Combustion zone propagation in a turbulent medium

COMBUSTION ZONE PROPAGATION IN A TURBULENT MEDIUM

P. V. Belousov and I. G. Dik

Combustion regimes of a homogeneous gas mixture under turbulence conditions are dis- tinguished by great diversity due to the specifics of the hydrodynamics in any engineering apparatus. In particular, the majority of the combustion regularities in a boundary layer is explained successfully on the basis of a model analogous to the description of laminar combustion with the replacement of the molecular by turbulent transport coefficients [1-4]. Such a model can turn out to be useful even for the development of a turbulent combustion theory by the mechanism of the motion of curved laminar flame sections since the presence of a high-frequency fluctuation component in turbulence results in a change in the velocity of the motion and the structure of the moving fronts. Other domains of application are also possible for the so-called models of volume gas combustion [5].

In the development of the papers [6, 7], the combustion of premixed gases is examined below within the framework of a model using equations for the mean value of the temperature <T> and single-point second correlation moments <w'T'> and <T'2>.

Let an unlimited homogeneous space be filled with a reactive gas in the turbulent state. The hydrodynamic characterstics of turbulence (r.m.s. velocity of the fluctuations b = ~ and the Lagrange scale L) are considered given. A plane layer is propagated in this space, where transformation of the initial substance into chemical reaction products occurs. It is assumed that the chemical reaction is exotherm with a sufficiently high activation energy and large thermal effect, so that the transformation layer is the flame front separating the initial mixture with temperature T_ from the combustion products of temperature T+.

Let us associate the coordinate system with the front being propagated and the x axis perpendicular to it. The velocity of combustion front propagation is determined by the physicochemical processes in the flame. The most important are the chemical reaction and the heat transfer from the high-temperature zone on whose behavior turbulence exerts sub- stantial influence.

We write the equation for the mean temperature <T>

~--~-~ <T> + ~=7~ Q <~ (r)> ( i )

and for the heat flux q = <w'T'>

dq D ~fq q _ b ~ a </'> ~ <w'CD w~ ~-~ ~ + (r)> (2)

as the mathematical model of combustion, where the usual closure hypotheses [8] are used in the writing.

In order of magnitude the diffusion coefficient for q should equal the characteristic time of the hydrodynamic fluctuation T I = L/b multiplied by the turbulence energy b 2 = <w'2> i.e., D = b2T1 = Lb. A matching factor [9, i0] can be introduced for combustion computations in jets, boundary layers, etc. It is recoE~nended to take this factor as 1.5 for a jet in [9], and on the order of 0.3 for a broad class of flows in [i0]. It is taken equal to 1 in this paper.

By the assumption of the existence of temperature-concentration similarity, the heat liberation function depends only on the temperature. Calculation of the mean value <~> re- quires knowledge of the magnitude of the temperature fluctuation. Without writing the equation for <T'2>, let us assume the existence of a rigid correlation

Tomsk. Translated from Fizika Goreniya i Vzryva, Vol. 21, No. 4, pp. 32-38, July- August, 1985. Original article submitted May 3, 1984.

410 0010-5082/85/2104-0410509.50 © 1986 Plenum Publishing Corporation

We write the temperature dependence of the heat liberation in the form

(1) (T) = [ ( r + - - T ) / ( T + - - T - ) l k o exp ( - - E / R T ) ,

(3)

(4)

which corresponds to a first-order reaction.

Introducing the dimensionless variables and parameters

a = (T+ - <T>)/(T+ - T_) , y = q/[(r+ - T_)J/DI~+],

= x/I[D~+, co = w]/x+/D,

E ( % - r_ , ) Sh = ~i/~+,~ Oo = /~r~_

where T+ = k-10exp (E/RT+), we write the problem (i), (2) as

du dy - - co ~ + - ~ = < ¢ Cu) >,~

dy d~y S h - 1 (y __ d~) + <to,cD (u) ) .

(5)

The boundary conditions are

du ~=--~: u = t , ~ = 0 , d~t ~=~: N=0.

( 6 )

To complete the formulation of the problem, there remains to determine the functions <¢> and <~'¢(u)>, for which certain statistical hypotheses must be relied upon.

The equations for the temperature fluctuation and velocity distribution functions require additional, often Competing, hypotheses for the closure [11-16]. A preliminary analysis of turbulent combustion is possible even at the level where the averaging method is postulated [6, 17, 18] and suitability of the averaging is seen in the final results. Used below for the averaging of the nonlinear functions is a hypothesis about the Gaussian temperature and velocity fluctuation distribution, i.e., a distribution density of the form

where o =VquT~; k = ¢q-jrr~; and r 0 final expressions, as it turns out.

The mean values

is a correlation factor which does not enter into the

a n d

= f[. ( , of)du'd¢o' <¢ (u)> ¢I, (u + u') ~ u'

= ( , ¢o') du'd¢o'

can be completely calculated analytically if the exponentials in (4) are transformed in conformity with Frank-Kamenetskii

(u) = u exp ( - O o u ) .

As a result of the averaging we obtain

411

] vu

7-"I A "' f 9> zl ~ A it--- ' ~ < Z \ 0,,4 t:, /,'r-

I - - j ; i . / ) ' , ' o 25 5o

I

9 0,5 a 0 0,5

Fig. I Fig. 2 Fig. 3

Fig. i. Range of variation of the temperature dispersion; Sh = 10.

Fig. 2. Temperature and heat liberation profiles in a flame for 00 = i0, Sh = 0 (i) and i00 (2).

Fig. 3. The ratio Vu = <¢>/¢(u) for O0 = i0 and Sh = 1 (I) and i00 (2).

( <@ (u)} = (u - - Ooo ~)exp - -Oou+ 2 Y'

<~o'¢ (u)> = - - y ( l + Oo2a °" -- Oou) ex p ( - - Oou + - - ~ ] . (7)

It is easy to verify that <~'~> = -yd<¢>/du. Requiring compliance with the rigid correlation condition (3), we write

a = -~Sh y. (8)

Therefore, in this formulation [the relationships (5)-(8)] the dimensionless propagation velocity ~ is a function of two parameters:

o = o(Oo, s h ) . ( 9 )

One of them, 00, is usual for combustion theory and reflects the strong response of a chemical reaction to the temperature. The number Sh was first introduced by Shchelkin [19] as a possible criterion for the realization of different combustion regimes in theory of turbulent flames [19]. Taking TI = 10-=-10 -3 sec, z+ : 10-4-10 -5 sec for the estimates, Sh = 101-102 can be expected.

In analyzing experimental data, the normal combustion velocity w n related to T+ is ordinarily observed. In the case of a first-order reaction [20, 21], we set

Wn = O~koexp -- = r+O~+"

We can now write (9) in the form%

~ r = r . 2× ~' Ooo 0o, ~ , w" = ~ \ × ' ~ ' r_ ' "

Therefore, according to the theory being developed, there are four governing parameters that should be checked in the experiment.

The problem (5)-(8) to determine the eigenvalue ~ was analyzed on a computer. As is usual in such cases [21], the heat liberation function was predetermined in the low-temperature domain. It was assumed that <¢> = (u - l)(8O/aU)lu=us for u > u s, i.e., u s is determined from the condition of smoothness of the heat liberation function at the point u s. In order of magnitude u s = 1 - 0-10. Using the results of the numerical computation, the distribution of different average characteristics in the combustion wave can be obtained.

The change in u together with the fluctuations is superposed in Fig. 1 within the framework of the dispersion deviation. The versionsrepresented are characterized by the fact that the domain of temperature variation does not anywhere emerge outside the scope of the thermo-

%The traditional notation w' = f~r~= b is used here and henceforth.

412

dynamically possible values. But as O o diminishes, the dispersion in the fluctuations of T grows and the temperature alternation in the flame must be taken into account [5, ii]. The domain of parameter variation was selected in this paper so that the alternation would not be felt.

The temperature profile of the flame and the heat liberation are shown in Fig. 2. It is seen that as Sh increases the heating zone becomes more populated. This is the result of a rise in the chemical reaction rate at low temperature. Judging by Fig. 2, stretching of the flame because of fluctuations in T results in the fact that even for large Sh the flame thickness remains on the order of the turbulence scale L. For Sh = i00, the flame thickness is approximately 50 Dv~+, while L -- ¢~ D~+ = 10v~-~+.

The behavior of the function Vu(u), the ratio between reaction rates in the turbulent and laminar regimes, is displayed in Fig. 3. The form of Vu(u) agrees qualitatively with that obtained in [6, 7, 22], where nonlocal terms in the equation for y- the second moment diffusion and convection- were not taken into account. Vulis [23] made the first estimates of the influence of temperature fluctuations, and later the effect of the influence of temperature and concentration fluctuations on the chemical reaction rate was studied in [24] for ignition processes. Such investigations were performed in [6, 25, 26] for combustion conditions.

Acceleration of the chemical reaction in the cold domain is related to the action of the temperature fluctuations. The turbulent heat flux IY! and, therefore, the temperature fluctuations ly S/~I shift toward the cold domain as Sh increases. The fluctuation amplitude of T grows. The pattern of these changes is given in Fig. 4.

The change in the temperature gradient is shown in Fig. 5 for different Sh. For large Sh the convective term I~0du/d~ I in the heat conduction equation becomes greater than the diffusion term dy/d~ in the combustion zone. For example, for Sh = I00 at the temperature where heat

Io) du-] dy This is associated with the influence of the tem- liberation is maximal, ~/~I 24

perature fluctuations on the effective coefficient of turbulent thermal diffusivity, which can be calculated as the ratio

A ..... Id<T> I..l eul

In the computations illustrated in Fig. 6, an almost complete cessation of heat conduction is observed in the zone of intensive chemical reactions. In the low-temperature domain A slightly exceeds the equilibrium level =b2~l.

Results of computing the propagation velocity m are represented in Fig. 7. It is seen that for any parameter 00 the curve m(Sh) has two characteristic sections. For Sh ~ 2, the quantity m is independent of Sh and it is easily verified that the turbulent flame velocity in this domain is determined by the combustion velocity and the ratio between turbulent and molecular intensity of turbulent heat transfer (in conformity with the volume combustion model )

w = Wn.~Lw'/×.

F o r Sh > 2 i n t h e l o g a r i t h m i c c o o r d i n a t e s l o g ~ - l o g Sh l i n e s a r e f o r m e d w h o s e s l o p e s depend on O o, but the curves are almost equidistant for sufficiently large 0 o. For large e 0 and Sh this permits proposing a power-law monomial approximating the numerical computation ~-: a Shnem0. For instance, for 00 ~ 15, a = 0.14, n = -0.36, m = -0.05. In dimensional form, the approximate formula has the form

1 +~n 1--2n l~2n ( T + ) 2 U), '2 I + 2 n r . . . . . . . . . w = aOo 1+~+2'~ ~ _ Wn ~ 2 ~ ( 1 0 )

and for the values found for the coefficients

~b //2 W n .

(ii)

413

3 G,E u

Fig. 4

r~M

i z

O o,s u Fig. 5

1

o O,S ,,.¢

Fig. 6

Fig. 4. Turbulent heat flux; O 0 = i0, Sh = 0 (I), 4 (2), 36 (3), and 100 (4).

Fig. 5. Dimensionless temperature gradient; ®0 = 10, Sh = i (i), 100 (2).

Fig. 6. Effective coefficient of thermal diffusivity; 00 = i0, Sh = 4 (i), 36 (2), IO0 (3).

%9c~ i !

I 4 ~a "-4

0,8 1,4 %gSh Fig. 7

3

o,4

Fig. 8

Fig. 7. Flame propagation velocity; 00 = i0 (I), 14 (2), 18 (3), 20 (4).

Fig. 8. Balance in the equation for y; e0 = I0, Sh = 100.

i) Generation and dissipation (,-~=); 2) concentration

[. d<~>h 4) (.Sh~); 3) action of the c h e m i c a l reaction ~Sn~},

For ®0 ~ 15, the exponents depend on 80, but coefficients can also be selected near ®0 = i0 (a = 0.25, n =-0.13, m ~-0.1), and the following formula can be used for the flame velocity:

( T ) o , 8 , . ~o + . ,o .e8. o.7+~o.~7 0"25=0"42-~-~_ w wn ~ " (12) w =

The power - l aw a p p r o x i m a t i o n fo rmula o f t h e t y p e (10) i s p r o p o s e d in [27] on t h e b a s i s of a similarity and dimensional analysis. It is shown there that for certain n different turbulent combustion models, known from the literature, are realized. In [28, 29] the turbulent combustion velocity is also obtained in the form of a power-law monomial of the type (I0) with n =-0.25 (there are no parameters 80 and T+/T_ in [28, 29]).

It follows from (II) and (12) that the exponent is generally speaking a function of the activation energy and the thermal effect of the combustion reaction. As 00 increases, Inl grows and Iml diminishes, and it can be assumed that lim Inl = 0.5 and !im Iml = 0 as 80 ÷ In this limit it follows from (10) that w - w' and the velocity of turbulent combustion ceasesto depend on the remaining parameters. Such a dependence is also cited in the literature [19, 20] as limiting within the framework of a surface model for gas combustion under strong turbulence conditions. Judging by the computations presented, the asymptoticity of this result is weak.

Comparison of (ii) and (12) with the experimental values of the sensitivity of the turbulent flame velocity to the change in different parameters (many are collected in [19, 20, 30]) is sufficiently satisfactory. In particular, according to numerous experimental results, usually d in w/d In w' = 0.7-0.8, which is close to what (ii) and (12) yield.

414

Let us also present a quantitative estimate of the turbulent flame velocity for the following values of the parameters: L = 2.5-10 "i cm, w' = 5 m/sec, w n = 0.8 m/sec, ~ = 0.6 cm2/sec, T_ = 440°K, T+ = 1800°K, E = 36 kcal/mole. Formula (12) yields w= 11.6 m/sec. The quantity w is measured in experiments to burn up a gasoline-air mixture in [31] in a 4-cm- diameter pipe for a 100 m/see fuel delivery rate. The experimental value is w = 12 m/sec. In a 9.8-cm-diameter pipe, the experiment yielded w ~ 16 m/sec. Considering the recommenda- tion of Khitrin and Gol'denberg [31] that L = 0.6 cm, we obtain w = 15.4 m/sec from (12), i.e., the computations yield a correct result in order of magnitude (taking into account a certain indeterminacy in the input parameters).

Let us turn to an estimate of the relative flame thickness. If it is defined as H/L = D/wL, it is easy to obtain that H/L = ~-ISh-°'~ For small Sh, the dimensionless velocity is independent of Sh. Accounting for ~ = w / ~ = Wn/~--7-~+ we obtain H/L. = /~@oSh -°'5 >> i.

For large Sh, it is possible to use (10). Then H/L =a-*o-m0Sh "(n+°'s). For large Sh, the growth of H/L with the increase in 00 is small (m < 0, Iml << I) and the dependence of H/L on Sh is also modified (n < 0). For 00 = 10, Sh = 100, we obtain H/L = 0.93, and for 0 o = 20, H/L = 4.46.

The presence of two turbulent flame propagation regimes is due primarily to the action of the temperature fluctuations on the heat liberation. From a comparison between Figs. 3 and 7 it is seen that growth of the heat liberation rate by several times in the low-temperature domain is not essential to the increase in the combustion velocity; much more effective is a moderate reduction in the reaction rate near the adiabatic combustion temperature, which, together with difficult heat elimination in the heating zone, alters the regularity ~(Sh) with the growth of Sh.

The temperature fluctuation level is added because of the different processes included in (2). The role of the corresponding terms in the balance equation for the second correlation moments is shown in Fig. 8. As follows from the computations, for large Sh the differ- ence between generation du/d~ and dissipation -y in the reaction zone is covered by the diffusion term Sh d2y/d~ 2. Because of the strong suppression of the reaction fluctuations

Sh<~'~> = --y Sh d<~>__~ in the reaction completion stage, the diffusion term becomes positive.

The authors are grateful to V. N. Vilyunov for useful discussions of the research.

LITERATURE CITED

i. Ya. B. Zel'dovich, Fiz. Goreniya Vzryva, [, No. 4, 463 (1971). 2. Ya. B. Zel'dovich, O. I. Leipunskii, and V. B. Librovich, Theory of Nonstationary

Powder Combustion [in Russian], Nauka, Moscow (1975). 3. V. N. Vilyunov, Dokl. Akad. Nauk SSSR, 136, No. 2, 381 (1961). 4. A. A. Belyaev, A. A. Zenin, V. V. Kuleshov, et al., Khim. Fiz., No. i0 (1982). 5. P. A. Libby and F. A. Williams (eds.), Turbulent Flows of Reacting Gases [Russian

translation], Mir, Moscow (1983). 6. V. N. Vilyunov and I. G. Dik, Zh. Prikl. Mekh. Tekh. Fiz., No. 5 (1976). 7. V. N. Vilyunov and I. G. Dik, Fiz. Goreniya Vzryva, 13, No. 3 (1977). 8. A. V. Lykov, Heat and Mass Transfer Handbook [in Russian], Energiya, Moscow (1978).

9. E. A. Meshcheryakov, Uch. Zap. TsAGI, ~, No. i (1974). 10. V. Llewellyn, Turbulence, Principles and Applications [Russian translation], W. Frost

and T. Mauldin (ads.), Mir, Moscow (1980). ii. V. R, Kuznetsov, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5 (1972). 12. V. R. Kuznetsov and V. A. Frost, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2 (1973). 13. V. A. Frost, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaze, No. 6 (1973). 14. C. Dopazo, Phys. Fluids, I_88, 397 (1975). 15. S. A. Nedorub and Yu. A. Shcherbina, VINITI, Dep. No. 3406-79. 16. S. A. Nedorub, V. A. Frost, and Yu. A. Shcherbina, VINITI, Dep. No. 3405-79. 17. K. N. C. Bray and P. A. Libby, Phys. Fluids, i_99, 1687 (1976). 18. V. L. Zimont, E. A. Meshcheryakov, and V. A. Sabel'nikov, Fiz. Goreniya Vzryva, 14,

No. 3 (1978). 19. K. I. Shchelkin and Ya. K. Troshin, Gasdynamics of Combustion [in Russian], Izd. Akad.

Nauk SSSR, Moscow (1963). 20. E. S. Shchetinkov, Physics of Gas Combustion [in Russian], Nauka, Moscow (1965).

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21. Ya. B. Zel'dovich, G. I. Barenblatt, V. B. Librovich, et al., Mathematical Theory of Combustion and Explosion [in Russian], Nauka, Moscow (1980).

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Nauk SSSR, Moscow (1960). 24. V. R. Kuznetsov, Trans. Second All-Union Symposium on Combustion and Explosion [in

Russian], Chernogolovka (1969). 25. V. S. Baushev and V. N. Vilyunov, Zh. Prikl. Mekh. Tekh. Fiz., No. 3 (1972). 26. K. N. C. Bray, Seventeenth Symposium (Int.) on Combustion, Pittsburgh (1979). 27. V. N. Vilyunov, Fiz. Goreniya Vzryva, ii, No. 1 (1975). 28. V. R. Kuznetsov, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 5 (1976). 29. V. I. Zimont and V. A. Sabel'nikov, All-Union School Conference on Combustion Theory

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Izd. Akad. Nauk SSSR, Moscow (1959).

THEORY OF COMBUSTION OF A LIQUID WITH A FREE SURFACE.

IV. NONSTEADY-STATE LIQUID COMBUSTION

S. N. Mil'kov, G. S. Sukhov, and L. P. Yarin

It was noted in [I] that effects caused by the nonsteady-state nature of the combustion of a finite layer of liquid are manifested more markedly the smaller the ratio between the liquid layer thickness 6 and the thickness of the thermal reaction zone L r. This makes it necessary to consider nonsteady-state phenomena when considering the final stages of burnup of thick liquid layers (6/L r >> I) and also in the combustion of liquid films (6/L r << I). Below we will study nonsteady-state combustion of thin liquid layers with a free surface (Fig. i) on the basis of the models developed in [I, 2].

In the case where 6/L r << i, and, correspondingly, L/6 >> 1 (where L is the characteristic dimension of the free surface), temperature field inhomogeneity in the transverse direction may be neglected. This permits use of a systean of quasi-one-dimensional transport equations [i-3]. Considering that the characteristic time for completion of the pro- cess in the gas phase is significantly shorter than that in the liquid phase, we write these equations (in a coordinate systean fixed to the free surface) in the following form%:

OTg 8T 2 OZT~ ~%~ -~ + p,c~u, j-~ = ;~, 7~" ~ (2)

or 3 a% a'% (3) Ps%8 "~ + pscpsu, ~ = ;~, aS"

The s o l u t i o n o f sys tem ( 1 ) - ( 3 ) must s a t i s f y the fo l l owing c o n d i t i o n s on the " e x t e r n a l " boundary of the r eg ion occupied by the gas phase ( s ee Fig. 1) and the f r e e s u r f a c e :

~-+ + ~: TI-+TI., cb-+0, ea-+c=., (4)

~-~f: T, = Tf~, ca = ca = 0~ (5)

%In writing system (1)-(3) and below (with the exception of cases which will be noted spe- cially), we employ the same notation as in [1-3].

Leningrad. Translated from Fizika Goreniya i Vzryva, Vol. 21, No. 4, pp. 39-44, July- August, 1985. Original article submitted March 26, 1984; revision submitted August 13, 1984.

416 0010-5082/85/2104-0416509.50 O 1986 Plenum Publishing Corporation

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Combustion zone propagation in a turbulent medium