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Rate of Reaction Between Molecular Hydrogen and Molecular Oxygen
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NATIiNAL MEONAOTICS SPAtf^ WMHINGTOM, D. C^ •^ FEBRUARY^ W
https://ntrs.nasa.gov/search.jsp?R=19730008201 2020-04-20T02:55:56+00:00Z
N MOLECULAR HYDROGEN
Administration
Administration
The dissociation processes
H 2 + M - 2H + M
O 2 + M - 2O + M
are sufficiently endothermic that they are unimportant except, possibly, at very high temperatures. After small concentrations of H, OH, or O are formed from reactions (01), (02), or (03), the atom and radical concentrations grow exponentially via the well-known branched- chain scheme k 1 OH+ H 0 — i-H 9 O + H (I)
O (II)
k- o + H 2 — i OH + H (in)
At low temperatures and high pressures the chainbreaking reaction
k 4 H + O 2 + M -4- HO 2 + M (IV)
must also be considered. Recently Jachimowski and Houghton (ref. 2) studied the hydrogen- oxygen initiation process behind incident shocks. They analyzed their data to obtain rate constants as- suming initiation by reaction (01) or (03). Their analysis used an approximate and intui- tive formulation proposed to explain hydrogen- oxygen ignition delays (ref. 3). In this report the data of Jachimowski and Houghton are reexamined, using a rigor- ous formulation of the initiation and chain-branching kinetics. Rate constants are ob- tained by assuming initiation by reaction (01), (02), or (03). The values of rate con- stants indicate the more likely initiation processes. The solution of the differential equations is presented in detail to provide a guide as to how the kinetics of other similar ignition systems may be solved.
The differential equations governing the growth of radical concentrations during the induction period are as follows (ref. 3): 2
t/ 3 [O] + 2i 3 (2)
i (3)
d[H] i -i _ _ /., , ., ^["Hl + v TOHl -i- v FOl -i- i Cl} dt
d[OH]_ dt
d[0] dt
Here [H], [OH], and [O]are the concentrations of hydrogen atoms, hydroxyl radicals, and oxygen atoms. Also, i^sk^H,,], ^ 2 - k 2 [O 2 ], v^ = k 3 [H 2 ], v^ = kj[O 2 ][M], and
centrations of molecular hydrogen and oxygen, [M] is the total gas concentration, and the k's are the specific reaction rate constants for reactions (01) to (03) and (I) to (IV). During the induction period the concentrations of H, OH, and O build up rapidly, while the concentrations of H£i 0 and O£t 9 are scarcely depleted. Hence, the z/s and i's in equations (1) to (3) may be taken as constants. The initiation rates i.., in, and !„ can be eliminated from the differential equations by introducing new variables CH ^ [H] + aH, CQH = [OH] + aQH, and CQ = [O] + aQ, where an, a^tr, and ao are constants. If these new variables are substituted into equa- tions (1) to (3), the initiation rates are eliminated by equating the sums of the constant terms to zero:
' "o ""& 4 VA )^"a n. ~ (^) 1 Uil^laf-vTr - oV)&r\ + ii U 1 — U (^/
The new differential equations are
dC jj. <a •* ra i \JLI " 3 CO
dC, _ (8)
dC 0 —- dt = i/ 22 CHH - i/-C3 O 0 (9)
One can write similar expressions for hydrogen and oxygen atoms and then use any two of equations (11) to (13) to eliminate the A^ and A~ in favor of A.-,,.n \J (Jti The same result can be obtained more easily by successive differentiation of equations (2) and (16):
and
~II~ "X 1 A1,OH + X 2 A2,OH + X 3 A3,OH = 2i 3 \ dt^ /t=
d^2 [OH] 2 2 2 T~ = XlAl, OH+ X 2 A2, OH+ X 3 A3, OH t=
= V^ /d[H]^ /d[OH]^ /d[0]
2 2(~^ I^ ~ "II -^ )^ +^ "31 -^ /
dt io
dt /t=o V dt /t=
"3*
Equations (17) to (19) are solved to obtain
A Xr.Xoa.-4Ti - 2iti o vJJtl 0 o(X tt^0 +
where the constant
is obtained by solving equations (4) to (6). Expressions for Ag QH and A, QH can be obtained by permitting the indices on Xj, Xg, and X« in equation (20). Thus, equations (16), (20), and (21) describe the growth of hydroxyl concentration until such time as the effects of depletion of molecular hydrogen and oxygen are impor- tant or until the temperature rises due to atom and radical recombination processes.
The induction times reported by Jachimowski and Houghton (ref. 2) correspond to the time at which the hydroxyl concentration has risen to 10 mole per cubic centi- meter. They also report experimental values of the growth constant. Thus, one can
obtain experimental values of A- QH from equation (16)
(22)
where X is the experimental growth constant and r is the induction time. Rate con- stants for initiation by reactions (01), (02), and (03) were obtained from equation (20), assuming that only one of the initiation reactions was occurring:
k 0101 = — - X [H 2 ][0 2 ]
(^2) "2A 2 X 3 "
[H 2 ][0 2 ]
2A
X - X. -
o^. - v. i
In these calculations the rate constants for reactions I, n, and III were taken from reference 5. The rate constants for reaction IV was taken from reference 6. And X.., Xg, and Xo were obtained by solution of equation (14). These rate constants were least-squares fitted to the Arrhenius equation
k - A exp/-?-\
Results are summarized in table I, where the Arrhenius equation parameters and their standard deviations are presented.
In this section the three candidate initiation reactions will be discussed in turn, with an indication as to which are the most likely initiation processes. The reaction
H 2 + O 2 - H + HO 2 (01)
is a simple abstraction or two center reaction involving the breaking and formation of
TABLE I. - RATE CONSTANTS OF POSSIBLE HYDROGEN-OXYGEN INITIATION REACTIONS
A, cm 3 mole" sec"1 1 Activation energy, E/R, K Standard deviations of - In k In A E/R, K
Reaction k H 2 + O 2 -^i H -i- HO 2
1.9xl0^13 24 100
1310
k Hft 9 + O, -^>H£> 9 £>O + O
4.1xl0^13 25 400
1250
kni IT f\ (^) M OH 4- OH
2.3X10^13 25 200
1310
NASA-Langley, 1973 33 E - 719 5
