On the surface and thernal effects on hydrogen oxidation

COM~[~USTION AND FLAME. 17, 223-235 (1971)

223

On the Surface and, Thermal Effects on Hydrogen. Oxidation K. K. Foo and C. H. Yang Department of Mechanics, State University of New York at Stony Brook. Stony Brook. New York

The kinetic scheme proposed by Baldwin for the H 2 and O z system ira acid coated vessels ~s modified to analyze data obtained in sak coated vessels. A thermokinetie model is ,developed to prescribe both the kinetic and "'self heating" characteristics of the system, With a single set of chosen values of rate ¢onstams and other parameters, all three explosion limits and slow reaction rates are calculated tbr vessels with different wall euat]ngs. Generally good agreement between the experimental data and theoretical predictions is obtained both qualitatively and quantitatively.

I. Introduction

O + H: ...-,OH + H

It is generally agreed that the basic kinetic mechanism of the oxidation of hydrogen generally involves an auto-catalytic branching chain reaction. The complete kinetic mechanism is, however, complicated by many possible additional reactions, One of the prominent un. certainties in the complete oxidation kinetics is the role played by the surface of the reaction vessel for different surface coatings [i-3]. The work of Eigerton and Warren 1-4], in which a boric acid c~ated vessel was used, showed characteristic new results in their measurement of the upper explosion limits [5, 6]. A "quadratic branching" kinetic scheme was formulated in addition to the basic branching reactions to explain the novel observations. Tile subsequent work of Baldwin 1"7] ct at. has modified and improved this theory to a point where all the kinetic data (upper limits, and slow reaction rates) from the boric acid coated vessels can be closely predicted. The Baldwin scheme [7] contains the following reactions*: OH + H z ~ H20 + H

H + O::-.-,OH + O

(1)

(2)

* Baldwin's numbering c,f the reactior~, is reu:ined for easy reference.

H + 02 + M ---,HO: + M H202 + M" ~ 2OH + M" H + '[{02 -..*2OH

(3) (4) (7) (8)

HO2 + li02 ~ H202 + Oz

(10)

H,. + 1:/O2-..',HzOz + H

(11)

H + H20 z H H20 + OH

(14)

O H + H202 H H20 + H02 H + H20:~"" HO2 + H:

(15) (14a)

In contrast, the detailed kinedes for systems in salt coated vessels are still regarded [l, 2, 4, 8] as uncertain. In examining the above,scheme, it is clear that many of the gas phase reactions that involve the HO 2 radical and hydrogen peroxide a r e postulated [4] )1:o explain the "'quadratic branching". Undoul:~,tedly, the3' also occur in salt coated vessels, where quadratic branching is not observed [4., 9"1. It should also be noted that in the Baldwin scheme active chain centers are not destroyed on the vessel surface, which is, of course, consistent with the experimental facts that the eyplosion limits are independent of vessel size. The chain centers are instead destroyed through reactions, [5"1 (14) and (14a), where threz~ centers (H20., Copyrigh~ ~ 1971 bYT~teCombustion Institute Published by American El~vler PublishingCompany, Inc.

2~

K. g. FO0 and

is regarded as two centers [5]) are reduced to one. Undoubtedly this type of gaseous phase chain center termination must also occur in salt coated vessels where heterogeneous chain center termination is always regarded as important. Warren [10] actually observed that after prolonged use, KCI coated .vessels did show "quadratic branching" characteristics. The question is whether the gaseous phase kinetics derived from the boric acid-coated vessel can be modified with the addition of heterogeneous chain termination reactions to explain as well the kinetic data from salt-coated vessels. To provide an answer to this question is one of the aims of the present work. A generally accepted mechanism for the third limits has not been available in the literature. Voevodsky [11] proposed that in KCI vessels the third limit is the chain type of explosion, while in clean, vessels (acid washed pyrex and silica vessels) it is the thermal type. The surfaces of the clean vessels are known [12] to be reflective to chain centers. Surface reactions, therefore, must be excluded. It can be shown that the third limit does not theoretically exist, if the Baklwin sche,~le is considered alone without further modification. Third limits for clean vessels were. of course, widely measured [11-16]. Any attempt to determine a general kinetic model must account for their existence. A thermokinetic theory was proposed previously [17-22] to deal with kinetic systems where thermal and chain mechanisms are interwoven. For the present system it is most appropriate that the unified thermal and chain theory is adopted. The mathematical formulation consists of the kinetic equatior, s which are based on the Baldwin scheme. Terms for chain center dcstruction on vessel sarfaces are added to accomodate the salt coated or other types of

c.

n. W , N G

vessels. For clean or boric acid coated vessels the efficiency factor for these chain termination terms is regarded as zero. In addition to the kinetic equations the energy conservation equation is introduced to couple with them. All the results are calculated with this system of differential equations, It must be emphasized, however, that once the rate constants for both the gaseous phase and surface reactions are selecte8 they are consistently applied throughout the calculations of all three explosion limits ~nd slow reaction rates in the entire temperat cre range where experimental data are available for comparison. Our calculated results show that all the limits, "quadratic branching" phenomenon, "'self heating", vessel surface effects, and slow reaction rates of the H2-O 2 system are closely prescribed and predicted by the present unified thermokinetic theory. If. Kinetic Scheme and the Thermokinetic Model All gaseous phase reactions in the previous section proposed by Baldwin et al. for the H2-O 2 system in boric acid coated vessels are adopted. Only the fo!lowing chain center termination reactions on the vel~sel surfaces are added to form the general kinetic scheme in the present work.

H w.tl~ destruction

(i6)

O ~'", destruction

(I 71:

OH ~.,it, destruction

(I 8 I

HO 2 w'a-kdestruction

(19!

H202 ~

destruction

(20)

From the above scheme, a thermokinetic model is formulated containing all kinetic equationq and the energy conservation equation of the system (with the depletion of the concentration of primary reactants neglected):

d[H]/dt =

kt[OH][H~] + l : s [ o ] [ n 2 ] + k, ,[HO2][H2] - k , [ H ] [ O , ] [ M ] - k2[H][O23 - k . [ H ] [ n o 2 ] - k , , [ H ] [ H 2 0 2 ] - kt,,,[U][.kl202 ] - k,[H].

(21)

d[O]/dt =

k , [ H ] [ O , ] - k , [ O ] [ H , ] - k,,[O],

(22)

SURFACETHERMALEFFECTSON HYDROGENOXIDATION

225

d[OH]/dt = 2k,[H2]]'O~ + k2[H][O2"l + kz[O'][H2 ] + 2kv[HzO2J[M' ] + 2ks[HJ[HO2"] +

+ kv~[H][H202] - kt[OH][H2"] - kts[Ol-I][H202-1 - kta[OH'l, d[HO2"]/dt = k,,[H][O2]rM ] + k,,~[H]['H202 ] + k,s[OH'lt'H202"l - 2k,o[HO2"l 2

- k~[H][.O~] - k,,[HO:][H=]

(23) -

-: k,.[HO~],

(24)

a [ H 2 0 = ] / a ~ = kto[H02] 2 + k t , [ H O 2 ] [ H z ] - k..,[H202][M' ] - k,~.[H][H202]

- k t , . [ H ] [ H 2 0 2 ] - k,s[OH][H~O2] - kts[H202],

(25)

C~(dT/dt) = R(T) - L(T)

= k,[H2][O:]h , + kt[OH][H2]ht + k2[H][Oz]h2 + ka[O][H:]ha + + k,,[H][O2][M]h,, + k,[H202][M']h7 + ks[H][HO2]hs + k~o[HO2]2h,o + +

k, t[H2][HO:]h,,

+

kt,t[H][H202]h,,,

+

kt.,,,[H][H:O2.]lh~

+ kts[OH][H202]hts- ~ (T- To), where

To = T= h~ = F= 2= k,~ = k~ =

+ ~ (26)

bath temperature °K, average temperature of reactants, °K, heats of reaction, kcal tool- t heat exchange coefficiem, conductivity of the mixture, cal cm- t ~K- t sec- t surface termination rate constant, sec- t, initiation reaction rate constant tool., cc - t , see- t

The above system of equations, (21)-(26), will now be used for calculating all kinetic behavior of the H2-O 2 system, including the three explosion limits, "'quadratic branching" characteristics, and slow reaction rates in vessels of different sizes and with different coatings. A stable state is related to a stable nodal singularity [213 that exists in the positive and finite domain of the si:~-dimensional phase space formed by the above system of equations. Singularities in the phase space are obtained by letting the time derivatives of Eqs. (21)-(26) be zero. The dependent variables at the singularities are called steady state values ([H2"]~, [O]~, etc,). Slow reaction rates at nodal singularities are characterized by the steady state values. When some parameters in the differential system, such as the bath temperature, ir~. and

partial pressures of the reactants (H2, 0 2, inerts, etc.) ate varied, they may pass through some critical values which induce bifurcations [17, 18] in the phase space. A nodal sing~lari~ may disappear from the phase space if one of the parameters passes its critical bifurcation value. The system becomes explosive as stable states of stable" singularities no longer exist. The bifurcation values of the parameters, are identified* as the measured explosion limits. Values of basic physical and kinetic para:meters such as wall efficiencies for chain centers with respect to specific vessel surfaces, diffusion coefficients, specific heats, pre-expo* Bifurcationsof nonlinearkineticsystera~are discussed in some detail in Pet*. 17 and 18. Gray discussedthe case of more generalphase spacesin Ref. 21.

226

K. K. rO0 and C. n. YANG Table 1, Values for h~-h,~

Reactions

Activationenergies kcal tool- a

Arrheniusfactors ec rao~- ~ scc- t

Heat of reaction keal mol- t ÷

5.2 16.5 10.2 -L3 47 0 0 25

2,3 × l0 la 2,04x t0 I" 4,0 × l0 ta 5,0 x 10Is 3,14× 10t~ 7,0 x I0sa 1.8 x I012 1.66 x l0 ts 4.16 x 10n4 ¢.16 × 10I" 3.6 x 1012 1014

14.8 -16.0 -1.8 47.2 -50.3 39.2 +42.3 - 14.7 68.5 14.7 29.3 - 1[,.6

{1) (2) (3) (4) (7) (8) (10)

OH+Ha H+O 2 O+H2 H+O2+M H:Oz+M' H+HO2 H02+I'lO2 (II) H Z+ HO2 (14) H + HzO2

(14a) H + H202 (15) OH + H~O~ (I)" H z + 0 2

9

II.8 0 70

~"The valuesare blken from G. Dixon-Lewis. Proe. Roy. Soe.. Ser. A, A298. 495 (1967) and V. N, Kandratiev, Chem/cal Kinetics of Gas Reactions. Pergamon Press (1964). * The rate constant is taken from Duff [23].

nential factors and activation energies of specific reactions, etc., once selected are used throughout the calculations. Exceptions, if any, are n~ted.

cylindrical vessels), V is the thermal velocity of the chain center, and d is the diameter of the vessel. For efficient surfaces, the rate is approximated [24"] by

III.

where B z = 4• z for spherical vessel, and 23 for cylindrical vessel. D is the diffusion constant, approximated by the hard sphere theory [25]

K, = BzD/d 2, Selection

of Rate Constants

ang

Methods of Computation The set of Arrhenius factors and activation energies of homogeneous reactions are taken from Baldwin's work. Warren [10] examined the effects of surface coating on the kinetics of the H , and Oa system and concluded that surfaces which have high efficiency in terminating one kind of chain centers usually have high efficiencies in terminating others. The rate of chain center termination on the surface is usually affected by its diffusion velocity te the surface and its probability of being absorbed by the wall during a collision. The slower of the two processes controls the overall rate. For inefficient surfaces, the rate can be approximated [24] by

K, = cBl VId, where e is the wall efficiency, B i is a constant depending on geometrical shape of the vessel (B 1 = 1.5 for spherical vessels and 1.0 for

DI. 2 = 2.62 X 10 - a [Ta(MI + M2)/2MtMz] llz [I/2(~ri + oz)]zP.

The molecular diameters a are : ax~ = 2.74 A, ao~ = 3.75 A, ~N2 = 3.61 A, ~'H = 0.66 A, ao = 1,2A, a o a = 3 A , ann 2 = 4 , 0 A . M l and M z are the molecular weights. For a mixture, the diffusion constant is given by

Dt

Dt.n~

Dl.o2

Dl.~z "

where .~h~, fo:, fN~ are the mote fractions of hydrogen, oxygen, and nitrogen, respectively. Due to, the difficulty encountered in reproducing identical coatings in different experiments, there is a rather wide range of wall efficiencies given in the literature for a specific coating with

SURFACETHERMALEFFECTSON HYDROGENOXIDATION respect to a specific chain center. In the present work, these efficieneies are chosen to be the best values that fit the experimental data. It turns out that the values employed are within the range giYe,r by other workers [7, 12, 24, 26].

R. = 2k,l'H2][O,.'], R5 = k l s a , ,

R = kt,*/k2,

R 7 = k!. ~/kl{., ,

Rio = ktl/k2,

Rt3 = kt4/kVo2 ,

Baldwin's [7] numerical method has been followed closely for solving the steady state equations [Eq. (21)-(26) ~vith all time derivatires taken to be zero]. We introduce the following velocity constant parameters Rt " R~,~ :

R3 = kz/k,,

R2 = kT, R 6 =0,

R9 = klan:k:,

227

R a = ks:k2kl ~ ,

RI l = ktzlka,

RI2 = k131kt,

and Rz,~ = kt$;

and replace the concentrations of the five chain centers by the parameters O = kz[n][oz],

K = k,[On][nz],

J = k.~[o][n2]

G = ;cl:02[no,],

and

1 = [n2oz].

We simplify as well the quantities AI - As, As0 - Az4:

and

A 1 = M / R 3,

A2 = Ra/lOzl,

,43 = Rs/iO~l,

A4 = R6/Itt~ ] ,

A5 = Rs/In~I,

A6 = RT[H2],

AT = 2R2[M'],

As = Rg/tO:l,

A2o = Rl0/tofl •

A21 = R t ilia,,1•

A22 = R : z/tun],

A23 = Rla,

A,,a = Rt4.

The steady state equatior, s ~hen become

K + A~G +J =Q(1 + At + Aft + Aal + AaG 4. Azo),

(27)

Q =J(l + A J + A2a ),

(28)

Rt + ATI + Q + J + 2A3QG + A21Q =K(I + Aft + As:),

(29)

A IQ + AslK + AslQ = A3QG + AoG + 2G" 4 Az3G,

(30)

G2 + A6G =0.5A71 + A2I Q + Asl Q +~A,,I'J + AflK + A2¢I.

(31)

0.5Rtth + Khl + Qh2 + Jh 3 + AIQha + 0.5A~lh7 + A3QGhs + GZhlo + A6GhII + AzlQht4 +

AslQht4~, + AflKhls - F A ( T -

Values for hi-hi5 are listed in Table 1. R(T) is the rate of heat generation inside the vessel and L(T) is the heat loss rate through conduction. The effective heat exchange coefficient F [27]. i~ taken to be 35.4 for a spherical vessel and 21.6 for a cylindrical vessel. The heat conductivity 2 of the mixture is taken to be 8.0 x 10 - " for hydrogen and 1.0 × 10 -~ for oxygen molecules.

To)/d 2 =0 =

R(T) - L(TL

(32)

Eliminating K, Q. J from Eq. (27)--(30). a cubic equation of G with the coefficients are functions of I is obtained:

S3G3+S2G2+StG+So=O.

(33)

The calculation process can be divided into six steps. Step 1: set the bath temperature to be the temperature at which the reaction .,starts.

228

K. K.

Step 2: a value of H202 concentration 1 is assumed, and the solution of G from Eq, (33) is obtained. Step 3; compute Q, J, K, and substitute these values into Eq. (31) to obtain a new I. Step 4; and iteration process on I can be repeated by returning to step 2 until a preselected criterion of convergence is satisfied. Step 5: substitute I, J, K, Q, and G into Eq. (32) and R(T) - L(T) is obtained. Step 6; temperature T is increased by an increment, AT, and the ca!culation goes back to step 2 until R(T) - L(TI changes sign. The cubic equation of HO., concentration, Eq. (33), always yields one re,I root which can be factorized out to redace to a quadratic equation. The explosion limit is obtained at the point where the discriminant becomes zero. In the slow reaction regions, the quadratic yields two positive real roots. The smaller of the two is taken as the steady state concentration of HO2. Steady s~.ate concentrations of H, OH. O. and H202 are then computed. The gross slow reaction rate is represented by half the rate of water formation :

F00 and C. l-I. YANG

Figure I. First limitsfor KCI coated vessel.Molefractionof 02 variedin the rangefrom0 to 0.72and H2 fixedat 0.28with N2 usedas the inertgas.

Vessel surfaces are divided into two groups; the chain center reflective type and the chain breaking type. The former group [6] includes silica and pyrex vessels washed with acid and vesseUs coated with boric acid. The chain breaking ability of these surfaces is almost nonexistant [28]. Accordingly, the rate constants for surface reactions (16)-{20) are all taken to be zero for calculation in this group of vessels. The second group includes salt and metal oxide coated vessels as well as pyrex or silica vessels washed with alkali. Different values of wall efficiency are selected for different surfaces depending upon their ability as chain carrier terminators.

sels. Experimental d-~ta for this group of vessels are extensive, especially for the B20 3coated vessels. First limits are usually unmeasurable in this group. The present theoretical model is consistent with this behavior. When all wall termination rate constants are taken to be zero, nodal singularities corresponding to the first limit no longer exist in the differential system of equations (21)-(26), Second limits have been computed* and compared with experimental data by Baldwin e' al. [6]. Good agreement was achieved. For the KCI coated vessels, wall efficiencies for H, HOz, and H 2 0 2 are selected to be 1,5 x 10 -3 , 1.50 x 10 -3, and 1.25 x 10-'=, respectively. These values are close to the values previously used by other authors i"24, 29]. The termination of OH and O centers on the wall is assumed to be diffusion controlled, and the diffusion coefficients are estimated according to procedures outlined in section Ill. in Fig. 1 the first limits in KCI coated vessels are plotted against the oxygen mole fraction (varied in the range from 0 to 0.72) for a hydrogen mole fraction fixed ++it0.28. The bath temperature is at 520°C and the diameters of the vessels are, 5.1 cm, 3.6 cm, 2.4 cm, and 1.4 cm. Experimental data (dotted curves) are taken from Ref. 29. The variation of first limits over the range of hydrogen mole frac-

First and Second Explosion Limits The appearance of "quadratic branching" is the common feature among the reflective ves-

• Baldwinet al calculatedthe limits without the energy equation, Our calculations with or without the energy equation showed only trivk~differences.

d[HzO]/dt = K + A,dd + A21Q + A slK,

(34)

IV. Calculation Results

S U R F A C E T H E R M A L EFFECTS O N H Y D R O G E N O X I D A T I O N

d --5,1 c m

t~ Figure 2. First limitfor KCI coated vessel.Mole fraction of Hz varied in the range from 0 to 0,72 and 02 fixedat 0,14 with N~ usedas the inert gas. tion from 0 to 0.86 is shown in Fig. 2. Oxygen mole fraction ~s held fixed at 0.14. Both the predicted curve ~nd the experimental data (dotted curves) ,,;how the first limits as almost straight lines which tend to bend upwards at nearly zero hydrogen mole fraction. The first limits are insensitive to the variations of termination effi¢imcies of H e z and H202 . The steady state concentrations of H e 2 and H202 at the low pressures are too small to affect the limits. The results computed for both ~he first and second limits of mixtures (in a 7.3-cm diameter spherical vessel) with varied H2 and Oz partial pressures at the temperature of 460°C is shown in Fig. 3, The experimental data were reported by Warren ['10.1 (dotted curves). The termination efficieneies for KH2PO4 coating are assumed to be 5.0 × 10 -a, 5.0 x 10 -4, 5.0 × 10 :'4, ' 5.0 × I0- 4 and 1.0 x IfJ" 5 for H, OH, O, HO2,

PO.2

mm

Figure 3. First and secondlimitsfor coated sphericalvessels (diameter~ 7.3 ¢m) at b~th temperalure of 460°C.

229 and H202, respectively. According to our calculations, the wall efficiencies of H-atom and H e radical termination are more important than the wall efficiency of H202 for determining the upper and lower limits. The explosion limits of a hypothetical surface that has wall termination efficiencies 1.0 :< 10-4 for H, 1,0 x 10 -4 for OH, 1.0 × 10-4 for O, 1.0 x 10 -4 for HOz, and 1.0 x 10 -5 for H202 are calculated (carve AB in Fig. 3). The termination efficienties of this surface lie between those of B20 3 coated vessels (zero termination efficiencies) and KH2POa coated vessels. A natural transition can be seen between the quadratic branching type limits (B203 coated vessel) and saltcoated type limits. Howson and Simmons [2] showed that BaBr coated surfaces exhibit such behavioJ~. For KOH coated surface calculations, O H and O are taken to be diffusion controlled. The termination efficiencies for H, He2, and H20 z are taken to be 5.0 x 10-3, 5.0 x 10 -3, and 5.0 × 10-4. respectively. This is not completely consistent with the conclusion of Lewis and yen Elbe ['121] that the KOH coated surface is efficient in destroying H. He2 radicals but inefficient towards H:O2. The behavior associated with quadratic branching is obtained in this model for the coated vessels by letting the wall termination efficiencies gradually approach zero. The separation between the limits of different coating materials is more noticeable at low oxygen partial pressures and becomes small at the high oxygen pressures. It is a natural consequence of the branching reaction (2) which is proportional to oxygen partial pressure and dominates the total reactivity of the system. At low oxyg,:n partial pressures, the gaseous phase reactivity of the system is low. The heterogeneous ~ermination rate becomes effective in determin~ing the explosion limits. At higher oxygen partial pressures the gaseous phase reactivity is high while the surface termination rate remains unchanged. Its effect on explosion limits, therefore, becomes less significant,

230

K. K. FOO

~ d

~ ~..

m~4Cm~

"-..

Figure 4. The effect of vessel size on the second limits [10] (KCI coated vessel at 500°C).

Figure 4 shows how the limits (KCI coated vessels) vary with different 'vessel diameters. The experimenta~ data are reported by Warren [10] (dotted curves). The calculated limits show quadratic branching behavior when the vessel diameter approaches infinity. It is compared with the experimental data [4] taken from a B::O3 coated vessel at the same bath temperature (T = 500°C). The explosion limits calculated and plotted in Figs. 1-4 follow very closely the prediction of the isothermal chain theory. When the same limits were recalculated with the energy equation left out results showed only trivial differences. This is consistent with the experimental fact that "'self heating" effects within this temperature range have not been observed.

and c . H. YANG

scheme, reaction (19) is crucial for the existence of bc~h the second and third limits. In absence of this reaction HO2 concentration will reach a steady state value between reactions (4) and (35). The chain carrier, H, terminated by reaction (4), is converted to HOz and returned to the reacting medium through reaction (35). Consequently, both second and third limits can not logically exist without reaction 09). Thus the theory of Willbourn and Hinshelwood may explain the third limits for salt coated vessels, but it cannot account for the existence of the same limits in reflective vessels. The thermokinetic model, on the ocher hand, predicts the third limits in both the reflective and chain breaking type of vessels. Results computed with this model are compared with experimental data plotted in Figs. 5-8. Figure 5 shows the calculated third limits in a 5.5.cm KCI coa':ed vessel at a bath temperature of 586°C with the partial pressures of H 2 and O~ varied. The theoretical curve A is compared with the data of Willboura and Hinshelwood [16] (dotted curve). In an attempt to achieve a better fit of the data, a second theoretical curve B is calculated by increasing the preexponential factor A4 by 5 per cent,

8

Third Explosion Limit Willbourn and Hinshelwood [I6] measured the third limits of H2-Oz mixtures in KCI coated vessels. They calculated the third limits based on an isothermal chain theory which includes reactions (1)-(4), (19), and the following reaction in addition : Hz + HQz-'+ H,O + OH.

(35)

The third limit is explained in terms of the second limit mechanism in which the chain centers HOz are destroyed on "the wall and propagated in the gaseous ph=se again at higher pressure through reaction (35). In this kinetic

f~ Figure 5. Third limits [16] for KCI coated vessel:d = 5.5 era and T = 570°C.

SURFACE THE~LMALEFFECTS ON HYDROGEN OXIDATION

,,\ i

~%

'

d--39 an

x

-/o

'

~

BATH TEM~RATU~OC

'

&

Figure 6, The effect of vessel size on third limits (K,CI coated vessels).

The vessel size effects on the third limits are presented in Fig. 6. Stoiehiomet,ric mixture and KCl coated vessels were employed. The slopes of experimental 1"12"] curves (dotted curve)

231

differ sighificantly from those of tlae theoretical curvcs. The third limits for KCI vessels are clearly affected by the vessel size. If both the third limit and the second limit are explained by a chain theory 1,,16] it is difficult to understand why second limits are not observed 1-10, 12-1 to be equally sensitive to the size factor. In the more complete thermokinetic model the size effect at the third limit is amplified by the accelerated rate of heat exchange between the reacting medium and the bath. The dotted curves in Fig. 7 show the measured second and third limits for constant temperatures equal to and above 540°C with the partial pressures of H 2 and 02 varied. The vessel is KCI coated and its diameter is 7.3 era. The calculated curves match the experimental data 1,10] for a slightly different temperature. It is, however, quite obvious that the qualitative characteristics of the theoretical prediction and the experimental measurement are identical. Figure 8 shows t,rie calculated first, second and third limits for a stoichiometric mixture in reflective and KCI coated vessels (d = 7.4 cm).

",\ ~.-., l~/l

t

I

r-~

/ / (~".,.c

BATH TEMr~RA'n~°C Figure 7. Second and !hird limits at high bath temperature, KCI coated vesscl~; d = 7.3 cm.

Figure 8. First. second, and ~:hird limits for reflective and KCI v,.'ssels.

232 The calculated results (curve A) for the reflective vessel show no first limits. Egerton and Warren's [4] data for the boric acid coated ve%sel are plotted (curve G) for comparison. The second limits for reflective vessels have also been c~alculated with the chain thegry which excludes the energy equation (26). Results are show.,, as the dotted curve H, which'extends to high temperature without bound. It clea,rly shows that the chain theory, by itself, cannot account for the third limits. Voevodsky [11] measured third limits of the stoichiometric mixture in a 4-cm pyrex vessel. His data are also plotted (curve B). The calculated second and third limits in KCi coated vessels (curve C) agree quite well with measurements (curve D) of Lewis and , o n Elbe [12] (curve E is for heavy KCI co~ ring). The first limits show rather wide differenceg largely because a log scale is chosen for the pressure and the errors are, therefore, exaggerated. Measurements of Voevodsky [11] in 4-.cm KCI coated vessels are also plotted (curve F) for reference. Explosion limits of the KCI coated vessel are calculated again with the isothermal theory. They are plotted as a dotted curve, J, which has a larger slope than the one calculated for reflective vessels: The iosthermal theory will probably pred i c t t h e third limits for the KCI vessels if unr~,~alisticaUy high surface efficiencies are used, but then the fit of the first limit data will be drastically poorer. In general the ~urt'ace reactions increase the slopes of all three limits in P- T plane. i The calculated second limits for the two types o'l" vessels tend to merge as pres'3ure and temperature increase. This indicates that the heterogeneous termination reactions gradually become less important when the gaseous phase reactivity increases. The plotted experimental third limit data o f Voevodsky [I1] show two curves with different slopes (B and F). It is in:cresting to note that the calculated curves A and C for the reflective and KCI coated vessels, respectively, also show similar difference in their slopes,

K. K. FOO and C. n. YANG Slow Reaction and Self l-l[eatlng Slow reaction rates are determined by product forming rate ',i.e., d[H20]/dt) or the pressure changing rate (dp/dt) of the reacting system. These two measurements are equivalent even if the reaction is not under isothermal conditions. The calculated slow reaction rates in the present work are all represented by or deduced from the water formation rate. Pease [30] noted th.',t the slow reaction rates in self-coated vessels were reduced by factors of 50-2000. The computed reaction rates in B203 coated and KCI coated vessels at P,2 = 300 mm, Po~ = 50 ram, and d = 5.5 cm are compared in Table 2. The,.,, are consistent with the observation made by Pease The ratio between t,.," reaction rates is sensitive to the bath temperature. At still lower temperatures it easily approaches a thousand fold. Willbourn and Hinshelwood [16] measured the slow reaction rates. Results are replotted ia Fig. 9. In general, the slow reaction rates and the lower explosion limits are both sensitive to the surface efficiency for terminating HO e radicals. The surface efficiency for terminating H atoms is important in determining the lower limits, while the slow reaction rate is insensitive to the surface effiek:ncy. The slow reaction rate is, however, very sensitive to the wall efficiency of H , O 2. "'Self heating" effects ha~e always b ~ n observed at pressures and temperatures between the second and third limits in reflective vessels. The temperature of the reactants may rise significantly above the bath temperature as a result of this phenomenon. The degree of self heating or thermal effect can be measured by the temperature difference. One may regard Table 2. Calculated ~low reaction rat~ in B,~O:~and

KCI

Coated Vessel~ B20 ) vessel

KCI vessel

Rate {B2Oa)/

TO

mm/min

mm/min

Rate (KCI)

560°C 566°C

68.3 99.6

0.426 1,7

158 59

570°C

127.5

3.91

33

233

SURFACE THERMAL EFFECTS ON HYDROGEN OXIDATION *C ~ar

d --Tac

F

i

./

~,

g

\';x-... -

;,a,

~t

/

["

'~

~

,

D~,

--~

Figure 10. Slow reaction rat~ in KCI and quartz v~,;sels, 21-I: + Oz mixtu~.- and P =600 ram.

~

Figure 9. Slow reaction rate~ in KCI and silica vessels, ,Pa2 = 300 rnm, Pu: = 50 ram. and d = 5.5 cm.

the point where this temperature difference (AT = T - To) reaches I°C as the starting point of a thermally significant region. The calculated resuks in Table 3 (2H 2 + 0 2 and d = 7.4 cm) show that the thermal effect becomes significant at different bath temperatures and pressures, depending on the type of vessel surface. In KCI coated vessels the thermal effect becomes significant at bath temperatures of 70°C higher. than in reflectbte vessels at the same pressure. The much lower thermal effect in the KCI coated vessels was probably the basis for some authors to regard the third limits measured ~n such vessels as chain in nature, while the third limits measured in quartz vessels have been widely accepted as ~:he thermal explosions. Slow reaction data of Holt and Oldenberg [31] are plotted in Fig. 10. The siz:: of the vessel

they used is, however, not available. Expe/imental curves are compared with arbitrarily chc,sen vessel sizes as labelled. The starting point (AT = I°C) of the calculated tl~ermally significant :region in each case is indicated with an ;arrow on the curves in Figs. 9 and 10. Before entering the thermally significant region, the theoretical and experimental curves of the reaction rates plotted against 1/T have constant slopes and approach to straight lines. The thermal effect is gradually visible a.s the slopes o f the curves gra~tually increase to reach explosion limits. Concentrations of H20 z were measured by Pease [30], Linnett [32], Holt and Oldenberg [31], and Baldwin ['33] et al. under very shnilar conditions in reflective vessels. The calculated values are compared with their measurement m Table 4. The calculated values in KCI coated

Table 4. Experlment~l and Calculated H~O~ Concentrations uuder Slow '~eaction Conditions

Table 3. Bat)~ Temperatures and P r a t e s ~ which Thermal Effects Begin To Be Sigeifie~ .~.

Pressure, ram

Reflective vessel Bath temlmrature °C

g.Cl '.'.ssei i'),,~J '. re, a~L,~t;~ °C

p ~ 400 P = 500 P = 600 P = 700

510 498 488 480

572 .~70 566 566

T~perature Pressure °C Rlm

Pease C3o] Holt & Oldenberg[31] Linnet& Tootal Baldwin['33~ calculated Not available•

540 540 567 500 540

~ 500 500

HzO~ mole, cc- 1

1,0 × tO-" 1,0 x I0 -a 5.7 x 10- ao 1.5 x 10-6 2.0 x 10=8

Z. K. ~OO and c. H. YANG

234,

/

5'

/ //

s

P~qESSU~Emm FigtJrc 11. Slow tcactio~ rates ~'~ KCI vessels with PH2 varied. Po~ = 100 ram. d = 5,5 c m , a~ld To = 570°C,

vessels are more than two orders of magnitude lower. This explains, perhaps, why they were not measured. The lower total steady state: chain c,-'nter concentration in KCI coated vessels also confirms the lower thermal effects in these ves,sels. Thus, it is clear that a sharp division between thermal and chain mechanisms ~,:loes not exist. In order to deal with surfaces that vary from perfectly reflective to extremely effective in chain t,.rrnination one requires a unified thermal and chain theory. Willboum and Hinshelwood [16] measured the slow reaction rate in KCI vessels with the partial pressare of hydrogen varied. The calculated curve agreed ,:losely With the measurements as shown in Fig. 11. At higher pressures thermal effects are a ore intense even in KCI vessels. The difffrence between the bath and r~aeting medium temperatures CxT) becomes very large at points righ~ below th~ third limit. Calculated results are shown in T~ble 5 for a mixture of 2Ha + O , in KCI and reflective vessels with d = 7.4 cm. Table 5. Cal,eulated Differences ~AT) Between the T e m ~ r atures of the Bath a~d the Reacting Mixture KCI vessel

Relleetive vessel

To{°C) P(mm) AT(°,C)

To(°C) P{ram) AT(°C)

565 5"t5 577 571

200 300 400 700

3 20 38 85

555 561 557 53'~

200 300 400 700

13 34 58 117

V. Discussion The modified Baldwin scheme of the H2 and O z system is stronglysupported by the wide range of experimental data obtained "in B z O a coated vesselsas well as in saltcoated vessels.S o m e of the very interesting kinetic characteristics can be traced to the chain termination process in this scheme. At pressures below the second limits chain centers are mainly terminated on vessel surfaces. At pressures above the second limit the major chain breaking step in all vessels regardles~'~ of coating shifts to the gaseous phase reaction (14) through which three chain ,:enters ( H a 0 z is equivalent to two centers) J:s converted to one. Reaction (4) should not be considered as a termination step as it merely converts an H center to an HO2 center. Reaction (8) was regarded to be responsible for "quadratic branching" by Egerton and Warren [4"1. Actually it is merely a straight chain step which converts one H and one HO2 into two OH centers. However, reaction (8) competes with ~-eaction (4) for H centers and diverts the chain centers from going through the termination step, reaction (14). At low O 2 partial pressures reaction (8) is more effective than rent.. tion (4), and a higher total pressure is necessary for suppressing the explosions. Consequently, the "quadratic branching" behavior rcsalts. Active chain termination on the wall of a salt coated vessel tends to suppress explosions. In such a case. the "'quadratic branching" phenomenon does not occur. h has been shown that in order to account for the third limits both for the reflective and salt coated vessels, a thermokinetic theory must be invoked Basically, lhe thermal mechanism always plays a role in explosion,~-- one of the necessary conditions of a chain explosion is that the overall process must be exothermic. HoweveL under some conditions, either the chain or the thermal factor may appear to dominate. In the Hz and 0 2 system the dominating role shifts from. chain to thermal when temperatures, and pressures are shifted from second to third limits. Acconlingly, it is interesting to

SURFACi-'~TH-I~MAL EFFECTS ON HYDROGEN OXIDATION note theft the surface reactions in salt c o a t e d vessels retard such a shift, so t h a t the third limit a p p e a r s to be chain d o m i n a t e d . T h e production o f chain centers a m o n g molecules o f the initial reactants a r e considered to be the result o f the initiation reactions. In a linear kinetic scheme, where interaction between c h a i n centers is absent, the slow reaction rate is f o u n d to be always p r o p o r t i o n a l to the initiation reaction rate. T h e a c t i v a t i o n energies o f initiation reactions are usually v e r y high. T h e predicted slow reaction rates by the linear s c h e m e are usually unsatisfactorily low w h e n c o m p a r e d with the m e a s u r e d values, T h e slow reaction rate for a nonlinear kinetic s c h e m e such as the present one can be independent o f the initiation reactions. In the present system, initiation reactions o f all chain centers a r e neglected. T h e slow reaction rate a b o v e the second limits is characterized by a nodal singularity in the p h a s e space wl'tich involves the reactions o f the chain centers instead. These reactions, in general, correspond to m u c h lower activation energies, which can easily explain the higher m e a s ured rates. Slow reactions at temperat,ures a n d pressu: ~s below the first limits, on the o t h e r h a n d , ,;te theoretically characterized by the initiation reactions, a n d they are indeed u n m e a s u r a b l e . The authors are indebted to Professor .R. R . Baldwin o f the University o f Hull. England. and to Professor A, L. Berlad o f the State University o f N e w York, U . S . A . . f o r m a n y valuable suggestions and discussions on the contents o f the manuscript.

References I. WARREN,D. R.. Elel,enth S),mp. (International) Com, bastien, p. 1088. The Combustion Institute. Pitt;,burgh (1967). 2. HowsoN. A. C,, and SIMMONS,R. F.. ibid. p. 1081. 3. BLACKMOt~.E,R. D., O'DONNEL.B., and SIMMONS,R, F., Tenth S)wtp. (International) Combustion. p. 303. The Combustion Institute: Pittsburgh tI1965). 4. EGER1ON,SIR A. C.. and WARREN. D. g., Proc. Roy. Soc., See. A, 204. 465 (1951).

235 5. BAI~DWlN.R. R., and MAYOR.L, Trans. Faraday Soc. 56. R~,, 103 (1960). 6. BALDWIN.R. R.o MAYOR,L. and DORAN.P.. Trans. Faraday Soc. 56, 93 (1960). 7. BALDWIN.R. P~.. JACKSON,D.. WALKER, R. W., and WEBSTER, S. J.° Trans. Faraday 8o¢. 63, 1665. 1676 (1967). 8. H!NSHELWOOD,SIR C. N., Proc. Roy. Soc.. Ser. A 188, I (1946). 9. SIMMONS,R. F.. Ref. I. p. 1088. 10. WARREN,D. R.. Proe. Roy. Soc.. Ser. A 211, 86. 96 (1956). I 1. VOEYOOSKY,V. V., Seventh Syrup. (/nternational) Combustion, p. 34. Bunerworths: London (1958). 12. LEWIS.B.° and VONELI~E,G. J.. Combustion, Flames and Explosions. Academic Press: New York (1961). 13. DIXON-LEWIS,G., LINNI~r. J. W, and HEATH,D. F,, Trans. Faraday Soc. 49. 766 (1953). 14. BALDWIN.R. R., Trans. Faraday Soc. 52 1344 (1956). 15. FROST, A. A., and ALYEA. N. H.° J. Amer. Chem. Soc. 55. 3227 (1933); f~i, 1251 (1934). 16. WILLBOURN,A. H,, and HINSHELWOOD.SIR '~. N., Proc. Roy, Sot.. Set. :1 18.5, 353. 369° 376 (1949). 17. YANG,C. H.. and GR/-Y,B. F, J. Phys. Chem. 73. 3395 (1969). 18. YANO.C. H., J. Phys. Chem. 73, 3407 (1969). 19. YANGoC. H.. and GRAY.B. F., Ref. I.p. i099. 20. GRAY,B. F., and YANC,.C. H., J. Phys. Chem. 69. 2747 (1965). 21. GRAY, B. F., Trans. Faraday Soc. 65, 1~-02, 2137' (1969). 22. YANO.C. H.. and GRAY, B. F., Trans. Faraday Sac. 6,5, 1614 (1969). 23. DOVE.R. E, J. Chem. Phys. 28. 1193 (1958). 24. SEMEUOY.N. N , Some Problems in Chemical Kinetics and Reactivity. Vol. I and tl. Princeton Univ. Press: Princeton, NJ. (1959). 25. HIRSCtIFELDER,J. O.o CURTISS.D. F.° and BraD, R. B.. Molecular Theory of Gases and Liquids. Wiley : New York (1966). 26. SMITH.W. V.. 3. Chem, Phys. II. II0 (1943). 27. FRANK-KAMENI~TSKI,I. A., Diffusion and Heat E.rchange in Chemical Kinetics. Princeton Univ. Press: Prim:eton. N,J. (1955). 28. WARREN. O. I~.. Trans. Faraday Soc. 53,'~199, 206 (1957). 29. BALDWIN,R. ~... Trans. Faraday Soc. 52, ~344 (1956). 30. PEASe. R. N.. J. Amer, Chcm. Soc. 52, 5107 (!935). 31. HoL'r, R. B., and OLDENEERa.G, O.. J. Chem. Phys. 17. 1091 (1949). 32. LINNETT.J. W., and TCOTAL.C P., Seventh Syrup. (InternationaL~ Combat,ion. p. 23. Butterworths: London (1959). 33. BALDWIN.R. R., ROSS!TEe,.B. N., and WALKER,R. W., Trans. Fa,aday Soc. 65, 1044 (1969).

(Received February 1971; reprised version received M a y 1972 )