Study of a Burning Laminar-Diffusion Gas Jet

N. N. Norkin

I. Introduction

The principal forms of gas combustion at constant pressure are the following three: 1) flame laminar-diffusion combustion; 2) flame turbulent-diffusion combustion; and 3) surface catalytic (flameless) combustion.

The first form of combustion can be realized when the following conditions are observed: a) when the outflow of gas and air from the burner occurs in parallel or concentric layers; preliminary mixing of gas and air in the burners is not carried out; the mixing of the gas and air streams takes place in the working space of the heat installation (in the furnace, kiln) as a result of molecular diffusion of the components of the gas and air; b) the velocities of outflow of the gas and air from the burner are the same, their magnitude being limited by the bounds of the laminar regime. As a result of such organization of the combustion process, the longest flame is obtained, with the most uniform distribution of temperature along its length. When burning a gas containing significant quantities of hydrocarbons, the diffusion flame has the greatest luminosity and a high degree of blackness of radiation.

The second form—flame turbulent-diffusion combustion, which is the most widespread in practice—is characterized by the following: a) the mixing of gas and air is carried out wholly or partially in the burner; b) the velocities of outflow of the gas and air, or of the combustible mixture, reach significant magnitudes and are in the region of the turbulent regime. As a result of this kind of organization of the combustion process, its rate depends on the conditions of mixing the gas with air and on the outflow velocities.

The third form—surface (flameless) combustion—is characterized by the fact that: a) the gas and air, before entering the burner, are preliminarily and thoroughly mixed to the state of a homogeneous mixture; b) the combustible mixture is burned on the surface of a burner that has a high temperature and plays the role of a catalytic surface.

The three indicated forms of combustion differ from one another in the degree of preparation of the combustible mixture and in the influence of diffusion and kinetic chemical factors on the combustion process. In flame laminar-diffusion combustion the determining factors are diffusion ones; in flameless catalytic combustion—kinetic chemi-

...physical factors; in ordinary flame turbulent combustion, both these and other factors have a greater or lesser influence.

Interest in the study of diffusion-combustion processes is due to practical considerations (the possibility of obtaining a long flame plume with a uniform temperature) and to their theoretical significance, since in purely diffusion combustion the equations of mathematical physics can be used for theoretical analysis.

Diffusion combustion is usually carried out by means of slot or annular burners. In the first case, alternating streams of air and gas flow from parallel slots separately along the height of the jet; in the second case, gas and air flow in concentric layers from tubes inserted one into another; usually gas flows through the inner tube, and air through the outer, surrounding one.

Gas combustion takes place at the surface separating the air and gas streams; this surface is the flame front. Combustible gas and oxygen from the air diffuse toward the flame front, while the combustion products formed diffuse into the surrounding medium. Diffusion combustion is used for the most part to burn gases containing considerable quantities of hydrocarbons. When such gases are burned by the diffusion-combustion method, a luminous flame is formed, owing to the release of sooty carbon, which is a product of the pyrolytic decomposition of methane and other gaseous hydrocarbons.

The field of practical application of diffusion combustion is quite broad and varied.

Professor I. I. Kitaigorodskii [4] draws attention to the special importance of the diffusion flame of hydrocarbon gases. The thermal energy emitted by a luminous flame is absorbed not only by the surface of the material (for example, glass in tank furnaces), but the heat rays also penetrate directly into the liquid material.

A number of practical advantages of diffusion combustion are noted by Hepburn [19] in a review article, giving descriptions of many diffusion-combustion heat installations.

A small number of experimental-theoretical studies have been published in the literature, devoted chiefly to determining the length of a diffusion flame. One of the first attempts in this regard is the work of N. S. Makhov [9], published in 1934, whose purpose was to develop a theory of the simplest gas burner. Then, in 1940, N. S. Makhov developed an approximate theory of the diffusion flame, based on the analogy between the equations of diffusion and heat transfer [10], taking as a basis the Nusselt–Gröber solution for the mean temperature head in laminar flow [18]. N. S. Makhov carried out experiments on the study of diffusion combustion, using blast-furnace gas as the fuel. The experiments, in general, confirmed his approximate theory, although the experimental conditions were not distinguished by thoroughness.

In 1936 G. P. Ivantsov [2] proposed a solution of the differential equation of diffusion as applied to the outflow of plane-parallel streams, formulating in the corresponding manner the conditions for the flame front; subsequently we used G. P. Ivantsov’s solution in studying a plane-parallel flame and expressed certain judgments on the advantages of G. P. Ivantsov’s method. Experimental studies confirming the method of solution (as far as we know) were not carried out by G. P. Ivantsov.

Burke and Schumann [17] published a theoretical-experimental study devoted to the problem of calculating the length and form of a diffusion flame. Having formulated in a definite way the initial and boundary conditions, they gave a solution of the differential equation of diffusion,

making it possible to determine the length of the flame as a function of the principal factors characterizing the phenomenon of diffusion, such as: the diffusion coefficient and the rate of outflow, which determine the dimensions of the air and gas streams. In the experiments of Burke and Schumann the following combustible mixtures were used: methane–air, hydrogen–air, carbon monoxide–air, methane–nitrogen–air; their combustion was carried out with the aid of plane-parallel and tubular burners. They also investigated the influence of preliminary heating of the gas and air on the length of the diffusion flame. It may be considered that the solution of the diffusion flame which they proposed found confirmation in their experiments. The shortcomings of the theoretical-experimental study of Burke and Schumann include: a) the complexity of the calculation; b) the experiments were carried out for mixtures at low excess-air coefficients \(a = 0.4 \div 0.6\); c) the outlet velocities of the streams were small (\(0.15\ \text{m/sec}\)) and not always identical; d) no gas analysis along the length of the flame was performed.

V. F. Kopytov and V. K. Ivanov, in reviewing the theoretical-experimental work of Burke and Schumann and using their method for solving the diffusion flame, formulated more clearly the dependence of the shape of the diffusion flame on the excess-air coefficient. Then their work gives a brief description of an experimental furnace built in the furnace laboratory of TsNIIMASH, equipped with a tubular diffusion burner; the furnace was fired with city gas. However, Kopytov and Ivanov did not attempt to determine theoretically the length of the diffusion flame and compare it with that observed in practice [6].

In the works discussed above, the diffusion flame was studied directly by burning gas. But since in diffusion combustion the degree of mixing of gas and air along the length of the flame is of decisive importance, the use of the modeling method is quite appropriate. This method was applied in 1939 by D. N. Lyakhovskii [7].

D. N. Lyakhovskii modeled the diffusion process by determining the degree of mixing of cold and heated air issuing in the form of plane-parallel streams; temperature measurement served as the criterion for judging the degree of mixing of the streams. D. N. Lyakhovskii did not process the results of his investigations from the standpoint of solving the diffusion equations, but the results of his investigations can be used for an approximate judgment of the path length of the flame at one or another degree of mixing.

The works of D. N. Lyakhovskii and Syrkin [8], F. P. Kazakevich and N. P. Dyundin [3], M. A. Glinkov and N. A. Kaloshin [1], P. I. Sychev [13], and K. Rummel [21] should also be assigned to this same category of experimental studies.

The experimental procedure used in investigations of this kind usually consisted in measuring the velocity and temperature fields of streams of cold and heated air issuing from nozzles whose axes are parallel.

With the aid of such experiments, the influence on the mixing process of the direction of the streams, the ratio of the stream velocities, the flame configuration, etc., is determined.

A number of experimental works have also been published devoted to the question of mixing of streams in the combustion process, but the nature of the experimental setup did not pursue the goal of studying diffusion combustion, and the results of the investigations were not processed in terms of the application of the theory and equations of diffusion.

Studies of this kind were usually carried out according to the following procedure: in laboratory gas furnaces, gas supplied separately from the air is burned, and the velocity, temperature, and concentration fields in the burning flame are measured. As a result, an understanding is obtained of the in-

intensity of mixing of gas and air, on gas burnout, and on the temperature distribution along the length of the flame.

Among the most thorough works in this area, the following should be noted: the works of Pistor [20], Rummel [21], V. N. Timofeev and P. I. Sychev [14], L. V. Podgurskii [12], and others. Questions concerning the interrelation of various forms of combustion and the influence of aerodynamic factors on the organization of the combustion flame are elucidated in the author’s work [11].

In our studies of diffusion combustion we set the following aims: first, to provide a more or less comprehensive treatment of the question of diffusion combustion, i.e., of the length of a diffusion flame, the temperature distribution along its length, and the composition of the combustion products; unfortunately, for a number of reasons, the heat-transfer phenomena of a diffusion flame were not subjected to detailed investigation, and this question is touched upon only approximately; second, to subject the physico-mathematical theories of the diffusion flame to detailed analysis and to compare the experimental data obtained with the calculated data; this will enable each reader to form a clear idea of the conditions of application of one or another theory and of the degree of deviation of experimental data from computational-theoretical data; third, to attempt to determine the degree of influence of unequal velocities of gas and air efflux on the purely diffusive process, since it is quite obvious that the practical realization of the basic condition of a diffusion process, \(w = \mathrm{idem}\), is rather difficult.

In connection with the tasks set, the investigation of diffusion combustion was carried out in the burning of illuminating gas and water gas by means of flat-slot and tubular burners.

The work was performed in the laboratory of thermal apparatus of the chemical-technological faculty of the Tomsk Polytechnic Institute. At various times the following members of the department took part in the work: V. V. Sigov, A. A. Starikov, N. D. Vylegzhanin, G. D. Spetsii, T. Ya. Vaganova.

II. Equations of the diffusion process in plane-parallel and concentric layers

A nonstationary diffusion process can be expressed mathematically by an equation in which the concentration of the diffusing substances, as a function, depends on the time coordinate and the spatial coordinates taken as independent variables, i.e.,

\[ c = f(\tau, x, y, z) \]

in Cartesian coordinates, and

\[ c = f(\tau, r, \psi, z) \]

in cylindrical coordinates.

For a one-dimensional concentration field,

\[ c = f(\tau, x) \]

or

\[ c = f(\tau, r). \]

It is known from mathematical physics that the process of distribution and equalization of concentrations in a nonstationary one-dimensional concentration field is described by Fick’s differential equation in Cartesian coordinates

\[ \frac{\partial c}{\partial \tau} = K \frac{\partial^2 c}{\partial x^2} \tag{1} \]

and in cylindrical coordinates

\[ \frac{\partial c}{\partial \tau} = K\left( \frac{\partial^2 c}{\partial r^2} + \frac{1}{r}\frac{\partial c}{\partial r} \right). \tag{2} \]

In these equations the following are denoted: \(c\)—the concentration of the combustible-gas component at any point of the field; \(\tau\)—the diffusion time; \(K\)—the diffusion coefficient; \(x\)—the spatial coordinate in the direction of diffusion; \(r\)—the radial coordinate.

To find the form of the function \(c=f(\tau,x)\) or \(c=f(\tau,r)\), it is necessary that: (a) the function satisfy the differential equation, (b) it correspond to the prescribed surface conditions, and (c) at \(\tau=0\) it give the required initial concentration distribution.

Equation (1) is used for the study of diffusion in plane-parallel layers, and equation (2) is used for the study of diffusion in concentric layers.

To study the phenomenon of diffusion in plane-parallel layers, let us present the following scheme of flow issuing. Gas passes through the inner slit and air through the two outer slits. We choose such a rectangular coordinate system that the plane \(yz\) passes through the middle of the gas layer; then the axis \(ox\) at every point will be perpendicular to the surfaces of the gas and air layers. This is shown in Fig. 1.

Fig. 1. Scheme of issuing plane-parallel layers

Mathematical formulation of the problem:

At the initial instant of time, for \(\tau=0\), the distribution of the gas concentration, expressed through its partial pressures, is determined by the function

\[ (p_A)=f(x), \]

\[ \tau=0, \]

for \(a>x>0\)

\[ (p_A)=1, \]

\[ \tau=0 \]

and for \(s>x>a\)

\[ (p_A)=0 \]

\[ \tau=0. \]

The spatial conditions are

\[ \left(-\frac{\partial p_A}{\partial x}\right)_{x=0}=0 \]

and

\[ \left(\frac{\partial p_A}{\partial x}\right)_{x=s}=0. \]

The differential equation for the diffusion of gas \(A\) will have the form

\[ \frac{\partial p_A}{\partial \tau} = K_A \frac{\partial^2 p_A}{\partial x^2} \tag{1'} \]

The solution of equation \((1')\) must satisfy the differential equation and the conditions on the surface; it was obtained by G. P. Ivantsov [2] for the above initial and boundary conditions:

\[ p_A=\frac{1}{2}c_0+\sum_{n=1}^{n\sim\infty} c_n \cos\left(n\pi\frac{x}{s}\right)e^{-\left(n\frac{\pi}{2}\right)^2\frac{K\tau}{s^2}} \tag{3} \]

or

\[ \begin{aligned} p_A={}&\frac{1}{2}c_0 +c_1\cos\left(\pi\frac{x}{s}\right)e^{-\left(\frac{\pi}{2}\right)^2\frac{K\tau}{s^2}} +c_2\cos\left(2\pi\frac{x}{s}\right)e^{-\left(\frac{2\pi}{2}\right)^2\frac{K\tau}{s^2}} \\ &+c_3\cos\left(3\pi\frac{x}{s}\right)e^{-\left(\frac{3}{2}\pi\right)^2\frac{2K\tau}{s^2}}+\ldots \end{aligned} \tag{4} \]

This equation characterizes the variation of the partial pressure of gas \(A\) in an unsteady concentration field.

At \(\tau=0\) the terms \(e\ \mathrm{expon}=1\), and then the resulting trigonometric series represents the equation for the concentration of gas \(A\) at the initial instant, with limiting values \(p_A=1\) for \(a>x>0\) and \(p_A=0\) for \(s>x>a\).

According to the rules of harmonic analysis, the coefficients \(c_0, c_m\) are calculated.

\[ \frac{1}{2}c_0=\frac{1}{2s}\int_0^{2s} f(x)dx =\frac{1}{2s}\,2\int_0^a dx =\frac{2a}{2s}=\frac{a}{s}, \tag{5} \]

\[ \begin{aligned} c_m&=\frac{1}{s}\int_0^{2s} f(x)\cos\left(m\pi\frac{x}{s}\right)dx =\frac{1}{s}\,2\int_0^a \cos\left(m\pi\frac{x}{s}\right)dx= \\ &=\frac{2}{s}\cdot\frac{s}{m\pi}\sin\left(m\pi\frac{a}{s}\right) =\frac{2}{m\pi}\sin\left(m\pi\frac{a}{s}\right). \end{aligned} \tag{6} \]

Consequently,

\[ c_1=\frac{2}{\pi}\sin\left(\frac{\pi a}{s}\right), \]

\[ c_2=\frac{1}{\pi}\sin\left(\frac{2\pi a}{s}\right), \]

\[ c_3=\frac{2}{3\pi}\sin\left(\frac{3\pi a}{s}\right)\ \text{etc.} \]

After substituting the coefficients \(c_0,c_1,c_2,c_3\ldots\) into the characteristic solution (4), the equation for the variation of the partial pressure of gas \(A\) in the diffusion process is obtained:

\[ \begin{aligned} p_A={}&\frac{a}{s} +\frac{2}{\pi}\sin\left(\pi\frac{a}{s}\right)\cos\left(\pi\frac{x}{s}\right)e^{-\left(\frac{\pi}{2}\right)^2\frac{K\tau}{s^2}}+ \\ &+\frac{1}{\pi}\sin\left(2\pi\frac{a}{s}\right)\cos\left(2\pi\frac{x}{s}\right)e^{-\pi^2\frac{K\tau}{s^2}}+ \\ &+\frac{2}{3\pi}\sin\left(3\pi\frac{a}{s}\right)\cos\left(3\pi\frac{x}{s}\right)e^{-\left(\frac{2}{3}\pi\right)^2\frac{K\tau}{s^2}}+\ldots \end{aligned} \tag{7} \]

For \(x=0\) the following equation is obtained for the change of partial pressure with time in the middle of the gas layer

\[ (p_A)_{x=0}=\frac{a}{s}+\frac{2}{\pi}\sin\left(\pi\frac{a}{s}\right)e^{-\left(\frac{\pi}{2}\right)^2\frac{K\tau}{s^2}} +\frac{1}{\pi}\sin\left(2\pi\frac{a}{s}\right)e^{-\pi^2\frac{K\tau}{s^2}} \]
\[ +\frac{2}{3\pi}\sin\left(3\pi\frac{a}{s}\right)e^{-\left(\frac{3}{2}\pi\right)^2\frac{K\tau}{s^2}}+\ldots \tag{8} \]

If it is assumed that the diffusion flame has a constant translational velocity of motion \(w\), then during the diffusion time \(\tau\) the path traveled by the gas stream, or, what is the same thing, the length of the flame, will be \(z=w\tau\).

Substituting the value \(\tau=\dfrac{z}{w}\) into equation (8), we obtain

\[ (p_A)_{x=0}=\frac{a}{s}+\frac{2}{\pi}\sin\left(\pi\frac{a}{s}\right)e^{-\left(\frac{\pi}{2}\right)^2\frac{Kz}{ws^2}}+ \]
\[ +\frac{1}{\pi}\sin\left(2\pi\frac{a}{s}\right)e^{-\pi^2\frac{Kz}{ws^2}}+ \]
\[ +\frac{2}{3\pi}\sin\left(3\pi\frac{a}{s}\right)e^{-\left(\frac{2}{3}\pi\right)^2\frac{Kz}{ws^2}}+\ldots \tag{9} \]

This equation for the diffusion process in plane-parallel layers was derived, as already indicated earlier, by Ivantsov [2]. From the equation one can calculate the length of the diffusion flame \(z\) for a specified degree of completion of diffusion, checked by means of gas analysis along the length of the flame.

Taking \(\tau\) to be an infinitely large quantity, \((p_A)_{x=0}\) at the end point of the flame is

\[ (p_A)_{\substack{x=0\\ \tau=\infty}}=\frac{a}{s}. \tag{10} \]

But the quantity \(\dfrac{a}{s}\) for the process of diffusion combustion is related in a definite way to the amount of air required for burning the gas.

Let us establish the relation between the dimensions of the air and gas slots, proceeding from the following conditions characteristic of diffusion combustion:

a) let the velocities of outflow of gas and air be equal and within the limits of the laminar regime, i.e.,

\[ w_{\text{gas}}=w_{\text{air}}, \]

\[ Re_{\text{gas}}<2320, \]

\[ Re_{\text{air}}<2320; \]

b) combustion of the gas must occur with a certain excess of air; the excess-air coefficient \(\alpha>1\), i.e., per \(m^3\) of gas there is required a volume of air

\[ V_{\text{air}}=\alpha V_0. \]

Then, for the cross section of the air slot \(F_{\text{a.s.}}\) and the gas slot \(F_{\text{g.s.}}\),

\[ \frac{V_{\text{air}}}{F_{\text{a.s.}}}=w_a=w_g=\frac{1}{F_{\text{g.s.}}}. \]

or

\[ \frac{\alpha V_0}{F_{\text{в.щ.}}}=\frac{1}{F_{\text{г.щ.}}}; \qquad \frac{F_{\text{в.щ.}}}{F_{\text{г.щ.}}}=\alpha V_0, \]

but

\[ F_{\text{в.щ.}}=(2s-2a)b=2(s-a)b, \]

\[ F_{\text{г.щ.}}=2ab. \]

Therefore,

\[ \frac{2(s-a)b}{2ab}=\alpha V_0 \]

or

\[ \frac{s-a}{a}=\alpha V_0. \tag{11} \]

It follows from equation (11) that the greater the theoretically required amount of air, and hence the higher the calorific value of the gas, the greater is the ratio of the width of the air slots to the gas slot when burning gas with the same coefficient of excess air.

From equation (11), \(\frac{s-a}{a}=\alpha V_0\), it follows that \(\frac{a}{s}=\frac{1}{1+\alpha V_0}\), therefore

\[ (p_A)_{\substack{x=0\\ \tau=\infty}}=\frac{1}{1+\alpha V_0} \tag{10'} \]

Let us introduce the concept of the coefficient of the degree of completion of diffusion \(E\), defining it as the ratio of the amount of gas that has diffused by a given time to its maximum possible amount, i.e.,

\[ E=\frac{1-(p_A)_\tau}{1-(p_A)_{\tau=\infty}} \]

Substituting the value \((p_A)_{\substack{x=0\\ \tau=\infty}}\), we obtain

\[ E=\frac{1-(p_A)_\tau}{1-\dfrac{1}{1+\alpha V_0}} \tag{12} \]

Using this equation, by assigning the value of the degree of completion of diffusion \(E\) for definite values of \(\alpha\) and \(V_0\), we determine \((p_A)_\tau\), and from equation (9) the length of the diffusion flame \(z\) corresponding to the magnitude of the degree of completion of diffusion; or, from the observed flame length, we determine the degree of completion of diffusion.

In what follows, equations (9) and (11) were used by us in the mathematical processing of experimental data from observations of the length of the diffusion flame. As far as we know, there are no experimental data in the literature with which it would be possible to verify the solution of the diffusion equation proposed by G. P. Ivantsov as applied to the combustion of gases.

Another method of solving the differential equation of diffusion was proposed by Burke and Schumann [17], in accordance with different initial and boundary conditions.

Defining the flame front as the surface on which, as a result of combustion, the concentration of the combustible is equal to zero, Burke and Schumann reduce the complicated problem of mutual diffusion of gas into air and air into gas to a simple phenomenon—the diffusion of only one combustible gas.

If we denote by \(c\) the concentration of combustible gas at any point of the flame, by \(c_1\) the initial concentration of combustible gas, by \(c_2\) the initial concentration of oxygen in the air, by \(i\) the number of volumes of oxygen required for the combustion of one volume of gas, and by \(\dfrac{c_2}{i}\) the number of volume fractions of oxygen equivalent to one volume of combustible gas, considered as negative combustible gas, then the problem of solving the diffusion equation is formulated as follows: combustible gas with initial concentration \(c_1\) in the original mixture diffuses into oxygen taken in the equivalent quantity \(-\dfrac{c_2}{i}\); the initial conditions are:

\[ \left. \begin{array}{c} \text{when } \tau = 0 \\ a > x > 0 \end{array} \right\} \quad c = c_1, \]

and

\[ \left. \begin{array}{c} \text{when } \tau = 0 \\ s > x > a \end{array} \right\} \quad c = -\frac{c_2}{i}. \]

The solution satisfying the differential equation of diffusion and the initial and boundary conditions for plane-parallel layers will be:

\[ c = c_0 \frac{a}{s} - \frac{c_2}{i} + \frac{2}{\pi} c_0 \sum_{n=1}^{n=\infty} \frac{1}{n} \sin \frac{n\pi a}{s} \cos \frac{n\pi x}{s} e^{-\frac{K n^2 \pi^2}{w s^2} z}. \tag{13} \]

Here

\[ c_0 = c_1 + \frac{c_2}{i}. \]

At the flame front \(c = 0\); hence

\[ E = \frac{\pi}{2}\left(\frac{c_2}{i c_0} - \frac{a}{s}\right) = \sum_{n=1}^{n=\infty} \frac{1}{n} \sin \frac{n\pi a}{s} \cos \frac{n\pi x}{s} e^{-\frac{K n^2 \pi^2}{w s^2} z}. \tag{14} \]

The characteristic parameter of the solution of the differential equation of diffusion, \(E\), may have a zero, positive, or negative value:

\[ \text{when } \frac{c_2}{i c_0} = \frac{a}{s} \quad \text{the value } E = 0, \]

\[ \text{when } \frac{c_2}{i c_0} > \frac{a}{s} \quad \text{the value } E > 0, \]

\[ \text{when } \frac{c_2}{i c_0} < \frac{a}{s} \quad \text{the value } E < 0. \]

Let us express \(\dfrac{c_2}{i c_0}\) and \(\dfrac{a}{s}\) in terms of the theoretically required quantity of air and the excess-air coefficient under diffusion-combustion conditions.

Suppose that for the combustion of 1 volume of fuel, \(i\) volumes of oxygen are required and, correspondingly, \(\dfrac{i}{c_2}\) volumes of air, calculated from the stoichiometric equations of the combustion reactions. If the concentration of the fuel is \(c_1\), then for the combustion of 1 volume of gas the theoretically required quantity of air will be

\[ V_0 = \frac{i}{c_2} c_1. \]

The actual amount of air for combustion is equal to

\[ \alpha V_0=\alpha \frac{i}{c_2}c_1. \tag{15} \]

For the conditions of diffusion combustion to be satisfied, according to equation (11),

\[ \frac{s-a}{a}=\alpha V_0. \]

From comparison of (11) and (15) it follows that

\[ \alpha \frac{i}{c_2}c_1=\frac{s-a}{a} \]

or

\[ \frac{a}{s}=\frac{c_2}{c_2+\alpha i c_1}. \tag{16} \]

If \(\alpha=1\), then

\[ \frac{a}{s}=\frac{c_2}{c_2+i c_1}=\frac{c_2}{i c_0} \]

and \(E=0\).

If \(\alpha>1\), then

\[ \frac{a}{s}>\frac{c_2}{i c_0} \]

and \(E>0\).

If \(\alpha<1\), then

\[ \frac{a}{s}<\frac{c_2}{i c_0} \]

and \(E<0\).

Thus, the sign for \(E\) depends on the excess-air coefficient, but this ultimately determines the shape of the diffusion flame. If \(E=0\), then the vertical projection of the flame will have a rectangular shape; if \(E>0\), then the projection of the flame will have approximately the shape of an isosceles triangle of height \(z\) with its vertex at \(x=0\); and if \(E<0\), then the projection of the flame will have approximately the shape of a trapezoid with its largest upper base at height \(z\) when \(x=s\).

In what follows, equations (16), (14) were used by us to calculate the theoretical length of the diffusion flame in processing the experimental data obtained in the study of a diffusion plane-parallel flame.

For the study of a concentric diffusion flame, we take the equation of the diffusion process expressed in cylindrical coordinates,

\[ \frac{\partial c}{\partial \tau} = K\left( \frac{\partial^2 c}{\partial r^2} + \frac{1}{r}\frac{\partial c}{\partial r} \right). \]

Fig. 2. Diagram of the outflow of concentric layers

The outflow diagram of concentric layers is shown in Fig. 2.

Let us establish the relationship between the dimensions of the gas and air tubes, proceeding from the following conditions:

a) let the discharge velocities of the gas and of the air be equal and lie within the limits of the laminar regime, i.e.

\[ w_{\text{gas}}=w_{\text{air}}, \]

\[ Re_{\text{gas}}<2320, \]

\[ Re_{\text{air}}<2320; \]

b) the combustion of the gas must take place with a certain excess of air, the excess-air coefficient \(\alpha>1\), i.e. for \(1\ \mathrm{m}^3\) of gas, the required amount of air is

\[ V_{\text{air}}=\alpha V_0, \]

then, with the cross section of the air tube \(F_{\text{a.t.}}\) and of the gas tube \(F_{\text{g.t.}}\),

\[ \frac{w_{\text{air}}}{F_{\text{a.t.}}}=w_{\text{a}}=w_{\text{g}}=\frac{1}{F_{\text{g.t.}}}, \]

or

\[ \frac{\alpha V_0}{F_{\text{a.t.}}}=\frac{1}{F_{\text{g.t.}}}, \qquad \frac{F_{\text{a.t.}}}{F_{\text{g.t.}}}=\alpha V_0, \]

but

\[ F_{\text{a.t.}}=\pi(R^2-\rho^2), \]

\[ F_{\text{g.t.}}=\pi\rho^2, \]

therefore,

\[ \frac{\pi(R^2-\rho^2)}{\pi\rho^2} =\alpha V_0 =\frac{R^2-\rho^2}{\rho^2} =\frac{R^2}{\rho^2}-1. \tag{17} \]

It follows from the equation that the greater the calorific value of the gas, the larger the ratio \(\dfrac{R}{\rho}\).

Defining the flame front as the surface on which, as a result of combustion, the concentration of the fuel is equal to zero, Burke and Schumann formulate the initial and boundary conditions as follows:

\[ \left. \begin{array}{r} \text{at } \tau=0\\ \text{and}\\ \rho>r>0 \end{array} \right\} \quad c=c_1, \]

\[ \left. \begin{array}{r} \text{at } \tau=0\\ \text{and}\\ R>r>\rho \end{array} \right\} \quad c=-\frac{c_2}{i}, \]

\[ \left. \begin{array}{r} \text{and at } r=0\\ r=R \end{array} \right\} \quad \frac{dc}{dr}=0. \]

If we express the diffusion time in terms of the length and velocity of the diffusion flame, and assume that along the entire length of the flame the velocity of its displacement remains constant,

\[ z=w\tau, \]

\[ d\tau=\frac{1}{w}\,dz, \]

then the differential equation of diffusion will take the form:

\[ \frac{dc}{dz}=\frac{K}{w}\left(\frac{d^2c}{dr^2}+\frac{1}{r}\frac{dc}{dr}\right). \tag{18} \]

The characteristic solution of the differential equation satisfying the initial and boundary conditions, derived by Burke and Schumann with the aid of cylindrical Bessel functions, has the form:

\[ c=c_0\frac{\rho^2}{R^2}-\frac{c_2}{i} +2\,\frac{\rho c_0}{R^2} \sum_{n=1}^{n=\infty}\frac{1}{\mu_n}\cdot \frac{I_1(\mu_n\rho)I_0(\mu_n r)} {\left[I_0(\mu_n R)\right]^2} e^{-\frac{K\mu_n^2 z}{w}} . \tag{19} \]

In this equation

\(c\) — the variable concentration of the fuel at any point \((r,z)\) of the nonstationary concentration field;

\[ c_0=c_1+\frac{c_2}{i}, \]

i.e., the sum of the equivalent initial concentrations of the fuel and of the oxygen of the air;

\(I_0\) — Bessel function of the first kind and of zero order;

\(I_1\) — Bessel function of the first kind and of first order;

\(\mu_n\) — positive roots of the transcendental equation,

\(I_1(\mu_n R)=0\), for values of \(n\) from 1 to infinity.

For the surface of the flame front, where \(c=0\), equation (19) will take the form:

\[ \sum_{n=1}^{n=\infty}\frac{1}{\mu_n}\cdot \frac{I_1(\mu_n\rho)I_0(\mu_n r)} {\left[I_0(\mu_n R)\right]^2} e^{-\frac{K\mu_n^2 z}{w}} = \frac{R^2c_2}{2\rho i c_0}-\frac{\rho}{2} \equiv E. \tag{20} \]

The characteristic parameter \(E\) may have a positive, negative, or zero value depending on the excess-air coefficient. To verify this, let us express the values of \(c_2\) and \(c_0\) in terms of the theoretically required volume of air for combustion and the excess-air coefficient.

Let the combustion of 1 volume of gas require \(i\) volumes of oxygen and, correspondingly, \(\dfrac{i}{c_2}\) volumes of air. If the concentration of combustible gas is \(c_1\), then for the combustion of a unit volume of gas the theoretically required volume of air is

\[ V_0=c_1\frac{i}{c_2}. \]

The actual amount of air will be:

\[ \alpha V_0=\alpha\frac{i}{c_2}c_1, \]

but according to equation (17)

\[ \alpha V_0=\frac{R^2}{\rho^2}-1. \]

Comparing the preceding equalities, we obtain:

\[ \alpha\frac{i}{c_2}c_1=\frac{R^2}{\rho^2}-1. \]

In the equation for \(E\) (20) let us multiply the numerator and denominator of the first term by \(\rho\), then:

\[ E=\frac{R^2 c_2}{2 \rho i c_0}\cdot \frac{\rho}{\rho}-\frac{\rho}{2}. \]

Substituting the value \(c_0=c_1+\dfrac{c_2}{i}\), and after simple transformations, we obtain:

\[ E=\frac{\rho}{2}\left[-\frac{V_0(\alpha-1)}{V_0+1}\right]. \tag{21} \]

From this equation it is clear that the sign of \(E\) is determined by the coefficient of excess air; if \(\alpha=1\), then \(E=0\), i.e., combustion with the theoretically required amount of air; if \(\alpha>1\), then \(E>0\) (a positive value) for combustion with an excess amount of air; if \(\alpha<1\), then \(E<0\) (a negative value) for combustion with a deficiency of air.

By means of particular solutions it can be seen that if the sign of \(E\) is minus, i.e., during combustion with insufficient air, then the flame front, arising at the mouth of the inner gas tube on the surface separating the gas and air streams, subsequently expands and ends at the radial distance \(r=R\) (curve \(AB\) in Fig. 3); if the sign of \(E\) is plus, i.e., during combustion with an excess of air, then the flame front, arising at the mouth of the inner gas tube on the surface separating the gas and air streams, narrows and ends at the radial distance \(r=0\) (curve \(AC\) in Fig. 3); if \(E=0\), then the flame front will have a cylindrical form (straight line \(AF\) in Fig. 3).

In Fig. 3 a diagram of the shape of the diffusion flame is shown as a function of the coefficient of excess air. To construct the curves, the ordinate axis \(oz\) is directed along the axis of the gas tube, and the abscissa axis \(or\)—along the radius. On the abscissa axis is plotted the relative distance \(\dfrac{r}{R}\) from the axis of the gas tube; on the ordinate axis—the length of the diffusion flame (conventionally).

The curves \(AB\) and \(AC\) in reality have a more complex form; however, for \(E>0\) point \(C\) has \(\dfrac{r}{R}=0\), and for \(E<0\) point \(B\) has \(\dfrac{r}{R}=1\).

Fig. 3. Diagram of the shape of a diffusion flame as a function of the coefficient of excess air

In each specific case, the length of the diffusion flame is determined by calculation from equation (20); moreover, if the diffusion flame expands toward the end, i.e., during combustion with insufficient air (curve \(AB\) in Fig. 3), then the maximum flame length \(z_{\max}\) is determined from the condition

\[ \frac{r}{R}=1; \]

if the diffusion flame narrows toward the end, i.e., during combustion with an excess of air (curve \(AC\) in Fig. 3), then the maximum

the flame length \(z_{\max}\) is determined from the condition \(\frac{\bar{c}}{R}=0\). In combustion with the theoretically necessary amount of air, when the diffusion flame has a cylindrical form, the flame length is determined by the degree of completion of the diffusion process.

In contrast to the Burke and Schumann method, for the analysis of a tubular diffusion flame N. S. Makhov [10] proposed an approximate theory of the diffusion flame based on the analogy between the laws of diffusion and convective heat transfer in laminar flow under conditions of internal flow. For this purpose N. S. Makhov used the Nusselt–Graetz solution [18] for the mean temperature excess in any cross section of a flow moving laminarly in a tube.

\[ \frac{\bar{\vartheta}}{\vartheta_0} = 0.821 e^{-14.63 \frac{1}{\overline{Pe}}\cdot\frac{z}{d}} + 0.0987 e^{-88.7 \frac{1}{\overline{Pe}}\cdot\frac{z}{d}} + 0.0135 e^{-212.2 \frac{1}{\overline{Pe}}\cdot\frac{z}{d}} +\ldots \tag{22} \]

In this equation \(\vartheta_0\) is the temperature excess in the initial section; \(\bar{\vartheta}\) is the mean temperature excess in the given section; \(d\) is the tube diameter; \(z\) is the distance of the point under consideration from the initial section; \(\overline{Pe}=\frac{\bar{w}d}{a}\), the averaged Peclet similarity criterion; \(\bar{w}\) is the mean flow velocity in the tube; \(a\) is the coefficient of thermal diffusivity.

Provided that the coefficient of thermal diffusivity \(a\) and the coefficient of diffusion \(K\) are equal to each other, the temperature field and the concentration field are described by one and the same equation, in accordance with which the equation for the variation of the mean concentration \(c\) of combustible components of the gas along the length of the diffusion flame takes the form

\[ \frac{\bar{c}}{c_0} = 0.821 e^{-14.63 \frac{1}{\overline{Pe}}\cdot\frac{z}{d}} + 0.0987 e^{-88.7 \frac{1}{\overline{Pe}}\cdot\frac{z}{d}} + 0.0135 e^{-212.2 \frac{1}{\overline{Pe}}\cdot\frac{z}{d}} +\ldots \tag{23} \]

In this equation \(c_0\) is the initial content of combustible components of the gas, expressed in volume percent; \(d\) is the diameter of the inner (gas) tube; \(\overline{Pe}=\frac{\bar{w}d}{K}\) is the averaged Peclet criterion for the diffusion process.

For approximate calculations N. S. Makhov discards the second, third, and subsequent terms of the series in equation (23), and instead of \(0.821\) takes the coefficient of the first term of the series equal to 1; as a result, equation (23) takes the form

\[ \frac{\bar{c}}{c_0} = e^{-14.63 \frac{1}{\overline{Pe}}\cdot\frac{z}{d}}, \tag{24} \]

whence

\[ z=0.1574\,\overline{Pe}\,d\lg\frac{c_0}{\bar{c}} \]

or

\[ z=0.1574\frac{\bar{w}d^2}{K}\lg\frac{c_0}{\bar{c}}. \tag{25} \]

The concentration of fuel \(c_0\) and \(\bar{c}\) can be expressed through the concentration of oxygen.

If we denote by \(i\) the number of volumes of oxygen \(O\) theoretically required for complete combustion of one volume of combustible constituents of the gas, by \(\alpha\) the excess-air coefficient, and by \(V_0\) the theoretically required amount of air, then we may write

\[ c_0=\frac{\alpha V_0 O_0}{i} \]

and

\[ \bar c_k=\frac{\bar O_k}{i}, \]

whence

\[ \frac{c_0}{\bar c_k}=\alpha V_0\frac{O_0}{\bar O_k}. \tag{26} \]

Substituting the value of \(\dfrac{c_0}{\bar c_k}\) into equation (25) for the final state of the diffusion flame \((\bar c=\bar c_k)\), we obtain:

\[ z=0.1574\frac{\bar w d^2}{K}\lg\left(\alpha V_0\frac{O_0}{\bar O_k}\right). \tag{27} \]

If the flame front has a cylindrical form (for \(\alpha=1\)), then the quantity \(d\) has a constant value along the length of the flame, equal to the diameter of the gas tube. If the flame front has an expanding or contracting form (for \(\alpha \gtreqless 1\)), then \(d\) acquires a variable value along the length of the flame. In these cases (according to N. S. Makhov) the variable value of \(d\) is determined as the root-mean-square of the initial \(d\), equal to the diameter of the gas tube, and the final \(d_k\)

\[ d_m^2=\frac{d^2+d_k^2+dd_k}{3}. \]

The quantity \(d_k\) can be found from the following relations: for a cylindrical diffusion flame, when \(w_2=w_в\),

\[ V_0=\frac{R^2}{\rho^2}-1, \]

for an expanding or contracting flame

\[ \alpha V_0=\frac{R^2}{\rho_k^2}-1; \]

whence

\[ d_k=2\rho_k=2\rho R\sqrt{\frac{1}{\alpha(R^2-\rho^2)+\rho^2}} \]

and

\[ d_m^2= \frac{ d^2+4\rho^2R^2\left[\frac{1}{\alpha(R^2-\rho^2)+\rho^2}\right] +d\,2\rho R\sqrt{\frac{1}{\alpha(R^2-\rho^2)+\rho^2}} }{3} \]

or

\[ d_m^2=\frac{1}{3}d^2\left[ 1+R^2\left(\frac{1}{\alpha(R^2-\rho^2)+\rho^2}\right) +R\sqrt{\frac{1}{\alpha(R^2-\rho^2)+\rho^2}} \right] \]

or

\[ d_m^2=\beta d^2, \]

where

\[ \beta=\frac{1}{3}\left[ 1+R^2\left(\frac{1}{\alpha(R^2-\rho^2)+\rho^2}\right) +R\sqrt{\frac{1}{\alpha(R^2-\rho^2)+\rho^2}} \right]. \tag{28} \]

Thus, equation (27) for determining the length of the diffusion flame takes the form

\[ z = 0.1574 \beta \frac{\overline{w} d^{2}}{K} \lg \left( \alpha V_{0} \frac{O}{O_{k}} \right). \tag{29} \]

This equation was subsequently used by us to determine the length of a tubular diffusion flame. It should be borne in mind that, for determining \(z\), an arbitrary (or experimentally determined) final content of undiffused oxygen is specified.

III. Experimental Study of a Concentric Diffusion Flame

The experimental study of a concentric diffusion flame was carried out using a three-ring cylindrical burner. A drawing of the burner is shown in Fig. 4.

For combustion, illuminating gas obtained at the institute’s gas plant was used. The gas composition according to chemical analysis was: \(CO_{2}\)—7.4%, \(O_{2}\)—5.9%, \(C_{2}H_{4}\)—1.4%, \(CO\)—14.0%, \(H_{2}\)—37%, \(CH_{4}\)—27.5%, \(N_{2}\)—6.8%. The calorific value of the gas was

\[ Q_{n}^{p} = 3900 \frac{\text{kcal}}{\text{m}^{3}}. \]

The content of combustible components in the gas was \(0.799 \frac{\text{m}^{3}}{\text{m}^{3}}\); the theoretically required amount of oxygen for combustion was \(0.788 \frac{\text{m}^{3}}{\text{m}^{3}}\), and the theoretically required amount of air was \(3.76 \frac{\text{m}^{3}}{\text{m}^{3}}\).

Fig. 4. Three-ring cylindrical burner

To determine the principal dimensions of the burner (the diameter and length of the cylindrical part), the following conditions were adopted: 1) that the coefficient of excess air be \(\alpha = 1.3\), and 2) that the length of the cylindrical part of the burner, measured from the point of introduction of the gas and air, be about 20 diameters. In accordance with these conditions, the following were calculated:

a) the outer radius of the annular space for air \(R\), with the radius of the inner gas tube \(\rho = 4.5\) mm, from equation (17)

\[ \alpha V_{0} = \frac{R^{2}}{\rho^{2}} - 1, \]

\[ R = \rho \sqrt{\alpha V_{0} + 1} = 4.5 \sqrt{1.3 \cdot 3.76 + 1} = 11 \text{ mm}. \]

The wall thickness of the internal gas tube is \(0.5\) mm.

Cross section of the internal gas tube

\[ \frac{\pi d^{2}}{4}=0.785(0.009)^{2}=0.64\cdot 10^{-4}\ \text{m}^{2}; \]

cross section of the air annulus

\[ \frac{\pi}{4}\left[D^{2}-(d+0.0005)^{2}\right]=0.31\cdot 10^{-3}\ \text{m}^{2}; \]

b) length of the cylindrical part of the burner above the gas and air inlet

\[ H=20D=20.22=440\ \text{mm}. \]

Illuminating gas was supplied through the inner tube with a diameter of 9 mm; the air required for combustion was supplied through the first annular space, with a diameter from 10 to 22 mm; then, through the second annular space, with a diameter from 23 to 27 mm, air was likewise supplied at the same velocity as in the first annular space, in order to isolate the flame jet from the influence of the stationary external medium; the third ring was not used in this series of experiments. To avoid curvature of the flame axis under the influence of buoyancy, the burner was placed vertically. The gas and air flow rates were measured by means of standard orifices and differential manometers. The aim of the investigation was to determine the flame-jet lengths at various efflux velocities within the laminar regime and to compare the experimental data with theoretical calculations.

The experiments were carried out at the following efflux velocities of the gas and air: 0.15 m/sec, 0.2 m/sec, 0.25 m/sec, and 0.3 m/sec. The visible flame jet was measured. The results of the measurements are presented in Table 1.

Table 1

Results of observations of the length of the diffusion jet of the annular burner

No.	Name of quantity	Dimension	Experiment 1	Experiment 2	Experiment 3	Experiment 4
1	Gas flow rate	\(\dfrac{\text{m}^{3}}{\text{hr}}\)	0,0345	0,0460	0,0576	0,0690
2	Air flow rate	\(\dfrac{\text{m}^{3}}{\text{hr}}\)	0,1695	0,2260	0,2826	0,3390
3	Efflux velocity of gas and air	\(\dfrac{\text{m}}{\text{sec}}\)	0,15	0,20	0,25	0,30
4	Excess-air coefficient	—	1,3	1,3	1,3	1,3
5	Reynolds number	\(\dfrac{\text{gas}}{\text{air}}\)	\(\dfrac{67,8}{120}\)	\(\dfrac{90}{161}\)	\(\dfrac{113}{201}\)	\(\dfrac{135}{241}\)
6	Length of diffusion jet according to observations	cm	10,5	12,4	15,2	16,8

Calculation of the Flame Length from the Diffusion Equations

(Burke and Schumann method)

1. Characteristic parameter of the solution of the differential equation.

According to equation (21),

\[ E=\frac{\rho}{2}\left[\frac{V_0(\alpha-1)}{V_0+1}\right]; \]

here

\[ \begin{aligned} \rho &= 0.45\ \text{cm},\\ V_0 &= 3.76\ \text{m}^3/\text{m}^3,\\ \alpha &= 1.3, \end{aligned} \]

\[ E=\frac{0.45}{2}\left[\frac{3.76(1.3-1)}{3.76+1}\right]=+0.053. \]

According to the equation

\[ E=\frac{R^2 c_2}{2\rho i c_0}-\frac{\rho}{2}; \]

here

\[ \begin{aligned} R &= 1.1\ \text{cm},\\ \rho &= 0.45\ \text{cm},\\ c_2 &= 0.21,\\ c_0 &= c_1+\frac{c_2}{i}. \end{aligned} \]

\(c_1\) (concentration of combustibles in the gas) \(=0.799\),

\(i\)—the number of volumes of oxygen per 1 volume of combustibles in the gas—is calculated as follows: the constituent combustible gases are recalculated to combustible mass, i.e., for example,

\[ (\mathrm{H}_2)^2=\frac{\mathrm{H}_2}{1-B}, \]

where \(B\) is the ballast in the gas.

Thus, the composition of the combustible mass is as follows:

\[ (\mathrm{H}_2)^2=\frac{37}{1-0.201}=46.4\%,\quad (\mathrm{CO})^2=17.5\%,\quad (\mathrm{CH}_4)^2=34.5\%,\quad (\mathrm{C}_2\mathrm{H}_4)^2=1.6\%. \]

Then, from the stoichiometric combustion equations, the quantity of oxygen required to burn 1 volume of combustible mass is determined, after subtracting the oxygen present in the gas; in our case

\[ i=1.057-0.059=0.998, \]

\[ c_0=c_1+\frac{c_2}{i}=0.799+\frac{0.21}{0.998}=1.01, \]

then

\[ E=\frac{(1.1)^2\,0.21}{2\cdot 0.45\cdot 0.998\cdot 1.01}-\frac{0.45}{2}=+0.055. \]

The positive value of \(E\) shows that the shape of the diffusion flame must be conically narrowing.

2. Calculation of the number \(Re\)

\[ Re=\frac{wd\rho}{\mu}=\frac{wd\gamma}{g\mu}. \]

For the gas: \(w=(0.15—0.20—0.25—0.30)\ \text{m/sec}\); \(\rho\), according to the additivity rule for the mixture, \(=0.0753\ \dfrac{\text{kg}\cdot \text{sec}^2}{\text{m}^4}\); \(\mu\), according to the additivity rule for the mixture, \(=0.01396\) centipoise,

\[ \gamma=0.7\ \text{kg}/\text{m}^3, \]

\[ d=0.009\ \text{m}. \]

For air: \(w\left(\dfrac{\text{m}}{\text{sec}}\right)\)—the same values as for the gas;

\[ \rho=0.132\ \text{kg}\cdot \text{sec}^2/\text{m}^4; \]

\[ \mu=0.0183\ \text{centipoise}; \]

\[ \gamma=1.226\ \text{kg}/\text{m}^3; \]

\[ d_{\text{pr}}=4\frac{F}{\Pi}=4\frac{\pi}{4}\left(D^2-d^2\right)\frac{1}{2\pi(R+r)}=0.013\ \text{m}. \]

3. Diffusion coefficient

The calculation was made by the inverse mixing formula

\[ \frac{1}{k}=\left(\frac{n_1}{k_1}+\frac{n_2}{k_2}+\ldots\right)0.01, \]

where \(n_1, n_2 \ldots\) are the volumetric contents of the constituent parts of the gas in %,
\(k_1, k_2 \ldots\) are the diffusion coefficients of the constituent gases in the gas–air system.

From the handbook of the Technical Encyclopedia, vol. VII, p. 247, for
\(\mathrm{H_2}\)—air—\(0.611\ \text{cm}^2/\text{sec}\);

\(\mathrm{CO}\)—oxygen—\(0.185\) ”; we also adopt this for the \(\mathrm{CO}\)—air system;

\[ \begin{aligned} \mathrm{C_2H_4}&\text{—air—}0.101\ \text{cm}^2/\text{sec},\\ \mathrm{O_2}&\text{—air—}0.178\ ”,\\ \mathrm{N_2}&\text{—air—}0.171\ ”,\\ \mathrm{CO_2}&\text{—air—}0.139\ ”,\\ \mathrm{CH_4}&\text{—air—}0.476\ \text{according to }[17]. \end{aligned} \]

Consequently,

\[ \frac{1}{k}=\left(\frac{37}{0.611}+\frac{14}{0.185}+\frac{1.4}{0.101}+\frac{5.9}{0.178}+\frac{6.8}{0.171}+\frac{7.4}{0.139}+\frac{27.5}{0.476}\right)0.01=3.3, \]

\[ k=0.3\ \frac{\text{cm}^2}{\text{sec}}. \]

4. Determination of the roots of the equation \(I_1(x)=0=I_1(\mu R)\) and of the Bessel functions of the first kind of orders zero and one.

From Table XXIV of special functions [16]

for root No. 1 its value is 3.8317 and \(I_0(x_1)=-0.4028\),

No. 2 — 7.0156 and \(I_0(x_2)=+0.3001\),

No. 3 — 10.1735 and \(I_0(x_3)=-0.2497\).

Since \(x=\mu R_1\), for \(R=1.1\ \text{cm}\) we obtain \(\mu_1=3.48\); \(\mu_2=6.37\) and \(\mu_3=9.25\); for \(\rho=0.45\ \text{cm}\): \(\mu_1\rho=1.57\); \(\mu_2\rho=2.88\); \(\mu_3\rho=4.15\).

From Table 1 of special functions [16], the quantities are \(I_1(\mu_1\rho)=+0.56674\), \(I_1(\mu_2\rho)=+0.38246\), \(I_1(\mu_3\rho)=-0.12121\). Since the shape of the flame is contracting, one should take \(r=0\) in equation (19), and then \(I_0(\mu r)=1\), i.e., the calculation of the length of the contracting flame should be carried out from the equation

\[ E=\frac{R^2 c_2}{2\rho i c_0}-\frac{\rho}{2} =\sum_{n=1}^{n=\infty}\frac{1}{\mu_n}\cdot \frac{I_1(\mu_n\rho)}{I_0^2(\mu_n R)}\cdot e^{-\frac{k\mu_n^2 z}{w}} \tag{30} \]

It is appropriate to note that, when solving this equation, one may restrict oneself to substituting the values of only the first three roots, owing to the sharp increase of the negative exponent.

1. The length of the diffusion flame as a function of the gas-flow velocity

for \(w_1=15\ \dfrac{\text{cm}}{\text{sec}}\),

\[ 0.055= \frac{1}{3.48}\cdot \frac{0.56674}{(-0.4028)^2}\cdot e^{-0.3\frac{(3.48)^2}{15}z_1} + \frac{1}{6.37}\cdot \frac{0.38246}{(0.3)^2}\cdot e^{-0.3\frac{(6.37)^2}{15}z_1} + \frac{1}{9.25}\cdot \frac{-0.12121}{(-0.2497)^2}\cdot e^{-0.3\frac{(9.25)^2}{15}z_1} \]

or

\[ 0.055=0.998e^{-0.224z_1}+0.67e^{-0.815z_1}+0.207e^{-1.7z_1}. \]

Hence \(z_1=11.8\ \text{cm}\),

for \(w_2=20\ \text{cm/sec}\),

\[ z_2=15.6\ \text{cm}, \]

for \(w_3=25\ \text{cm/sec}\),

\[ z_3=19.6\ \text{cm}, \]

for \(w_4=30\ \text{cm/sec}\),

\[ z_4=23.6\ \text{cm}. \]

When calculating a diffusion concentric flame by the method of N. S. Makhov, it is necessary first to specify the oxygen content at the end of the diffusion flame, referring it to the initially taken air. In combustion with excess air, it should be assumed that at least such an amount of oxygen remains which has not entered into combination with the combustible gases and which corresponds to the coefficient of excess air. Then, for calculating the length of the diffusion flame, the quantities entering equation (29) will be as follows:

\[ w=15\ \text{cm/sec}—20\ \text{cm/sec}—25\ \text{cm/sec}\ \text{and}\ 30\ \text{cm/sec}, \]

\[ d=0.9\ \text{cm}, \]

\[ k = 0.3\ \text{cm}^2/\text{s}, \]

\[ \alpha = 1.3, \]

\[ V_0 = 3.76\ \text{m}^3/\text{m}^3, \]

\[ O_0 = 21\%, \]

\[ O_k = 4.89\%, \]

\[ R = 1.1\ \text{cm}, \]

\[ \rho = 0.45\ \text{cm}. \]

Substituting these quantities into equation (29), we obtain, for the velocity \(w_1 = 15\ \text{cm}/\text{s}\), the length of the diffusion flame

\[ z_1 = 0.1574 \cdot 0.9 \frac{15(0.9)^2}{0.3} \lg \left(1.3 \cdot 3.76 \frac{21}{4.89}\right), \]

\[ z_1 = 8.45\ \text{cm}, \]

for the velocity \(w_2 = 20\ \text{cm}/\text{s}\)

\[ z_2 = 11.3\ \text{cm}, \]

for the velocity \(w_3 = 25\ \text{cm}/\text{s}\)

\[ z_3 = 14.1\ \text{cm}, \]

for the velocity \(w_4 = 30\ \text{cm}/\text{s}\)

\[ z_4 = 16.9\ \text{cm}. \]

For greater clarity, Table 2 gives comparative data from observations and calculations of the diffusion-flame length.

Table 2

Comparative data on the length of a concentric diffusion flame

No.	Name of quantities	Gas and air outflow velocity	Gas and air outflow velocity	Gas and air outflow velocity	Gas and air outflow velocity
		\(15\ \text{cm}/\text{s}\)	\(20\ \text{cm}/\text{s}\)	\(25\ \text{cm}/\text{s}\)	\(30\ \text{cm}/\text{s}\)
1	Flame length by observation \(z_k\)	10.5	12.4	15.2	16.8
2	Flame length by calculation using the Burke and Schumann method \(z_{BSh}\)	11.8	15.6	19.6	23.5
3	Ratio \(\dfrac{z_k}{z_{BSh}}\)	0.89	0.89	0.77	0.72
4	Flame length by calculation using Makhov’s method \(z_M\)	8.45	11.3	14.1	16.9
5	Ratio \(\dfrac{z_k}{z_M}\)	1.24	1.10	1.08	0.995

As the outflow velocity increases, the discrepancy between the observed diffusion-flame length and the length calculated by the Burke and Schumann method increases; in this case the observed length is always smaller than the calculated one. Conversely, as the outflow velocity increases, the discrepancy between the observed diffusion-flame length and the length calculated by the method of N. S. Makhov decreases. The discrepancy between the diffusion-flame lengths calculated by the method of N. S. Makhov and the observed lengths is considerably smaller than for those calculated by the Burke and Schumann method. It should also be noted that, for practical use, the calculation equations according to the method of N. S. Makhov are considerably simpler, and this is the advantage of the method.

IV. Experimental Study of a Plane-Parallel Diffusion Flame

A. The Length of a Diffusion Flame According to Observations

The experimental study of a plane-parallel diffusion flame was carried out in two ways: first, with the gas and air streams issuing into the atmosphere and, second, with the gas and air streams issuing into a shielded furnace chamber.

For the study of the diffusion flame under atmospheric conditions, three burners were made: three- and seven-slot gas burners, 3ShG,

Fig. 5. Drawings of the burners tested

7ShG—for burning illuminating gas—and a three-slot burner of small dimensions for burning water gas. Drawings of the burners are shown in Fig. 5.

For combustion, illuminating gas of the same composition as in the study of the concentric diffusion flame was used, and water gas of the following composition: \(\mathrm{CO_2}\)—4.8%, \(\mathrm{O_2}\)—0.4%, \(\mathrm{CO}\)—29%, \(\mathrm{H_2}\)—52.2%, \(\mathrm{CH_4}\)—6.2%, \(\mathrm{N_2}\)—7.3%. The dimensions of the three-slot burner for burning illuminating gas are as follows:

slot: width of the inner gas slot 2 mm, width of each outer slot 5 mm; total width of the air and gas stream \(5+2+5=12\) mm. Slot length 60 mm. Cross section of the gas slot \(0.0020.0.06=0.00012\ \text{m}^2\), cross section of the air slots \(2.0.005.0.06=0.0006\ \text{m}^2\).

With equality of the velocities of gas and air outflow, the ratio between their quantities (the excess-air coefficient) is determined from equation (11)

\[ \frac{s-a}{a}=\alpha V_0, \]

\[ 2s=12\ \text{mm}, \]

\[ 2a=2\ \text{mm}, \qquad \alpha=\frac{2s-2a}{2aV_0}=\frac{12-2}{2\cdot 3.76}=1.33. \]

\[ V_0=3.76\ \text{m}^3/\text{m}^3. \]

The burner is demountable and is assembled from three separate elements forming the slots. This was done so that subsequently it would be possible to investigate the effect of inclining the air streams toward the gas stream on the length of the diffusion flame. The metal walls forming the slots were tapered down to 0.1 mm before the gas and air outlet.

The seven-slot burner for burning illuminating gas had the following dimensions: an inner gas slot 2 mm wide, then 2 slots enclosing the inner one, 10 mm thick each, then 2 gas slots 2.2 mm wide each, and finally 2 outer air slots 5 mm wide each. The total width of the air-gas stream was \(5+2.0+10+2.0+10+5=34\) mm. Slot length 60 mm. Cross section of the gas slots \(3.0.002.06=0.00036\ \text{m}^2\); cross section of the air slots

\[ 2.0.01.0.06+2.0.005.0.06=0.0018\ \text{m}^2. \]

With equality of the velocities of gas and air outflow, the ratio between their quantities is determined from the equation

\[ \alpha V_0=\frac{F_{\text{a.sl.}}}{F_{\text{g.sl.}}}, \]

\[ F_{\text{a.sl.}}=0.0018\ \text{m}^2, \]

\[ F_{\text{g.sl.}}=0.00036\ \text{m}^2 \qquad \alpha=\frac{0.0018}{3.76.0.00036}=1.33. \]

\[ V_0=3.76\ \text{m}^3/\text{m}^3. \]

The same ratio will hold between air and gas if one considers not the whole burner, but its separate elements. The seven-slot burner is also demountable. Investigations of this burner were carried out in order to trace the influence of neighboring enveloping flames on the central flame. When studying a flame issuing into the atmosphere, the burners were arranged vertically in order to avoid curvature of the flame axis under the influence of buoyancy. The flow rate of gas and air was measured by means of standard orifices and differential manometers. The experiments were carried out at gas and air outflow velocities of 0.9 m/sec—1.5 m/sec—2 m/sec. The visible flame was measured. The measurement results are presented in Table 3, and a photograph of the experimental setup in Fig. 6.

Fig. 6
Photograph of the experimental setup for diffusion combustion

Table 3

Results of observations of the length of a flat diffusion flame of illuminating gas

No.	Name of quantities	Dimension	Three-slot burner, gas and air discharge velocity 90 cm/sec	Three-slot burner, gas and air discharge velocity 150 cm/sec	Three-slot burner, gas and air discharge velocity 200 cm/sec	Seven-slot burner, gas and air discharge velocity 90 cm/sec	Seven-slot burner, gas and air discharge velocity 150 cm/sec	Seven-slot burner, gas and air discharge velocity 200 cm/sec
1	Gas consumption	m³/hour	0,389	0,650	0,865	1,17	1,94	2,59
2	Air consumption	m³/hour	1,92	3,25	4,32	5,83	9,73	12,96
3	Excess-air coefficient	—	1,33	1,33	1,33	1,33	1,33	1,33
4	Reynolds number: gas / air	—	174 / 555	290 / 925	388 / 1230	174 / 1025	290 / 1710	388 / 2280
5	Reduced diameter: gas / air	m	0,00387 / 0,0092	0,00387 / 0,0092	0,00387 / 0,0092	0,00387 / 0,0092 and 0,017	0,00387 / 0,0082 and 0,017	0,00387 / 0,0092 and 0,017
6	Length of the diffusion flame according to observations	cm	17,5	31	44	19	33	46

Note: \(d_{\mathrm{pr.}} = 0,0092\) for the outer air slots with dimensions \(5 \times 60\),
\(d_{\mathrm{pr.}} = 0,017\) for the inner air slots with dimensions \(10 \times 60\).

The three-slot burner for burning water gas had the following dimensions: the inner gas slot had a width of \(1\) mm, the width of each outer slot was \(1,1\) mm; the total width of the gas and air streams was \(1,1 + 1 + 1,1 = 3,2\) mm. The length of the slots was \(19,5\) mm. The cross section of the gas slot was \(0,0000195\) m²; the cross section of the air slots was \(0,0000429\) m². Given equality of the discharge velocities of gas and air, the ratio between their quantities (the excess-air coefficient) is determined from equation (11)

\[ \frac{s-a}{a}=\alpha V_0, \]

\[ 2s=3,2\,\text{mm}, \]

\[ 2a=1\,\text{mm}, \qquad \alpha=0,88 \]

\[ V_0=2,5\,\text{m}^3/\text{m}^3. \]

Gas and air consumption was measured by means of gasometers, each with a capacity of 15 liters; the discharge of gas and air from the gasometers took place at a constant water head: when the water level in the lower part of the gasometer rose, the water level in the upper part of the gasometer rose by the same height. The inflow of water was regulated by means of a float placed in the upper part of the gasometer; the float was equipped with pointer arrows indicating the water levels in the lower and upper parts of the gasometer. The distance between the water levels above and below was maintained constant, equal to \(1015\) mm. Depending on the discharge velocity, the exposure of the observations lasted from 40 to 350 seconds. The uniformity of discharge throughout the entire exposure time was ensured by preliminary adjustment of the valves on the air-

air–gas pipeline according to a special scale. The results of the measurements are presented in Table 4. The volumes of gas and air in the table are reduced to atmospheric pressure under conditions of isothermal expansion.

Table 4

Results of observations of the length of a plane diffusion flame of water gas

No.	Name of quantities	Dimension	Experiment 1	Experiment 2	Experiment 3	Experiment 4	Experiment 5	Experiment 6	Experiment 7
1	Exposure time	sec	337,5	151,5	188,5	53	55	64	62
2	Gas flow rate	liter	5,92	5,42	6,52	5,7	5,51	5,14	5,02
3	Air flow rate	"	13,62	12,16	14,90	13,4	12,97	13,5	12,04
4	Excess-air coefficient	—	0,91	0,89	0,91	0,93	0,93	0,93	0,952
5	Efflux velocity, gas / air	m/sec	0,9 / 0,94	1,83 / 0,87	1,77 / 1,84	5,5 / 5,9	5,15 / 5,5	4,11 / 4,38	4,15 / 4,52
6	Reynolds number, gas / air	—	70,6 / 138	144 / 275	139 / 271	432 / 868	405 / 735	323 / 645	326 / 666
7	Length of the observed flame	mm	21	42	41	120	110	99	101

B. Calculation of the flame length from the diffusion equations

For a flame of illuminating gas

Characteristic parameter of the solution of the differential diffusion equation for the combustion of illuminating gas.
According to equation (14)

\[ E=\frac{\pi}{2}\left(\frac{c_2}{ic_0}-\frac{a}{s}\right). \]

Here

\[ c_2=0,21,\qquad i=0,998,\qquad c_1=0,799. \]

From the preceding, for illuminating gas of the previously specified composition,

\[ c_0=c_1+\frac{c_2}{i}=0,799+\frac{0,21}{0,998}=1,01, \]

\[ a=0,1\ \text{cm}, \]

\[ s=0,6\ \text{cm}, \]

\[ E=\frac{3,14}{2}\left(\frac{0,21}{0,998\cdot1,01}-\frac{0,1}{0,6}\right)=0,065. \]

The positive value of \(E\) shows that the form of the diffusion flame is convergent at \(x=0\). The length of the diffusion flame as a function of the flow velocity:

for \(w_1=90\ \text{cm/sec}\).
According to equation (14)

\[ E=\sum_{n=1}^{n=\infty}\frac{1}{n}\sin\frac{n\pi a}{s}\cos\frac{n\pi x}{s}\,e^{-\frac{kn^2\pi^2}{ws^2}z}. \]

The calculation is carried out with respect to \(z\) at \(x=0\), i.e., the maximum length of the converging flame is calculated. In view of the rapid convergence of the series, it is possible to restrict oneself to the first three terms of the series. Then

\[ \begin{aligned} 0.065={}&\frac{1}{1}\cdot \sin \frac{1\cdot 3.14\cdot 0.1}{0.6}\cdot 1\cdot e^{-\frac{0.3\cdot 1^2(3.14)^2}{90\cdot(0.6)^2}z_1} \\ &+\frac{1}{2}\cdot \sin \frac{2\cdot 3.14\cdot 0.1}{0.6}\cdot 1\cdot e^{-\frac{0.3\cdot 2^2(3.14)^2}{90\cdot(0.6)^2}\cdot z_1} \\ &+\frac{1}{3}\cdot \sin \frac{3\cdot 3.14\cdot 0.1}{0.6}\cdot 1\cdot e^{-\frac{0.3\cdot 3^2(3.14)^2}{90\cdot(0.6)^2}z_1}, \end{aligned} \]

\[ z_1=21.9\ \text{cm}. \]

For \(w_2=150\ \text{cm/sec}\), \(z_2=36.5\),

for \(w_3=200\ \text{cm/sec}\), \(z_3=48.7\).

Let us compare the observed heights with the calculated ones. For a three-slot burner at velocity \(w_1=0.9\ \text{m/sec}\), the ratio

\[ \varphi_1=\frac{z\ \text{observed}}{z\ \text{calculated}}=\frac{17.5}{21.9}=0.8; \]

at velocity \(w_2=1.5\ \text{m/sec}\),

\[ \varphi_2=\frac{31}{36.5}=0.85 \]

and at velocity \(w_3=2.0\ \text{m/sec}\),

\[ \varphi_3=\frac{44}{48.7}=0.90. \]

For a seven-slot burner at velocity \(w_1=0.9\ \text{m/sec}\), the ratio

\[ \varphi_1=\frac{19}{21.9}=0.87, \]

at velocity \(w_2=1.5\ \text{m/sec}\),

\[ \varphi_2=\frac{33}{36.5}=0.9 \]

and at velocity \(w_3=2.0\ \text{m/sec}\),

\[ \varphi_3=\frac{46}{42.7}=0.95. \]

The results indicated provide grounds for asserting that the seven-slot burner gives a longer flame than the three-slot burner at identical outflow velocities and identical slot widths, and that this elongation of the flame apparently occurs not as a result of an increase in burner capacity, but as a result of the influence of adjacent enveloping moving layers.

We shall use the equations of P. G. Ivantsov, derived for the diffusion process (9), to judge the degree of completion of diffusion from the observed lengths of the diffusion flame. For greater clarity and comparison, the data calculated by P. G. Ivantsov’s method, by the Burke and Schumann method, and those observed in experiments are presented in Table 5.

For a water-gas flame

By the method set forth above, the length of the diffusion flame of water gas was calculated.

The characteristic parameter of the solution of the differential diffusion equation, according to equation (14), is

\[ E=\frac{\pi}{2}\left(\frac{c_2}{i c_0}-\frac{a}{s}\right). \]

Table 5

Degree of completion of diffusion in the flame front, calculated by Ivantsov’s equation (9) in comparison with the calculated, according to Burke and Schumann, and observed lengths of the diffusion flame. Burner—flat, three-slot; illuminating gas.

No.	Name of quantities	Symbol and dimension	Gas and air outflow velocity: 90 cm/sec	Gas and air outflow velocity: 150 cm/sec	Gas and air outflow velocity: 200 cm/sec
1	Length of the diffusion flame, calculated by the method of Burke and Schumann	cm	21.9	36.5	48.7
2	Length of the diffusion flame according to the author’s experimental data	cm	17.5	31.0	44.0
3	Concentration of the diffusing gas in the flame front according to Ivantsov’s equation at: \(\alpha = 1.33\) \(V_0 = 3.76\) \(a = 0.1\ \text{cm}\) \(s = 0.6\ \text{cm}\)	\((P_A)_{\tau}\) \(x=0\)	0.359	0.305	0.271
4	Degree of completion of diffusion		0.78	0.84	0.88

\[ c_2 = 0.21, \]

\[ i = 0.61, \]

\[ c_1 = 0.874, \qquad E = \frac{3.14}{2}\left(\frac{0.21}{0.61 \cdot 1.218} - \frac{0.05}{0.16}\right) = -0.0471 \]

\[ c_0 = 1.218, \]

\[ a = 0.05\ \text{cm}, \]

\[ s = 0.16\ \text{cm}. \]

The negative value for \(E\) shows that the shape of the flame is divergent, and in equation (14) one should take \(x=s\).

The diffusion coefficient, calculated by the inverse mixing formula, is equal to \(K = 0.3\ \text{cm}^2/\text{sec}\). The results of calculating the length of the diffusion flame of water gas for different outflow velocities are given in Table 6.

Table 6

Calculated length of the diffusion flame of water gas

No.	Name of quantities	Dimension	Experiment 1	Experiment 2	Experiment 3	Experiment 4	Experiment 5	Experiment 6	Experiment 7
1	Outflow velocity: gas	m/sec	0.9	1.83	1.77	5.5	5.15	4.11	4.15
1	Outflow velocity: air	m/sec	0.94	1.87	1.84	5.9	5.5	4.38	4.52
2	Calculated length of the diffusion flame	mm	23.3	47.5	45.8	142	133	106	107
3	Ratio: observed length / calculated length	—	0.9	0.89	0.9	0.85	0.83	0.93	0.95

V. Investigation of the Effect of Preliminary Heating of Gas and Air on the Length of the Diffusion Flame

Preliminary heating of the gas and air affects the development of the diffusion process because: a) the volume of the gases increases according to the binomial expansion law \(v_t = v_0(1+\beta t)\), and, with the same outlet cross section, the outflow velocity also increases, \(w_t = w_0\left(\dfrac{T}{T_0}\right)\), and

b) the diffusion coefficient increases approximately according to the law
\(K_t = K\left(\dfrac{T}{T_0}\right)^{1.75}\). According to the diffusion equations (9, 14), the length of the diffusion flame is determined from the dimensionless complex \(\dfrac{Kz}{ws^2}\) and varies in proportion to the outflow velocity and inversely proportional to the diffusion coefficient; therefore the joint influence of both factors will be determined by the equation

\[ z = f\left(\frac{w_0 T}{T}\cdot \frac{T_0^{1.75}}{K T^{1.75}}\right) = f\left[\frac{w_0}{K}\left(\frac{T_0}{T}\right)^{0.75}\right]. \tag{31} \]

It follows from this equation that, with an increase in the temperature of the diffusing gases, the length of the diffusion flame decreases in the ratio

\[ \left(\frac{T_0}{T}\right)^{0.75}. \]

The investigation of the effect of preliminary heating of gas and air on the length of the diffusion flame was carried out with the aid of the flat slot burner described above. The pipelines supplying gas and air to the burner were placed in two tubular electric furnaces. The temperature of heating of the gas and air, measured at their exit from the burner, was 220–260°C. The results of measurements and of calculation, according to equation (14), of the length of the diffusion flame are presented in Table 7.

Table 7

Results of observation and calculation of the length of the diffusion flame of a flat slot burner with heating of gas and air to 220°C

No.	Quantity	Value
1	Gas flow rate, \(\dfrac{\mathrm{nm}^3}{\mathrm{h}}\)	0.865
2	Air flow rate, \(\dfrac{\mathrm{nm}^3}{\mathrm{h}}\)	4.32
3	Excess-air coefficient	1.33
4	Outflow velocity at \(NT\), \(\left\{\dfrac{\text{gas}}{\text{air}}\right.\)	\(=\dfrac{2.00}{2.00}\ \dfrac{\mathrm{m}}{\mathrm{s}}\)
5	Outflow velocity at \(220^\circ\mathrm{C}\), \(\left\{\dfrac{\text{gas}}{\text{air}}\right.\)	\(=\dfrac{3.62}{3.62}\)
6	Diffusion coefficients	\(\left\{\begin{array}{l} K=0.3\ \mathrm{cm}^2/\mathrm{s},\\ K_{220^\circ}=0.861\ \mathrm{cm}^2/\mathrm{s}. \end{array}\right.\)
7	Ratio \(\left(\dfrac{T_0}{T}\right)^{0.75}\cdot \dfrac{w^0}{K}\)	\(=0.644\cdot 668=430\)
8	Length of the diffusion flame, \(\dfrac{\text{observed}}{\text{calculated}}\)	\(=\dfrac{30\ \mathrm{cm}}{32\ \mathrm{cm}}=0.93\)

Let us compare the results for the observed and calculated length of the diffusion flame when “cold” gas and air, and gas and air preheated to 220°C, are discharged. From Table 3 it is seen that at a velocity of 200 cm/sec the observed flame length is 44 cm, and in the present case 30 cm; the relative decrease in length is

\[ \frac{44-30}{44}=0.32. \]

For the discharge of “cold” gas and air the calculated flame length at the same velocity is 48 cm; for the discharge of preheated ones, 32 cm; the relative decrease in length is

\[ \frac{48.7-32}{48.7}=0.345. \]

From comparison of the quantities characterizing the calculated and actual relative decrease in the length of the diffusion flame, it may be concluded that, apparently, the increase in the diffusion coefficient with increasing temperature does not occur proportionally to

\[ \left(\frac{T}{T_0}\right)^{1.75}, \]

but to a greater degree. Incidentally, it should be noted that the exponent is usually given in handbooks within the limits from 1.5 to 2.0.

G. Study of the influence of unequal discharge velocities of gas and air on the length of the diffusion flame

The basic condition for a purely diffusion process is the presence only of molecular transfer of substance, which is possible only in the diffusion of stationary gases or gases in a state of laminar motion at equal flow velocities of gas and air. The basic diffusion equation was derived precisely for the indicated conditions. It is of interest to determine how inequality of the discharge velocities affects the diffusion process. Experiments show that if there is inequality in the velocities of diffusing parallel streams, then the diffusion process of parallel streams and the length of the diffusion flame change: if the gas velocity is greater than the air velocity, then the length of the diffusion flame increases; if the air velocity is greater than the gas velocity, then the length of the diffusion flame decreases.

Table 8 presents the results of studying the influence of unequal discharge velocities of gas and air on the flame length during the parallel discharge of gas and air from a three-slot flat burner.

Table 8

Results of observations of the length of the diffusion flame of a flat three-slot burner at unequal discharge velocities of gas and air

No.	Name of quantities	Dimension	170/200	200/200	225/200	240/200	260/200	300/200	Note
1	Gas consumption . . . .	m³/hour	0.865	0.865	0.865	0.865	0.865	0.865	Purely diffusion combustion at an air and gas velocity equal to 200 cm/sec.
2	Air consumption . .	m³/hour	3.66	4.32	4.86	5.17	5.61	6.48	Purely diffusion combustion at an air and gas velocity equal to 200 cm/sec.
3	Excess-air coefficient . . .	—	1.12	1.33	1.50	1.59	1.72	1.59	Purely diffusion combustion at an air and gas velocity equal to 200 cm/sec.
4	Length of the diffusion flame . .	cm	48.0	44.0	43.0	42.0	41.0	40.0	Purely diffusion combustion at an air and gas velocity equal to 200 cm/sec.

We shall attempt to calculate the length of the diffusion flame at unequal velocities of outflow of the gas and air, but within the limits of the laminar regime. We shall adopt the following assumptions: when the flame is purely diffusional, i.e., when

\[ \frac{w^{\mathrm{g}}}{w^{\mathrm{a}}}=1, \]

then the shape of the flame is determined by the sign, for

\[ E=\frac{\pi}{2}\left(\frac{c_2}{i c_0}-\frac{a}{s}\right)\gtreqless 0, \]

and the quantity \(\frac{a}{s}\) is determined in turn from the equation

\[ \alpha V_0=\frac{s-a}{a}; \]

at constant values of \(a\) and \(s\), a change in the velocity of the air and gas streams will affect \(\alpha\): with an increase in \(\frac{w^{\mathrm{g}}}{w^{\mathrm{a}}}\), \(\alpha\) will decrease, and conversely, with a decrease in \(\frac{w^{\mathrm{g}}}{w^{\mathrm{a}}}\), \(\alpha\) will increase, as is evident from the following relations:

\[ \alpha V_0=\frac{F^{\mathrm{a}}}{F^{\mathrm{g}}}; \qquad \text{but}\qquad \alpha' V_0=\frac{w^{\mathrm{a}}F^{\mathrm{a}}}{w^{\mathrm{g}}F^{\mathrm{g}}}, \]

whence

\[ \frac{\alpha'}{\alpha}=\frac{w^{\mathrm{a}}}{w^{\mathrm{g}}} \qquad \text{or} \qquad \alpha'=\alpha\,\frac{w^{\mathrm{a}}}{w^{\mathrm{g}}}; \]

when \(w^{\mathrm{a}}=w^{\mathrm{g}}\), \(\alpha'=\alpha\).

Let us introduce the concept of an equivalent layer thickness \(s'\), i.e., such an assumed layer thickness at which the air flow rate changes (as a result of a change in its outflow velocity), but at the same time the equality \(w^{\mathrm{g}}=w^{\mathrm{a}}\) is ensured.

The value of the equivalent layer thickness \(s'\) may be determined from a relation obtained on the basis of the following considerations: since the thickness of the gas layer \(a\) remains unchanged, as does the gas outflow velocity \(w^{\mathrm{g}}\), while the air outflow velocity changes, the coefficient of excess air will change.

For the new value of the coefficient of excess air \(\alpha'\), in order that the gas and air outflow velocities be equal, the quantity \(\frac{a}{s}\) must be determined from the relation

\[ \alpha' V_0=\frac{s'-a}{a}, \]

but

\[ \alpha'=\alpha\,\frac{w^{\mathrm{a}}}{w^{\mathrm{g}}}, \qquad \text{therefore} \qquad \alpha\,\frac{w^{\mathrm{a}}}{w^{\mathrm{g}}}\,V_0=\frac{s'-a}{a}, \]

since

\[ \alpha V_0=\frac{s-a}{a}, \qquad \text{then} \qquad \frac{w^{\mathrm{a}}}{w^{\mathrm{g}}}\cdot\frac{s-a}{a}=\frac{s'-a}{a}. \]

Hence

\[ s'=(s-a)\frac{w^{\mathrm{a}}}{w^{\mathrm{g}}}+a, \tag{32} \]

when

\[ w^{\mathrm{a}}=w^{\mathrm{g}} \]

\[ s'=s. \]

Table 9

Calculated length of the diffusion flame at unequal gas and air outflow velocities; flat three-slot burner

No.	Name of quantities	Designation and dimension	Outflow velocity, gas/air, cm/sec: 200/170	Outflow velocity, gas/air, cm/sec: 200/200	Outflow velocity, gas/air, cm/sec: 200/225	Outflow velocity, gas/air, cm/sec: 200/240	Outflow velocity, gas/air, cm/sec: 200/260	Outflow velocity, gas/air, cm/sec: 200/300
1	Equivalent layer thickness \(s'\)	mm	5.25	6.0	6.63	7.0	7.5	8.5
2	Characteristic parameter	\(E=\dfrac{\pi}{2}\left(\dfrac{c_2}{i c_0}-\dfrac{a}{s'}\right)\)	0.03	0.065	0.091	0.104	0.118	0.143
3	exponent \(CZ\)	\(\dfrac{k\pi^2 z}{w'(s')^2}\)	0.053	0.0412	0.0337	0.0302	0.0262	0.0205
4	Length of the diffusion flame (calculated)	cm	53.8	48.7	47.5	47.0	46.5	44.0
5	Ratio \(\dfrac{\text{Observed length}}{\text{Calculated length}}\)	—	\(\dfrac{48.0}{53.8}=0.89\)	\(\dfrac{44.0}{48.7}=0.9\)	\(\dfrac{43.0}{47.5}=0.9\)	\(\dfrac{42.0}{47.0}=0.89\)	\(\dfrac{41.0}{46.5}=0.88\)	\(\dfrac{40.0}{44.0}=0.91\)

Thus, from relation (32) one can determine the value of the equivalent thickness from the actual dimensions of a flat-slot burner and the efflux velocities of the gas and air.

The introduction of the concept of the equivalent layer thickness makes it possible to use, for calculating the length of the diffusion flame, the equations derived earlier, if in these equations one substitutes the value of the equivalent layer thickness \(s'\) and the gas efflux velocity \(w'\).

Fig. 7. Length of the diffusion flame at unequal gas and air efflux velocities

Below is given a calculation of the length of the diffusion flame for the observed cases (Table 8); the calculated quantities are presented in Table 9, and for comparison of the observed and calculated quantities a graph is given in Fig. 7.

D. The combustion process and the temperature distribution along the length of the diffusion flame

A rather widespread opinion is that the processes of combustion and heat release along the length of a diffusion flame are uniform, as a result of which it is assumed that the temperature along the length of a diffusion flame is more or less constant. To test experimentally assumptions of this kind, we investigated the partial chemical composition of the combustion products and the temperature of the flame.

To determine the composition of the combustion products along the length of the flame, a flat three-slot burner was placed in a vertical shaft 80 cm high, having the form of a truncated cone and open at the top for the free exit of the combustion products. On the side surface of the shaft two openings were made: the lower one at a height of 210 mm and the upper one at a height of 570 mm from the burner mouth. Through these two openings, samples were drawn off for analysis by introducing into the flame a thin porcelain tube along the vertical axis of the flame. The analysis determined only the content in the combustion products of CO, \(O_2\), and \(CO_2\), using an Orsat-type gas analyzer. The data of the gas analysis should be regarded as approximate, since a complete gas analysis was not carried out; nevertheless, from the \(O_2\) content one may judge the degree of completion of combustion, and from the content of the slowly burning carbon monoxide CO one may form an idea of the completeness of combustion. The results of the gas analysis were processed for the following conditions of diffusion combustion: the efflux velocities of gas and air from the three-slot burner are equal to 2 m/sec, which corresponds to purely diffusion combustion, theore-

theoretical length of the flame (according to Table 5) is 48.7 cm, while the actual length according to Table 3 is 44 cm; the excess-air coefficient is \(\alpha = 1.33\). Table 10 gives the gas-analysis data for the above-mentioned conditions of diffusion combustion.

Table 10

Gas-analysis data along the length of the diffusion flame

Gas constituents	Number of volumes in gas	Number of volumes from air	Combustible mixture: number of volumes	Combustible mixture: %	Products of complete combustion by calculation (without water vapor): number of volumes	Products of complete combustion by calculation (without water vapor): %	Gas analysis: lower sample, %	Gas analysis: upper sample, %
\(\mathrm{CO_2}\)	7.4	—	7.4	1.20	51.7	10.80	4.0	10.6
\(\mathrm{O_2}\)	5.9	105.0	110.9	18.50	26.2	5.47	12.0	6.0
\(\mathrm{CO}\)	14.0	—	14.0	2.34	—	—	2.5	0.2
\(\mathrm{H_2}\)	37.0	—	37.0	6.17	—	—	—	—
\(\mathrm{CH_4}\)	27.5	—	27.5	4.56	—	—	—	—
\(\mathrm{C_2H_4}\)	1.4	—	1.4	0.23	—	—	—	—
\(\mathrm{N_2}\)	6.8	395.0	401.8	67.0	401.8	83.73	—	—
	100	500	600	100	479.7	100

For greater clarity, in Fig. 8 lines and points have been plotted which characterize the gas-analysis data; on the abscissa axis are laid off the theoretical length of the diffusion flame, equal to 487 mm, and the points of gas sampling for analysis—at distances of 21 cm and 57 cm from the burner mouth.

On the ordinate axis is the percentage content of gases by volume. The straight dashed line for \(\mathrm{O_2}\) begins on the ordinate axis at 18.5% (oxygen content in the initial mixture) and ends at 5.47% (oxygen content in the products of complete combustion), it being assumed that the burning-out of oxygen is uniform along the length of the flame. At the lower gas-sampling point, 12% oxygen (\(\mathrm{O_2}\)) was found instead of 12.8%, and at the upper sampling point 6.0% instead of 5.47%. The second dashed straight line for \(\mathrm{CO_2}\) has the coordinates: at the beginning 1.2% (the \(\mathrm{CO_2}\) content in the initial mixture) and at the end 10.6% (the \(\mathrm{CO_2}\) content in the products of complete combustion), it being assumed that the formation of carbon dioxide is uniform along the length of the flame. At the lower sampling point, 4% \(\mathrm{CO_2}\) was found instead of 5.4%, and at the upper one—10.6% instead of 10.8%. The third dashed line for \(\mathrm{CO}\) has the coordinates: at the beginning 2.34% (the \(\mathrm{CO}\) content in the initial mixture) and at the end 0%. At the lower sampling point, 2.5% was found instead of 1.4%, and at the upper one—0.2% instead of 0%.

Although, as was indicated above, the results of the analysis do not make it possible to judge quite clearly the development of the combustion process along the length of the flame because of the small number of gas-sampling points and the incomplete gas analysis, some idea of this can nevertheless be formed. Thus, it may be considered that in practice the combustion process ends at a length of 570 mm, since the \(\mathrm{CO}\) content is 0.2%; it may be asserted without doubt that at this length \(\mathrm{H_2}\) and \(\mathrm{CH_4}\) must be absent from the combustion products, since \(\mathrm{H_2}\) has the largest diffusion coefficient and since \(\mathrm{CH_4}\) in

in the combustion process decomposes into CO and H₂, which also enter into the combustion reaction. It may then also be assumed that oxygen decreases in the flame considerably faster than under the development of uniform combustion, because in the initial stage of combustion hydrogen burns out predominantly and at a high rate for the reason indicated above; this is also evidenced by the analysis for CO₂—the increase in CO₂ occurs more slowly than in the assumed uniform combustion. In view of the fact that the conclusions indicated are tentative and are only partially supported by the present experiments, it becomes necessary to carry out a more thorough study of the burning-out of the individual gases from the mixture along the length of the flame; this has been done in a special work on the structure of the flame.

Fig. 8. Gas analysis along the length of the diffusion flame

The temperature along the length of the flame was measured with 10 Pt—PtRh thermocouples arranged on a stand along the height of the flame. The readings of the thermocouple e.m.f. were recorded on a single galvanometer, to which a common panel with 10 contacts was connected. The thermocouples were introduced into the flame with bare (unprotected) junctions. Undoubtedly, a bare junction has a catalytic influence on the combustion process, but we believe that this influence is to a certain extent compensated by the inaccuracy of measuring the true temperature by other methods. From this point of view, the statement of the Leningrad Scientific-Research Institute of Heat Engineering is valuable: “Bare thermocouples, which, owing to their low radiation coefficient and small diameter, ensure good heat transfer, are incomparably more convenient for measuring flame temperature and give a temperature close to the true one. In the case of short-term measurements, platinum thermocouples can serve for a fairly long interval of time without any changes in their properties. In operation, a bare Pt—PtRh thermoelement, after a number of measurements (10–15) in a strongly reducing atmosphere and at temperatures up to 1500°C, showed no changes in e.m.f. A check calibration, carried out very carefully, coincided with the initial one to an accuracy of up to 0.02 μV” [15]. V. F. Kopytov points to the strong catalytic influence of platinum: “At temperatures above 500°C platinum is a strong catalyst for combustion reactions and the thermocouple does not give correct readings”; therefore, in his studies of gas combustion V. F. Kopytov measured the temperature with a Ni—NiCr thermocouple [5]. Od-

It is, however, known that these metals also have a catalytic effect on the combustion process, although to a lesser degree than platinum.

In our experiments on flame height, the thermocouples were arranged in the following order: the lowest thermocouple was at a distance of 5 mm from the burner mouth, then every 45 mm, and the last, the 10th thermocouple, at a height of 410 mm. The results of the flame-temperature measurements are presented in Table 11 under the following experimental conditions:

a) flat three-slot burner; gas and air exit velocities equal to 200 cm/sec, excess-air coefficient \(\alpha = 1.33\); gas and air cold (without preliminary heating); theoretical flame length 48.7 cm, actually measured length 44 cm;

b) flat three-slot burner; gas and air exit velocities equal to 200 cm/sec; excess-air coefficient \(\alpha = 1.33\); gas and air preheated to 220°C; theoretical flame length 32 cm, actually measured length 30 cm;

c) flat seven-slot burner; gas and air exit velocities equal to 200 cm/sec; excess-air coefficient \(\alpha = 1.33\); gas and air cold; theoretical flame length 48.7 cm, actually measured 46 cm.

Fig. 9. Curves of temperature distribution along the length of a diffusion flame

Figure 9 shows the curves of temperature distribution along the length of the diffusion flame, according to the data of Table 11.

Table 11.

Temperature distribution along the length of a diffusion flame

No. of measurement point	Height from burner mouth, mm	Experiment (a): three-slot burner without preheating, °C	Experiment (b): three-slot burner with preheating, °C	Experiment (c): seven-slot burner without preheating, °C
1	5	530	630	810
2	50	790	930	1100
3	95	910	990	1210
4	140	1010	1200	1060
5	185	1050	1120	1020
6	230	970	1080	1000
7	275	930	1010	950
8	320	850	1020	900
9	365	720	1000	850
10	410	650	850	800

On the basis of the data given in Table 11 and in Fig. 9, it may be concluded that the temperature distribution along the flame length is not-

uniform; the temperature maximum is located approximately at a distance from one-third to one-half of the flame length.

V. Experimental investigation of a diffusion flame in the combustion of water gas in a shielded combustion chamber

A. Description of the experimental apparatus

All the previously described experiments on the investigation of the diffusion flame were carried out under conditions in which the flame discharged into the atmosphere; the direction of the flame was vertical. Taking into account that under production conditions the direction of the flame is predominantly horizontal, an experimental shielded combustion chamber was constructed for the study of a plane horizontal flame. The internal dimensions of the combustion chamber were: width 320 mm, height 500 mm, length 2000 mm. Along the inner side walls of the chamber 10 vertical flat screens were installed, 5 on each side.

Fig. 10. Drawing of the shielded combustion chamber

1—screens; 2—flat-slot diffusion burner; 3—pipes supplying water; 4—pipes removing heated water; 5—smokestack; 6—openings for taking gas samples for analysis; 7—openings for thermocouples.

The screens are made of sheet steel, 2 mm thick; the dimensions of a screen are: width 65 mm, height 465 mm, and length 400 mm. A drawing of the shielded combustion chamber is presented in Fig. 10.

Each screen has an independent supply of cold water from the collector. Heated water is likewise drawn off from each screen. The following measurements were made on the experimental apparatus: 1) the temperature of the cold water entering the screens; 2) the temperature of the heated water leaving the screens; the water temperature was measured with mercury thermometers; 3) the quantity of water supplied, by weighing; 4) the temperature in the combustion chamber; for this purpose the following were installed: at a distance of 20 cm from the burner mouth, Pt—PtRh thermocouples, then three Fe—Const thermocouples at

at distances of 45 cm from one another, and at the last fifth screen—a mercury thermometer. For the Pt—PtRh thermocouple there is a separate galvanometer, and for the 3 Fe—Const thermocouples—a common galvanometer with a switch; 5) taking gas samples for analysis; for this purpose, metal tubes are embedded in the masonry of the furnace chamber between the screens, through which gas was drawn off for analysis; along the height of the chamber the suction tubes were located at half the height of the chamber in the light; 6) the gas flow rate was measured by means of a sharp-edged orifice in the gas pipeline and a differential manometer; 7) the air flow rate was measured by means of a sharp-edged orifice in the air duct and a differential manometer; the air was supplied to the system by a centrifugal fan.

The general layout of the experimental installation, with indication of the measuring points, is presented in Fig. 11.

Before measurements begin, a steady state is established; the supply of water to the screens is regulated so that the same quantity enters each screen; this is achieved by weighing the water and adjusting the corresponding valves; then the exhaust fan is switched on and gas is supplied to the chamber, where it is ignited, and the forcing air fan is switched on. The ratio between the quantities of air and gas is set in advance and regulated by valves. Beginning from the moment when the temperature of the water flowing out of each screen becomes constant, all measured quantities are recorded.

B. Results of the study of the diffusion flame of water gas in a screened furnace chamber

On the experimental installation, two series of experiments were carried out in accordance with the tasks posed: the first series of experiments—with equal velocities of gas and air discharge from 0.973 m/sec to 2.08 m/sec, and the second series of experiments—with unequal velocities of gas and air discharge. In each series of experiments up to 7 complete tests were performed.

To organize the diffusion flame, a flat seven-slot burner was used, having the following dimensions: an inner gas slot 2 mm wide, then two slots surrounding the inner one, each 7.5 mm wide, then 2 gas slots each 2 mm wide, and finally two outer air slots each 3.75 mm wide. The total width of the air-gas flow is

$3.75 + 2 + 7.5 + 2 + 7.5 + 2 + 3.75 = 28.5$ mm;

the length of the slots is 60 mm. The cross section of the gas slots is

$3.0,002.0,060 = 0,00036\ \text{m}^2$;

the cross section of the air slots is

$2.0,00375.0,060 + 2.0,0075.0,060 = 0,00135\ \text{m}^2$.

When the velocities of gas and air discharge are equal, the ratio between their quantities is determined from the equation

\[ \alpha V_0 = \frac{F_{\text{a.s.}}}{F_{\text{g.s.}}}. \]

Here

\[ F_{\text{a.s.}} = 0,00135\ \text{m}^2, \]

\[ F_{\text{g.s.}} = 0,00036\ \text{m}^2, \]

\[ V_0 = 2,5\ \text{m}^3/\text{m}^3 \]

is the theoretically necessary quantity of air for the combustion of water gas,

hence

\[ \alpha = \frac{0,00135}{2,5 \cdot 0,00036} = 1,5. \]

Fig. 11. General view of the shielded combustion chamber.

1—Fe—Const thermocouple; 2—switch; 3—galvanometer for Fe—Const thermocouples; 4—galvanometer for Pt—PtRh thermocouples; 5—Pt—PtRh thermocouple; 6—water outlets from the shields; 7—diaphragm on the air duct; 8—plane-slotted diffusion burner; 9—shields; 10—diaphragm on the gas pipeline; 11—water inlets to the shields; 12—thermometers for measuring the water temperature; 13—gas sampling points for analysis.

Water collectors and balances are not shown in the drawing.

Table 12

Results of the investigation of diffusion combustion of water gas in a screened chamber.
Series I. At equal rates of discharge of gas and air

Test No.	Flow rate: gas, m³/hour	Flow rate: air, m³/hour	Excess-air coefficient	Discharge velocity: gas, m/sec	Discharge velocity: air, m/sec	Temperature distribution along the flame length at distance from burner mouth: 20 cm, °C	65 cm, °C	110 cm, °C	155 cm, °C	200 cm, °C	In flue, °C	Water consumption and temperature rise in screens: I and II screens—consumption, liters/hour	I and II screens—Δt°	III screen—consumption, liters/hour	III screen—Δt°	IV and V screens—consumption, liters/hour	IV and V screens—Δt°	Total water consumption, liters/hour	Analysis of combustion products in flue: CO₂, %	CO, %	O₂, %
1	1,26	4,76	1,51	0,97	0,98	730	550	440	325	150	80	25,0	70	15	30	30	25	70	10,6	0,1	7,00
2	1,60	6,00	1,50	1,24	1,23	800	690	510	420	190	165	27,0	80	14,8	35	31	28	75,8	10,7	0,0	7,10
3	1,78	6,65	1,49	1,37	1,35	[[unclear: temperature, appears as “8 0”]]	720	600	465	200	118	30,0	82	18	39	36	29	84,0	10,6	0,1	7,15
4	1,98	7,30	1,48	1,53	1,50	960	690	580	440	190	115	61,0	50	15	50	31	29	107	10,6	0,1	7,20
5	2,18	8,25	1,51	1,68	1,70	820	670	560	450	200	115	62,0	50	15	59	32	30	109	10,7	0,0	7,00
6	2,52	9,50	1,50	1,94	1,95	750	840	700	515	260	130	60,0	62	18	70	25	35	103	10,55	0,1	7,1
7	2,70	10,0	1,49	2,08	2,06	740	860	730	535	270	140	65	55	19	80	26	36	110	10,55	0,1	7,2

In the shielded combustion chamber, water gas was burned, obtained in gas generators during the gasification of Aralichevsk bituminous coal; the composition of the dry water gas was as follows: CO₂—4%, CO—30.5%, H₂—53.4%, CH₄—5.2%, and N₂—6.9%. The theoretically required amount of air was \(V_0 = 2.5\ \text{m}^3/\text{m}^3\). With an excess air factor \(\alpha = 1.5\), the composition of the combustion products should be as follows: CO₂—10.7%, O₂—7.1%, N₂—82%.

To calculate the theoretical length of the diffusion flame, it is necessary to know the diffusion coefficient. The diffusion coefficient was calculated to be

\[ k = 0.3\ \text{cm}^2/\text{sec}. \]

Table 12 presents the results of the first series of tests, i.e., with equal outflow velocities of gas and air.

From the data of Table 12 it is seen that combustion of the gas in the chamber is almost complete, since the CO content in the exhaust gases is insignificant; as the outflow velocity of the gas and air increases, the diffusion flame lengthens and the zone of high temperatures shifts along the length of the flame.

Table 13 gives the heat balance of the shielded combustion chamber obtained in this same series of tests.

Table 13

Heat balance of the shielded combustion chamber for diffusion combustion.
Series 1

Test No.	Heat of water heating in the shields I and II, kcal/hour	Heat of water heating in the shields III, kcal/hour	Heat of water heating in the shields IV and V, kcal/hour	Heat of water heating in the shields, total, kcal/hour	Heat of exhaust gases, kcal/hour	Volume of combustion products, m³/hour	Heat of combustion of gases, kcal/hour	Balance discrepancy (losses), kcal/hour	Balance discrepancy (losses), %
1	1750	450	750	2950	150	5.45	3400	300	8.8
2	2160	520	950	3630	232	6.90	4320	458	10.5
3	2460	700	1050	4210	290	7.70	4800	300	6.3
4	3050	750	900	4700	324	8.56	5450	426	7.8
5	3100	885	960	4945	348	9.42	5900	607	10.1
6	3720	1260	880	5860	452	10.9	6900	588	8.5
7	3575	1520	936	6031	540	11.7	7280	709	9.2

From the heat-balance data it is seen that, as the diffusion flame lengthens and the zone of higher temperatures shifts, the heat-release intensity of the middle III shield increases; thus, if the heat-release intensity of the last V shield is taken as unity, then the heat-release intensity of the middle III shield is characterized by the following ratios (from the first test to the last):

\[ \frac{1.2}{1};\quad \frac{1.09}{1};\quad \frac{1.3}{1};\quad \frac{1.67}{1};\quad \frac{1.83}{1};\quad \frac{2.86}{1};\quad \frac{3.25}{1}. \]

The second series of tests for studying the diffusion flame of water gas in a shielded combustion chamber was carried out with unequal outflow velocities of gas and air. The gas outflow velocities varied from \(0.973\ \text{m/sec}\) to \(2.08\ \text{m/sec}\), and the air outflow velocities from \(0.715\ \text{m/sec}\) to \(1.42\ \text{m/sec}\), respectively. The burner design was the same as before, a flat seven-slot burner.

Table 14

Results of the investigation of diffusion combustion of water gas in a screened furnace chamber.
Series II — at unequal gas and air exit velocities

Test No. (1)	Gas consumption, m³/h (2)	Air consumption, m³/h (3)	Excess-air coefficient (4)	Gas exit velocity, m/s (5)	Air exit velocity, m/s (6)	Temperature distribution along flame length at 20 cm from burner mouth, °C (7)	Temperature distribution along flame length at 65 cm from burner mouth, °C (8)	Temperature distribution along flame length at 110 cm from burner mouth, °C (9)	Temperature distribution along flame length at 155 cm from burner mouth, °C (10)	Temperature distribution along flame length at 200 cm from burner mouth, °C (11)	Temperature distribution along flame length in the stack, °C (12)	Water consumption and temperature rise in screens I and II, liter/h (13)	Water consumption and temperature rise in screens I and II, Δt° (14)	Water consumption and temperature rise in screen III, liter/h (15)	Water consumption and temperature rise in screen III, Δt° (16)	Water consumption and temperature rise in screens IV and V, liter/h (17)	Water consumption and temperature rise in screens IV and V, Δt° (18)	Total water consumption, liter/h (19)	Analysis of combustion products in the flue: CO₂, % (20)	Analysis of combustion products in the flue: CO, % (21)	Analysis of combustion products in the flue: O₂, % (22)
1	1,26	3,47	1,10	0,973	0,715	400	800	510	355	102	53	24	61	12,6	27,5	24,7	22	60,9	11,3	0,5	1,62
2	1,54	4,26	1,11	1,18	0,88	410	820	590	465	200	63	35	70	15,8	31,5	28,9	25	79,7	11,5	0,4	1,60
3	1,78	4,95	1,11	1,37	1,02	600	650	600	440	232	110	86	35	44	15	40	15	170	11,0	0,8	1,7
4	1,97	5,50	1,12	1,52	1,13	690	880	632	470	230	115	90	38	50	17	40	15	180	11,2	0,6	1,65
5	2,18	6,00	1,10	1,68	1,23	700	820	654	490	210	120	90	38	60	17	50	15	200	11,1	0,7	1,69
6	2,34	6,55	1,12	1,80	1,35	700	830	678	510	220	116	90	40	60	20	50	18	200	11,8	0,2	1,59
7	2,54	6,90	1,08	1,96	1,42	730	850	710	520	150	112	93	41	61	21	50	19	204	11,25	0,55	1,60

In carrying out this series of tests, it was desirable to determine the effect of unequal velocities of gas and air outflow on the temperature distribution in the shielded combustion chamber and on the length of the diffusion flame. The measurement procedure used was the same as in carrying out the first series of experiments.

Table 14 presents the results of the second series of tests. With an excess air coefficient \(\alpha = 1.1\), the composition of the combustion products should be as follows: \(\mathrm{CH_2}—11.9\%\), \(\mathrm{O_2}—1.57\%\), and \(\mathrm{N_2}—65.5\%\).

Table 15 gives the heat balance for the second series of experiments.

Table 15

Heat balance of the shielded combustion chamber for diffusion combustion.
Series II

Test No.	Heat for heating the water in shields I and II, kcal/hour	Heat for heating the water in shield III, kcal/hour	Heat for heating the water in shields IV and V, kcal/hour	Total, kcal/hour	Heat of outgoing gases: volume, m³/hour	Heat of outgoing gases, kcal/hour	Heat of gas combustion, kcal/hour	Balance discrepancy (heat losses), kcal/hour	Balance discrepancy (heat losses), %
1	1470	346	542	2358	4.2	74	3400	168	4.95
2	2450	500	722	3672	5.1	106	4150	378	9.10
3	3000	660	600	4260	5.9	214	4800	326	6.9
4	3420	850	600	4870	6.55	250	5320	200	3.8
5	3420	1020	750	5190	7.25	286	5900	424	7.2
6	3600	1200	900	5700	7.8	300	6320	320	5.1
7	3820	1280	949	6049	8.45	312	6850	489	7.1

B. Calculations of the Length of the Diffusion Flame of Water Gas

A. For the first series of experiments—when gas and air flow out at equal velocities—the characteristic parameter of the solution of the differential equation of diffusion is

\[ E=\frac{\pi}{2}\left(\frac{c_2}{ic_0}-\frac{a}{s}\right). \]

Here \(C_2=0.21\),

\[ i=0.587. \]

The method for calculating the value \(i\) was indicated earlier on p. 62,

\[ c_1=0.891, \]

\[ c_0=c_1+\frac{c_2}{i}=0.891+\frac{0.21}{0.587}=1.249, \]

\[ a=0.1\ \text{cm}, \]

\[ s=0.375\ \text{cm}, \]

\[ E=\frac{3.14}{2}\left(\frac{0.21}{0.587\cdot 1.249}-\frac{0.1}{0.375}\right)=0.033; \]

from equation (14)

\[ E=\sum_{n=1}^{n=\sim}\frac{1}{n}\sin\frac{n\pi a}{s}\cos\frac{n\pi x}{s}e^{-\frac{kn^{2}\pi^{2}z^{2}}{ws^{2}}}, \]

for \(x=0\) (since \(E\) has a positive value) we find:

for \(w_1=0.98\ \dfrac{m}{sec}\)
\[ z_1=14\ cm; \]

for \(w_2=1.23\ \dfrac{m}{sec}\)
\[ z_2=18.2\ cm; \]

for \(w_3=1.36\ \dfrac{m}{sec}\)
\[ z_3=19.7\ cm; \]

for \(w_4=1.53\ \dfrac{m}{sec}\)
\[ z_4=22.8\ cm; \]

for \(w_5=1.68\ \dfrac{m}{sec}\)
\[ z_5=25.0\ cm; \]

for \(w_6=1.94\ \dfrac{m}{sec}\)
\[ z_6=29.0\ cm; \]

for \(w_7=2.08\ \dfrac{m}{sec}\)
\[ z_7=31.0\ cm. \]

B. For the II series of experiments—when gas and air flow out at unequal velocities.
The calculation of the flame length was made according to equation (14), and the equivalent layer thickness \(s'\) according to formula (32); the results of the calculation are given in Table 16.

Table 16

Length of the diffusion flame of water gas at unequal velocities of gas and air outflow, flat slit burner

No.	Name of quantities	Designation and dimension	Gas/air outflow velocity, cm/sec 97,5 / 71,5	Gas/air outflow velocity, cm/sec 118 / 88	Gas/air outflow velocity, cm/sec 137 / 102	Gas/air outflow velocity, cm/sec 152 / 113	Gas/air outflow velocity, cm/sec 168 / 123	Gas/air outflow velocity, cm/sec 180 / 135	Gas/air outflow velocity, cm/sec 196 / 142
1	Equivalent layer thickness	\(s'\ cm\)	0,31	0,305	0,305	0,305	0,302	0,306	0,299
2	Characteristic parameter	\(E\)	0,565	0,066	0,066	0,066	0,706	0,645	0,767
3	Length of the diffusion flame at \(x=s'\) \(K=0.3\ cm^{2}/sec\) \(w_2=var\)	\(z\ cm\)	8,9	9,7	11,6	12,5	13,0	14,8	14,9

VI. Conclusions

In the present work, physicomathematical theories of the diffusion flame have been considered, and the conditions of their applicability to the study of diffusion-combustion processes have been determined in detail. A number of specific numerical

of solutions clearly shows the nature of the difficulties that arise in the practical use of the mathematical apparatus of diffusion theory and the necessity of certain simplifying assumptions; however, despite these difficulties and assumptions, the conclusion should be drawn that a preliminary calculation of a diffusion flame is necessary and quite possible, since the quantities actually observed in experiments do not differ so very sharply from the theoretically calculated ones.

Strict observance of the conditions of purely diffusion combustion is hardly possible in practice; in particular, it is difficult to observe the principle of equal velocities of gas and air efflux. Therefore, in this work the author has introduced the concept of the equivalent thickness of the diffusion layer and has given a mathematical expression for it in terms of the actual dimensions of the diffusing layers. This makes it possible to use, with a sufficient degree of reliability, the mathematical apparatus of diffusion theory in the study of diffusion flows with unequal efflux velocities.

The extensive experimental investigations of diffusion combustion of illuminating gas and water gas under various conditions carried out by the author, although not distinguished by great accuracy, are the most comprehensive in comparison with the experimental investigations of other authors (it should be noted, incidentally, that the number of experimental works in this field is very small).

In the experimental investigations carried out by the author, observations were made of:

1) the influence of the efflux velocity on the length of a diffusion plane-parallel and concentric flame;

2) the influence of unequal velocities of gas and air efflux on the length of a diffusion plane-parallel flame;

3) the influence of the preheating temperature of gas and air on the length of a diffusion flame;

4) the temperature distribution along the length of a diffusion flame;

5) incomplete analysis of combustion products along the length of a diffusion flame;

6) a diffusion flame in a shielded furnace chamber (temperature, heat transfer, heat balance).

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