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RATE OF REACTIONf BETWEEN MOLECULAR HYDROGEN

AND MOLECULAR OXYGEN ,

' by Richard S. Brokaiv :

Leivis Research Center6 J o

-Cleveland, Ohio

NATIiNAL MEONAOTICS SPAtf WMHINGTOM, D. C • FEBRUARY W73

https://ntrs.nasa.gov/search.jsp?R=19730008201 2020-04-01T07:10:24+00:00Z

1. Report No.

NASA TM X-27074. Title and Subtitle

RATE OF REACTION BETWEE

AND MOLECULAR OXYGEN

2. Government Accession No.

N MOLECULAR HYDROGEN

7. Author(s)

Richard S. Brokaw

9. Performing Organization Name and Address

Lewis Research CenterNational Aeronautics and SpaceCleveland, Ohio 44135

12. Sponsoring Agency Name and Address

National Aeronautics and SpaceWashington, D.C. 20546

Administration

Administration

3. Recipient's Catalog No.

5. Report Date

February 19736. Performing Organization Code

8. Performing Organization Report No.

E-719510. Work Unit No.

502-04

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

The shock tube data of Jachimowski and Houghton have been rigorously analyzed to obtain rate

constants for the candidate initiation reactions H2 + O2 > H + HO,, kQ. = 1.9x10

exp(-24 100/T); H2 + O2— -H2O + O, kQ2 = 4. IxlO13 exp(-25 400/T); and H2 + O2-^OH + OH,1 3 3 1 1kgg = 2.3x10 exp(-25 200/T). (Rate constants in cm mole" sec" , temperature in degrees

kelvin. ) Reaction (01) is probably not the initiation process because the activation energy ob-tained is less than the endothermicity and because the derived rates greatly exceed values in-ferred in the literature from the reverse of reaction (01). Reactions (02) and (03) remain aspossibilities, with reaction (02) slightly favored on the basis of steric and statistical considera-tions. The solution of the differential equations is presented in detail to show how the kineticsof other ignition systems may be solved.

17. Key Words (Suggested by Author(s))

Reaction kineticsIgnitionCombustionInitiation

19. Security Classif. (of this report)

Unclassified

18. Distribution Statement

Unclassified - unlimited

20. Security Classif. (of this page)

Unclassified21. No. of Pages 22. Price"

10 $3.00

'Mff'N*a^oria']'f6ch?tcIi"ihformaJlo'n"Service, Springfield, Virginia 22151

RATE OF REACTION BETWEEN MOLECULAR HYDROGEN

AND MOLECULAR OXYGEN

by Richard S. Brokaw

Lewis Research Center

SUMMARY

The shock tube data of JachimowsM and Houghton have been rigorously analyzed to01obtain rate constants for the candidate initiation reactions H2 + Og—^H + HO2,

kQ1 = 1.9X1013 exp(-24 100/T); H2 + Og-^-HgO + O, kQ2 - 4. IxlO13 exp(-25 400/T);

and H2 + O2 -22- OH + OH, kQ3 = 2.3xl013 exp (-25 200/T). (Rate constants in3 -1 -1cm mole sec , temperature in degrees kelvin.) Reaction (01) is probably not the

initiation process because the activation energy obtained is less than the endothermicityand because the derived rates greatly exceed values inferred in the literature from thereverse of reaction (01). Reactions (02) and (03) remain as possibilities, with reac-tion (02) slightly favored on the basis of steric and statistical considerations. The solu-tion of the differential equations is presented in detail to show how the kinetics of otherignition systems may be solved.

INTRODUCTION

When a mixture containing hydrogen and oxygen is subjected to a temperature andpressure pulse in a shock tube, small concentrations of atoms and free radicals arefirst formed from molecular hydrogen and oxygen. Reactions which have been suggestedinclude (ref. 1)

O2-2i-H + HO2 AH298/R - 28 700 K (01)

knot- O9

u^ H9O + O AH9QO/R = 900 K (02)A £* £t y o

H2 + p2-li£OH + OH AH298/R = 9400 K (03)

The dissociation processes

H2 + M - 2H + M

O2 + M - 2O + M

are sufficiently endothermic that they are unimportant except, possibly, at very hightemperatures.

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

k1OH+ H0— i-H9O + H (I)

O (II)

k-o + H2— i OH + H (in)

At low temperatures and high pressures the chainbreaking reaction

k4H + O2 + M -4- HO2 + 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 differentialequations is presented in detail to provide a guide as to how the kinetics of other similarignition systems may be solved.

THEORETICAL CONSIDERATIONS

The differential equations governing the growth of radical concentrations during theinduction period are as follows (ref. 3):

2

t/3[O] + 2i3 (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 -

k2[O2], v^ = k3[H2], v^ = kj[O2][M], and

centrations of molecular hydrogen and oxygen, [M] is the total gas concentration, andthe 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 H0 and O9 are scarcely depleted. Hence, the z/s and i's

£i £t

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 constantterms to zero:

' "o "*" VA ) "a ~ ^laf-vTr - V*)&r\ + ii — U (^/& 4 n. 1 Uil o U 1

- ^2aH + i/3a0 + i2 = 0 (6)

The new differential equations are

dC" C

jj. <a •* ra i \JLI 3 O

dC, _(8)

dC0—- = i/2CH - i/-C0 (9)

dt 2 H 3 O

A particular solution to equations (7) to (9) is (refs. 3 and 4)

Cj = Aj exp(Xt), i = H, OH, O (10)

Substitution of equations (10) into equations (7) to (9) yields the relations

-(i>2 + ^4 + X)AH + "IAQH + ^3Ao = ° (^

= 0 (12)

= 0 (13)

A nontrivial solution requires that the determinant of the coefficients of the A. be zero,which leads to the cubic equation

3 2X + (v* + i>2 + Vn + ^)X + [i>* Vn + V \ V A + ^3^4]^ ~ vi ^3(2^2 ~ ^4) = 0 (14)

In the chain-branching region (2^ > v^> Descartes' rule of signs indicates that, if theroots of equation (14) are real, there are one positive and two negative roots. It can beshown that the roots are always real provided k1 > 2k,, which is true for temperatures

i. «5below about 2280 K, and hence true for the data of Jachimowski and Houghton (ref. 2),which are in the range 1100 to 1900 K.

The general solution to equations (7) to (9) is

Ci = Aj i exp(X1t) + A2 ,. exp(X2t) + AS i exp(Xgt) i - H, OH, O (15)

where Xj, Xg, X, are the roots of equation (14). We are specifically interested in thegrowth of hydroxyl concentration, which was the quantity monitored in reference 2. Ifwe designate Xj as the positive root and X2, X^ as negative roots, then

[OH] = A l jQH exp(X1t) + A2jQH exp(X2t) + A3>OH exp(X3t) - aQH

= A l jQHexp(X1t) (16)

The approximation shown is a good one, except very early in the reaction, because a^jris small and the negative exponentials soon die out.

We will now evaluate the coefficient A^ OTT. Initially, at t - 0, the hydroxyl radicalconcentration is zero, and equation (16) gives

~ aOH

One can write similar expressions for hydrogen and oxygen atoms and then use any twoof equations (11) to (13) to eliminate the A^ and A~ in favor of A.-,,. The same result

n \J (Jtican be obtained more easily by successive differentiation of equations (2) and (16):

and

~II~ "X1A1,OH + X2A2,OH + X3A3,OH = 2i3\ dt /t=0

d2[OH] 2 2 2T~ = Xl Al, OH + X2A2, OH + X3 A3, OH

t=0

/d[H]\ /d[OH]\ /d[0]\= V2(~^ I ~ "II - ) + "31 - /2\ dt io \ dt /t=o V dt /t=

"3*2

Equations (17) to (19) are solved to obtain

Xr.Xoa.-4Ti - 2i0(X0 +A ti o vJJtl o tt1}0n (x2-

where the constant

is obtained by solving equations (4) to (6). Expressions for Ag QH and A, QH can beobtained by permitting the indices on Xj, Xg, and X« in equation (20).

Thus, equations (16), (20), and (21) describe the growth of hydroxyl concentrationuntil 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.

ANALYSIS OF EXPERIMENTAL INDUCTION TIMES

The induction times reported by Jachimowski and Houghton (ref. 2) correspond tothe 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:

k01 = —01 - X

[H2][02]

2"2A2X3"2

-1(23)

[H2][02]

2A

+ ^

X - X. -

-1(24)

. - v.o i

-1(25)

In these calculations the rate constants for reactions I, n, and III were taken fromreference 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/-?-\VRT/

(26)

Results are summarized in table I, where the Arrhenius equation parameters and theirstandard deviations are presented.

DISCUSSION OF RESULTS

In this section the three candidate initiation reactions will be discussed in turn, withan indication as to which are the most likely initiation processes.

The reaction

H2 + O2 - H + HO2 (01)

is a simple abstraction or two center reaction involving the breaking and formation of

one bond. Hence it is the most likely candidate based on steric considerations. How-ever, as Jachimowski and Houghton (ref. 2) have already observed, the activation en-ergy, 24 100 K, is substantially less than the endothermicity of 28 700 K. This amountsto more than three standard deviations in the activation energy and would seem to elimi-nate reaction (01), barring a temperature dependent systematic error in the experimen-tal data.

Further, in reference 7 the rate of reaction (01) is reported to be

kQ1 = 5.5X1013 exp(-29 100/T) (27)

in the range 290 to 800 K, with an uncertainty of a factor of 2. 5. Equation (27) predictsa rate at 1100 K which is less than one-thirtieth of the rate at that temperature given bythe constants in table I. Thus it is very unlikely that reaction (01) is the initiation pro-cess in these mixtures.

The reaction

H2 + O2 - H2O + O (02)

might be termed a three center reaction inasmuch as new bonds are formed among threeatoms. This would seem sterically more probable than the four center reaction

H2 + O2 - OH + OH (03)

In addition, reaction (02) is less endothermic than reaction (03), although in either casethe endothermicities are substantially smaller than the activation energies. Finally,the standard deviations in In k, In A, and E/R are slightly smaller for reaction (02)than for either reaction (01) or reaction (03). Initiation in these mixtures may be due toeither reaction (02) or reaction (03) (or both), with reaction (02) the slightly more prob-able candidate.

Lewis Research Center,National Aeronautics and Space Administration,

Cleveland, Ohio, November 10, 1972,502-04.

REFERENCES

1. Ripley, Dennis L.; and Gardiner, W. C., Jr.: Shock-Tube Study of the Hydrogen-Oxygen Reaction, n. Role of Exchange Initiation. J. Chem. Phys., vol. 44, no. 6,Mar. 15, 1966, pp. 2285-2296.

2. Jachimowski, Casimir J.; and Houghton, William M.: Shock-Tube Study of theInitiation Process in the Hydrogen-Oxygen Reaction. Combustion and Flame,vol. 17, no. 1, Aug. 1971, pp. 25-30.

3. Brokaw, Richard S.: Analytic Solutions to the Ignition Kinetics of the Hydrogen-Oxygen Reaction. Tenth Symposium (International) on Combustion. CombustionInstitute, 1965, pp. 269-277.

4. Brokaw, Richard S.: Ignition Kinetics of the Carbon Monoxide-Oxygen Reaction.Eleventh Symposium (International) on Combustion. Combustion Institute, 1967,pp. 1063-1072.

5. Brabbs, T. A.; Belles, F. E.; and Brokaw, R. S.: Shock Tube Measurements ofSpecific Reaction Rates in the Branched-Chain H2-CO-O2 System. ThirteenthSymposium (International) on Combustion. Combustion Institute, 1971, pp. 129-135.

6. Baulch, D. L.; Drysdale, D. D.; and Lloyd, A. C.: Critical Evaluation of RateData for Homogeneous, Gas-Phase Reactions of Interest in High-Temperature Sys-tems. Rep. 3, Leeds Univ., Apr. 1969, pp. 18-27.

7. Baulch, D. L.; Drysdale, D. D.; Home, D. G.; and Lloyd, A. C.: EvaluatedKinetic Data for High Temperature Reactions. Vol. 1. Homogeneous Gas PhaseReactions of the H2-O2 System. CRC Press, 1972.

TABLE I. - RATE CONSTANTS OF POSSIBLE HYDROGEN-OXYGEN

INITIATION REACTIONS

3 1 1A, cm mole" sec"Activation energy, E/R, KStandard deviations of -

In kIn AE/R, K

Reaction

k

H2 + O2-^i H -i- HO2

1.9xl013

24 100

1.220.951310

k

H9 + O, -^>H9O + Oft £> £>

4.1xl013

25 400

1.150.901250

kniIT f\ M OH 4- OH

2.3X1013

25 200

1.210.94

1310

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