THE FLAME PROCESS OF GAS COMBUSTION UNDER CONDITIONS OF A FREE TURBULENT JET

N. N. NORKIN

The development of the flame process of combustion of a gas and of a combustible gas mixture as they issue into the surrounding motionless air depends on the interaction of kinetic physicochemical and hydrodynamic phenomena. A macroscopic study of the one-sided influence of factors of any one category does not make it possible to reveal the characteristics of the flame process, and the attempts at generalization undertaken in this way lead to abstract schemes of processes that are not realized in practical experience.

The first comprehensive scientific study of the flame process of gas combustion was carried out by the outstanding Russian physicist Vladimir Aleksandrovich Mikhelson in 1888; he developed the theoretical foundations of the dynamic method for determining the velocity of flame propagation [1]. V. A. Mikhelson was the creator of a new chapter in science—the “chapter of combustion physics” (A. S. Predvoditelev) [2]. He established the combined influence of chemical kinetics (the velocity of flame propagation) and hydrodynamic kinetics (the velocity of outflow of the combustible mixture) in the form of the equation

\[ \frac{l}{d}=0.5\sqrt{\left(\frac{W}{U}\right)^2-1}, \tag{1} \]

where
\(l\) — length of the kinetic flame (or of its kinetic part),
\(d\) — burner diameter,
\(W\) — initial mean cross-sectional outflow velocity,
\(U\) — flame propagation velocity.

The present work is devoted to the development of V. A. Mikhelson’s ideas in the direction of practical application.

The simplest convective-diffusion burning flame

The simplest convective-diffusion burning flame under conditions of a free turbulent jet is a flame formed during the combustion of a gas jet flowing from a burner into the atmosphere. There is no primary air in the jet (the coefficient of excess primary air in the jet \(\alpha_{\mathrm{p}}=0\)); combustion occurs on the surface of the gas jet by means of convection and diffusion of air from the surrounding medium. As experience shows, the length of the burning flame depends on the nature of the gas, the outflow velocity, and the burner diameter.

In Fig. 1, for burners of diameter \(2.32\ \mathrm{mm}\) and \(7.6\ \mathrm{mm}\), experimental curves are shown that we obtained in the investigation of the combustion of water gas of the following composition: \(\mathrm{CO_2}\)—4.2%, \(\mathrm{CO}\)—29.4%, \(\mathrm{H_2}\)—52.3%, \(\mathrm{CH_4}\)—6.4%, \(\mathrm{O_2}\)—0.2%, and \(\mathrm{N_2}\)—7.5%. In the experiments the outflow velocity of water gas varied from

\[ W=0.6\ \mathrm{m/sec}\quad \text{to}\quad W=9.2\ \mathrm{m/sec}. \]

In this same figure, for comparison with the experimental data, straight lines are shown, parallel to the velocity axes, calculated from equations derived from the theory of a free turbulent “cold” (non-burning) jet:

Fig. 1. Dependence of the relative length of the burning flame on the nozzle diameter and the efflux velocity

according to the equation of G. N. Abramovich [3]

\[ \frac{V}{V_0}=2.18\left(2a\frac{l}{d}+0.29\right), \tag{2} \]

according to the equation of G. P. Ivantsov [4]

\[ \frac{V}{V_0}=0.82\left(0.32\frac{l}{d}+1\right), \tag{3} \]

according to the equation of D. N. Lyakhovsky and S. N. Syrkin [5]

\[ \frac{V}{V_0}=1+0.207\frac{l}{d}. \tag{4} \]

In these equations: \(a\) is the coefficient of jet structure,
\(V\) — m\(^3\)/sec, the volume of mixture in the jet at a distance \(l\) from the nozzle,
\(V_0\) — m\(^3\)/sec, the initial volume of gas in the jet.

The initial direction of the jet has a considerable influence on the development of combustion processes under the conditions of the simplest flame.

In Fig. 2 experimental curves are shown characterizing the change in the length of the burning flame of water gas for vertical, horizontal, and inclined directions of the jet.

At identical efflux velocities the vertical flame is the longest, the inclined flame the shortest; the length of the horizontal flame differs little from that inclined at an angle of \(20^\circ\). The ratio of the lengths of the vertical and horizontal flames is expressed approximately by the relation

\[ \left(\frac{l}{d}\right)_{\text{vertical}} =1.31\left(\frac{l}{d}\right)_{\text{horizontal}}. \tag{5} \]

The influence of the direction of the jet on the length of a free burning flame is explained by the different conditions of mixing of the gas with air.

For the vertical direction of a free flame without primary air, the conditions for the inflow of air from the surrounding medium to the front

flame are the least favorable, since around the flame front there is a layer of combustion products which hinders the diffusion of oxygen from the air; the supply of oxygen to the flame front is most accessible at the burner mouth. Owing to the difference in aerostatic pressures at the burner mouth, between the combustion products and the outside air, the latter, as it were, “washes” the combustion products from the surface of the flame front along the height of the torch.

With a horizontal direction of the torch, owing to the difference in aerostatic pressures, air flows to the surface of the front of the burning torch along its entire length; consequently, the conditions for access of air are more favorable than in the vertical arrangement, and therefore a horizontally directed torch is shorter than a vertical one. At a small angle of inclination of the torch axis, the conditions of access

Figure text: ordinate: “Relative torch length \(l/d\)”; abscissa: “Outflow velocity \(W\), m/sec.” Legend: “Water gas; \(t = 0^\circ\); vertical torch; horizontal; inclined \(< 20^\circ\).”

Fig. 2. Influence of the direction of the jet on the length of the burning torch

of air to the flame front remain more or less similar to those in the case of the horizontal direction; therefore the torch length is shortened only slightly in comparison with the horizontal torch. But as the angle of inclination increases, the accessible surface increases, and at the limit, when the angle of inclination is equal to \(90^\circ\), i.e., when the torch is directed vertically downward, the inflow of air will be facilitated over the entire surface of the burning torch, as a result of which the vertically downward torch will be the shortest, as is also confirmed by experiment.

Certain generalized relations for the length of the simplest torch were obtained by B. I. Kitaev and P. V. Levchenko [6] as a result of an experimental investigation of gas combustion without primary air. They established: 1) that the transition of the flow from laminar to turbulent conditions under torch combustion is not characterized by the value \(Re_{\text{crit}} = 2320\), calculated from the parameters of the nonburning jet in the nozzle, and, consequently, does not depend on one or another regime in the nozzle; 2) that in a calmly burning torch without eddies (laminar regime) the torch length varies proportionally to the gas outflow velocity; whereas in a vortex (turbulent) torch the torch length for nozzles of large diameters increases with increasing outflow velocity and is almost independent of the velocity for nozzles of small diameters. The authors proposed a generalized empirical equation for calculating the length of a freely burning torch in an atmosphere of air:

\[ l_{\phi} = (13.5 \div 14.0) K W^{0.34} d^{0.83}; \tag{6} \]

Here, for coke-oven gas the value \(K=1\), and for peat-generator gas \(K=0.65\). Figure 3 shows curves indicating the dependence of the relative length of the burning flame of water gas (without primary air) according to the equation of B. I. Kitaev and P. V. Levchenko and according to our experimental data [7]; in calculating \(l_f\) by equation (6), the value \(K=0.8\) was adopted for water gas. From comparison of the curves it is evident that, at low outflow velocities, there is a considerable discrepancy between the calculated values and the experimental ones for water gas, but as the outflow velocity increases these discrepancies diminish. For flames with primary air in the combustible mixture, equation (6) is not applicable.

Fig. 3. Dependence of the relative length of a burning flame on the nozzle diameter and the gas outflow velocity

Complex diffusion-kinetic burning flame

A complex diffusion-kinetic flame is formed in the combustion of combustible gas mixtures containing primary air in an amount insufficient for complete combustion; under these conditions two combustion fronts are formed—on the surface of the inner kinetic front, gas burns in an amount equivalent to the content of primary air, and the primary air is entirely consumed; on the surface of the outer diffusion front, the remaining part of the gas burns with secondary air diffusing from the surrounding medium. This is how partially prepared combustible mixtures burn, in contrast to the combustion of completely prepared combustible mixtures, for which \(\alpha_{\mathrm{p}}>1\) and the outer combustion front is absent, and also in contrast to the combustion of an issuing gas without primary air, when the inner combustion front is absent.

In studying the combustion of partially prepared combustible mixtures, the following questions arise: a) what factors influence the development of the inner kinetic flame; b) how does the kinetic flame influence the development of the outer diffusion flame; and c) what is the process of burnout of the constituents of the combustible mixture? The investigations carried out by us in this direction make it possible to draw certain general conclusions.

Repeated unsuccessful attempts have been made to extend to a freely burning gas flame the generalized dependences of the theory of a free turbulent jet [8] or other, purely hydrodynamic propositions [4; 5]. However, practical experience shows that these theories do not reflect the processes of flame combustion and, as in the case of the simplest flame, cannot be used for a quantitative assessment.

because they lack the principal kinetic quantities characterizing the process of flare combustion.

In the theory of a free turbulent jet by G. N. Abramovich [8], two variants of equations were developed for calculating the length of a burning flare. According to the first variant of 1948, the relative length of the burning flare is determined from the equation

\[ \frac{l_{\phi}}{d_0}=0.35\,\frac{K_0}{aK_{kp}} \tag{7} \]

and according to the second variant of 1951—from the equation

\[ \frac{l_{\phi}}{d_0}=0.35\,\frac{K_0}{aK_{kp}}\sqrt{\frac{1+K_0}{1+0.77K_{kp}}}. \tag{8} \]

In these equations: \(K_0\)—the initial concentration of fuel in the mixture

\[ \frac{\text{weight of fuel}}{\text{weight of air}}; \]

\(K_{kp}\)—the final concentration of fuel in the mixture according to stoichiometric calculations at \(\alpha_n=1\)

\[ \frac{\text{weight of fuel}}{\text{weight of air}}; \]

\(a\)—the coefficient of jet structure: for a burner of circular cross section \(a=0.076\).

If we take the example given by G. N. Abramovich [8] for an oil flare, and assume that the initial concentration \(K_0=\frac{1}{1.5}=0.66\) and the final concentration \(K_{kp}=\frac{1}{15}=0.066\), then from equation (7) we obtain

\[ \left(\frac{l_{\phi}}{d_0}\right)_{\mathrm{I}}=34.7 \]

and from equation (8)

\[ \left(\frac{l_{\phi}}{d_0}\right)_{\mathrm{II}}=43.6. \]

The discrepancy is

\[ \frac{(l_{\phi})_{\mathrm{II}}}{(l_{\phi})_{\mathrm{I}}} = \frac{43.6}{34.7} = 1.25. \]

With an increase in the initial concentration of fuel in the mixture, for example, to \(K_0=\frac{1}{1}=1\), the discrepancy increases to

\[ \frac{(l_{\phi})_{\mathrm{II}}}{(l_{\phi})_{\mathrm{I}}} = \frac{72.5}{52.5} = 1.37. \]

We investigated the combustion of combustible mixtures of water gas–air; the gas content in the combustible mixtures varied from 15.8 to 73.2%, which corresponded to the fraction of primary air in the mixture from \(\alpha_n=0.14\) to \(\alpha_n=2.13\). The efflux velocities of the combustible mixtures varied from \(1.58\ \text{m/sec}\) to \(123.3\ \text{m/sec}\); the values of \(Re\) for the flow conditions in the nozzle varied from \(Re=738\) to \(Re=9260\). The nozzle diameters were: 0.8—1.18—1.25—1.54—2.32—2.9—3.7—5—8 mm.

In Fig. 4 are shown: the experimental curve showing the influence of the efflux velocity of a combustible mixture of composition \(\alpha_n=0.31\) on the length of the burning fla-

body; nozzle diameter 2.32 mm; in the experiments the velocity varied from \(W=5.26\ \text{m/sec}\) to \(W=11.85\ \text{m/sec}\). The relative length of the flame in this case ranged from \(\dfrac{l_{\phi}}{d_0}=21.7\) to \(\dfrac{l_{\phi}}{d_0}=49\); the straight line constructed from equation (8) of the theory of the free turbulent jet does not depend on the outflow velocity and is equal to \(\dfrac{l_{\phi}}{d_0}=27.4\). In constructing the straight line according to equation (8), the following were calculated: \(K_0=0.627\), \(K_{\text{cr}}=0.127\). As is seen from Fig. 4, agreement of the length of the experimental flame with the calculated length was obtained only at the point of intersection of the curve and the straight line, which corresponds to an outflow velocity \(W=7.2\ \text{m/sec}\).

Fig. 4. Dependence of the relative length of the burning flame on the outflow velocity of the combustible mixture

In Fig. 5 the experimental points of the relative flame lengths are plotted for \(\alpha=1\) for burners of diameter 3.7 mm at \(W=8.5\ \text{m/sec}\) and \(W=9.6\ \text{m/sec}\), of diameter 2.9 mm at \(W=14.9\ \text{m/sec}\) and \(W=16.2\ \text{m/sec}\), and of diameter 1.54 mm at \(W=48\ \text{m/sec}\) and \(W=51\ \text{m/sec}\). These experimental points show—

Fig. 5. Dependence of the relative length of the burning flame on the nozzle diameter and the outflow velocity of the combustible mixture

—that the relative length of the kinetic flame depends both on the burner diameter and on the outflow velocity. The straight line, however, constructed according to equation (8), represents the flame length \(\dfrac{l_{\phi}}{d_0}=7.8=\text{const}\), independent of both the burner diameter and the outflow velocity. In constructing the straight line

according to equation (8) it was calculated: \(K_0=0.194\) for \(\alpha_p=1\) and \(K_{cr}=0.127\) for \(\alpha_p=1.52\).

Thus, experiment shows that both in the case of the simplest and in the case of a complex diffusion-kinetic burning flame, the theory of the free turbulent jet and the analogous hydrodynamic propositions associated with it cannot serve as a measure of qualitative and quantitative relations for a burning flame.

Generalized relationships of a complex diffusion-kinetic burning flame

Length of the internal kinetic flame

The length of the internal kinetic flame for combustible mixtures containing air in an amount \(\alpha_p<1\), or the total flame length for \(\alpha_p>1\), is uniquely determined by the ratio \(\dfrac{W}{U}\), as follows from equation (1) of V. A. Mikhelson,

\[ \frac{l}{d}=0.5\,\frac{W}{U}, \tag{9} \]

since

\[ \left(\frac{W}{U}\right)^2 \gg 1. \]

We consider it expedient to call the ratio \(\dfrac{W}{U}\) the criterion determining the chemical and hydrodynamic kinetics of the process of flame combustion, and to denote this criterion, in honor of V. A. Mikhelson, by

\[ Mi=\frac{W}{U}. \tag{10} \]

In Fig. 6 a straight line is shown that characterizes the dependence of the relative length of the kinetic flame \(\dfrac{l}{d}\) on the criterion \(Mi\), verified experimentally for combustible mixtures of water gas with air over a wide range of variation in the composition of the combustible mixtures and for different burner diameters.

Using V. A. Mikhelson’s method in processing the results of experimental investigations, we establish the dependence between the height (length) of the internal kinetic flame \(l\), the velocity of discharge of the combustible mixture from the burner \(W\), the burner diameter \(d\), and the kinematic viscosity of the combustible mixture \(\nu\) in the form of the equation

\[ l=K W^{0.5} d^{0.15} \nu^{0.5}. \tag{11} \]

Here \(K\) is an empirical coefficient depending on the fraction of primary air in the combustible mixture, \(K=f(\alpha_p)\).

The total length of the external diffusion flame depends on the length of the internal kinetic flame. At \(\alpha_p=0\), the flame becomes diffusion-controlled, and its length \(L\) is greatest; when primary air is introduced into the mixture, when \(\alpha_p>0\), the lengths of the external and internal flames decrease, but the internal flame decreases considerably faster than the external one; at \(\alpha_p=0.5\) a minimum value of \(\dfrac{l}{L}\) is established. With a further increase in \(\alpha_p\), further-

the reduction of \(l\) and \(L\), but the length of the inner kinetic flame decreases more slowly than that of the outer one, as a result of which the ratio \(\frac{l}{L}\) increases. When \(\alpha_{\mathrm{p}}\) approaches unity, the inner and outer flames merge.

Fig. 6. Dependence of the relative length of the kinetic flame \(\frac{l}{d}\) on the criterion \(Mi\) for the vertical direction of the flame

At \(\alpha_{\mathrm{p}} = 0.5\), the total length of the flame is 2.5 times greater than the height of the inner kinetic flame.

Pyrometric characteristics of the kinetic flame

The temperature of the inner kinetic flame over a wide range of values of the coefficient \(\alpha_{\mathrm{p}}\) is almost uniquely determined by the ratio of the heat density of combustion \(q\), kcal/m\(^2\)·sec, to the flame-propagation velocity \(U\), in the form of the following empirical equation

\[ t = (1.75 \div 2.2)\frac{q}{U}. \tag{12} \]

For combustible mixtures with \(\alpha_{\mathrm{p}}\) from 0.6 to 1.0, the numerical factor is an almost constant quantity, equal to 1.8. This means that as the intensity of the combustion process increases, the pyrometric combustion coefficient \(\eta_{\mathrm{pir}}\) increases, since the following relation holds:

\[ \frac{q}{U} = C_{\mathrm{pr.g}} = V_{\mathrm{pr.g}}\frac{t}{\eta_{\mathrm{pir}}}. \tag{13} \]

Here: \(C_{\mathrm{pr.g}}\), kcal/m\(^3\)·deg—the heat capacity of the combustion products,
\(V_{\mathrm{pr.g}}\), m\(^3\)/m\(^3\)—the volume of the combustion products,
\(\eta_{\mathrm{pir}}\)—the pyrometric combustion coefficient, representing the ratio of the measured temperature \(t\) to the calorimetric temperature \(t_{\mathrm{kal}}\).

The volumetric thermal intensity of the kinetic flame reaches values on the order of \(2.8 \cdot 10^{10}\) kcal/m\(^3\)·hr; this indicates that kinetic flame combustion possesses greater thermal intensity than so-called “flameless” combustion.

Order of Burnout of the Components of a Combustible Mixture

To solve the question of the order of burnout of the constituent parts of a combustible mixture, the method of splitting the combustion flame into an inner and an outer flame was applied, followed by chemical analysis of the composition of the interfront gases. As a result it was established: a) that during the combustion of water gas containing $\mathrm{H_2}$, CO, and $\mathrm{CH_4}$, burnout of the components occurs not selectively and successively, but simultaneously, though with different intensities; b) the intensity of burnout of the components changes under the influence of the amount of primary air in the combustible mixture—at small values of $\alpha_{\mathrm{p}}$ the greatest burnout intensity is possessed by $\mathrm{H_2}$; at values $\alpha_{\mathrm{p}}>0.5$ the burnout intensity of hydrogen decreases. Under the catalytic action of platinum on the combustion process, at any amount of primary air in the combustible mixture, the burnout intensity of $\mathrm{H_2}$ is the greatest; c) free oxygen was not detected in the interfront gases; this confirms the circumstance that all oxygen is consumed in the inner kinetic flame.

Kinetic Significance of the Michelson Criterion

The criterion $Mi$ is a measure of the ratio of chemical and hydrodynamic kinetics in flame combustion. The criterion $Mi$ characterizes the stability of the flame process—at $Mi<1$ the flame process, being unstable, is not stabilized. The critical value is $Mi_{\mathrm{cr}}=1$.

In homogeneous-heterogeneous combustion on an incandescent surface (thin channels, a granular bed, porous bodies), as experiment shows, a stable combustion process is possible at a value of $Mi$ approaching unity. The upper limit of stability of the flame-combustion process in experiments with water gas was at $Mi=70.5$.

The relative length of the kinetic flame of combustible mixtures of a given composition is uniquely determined by the criterion $Mi$, and with variable composition of combustible mixtures is a function of this criterion and of the coefficient of excess primary air.

The criterion $Mi$ resolves the question of the possibility of modeling the kinetic combustion of gases differing in composition and nature and expands the field of application of flame models.

Conclusion

1. The theory of the free turbulent jet, or other purely hydrodynamic propositions, do not reflect the essence of the processes of flame combustion of gas under conditions of a free jet, since the basic kinetic quantities are absent from them.

2. To characterize the flame process of gas combustion under conditions of a free jet, a new Michelson criterion is proposed, taking into account the influence of chemical and hydrodynamic kinetics.

3. With the aid of the Michelson criterion, a number of generalized relations of flame combustion have been established in experiments on the burning of water gas.

CITED LITERATURE

1. Mikhelson V. A. Collected Works, vol. 1, 1930.

2. Essays on the History of Physics in Russia. Edited by A. K. Timiryazev, 1949.

3. Abramovich G. N. Proceedings of TsAGI, 293, 1936.

4. Ivantsov G. P. Collection “Problems of Gas Motion in Furnaces,” 1936.

5. Lyakhovskii D. N. and Syrkin S. N. Soviet Boiler and Turbine Construction, 2, 1938.

6. Kitaev B. I. and Chenko P. V. Heat Exchange and Questions of Fuel Economy in Metallurgical Furnaces. Metallurgizdat, 1951.

7. Norkin N. N. Izv. Tomsk Polytechnic Institute, vol. 68, no. 1, 1951.

8. Abramovich G. N. Turbulent Free Jets of Liquids and Gases, 1948. Applied Gas Dynamics, 1951.