INDUSTRIAL AND ENGINEERING CHEMISTRY
April 1955
preach t o the carbon hydrogen equilibrium. Under this assump-
where K is t h e equilibrium constant for the reaction C (graphite) +2H2 = CHI and is expressed as (PH,)~/PcH,. Ro is t h e differential reaction rate for gasification with pure hydrogen. T h e experimental points are fairly close to those predicted b y the equation. Obviously, therefore, methane is not an inhibitor in the usual sense. Role of Diffusion. It is likely that the experimentally observed rates are characterized by the chemical reaction rate between the internal carbon surface and the hydrogen and are not significantly effected by slow diffusion into the interior of the particle. T h e argument for.this conclusion is based on two sets of experimental facts-namely, the effect of particle size and the observed temperature coefficient of the reaction. If diffusion into the interior of the particle were of importance in the over-all rate, the specific reaction rate would decrease with increasing particle size. However, results shown in Table IV do not bear this out. Dent also observed no effect ,of particle size or rate. On the second point, both Thiele (8) and Wheeler (9) considered theoretically the role that diffusion into the pores of catalysts plays in the kinetics of heterogeneous reactions. Let us consider a heterogeneous reaction whose chemical kinetics can be described b y the relationship
R
825
It is possible to show quite generally on the basis of the treatment given b y these authors that the measured activation energy in the diffusion controlled range would be given b y E / 2 while in the range where chemical reaction rate is controlling, the activation energy would be given b y E. E is, of course, the activation energy corresponding to the rate constant, k. Therefore, on a plot of the log of the reaction rate as a function of the reciprocal temperature, the slope presumably would decrease in a temperature range where the rate of diffusion begins to play a role in determining the over-all reaction rate. The fact t h a t the slope in Figure 7 is constant over the temperature range 1500” t o 1700” F. strongly suggests that diffusion is not an important factor over this range. LITERATURE CITED
Barthauer, G. L., Haggerty, Alice, and Friedrich, R. J., Anal. Chem., 25,256 (1953). Dent, F. J., Gas Research Board Communication, G.R.B. 13/3, 1950. Dent, F J., Blackburn, W. H., and Millet, H. C., Joint Research Committee of Institution of Gas Engineers and University of Leeds, 41st Rept., 1937. Ibid., 43rd Rept., 1938. Goring, G. E., Curran, G. P., Tarbox, R. P., and Gorin, E., IND. ENG.CHEM.,44,1051 (1952). Ibid., p. 1057.
Goring, G. E., Curran, G. P., Zielke, C. W., and Gorin, E., Ibid., 4 5 , 2 5 8 6 (1953).
Thiele, E. W., Ibid., 31, 916 (1939). Wheeler, A., in “Advances in Catalysis,” Coll. Vol. 111, Academic Press, New York, 1951.
p. 249,
kf(PH1)
where ~ ( P H is ~ any ) arbitrary function of the hydrogen pressure that does not contain temperature sensitive parameters.
RECEIVED for review May 1, 1954. ACCEPTED November 1, 1954, Presented at the 126th Meeting, ACS, Kansas City, Mo., March-April 1954.
Burning Velocity of Unconfined Turbulent Flames THEORY OF TURBULENT BURNING VELOCITY KURT WOHL D e p a r t m e n t of Chemical Engineering, University of Delaware, Newark, Del.
T
HE theories of the burning velocity of turbulent flames which have recently been advanced b y Karlovitz, Denniston, and Wells ( 1 ) and Scurlock and Grover (3, 4)represent a great step forward, though many problems are still unsolved. These theories are restricted t o continuous flame fronts distorted b y turbulence, the scale of which is large compared with the thickness of the laminar flame front. They do not apply to the disruption of the flame front which is realized in highly turbulent flames at a large distance from the flame base (6, 8). Either theory deals mainly with two extreme cases of flame front distortions: I n the one case the approach stream turbulence is not augmented in the flame front, so that the flame front may be said t o be “passive,” or to respond passively to the approach stream turbulence. I n the other case all the available kinetic energy of the expanded combustion products is utilized for augmenting the flame front distortions. An empirical judgment as t o the actual amount of “flame-generated turbulence” depends on a reliable equation for the turbulent burning velocity of a passive flame front.
PASSIVE FLAME FRONT
The concept of turbulent burning velocity pertains to a smooth area which is supposed to represent the average position of the turbulent flame front. This description of t h e flame has t o be supplemented by information on the thickness of the flame brush which, for strongly turbulent flames, may acquire a higher technical importance than burning velocity (8). I n the theory of Karlovitz and coworkers it is assumed that the flame front elements undergo Lagrangian fluctuations for a period 7
12 =-
so
where 12 is the Eulerian scale and SOis normal burning velocity, and that after this period the fluctuations start afresh from zero. When this assumption is adopted, the equation of Karlovita and coworkers for the turbulent burning velocity can be derived in a manner different from that used by the authors themselvesnamely, from the viewpoint of an increase of flame front area
INDUSTRIAL AND ENGINEERING CHEMISTRY
826
(8). If the flame front distortions are assumed t o be of rectangular cross section, one obtains
Vol. 47, No. 4
values of Y/lz. The advantage gained by this substitution is that the equation can now easily be integrated in a closed form, so that the doubtful mathemat,ical treatment described above can be avoided. The result is
N
where S t is turbulent burning velocity, h is mean amplitude, N
d is mean base width of flame front distortions, Y is root mean square displacement of a flame front element, and K1 and K Zare proportionality coefficients. Introduction of the expression of
%]
N
Karlovitz and coworkers for Y yields
-
tl
For very small values of t there follows where 11 is Lagrangian scale of turbulence and V I is intensity of approach stream turbulence (isotropy assumed). The equation of Karlovitz and coworkers follows with K I / K z = ‘ / z and 11 = 12. Scurlock and Grover in their equations assume K1/K2 = 1 and 11 = 12/2. It seems t o the author that K1/K2 = ‘/z and 11 = 1 2 / 2 are closest to the truth. Scurlock and Grover operate with a flame front model which has distortions of triangular cross sections-i.e., they use Shelkin’s equation
N
-
llVl
-
-
(U’t/Zl)
[l
+ (ZISO/lZU’)l
[I-e
(7)
(v’/So) KP2
y
dl
12
+
Z2V’/hSO
Values of St for the rectangular flame model are obtained by inserting Equation 6 into Equation 2. It is seen that, for small values of t, the expression ( S t / S o )- 1 is inversely proportional t o the Eulerian scale, while in the steady-state equation, just a s in the equation of Karlovitz and coworkers, the absolute value of the scale does not occur. The calculated values of St depend not only on the basic concepts but also on the choice of the geometry of flame front distortions and of the values of 11/12. At our present state of knowledge this choice is arbitrary, so that the theoretical results are somewhat uncertain. A numerical comparison of the steady-state equation derived from Scurlock and Grover’s theory t o the original equation of Karlovitz and coworkers and its modifications suggested here is given for a few cases in Table I. With the same suppositions, the two theories lead to similar results for the steady state of the passive flame front.
where KJK2 is made equal t o 1. An important contribution of Scurlock and Grover to theory is that they draw attention t o the fact that the limitation of fluctuations by a period T (Equation l) is not realistic. They consider the transition of the propagating flame front element from one eddy t o another as an event which leads to Eulerian fluctuations. Both fluctuations first increase with time but approach a finite value owing to the tendency of the distortions t o smooth out through the process of flame propagation (Karlovitz and coworkers). The general equation of Scurlock and Grover ( 3 , Equation 13b) can be written
dt
12
N
- =
(4)
N
= v’t -
For very large values of 1, Equation 6 yields the steady-state equation
f
E -
Y
12
-
1-
OBLIQUE PASSIVE FLAME FRONT
2K1 The first term is independent of the specific flame front model; the second, which takes care of flame front smoothing, corresponds t o the triangular model (Equation 4). Scurlock and Grover evaluate Equation 5 by first writing it without the second term. The incomplete equation can be N
integrated, and is used t o express 1 as a function of Y and the parameters of the equation. This expression is inserted in the complete equation in order t o evaluate the latter with respect
I n all practical cases the average front of a turbulent flame is obliquely inclined to the direction of flow. In the theory of Scurlock and Grover the time which is available for the development of distortions is determined by the accepted fact, explained by Markstein (a), that the distortions travel down the flame front with a velocity U1 cos p, so that the time t o be inserted into Equations 5 t o 7 becomes
N
to y. It seems that this procedure is not rigorous; it may be permissible, but this is difficult to judge. -1
The value finally obtained for Y / l z is inserted, of course, in Equation 4 for calculating St. A comparison of Equations 4 and 5 shows that the second term in Equation 5 can be written (St S0)/2Ki. If this term is combined with Equation 2 (which is based on a rectangular flame
I , Generalized Equation of Karlovitz and Coworkers, E q . 3a b C K I = I/zKz Ki K2 KI =
-
N
model), the simple linear expression SOY/ Kzl~is obtained for the second term on the right side of Equation 5. It shows the same trend as the corresponding, more complicated, term given ’in Equation 5 and has the same limit for high
Theoretical Values of ( S t / & ) - 1 for Steady State of Passive Flame Front
Table I.
9
11
-< V’ 80
>I so a
=
1/212
Z’ so
2 4 2
11
=
12
-
U’
SO
4GQ
11
= l/dZ
V’
SO
q;
11, Modified Equation of Scurlock and Grover, Eqs. 2 and 8 a b C K I = K I = 2 K I = 1; K2 = 2 K I = 1 ; Kz = 2 11
= 1/2h
2 4 / 2 2 2
&
11 = 12
5 U‘G
4 2
11
=
45
so
1/212
4;
Derived by flame front area concept. a. Suppositions of Scurlock and Grover. b . Suppositions with which Eq. 3 becomes identical with original equation of Karlovitz and coworkers. C . Suppositions considered most plausible.
April 1955
INDUSTRIAL AND ENG INEERING CHEMISTRY
827
semblance to instantaneous flame front fluctuations. It has been transferred by the authors to the flow components normal to an instantaneous oblique unconfined turbulent flame front. On this base, Scurlock and Grover propose the following equation for the maximum intensity of turbulence, vfm, created by an unconfined flame [4, Equation 3, with E,,, = ( 3 / 2 ) ( ~ ’ and )~ K, = PU/100 P a l
where pv equals density of the unburned gas and density of the burned gas.
pb
equals
The final value of turbulent intensity, vi, which is inserted into Equation 5 instead of v‘ is Figure 1. Theoretical values of burning velocity for passive flame fronts Scurlock and Grover, steady state (4, Figure 19) Scurlock and Grover, average,values a t qonditions of Williams and Bollinger’s experiments (4, Figure 21) 3. Modified equation of Scurlock and Grover, steady state (Equations 2 and 8, case IIa, Table I) 4. Original equation of Karlovitz and coworkers (Equation 3, case I b , Table I) 5. Equation of Karlovitz and coworkers with 21 = 1 / d z (Equation 3, case IC. Table I) 1. 2.
where Ut equals local flow velocity, p equals angle between flame front and vector U I , and L equals length of mean flame front. As a result, S t increases with increasing distance from the flame base. A number of St curves are given in Figure 1. The significance of most of them is sufficiently explained in the legend. Curve 2 shows burning velocities, averaged over the whole flame front for a time t of travel from base to tip, corresponding to Williams and Bollinger’s (6) experimental conditions. This curve practically coincides with curve 5, which represents the equation of Karlovitz with ll = 1/2 1 2 . Very close t o these two is the curve which corresponds to case I1 c of Table I (Equations 2 and 8 with Kl = 1, K 2 = 2, and 11 = 1/2 1 2 ) ; the two latter suppositions are considered the most plausible. It is concluded from this figure that, in view of the uncertainties of the time effect, it is reasonable t o use Karlovitz’s original equation (curve 4) for roughly estimating the average values which would be expected from passive flame fronts a t laboratory conditions. One question may be raised in connection with the oblique flame front. If Equation 9 is used for the time during which distortions grow under the cumulative action of approach stream turbulence, the action of the approach stream is not considered on one and the same material flame front element, but on the element of a traveling wave. It follows that the characteristic time, T , of the Eulerian fluctuations is no longer the time during which a flame element propagates through an eddy-this gave T = 1 1 / 8 0 according to Equation 1-but the time which it takes for the wave t o travel the distance of one eddy diameter. Since the relative velocity of the wave with respect to the mean flow is U1 sin p = St, it seems that the characteristic period should now be T = 12/St. Then S t instead of Sowould be written in the first term of Equation 5 . As a consequence, the steady state would be reached after a shorter time or length of travel.
st,
sl
FLAME-GENERATED TURBULENCE
Scurlock and Grover, like Karlovitz and coworkers, assume that the available kinetic energy of combustion gases behind the flame front can be converted into turbulence which distorts the flame front through back-action in the same manner as approach stream turbulence. Full utilization of the energy produces the maximum intensity of flame-generated turbulence, v ’ ~ . I n order t o estimate dm,Scurlock and Grover make use of a flame front and flow pattern which describe directly the average behavior of a flame burning in a duct of constant cross section from a set of flame holders which are arranged in a plane perpendicular to the direction of flow (3, 7). This pattern bears a remote re-
This procedure presupposes that the scale which is characteristic of flame-generated turbulence equals that of the approach stream. More is said on this point, and on Equation 11, below. Markstein (2) claims that the fluctuations of the approach stream are augmented in the flame front directly, without the intermediate stage of turbulence behind the flame front. According t o him, flame front distortions grow during their travel, quite apart from the effect of turbulence. At large distances from the flame base the turbulent flame front will be disrupted through excessive oscillations (8). These last remarks emphasize that a t present theory can serve only as a semiquantitative guide through experience. NOMENCLATURE
d = mean base width of flame front distortions h = mean amplitude of flame front distortions K1 = ratio of mean amplitude of flame front distortion to root mean square displacement of a flame front element KP = ratio of mean base width of flame front distortions to the Eulerian scale L = total length of mean flame front 11 = Lagrangian scale of turbulence 12 = Eulerian scale of turbulence SO = normal burning velocity St = turbulent burning velocity t = time U1 = local flow velocity = intensity of approach stream turbulence = maximum intensity of flame-created turbulence u t = total intensity of turbulence Y = root mean square displacement of a flame front element p = angle between flame front and vector U1 pb = density of burned gas pu = density of unburned gas T = time of travel of a combustion wave through a “turbulent eddy”
ii N
LITERATURE CITED (1) Karlovitz, B., Denniston, D. W., Jr., and Wells, F. E., J. Chem. Phys., 19, 541-7 (1951). (2) Markstein, G. H., “Selected Combustion Problems,” Combustion Colloquium, Cambridge, England, AGARD, NATO, pp. 263-5, Butterworths Scientific Publications, London, 1954. (3) Scurlock, A. C., and Grover, J. H., Fourth Symposium on Combustion, pp. 645-58, Williams & Wilkins, Baltimore, 1953. (4) Scurlock, A . C., and Grover, J. H., “Selected Combustion
Problems,” Combustion Colloquium, Cambridge, England, AGARD, NATO, pp. 215-47, Butterworths Scientific Publications, London, 1954. ( 5 ) Shelkin, K. I., 2. tech. Phys., Moskow, 13, 520-30 (1943); Natl. Advisory Comm. Aeronaut., Tech. Memo. 1110 (1947) (English tr.). (6) Williams, D. T., and Bollinger, L.
RI., Third Symposium on Combustion, pp. 176-85, Williams & Wilkins, Baltimore, 1949. (7) Williams, G . C., Hottel, H. C., and Scurlock, A. C., Third Symposium on Combustion, pp. 21-40, Williams & Wilkins, Baltimore, 1949. (8) Wohl, K., Shore, L., von Rosenberg, H., and Weil, C. W., Fourth Symposium on Combustion, pp. 620-35, Williams & Wilkins, Baltimore, 1953.
