Study of Flare Convective-Turbulent Combustion

N. N. Norkin

The formation and development of a stable process, and the intensity of flame convective-turbulent combustion, are influenced predominantly by aerodynamic phenomena: the velocity of outflow of gas and air, the direction of the gas and air streams and their mixing, the curvature of the flame axis, the ejection of combustion products by the flame, etc. Experimental and theoretical studies in this field are usually carried out by two methods: 1) investigation of the aerodynamics of the flare process under isothermal or nonisothermal conditions in the absence of combustion, and 2) investigation of the aerodynamics of the flame during the combustion process. The most generalizing conclusions and mathematical relationships have been obtained in studies by the first method, but the degree of their applicability to actual combustion processes is rather approximate, and the limits of applicability are restricted by the necessity of introducing a number of simplifying assumptions. As for the results of studies carried out by the second method, in most cases they are limited by the specific conditions of the experimental investigation, while generalizing conclusions are formulated only in the most general form; physicomathematical relationships are almost always absent.

Experimental studies by D. N. Lyakhovskii and S. N. Syrkin [1], F. P. Kazakevich and N. P. Dyundin [2], M. A. Glinkov and N. A. Kaloshin [3], P. I. Sychev [4], K. Rummel [5], and others are devoted to the study of phenomena of gas mixing during their outflow without chemical interaction. Usually the following methodology is applied in these investigations: during the outflow from two parallel nozzles of streams of cold and hot air, the relation between the outflow velocities of the streams and the temperature distribution along the length of the flame is established. During the outflow of two air streams of the same or different temperature at certain angles of one stream to the other, the influence of the conditions of impingement of the streams on the temperature distribution is established; and during the outflow of concentric streams of the “pipe-in-pipe” type, the relation between the outflow velocities and the temperature distribution along the mixing path is established. Lyakhovskii and Syrkin [1], using Reynolds’ temperature-velocity analogy, developed a theory of the experiment for modeling mixing processes, introducing the concept of degree of mixing. The experimental material obtained by them and the mathematical relationships established by them are of great practical value in solving questions concerning the mixing of gases of different density and temperature under diverse conditions of organizing the mixing process, and may be used in the study of convective-turbulent flame combustion of gases.

A direct investigation of the aerodynamics of the flare flame combustion process was carried out by V. N. Timofeev and P. I. Sychev

[6], L. V. Podgurskii [7], Pistor [8], Rummel [5], N. N. Dobrokhotov and collaborators [9], B. I. Kitaev [10], G. Ya. Vyugova [11], and others. The procedure used in these investigations was as follows: in laboratory or industrial gas furnaces equipped with burners of one design or another, producing different degrees of mixing, the velocity, temperature, and concentration fields in the burning flame are measured. Investigations of this kind make it possible to judge the influence on the flame length, on the completeness of combustion, and on the temperature distribution in the flame of such factors as the relative velocity of the air and gas streams, the angle at which the streams meet, the structural design of the burners, etc. In accordance with this, Pistor [8] developed five typical schemes of gas burners in order of increasing mixing intensity and investigated them. Timofeev and Sychev [6] gave a mathematical expression for the coefficient of turbulent diffusion which, in the opinion of these authors, takes into account molecular and molar diffusion, relating this coefficient by a certain relation to the initial mean efflux velocity. Of interest and value are the results of measurements of temperature fields in the flame as a function of burner type, carried out by P. I. Sychev [6].

In addition to the study of the concentration, velocity, and temperature fields in a flame by experimental methods, a number of theoretical physicomathematical investigations have been published whose aim is to discover the regularities and to express them by equations that make it possible to find the flame length and the temperature distribution along the flame length. The most complete works in this direction are those of V. A. Shvab [12], Hailigenstedt [13], and M. A. Glinkov [14].

As the basis for his conclusions, V. A. Shvab adopted, as an initial premise, the Reynolds temperature-velocity analogy developed by Lyakhovskii and Syrkin [1]. V. A. Shvab extended the temperature-velocity analogy to the field of combustion phenomena. In doing so he adopted the following assumptions: 1) that the combustion process proceeds practically instantaneously; 2) that, in an instantaneously occurring combustion process, at any point of the flame there exists a mathematical combustion surface on which, as a result of turbulent diffusion, the fuel gas and oxygen interact. The coordinates of this surface are determined by the stoichiometric ratio between the gas and the oxygen necessary for combustion that penetrate to the combustion surface; 3) that there is a direct proportionality between the rate of heat release and the rate of formation of combustion products; 4) that there is similarity between the temperature field and the concentration field and that, consequently, the temperature and velocity fields will be similar; this makes it possible to find the temperature distribution in the flame if the velocity field in the flame is known (the latter is considered to be readily measurable or calculable on the basis of the laws of flame aerodynamics). To take into account the heat transfer of the flame to the surrounding medium, according to Shvab, it is necessary to draw up a heat balance for any cross section of the flame, and then the value of the working temperature at any point of the flame can be found as a function of the velocity field and the intensity of heat transfer. V. A. Shvab illustrates his theoretical propositions with practical examples: combustion of stoichiometric mixtures, combustion with a deficiency of air. It should be noted, however, that for practical application the relations derived by Shvab are extremely complicated and require the adoption of many simplifying assumptions.

Another method of solving the question of the temperature distribution in a flame is the method of Hailigenstedt [13], subsequently somewhat modified by M. A. Kuzmin [15] and E. V. Virozub [16]. This method may be suitable for approximate calculations and in the design of

burners; it was used by us in considering combustion processes in the heating system of coke ovens when they are fired with coke-oven and blast-furnace gases. The method of Heiligenstedt and other authors of this school is based on the assumption that, in the study of flare combustion, the mathematical expression of the law of mass action is applicable, but the physicochemical constant of the reaction rate is replaced by a new constant depending chiefly on the conditions of mixing of the fuel with air, i.e., on the burner design and on the type of gas-combustion chamber, and only to a limited extent on the nature of the gas. Under these conditions the rate of the heat-release process can be described by an equation formally analogous to the equation of the law of mass action,

\[ \frac{dq}{d\tau}=-k(q_0-q_1), \tag{1} \]

where \(q_0\) is the initial heat content of \(1\ \text{m}^3\) of combustion products, i.e.,

\[ q_0=\frac{Q_n^p}{V_{\text{pr.gor.}}}; \]

\(q_1\) is the amount of heat converted from the potential form into physical heat content up to the time under consideration;

\(k\) is the rate coefficient of the heat-release process, depending chiefly on the burner design.

If the intensity of heat transfer from the flame is taken to be the same over the entire length of the flame, then the total amount of heat given up by the flame can be expressed by the equation

\[ dq_2=i\,dx; \tag{2} \]

where \(i\) is the intensity of heat transfer of \(1\ \text{m}^3\) of combustion products along a path of the flame equal to \(1\ \text{m}\). Consequently,

\[ dx=W_{\text{avg.}}\cdot d\tau;\qquad dq_2=iW_{\text{avg.}}\cdot d\tau. \tag{3} \]

The simultaneous solution of differential equations (1) and (3) makes it possible to calculate the amount of heat converted from the potential form into physical heat content at a given moment of time, i.e.,

\[ q_\tau=\frac{k}{k-m}\,q\left(e^{-m\tau}-e^{-k\tau}\right)-273\,C_p\left(1-e^{-m\tau}\right). \tag{4} \]

Here \(W_0\) is the initial velocity of the gas stream,

\(C_p\) is the mean heat capacity of the combustion products; but \(q_\tau\) can be expressed through \(tC_p\), then

\[ q_\tau=C_p t;\qquad m=\frac{iW_0}{273C_p}. \tag{5} \]

After substituting the value of \(q_\tau\), an equation is obtained by which the temperature distribution along the length of the flame can be calculated:

\[ t=\frac{k}{(k-m)C_p}\,q\left(e^{-m\tau}-e^{-k\tau}\right)-273\left(1-e^{-m\tau}\right). \tag{6} \]

To determine the maximum temperature, the first derivative of \(q_\tau\) with respect to \(\tau\) is taken and set equal to zero.

The heat-release-rate coefficient, on the basis of experimental data appropriately processed by Heiligenstaedt, is characterized by the following values:

1) a burner built of brick, having one channel for gas and another channel for air, the channels having a slight inclination toward one another,

\[ k = \text{from } 1.5 \text{ to } 2.0; \]

2) the same burner, but with a stronger inclination of the streams toward one another,

\[ k = 2.5; \]

3) when mixing nozzles are used,

\[ k = 3.5; \]

4) nozzle burners with weak turbulence,

\[ k = \text{from } 3.0 \text{ to } 4.0; \]

5) burners with strong turbulence,

\[ k = \text{from } 4.0 \text{ to } 6.0; \]

6) burners with premixing,

\[ k = \text{from } 8.0 \text{ to } 10.0. \]

The influence of heat transfer on the flame temperature is characterized by the coefficient

\[ m = \frac{i W_0}{273 C_p}. \]

The heat-release-rate coefficient \(k\), the coefficient characterizing the intensity of heat transfer along the flame length \(m\), and the pyrometric combustion coefficient are interrelated by the relation:

\[ \eta = \frac{t_{\max\ \mathrm{effective}}}{t_{\mathrm{calorimetric}}} = \varphi \frac{k}{m}. \tag{7} \]

Heiligenstaedt calculated this functional dependence, and M. A. Kuzmin [15] processed the results of the calculation graphically in the coordinates \(\eta\) and \(\frac{k}{m}\). Using the indicated data, for a given coefficient \(\eta\) one can find the ratio \(\frac{k}{m}\) and the temperature distribution in the flame, as well as the point with the maximum flame temperature. In the event that the obtained temperature distribution and the position of the maximum do not satisfy the technological or other operating conditions of the firebox or furnace, then, by varying the burner index \(k\), one can achieve the desired flame length and the desired temperature distribution along it.

The shortcomings of Heiligenstaedt’s method for calculating the flame temperature in the combustion process are: 1) the use of the law of mass action to characterize the kinetics of the heat-release process in the form of equation (1) assumes the heat-release process to be analogous to a first-order combustion reaction; 2) the value \(k\) is taken as a constant independent of temperature; 3) heat transfer from the flame is assumed proportional to the flame path length, and not to the temperature.

E. V. Virozub [16], developing Heiligenstaedt’s method but not resolving the question of the order of the reaction and of the dependence of \(k\) on temperature, determines

the amount of heat given off by the flare to the surrounding space as a function of the temperature difference, according to the equation

\[ dq_z=\frac{1}{G}\left[\alpha (t-t_0)F\right]dx; \tag{8} \]

here \(\alpha\) is the total coefficient of heat transfer by convection and radiation;

\(t\) is the variable temperature of the flame flare;
\(t_0\) is the temperature of the heat-absorbing surface;
\(F\) is the specific heat-absorbing surface per \(1\ \text{m}\) of flame-flare length;
\(G\) is the hourly volume of combustion products.

Solving the differential equations of heat release and heat transfer, E. V. Virozub obtains the following equation for the mean temperature along the length of the flame flare

\[ t_m=t_0+\frac{q}{(m-k)C_p\tau}\left(e^{-m\tau}-e^{-k\tau}\right) +\frac{1-e^{-m\tau}}{mC_p\tau}(q+q_0+C_p t_0). \tag{9} \]

To calculate the temperature it is necessary to know the quantities \(k\) and \(\tau\). According to Virozub, these quantities are determined as follows: from the geometric volume of the furnace space \(V_{m.pr}\) and from the per-second volume of combustion products \(V_{pr.gor.}\), reduced to the temperature sought (in doing so, the desired temperature has to be assigned approximately), the residence time of the flame flow in the furnace space is determined:

\[ \tau=\frac{V_{m.pr}}{V_{pr.gor}}. \]

Assuming the combustion and heat-release process to be completed by a definite fraction of the potential heat in the initial combustible gas, according to the equation

\[ \beta=\frac{q_1}{q}=1-e^{-k\tau} \tag{10} \]

and specifying the degree of completeness of combustion (heat release) \(\beta\), the value of \(k\) is determined from the residence time of the flame flow \(\tau\). Then the value of \(t_m\) is determined, if the heat-transfer coefficients are known beforehand or have in some way been calculated.

However, calculation of the temperature distribution along the length of the flame flare by equations (6, 9) requires cumbersome preliminary computations and does not give a clear picture of the development of temperature along the length of the flare; therefore the author proposes a graphical-analytical method of calculation in the form of a \(t—k—\beta\) nomogram, especially convenient for variant calculations. In what follows, with the aid of this nomogram, the combustion process in the heating system of coke ovens is considered.

M. A. Glinkov [14], generalizing experimental data from many studies of a burning gas flare, came to the conclusion that, for calculating the length of the path over which gas and air streams mix—identified with the length of the burning flare—the type of equation of Williamson and Adams [17], formulated by them for the temperature field in an infinite plate when it is heated with a sudden change in the surface temperature at the initial instant, is entirely applicable. Having made a number of justified assumptions and taking into account the influence of the angle at which the streams meet, of confining walls, and so forth, on the distribution of the concentration of reacting substances in the burning flare, M. A.

Glinkov, while retaining the type of solution according to [17], proposed the following equation for calculating the length of a burning gas flare:

\[ \frac{E-E_k}{E_n-E_k} = 1.275\, e^{-0.145\left(\frac{x}{k_z d}\right)^{0.75}} \cdot \cos\left(\frac{W_2}{W_1}\right)^{0.434} \tag{11} \]

In this equation, \(E\) is the concentration of the substance along the jet axis at a distance \(x\) from the nozzle;

\(E_n\) is the initial concentration;

\(E_k\) is the final concentration of the substance after complete mixing;

\(W_2\) is the velocity of the enveloping flow;

\(W_1\) is the velocity of the central flow;

\(\lambda\) is the length of the mixing path;

\(d\) is the nozzle diameter;

\(k_z\) is a coefficient introduced by M. A. Glinkov to take into account the influence of various factors on the burning flare, where

\[ k_z = k_0 k_1 k_2 \ldots \ldots \ldots k_n, \tag{12} \]

where \(k_0\) is the coefficient allowing for the influence of the thicknesses of the central and enveloping flows, with

\[ k_0 = \sqrt{\frac{d_1}{d_2}}, \tag{13} \]

\(k_1\) is the coefficient allowing for the influence of the angle at which the flows meet, which, according to the results of research by M. A. Glinkov and N. A. Kaloshin [3], is recommended to be calculated by the formula

\[ k_1 = e^{-1.25\alpha\,0.575}; \tag{14} \]

\(\alpha\) is the angle between the axes of the flows, expressed in fractions of a right angle;

\(k_2\) is the coefficient allowing for the influence of the height of the space in which mixing takes place; according to Rummel’s data [5], M. A. Glinkov found that

\[ k_2 = e^{-4\left(\frac{d}{H}\right)^{1.45}}, \tag{15} \]

\(H\) is the height of the space in which mixing takes place;

\(k_3\) is the coefficient allowing for the influence of the temperature in the burning flare on the intensity of mixing, where

\[ k_3=\left(\frac{T}{T_0}\right)^{\frac{1}{3}}, \tag{16} \]

where \(T\) is the mean temperature of the flare,

\(T_0\) is the temperature of the flows at the outlet from the burner.

M. A. Glinkov calls the ratio

\[ \frac{E-E_k}{E_n-E_k} \]

the degree of burnout and, comparing experimental data from other studies with the calculated length of the burning flare \(x\), obtained from equation (11), considers that a satisfactory agreement is obtained between the actually observed length of the burning flare and the calculated one. Apparently, the inevitable prerequisite for such an assertion was the assumption of an identical degree

burnout in all experiments, while in the work of M. A. Glinkov this is not indicated, nor are data given on the actually observed degrees of burnout.

With regard to the equation proposed by M. A. Glinkov for calculating the length of the burning flare (11), we consider it necessary to state the following considerations: in order to calculate the length of the burning flare by equation (11), proposed by M. A. Glinkov, it is absolutely necessary first to choose (assign) the degree of burnout. What should guide the choice of the degree of burnout? Obviously, some permissible amount of unburned gas—for example, such an amount at which the content of combustible elements becomes less than the possible error of instruments analyzing the gas, or at which a practically permissible incompleteness of gas combustion is obtained (0.5–1.0%). But does this mean that we can visually observe the full length of the burning flare at a given degree of burnout? This is doubtful! At very high degrees of burnout, when an insignificant amount (1.0–1.5%) of combustible remains in the combustion products, we undoubtedly are not able visually to observe the “tail” of the flare with such a small concentration of combustible. In addition, at high degrees of burnout the length of the burning flare calculated by equation (11) turns out to be excessively and unrealistically large; this is not difficult to verify by making a calculation: for example, for simplicity let us calculate the length of the burning flare of hydrogen with oxygen at \(W_2 \simeq W_1\), under the condition that the amount of unburned \(H_2\) does not exceed \(1\%\). Then, according to equation (11), we obtain

\[ \frac{x}{d}=2560 \]

and if we take \(d=5\ \text{mm}\), then

\[ x=12800\ \text{mm}! \]

Finally, in view of the fact that

\[ \frac{x}{d}=f\left(\frac{E-E_k}{E_n-E_k}\right) \]

varies along an asymptotically decaying curve, at small values of the concentration of combustible in the combustion products an insignificant change in \(E\) causes a sharp change in \(\frac{x}{d}\); if in the preceding example the amount of unburned \(H_2\) is taken to be not \(1\%\), but \(1.5\%\), then we obtain

\[ \frac{x}{d}=2320 \]

and \(x=11600\ \text{mm}\), i.e., the flare length is reduced by \(10\%\).

The foregoing convinces us that an arbitrary choice of the degree of burnout—and still more the choice of such a degree of burnout at which the concentration of combustible in the flare represents an insignificant, although practically undoubtedly desirable and actually attainable and observable, quantity—when solving equation (11), will lead to an excessively and unrealistically large flare length. Therefore it may be assumed that equation (11) can expediently be used not for calculating

\[ \frac{x}{d}, \]

but for calculating

\[ \frac{E-E_k}{E_n-E_k} \]

from the value of

\[ \frac{x}{d}; \]

in other words, from the value of the flare length actually attainable under one or another specific set of conditions, with the aid of equation (11) one can approximately form an idea of the maximum possible degree of burnout.

We consider it possible to apply equation (11) to calculate the length of the burning flare under the condition when the amount of air in the stream exceeds that theoretically required, i.e., in combustion with excess air. In this case there is no need for an arbitrary choice of the magnitude of the degree of burnout; combustion in the flare will end when the combustible particles

gas and air mix in the stoichiometric ratio, i.e. at $\alpha=1$. In accordance with this, the left-hand side of the equation is expressed in an explicit form depending on the excess-air coefficient $\alpha$.

Indeed, let $E_n$ be the initial concentration of fuel, i.e. $E_n=1$; then, with complete mixing, if air $\alpha V_{\text{theor}}$ is introduced,

\[ E_k=\frac{1}{1+\alpha V_{\text{theor}}}; \]

$E$ is the concentration of fuel on the axis of the jet at distance $x$, i.e. at the end of the burning flare; obviously, for complete combustion

\[ E=\frac{1}{1+V_{\text{theor}}}. \]

Substituting the values $E$, $E_n$, $E_k$, we obtain the value of the characteristic parameter (the left-hand side of the equation)

\[ P=\frac{\alpha-1}{\alpha(1+V_{\text{theor}})} . \tag{17} \]

With this interpretation of the expression on the left-hand side of equation (17), three cases are possible:

1) when $\alpha=1$, then $P=0$ and $\dfrac{x}{d}$ from equation (11) is equal to infinity. This was to be expected; and in order that $\dfrac{x}{d}$ be finite, it is necessary to prescribe the degree of burnout. However, from the preceding considerations it was clear to what unrealistic results an arbitrary specification of the degree of burnout leads;

2) when $\alpha<1$, i.e. in combustion with a deficiency of air, $P$ will have a certain negative value. The minus sign indicates that combustion is incomplete, while the absolute value $\dfrac{x}{d}$ obtained in this case will always be less than the actual length of the flare during combustion in an open burner in atmospheric air, since the burnout of the flare will occur due to additional diffusion of air from the surrounding medium. For the conditions of a closed burner, the value $\dfrac{x}{d}$ will denote that finite length of the flare at which the primary air will be completely exhausted;

3) when $\alpha>1$, a value of $\dfrac{x}{d}$ is obtained at which the combustion process will be completely completed.

Finally, it should be pointed out that it would be absurd to apply equation (11) to the case in which the flare issuing from the burner of only a single gas stream is considered.

We allow ourselves to consider that the above reasoning on the form and value of the characteristic parameter more fully and more accurately reflects the qualitative and quantitative aspect of the phenomenon of a burning flare.

In what follows, equations (11 and 17) are used by us for comparison with data from experimental studies of a burning gas flare.

Our studies of flare convective-turbulent combustion were carried out by observing a burning flare; we believe that, although complex processes occur in a burning flare as a result of the interaction of the phenomena of hydrodynamics, heat transfer, and chemical kinetics, nevertheless

each of these phenomena separately has been studied and generalized to a sufficient degree, whereas the published experimental material on the observation and study of a burning flare is not so extensive that generalizations could be made. Our investigations of a burning flare were carried out in the combustion of illuminating, water, and blast-furnace gases both under laboratory conditions and, in part, in industrial thermal installations, and concern the following questions:

1) the length of the burning flare, the composition of the combustion products, and the temperature distribution in the flare as functions of the velocities of the gas and air streams and of their angle of intersection;

2) violation of the stability of flare combustion as a result of separation of the flare from the burner mouth;

3) the influence of recirculation of combustion products on the burning flare.

It should be noted that the methodology used in studying the burning flare was conventional and simple, but we permit ourselves to consider that it satisfies the requirements of engineering practice, while certain generalizations, sometimes formulated in the form of a mathematical dependence, may be useful in regulating the flare and in designing gas-burning devices of the flame convective-turbulent combustion type.

1. Influence of the discharge velocity and the angle of intersection of the streams on the length and temperature distribution in the burning flare. Separation of the flame flare

The experimental study of the burning flare was carried out with the aid of three-slot and seven-slot sectional flat burners.

The three-slot burner had an inner slot of rectangular cross section, 60 by 2 mm, for the discharge of gas, and two outer side slots of rectangular cross section, 60 by 5 mm, for the discharge of air; the flare length was measured directly with the burners arranged in a vertical plane. Temperature was measured simultaneously at 10 points along the length of the flare by Pt—PtRh thermocouples with an unprotected junction. Along the length of the flare the thermocouples were placed at the following distances: the 1st thermocouple—55 mm from the burner mouth, and the subsequent 9 thermocouples at intervals of 45 mm, so that the length of the flare measured by them was 410 mm.

In determining the degree of completeness of mixing and combustion of the gas, the vertical flame flare was covered with a metal hood with the flare outlet upward. Gas samples for analysis were taken through holes in the side surface of the hood.

Illuminating gas was supplied to the burners under pressure in the gas pipeline; its flow rate was measured with a standard orifice,

\[ \frac{d}{D}=\frac{12.5}{17}\simeq 0.736 \]

and with a differential manometer of the “Teplopribor” factory. Air was supplied to the burners by a fan, and the air flow rate was likewise determined by orifices

\[ \frac{d}{D}=0.5 \]

and differential manometers. The orifices were calibrated before the experiments.

Variation of the discharge velocity was effected by changing the flow rate of gas and air; the angle of intersection of the streams was changed by the corresponding arrangement of the burner slots.

The results of the investigation of the length of the burning flame flare of illuminating gas and of the temperature distribution in the flare during the discharge of gas and air at various angles of intersection of the streams and at various discharge velocities are presented in Table 1; the velocity of the gas stream was constant and equal—

Table 1

Length of the burning flare of illuminating gas flame and temperature distribution in the flare during the outflow of gas and air at various angles of intersection of the streams. Slotted, flat burner

No.	Angle of intersection of streams, \(\beta\)	Gas velocity \(w_2\)	Air velocity \(w_{\mathrm{в}}\)	Length of burning flare \(L\), mm	Length of burning flare \(l\), mm	Temperature along the burning flare, °C: measurement point 1	2	3	4	5	6	7	8	9	10	Excess-air coefficient \(\alpha\)
1	0°	2,38	0	510	0	550	925	950	860	1000	1060	1080	925	890	820	0,0
2	0°	1,8	1,8	485	80	610	1060	825	970	1040	890	910	620	720	410	1,0
3	0°		2,2	190	75	490	540	950	1010	1050	1000	950	900	720	450	1,23
4	0°		2,6	140	70	460	350	970	1020	1070	970	950	920	740	500	1,45
5	0°		3,1	110	69	650	770	970	870	1120	970	925	850	750	540	1,72
6	5°		1,3	375	78	75	375	440	460	910	825	975	1000	975	825	0,74
7	5°		1,8	325	72	320	375	225	600	1070	1060	1060	1070	990	750	1,0
8	5°		2,2	170	70	320	320	225	590	590	975	1125	1200	980	740	1,24
9	15°		1,3	320	75	525	610	870	1200	1180	1200	1140	1120	925	750	0,74
10	15°		1,8	260	67	525	590	675	980	1230	1200	990	760	600	375	1,0
11	15°		2,0	185	58	525	600	880	980	1275	1190	1190	1190	880	600	1,13
12	30°		1,3	260	64	150	375	600	750	920	930	1060	1210	900	590	0,74
13	30°		1,80	200	51	225	225	225	300	300	530	825	1060	750	375	1,00
14	30°		2,0	150	51	150	200	270	280	350	600	820	1280	740	370	1,13

2.38 \(\dfrac{m}{sec}\), while the velocity of the enveloping air stream varied from 0 to 3.1 \(\dfrac{m}{sec}\); the angle of incidence of the streams varied from 0 to 30°. The outflow regime of the gas and air was formally laminar, since the values of \(Re\) calculated in the usual way did not exceed 2000. Undoubtedly, the burning flame investigated, formed when streams with different velocities issue and at angles of incidence of the streams \(>0^\circ\), must be called conditionally laminar, since although the outflow velocities of the streams were below the critical ones, mixing of the streams occurred not only under the influence of microdiffusion but, to a large extent, under the influence of external turbulizing factors. It should be supposed that Prof. B. I. Kitaev, who published the results of his investigations on the study of a burning flame under the title “laminar flame regime” [10], undoubtedly had in mind a conditionally laminar regime, and in this connection G. P. Ivantsov’s [18] rather sharp objection to such a name is hardly appropriate.

Fig. 1

According to the data given in Table 1, the curves in Fig. 1 have been plotted: on the ordinate axis are laid off the values of the relative velocity \(\dfrac{W_2}{W_1}\), and on the abscissa axis—the relative flame length \(\dfrac{L}{d_1}\), where \(L\) is the visually observed length of the burning flame, \(d_1\) is the reduced diameter of the slot through which the gas flowed from the burner; for a three-slot burner \(d_1\) was determined as twice the width of the gas slot, i.e. \(2.2 = 4\) mm. Curve (1) characterizes the change of \(\dfrac{L}{d_1}\) for parallel outflow of the streams; curve (2)—

change in \(\dfrac{L}{d_1}\), for the angle of meeting of the streams \(\beta = 5^\circ\); curve (3) is the change in

\[ \frac{L}{d_1} \]

for \(\beta = 15^\circ\), and curve (4) for \(\beta = 30^\circ\).

In Fig. 2 the curves are plotted for the change in the geometric factor of the flame structure, \(\dfrac{l}{L}\), i.e., the ratio of the length of the inner dark cone of unignited gases to the length of the enveloping burning flare, as a function of the outflow velocity and for different angles of meeting of the streams: \(0^\circ\), \(5^\circ\), \(15^\circ\), \(30^\circ\).

From the results of the study presented in Table 1 and the curves shown in Figs. 1 and 2, it follows:

1. If one compares the length of the burning flare formed during the outflow of gas alone, when \(W_2 = 0\), with the length of the burning flare formed during the outflow of parallel gas and air streams, then, at the same gas outflow velocity in both cases, the greatest length of the burning flare is obtained at \(W_2 = 0\), i.e., when the air necessary for combustion diffuses to the combustion surface from the surrounding atmosphere. Thus, at a gas outflow velocity \(W_1 = 2.38\ \text{m/sec}\) and \(W_2 = 0\), \(L = 510\ \text{mm}\), while at \(W_2 > 0\), \(L\) decreases; at \(W_1 = 5\ \text{m/sec}\) and \(W_2 = 0\), \(L = 570\ \text{mm}\), while at \(W_2 > 0\), \(L\) decreases. The mechanism of mixing

Fig. 2

of gas with air during the outflow of a single gas stream, when \(W_2 = 0\), is due partly to molecular diffusion and partly to the mechanical entrainment into motion caused by the mutual friction of stationary layers of the surrounding air. In the forced outflow of both streams, the role of molecular diffusion in the mixing process decreases, while the influence of the mutual mechanical interaction of the streams increases, as a result of which the range of the flare decreases.

A vertical flare burning in an unbounded space, with the outflow of gas alone, is outwardly to a certain extent similar to the phenomenon of the outflow of a free jet; therefore some authors draw an analogy between these phenomena. Thus, B. I. Kitaev believes that in his investigations the gas burned in the surrounding air with the formation of a flare under the conditions of a free jet [10]; N. N. Dobrokhotov and his collaborators, in experiments on a fire model, established that ignition of the flare does not change the regularities observed in a cold flare [19]. Therefore it is

a tempting attempt to extend the regularity of a free jet to the determination of the length of a burning flare. However, such an attempt leads to results that diverge sharply from experimental data.

Let us take the flow-rate equation in the form proposed by G. P. Ivantsov on the basis of a generalization of experimental data on the study of the phenomenon of the free jet,

\[ V = 0.82\, V_0 \left(0.32 \frac{L}{d} + 1\right), \tag{18} \]

where \(V\) is the volume of gas in the jet at a distance \(L\) from the nozzle;
\(V_0\) is the same at the nozzle exit;
\(L\) is the range of the flare;
\(d\) is the nozzle diameter.

Applying this equation to the conditions of a burning flare for the purpose of calculating its length, it would seem permissible to regard the ratio \(\frac{V}{V_0}\) as the ratio of the volume of combustion products to the volume of gas. Then, for the conditions of the 1st experiment, taking into account the stoichiometric calculation of combustion and the maximum temperature of the flare, we obtain

\[ \frac{V}{V_0} = 25; \]

substituting the value \(\frac{V}{V_0}\) into equation (18), we obtain the calculated

\[ \left(\frac{L}{d}\right)_{\text{calc}} = 92. \]

The actually measured relative length is

\[ \left(\frac{L}{d}\right)_{\text{exp}} = 127. \]

The discrepancy between the experimental data and the calculated data is approximately 37%.

Let us take the flow-rate equation in the form proposed by D. I. Lyakhovskii and S. N. Syrkin [1] on the basis of a generalization of the authors’ experimental data on the study of the phenomenon of the free jet,

\[ V = V_0 \left(1 + \frac{L}{4.83\, d}\right). \tag{19} \]

Applying this equation to the conditions of a burning flare and substituting the value \(\frac{V}{V_0} = 25\), we obtain

\[ \left(\frac{L}{d}\right)_{\text{calc}} = 116. \]

The discrepancy between the experimental data and the calculated data is smaller than according to the equation proposed by G. P. Ivantsov, but nevertheless amounts to about 10%.

Thus, it would seem that, for an approximate judgment of the length of a burning flare when only one gas issues, one may take as a basis the flow-rate equations for a free jet, making the assumption that the length of the burning flare is equal to the range of the free jet when the flow rate in the jet \(V\) is equal to the volume of the products formed during combustion.

combustion. However, we believe that such a formal extension of the equations of a free jet to a burning flare, on the basis of a greater or lesser convergence of calculated quantities with actual ones, is hardly legitimate.

A confirmation of the correctness of our assertion is at least the well-known and easily observed fact that the length of a burning flare issuing from one gas depends on the gas outflow velocity: the greater the gas outflow velocity, the longer the burning flare, whereas in the equations discussed above the gas velocity does not enter and, consequently, the value \(\left(\dfrac{L}{d}\right)_p\) of the burning flare should remain the same at any gas outflow velocities. An attempt to apply the flow-rate equations of a free jet in order to calculate the length of a burning turbulent flare, using for this the experimental data of B. I. Kitaev [10], likewise remained unsuccessful. In a turbulent flare, as B. I. Kitaev showed, the length of the burning flare does not depend on the gas outflow velocity; however, the values \(\left(\dfrac{L}{d}\right)_p\) calculated by us from the flow-rate equations of a free jet diverge still more considerably from Kitaev’s experimental data.

1. According to the experimental data of our investigation, the length of the burning flare formed by parallel flows decreases as the velocity of the enveloping air outflow increases; it is interesting to note that, beginning from point \(A\), there occurs a sharp change in the rate of shortening of the flare. In the portion of the curves above \(A\), a rather significant rate of shortening of the flare is observed with increasing \(\dfrac{W_2}{W_1}\), while in the portion of the curves below \(A\) the rate is more retarded. The point \(A\), at which the change in the rate of shortening of the flare occurs, corresponds to such a ratio between the gas and air flows at which the excess-air coefficient is approximately equal to unity. Consequently, an increase in the excess-air coefficient in the enveloping air flow sharply shortens the length of the burning flare and, conversely, in the combustion of mixtures with a deficiency of air in the enveloping air flow, with increasing \(\dfrac{W_2}{W_1}\) there occurs a slow shortening of the lengths of the burning flare. It seems to us that this observation should be taken into account as a certain new determining factor in formulas and equations by which one attempts to solve the question of calculating the length of a burning flare as a function only of the value \(\dfrac{W_2}{W_1}\).

However, the experimental data at our disposal are insufficient to reveal the quantitative value of the influence of the excess-air coefficient.

1. As was already indicated above, for calculating the length of a burning flare at different outflow velocities and angles of intersection of the flows, M. A. Glinkov proposed equations. Let us use equations (11, 17) and compare the calculated lengths of the burning flare with the observed ones. The results of the calculations are given in Table 2.

The calculation of the length of the burning flare \(x_{\text{calc}}\) was carried out for cases in which the excess-air coefficient in the flows \(\alpha > 1\) or \(\alpha < 1\); at the same time \(V_{\text{theor}} = 3.74 \dfrac{m^3}{m^3}\); calculation of the length of the burning flare was not carried out for cases when \(\alpha = 1\) and when \(W_2 = 0\), for the reasons indicated above.

Table 2

Comparison of experimental data with calculated data for measuring the length of the burning flare for a three-slot burner

No. of experiments	$k_0$	$k_1$	$k_2$	$k_3$	$k_z$	$x_{\text{calc}}$, mm	$l$, mm	$\dfrac{l}{x_{\text{calc}}}$
3	0.5	1	1	1.65	0.83	180	190	1.05
4	0.5	1	1	1.65	0.83	126	140	1.10
5	0.5	1	1	1.71	0.86	100	110	1.10
6	0.5	0.80	1	1.63	0.65	116	375	3.24
8	0.5	0.80	1	1.77	0.71	152	170	1.12
9	0.5	0.63	1	1.77	0.56	100	320	3.20
11	0.5	0.63	1	1.80	0.57	165	185	1.12
12	0.5	0.50	1	1.77	0.45	80	260	3.25
14	0.5	0.50	1	1.81	0.46	133	150	1.13

Comparing the visually observed length of the burning flare $L$ with the calculated $x_{\text{calc}}$ for such a ratio between air and gas, when $a > 1$, we note that the discrepancy between them lies within the range from 5 to 13%, and, as a rule, the observed $L$ is greater than $x_{\text{calc}}$. This gives grounds to assert that the approximate calculation of the length of the burning flare by the equation proposed by M. A. Glinkov and modified by us with respect to the characteristic parameter is quite possible and expedient; moreover, owing to the modification of the characteristic parameter proposed by us, the necessity of arbitrarily choosing the degree of burnout is eliminated.

As was to be expected, a sharp difference is observed between the actual length of the burning flare and the calculated length when $a < 1$, and the discrepancy of these quantities is the greater the smaller $a$ is. This discrepancy is explained, as was already indicated above, by the fact that $x_{\text{calc}}$ determines the length of the flare where the oxygen necessary for combustion disappears; however, in combustion under atmospheric-air conditions, oxygen diffuses from the surrounding medium, and not from the air in the stream, and thereby the combustion process is prolonged.

The calculated and experimental data on the length of the burning flare confirm the great influence of the angle of intersection of the streams on the length of the flare: the larger the angle, the shorter the flare.

The curves in Fig. 2 show that the relative length of the inner dark cone depends strongly on $\dfrac{W_2}{W_1}$ (more precisely, on the excess-air coefficient) when the streams issue in parallel; the larger the excess-air coefficient $a$, the larger $\dfrac{l}{L}$; whereas for streams directed toward one another at one angle or another, the value $\dfrac{l}{L}$ depends little either on the angle of intersection of the streams or on the excess-air coefficient, which indicates the increasing role of turbulent mechanical mixing of the streams in comparison with the influence of microdiffusion.

A series of experiments was also carried out to measure the length of the burning flare when the gas and air were preheated to 70, 220°C. These experiments confirmed, as in the case of the study of diffusion combustion, that the burning flare is considerably shortened with preliminary heating.

Gas analysis for CO, CO₂, and O₂ in the combustion products, sampled along the height of the flare at distances of 210 mm and 450 mm from the burner mouth, confirms the increase in the intensity of mixing and the shortening of the length.

of the burning flare with an increase in the angle at which the flows meet, the difference in their outflow velocities, and with an increase in the preliminary heating of the gas and air.

When organizing flare convective-turbulent combustion, cases are often observed in which the flare separates from the burner mouth, entailing extinction of combustion or transfer of combustion to some distance from the burner mouth.

In order to determine the conditions for flare separation and to explain the causes of flare separation and of its burning at a distance, experiments were carried out in burning illuminating gas by means of a four-ring cylindrical burner; the burner had the following dimensions: an inner gas tube 9 mm in diameter, then an enveloping air ring with a diameter from 11 to 21 mm, then a gas ring with a diameter from 23 to 28 mm, and, finally, an outer air ring with a diameter from 30 to 35 mm. With such a burner it is possible to organize two-flare combustion: the inner flare is formed when gas flows out through the inner tube and air through the air ring enveloping it; the outer flare is formed when gas flows out through the gas ring (diameter 23–28 mm) and air through the outer air ring. By regulating the gas and air flow rates, the dimensions of the inner and outer flares can be changed. Fig. 3 shows flare configurations at various flow rates.

In Fig. 3A, at equal outflow velocities of gas and air, a continuous flare is formed without a distinct demarcation of the surfaces of the inner and outer flares. In Fig. 3B there is a sharp separation of the preceding continuous flare into two—the outer and the inner; the inner flare is elongated (its length is 585 mm), the outer one is shortened (its length is 157 mm) owing to a reduction in the supply of gas and air for the outer flare.

With a further reduction of the gas and air flow rate for the outer flare, its length decreases to 75 mm (Fig. 3C) and to 28 mm (Fig. 3D); in the latter case the outer flare plays the role of an igniting, pilot fire belt.

Leaving unchanged the gas and air flow rate for the outer pilot belt, let us increase the outflow velocity of gas and air for the inner flare. Then, as shown in Fig. 3E, the inner flare separates from the burner mouth and burns steadily at a distance of 60 mm from the outer flare; the outflow velocity of gas and air for the inner cone is about 5 m/sec. When the outflow velocity of gas and air for the inner cone is increased to 10.1 m/sec, the separated inner flare becomes elongated, as shown in Fig. 3F; the distance between the separated inner flare and the outer one increases to 140 mm; a further increase in the outflow velocity of gas and air for the inner flare extinguishes the flame flare. But if a refractory tile is placed in the path of the flare, as was done in the experiments, then stable combustion of the separated flare occurs on the surface of the refractory tile, despite the fact that the distance from the burner to the tile is 200 mm. Photographs are shown in Fig. 4A and Fig. 4B.

To explain the causes that determine the separation of the inner flame flare from the burner, the stable combustion of the separated flare at some distance from the burner, and then the extinction of the flame flare, in our opinion the following proposition is applicable: we shall use the concept, introduced by us, of the apparent (conditional) flame-propagation velocity, which we used in describing the phenomena of curved and forced flame flares [20]. The magnitude of the apparent flame-propagation velocity is determined primarily by the magnitude of the diffusion factor (mixing coefficient)

\[ U_k = UK, \tag{20} \]

where \(U_k\) is the apparent velocity of flame propagation;
\(U\) is the normal (according to Michelson) velocity of flame propagation;
\(K\) is the diffusion factor (mixing coefficient).

Fig. 3A

Fig. 3B

Fig. 3C

It is obvious that \(K\) varies within the limits:

\[ \begin{aligned} &\text{from} \quad K = 0\\ &\text{to} \quad K = 1. \end{aligned} \]

For \(K = 0\)

\[ U_k = 0, \]

there is no combustion, owing to the absence of mixing;

for \(K = 1\)

\[ U_k = U, \]

Fig. 3D

Fig. 3E

Fig. 3F

i.e., for premixed mixtures the flame propagation velocity depends on the composition of the combustible mixtures.

If mixing occurs simultaneously with the combustion process, then \(K < 1\), and the worse the conditions for mixing, the smaller \(K\) is.

As is known, for a stable combustion process it is necessary that the outflow velocity of the combustible mixture be greater than the flame-propagation velocity, i.e.,

\[ \frac{W}{U_k} > 1. \]

Fig. 4A

However, at large values of this ratio, blowoff and extinction of the flare are possible owing to the fact that the quantity of combustible mixture arriving at the flame front is so large in comparison with the quantit-

Fig. 4B

…of the combustible mixture, that the heat released during combustion is insufficient to heat the combustible mixture to the ignition temperature, or, in other words, the rate of heat transfer to the incoming combustible mixture is less than the rate at which the combustible mixture enters the flame front.

Under our conditions, gas and air are supplied to the flame front of the internal torch as parallel streams; even at unequal gas and air outflow velocities, when it may be assumed that turbulent mixing takes place, the mixing intensity is small, the diffusion factor (mixing coefficient) \(K\) is a rather small quantity, and therefore the apparent flame-propagation velocity \(U_k = UK\) is also a small quantity; consequently, blow-off of the torch may occur even at comparatively low outflow velocities. In our experiments, blow-off of the torch occurs at a gas outflow velocity \(W_g = 4.5\ \mathrm{m/sec}\) and at air outflow velocities \(W_a\) ranging from \(5.5\ \mathrm{m/sec}\) to \(10\ \mathrm{m/sec}\).

The numerical values of the diffusion factor and of the apparent flame-propagation velocity are unknown. However, the order of magnitude can be determined in the following way: in investigations of the combustion of producer gas in partially prepared combustible mixtures with burners with ignition [21], the limiting ratio

\[ \frac{W}{U} = 70, \]

above which combustion is unstable. If the limiting value is taken as

\[ \frac{W}{U} = 70 \]

(although for an illuminating-gas flame this ratio should be smaller, since, under equal conditions, diffusion in the illuminating-gas–air system is less intense than diffusion in the producer-gas–air system), then the value \(U_k\) is determined from the relation:

\[ \frac{W}{U_k} \simeq 70; \qquad U_k \simeq \frac{10}{70} = 0.14\ \mathrm{m/sec}. \]

It is appropriate to recall that the normal (according to Michelson) flame-propagation velocity in limiting combustible illuminating-gas–air mixtures is expressed by a value of the order of \(0.2\ \mathrm{m/sec}\).

Why does a detached torch ignite at some distance from the burner? Owing to the decrease in the axial and mean discharge velocity of the stream, according to the principle of constancy of momentum \(mW_0 = \mathrm{const}\), at some distance from the burner the stream velocity, and also owing to the increase in the apparent flame-propagation velocity due to greater mixing, the ratio

\[ \frac{W}{U_k} \]

decreases; these velocities come into a corresponding state, in which the heat release during combustion corresponds to heating the mixture to the ignition temperature and, in the presence of an ignition source (an external igniting torch or a heated surface), stable combustion of the detached torch occurs. In our experiments, the path length for transition of the detached torch into the corresponding state is \(60\text{--}140\text{--}200\ \mathrm{mm}\), depending on the outflow velocity. Naturally, in the absence of ignition carried out in one way or another (external igniting torch, heated surface), the detached torch does not ignite. Under chamber-combustion conditions, ignition of the detached torch is ensured by radiation from the lining. However, observance of only one condition for stable combustion of the detached torch—the presence of an ignition source—is insufficient; it is necessary that, in the corresponding state, the composition of the combustible mixture be within the ignition limits. Under the conditions of our experiments, at an outflow velocity of the enveloping air exceeding \(15\ \mathrm{m/sec}\), with any ignition sources…

the detached flare did not ignite. The reason is that, at \(W_{\mathrm{v}}=15\ \text{m/sec}\) and a gas outflow velocity \(W_{2}=4.5\ \text{m/sec}\), the flow rate of the combustible mixture is equal to: gas—

\[ W_{2}\frac{\pi d^{2}}{4}\,3600 = 4.5 \cdot 0.785(0.009)^{2}\,3600 = 1\ \text{m}^{3}/\text{h}; \]

air—

\[ W_{\mathrm{v}}\frac{\pi}{4}(D^{2}-d_{1}^{2})\,3600 = 15 \cdot 0.785\bigl[(0.021)^{2}-(0.011)^{2}\bigr]\cdot 3600 = 13.5\ \text{m}^{3}/\text{h}. \]

Composition of the combustible mixture:

\[ \text{gas—}6.9\%;\quad \text{air—}93.1\%. \]

Thus, the amount of gas is less than the lower ignition limit permissible for illuminating gas; therefore the detached flare does not ignite.

2. Influence of the Conditions of Mixing of Gas and Air and Recirculation of Combustion Products on the Temperature Distribution along the Length of the Flare

The temperature in a flame flare is not the same throughout the entire volume occupied by the flare; it varies both along the length of the flare and across its cross-section. Detailed studies by V. N. Timofeev and P. I. Sychev [6], K. Rummel [5], and Pistor [8] have shown that the temperature field of a flame flare consists of a series of isotherms; the character of the isotherms depends on the conditions of mixing of gas and air. The highest isotherms are located along the axis of the flare; as the distance from the flare axis increases, the temperature decreases. With an increase in the angle at which the streams meet, the isotherms along the length of the flare become shorter and the high-temperature zone is concentrated in the core of the flare; thus, for example, according to the data of V. N. Timofeev and P. I. Sychev, with parallel discharge of the streams the \(900^\circ\text{C}\) isotherm has an extent of \(650\ \text{mm}\) along the length of the flare; with inclined discharge of the streams at an angle of \(45^\circ\), the length of the isotherm is reduced to \(575\ \text{mm}\). The nonuniformity of the temperature in the flare along its length and across its section is due to the influence of the rate of heat release and of direct heat transfer by radiation.

Under these conditions, an unambiguous determination of the temperature along the length of the flare throughout its entire volume is impossible. However, in engineering practice it is absolutely necessary to operate with the determination of some mean temperature over the flare cross-section, varying unambiguously along the length of the flare; without such a determination it is impossible to carry out any calculation of heat exchange in the working space of a combustion chamber. An admissible justification for such a representation of the mean flame temperature varying along the length of the flare is the proposition that the heat released during combustion is distributed uniformly over the volume of the flare at a given segment of the flare length; at another segment of the flare length, depending on the conditions of heat release (combustion) and heat exchange, another temperature, averaged over the volume, is established, and so on. As Heiligenstaedt [13] indicates, even with a 30% deviation across the flare section of the mixing (combustion) time of individual gas particles, the deviation of the heat content of the flare over the section does not exceed 3% of its mean value; with such a representation of changes in the mean heat content over the flare section, it becomes possible to calculate the distribution of temperature along the length of the flare.

An analysis of the calculation equations proposed by various authors was given earlier. For greater clarity of the representations of the temperature distribution along the length of the flare, a grapho-analytical method of calculation and representation in the form of a \(t-k-\varphi\) diagram is proposed. The basic assumptions adopted—

for constructing the diagram, in accordance with the heat-release and heat-transfer equations along the length of the flare proposed by Heiligenstaedt (4.6), we have: the equation for the change in the flare temperature in the process of heat release without heat transfer, in the presence of regeneration of heat from the combustion products,

\[ t=-\frac{q_x}{C_p}=\frac{1}{C_p}\left[rq_0+q_0\left(1-e^{-k\tau}\right)\right], \]

but \(1-e^{-k\tau}=\beta\) is the degree of completeness of heat release, depending on the conditions of mixing of gas and air, i.e., on the burner characteristic \(k\) and the combustion time \(\tau\).

Consequently,

\[ t=\frac{(r+\beta)q_0}{C_p}. \tag{21} \]

For definite values of \(r\), \(q_0\), \(C_p\), the temperature \(t\) is a function of the completeness of heat release \(\beta\).

For \(\beta=0\), when \(\tau=0\), the temperature of the non-ignited flare is determined by the degree of heat regeneration

\[ t_0=\frac{rq_0}{C_p}. \]

For \(\beta=1\), when \(\tau=\infty\), the flare temperature upon complete completion of combustion will be equal to the calorimetric combustion temperature

\[ t=\frac{(1+r)q_0}{C_p}. \]

However, at the value \(k\tau=6\), the degree of completeness of heat release already becomes equal to \(\beta=0.998\), i.e., practically the process of heat release (combustion) may be considered completed.

The equation of temperature losses due to heat transfer of the flare along the path of motion, according to [13], has the form:

\[ \Delta t=\frac{1}{C_p}\left[273\,C_p+(1+r)q_0\right]\left(1-e^{-m\tau}\right)- \]

\[ -\frac{m}{(k-m)C_p}q_0\left(e^{-m\tau}-e^{-k\tau}\right). \tag{22} \]

The equation of the actual temperature according to [13] is

\[ t_d=t-\Delta t. \]

\[ t_d=\frac{1}{C_p}\left[\frac{1}{k-m}q_0\left(e^{-m\tau}-e^{-k\tau}\right)+rq_0e^{-m\tau}-273\,C_p\left(1-e^{-m\tau}\right)\right]. \tag{23} \]

In the absence of heat transfer, equation (23) reduces to (21).

Fig. 5 shows a \(t-k-\beta\) diagram. The diagram is constructed in 3 quadrants: for the curves of the 1st and 2nd quadrants the common ordinate axis is \(\beta\), with values from \(\beta=0\) to \(\beta=1\); the abscissa axis in the first quadrant is the parameter \(k\tau\). This abscissa axis is common to the 2nd and 3rd quadrants; along the ordinate axis in the 3rd quadrant are plotted the values of \(k\)—the burner characteristics, whose magnitude varies from 0 to 10.

In the 3rd quadrant straight lines of equal time intervals \(\tau\) are constructed; in the 2nd quadrant, the curve \(\beta=1-e^{-k\tau}\) is constructed as a single-valued exponential function of the parameter \(k\tau\).

The following procedure is recommended for using the diagram.

1) From the burner characteristic “\(k\)” and the residence time of the combustible gas mixture in the given section of the heating system, the value of \(\beta\)—the coefficient of completeness of heat release—is found on the curve \(\beta=f(k\tau)\) (right-hand part of the diagram). The residence time of the combustible gas mixture must be coordinated with its velocity of motion and with the length of the given section of the heating system under consideration, according to the relation:

\[ \tau=\frac{L}{W_t}, \]

2) From the degree of completeness of heat release \(\beta\) and from the curve in the left-hand part of the diagram, the temperature of the flame flare at the given point of the heating system is found.

t–k–\(\beta\) diagram.

Fig. 5

of the heating system. The above calculation procedure is convenient to use for characterizing the temperature regime of an already operating heating system (the working space of a furnace).

In the case of designing a new heating system, another procedure for calculations and for using the diagram is recommended, namely: from the previously compiled heat balance of the individual sections of the heating system and from the temperature curve in the left-hand part of the diagram, the value of \(\beta\) is found on the ordinate axis. This point is transferred to the curve in the upper right-hand part of the diagram and, projecting it onto the abscissa axis, the value of \(k\tau\) is determined. By varying \(\tau\) or, correspondingly, \(W_t\), one can find the optimal burner index “\(k\).” A burner having such an index “\(k\)” will provide the distribution of heat and temperature specified by the heat balance.

To illustrate the use of the diagram, a temperature curve was constructed for the heating system of coke ovens (verticals) of the Becker type when heated with coke-oven gas.

The following initial data were adopted for constructing the temperature curve:

a) burner index \(\kappa = 1.5\); this index, according to the classification given earlier, is most applicable to burners of the heating system of coke ovens. Deshalit [22], in calculating blast-furnace-gas combustion, found that \(\kappa = 1.75\);

b) the total residence time of the flame gases in the heating system of the oven was calculated from the condition that the velocity of motion of the flame gases is \(W_0 = 0.242\ \text{m/sec}\);

c) the intensity of heat transfer of \(1\ \text{m}^3\) of combustion products along the path of the flame gases in \(1\ \text{m}\) was calculated from their temperature and heat content at the outlet from the ascending vertical and checked by the magnitude of the heat-transfer coefficients in the combustion zone. The heat-transfer coefficients adopted for the calculation

\[ \alpha = \alpha_l + \alpha_k = 122\ \frac{\text{kcal}}{\text{m}^2\text{hr deg}} \]

were determined by us at one time (1936) by the heat-balance method and almost coincide with the data of Deshalit [22]. According to Deshalit,

\[ \alpha = \alpha_l + \alpha_k = 118.5\ \frac{\text{kcal}}{\text{m}^2\text{hr deg}}. \]

Table 3

Coordinates of points for constructing the temperature curve of the heating system of coke ovens

No.	Designation	O	A	B	C	D	E	F
1	Combustion time \(\tau\), sec	0	0.5	1.0	1.25	1.50	2.0	2.17
2	Along the height of the vertical \(L\), m	0	0.765	1.53	1.90	2.30	3.06	3.30
3	Parameter \(\kappa \tau\)	0	0.75	1.5	1.88	2.25	3.0	3.26
4	Parameter \(m\tau\)	0	0.18	0.358	0.444	0.538	0.716	0.78
5	Degree of completion of combustion—heat release \(\beta\)	0	0.528	0.777	0.85	0.89	0.95	0.97
6	Flame temperature, °C	815	1470	1560	1570	1460	1360	1350

From Table 3 and from the temperature curve (Fig. 5) it is seen that the maximum flame temperature is located at a distance of \(1.53\)—\(1.90\ \text{m}\) from the hearth of the vertical; here the degree of completion of combustion is expressed by the values: \(\beta = 0.777\)—\(0.85\). A. A. Agroskin [23] determines, from theoretical and experimental data, the position of the maximum at a height of \(1.5\)—\(1.8\ \text{m}\).

The character of the temperature distribution along the length of the flame is most substantially influenced by the conditions of mixing of gas and air, represented by the burner index \(\kappa\); with an increase in \(\kappa\), at constant \(\tau\), the degree of heat release increases. For example, at

\[ \tau_1 = 1\ \text{sec} \quad \text{and} \quad \kappa_1 = 1.5 \qquad \text{the value } \beta_1 = 0.777 \]

and at

\[ \tau_2 = 1\ \text{sec} \quad \text{and} \quad \kappa_2 = 2.5 \qquad \text{the value } \beta_2 = 0.920, \]

i.e. the intensity

the heat release per unit time increases considerably, as a result of which the maximum temperature of the flame shifts toward the burner, thereby increasing the nonuniformity of the temperature distribution and of heat transfer along the length of the flame. In many cases of industrial heat engineering a long flame is required, with a temperature uniformly distributed along its length or height. This is achieved by various methods. One of the most successful methods of organizing a long flame with a more or less uniformly distributed temperature is the recirculation of combustion products—the entrainment of flame combustion products into the flame.

V. E. Grum-Grzhimailo was the first to draw attention to the importance of recirculation of combustion products. At the Third All-Union Heat-Engineering Congress on November 11, 1926, V. E. Grum-Grzhimailo presented a report on “A New Chamber Furnace with Circulation of Furnace Gases Equalizing the Temperature” [24]. Figure 6 shows a diagram of gas recirculation in a chamber furnace.

Fig. 6

The ultramarine furnace is a periodically operating chamber furnace. Its dimensions are: floor—\(7000 \times 4250\) mm, height—2880 mm. Below the level of the furnace floor two fireboxes are placed. On the floor there is a setting of pots for ultramarine. Part of the cooling furnace gases is directed to the firebox through horizontal channels somewhat above the level of the grates. The mechanism of operation of the furnace and of the motion of the furnace gases is as follows: the combustion gases, leaving the firebox at a temperature of \(1200^\circ\mathrm{C}\), rise upward under the crown. The cooling gases that have filled the furnace descend along the furnace walls and between the pots, and part of them is directed into the firebox, while the greater part goes into the flues and the stack.

As a result of the systematic admixture of the cooling furnace gases to the incandescent combustion gases, the temperatures between the top and the bottom are equalized, the cooling furnace gases are vigorously replaced by fresh combustion gases, the cooled gases are heated at the expense of the combustion gases, and the temperature throughout the furnace is generally equalized. The process of systematic admixture of furnace gases to the combustion gases is a self-regulating process.

The uniformity of temperature distribution during the test was achieved as follows:

\[ \text{at the top—}945\text{ to }940^\circ\mathrm{C}, \]

\[ \text{at the bottom—}930\text{ to }920^\circ\mathrm{C}. \]

In carrying out the recirculation of furnace gases, V. E. Grum-Grzhimailo considered that the principles of the hydraulic theory of the motion of furnace gases, based on the laws of hydrostatics, were insufficient for the correct design of furnaces.

At the congress he stated: “I had to pass from hydrostatics, which I taught, to hydrodynamics, since in furnaces built on

on a purely hydrostatic principle, the temperature at the top was higher, and at the bottom always lower. This is a major shortcoming of furnaces built on a purely hydrostatic principle, and it must be eliminated.”

The 3rd All-Union Heat-Engineering Congress, in connection with the report by V. E. Grum-Grzhimailo, confirmed that “hydraulic theory is entering a new stage of development, being based not only on the laws of statics, but also on the laws of hydrodynamics.”

A. A. Agroskin [23] also attaches great importance to the recirculation of furnace gases for equalizing the temperature in the verticals of the heating system of coke ovens with a circular flow. The extensive experimental investigations carried out by A. A. Agroskin revealed the quantitative aspect of the recirculation phenomenon and the causes of this phenomenon.

On the question of the causes that determine the circulation of furnace gases, there are different opinions. Some consider that the circulation of furnace gases (the circular flow) is caused by the ejecting action of the stream of combustion gases from the combustion zone (from the firebox or burner). Others believe that the circulation of gases is a consequence of the difference in static pressures in the ascending and descending gas streams. Some, analyzing the velocity of gas issuing from a burner into the ascending vertical, come to the conclusion that circulation is caused by the ejecting influence of the ascending stream and by the difference in static pressures simultaneously. Undoubtedly, the simultaneous influence of both factors occurs, and the injection factor has the greater significance, the greater the velocity of discharge of gas or flame from the burner or firebox. In those cases, however, when the velocity of discharge of the furnace gas is insignificant (for example, in the ultramarine furnace mentioned), the ejecting influence of the ascending gas stream is negligible.

V. E. Grum-Grzhimailo gives the following indications for analyzing the causes of circulation of furnace gases and for determining the circulation velocity: “The column of gases rising upward has the maximum temperature; the column descending downward has the minimum temperature. The difference between the maximum temperature and the minimum temperature gives the impulse to the motion. If we take the difference in weights of a cubic meter at the maximum \(t^\circ\) and minimum \(t^\circ\), we obtain the pressure difference per 1 sq. m. From this pressure difference per 1 sq. m one determines the head that drives the furnace gases. The head creates the circulation velocity. The greater the difference between the maximum \(t^\circ\) and minimum \(t^\circ\), the greater the circulation velocity and the better the temperature at the top and bottom of the furnace will be equalized.”

In order to settle the question, when studying the operation of a furnace or when designing anew, it is first of all necessary to clarify the influence of the difference in hydrostatic pressures in the ascending and descending streams; and, depending on how great or small the action of this factor is, it is necessary also to decide the further question, namely that of the magnitude of injection, of the initial velocity of the gaseous flame stream as a function of the amount of combustion products that must be systematically mixed with the furnace gases.

In 1934 the author undertook an attempt to develop a method for calculating the recirculation of combustion products [25] for the scheme shown in Fig. 7. In addition to this, by applying the method for calculating the temperature distribution along the length of the flame in the form of a \(t - k - \beta\) diagram, let us determine what consequences are caused by recirculation. The entrainment of part of the combustion products into the flame increases the degree of heat regeneration \(r\) and the total heat content of the flame gases \((r + \beta) q_0\), as a result of which the flame temperature calculated from the heat content increases, which also causes an increase in the average intensity of heat transfer of the flame. The combustion products entrained into the flame stream decrease the concentration of the active combustible constituent parts of the gas and of oxygen; this causes a slowing of the process

mixing of the fuel with oxygen, i.e., it lowers, as it were, the numerical value of the burner index \(k\), without requiring a change in the design of the burner; with a certain structural design of the burner, its index under recirculation \(k_p\) is smaller than the index of the burner without recirculation \(k\), i.e., \(k_p < k\). As is evident from the course of the heat-release curve on the \(t—k—\beta\) diagram, at one and the same value of the time \(\tau\), a decrease in \(k\) causes a decrease in \(\beta\), i.e., the process of heat release is slowed down and extended

Fig. 7

over a large length of the flame path. It is also not difficult to arrive at the conclusion that the temperature curve (in the first quadrant of the diagram) must be smoother, the sharpness of the temperature maximum more smoothed out, owing to the fact that the sharpness of the maximum on the temperature curve is determined not only by the rate of heat release \(\beta\), but also by the ratio between the amount of heat released during combustion, \(\beta q_0\), and the amount of heat introduced by regeneration, \(r q_0\). With recirculation of the combustion products, the regenerated heat \(r q_0\) is considerably increased and the heat released during combustion, \(\beta q_0\), is relatively decreased; but since the regenerated heat uniformly affects the temperature distribution along the entire length of the flame, creating, as it were, a certain temperature inertia of the flame, the sharpness of the maximum on the temperature curve is correspondingly smoothed out.

These general propositions concerning the influence of recirculation on the temperature distribution along the length of the flame, in each individual case, are made quantitatively specific; the proportion of recirculated combustion products must depend on this.

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