Effect of Axis Curvature and Wall-Hugging of a Burning Gas Flame on Its Range

N. N. Norkin

I.

The founder of the hydraulic theory of flame furnaces, V. E. Grum-Grzhimailo, was the first to draw attention to the essential significance of the phenomenon of wall-hugging of a flame in furnaces. Describing in his work Flame Furnaces [1] the experiments of engineer V. A. Slesarev on the extinction of an air jet from a fan at a distance of 4 m (in air) and more than 10 m (the jet directed along a wall), V. E. Grum-Grzhimailo explains the phenomenon of wall-hugging of the jet in the following words: “The forward velocity of the jet creates, in the motionless air, a series of vortex rings, slowly moving in the direction of the jet, with their diameter continually increasing, and in this way it is rapidly extinguished. If the jet from the fan is directed along a wall, then it will produce vortices only on the air side; friction on the side of the smooth wall is negligible. The jet is pressed tightly against the wall, and the range of its flight increases. Moreover, the jet follows the bends of the wall without separating from it... The jet licks the wall.” Turning to the description of the phenomenon of wall-hugging of a flame in open-hearth furnaces, V. E. Grum-Grzhimailo writes: “Were it not for the wall-hugging of the flame, it would be impossible to heat the metal: the flame would tear away from its surface, and the bath would find itself in a sack of cold gases.”

A similar explanation of the phenomenon of wall-hugging of jets is given by G. P. Ivanov [2]: “A jet directed along a wall interacts with the surrounding gaseous medium over a considerably smaller perimeter, entrains a smaller amount of it, and for this reason preserves its velocities over a greater distance—its ‘range’ increases.”

M. A. Glinkov [3], without analyzing the reasons for the increase in the range of a jet during wall-hugging outflow, considers that in open-hearth furnaces the role of wall-hugging is especially important during two characteristic periods—in the period of fettling (charging) the hearth and in the period of melting after formation of the bath mirror.

No direct experiments on the study of wall-hugging of a burning gas flame have been described in the literature; likewise, there have been no attempts to establish quantitative relations characterizing the influence of wall-hugging of the flame. To fill this gap, we undertook an experimental investigation of the wall-hugging of a burning gas flame of water gas.

The phenomenon of wall-hugging of a burning gas flame is closely interwoven with the phenomena of nonisothermal outflow of a free jet and of curvature of the axis of a burning (nonisothermal) jet.

In industrial furnaces of the bath-type melting-furnace kind, an elongated gas flame, having a temperature higher than that of the combustion products filling the working space of the furnaces, is usually directed at a certain angle to the surface of the bath mirror in order to “wash” from the surface—

the surface the cooled combustion products and thereby improve the conditions of heat transfer from the flame flow to the bath mirror.

V. E. Grum-Grzhimailo was also the first to draw attention to the significance of the curvature of the axis of a nonisothermal flame, calling this phenomenon a gas fountain [1]. V. E. Grum-Grzhimailo wrote: “The bath of an open-hearth furnace is a sack from which the cooled gases have no natural outlet and therefore must stagnate. In order to dislodge them from the hearth, to mix the gas and air entering the working space of the open-hearth furnace with the incandescent gases of the flame, they are directed onto the hearth... For welding on the hearth it is necessary that the flame, directed by inclined ports, be able to touch its very lowest point. Hence follows the need first of all to determine to what depth below the normal level of the gas flow the jet of incandescent combustion products must be lowered.”

I. G. Esman [4], subordinating the laws of motion of a nonisothermal gas flame to the laws of hydrodynamics and assuming that the outflowing gas jet, under the influence of the force of gravity directed downward and the lifting force directed upward, moves along a parabola, derived a formula for the depth of immersion of the gas fountain (the vertex of the parabola)

\[ H=\frac{w_0^2 T_0}{2g\Delta T}\sin^2\alpha, \tag{1} \]

where

\[ \begin{aligned} H&\text{— m—depth of immersion of the gas fountain (the vertex of the parabola),}\\ w_0&\text{— m/sec—initial velocity of discharge of the gas flow,}\\ T_0&\text{— temperature of the stationary gaseous medium filling the furnace,}\\ \Delta T&=T_1-T_0\text{— temperature difference between the gas flow and the stationary medium at the beginning of discharge,}\\ \alpha&\text{— angle between the initial direction of the flow and the horizontal surface,}\\ g&\text{— }9.81\ \text{m/sec}^2. \end{aligned} \]

The scheme of the axis of a nonisothermal jet is shown in Fig. 1.

I. G. Esman, recommending equation (1) for practical calculations, points to the necessity of introducing a velocity coefficient depending on the shape of the opening. V. E. Grum-Grzhimailo introduces into I. G. Esman’s formula a coefficient that takes into account the influence of vortices, whose value, according to V. Slesarev’s experiments [5], is equal to 0.7 for an air flow. Consequently, the final form of the formula for the depth of a gas fountain according to Esman–Grum-Grzhimailo will be

\[ H=k\,\frac{w_0^2 T_0}{2g\Delta T}\sin^2\alpha \tag{2} \]

Fig. 1. Scheme of the trajectory of the axis of an inclined nonisothermal flame.

Following I. G. Esman’s method, the equation of the trajectory of the gas fountain is derived as

\[ y=\frac{g\Delta T}{2w_0^2 T\cos^2\alpha}\,x^2 \tag{3} \]

The length of the horizontal segment of the trajectory to the vertex of the gas fountain is equal to

\[ x=k\,\frac{w_0^2 T}{2g\,\Delta T}\,\sin 2\alpha . \tag{4} \]

If the initial direction of the gas stream is horizontal, i.e. \(\alpha=0\), then from the equation of the stream trajectory (3) it follows that

\[ y=\frac{g\,\Delta T}{2w_0^2 T}\,x^2 . \tag{5} \]

For convenience in comparing the Groume-Grzhimailo–Essmann formulas with the results of studies by other authors, let us reduce the above formulas to a dimensionless criterion form. The complex of quantities

\[ \frac{g\,\Delta T d}{T w_0^2} \]

is customarily called the Archimedes criterion; therefore, in the new form, the above equations will be: the trajectory of a stream issuing through an orifice of diameter \(d\) at an angle \(\alpha\) to the abscissa axis, from (3),

\[ \frac{y}{d}=\frac{1}{2}\,k\,Ar\left(\frac{x}{d\cos\alpha}\right)^2, \tag{6} \]

and, for a stream having a horizontal initial direction at \(\alpha=0\),

\[ \frac{y}{d}=\frac{1}{2}\,k\,Ar\left(\frac{x}{d}\right)^2, \tag{7} \]

the greatest depth of the gas fountain according to the scheme in Fig. 1 is

\[ \frac{H}{d}=\frac{1}{2}\,k\,\frac{\sin^2\alpha}{Ar}. \tag{8} \]

The horizontal distance to the point of greatest depth of the gas fountain according to the scheme in Fig. 1 is

\[ \frac{x}{d}=k\,\frac{\sin 2\alpha}{2Ar}. \tag{9} \]

For a vertical fountain directed downward at \(\alpha=90^\circ\),

\[ \frac{H}{d}=\frac{1}{2}\,\frac{k}{Ar}. \tag{10} \]

In the works of Groume-Grzhimailo and Essmann there are no experimental data that would confirm the possibility of confidently using their equations in studying the propagation of a burning gas flame; however, V. E. Groume-Grzhimailo repeatedly applies the equation for the greatest depth of a gas fountain in analyzing the operation of bath-type melting furnaces (in particular, open-hearth furnaces).

In 1936 G. N. Abramovich [6] developed the theory of a free turbulent jet and derived an equation for the aerodynamic axis of a jet having an initial horizontal direction and curved by gravitational forces during nonisothermal discharge of the jet into a medium of different density. Abramovich’s original equation has the form

\[ \frac{y}{d}=Ar\left(\frac{x}{d}\right)^2\left[\,1.48\,a\,\frac{x}{d}+0.32\,\right] \tag{11} \]

In this equation the quantity \(a\) is the coefficient of the structure of the jet, dependent on the shape of the cross section of the nozzle from which the jet issues. In particular, for round nozzles the value of \(a\) varies from \(a = 0.066\) to \(a = 0.076\), and on average \(a = 0.07\) is adopted; for rectangular nozzles \(a = 0.09 \div 0.12\). A summary of values of \(a\) was published in the article by I. A. Shepelev [7].

After D. N. Lyakhovskii and S. N. Syrkin [8, 9] published in 1938–1939 the results of extensive experimental studies they had carried out on the curvature of a flame discharging into a medium of another temperature, and pointed out the large discrepancy between the experimental data and Abramovich’s theory, owing to the latter’s neglect, in deriving the equations, of the vertical component of the drag force, G. N. Abramovich, in his work published in 1948 [10], gave a new equation for calculating the trajectory of a nonisothermal jet having an initial horizontal direction, in the form

\[ \frac{y}{d} = Ar \left( \frac{x}{d} \right)^2 \left[ 0.72a \frac{x}{d} + 0.31 \right]. \tag{12} \]

From a comparison of equations (11) and (12) it is seen that the trajectory of rise of the flame axis, calculated from equation (12), is less steep than that obtained from equation (11); G. N. Abramovich explains the inaccuracy of his first equation (11) by the fact that, in deriving equation (11), it was assumed that the lifting Archimedean force was expended entirely on imparting vertical acceleration, and the expenditure of part of it on imparting vertical velocity to the gas entrained by the flame from the surrounding medium was not taken into account. In equation (12) this is taken into account. G. N. Abramovich compared the calculated data according to equation (12) with the experimental data of Lyakhovskii and Syrkin and came to the conclusion that the equation (12) proposed by him fully satisfies the experimental data.

For the convenience of comparing experimental data with calculations, I. A. Shepelev proposed processing the results of experimental data in the form of a generalized curve in coordinates—the abscissa axis \(x/d\) and the ordinate axis \(\dfrac{y}{d} \cdot \dfrac{1}{Ar}\).

As already mentioned, D. N. Lyakhovskii and S. N. Syrkin carried out an extensive experimental investigation of the propagation of a nonisothermal jet [8, 9]. They investigated 22 jets, measured more than 100 temperature and velocity fields, and in all made more than 2000 measurements, which were painstakingly processed. The experiments were performed with nozzles of diameter 50, 100, 150, and 200 mm; the temperature of the issuing air flame reached \(270^\circ\text{C}\); the discharge velocity varied from \(1.37\ \text{m/sec}\) to \(6.4\ \text{m/sec}\), and the value of the Archimedes criterion from 0.0055 to 0.275. Lyakhovskii and Syrkin processed the results of their studies in the form of curves in the coordinates \(x/d\), \(y/d\) for different values of \(Ar\), but they did not propose a generalized equation, explaining that for an analytical solution of the problem of nonisothermal discharge there is insufficient information on the drag force arising during the rise of the flame, and that a more fruitful method of solving such a problem is experimental investigation. In processing experimental data by the methods of similarity theory, they recommend as the governing criterion the Archimedes criterion, characterizing the ratio between the lifting force per unit volume \(g(\rho_0 - \rho)\) and the inertial force proportional to the quantity \(\rho \dfrac{w_0^2}{d}\). As a practical example of the application of the experimental ...

dependences, they solve the problem—finding the downward deflection of the axis of a jet of air issuing with an initial temperature of \(100^\circ\text{C}\) through a rectangular opening \(1.2 \times 0.8\ \text{m}\) at a velocity \(w_0 = 5.1\ \text{m/sec}\). With regard to this example it must be noted that the following circumstances raise considerable doubt: a) the example is solved for a jet of rectangular cross section, while experimental data for jets of circular cross section are used, although the structure coefficients of the jets are different in the two cases; b) the dimensions of the opening in the example differ substantially from the dimensions of the nozzles investigated. With regard to the curvature of a nonisothermal plume having an initially inclined direction, Lyakhovskii and Syrkin believe that the experimental data obtained by them can, with a known approximation, also be used for nozzles only slightly inclined to the horizon, if \(\dfrac{y}{d}\) is measured from the inclined geometric axis of the plume. For nozzles strongly inclined, at an angle \(>20^\circ\), and for vertical jets directed upward or downward, in their opinion it is necessary to introduce one more determining criterion—the angle of inclination.

As for the curvature of the axis of a burning plume, Lyakhovskii and Syrkin warn that it is unjustified to apply to a burning plume experimental data obtained in the investigation of heated jets; they believe that a burning plume will curve more strongly than a plume without heat release; they planned to carry out a study of the curvature of a burning plume in 1939, but the results of the study have not yet been published.

In 1940 I. A. Shepelev published a paper [11] on the propagation of warm and cold air jets; in this work he derived the equation of the aerodynamic axis of a nonisothermal jet issuing from a nozzle located at an angle \(\alpha\) to the horizon,

\[ -\frac{y}{d} = Ar\left(\frac{x}{d\cos\alpha}\right)^2 \left[ 0.51a\,\frac{x}{d\cos\alpha}+0.35 \right] \pm \frac{x}{d}\tg\alpha \tag{14} \]

The plus sign before the last term of the equation is taken when the direction is upward, and the minus sign when the direction is downward. For horizontal discharge, when \(\alpha=0\), the equation of the flow axis will be

\[ -\frac{y}{d} = Ar\left(\frac{x}{d}\right)^2 \left[ 0.51a\,\frac{x}{d}+0.35 \right] \tag{15} \]

Comparing (15) and (12), we note that the rise of the flow axis is less steep according to Shepelev’s equation (15) than according to Abramovich’s equation (12), and the results obtained from equation (15) are in close agreement with the experimental data of Lyakhovskii and Syrkin, which is especially clearly revealed when plotting the generalized curve in the coordinates \(\dfrac{y}{dAr}\), \(\dfrac{x}{d}\).

The coordinates of the vertex of the parabolic trajectory of the jet (gas fountain) issuing obliquely in accordance with the scheme in Fig. 1, according to Shepelev’s equation, are obtained by the method of finding the maximum value of the function \(f'\!\left(\dfrac{x}{d}\right)=0\), i.e.

\[ \text{abscissa of the vertex}\quad \left(\frac{x}{d}\right)^{\mathrm{v}} = 0.583\cos\alpha\sqrt[3]{\frac{\sin^2\alpha}{aAr^2}}, \tag{16} \]

\[ \text{ordinate of the vertex}\quad \left(-\frac{y}{d}\right)^{\mathrm{v}} = 0.35\sin\alpha\sqrt[3]{\frac{\sin^2\alpha}{aAr^2}}. \tag{17} \]

The greatest height of the vertical fountain \((\alpha = 90^\circ)\)

\[ \frac{y_{\max}}{d}=\frac{0.35}{\sqrt[3]{aAr^2}}, \tag{13} \]

I. A. Shepelev illustrates the application of the relations he has derived with two examples of the outflow of cold and warm air.

With regard to the relations derived by Shepelev, in the collection edited by V. A. Turkus [12] the opinion is expressed that the equations derived cannot be extended to all cases encountered in practice, and, in particular, to systems for the concentrated supply of considerable quantities of air into rooms.

A large theoretical study by M. A. Glinkov [3] is devoted to questions of the curvature of the axis of a nonisothermal jet and the flatness of a flame in open-hearth furnaces. In his study M. A. Glinkov, as do Lakhovskii and Syrkin [8;9], criticizes the equations of Grum-Grzhimailo–Esman (2;4) and subsequently uses, as the initial equation of a free turbulent jet, G. N. Abramovich’s equation in its earlier form, i.e., the uncorrected (11), whose inaccuracy, as indicated above, was acknowledged by the author himself, who later gave in its place a new, refined equation (12). Wishing to apply Abramovich’s earlier equation (11) to a study of the propagation of a burning flame, in which there is heat release and radiant heat exchange with the enclosing walls, M. A. Glinkov introduces a correction coefficient with an average value of 0.7 and subsequently uses Abramovich’s equation (10): for a horizontal jet in the form

\[ -\frac{y}{d}=kAr\left(\frac{x}{d}\right)^2 \left[1.48\,a-\frac{x}{d}+0.32\right] \tag{19} \]

and for an inclined jet in the form

\[ -\frac{y}{d}=kAr\left(\frac{x}{d\cos\alpha}\right)^2 \left[1.48\,a-\frac{x}{d\cos\alpha}+0.32\right] +\frac{x}{d}\operatorname{tg}\alpha. \tag{20} \]

To find the coordinates of the apex of the gas fountain (in accordance with the diagram in Fig. 1), M. A. Glinkov determines the maximum of the function from equation (20). The formulas obtained in this way are extremely cumbersome; although M. A. Glinkov in the cited work applies his equations to solve a number of practical problems from the operation of open-hearth furnaces, it must nevertheless be admitted that the basis for deriving the equations is inaccurate, since Abramovich’s earlier equation was used in deriving them.

For a visual presentation, let us compare the calculated data for determining the trajectory of an inclined gas jet according to the equations of Grum-Grzhimailo–Esman (equations 8 and 9), Shepelev (equations 16 and 17), and Glinkov (20) for different values of the Archimedes criterion. The results of the calculations and a graphic representation of the trajectory of the flow axis inclined at an angle \(\alpha = 12^\circ\) are given in Fig. 2. From the data in the table and from the nature of the curves it is evident, first, that a sharp difference is found in the values of the coordinates of the apex of the curved axis, especially the coordinate \(x_{\max}\), according to the data of the indicated authors, and, second, that the flattest trajectory of the flow axis is obtained from the equations of Grum-Grzhimailo–Esman, and the steepest from Glinkov’s equations. This is explained by the fact that in the theory of gas fountains of Grum-Grzhimailo–Esman the entrainment into the flow of gases from the surrounding medium is not taken into account, while in Glinkov’s equation the expenditure of energy of the vertical component of velocity on accelerating the entrained ...

... flow; attention was drawn to these aspects already by Lyakhovskii and Syrkin [8;9].

I. K. Neimark [13] attempted to derive the equation of the trajectory of jets of sharp secondary blowing of cold air supplied into a furnace, taking into account heat release in the jet due to combustion of volatiles and entrainment into the jet. For this purpose I. K. Neimark used the basic propositions of Abramovich’s theory of a free jet [6], adding to them an additional condition—the release of heat due to combustion. However, it does not appear possible to use the equations derived by I. K. Neimark for analyzing phenomena in a burning gas flame, since the equations include a poorly defined quantity \(x_m\)—the path over which combustion occurs; the use of a simplified formula in which this quantity is absent, as the editors of the journal Soviet Boiler and Turbine Engineering [13] point out, gives very substantial errors. In addition, as is evident from the calculated data cited by Neimark in Table 2, but not verified by experiment, the magnitude of the correction for the change in ordinate of the jet trajectory due to afterburning of 20% of the volatiles amounts to only about 4%.

Fig. 2. Curvature of the axis of a nonisothermal flame
(according to the formulas of various authors).

G. N. Abramovich, in the work already cited [10], in a separate paragraph, “Some characteristics of a burning flame,” considers the question of the length of a burning flame. For calculating the length of a burning flame, G. N. Abramovich derives the equation

\[ \left(\frac{\alpha x}{R_o}\right)_{\max}=0.7\,\frac{K_o}{K_{kr}} \tag{21} \]

Here \(K_o\) is the initial concentration of fuel in the mixture, equal to the ratio of the weight of fuel to the weight of air in the mixture;

\(K_{kr}\) is the final concentration of fuel in the mixture, equal to the ratio of the weight of fuel to the weight of air for a stoichiometric mixture. In our opinion, Abramovich’s equation is inapplicable to the real process of combustion, since the equation lacks the principal physicochemical quantity characterizing the combustion process—the flame-propagation velocity, which depends on the nature of the fuel and the composition of the combustible mixture. According to Abramovich’s equation, the flame length will be one and the same both for combustion of \(\mathrm{H}_2\),

both for the combustion of CO and for the combustion of CO, since \(K_{cr}\) is identical in both cases; meanwhile, it is well known that, under identical conditions, the length of the burning flame of \(\mathrm{H}_2\) is considerably shorter than the length of the burning flame of CO, owing to the greater rate of flame propagation in combustible mixtures containing \(\mathrm{H}_2\).

This equation of G. N. Abramovich is completely inapplicable for characterizing a burning gas flame in general; indeed, when primary air is absent in a burning flame and all the air required for combustion flows to the flame from the surrounding medium, in order to solve equation (21) one would have to take \(K_0\) as an infinitely large quantity, and then

\[ \frac{x}{R_0}=\infty \]

When a fully prepared gas mixture is burning, however, one would have to take \(K_0=K_{cr}\); in this case, regardless of the nature of the fuel, a flame is always obtained of the same length, equal to

\[ \frac{x}{R_0}=\frac{0.7}{a} \]

An attempt to apply equation (21) to the study of the combustion of partially prepared combustible mixtures also does not lead to results that agree with experimental data. For example, for a partially prepared combustible mixture of water gas–air, with the amount of primary air in the mixture \(\alpha=0.3\), one should substitute in G. N. Abramovich’s equation: \(K_0=0.72\) and \(K_{cr}=0.216\), and then

\[ \frac{x}{R_0}=0.7\,\frac{0.72}{0.216\,a}=\frac{2.24}{a} \]

For a tubular burner the jet-structure coefficient \(a=0.076\), and then

\[ \frac{x}{R_0}=\frac{2.24}{0.076}=29.4, \]

i.e., it follows that, irrespective of the initial efflux velocity, the flame length will always be one and the same. Meanwhile, as is evident from the table of our experiments, the relative flame length varies from \(\frac{l}{d}=21.7\) at an efflux velocity of \(5.26\ \frac{\mathrm{m}}{\mathrm{sec}}\) to \(\frac{l}{d}=49\) for an efflux velocity of \(11.85\ \frac{\mathrm{m}}{\mathrm{sec}}\).

Then G. N. Abramovich, considering that at stoichiometric ratios between the amount of fuel and the air required for combustion in the flame complete uniformity of mixing is not ensured, establishes, on the basis of the law of jet formation, a dependence between the length of the mixing path and the amount of air entrained into the flame by the end of the combustion process. From this dependence it follows that complete mixing of the fuel with air will occur in all cases when the excess-air coefficient is equal to \(\alpha=1.52\).

Such a generalization is clearly untenable and contradicts generally known facts.

G. Ya. Vyugova [14] carried out an extensive experimental investigation of a vertical freely burning flame of a gas–air mixture of producer gas with a calorific value of \(4500\ \dfrac{\text{kcal}}{m^3}\); she studied the distribution of temperature and velocity fields in three zones of the burning flame and established generalized dependences for the variation of the relative velocity and relative temperature along the length of the flame. The investigation did not touch upon the following questions: a) the position of the vertex of the combustion cone as a function of the outflow velocity and the composition of the combustible mixtures, i.e., the question of the flame length of the burning flame was not studied; b) since in the experiments the flame was vertical, naturally the questions of the influence of curvature of the flame axis on the flame length were not touched upon at all; c) the practical example given by G. Ya. Vyugova, illustrating the application of the relations she found, concerns the third zone of the flame—the zone of cooling of the combustion products at a large distance from the vertex of the combustion cone—the calculated section is located at a distance of \(242\ \text{mm}\) with the height of the vertex of the combustion cone equal to \(80\ \text{mm}\); in this region, undoubtedly, the relations established for hot (but not burning) jets are applicable.

Thus, in summing up the literature data on the state of the question of the nonisothermal outflow of a burning flame, it is necessary to come to the conclusion that at present, without carrying out a special experimental investigation, this question cannot be satisfactorily resolved. In connection with this, the author undertook an experimental study of the range of a burning flame, varying under the influence of curvature of the flow axis and flatness.

II.

The method used to study the influence of curvature of the axis and flatness of a burning gas flame on its range was very simple: a jet of gas or combustible mixture (gas and primary air) was discharged from a tube into the atmosphere at various angles of inclination and was ignited; in the resulting flame the temperature and length were measured, and the contours of the flame were traced from its image on a screen; at the same time the flow rate of gas and primary air was measured by means of constant-pressure gas meters. To study the phenomenon of flatness, the jet of burning gas was directed along the surface of a smooth metal plate and the contours of the burning flame were traced.

The data obtained in the investigation are qualitative in character; nevertheless, it is possible from them to judge approximately the degree of influence of curvature of the axis and flatness of the flame on its range and to establish approximate quantitative relationships.

In the experiments with water gas, a tubular burner of diameter \(2.32\ \text{mm}\) was used; the outflow velocity varied within the limits from \(w_o = 3.5\ \dfrac{m}{\text{sec}}\) to \(w_o = 11.85\ \dfrac{m}{\text{sec}}\); the amount of primary air in the combustible mixture varied from \(\alpha = 0\) to \(\alpha = 0.31\).

The composition of the water gas used for the investigation was:

\[ \mathrm{CO_2}—4.2\%,\quad \mathrm{CO}—29.4\%;\quad \mathrm{H_2}—52.3\%;\quad \mathrm{CH_4}—6.4\%;\quad \mathrm{O_2}—0.2\%;\quad \mathrm{N_2}—7.5\%. \]

For complete combustion of this gas the theoretically required amount of air is \(2.55\ \dfrac{m^3}{m^3}\). The measurement results are presented in Table 1.

Results of the investigation of the effect of axis curvature and flatness

Effect of a burning gas jet on its range

Table

with smoky air

The following notation is used in the table:

\[ \frac{l}{d_o} \]

— the total relative length of the visible burning flame, measured along the curved axis (axis trajectory);

\[ \frac{x}{d_o} \]

— the projection of the trajectory of the axis of the burning flame onto the abscissa axis;

\[ \frac{y}{d_o} \]

— the same onto the ordinate axis;

\[ \left(\frac{x}{d_o}\right)^{в} \]

— the length of the horizontal projection from the origin to the tip of the burning flame;

\[ \left(\frac{y}{d_o}\right)^{в} \]

— the length of the vertical projection from the origin to the tip of the burning flame; the sign \(+\), when the tip is above the abscissa axis; the sign \(-\), when the tip of the flame is below the abscissa axis.

“Calculated”: \(\left(\frac{x}{d_o}\right)^{в}\), \(\left(\frac{y}{d_o}\right)^{в}\) — coordinates of the flame tip, calculated by Shepelev’s formulas (40 and 41).

The Archimedes criterion was calculated by the formula:

\[ Ar=\frac{g\Delta T d}{T w_o^2}, \]

\[ g=9.81\ \frac{m}{sec^2}; \]

\(\Delta T\) — according to the flame temperature (variable) and the temperature of the surrounding medium (constant);

\(d_o\) — diameter of the burner orifice, equal to \(0.00232\ m\).

\(w_o\) — initial discharge velocity of the efflux.

Experiments were also carried out on a burning gas flame of illuminating gas; for this purpose 3- and 7-slot gas burners were used; since the investigation of the burning flame of illuminating gas was carried out under conditions close to the diffusion method of gas combustion, the results of the experiments are described in a special work devoted to the investigation of a burning laminar-diffusion gas flame [15].

The graphical dependence between the length of free and wall flames without primary air and the velocity of gas efflux from the burner is presented in Fig. 3 in the form of six straight lines: three straight lines for the free flame and three straight lines for the wall flame. The experimental points are located quite close to the straight lines. For both free and wall flames, at identical efflux velocities, vertical flames have the greatest length, while inclined (curved) flames have the smallest length. The ratio of the lengths of the vertical and curved flames is: for the free flame—from 1.20 to 1.48, and for the wall flame—from 1.20 to 1.38; moreover, with changes in the efflux velocity, the magnitude of the ratio does not change.

For horizontal and inclined directions of the flame axes, the latter curve; the curved axes of the flames approach parabolic curves. However, the lengths of the curved axes of flames directed

inclined, differ very little from the curved length of a horizontal flame, since the experimental points for horizontal and inclined flames are located close to one another and in some cases coincide. This circumstance indicates more or less identical conditions of mixing of the surrounding air with the gas in the burning flame, independently of the angle of inclination of the flame axis (excluding the vertical direction).

To explain the phenomena indicated above, the following scheme may be proposed.

With the vertical direction of a free flame without primary air, the conditions for the inflow of air from the surrounding medium to the flame front are least favorable, since around the flame front there is a layer of combustion products, which also impedes the diffusion

Fig. Length of a free and wall-adjacent burning gas flame during combustion without primary air:

1 — free horizontal flame; 2 — free vertical flame; 3 — free flame at an angle of \(20^\circ\); 4 — wall-adjacent horizontal flame; 5 — wall-adjacent vertical flame; 6 — wall-adjacent flame at an angle of \(20^\circ\).

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 flame. (The scheme of air inflow to the vertical flame is shown in Fig. 4, position 1.)

With the horizontal direction of the flame, because of the difference in aerostatic pressures, air flows to the surface of the front of the burning flame over its entire length; consequently, the conditions for access of air are more favorable than with a vertical arrangement, and therefore the horizontally directed flame is shorter than the vertical one. At a small angle of inclination of the flame axis, the conditions for access of air to the flame front remain more or less similar, as in the case of the horizontal direction; therefore the flame length is shortened only slightly. But as the angle of inclination of the flame axis increases, the accessible surface increases, and in the limiting case, when the angle of inclination is \(90^\circ\), i.e., when the flame

will be directed vertically downward; then the inflow of air will be toward the entire surface of the burning flame, as a result of which the flame will be the shortest, which is also confirmed by experiment (the scheme of air inflow is shown in Fig. 4, position 3).

The wall jet is always longer than the free jet (for identical efflux velocities). In the absence of combustion, the elongation of a wall jet is usually explained by the fact that the resistance to the motion of the flow spreading over the surface is reduced, in comparison with the resistance of the surrounding stationary gaseous medium. From the theory of constancy of the amount of motion as applied to the conditions of gas mixing, when

\[ m w_0 = \mathrm{const}, \tag{23} \]

it follows that, owing to the smaller entrainment of the surrounding medium into the main flow during wall motion, the initial velocity \(w_0\) decays less intensely and is preserved over a greater distance; therefore the range of the jet also increases.

The elongation of the burning wall flame, in comparison with the burning free flame, is affected by the impeded access of air required for combustion by diffusion from the surrounding medium.

In our experiments, the relative elongation of the wall flame in comparison with the free flame is from 32 to 41%—on average 37%; moreover, the ratio of the length of the wall flame to the length of the free flame remains approximately constant and thus does not depend on an increase in the efflux velocity. Thus, when the efflux velocity is increased from \(3.50\ \text{m/sec}\) to \(9.3\ \text{m/sec}\), i.e., by a factor of 2.8, the ratio of the length of the wall flame to the length of the free flame is: at the minimum (in the experiments) efflux velocity

Fig. 4. Scheme of air inflow to the burning flame.

\[ \frac{l_{\mathrm{н}}}{l_c} = 1.38 \quad \text{and at the maximum (in the experiments) efflux velocity} \quad \frac{l_{\mathrm{н}}}{l_c} = 1.35 \]

(for a vertical burning flame without primary air, i.e., experiments 1 to 25). If, as the initial quantity, one takes the length of a horizontal or slightly inclined flame (angle of inclination not more than \(20^\circ\)) and compares with it the lengths of the vertical free and wall flames, respectively, then the latter increase as a result of the influence of conditions that hinder the access (diffusion, mixing) of air from the surrounding medium to the flame front. For an approximate quantitative estimate of this influence, let us introduce the concept of the apparent (conditional) velocity of flame propagation. If the actual, normal (according to Michelson) velo-

the flame-propagation velocity depends on the composition of the combustible mixtures (previously prepared), then the apparent, conditional flame-propagation velocity introduced by us depends on the conditions of access of air to the flame front (diffusion, mixing). Therefore the apparent flame-propagation velocity may be represented as the product of the normal velocity by a certain coefficient characterizing the influence of the conditions of air access, i.e.,

\[ U_k = KU \tag{24} \]

where \(U\) is the kinetic factor (normal velocity),
\(K\) is the diffusion factor.

In all the cases considered of combustion without primary air, the kinetic factor \(U\) may be taken as the same, while the diffusion factor \(K\) is a variable quantity.

From Michelson’s definition of the flame-propagation velocity [16] it follows, approximately, that

\[ l = r_0 \frac{w_0}{U_k} = r_0 \frac{w_0}{KU} \]

or

\[ \frac{l}{d_0} = 0.5 \frac{w_0}{KU}. \tag{25} \]

If the product \(KU\) does not depend on the efflux velocity (within the limits of the experiments carried out), then graphically, in the coordinates \(\frac{l}{d_0}\) and \(w_0\), we obtain a straight line whose tangent of the angle of inclination will be equal to \(\frac{0.5}{KU}\). For horizontal or slightly inclined free flames, from Fig. 3 we obtain

\[ \frac{0.5}{K_1U} = 5.77,\quad \text{whence } K_1U = 0.0870; \]

for a vertical free flame,

\[ K_2U = \frac{0.5}{7.90} = 0.0667; \]

for a horizontal wall-attached flame,

\[ K_3U = \frac{0.5}{8.15} = 0.0615; \]

for a vertical wall-attached flame,

\[ K_4U = \frac{0.5}{10.4} = 0.048. \]

The relative value of the diffusion factor varies in the following order:

if for a horizontal free flame \(K_1 = 1\),
then for a vertical free flame \(K_2 = 0.76\),
for a horizontal wall-attached flame \(K_3 = 0.70\),
for a vertical wall-attached flame \(K_4 = 0.55\).

The flame length changes correspondingly.

Consequently, \(\left(\frac{l}{d_0}\right)\) of a vertical free flame \(= 1.31 \left(\frac{l}{d_0}\right)\) of a horizontal free flame; \(\left(\frac{l}{d_0}\right)\) of a horizontal wall-attached flame \(= 1.43 \left(\frac{l}{d_0}\right)\) of a horizontal free flame; \(\left(\frac{l}{d_0}\right)\) of a vertical wall-attached flame \(= 1.81 \left(\frac{l}{d_0}\right)\) of a horizontal free flame.

In Fig. 5 the relationships between the flame length \(\dfrac{l}{d_0}\) and the initial outflow velocity are shown graphically for a combustible mixture containing primary air in an amount corresponding to \(\alpha = 0.31\), for vertical, horizontal, and inclined directions. Under similar conditions of flame organization, its length for combustible mixtures at \(\alpha = 0.31\)

Fig. 5. Length of a free and impinging burning gas flame when burning with primary air:

1 — free horizontal flame; 2 — free vertical flame; 3 — free flame at an angle of \(20^\circ\); 4 — impinging horizontal flame; 5 — impinging vertical flame; 6 — impinging flame at an angle of \(20^\circ\).

is considerably shorter than for a flame without primary air. The experimental points lie fairly close to straight lines, the equations of which have the general form

\[ \frac{l}{d_0}=cw_0, \tag{26} \]

where, as before, the quantity

\[ c=\frac{0.5}{K_n U_n}; \]

in this case

for a horizontal free flame
\[ c=3.40;\qquad K_n U_n=0.147 \]

for a vertical free flame
\[ c=4.10;\qquad K_n U_n=0.122 \]

for a horizontal impinging flame
\[ c=4.87;\qquad K_n U_n=0.102 \]

for a vertical impinging flame
\[ c=5.70;\qquad K_n U_n=0.0875 \]

For similar conditions of flame organization, the diffusion factor \(K_n\) may be taken as the same both for a flame without primary air and for a flame with primary air, while the kinetic factor \(U_n\) will be different. The ratio of the kinetic factors \(\dfrac{U_n}{U}\) gives an idea of the increase in the velocity of flame propagation under the conditions of a flame with primary air in comparison with a flame without primary air.

For a horizontal free flame,

\[ \frac{U_n}{U}=1.69 \]

for a vertical “ ”

\[ \frac{U_n}{U}=1.83 \]

for a horizontal wall “ ”

\[ \frac{U_n}{U}=1.67 \]

for a vertical “ ”

\[ \frac{U_n}{U}=1.82 \]

Thus, for all four conditions of flame organization with primary air in an amount corresponding to \(\alpha=0.31\), the kinetic factor (the conditional velocity of flame propagation) is approximately \(1.80\) times greater than for a flame without primary air.

When an inclined burning flame issues, its axis is curved under the influence of the buoyant force, caused by the difference in densities of the gases in the flame and in the surrounding medium. The equations for determining the coordinates of the vertex of the parabolic trajectory of the flame, established and verified under the conditions of a nonisothermal nonburning flame, for example equations (16) and (17) according to Shepelev, are not suitable for characterizing a burning flame. A burning gas flame has a nonuniform thermal and chemical structure: the core of the flame (without primary air), consisting of unignited gas, has a low temperature, below the ignition temperature; the flame envelope—the burning surface—has a high temperature. In the presence of primary air in the combustible mixture, the temperature of the surface of the inner cone differs from the temperature of the surface of the outer cone of combustion, etc. Therefore an unambiguous characterization of a burning flame by the Archimedes criterion is clearly insufficient, as a result of which the experimental data for determining the coordinates of the vertex of the curved (parabolic) trajectory of a burning flame diverge sharply from the calculations by equations (16) and (17). (Compare the vertex coordinates \(\left(\frac{x}{d_0}\right)^e\) and \(\left(\frac{y}{d_0}\right)^e\) in the column “calculated” with the measurements \(\left(\frac{x}{d_0}\right)^e\) and \(\left(\frac{y}{d_0}\right)^e\) in the columns for free and wall flames).

The reason for these discrepancies can be identified in the following way: let us consider an isothermal inclined flame, for which \(\Delta T=0\) and \(Ar=0\); the flame is rectilinear, and the trajectory of its axis is described by the equation

\[ \frac{y}{x}=\operatorname{tg}\alpha \tag{27} \]

For a nonisothermal but nonburning inclined flame, \(\Delta T>0\) and \(Ar>0\); the flame is curved; for the vertex of the rectilinear trajectory, from Shepelev’s equations (16) and (17) we obtain

\[ \left(\frac{y}{x}\right)^e=0.6\,\operatorname{tg}\alpha \tag{28} \]

Here the coefficient \(0.6\) is a measure of the deviation of the flame axis at its vertex from the initial direction, owing to the influence of the buoyant force.

For a nonisothermal burning flame, according to our experimental data, the position of the vertex of the curvilinear trajectory is characterized by the equation

\[ \left(\frac{y}{x}\right)^e=m\,\operatorname{tg}\alpha, \tag{29} \]

where \(m\) is a measure of the deviation of the axis of the burning flame at its vertex from the initial direction owing to the influence of the buoyant force and chemical-

... and thermal nonuniformity of the flame; moreover, the value \(m\) does not remain constant, but depends chiefly on the content of primary air in the combustible mixture. For a flame without primary air the value \(m\) is approximately equal to 0.3; for a flame with an amount of primary air equal to 0.3 of that theoretically required, the value \(m\) is approximately equal to 0.45. It may be assumed that the more primary air there is in the combustible mixture, the more uniform the flame is thermally, and the greater the degree of approximation of the relative coordinate values of the vertex of the axis of the burning flame to the relative coordinate values of the vertex of the axis of the nonisothermal nonburning flame. But the absolute coordinate values of the vertex of the axis of the burning flame will not coincide with those for the vertex of the nonburning nonisothermal flame; the latter are much larger than the former, as is seen from a comparison of \(x_b\) and \(y_b\) in the column “calculated” and \(x_b\) and \(y_b\) from measurements. This circumstance indicates that the trajectory of the axis of the burning flame is not determined uniquely by the value of the criterion \(Ar\); the trajectory is also affected by the rate of flame propagation, i.e., by the nature of the fuel and the composition of the combustible mixtures. We have not investigated this question in detail.

Attempts at practical application of the phenomena of the gas fountain (without taking account of the influence of flame impingement) to tank glass-melting furnaces were undertaken by us as early as 1936 in work [17], where the relationship between the width of the flame span, the outflow velocity, and the angle of inclination of the burner was shown graphically and clearly. The experimental material presented in the present work considerably broadens and refines earlier ideas concerning the dynamics of a flame flow.

Brief Conclusions

1. The effect of curvature of the axis and impingement of a burning gas flame on its range has been experimentally investigated for various initial outflow velocities, various angles of inclination of the flame axis, and various compositions of combustible mixtures.

2. It has been shown that the actual trajectory of the axis of burning free and impinging flames cannot be satisfactorily described by equations with a unique dependence only on the buoyancy-force criterion (the Archimedes criterion).

3. An attempt has been made to explain the shortening of the range of horizontal and inclined flames, as compared with a vertical one, from the standpoint of the influence of kinetic and diffusion factors on the mixing of fuel with air and on the combustion process.

4. Relationships between the range of impinging and free flames have been established and generalized. The elongation of an impinging flame in comparison with a free one is explained by the influence of the diffusion factor, which reduces the intensity of mixing of the fuel with air.

5. The shortening of free and impinging flames in a combustible mixture containing primary air, as compared with a flame formed when only a single gas issues, is explained by the influence of kinetic factors (an increase in the flame-propagation velocity).

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