arguments for and against the hypotheses leading to the processes given in Eqs. (48). NH4CI: Since the pyrolysis data 2 yields the same activation energy as the vacuum sublimation data ~ and the low-pressure equilibrium appears at present almost certainly to involve entirely dissociated NH4C1 in the gas, 2s our interpretation will be valid only if hypothesis (2) of one of the previous paragraphs holds true. The large effect of H20 in catalyzing the decomposition,2s and the absence of It20 from the dried NH4C1 samples used in the pyrolysis experiments, I make it probable that the gas-phase dissociation is slow. NH4N03: In pyrolysis NH4N03 melts before gasification, and a change in the mechanism of gasification at the melting point causes the break in the experimental pyrolysis curve. 5 Since there exists some evidence that in equilibrium the gas is completely dissociated, ~~hypothesis (2) seems more plausible than (1) for the temperature range of the pyrolysis experiments. NH4C104: In spite of the large amount of recent work on NH4C104, [for example, The
Eighth International Symposium on Combustion, Williams and Williams Co., Baltimore, 1962, and also other papers of the present symposium] there appears to be no conclusive experimental information to eliminate hypothesis (1) or (2). A number of investigators (e.g., reference 14) have cited experimental evidence which is consistent with the assumption of dissociated vapor at equilibrium, but all of this evidence is equally consistent with the hypothesis of associated vapor. Since Galwey and Jacobs32concluded from theoretical considerations that the equilibrium vapor should exhibit a tendency toward association, hypothesis (1) may be valid. If Eq. (48c) is correct, then the pyrolysis experiments provided the first determination of any thermodynamic property of undissociated NH4C104(g).
D H
enthalpy per unit mass ]-cal/gm] hr
hg h,
p
• (v. 2 - v. 2) [cal/gm]; see Eq. (5) Thickness of gas film [cin] Total number of gaseous chemical species evolved from the solid Chemical symbol for solid Chemical symbol for gaseous species K Mass flux per unit area [gm/cm 2 sec] Total number of gaseous chemical species Permeability of porous hot plate to gaseous species K [gm/cm sec atm] Total pressure [atm]
p*
t
l M
M~(c) Mg(g) n~ N
PK
Y:. C~.KYK, mass-average specific heat at
M
p~
of
gas
E PK,e [atm]
K~I
pK
Partial pressure of species K at surface equilibrium [atm] Partial pressure of species K [ a t m ]
Pi.oo
Z PK,ooEatin]
p,
T t v WK
A constant in the momentum equation [atm]: see Appendix Universal gas constant, 1.986 [cal/ mole ~ Universal gas constant, 82.06 [atm cm~/mole OK] Universal gas constant, 8.317 X 107 [gm (cm/sec) 2/mole ~K] Temperature [ ~ Thickness of the porous plate [cm] Mass-average velocity [cm/sec] Molecular weight of species [gin/mole]
I~
[K=~t YK/WI~I; mass-average molecular
WE
weight Mass of molecules of species K leaving a unit area of surface per second [gm/cm 2
pK,~
M
R1
k~l
constant pressure [cal/gin ~
(dx/oD);
characteristic pressure
hr
5p
(eK/PK) K~I
R3
%,~
Z. Y,:,,hK(T,) -- h.(T,) + 89 K~I
R2
[(R3T)/(21rWIr ~, average velocity of molecules of species K in the --x direction [cm/sec] Specific heat at constant pressure of gaseous species K [cal/gm ~
[eal/gm] Total enthalpy per unit mass of gaseous species K [cal/gm] Total enthalpy per unit mass of solid [cal/gm] N
J
ACKNOWLEDGMENT
Cg
hK YK, total enthalpy per unit mass
h
This research was supported by the Air Force Office of Scientific Research, Office of Aerospace Research, under Contract AF 49(638)-412. Nomenclature
Binary diffusion coefficient Fcm2/sec]
h -4- (v2/2)(R1/R3), total stagnation
mixture
see 3
x
Spatial coordinate [cm]
ANALYSIS OF LINEAR PYROLYSIS EXPERIMENTS
YK
Mass fraction of gaseous species K
S
M
Y
~
K=I
YK, total mass fraction of gases
oc
355
At solid-gas interface In solid At downstream face of porous hot plate
evolved from solid
aK
Surface accommodation coefficient for species K (dimensionless) K=I
BE
ClK(pCK),
[dx/(pD)-],
EK
characteristic diffusion time to a characteristic recondensation time for species K Mass flux fraction of species K
AQ
Y~ eghK( Ts) -- h~( T,)
ratio
of
a
M K=I
(4/3)# -{- K, effective viscosity coefficient r g m / c m sec-] Coefficient of bulk viscosity [-gm/cm
K
see-]
),
rE
!
Thermal conductivity of the gas Fcal/cm see OK-] Ordinary (shear) viscosity coefficient [-gm/cm sec-] Stoichiometric coefficients for surface process, Eq. (28)
rh
f
[-dx/(pD) ~, dimensionless spatial
coordinate
~,
7h
Edx/ (pD) -], dimensionless mass
flow rate p (~ r o0 OJK
Density [-gm/cm ~] -~ p*/pp; dimensionless measure of pressure at upstream face of porous hot plate Dimensionless surface temperature; see Eq. (36) Dimensionless forward surface gasification rate; see Eq. (33) WK/[OtK(pCK)s], dimensionless forward surface gasification rate of species K
Subscripts e i K p r
Surface equilibrium A t upstream boundary of solid (x = --~) Integer index used only to denote chemical species At upstream face of porous hot plate Reference condition at which equilibrium vapor pressure is known
REFERENCES
1. SCHULTZ,R. D. and DEKKER, A. 0.: AerojetGeneral Corporation, The Absolute Thermal Decomposition Rates of Solids: Part I, The HotPlate Pyrolysis of Ammonium Chloride and the Hot-Wire Pyrolysis of Polymethylmethacrylate (Plexiglas IA). Technical Note No. 2, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)-1026, Azusa, November 1954. 2. BILLS, K., et al.: The Linear Vaporization Rate of Solid Ammonium Chloride. Technical Note No. 10, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)-1026, Azusa, July 1955. 3. SCHULTZ,R. D. and DEKKER, A. O. : Fifth Symposium (International) on Combustion, pp. 260267. Reinhold, New York, 1955. 4. BARSH, M. K., et al.: Rev. Sci. Instr. 29, 392 (1958). 5. SCHULTZ,R. D., GREEN, L. JR. and PENNER, S. S.: Studies of the Decomposition Mechanism, Erosive Burning, Sonance and Resonance for Solid Composite Propellants, Combustion and Propulsion, Third AGARD Colloquium, pp. 367-420. Pergamon Press, New York, 1958. 6. ANDERSON,W. H. and CHAIKEN,R. F.: AerojetGeneral Corporation, On the Detonahility of Solid Composite Propellants, Part I. Technical Memorandum No. 809, Azusa, January 1959. 7. FRIEDMAN, R., NUGENT, R. G., RUMBLE, K. E. and SCURLOCK, A. C.: Sixth International Symposium on Combustion, pp. 612-618. Reinhold, New York, 1957. 8. FRIEDMAN,R., LEVY,J. B. and RUMBLE, K. E. : Atlantic Research Corporation, The Mechanism of Deflagration of Ammonium Perchlorate. AFOSR TN-59-173, AD No. 211-313, Alexandria, Va., February 1959. 9. LEVY, J. B. and FRIEDMAN, R.: Eighth International Symposium on Combustion, pp. 663672. Williams and Wilkins Co., Baltimore, 1962. 10. JOHNSON, W. E. and NACHBAR, W. : Eighth Inter-
11.
12.
13. 14.
national Symposium on Combustion, pp. 678-689. Williams and Wilkins Co., Baltimore, 1962. ADAMS, G. K., NEWMAN, B. H. and ROBINS, A. B.: Eighth International Symposium on Combustion, pp. 693-705. Williams and Wilkins Co., Baltimore, 1962. ARDEN, E. A., POWLINO, J. and SMITH, W. A. W.: Combustion and Flame 6, 21 (1962). ROSEN, G.: J. Chem. Phys. 32, 89 (1960). BIRCUMSHAW, L. L. and PHILLIPS, T. R.: J. Chem. Soc. 1957, 4741.
356
COMBUSTION INVOLVING SOLIDS
15. PENNER, S. S.: Chemistry Problems in Jet Propulsion, pp. 234-241. Pergamon, New York, 1957. 16. KNUDSEN, M.: Ann. Physik 28, 75 (1909). 17. KENNARD, E. H.: Kinetic Theory of Gases, pp. 60-71. McGraw-Hill, New York, 1938. 18. PLESSET, M. S.: J. Chem. Phys. 20, 790 (1952). 19. BROOKFIELD,K. J., et al.: Proc. Roy. Soc. (Lon-
don) A190, 59 (1947). 20. KNUDSEN, M.: Ann. Physik, JT, 697 (1915).
21. Wvlmln, G.: Proc. Roy. Soc. (London) A197, 383 (1949). 22. KENNARD, E. H.: Kinetic Theory of Gases, pp. 320-324. McGraw-Hill, New York, 1938. 23. SCHULTZ,R. D. and DEKKER, A. O.: AerojetGeneral Corporation, The Absolute Thermal De-
composition Rates of Solids: Part III, The Vacuum Sublimation Rate of Ionic Crystals. Technical Note No. 7, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)1026, Azusa, July 1955. 24. BRADLEY, R. S. and VOLANN,P.: Proc. Roy. Soc. (London) A217, 508 (1953). 25. National Bureau of Standards, Circular 500, 1952. 26. SCHULTZ,R. D. and DEKKER, A. O.: AerojetGeneral Corporation, The Absolute Thermal
Decomposition Rates of Solids: Part IV, Transition-State Theory of the Linear Rates of Exchange and Decomposition of Crystalline Ammonium Chloride. Technical Note No. 8, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)-1026, Azusa, July 1955. 27. SPINGLER,H.: Z. Physik. Chem. B52, 90 (1942). 28. RODEBUStI, W. n . and MICHALEK, J. C.: J. Am.
Chem. Soc. 51, 748 (1929). 29. LUFT, N. W.: Ind. Chemist 31, 502 (1955). 30. FEICK, G.: J. Am. Chem. Soc. 76, 5858 (1954). 31. CANTRELL,R. H. Ja. : Gas Film Effects in Linear Pyrolysis of Solids. Ph.D. thesis, Harvard University, Cambridge, May 1961. 32. GALWEY, A. K. and JACOBS, P. W. M.: J. Chem. Soc. 1959, 837. Appendix
The M o m e n t u m E q u a t i o n In the gaseous region, the one-dimensional momentum equation is
dv._~R3dp rhdx ~'dx
d (de) dx ndxx ----- 0
the first integral of which is
(49)
p -~ rhv -- 77 dxx =
p~ = constant (50)
Equations (1), (18), and (50) determine the distributions of p, p, and v in the gas in terms of the composition and temperature distributions studied in the main body of the paper. The evaluation of the constant p~ in Eq. (50) depends upon the experimental arrangement. If the gas layer alone supports the solid (in whicb case 1 is not known a priori), then it can be shown that p~ is the axial compressive stress in the solid, which equals the atmospheric pressure plus any additional pressure applied to the cold end of the solid sample in order to increase "thermal contact" with the porous hot plate. On the other hand, if the solid is supported in such a way that the spacing 1 is fixed (e.g., by means of a specially designed sample holder, or by protuberances on the surface of the sample, as appears to be the case in the previous pyrolysis experiments), then the gas-phase equations given in the main body of the text determine p~. An interesting consequence of these observations arises when
rhv -- y(dv/dx) << (R~/R2)p
(51)
Equations (50) and (51) imply, approximately, p =
p~ = p . =
p.
(52)
In the discussion at the end of the section entitled Simple, Chainlike Surface Processes, it is assumed that Eqs. (51) and (52) are valid first approximations for an integral of Eq. (50). The dimensional form of Eq. (38b) is written for this case as rh =
(P,/t)
(p. -
p,~)
(53)
If p, is not measured, as in the second experimental arrangement discussed above, then it is determined through Eq. (53) by measurements of rh and p.~ (provided that Eq. (52) is valid). If p~ is easily measured, as in the first experimental arrangement, then we might ask whether this measurement could replace the more difficult measurement of l for this arrangement. Unfortunately, it cannot. In this case, 7h and p:, both easily measured quantities, satisfy Eq. (53) as a redundant equality independent of 1. I t would still be necessary to measure l for this first experimental arrangement in order to carry out the calculations indicated in the section mentioned above.
ANALYSIS
OF LINEAR
PYROLYSIS
357
EXPERIMENTS
Discussion DR. D. J. SIBBETT (Aerojet-General Corporation): The analysis of linear pyrolysis experiments by Dr. Nachbar and Prof. Williams indicates very nicely the critical factors and limitations of the technique. I t is obviously of p a r a m o u n t importance in the interpretation of data collected by this method t h a t the measured experimental temperature be identifiable with surface temperature. Recently in the course of a study of the rates of sublimation of the four ammonium halides, evidence was obtained which confirms this identity. In these experiments the rates of vacuum sublimation of the ammonium halides were determined gravimetrically by use of the quartz balance at low temperatures and by the linear pyrolysis technique at high temperatures. Surface temperatures during the gravimetric determinations were obtained from dual sets of experiments. In one set, rates of sublimation and the temperature of a thermowell in the quartz balance wall were obtained simultaneously. In the second set, thermowell temperatures and sample surface temperatures were correlated to give temperature corrections which were applied to the rate measurements. Linear pyrolysis temperatures were measured directly. The figure, which is a plot of the rates of sublimation of ammonium bromide between 153 and 574~ is a typical example of these data. Rates, which are indicated on a logarithmic scale on the ordinate, have been converted to linear regression units (cm/ sec). D a t a from 153 to 227~ were obtained gravimetrically; those at higher temperatures were obtained b y use of the linear pyrolysis or "hotplate" method. For all four halides, NF4F1, NH4CI, NH4Br, and NH4I, the data took the same form. The figure demonstrates t h a t : 1, The vacuum sublimation data are consistent with the linear pyrolysis data and give an Arrhenius plot with a single apparent activation energy value. 2, At very high rates, a departure from linearity was observed which corresponds to the range of conditions in which meaningful temperature measurements fail. However, b y increasing the loading force between the stand and the hot plate, which decreases the thickness of the gaseous layer, the linear plot can be extended. This effect is demonstrated b y the two sets of data at the upper limits of the rate measurements. From these data, it has been concluded: 1, The
10-2
9'~ 10-3 -2
v
~
lO "I
HOT PLATE DETERMINATIONS
--
g/cm2LoadingPressur~
o 4865 A 2148 g/cm2LoadingPressure [ ~
.J
10.5
I: []QUART"BALA,CE
\
DETERMIHATIONS
1.0
1.2
1.4
1.6
1.8
' ~
2.0
2.2
2.4
SURFACE TEMPERATURE, 1/Ts, 10.3 ( ~
FIG. 1. Rates of sublimation of ammonium bromide.
temperatures measured b y the nonporous hot-plate apparatus are valid measures of the surface temperature in these experiments. 2, Linear pyrolysis measurements are determinations of the rates of sublimation of the a m m o n i u m halides. T h a t is, a single reaction mechanism appears to be operating over the entire temperature range. Application of the linear pyrolysis technique requires experimental or theoretical analysis of the processes involved at the hot interface in order to establish the meaning of the measured temperature. Within these limitations it has proven a very useful tool. PROF. F. A. WILLIAMS (Harvard University): The only ammonium halide t h a t we have considered is NH4C1, and, as indicated in the paper, we agree with Dr. Sibbett t h a t the difference between the plate temperature and the surface temperature is negligible in this case. We found significant differences only for NH4NO3 and NH4CIO4.
Vice Chairman: Dr. P. Nichols
(United Technology Corp.)
(A erojet-General Corp.)
ON THE ANALYSIS OF LINEAR PYROLYSIS EXPERIMENTS W. N A C t I B A R A N D F. A. W I L L I A M S
Linear pyrolysis experimental data have been used as surface boundary conditions in the analytical treatment of solid propellant combustion problems. The surface boundary conditions required for the analysis are functional relationships between normal regression velocity, gaseous reactant density, total pressure, temperature, and possibly derivatives of these quantities as well, evaluated at the interface between condensed and gaseous phases during combustion. Available experimental data indicates t h a t a relation based upon equilibrium conditions prevailing at the interface is not generally valid. There are available for many substances, however, hot-plate linear pyrolysis data which give an explicit relation between regression velocity of the conden~d phase and a measured hot-plate temperature. This hot-plate temperature has L~en assumed to be also the temperature at the interface. Ilowever, this assumption is thought in some cases to be questionable. In addition, the use of such a rate law in combustion analysis would imply t h a t the total pressure and the gaseous reactant partial pressures or densities, whieh are greatly different for the pyrolysis test as compared to the eombustion, have a negligible effect on the rate. This implication again is questionable. The pre~ent paper seeks to answer in part these questions on the validity of using linear pyrolysis data. A one-dimensional hot-plate linear pyrolysis cxperimenf, is proposed which uses a porous heated plate in place of the present impervious plate. The proposed experiment, having simpler fluid dynamics than present hot-plate experiments, is Capable of a reasonably precise analytical description. The analysis is carried out for a general case and is then specialized to simple and chainlike surface gasification processes. Conditions are developed to determine whether the surface process is either a rate process or is one of Imar-e.quilibrium. A description is presented of the possible use of data from the proposed pyrolysis experiments to compute the accommodation c(x~tticient and the vacuum sublimation r a ~ as functions of surf'~ce temperature. With this information, the surface boundary condition for a given material will then be specified. The pyrolysis rate of potassium chloride is calculated as an example of the use of the derived formulas. If fuid dynamic effects are not too i m p o r t a n t in the present experiments, and in some respects it is estimated t h a t they are not, interpretation from the results of this analysis serves present pyrolysis data as well. Interpretations are carried out for a number of compounds which have interest in propcllant combustion studies. From published data on ammonium perchloratc, for example, revised values are estimated for the heat of sublimation, for the activation energy in linear pyrolysis, and for the difference between the hot-l)l.~te temperature and the surface temperature in linear pyrolysis. 345
346
COMBUSTION INVOLVING SOLIDS
Introduction Schultz et al. t-6 have reported experiments in which hot-plate pyrolysis techniques are used to measure the linear regression rates of solids undergoing surface heating. These experiments have yielded pyrolysis rate data for many different materials at pressures and temperatures approaching (but not yet equaling) pressures and temperatures encountered in solid propellant rockets. These data are of practical importance in helping to predict, for example, the deflagration rate of pure ammonium perchlorate, ~-12 and thereby in promoting further understanding of the factors controlling the burning rate of composite solid propellants containing ammonium perchlorate. I t has also been proposed ta to use pyrolysis data for actual rocket propellants (if such data is available) to predict burning rates. The pressure is sufficiently high in these pyrolysis experiments so that the continuum equations of fluid mechanics may be applied to the gas adjacent to the solid. The geometry of the gas flow is so complicated, however, that unquestionable interpretations oi the significance of the measurements appear to be difficult to obtain. Consequently, it is of interest to investigate the possibility of modifying these experiments in order to yield data which can be interpreted more easily. In the present paper there is suggested a modified hot-plate linear pyrolysis experiment which should not be difficult to perform. A mathematical analysis of this experiment is caried out under simplifying assumptions which, in the main, do not appear to be unduly restrictive. However, the reader's attention is directed to the following assumptions made in this analysis. Except for the mathematically thin, plane surface in contact with the hot gas, the condensed phase is assumed here to be in a uniform phase and to be homogeneous. (We will refer in particular to the solid phase in our discussion.) Both gaseous and condensed phases are assumed to exhibit no chemical reaction. These assumptions may not be valid for certain materials under all conditions. There may occur in the solid energetic bulk chemical reactions, at temperatures less than the surface temperature, which transform one solid-phase state into another solid-phase state. 14 There may also occur, as in ammonium perchlorate, for example, a partial decomposition of the solid into a reactant gas plus a solid residue. The plane surface in this latter case is more exactly a layer of finite thickness in which the two phases exist simultaneously. These mechanisms may be important if the linear regression rate of the solid is sufficiently small. There appears to be no
essential difficulty, however, in extending the present analysis to the more complex cases where these assumptions are relaxed. The usefulness of such an extension in particular cases will depend upon whether the kinetics of the condensed-phase reactions and the degree of nonhomogeneity are known. Conditions under which the surface process is either a rate process or is one of near-equilibrium are determined for the modified experiment from our analysis, and interpretations of the significance of the measurements are given for these two limiting cases. The pyrolysis rate of potassium chloride is computed as an example. The mathematical simplicity of the system described here provides a strong incentive to construct such an apparatus for measuring regression rates. A degree of similarity of the present model to the previous hot-plate linear pyrolysis experiments also makes it possible to draw some conclusions concerning the previous measurements.
The Modified Experiment It is suggested that the impervious hot plate used in previous pyrolysis experiments be replaced by a porous hot plate. If suitable shielding can be provided to prevent gas from escaping without passing through the porous plate, then the flow would be approximately one-dimensional, and the arrangement would appear as illustrated in Fig. 1. The solid is assumed to be converted to gas at x equals 0, and the gas flows through the porous plate at x equals l, where l is the normal distance between the surface and the inner face of the porous plate. Steady-state conditions are assumed. For convenience in the analysis, the solid is considered to be semiinfinite. At x equals -- ~ , the solid has a density p, and a velocity v, directed towards the interface, which remains stationary. In analyzing this experiment, the gas-phase and the interface conditions will be considered in that order. The momentum conservation equation is considered in the Appendix. The solid sample is assumed to be homogeneous and nonreactive for x < 0.
~'$~P'I~E'~ 0 -co=
1
s
e SOT PLATEr + x p
co
FIo. 1. Schematic diagram of modified pyrolysis experiment.
ANALYSIS OF L I N E A R PYROLYSIS E X P E R I M E N T S
The one-dimensional, steady-state equations on 0 <: x < l for conservation of total mass, mass of species K (in the absence of gas-phase chemical reactions), and energy of a mixture of N species of ideal gases are, respectively15:
pv = m = p.v,~ = constant .dYE m dx
(
d dx oD d~x ]'
(1)
347
Denoting the boundary values of YK and H at x equals 1 as Y~ and H~, one finds the solutions to Eqs. (3), (5), and (6) to be H(~) ---- H ~ -
J[exp (~p) -- e~3
(7)
and Yt;(~) = YK,p exp(~--~p)
K =- 1, " " , N
(2)
+ e K [ 1 - - exp (~--~p)3,
K=
1,-..,Y
(8)
where
and
dH m dx = d-x pD -~x
-- ~it
9
(3)
Equation (2) implies that the binary diffusion coefficients of all pairs of chemical species are equal. Equation (3) relies upon this assumption, the approximation that the Lewis number is unity (k = pDhp), and the assumption that either the mass diffusivity equals the effective coefficient of viscosity (~ = pD) or the ~kinetic energy is negligible in comparison to the thermal plus chemical energy:
([~v~( Rff R3) ] ( ( h ) It is to be noted 5p, ),, and pD may depend upon both T and YK in this formulation. When the energy flux at x = ~ oo is equated to the energy flux at x = 0-t-, and when the conditions T = T~ and (dH/dx) = 0 are imposed at x = - - ~ , the equation
~, ~ m
s
[dx/oD 3
(9a)
[dx/oD 3
(9b)
While the experimenter often has direct control over the plate temperature Tp, the values of YK.p [which are needed along with Tp to determine Hp and Yp in Eqs. (7) and (8)-]* will basically be determined by a permeability Pt: of the porous plate of species K and by the partial pressure pK.~ of species K on the downstream side of the plate. A reasonable phenomenological expression is obtained by assuming that the mass flux ?beE for each species is proportional to the pressure difference across the plate for each species, viz.
PK(
rheK = --t- \-
~
)
PK.~o_ K =
rh [hT( T~) "-{- ( RffRa) 89
(lO)
.= ~h [K~ YK.,hK( T,) --}- (RI/R3) 89 -- [pD(dH/dx)~,
(4)
is obtained. This relation is a boundary condition for Eq. (3), and it is written as
oD ~
1, " " , N
, = ThJ
Here Pv is the total pressure at x = l, t is the thickness of the porous plate, and the first term inside the bracket is the partial pressure pK of the species K at x = 1. The mass-average molecular weight of the gases at x = l is Wv:
(5)
Conservation of mass across the phase interface requires that the net flux (i.e., mass per unit area per second) of species K leaving the surface to be equal to ~heg, which is the mass flux of species K produced in the gasification process. The eK'S are the mass flux fractions of species K in the decomposition of the solid, and they are determined by the over-all stoichiometry of the gasification process. With this boundary condition at x equals 0, the following integrals of Eq. (2) are obtained
~hl"K -- pD(dYK/dx) = ~he~:, K ~- 1, " ", N (6)
Equation (10) can be put into a convenient, nondimensional form by introducing a characteristic pressure p*, defined as
p* - t z.. (~:/PK) K=I
/fzdx, pD
/Jo
(11)
and a ratio ~, which is a measure of p~ and which is defined as
= p*/p~
(12)
* Strictly speaking, either vp must also be measured or the momentum equation (see Appendix) must be solved in order to determine completely the constants in Eq. (7). However, the kinetic energy will usually be negligibly small.
348
COMBUSTION INVOLVING SOLIDS
In view of these definitions and of Eq. (9b), Eq. (10) is written as -l/V, -
WK
y
o"
K,p ~
-( e e- / P K )
tp "4-
Pe,oo l
N
we and a e may be functions of Tz, ps, and all the Ye.,'s for complex surface processes. Equation (161) may be written in the nondimensional form
P$ |
F_, ~/P~
~pee = ilK(we -- YK,8),
J=l
K = 1, " " , M
(19)
[dx/(pD) ]
(20)
where K=
(13)
1,-..,N
Although only N -- 1 of the YK,p are linearly independent, the N relations in Eq. (13) are independent, since summing Eq. (13) over all N species gives the nontrivial equation
fie ~ o~e(pce), re
-
(21)
weE-e(pce).]-'
(14)
From Eq. (19) it follows that, in general, the surface process with respect to any evolved species K will be a rate process, similar to that
where p,~ is the total pressure maintained on the downstream side of the plate, v/z.
encountered in vacuum vaporization or sublimation experiments l~m if the following condition holds:
1 = a (#, -]- P,co ~-)
(Ye,./~)
N
(15)
P,oo = ~_, PK,o~ e~l
A general boundary condition at the solid surface is now obtained for each of the species evolved in the gasification of the solid. Let the species numbering be so ordered that, if M is an integer, 0 < M < N, then the first M species in the gas are evolved from the decomposition of the solid. Consequently, K = M -{- 1, ---, N denotes the chemically inert species in the pressurizing gas which are present at the surface through diffusion. If M = N, there are no inert species. It follows that eK ---- 0, by definition, for the inert species K -- M -[- 1, . . - , N. The general boundary condition for the evolved species is obtained by equating the net surface mass flux ~hee of each species to the difference between the vacuum sublimation rate WK, which is the mass flux of the species leaving the surface, and the mass flux of the species entering the surface, 16-~9 rheg = WE -- a e Ye,~(pce)~
K -=- 1, . ' . , M
(161) eK = 0
K=
M-[-1,
"",N
(16b) Here a/~ is a surface accommodation coefficient for species K, cK = [ ' ( R 3 T , ) / ( 2 7 r W K ) -]~
(17)
is the average velocity of molecules of species K (of the ideal gas mixture) in the - - x direction at the surface, and p = pI~/'/(R2T)
=
(~.~e/~e&)
<<
1 (22)
At the opposite extreme, near-equilibrium conditions will prevail for an evolved species K at the surface if (Ye,Jwe) ~ 1, that is, if
(~,eK/wefle) << 1
(23)
General Equations for Determination of Pyrolysis Rate The simple geometry of the modified pyrolysis experiment, and the simplifying assumptions which have been made in the analysis, allow the temperature and concentrations at each point of the gas to be expressed in an elementary form by Eqs. (7) and (8). The constants or parameters appearing in these equations must be determined uniquely with the use of the boundary conditions, however, in order to determine a unique solution. In order to do this, certain of the parameters must be known either from predetermined physical properties or from measurements taken during the pyrolysis experiment. Which of the parameters fall into this class is somewhat arbitrary. We will assume at first that the unknown parameters are:
{ Yg,p}, [ fg,.}, IeKI, Ts, r ~p
(24)
The notation {( )g} stands for the set ( )g, K = 1, - - . , N. Since there are 3N -{- 3 unknown parameters in group (24), then the same number of independent equations must exist to determine them. Evaluating Eqs. (7) and (8) at $ equals 0 gives N ~ 1 of these equations, viz. YK,, = YK,p exp (--~p) -{-eKF1 -- exp (--~p)l K--- 1 , . . . , N
(18)
according to the ideal gas law. In general, the
1 -
H, = H p -
JFexp ($p) -
1]
(25) (26)
ANALYSIS OF LINEAR
PYROLYSIS
Note that the identity
geometry of the gas flow. The modified pyrolysis experiment suggested in the present paper has certain advantages in this application, and this will bc dcmonstrated in the following sections.
N
~'-~eK=l is implied by the summation of Eqs. (25) from K= ltoK= N. Equations (13), (16b), (19), (25), (26) and the two identities N
N
Y~. YK,. = E YK,~, = 1
K~I
K~I
349
EXPERIMENTS
(27)
constitute a set of (3N + 3) transcendental equations to determine the unknown parameters in (24). For this problem it is assumed that { P K } , { W K } , and t are predetermined constants for an experiment, that l, Ps, {PK.oo}, Ti, and Tp are parameters measured in the experiment, and that ~p, pD, J, {ilK}, and {WE} are known functions of these parameters and of the parameters in group (24). If all of this information is given, then one could, for instance, compute the pyrolysis rate v~ expected in a pyrolysis test as a function of the plate temperature Tp and the downstream pressure p.r A numerical example of such a computation is given in one of the following sections. The application of the hot-plate, linearpyrolysis test results to practical problems which have been its chief motivation is not nearly so straightforward, however. An important practical problem is to obtain, for solid propellant constituents, the data which enter into the general surface conditions, Eqs. (16), so that these equations can be used to compute propellant deflagration rates from a combustion theory. If {WE} and {aK} are known functions, however, as was implied in the paragraph just preceding, then it would not be necessary, for application to this particular combustion problem at least, to perform the pyrolysis test. The necessary parameters for the general surface conditions, Eqs. (16), would be determined already. However, the central difficulty of this problem is that the data necessary to evaluate Eqs. (16) for propellant constituents are not usually available, and especially not for the range of surface temperatures obtained in deflagration. To apply Eqs. (16) to the propellant combustion problem, we should ideally like to be able to calculate from the linear pyrolysis data the vacuum sublimation rates {WK} and the surface accommodation coefficients {aK} as functions of Ts, Ps, and { YK,S}. Data from previous pyrolysis experiments 1-6 (i.e., v, as function of Tp for fixed P.oo) has not been applied to the calculation of these functions, however. Such application would appear to be generally quite difficult to perform in an unequivocal way because of the complicated
Simple, Chainlike Surface Processes The equations which are developed in the preceding two sections are now reduced to a much simpler set by the introduction of the following additional assumptions. a. The surface process is simple, that is, the fig and wK are functions only of T~ (i.e., independent of p~ and { YK.8}). b. The surface process is chainlike, meaning that parallel, competing surface reactions do not occur. A consequence of this assumption is that the e g are constants--independent of Ta, pa, and { Yg,8}--and each eg will be assumed to have a known value. c. The molecular weights of the gaseous species are all equal to a common molecular weight W. d. Changes in kinetic energy are neglected in comparison to changes in thermal energy to a first approximation (higher approximation can be computed through integration of the momentum equation). e. The isobaric heat capacities of all species are equal to an average value of the mixture heat capacity 5p between T, and Tp. Let the process indicated in (b) above be represented by M
M,(c) ~ E vg'MK(g)
(28)
K~I
The VK'S are the fixed number of moles of species K produced per mole of solid; as a consequence of (b) and (c) above, EK = VK!
PK!
Consider Eq. (28) in equilibrium at a fixed T~; let Pg,e be the equilibrium partial pressure of species K, and let pe be the total equilibrium pressure of all evolved species: M
Pe =
~_, PK,e K~I
As a consequence of the chainlike assumption (b),
PK.e = egpe
K = 1, " " , M,
(29)
the enthalpy difference AQ(T~) corresponding to Eq. (28) is, per unit mass of solid, M
AQ(T,) = E eghK( Ta) -- h,( T,) K=I
350
COMBUSTION
INVOLVING
Hence, the Van't Hoff equation for the equilibrium (28) is written as d [ i n K~I~I(PK,.)vK'] = W~. AQ dTJ(R1T~ 2) (30) The ehainlike assumption means dO'K') = O, K = l, ---, M. Since mass conservation requires M
SOLIDS
as a measure of To. The last equality in Eq. (36) can be derived from Eqs. (5), (25), and (26). For typical endothermic surface reactions, it is usually the case that r << 1. It is the case in many of the pyrolysis experiments reported previously. The first two of the following set of equations are obtained by summing Eqs. (13) and (25) from K = 1, . - ' , M; the third is obtained from Eq. (34) ; the fourth from Eq. (14) and the fifth from Eq. (26).
Yp = o-[~p-~ (p~,Jp*)-] then the following integral of Eq. (30) expresses p~ as a function of T. only:
p~( T~) = p.( T~) exp [ - fT~ W AQ dT] I_ .,To R~T ~
~(T.) = p-' ~ ~K P~ KL__I
(32)
(33)
(1-- Yp)exp(--$p)
~. = 811 - (Y,p./p~
(31)
Here T~ is an arbitrary reference temperature at which p~ is known. These results are now applied to Eqs. (19). The quantities/~ and w, which are functions only of T. [cf., Eqs. (17), (18), (20), (21), (31), and assumption (a)], are defined as
8( T.) = pa p-~ /~=1 (eg/flK)
1-- Y.=
(37a) (37b) (37c)
1 = a [-~p+ (p,~/p*)-]
(37d)
7- = exp (~p) -- 1
(37e)
In Eq. (37a), M P M , c o ~--
~ K~I
PK,~
Equations (37a, b, c, d) may be combined to eliminate Y, and Y.; the result is the equations ~, -~- p .
The summation of Eqs. (19) over K = 1, -. ", M then gives the equation
1--
-X exp (--~p)
pJp* = ~p + (p,~/p*)
(38a) (38b)
An approximate solution of the preceding equations is now obtained in order to discuss the physical aspects of these results. It is now assumed that v << 1, and so it follows from Eq. (37e) that
where Y, defined as M
Y--~_~YK K~I
is the total mass fraction of the evolved species. But Yaps in Eq. (34) is the partial pressure of the evolved species at the surface. For fixed T,, ~ (orTh) varies only with Y,p~, and as Y,p~-~ p~, the equilibrium requirement is Sp --~ 0. I t must therefore follow (under the assumptions already made) that w(T.) ~ 1 for all T,. Using this result, together with Eqs. (17), (18), (21), and (33), we find the following equation to hold:
~R2'T. j ~] ,~ WE(To) . ~ ( n ) = p~(n)
7" - -
-
-
g.
g
5,(Tp--
~p << 1
(35)
T~)
AQ(T~) -]- h,~(T~) -- h~-(T,) (36)
(39)
As a first approximation we set p. equal to p, (see Appendix). Then, neglecting terms which are of higher order than the first power of 7-, Eq. (38a) may be shown to take the following form; it is understood that p~ and/~ are evaluated at the plate temperature T, in this expression: P~-
P~
$P = t-P* -F P,o~ -- PM,~ -'~ Pe(ff-~ -'F A)']
Returning now to the development of a simplified set of equations, we find it convenient, as a consequence of assumptions (d) and (e), to define the nondimensional parameter % _ Yp-
7- ~
(40)
where
A --
5pR1Tp 2
eghg(Tp) -- h,~(T~)
For simplicity of presentation, we will neglect A as small compared to if-l, and we will also neglect PM,~ with respect to both pe and p,~. The limiting criteria for the surface process (cf. Eqs. (22) and (23)) are (~p//~) ~ 1 for a rate
ANALYSIS OF LINEAR PYROLYSIS EXPERIMENTS
process and (~p/~) (~ 1 for a near-equilibrium process. We may then write from Eq. (40) :
(~,/~) = i-1 + (p* + P,oo)(~/P~)]--'
(41)
The limiting criteria are therefore
(p* + p.~) (~/p~) << 1 for a rate process, and (p* ~ p,~) (fl/p~) >> 1 for a near-equilibrium process. An expression for (~/p~) follows from Eqs. (17), (18), (20), and (32); again for simplicity, we restrict attention to the case M = 1:
/ R3W1 fo Zdx
(42)
In most present pyrolysis experiments, ~-6 experimental conditions are approximately equivalent to setting p* ~- p,~ equal to p.~ equal to 1 atm. Surface temperatures are assumed to be in the range 300~ ~ T, ~ 900~ and for reasonable pD and WI(pD ~ 5 X 10-a g/em see, W~ = 75 g/mole) it follows from Eq. (42) that
p~l
351
series as a rate process; the other has the surface process as near-equilibrium. From Eq. (41) and the subsequent discussion, it follows that, in the rate process ~ ~ ~p, while in the equilibrium process p~ ~- (p* -t- p,~)~l. Hence, if ~h is measured as a function of Tp, then we determine p~(T,) in the surface-equilibrium test series, and we determine ~(T~) in the surface rate-process test series. Using this data we can determine a(T.) from Eq. (42) and w(T,) from Eq. (35). Pyrolysis R a t e of P o t a s s i u m Chloride As an example of the application of the preceding analysis we shall consider KCI, one of the few materials whose properties are sufficiently well known to permit an a priori calculation of the pyrolysis rate. Sublimation of KCI is a simple chainlike surface process (reference 23). In Fig. 2, the theoretical pyrolysis rate is plotted as a function of the plate temperature Tp for various values of 1. The computation was made from Eq. (40) with p* -[- p,r equal to 1 arm, PM,~ equal to 0, and T; equal to 300~ Here (references 23, 24,
= 5 X 103 (era-1)
10"3 - ~ - i
within a factor of 10. Since it is expected that one mean free path ~ 1 0 -5 cm, it is therefore estimated that 10-5 cm ~ l ~_ 10-2 cm, and it follows that
~
~---~
I0"4
0.05~ _~ (p* -~- P,~o) ~- ~ 50a P, Hence, for simple, chainlike surface processes with r << 1, the surface process will probably be near equilibrium for a equals 1 and a rate process for ~ < 0.01. We now discuss the determination of accommodation coefficients and vacuum sublimation rates, of substances for which these functions are not known, by means of the modified hot-plate, linear-pyrolysis test proposed in this paper. For this purpose we restrict attention to simple, ehainlike surface processes for which M = 1, and base this discussion on the simplified set of Eqs. (41) and (42). There are to be varied at our disposal in this experiment the dimensions and porosity of the hot plate, which are incorporated into the characteristic pressure p*, the pressure p.~ maintained on the downstream face of the hot plate, the hot-plate temperature Tp, and possibly also the sample spacing 1. It is proposed that suitable ranges of values for these parameters be found so that, on any one given material, two series of tests may be performed: one has the surface process for the test
~
10"5
:_~ - _ _____
V
~
e= 10"2cm
v
107 ......... 0.9
1,0
\
...... 1 \
1.1 1.2 1.3 Tp'l x10-3(OK-l)
-
1,4
Fro. 2. Linear pyrolysis rate of potassium chloride.
352
COMBUSTION INVOLVING SOLIDS
and 25) G = 0.117 cal/gm~ h,(T,) -h~-(Ti) = 0 . 1 6 6 ( T , - Ti) cal/gm, AQ = 693 eai/gm, pD ---- 5 X l0 -4 gin/era sec, a = 0.63, and p~ = 6.4 X l0 T exp (--26,070/T~) atm. The computations were completed only for T~ less than the melting point (1049~ The results imply that surface equilibrium exists (provided l > 10-5 cm). Since it was always found that Tp -- T~ ( 5~ Eq. (40) is actually valid here, and a pyrolysis rate measurement yields the equilibrium sublimation pressure.
V given by Eq. (36), relies upon none of the aspects of our preceding model except the heattransfer equation. The pyrolysis results, 2'5,6 are m/pT = 120 exp [--13,500/(RIT~)] cm/see (629~
Schultz and co-workers have reported linearpyrolysis rate measurements for NH4C1,~''~ NH4NO3, 5 and NH4CI04, 6 and have given consistent interpretations of their results on the basis of absolute reaction-rate theory. 1,5,~6 These interpretations rely upon the assumption that the surface process is an unopposed rate process (i.e., surface conditions are far from equilibrium). Our previous analysis of the surface process indicates that, under certain conditions, surface equilibrium may be approached in these experiments. In this section, we apply some of our preceding results to the interpretation of these pyrolysisrate measurements, and we suggest an alternative hypothesis, concerning the nature of the surface process, which leads to a simple explanation of the pyrolysis results whether the surface process is near or far from equilibrium. Since our preceding results were derived for a modified pyrolysis experiment, we will find it necessary in the following discussion to consider the extent to which the results that we use are applicable to the previous pyrolysis experiments. Difference between Plate Temperature and Surface Temperature. In the previous interpretations of the pyrolysis experiments it has been assumed that the plate and surface temperatures are approximately equal in order to compute activation energies for the surface process. It appears reasonable that in the pyrolysis experiments heat is transferred from the hot plate to the solid surface across the gas film by a conduction process which is essentially one-dimensional. If this is the case, then the difference between the plate temperature Tp and the surface temperature T, can easily be estimated in terms of the observed pyrolysis rate. The appropriate one-dimensional heat conduction equation has, in fact, been solved in Eq. (7), and the result is given by Eq. (37e), since the pyrolysis rates are so small that kinetic energy and viscous dissipation are negligible. It can be shown that Eq. (37e), with
(43a)
for NH4C1,
,i~/p~ = 12o e:,p [ - 7100/ ( R1T,) J em/seo (453~
Remarks on the Interpretation of Pyrolysis Experiments on Ammonia Salts
< T. ( 8 0 7 ~
< Tp < 573~
(43b)
for NH4NO3, and ~h//p~ = 4600 exp [--20,O00/(R1T~,)] cm/sec (753~
< Tp < 903~
(43c)
for NH4C104. Since p~ is 1.527 gm/cm 3 for NHaC1, 1.725 gm/cm a for NH4NOa, and 1.95 gm/cm 3 for NH4C104, Eqs. (43) imply that 7h never exceeds 0.35 gm/cm 2 sec. Utilizing the reasonable values that pD ~- 5 X 10-4 gm/cm sec and l ~- 10-4 cm, we find ~, -- ~h
Edx/(pD)] << 1,
whence Eq. (39) is valid. With the reasonable estimate 5p = 0.2 cal/gm~ the dimensional form of Eq. (39) becomes (Tp -- T,) --= rh [AQ + h,~(T,) -- h,~(T,)3 (44) where use has been made of Eq. (36). The quantity in the square brackets in Eq. (44) may be computed from the relations AQ = AH/W,~
(45)
and h~( T.) -- hT( T,) = cp,T,l( T t -
T~) + Aht
+ e.,.,2(T~-- Tt)
(46)
where AH is the enthalpy change in the gasification of one mole of the salt, Tt is a temperature at which a phase transition of the sample occurs, Aht is the heat of transition per unit mass, and c,.,~.~ and cp.~.2 are the heat capacities of the sample below and above the transition temperature, respectively. For NH4C1,~5,2~-29 W~=53.5 gm/molc, Tt = 457.6~ (a solid phase transition), AhtW,~ = 1000 cal/mole, cp.,~,lW~ -~ 20.1 cal/mole ~ and cp,~.~lV~ = 22.3 cal/mole ~ for NH4N03 (references 25, 29, 30), W~ = 80 gm/mole, Tt = 442,8~ (melting), AhtW~ = 1300 cal/mole, Cp,~-A]4'~w = 43.5 cal/mole ~ and cp.,~.2W~ = 38.5 cal/mole ~ for NH4C104,25 W~ = 117.5 gin/mole, Tt = 513~ (a solid phase
A N A L Y S I S OF L I N E A R P Y R O L Y S I S E X P E R I M E N T S
transition) but since heat of transition data and heat-capacity data appear to be absent, it will be assumed that Ah~ = 0, cp.~,l ~ cp,~2 = 0.5 cal/gm ~ and Ti = 300~ in all cases. Utilizing this data and Eqs. (43)-(46), one can solve for (Tp -- T,) as a function of T~ for all three salts, provided AH values are known. Using the values of AH given in Eqs. (48) (based on our alternative hypothesis concerning the surface process) we find that (Tp -- T,) varies from I~ at Tp equals 629~ to 14~ at Tp equals 807~ for NH4C1, from 9~ at Tp equals 453~ to 80~ at Tp equals 573~ for NH4NOa, and from 3~ at Tp equals 753~ to 40~ at Tp equals 903~ for NH4CIOa. Thus, the assumption Tp = T, is approximately valid for NH4C1, invalid for NH4N0a, and poor for NH4CIOa. It is interesting that the variation in (Tp -- T,) is such that the experimental checks employed indicate that the approximation T, ~ T, is valid; e.g., (Tp -- T,) is relatively small at the melting point of NH4N03, so that the break in the pyrolysis curve would occur at a value of Tp near the melting point. If the larger AH values in Eqs. (47) (which are usually assumed to hold) are used instead of those in Eqs. (48), then the calculated values of (Tp -- T,) are roughly doubled; it is clear that they must increase since the term #zAH/W~ enters on the rigbt-hand side of Eq. (44). The estimate of pDgp used in this calculation should not be wrong by more than a factor of 5 and therefore cannot substantially modify our conclusion. Varying l does not alter the conclusion; if 1 is increased the estimate of (Tp -- T~) increases [-see Eq. (37e)-], while when / i s decreased free molecule heat conduction sets in and the estimate of (T~, -- T~) changes very little. The result that ( Tp -- T.~) increases substantially as Tp increases for NH4N04 and NHaCIO4 appears to be inescapable. A theoretical analysis by Cantrell, 31 in which a thin-film lubrication theory was used to account for the non-onedimensional flow effects in pyrolysis experiments, also supports the contention that (Tp -- T,) is not small. The main effect of the difference ( Tp -- T~) is to modify the computed activation energy for the interpretation corresponding to Eqs. (47), and to modify the AH values for the interpretation corresponding to Eqs. (48). For NH4C1, NH4NO3, and NH4CI04, respectively, the corrected activation energies would be roughly 14 kcal/mole, 30 kcal/mole, and 30 kcal/mole and the corrected heats of sublimation would be very roughly 14 kcal/mole, 25 kcal/mole, and 25 kcal/mole.
Alternative Interpretation of Pyrolysis Experiments. Other investigators have assumed that in
353
the pyrolysis of ammonia salts the over-all surface process is one of dissociation, 2'5'6 viz., NH4CI(s) ~ NH~(g) ~- HCI(g)
All = 40,3 kcal/mole
(47a)
NH4NOa(1) --~ NHa(g) -t- HNO3(g) AH = 39.8 kcal/mole
(47b)
and NH,CIO4(s) --~ NH3(g) + HCIOt(g) AH = 56 kcal/mole
(47c)
This hypothesis is supported by some evidence for dissociation in vacuum sublimation experiments. 1,5,~4,26-2s,~ The AH values listed here are obtained mainly from equilibrium vapor-pressure measurementsy -3~ except in the case of NH4C10, for which AH must be calculated from the heats of formation since no vapor-pressure measurements exist. In the pressure and temperature range of the pyrolysis experiments, it may be true that either (1) the equilibrium composition is one with a negligible amount of dissociation, or (2) dissociation occurs in a gaseous process, the rate of which is small compared with the rate of escape of the gas from the film between the plate and the sample. In either of these cases, the over-all surface process would be sublimation to the associated vapor. If the surface processes are simple chainlike processes such as that encountered in KC1 sublimation, then our previous analysis suggests that the temperature dependence of the pyrolysis rate would determine the heat of sublimation regardless of the importance of the fluid-transfer processes. The restriction to simple chainlike surface processes can be removed if surface equilibrium prevails in the pyrolysis experiments. When the correction for Tp ;~ T, is neglected, Eqs. (43) would then imply that NH4el(s) --~ NH4Cl(g) AH --- 13.5 kcal/mole
(48a)
NH4NO3(I) --~ NH4NO:~(g) AH = 7.1 kcal/mole
(48b)
NH4CIOt(s) --+ NH4CIO4(g) AH = 20 kcal/mole
(48c)
and
respectively, in the temperature ranges indicated in Eqs. (43). Although these conclusions seem reasonable because of the similarity between our proposed modified pyrolysis experiment and the experiments of Schultz and co-workers, the present conclusions lack the rigor of those in the previous subsection. Finally, we shall briefly discuss each material
354
COMBUSTION INVOLVING SOLIDS
separately, pointing out arguments for and against the hypotheses leading to the processes given in Eqs. (48). NH4CI: Since the pyrolysis data 2 yields the same activation energy as the vacuum sublimation data ~ and the low-pressure equilibrium appears at present almost certainly to involve entirely dissociated NH4C1 in the gas, 2s our interpretation will be valid only if hypothesis (2) of one of the previous paragraphs holds true. The large effect of H20 in catalyzing the decomposition,2s and the absence of It20 from the dried NH4C1 samples used in the pyrolysis experiments, I make it probable that the gas-phase dissociation is slow. NH4N03: In pyrolysis NH4N03 melts before gasification, and a change in the mechanism of gasification at the melting point causes the break in the experimental pyrolysis curve. 5 Since there exists some evidence that in equilibrium the gas is completely dissociated, ~~hypothesis (2) seems more plausible than (1) for the temperature range of the pyrolysis experiments. NH4C104: In spite of the large amount of recent work on NH4C104, [for example, The
Eighth International Symposium on Combustion, Williams and Williams Co., Baltimore, 1962, and also other papers of the present symposium] there appears to be no conclusive experimental information to eliminate hypothesis (1) or (2). A number of investigators (e.g., reference 14) have cited experimental evidence which is consistent with the assumption of dissociated vapor at equilibrium, but all of this evidence is equally consistent with the hypothesis of associated vapor. Since Galwey and Jacobs32concluded from theoretical considerations that the equilibrium vapor should exhibit a tendency toward association, hypothesis (1) may be valid. If Eq. (48c) is correct, then the pyrolysis experiments provided the first determination of any thermodynamic property of undissociated NH4C104(g).
D H
enthalpy per unit mass ]-cal/gm] hr
hg h,
p
• (v. 2 - v. 2) [cal/gm]; see Eq. (5) Thickness of gas film [cin] Total number of gaseous chemical species evolved from the solid Chemical symbol for solid Chemical symbol for gaseous species K Mass flux per unit area [gm/cm 2 sec] Total number of gaseous chemical species Permeability of porous hot plate to gaseous species K [gm/cm sec atm] Total pressure [atm]
p*
t
l M
M~(c) Mg(g) n~ N
PK
Y:. C~.KYK, mass-average specific heat at
M
p~
of
gas
E PK,e [atm]
K~I
pK
Partial pressure of species K at surface equilibrium [atm] Partial pressure of species K [ a t m ]
Pi.oo
Z PK,ooEatin]
p,
T t v WK
A constant in the momentum equation [atm]: see Appendix Universal gas constant, 1.986 [cal/ mole ~ Universal gas constant, 82.06 [atm cm~/mole OK] Universal gas constant, 8.317 X 107 [gm (cm/sec) 2/mole ~K] Temperature [ ~ Thickness of the porous plate [cm] Mass-average velocity [cm/sec] Molecular weight of species [gin/mole]
I~
[K=~t YK/WI~I; mass-average molecular
WE
weight Mass of molecules of species K leaving a unit area of surface per second [gm/cm 2
pK,~
M
R1
k~l
constant pressure [cal/gin ~
(dx/oD);
characteristic pressure
hr
5p
(eK/PK) K~I
R3
%,~
Z. Y,:,,hK(T,) -- h.(T,) + 89 K~I
R2
[(R3T)/(21rWIr ~, average velocity of molecules of species K in the --x direction [cm/sec] Specific heat at constant pressure of gaseous species K [cal/gm ~
[eal/gm] Total enthalpy per unit mass of gaseous species K [cal/gm] Total enthalpy per unit mass of solid [cal/gm] N
J
ACKNOWLEDGMENT
Cg
hK YK, total enthalpy per unit mass
h
This research was supported by the Air Force Office of Scientific Research, Office of Aerospace Research, under Contract AF 49(638)-412. Nomenclature
Binary diffusion coefficient Fcm2/sec]
h -4- (v2/2)(R1/R3), total stagnation
mixture
see 3
x
Spatial coordinate [cm]
ANALYSIS OF LINEAR PYROLYSIS EXPERIMENTS
YK
Mass fraction of gaseous species K
S
M
Y
~
K=I
YK, total mass fraction of gases
oc
355
At solid-gas interface In solid At downstream face of porous hot plate
evolved from solid
aK
Surface accommodation coefficient for species K (dimensionless) K=I
BE
ClK(pCK),
[dx/(pD)-],
EK
characteristic diffusion time to a characteristic recondensation time for species K Mass flux fraction of species K
AQ
Y~ eghK( Ts) -- h~( T,)
ratio
of
a
M K=I
(4/3)# -{- K, effective viscosity coefficient r g m / c m sec-] Coefficient of bulk viscosity [-gm/cm
K
see-]
),
rE
!
Thermal conductivity of the gas Fcal/cm see OK-] Ordinary (shear) viscosity coefficient [-gm/cm sec-] Stoichiometric coefficients for surface process, Eq. (28)
rh
f
[-dx/(pD) ~, dimensionless spatial
coordinate
~,
7h
Edx/ (pD) -], dimensionless mass
flow rate p (~ r o0 OJK
Density [-gm/cm ~] -~ p*/pp; dimensionless measure of pressure at upstream face of porous hot plate Dimensionless surface temperature; see Eq. (36) Dimensionless forward surface gasification rate; see Eq. (33) WK/[OtK(pCK)s], dimensionless forward surface gasification rate of species K
Subscripts e i K p r
Surface equilibrium A t upstream boundary of solid (x = --~) Integer index used only to denote chemical species At upstream face of porous hot plate Reference condition at which equilibrium vapor pressure is known
REFERENCES
1. SCHULTZ,R. D. and DEKKER, A. 0.: AerojetGeneral Corporation, The Absolute Thermal Decomposition Rates of Solids: Part I, The HotPlate Pyrolysis of Ammonium Chloride and the Hot-Wire Pyrolysis of Polymethylmethacrylate (Plexiglas IA). Technical Note No. 2, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)-1026, Azusa, November 1954. 2. BILLS, K., et al.: The Linear Vaporization Rate of Solid Ammonium Chloride. Technical Note No. 10, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)-1026, Azusa, July 1955. 3. SCHULTZ,R. D. and DEKKER, A. O. : Fifth Symposium (International) on Combustion, pp. 260267. Reinhold, New York, 1955. 4. BARSH, M. K., et al.: Rev. Sci. Instr. 29, 392 (1958). 5. SCHULTZ,R. D., GREEN, L. JR. and PENNER, S. S.: Studies of the Decomposition Mechanism, Erosive Burning, Sonance and Resonance for Solid Composite Propellants, Combustion and Propulsion, Third AGARD Colloquium, pp. 367-420. Pergamon Press, New York, 1958. 6. ANDERSON,W. H. and CHAIKEN,R. F.: AerojetGeneral Corporation, On the Detonahility of Solid Composite Propellants, Part I. Technical Memorandum No. 809, Azusa, January 1959. 7. FRIEDMAN, R., NUGENT, R. G., RUMBLE, K. E. and SCURLOCK, A. C.: Sixth International Symposium on Combustion, pp. 612-618. Reinhold, New York, 1957. 8. FRIEDMAN,R., LEVY,J. B. and RUMBLE, K. E. : Atlantic Research Corporation, The Mechanism of Deflagration of Ammonium Perchlorate. AFOSR TN-59-173, AD No. 211-313, Alexandria, Va., February 1959. 9. LEVY, J. B. and FRIEDMAN, R.: Eighth International Symposium on Combustion, pp. 663672. Williams and Wilkins Co., Baltimore, 1962. 10. JOHNSON, W. E. and NACHBAR, W. : Eighth Inter-
11.
12.
13. 14.
national Symposium on Combustion, pp. 678-689. Williams and Wilkins Co., Baltimore, 1962. ADAMS, G. K., NEWMAN, B. H. and ROBINS, A. B.: Eighth International Symposium on Combustion, pp. 693-705. Williams and Wilkins Co., Baltimore, 1962. ARDEN, E. A., POWLINO, J. and SMITH, W. A. W.: Combustion and Flame 6, 21 (1962). ROSEN, G.: J. Chem. Phys. 32, 89 (1960). BIRCUMSHAW, L. L. and PHILLIPS, T. R.: J. Chem. Soc. 1957, 4741.
356
COMBUSTION INVOLVING SOLIDS
15. PENNER, S. S.: Chemistry Problems in Jet Propulsion, pp. 234-241. Pergamon, New York, 1957. 16. KNUDSEN, M.: Ann. Physik 28, 75 (1909). 17. KENNARD, E. H.: Kinetic Theory of Gases, pp. 60-71. McGraw-Hill, New York, 1938. 18. PLESSET, M. S.: J. Chem. Phys. 20, 790 (1952). 19. BROOKFIELD,K. J., et al.: Proc. Roy. Soc. (Lon-
don) A190, 59 (1947). 20. KNUDSEN, M.: Ann. Physik, JT, 697 (1915).
21. Wvlmln, G.: Proc. Roy. Soc. (London) A197, 383 (1949). 22. KENNARD, E. H.: Kinetic Theory of Gases, pp. 320-324. McGraw-Hill, New York, 1938. 23. SCHULTZ,R. D. and DEKKER, A. O.: AerojetGeneral Corporation, The Absolute Thermal De-
composition Rates of Solids: Part III, The Vacuum Sublimation Rate of Ionic Crystals. Technical Note No. 7, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)1026, Azusa, July 1955. 24. BRADLEY, R. S. and VOLANN,P.: Proc. Roy. Soc. (London) A217, 508 (1953). 25. National Bureau of Standards, Circular 500, 1952. 26. SCHULTZ,R. D. and DEKKER, A. O.: AerojetGeneral Corporation, The Absolute Thermal
Decomposition Rates of Solids: Part IV, Transition-State Theory of the Linear Rates of Exchange and Decomposition of Crystalline Ammonium Chloride. Technical Note No. 8, U. S. Air Force Office of Scientific Research Contract No. AF 18(600)-1026, Azusa, July 1955. 27. SPINGLER,H.: Z. Physik. Chem. B52, 90 (1942). 28. RODEBUStI, W. n . and MICHALEK, J. C.: J. Am.
Chem. Soc. 51, 748 (1929). 29. LUFT, N. W.: Ind. Chemist 31, 502 (1955). 30. FEICK, G.: J. Am. Chem. Soc. 76, 5858 (1954). 31. CANTRELL,R. H. Ja. : Gas Film Effects in Linear Pyrolysis of Solids. Ph.D. thesis, Harvard University, Cambridge, May 1961. 32. GALWEY, A. K. and JACOBS, P. W. M.: J. Chem. Soc. 1959, 837. Appendix
The M o m e n t u m E q u a t i o n In the gaseous region, the one-dimensional momentum equation is
dv._~R3dp rhdx ~'dx
d (de) dx ndxx ----- 0
the first integral of which is
(49)
p -~ rhv -- 77 dxx =
p~ = constant (50)
Equations (1), (18), and (50) determine the distributions of p, p, and v in the gas in terms of the composition and temperature distributions studied in the main body of the paper. The evaluation of the constant p~ in Eq. (50) depends upon the experimental arrangement. If the gas layer alone supports the solid (in whicb case 1 is not known a priori), then it can be shown that p~ is the axial compressive stress in the solid, which equals the atmospheric pressure plus any additional pressure applied to the cold end of the solid sample in order to increase "thermal contact" with the porous hot plate. On the other hand, if the solid is supported in such a way that the spacing 1 is fixed (e.g., by means of a specially designed sample holder, or by protuberances on the surface of the sample, as appears to be the case in the previous pyrolysis experiments), then the gas-phase equations given in the main body of the text determine p~. An interesting consequence of these observations arises when
rhv -- y(dv/dx) << (R~/R2)p
(51)
Equations (50) and (51) imply, approximately, p =
p~ = p . =
p.
(52)
In the discussion at the end of the section entitled Simple, Chainlike Surface Processes, it is assumed that Eqs. (51) and (52) are valid first approximations for an integral of Eq. (50). The dimensional form of Eq. (38b) is written for this case as rh =
(P,/t)
(p. -
p,~)
(53)
If p, is not measured, as in the second experimental arrangement discussed above, then it is determined through Eq. (53) by measurements of rh and p.~ (provided that Eq. (52) is valid). If p~ is easily measured, as in the first experimental arrangement, then we might ask whether this measurement could replace the more difficult measurement of l for this arrangement. Unfortunately, it cannot. In this case, 7h and p:, both easily measured quantities, satisfy Eq. (53) as a redundant equality independent of 1. I t would still be necessary to measure l for this first experimental arrangement in order to carry out the calculations indicated in the section mentioned above.
ANALYSIS
OF LINEAR
PYROLYSIS
357
EXPERIMENTS
Discussion DR. D. J. SIBBETT (Aerojet-General Corporation): The analysis of linear pyrolysis experiments by Dr. Nachbar and Prof. Williams indicates very nicely the critical factors and limitations of the technique. I t is obviously of p a r a m o u n t importance in the interpretation of data collected by this method t h a t the measured experimental temperature be identifiable with surface temperature. Recently in the course of a study of the rates of sublimation of the four ammonium halides, evidence was obtained which confirms this identity. In these experiments the rates of vacuum sublimation of the ammonium halides were determined gravimetrically by use of the quartz balance at low temperatures and by the linear pyrolysis technique at high temperatures. Surface temperatures during the gravimetric determinations were obtained from dual sets of experiments. In one set, rates of sublimation and the temperature of a thermowell in the quartz balance wall were obtained simultaneously. In the second set, thermowell temperatures and sample surface temperatures were correlated to give temperature corrections which were applied to the rate measurements. Linear pyrolysis temperatures were measured directly. The figure, which is a plot of the rates of sublimation of ammonium bromide between 153 and 574~ is a typical example of these data. Rates, which are indicated on a logarithmic scale on the ordinate, have been converted to linear regression units (cm/ sec). D a t a from 153 to 227~ were obtained gravimetrically; those at higher temperatures were obtained b y use of the linear pyrolysis or "hotplate" method. For all four halides, NF4F1, NH4CI, NH4Br, and NH4I, the data took the same form. The figure demonstrates t h a t : 1, The vacuum sublimation data are consistent with the linear pyrolysis data and give an Arrhenius plot with a single apparent activation energy value. 2, At very high rates, a departure from linearity was observed which corresponds to the range of conditions in which meaningful temperature measurements fail. However, b y increasing the loading force between the stand and the hot plate, which decreases the thickness of the gaseous layer, the linear plot can be extended. This effect is demonstrated b y the two sets of data at the upper limits of the rate measurements. From these data, it has been concluded: 1, The
10-2
9'~ 10-3 -2
v
~
lO "I
HOT PLATE DETERMINATIONS
--
g/cm2LoadingPressur~
o 4865 A 2148 g/cm2LoadingPressure [ ~
.J
10.5
I: []QUART"BALA,CE
\
DETERMIHATIONS
1.0
1.2
1.4
1.6
1.8
' ~
2.0
2.2
2.4
SURFACE TEMPERATURE, 1/Ts, 10.3 ( ~
FIG. 1. Rates of sublimation of ammonium bromide.
temperatures measured b y the nonporous hot-plate apparatus are valid measures of the surface temperature in these experiments. 2, Linear pyrolysis measurements are determinations of the rates of sublimation of the a m m o n i u m halides. T h a t is, a single reaction mechanism appears to be operating over the entire temperature range. Application of the linear pyrolysis technique requires experimental or theoretical analysis of the processes involved at the hot interface in order to establish the meaning of the measured temperature. Within these limitations it has proven a very useful tool. PROF. F. A. WILLIAMS (Harvard University): The only ammonium halide t h a t we have considered is NH4C1, and, as indicated in the paper, we agree with Dr. Sibbett t h a t the difference between the plate temperature and the surface temperature is negligible in this case. We found significant differences only for NH4NO3 and NH4CIO4.