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Carbon Combustion*
L. N. Khitrin and O. A. Tsukhanova
Interest in the study of the process of coal combustion increased considerably in the 1930s of our century. From that time, systematic investigations of this question began in the Soviet Union. These investigations were connected above all with the new tasks posed by the developing technology of coal combustion and gasification. The problem of intensifying combustion, the creation of powerful fuel units, the use of low-grade fuels, chamber combustion, underground gasification, the use of solid fuel for gas turbines, etc., require, for their solution, a more close study of the laws governing the burning of solid fuel. The immediate task becomes the study of the mechanism of the combustion process and, first of all, the development of methods for investigating the process.
In 1932–1933 A. S. Predvoditelev set forth a series of investigations of the process of carbon combustion in the physico-technical laboratory of the All-Union Heat Engineering Institute, which were subsequently concentrated and completed at the Power Engineering Institute named after Academician G. M. Krzhizhanovsky of the Academy of Sciences of the USSR. By that time, thanks to the work of a large collective of scientists headed by Academician N. N. Semenov, major successes had been achieved in the Soviet Union in the development of theories of gas combustion and the kinetics of chain reactions.
The investigations of A. S. Predvoditelev and the group of his collaborators took as their task the study of the physicochemical mechanism of carbon combustion under conditions analogous to the technical process. They occupy an intermediate position between the study of the entire complex of the process of burning natural coals (the school of G. F. Knorre) and the line of work studying the elementary acts of the interaction of carbon with the gas phase.
* A presentation of the principal results contained in the monograph Carbon Combustion. An Experiment in Constructing the Physicochemical Foundations of the Process. Authors: A. S. Predvoditelev, L. N. Khitrin, O. A. Tsukhanova, Kh. I. Kolodtsev, M. K. Grozdovsky. Published by the Academy of Sciences of the USSR, 1949.
By decree of the Council of Ministers of the USSR (Pravda, March 4, 1950), the authors of the monograph were awarded the Stalin Prize.
L. N. KHITRIN AND O. A. TSUKHANOVA
§ 1. BASIC PREMISES
An analysis of various methods of combustion and gasification leads to the conclusion that, in this complex phenomenon, the basic, determining process is the combustion of the carbon of the fuel. Although in the combustion of fuel the preparation of the fuel, the composition and properties of the ash, the release and combustion of volatiles and, finally, the thermal conditions sometimes play a very important role, nevertheless in all cases the combustion of the solid carbonaceous residue of the fuel remains the most important stage of the process. The leading role belongs to the process of carbon combustion because, first, the solid carbon contained in the fuel is the main combustible constituent of almost all natural coals; second, the stage of combustion of the coke residue proves to be the longest of all the successive stages of the process; and, third, the process of coke combustion is of decisive importance in creating the thermal conditions for the development of other coupled processes, in particular reduction processes.
Therefore the study of the processes of combustion and gasification of solid fuel should begin with the study of the process of carbon combustion.
The basis of the process of carbon combustion is the chemical reaction of carbon with oxygen. The kinetics of this reaction has been studied under conditions of high vacuum. However, from these data one cannot obtain a definite conception of the chemical mechanism of carbon combustion. Moreover, under ordinary conditions combustion proceeds in an environment very different from the conditions of experiments in vacuum. It is therefore necessary to pose and solve the problem of investigating the mechanism of carbon combustion under normal conditions. The question is complicated by the fact that under these conditions the reaction of carbon combustion is accompanied, to one degree or another, by two secondary reactions: the reaction of reduction of carbon dioxide on carbon and the reaction of combustion of carbon monoxide.
But there is also another, more essential aspect of the matter. Two of the three possible reactions listed, namely the reactions \( \mathrm{C} + \mathrm{O}_2 \) and \( \mathrm{C} + \mathrm{CO}_2 \), are heterogeneous, i.e., reactions proceeding between substances that are in different phases. For such reactions, the purely physical mechanism of supplying gaseous reactants from the bulk to the solid surface by diffusion assumes special significance.
Indeed, if the reaction between carbon and oxygen takes place on the surface of a carbon body, then the concentration of oxygen near this surface must decrease as a result of combustion and as a result of the accumulation of reaction products. Owing to this, a difference is created between the oxygen concentrations in the medium adjacent to the carbon and far from it, and a diffusional flux of oxygen arises.
Diffusion processes are not confined to the region of free gas space, but usually extend farther, into the depth of the porous mass of carbon.
A. S. Predvoditelev outlined the following path for studying the process of carbon combustion. First of all, the region of greatest practical importance must be investigated: combustion at atmospheric pressure and at temperatures attained in the technical combustion of coals. The investigations must be arranged so that it would be possible to separate the influence of physical and chemical factors on the process, and then, having established their interrelation, to find the basic regularities of the process and construct its computational scheme. In order to study combustion under real conditions, such an experimental method must be developed in which the diffusion process would not be excluded, but would be accurately taken into account. Along with diffusion, the combustion temperature must be precisely determined and easily varied, as a factor of decisive importance for the chemical side of the process. These requirements are met by investigations of the combustion of bodies of the simplest forms: a carbon spherical particle under external flow, and a carbon channel under forced flow. In addition, the particle and the channel enter as constituent elements into all methods of technical combustion of coals: the combustion of a coal suspension includes the combustion of dust particles; a burning fuel bed is a combination of the combustion of individual particles under external flow with combustion inside curvilinear channels formed by the interlump spaces. Therefore the study of the combustion of bodies of the simplest forms not only makes it possible to establish the regularities of the combustion process, but also helps to outline calculation methods for more complex and technically important cases of combustion of a bed and a suspension. Undoubtedly, alongside the study of the combustion of simple bodies, investigations should be carried out of the combustion processes of a bed and a suspension, since these processes have their own specific features.
§ 2. BASIC CHARACTERISTICS OF THE COMBUSTION PROCESS
The basic characteristics of the combustion process are the combustion rate and the composition of the combustion products. The combustion rate is the quantity of carbon (or oxygen) burned per unit of visible surface per unit time. On the one hand, the combustion rate depends on the rate of the chemical reaction, whose rate constant, as is known, depends on temperature in the following way*):
\[ k=\frac{k'_0}{\sqrt{T}}\,e^{-\frac{E}{RT}}\simeq k_0 e^{-\frac{E}{RT}}. \tag{1} \]
*) For the list of symbols, see the end of the article (p. 329).
On the other hand, the rate of combustion is determined by the amount of oxygen arriving at a unit visible surface per unit time by diffusion. This amount can be calculated from the expression
\[ -D\left(\frac{\partial c}{\partial n}\right)_F . \tag{2} \]
The latter depends to a lesser degree on temperature than does the quantity \(k\), since the diffusion coefficient is a function of temperature of the form
\[ D = D_0\left(\frac{T}{T_0}\right)^m, \]
where \(1.5 \leq m \leq 2\). The principal influence on the magnitude of (2) is exerted by
\[ \left(\frac{\partial c}{\partial n}\right)_F, \]
i.e., by the concentration gradient at the wall, which is determined by the conditions of flow past the surface by the gas and depends, first of all, on the blowing velocity.
The starting point of the investigation is to establish the relation between the chemical process, which is described by equation (1), and the physical process of diffusion, whose rate is determined by the quantity (2). The first investigation of the process of carbon combustion, carried out at VTI on the initiative of A. S. Predvoditelev, was performed by V. I. Blinov. He proposed a simplified computational scheme for the process, consisting in the fact that the problem of combustion is reduced to the problem of diffusion of oxygen to the coal surface under a boundary condition of the following form:
\[ K_s = -D\left(\frac{\partial c}{\partial n}\right)_F = k c_F . \tag{3} \]
The surface concentration is assumed constant and equal to \(c_F\), and the reaction of carbon with oxygen is assumed to be first order, i.e., its rate is taken as proportional to the first power of the oxygen concentration at the coal surface.
To find the rate of carbon combustion, the rate of oxygen combustion \(K_s\) must be multiplied by the stoichiometric factor \(\beta\). Its magnitude is determined by the composition of the surface oxides and may vary from the value 0.375, if the reaction proceeds to carbon dioxide, to the value 0.75, if the reaction product is only carbon monoxide.
The convenience of this method of calculation consists in the fact that, owing to the assumption of constancy of the surface concentrations, the calculation of diffusive transfer is made by analogy with the calculation of heat transfer. Application of the simplified computational scheme to the analysis of experimental data made it possible to establish that the law of reaction of carbon with oxygen according to first order is close to reality. Further, an explanation was given for two distinct regions
combustion. At relatively low reaction temperatures the combustion rate is determined chiefly by the rate of the chemical reaction of carbon with oxygen and does not depend on the rate of blowing, because the diffusional capacity of oxygen toward the carbon surface exceeds the reaction capacity. This region has been called the kinetic region. Then follows a transition region, in which diffusion and the chemical reaction influence the process to an equal degree. With a further increase in temperature, the reaction capacity greatly outstrips the diffusional capacity, and the combustion rate is determined predominantly by the diffusion term.
This calculation method has been widely used by other investigators. The following relation for \(K_s\) is well known, in which the dependence of the combustion rate on the rate of chemical reaction and diffusion is clearly represented:
\[ K_s=\frac{c_0}{\frac{1}{k}+\frac{1}{\frac{D}{d}Nu_{\mathrm{diff}}}}, \tag{4} \]
where
\[ Nu_{\mathrm{diff}}=\frac{\alpha_{\mathrm{diff}}d}{D}. \]
However, the assumption of constancy of the concentration at the surface is not justified, for example, in an attempt to calculate the amount of oxygen burned in coal channels of various lengths. A. S. Predvoditelev formulated the problem of oxygen diffusion to the surface of a burning body in a more general form, assuming that \(c_F\) is variable, and with this boundary condition solved a number of problems: the problem of combustion of a carbon sphere under external flow, the problem of burnout of a carbon channel in the case of a plane and parabolic profile of the distribution of gas velocities over the channel cross-section, and the problem of burnout of a channel with one carbon wall. The application of A. S. Predvoditelev’s exact solutions to the analysis of experimental data made it possible to pass from a simplified scheme for calculating carbon combustion, which is based on considering the behavior of only one of the components of the reacting gas—oxygen—under the condition of its reaction only on the external surface of the coal, to the construction of a complete picture of the process, taking into account secondary reactions and internal reaction, and, at L. N. Khitrin’s suggestion, to introduce the new important concept of the coefficient of reaction gas exchange.
The principal part of the experimental data on the influence of various factors was obtained by A. S. Predvoditelev’s collaborators by studying the combustion of a carbon sphere and channel.
The technique of conducting the experiments was as follows:
A spherical carbon particle was placed in a furnace through which a gas of definite composition was blown at a known rate. During the experiment, at definite intervals of time, the temperature of the particle and its weight or dimensions were measured. These data
allowed the combustion rate to be calculated from carbon. In the study of combustion by the channel method, the combustion rate was calculated on the basis of data on the composition of the combustion products, the blowing rate, and the initial composition of the gas being blown, and could be determined both from carbon and from oxygen. Simultaneously with taking a gas sample at the outlet of the coal channel, measurements were made of the temperature of the channel walls and of the initial temperature of the gas being blown. By these methods, more than 1000 experiments were carried out with electrode and wood charcoals and various cokes, with variation of the coal temperature from 300 to 1600°C and of the linear blowing velocity—from the rates of natural diffusion up to 140 m/sec, for various geometrical sizes of bodies and various compositions of the gas being blown.
Fig. 1. Dependence of the combustion rate of anthracite (coke) on temperature. (Khitrin’s experiments).
Before proceeding to present the new results obtained, let us show to what extent the experimental data agree with relation (4).
According to relation (4), the combustion rate \(K_s\) in the kinetic region, i.e., when \(\frac{1}{k} \gg \frac{1}{\frac{D}{d} Nu_{\mathrm{diff}}}\), must depend on temperature in the same way as \(k\) according to equation (1). Consequently, the dependence of \(\lg K_s\) on \(\frac{1}{T}\) should be represented by a straight line. In reality, however, on the basis of Khitrin’s experimental data, a sharp break is observed (Fig. 1). The distortion of the linear dependence occurred as a result of the phenomenon of carbon combustion inside the coal particle.
Further, from relation (4) it follows that in the diffusion region, i.e., when \(\dfrac{1}{k} \ll \dfrac{1}{\dfrac{D}{d}Nu_{\text{diff}}}\), the rate of combustion should increase with increasing blowing velocity. In reality, however, a distinct retardation is observed, occurring when blowing velocities are reached that correspond to the separation from the frontal surface of the burning particle of the carbon-oxide flame, which until then had enveloped the particle on all sides (Fig. 2).
The slowing of the combustion rate is caused by the fact that, as a result of the separation of the film of burning carbon monoxide from the frontal part of the coal particle, the carbon-oxide flame is carried to the rear part and impedes the access of oxygen to it. The combustion of the particle becomes sharply asymmetric. In Figs. 3 and 4 successive photographs are compared of the burnout of a spherical particle at low and high flow velocities. In the latter case the “tail” of burning carbon monoxide is clearly visible. It has also been established that the retarding action of the carbon-oxide flame explains the negative course of the dependence of the combustion rate \(K_s\) on temperature, which is completely incomprehensible from the standpoint of relation (4) (Fig. 5).
Fig. 2. Dependence of the combustion rate of a spherical particle of electrode carbon on the flow velocity. (Khitrin’s experiments.)
Under some conditions, in combination with the reduction reaction, the afterburning reaction of carbon monoxide may lead to a considerable acceleration of the combustion process. Fig. 6 presents the dependence of the composition of the combustion products at the exit from the coal-
channel on temperature. The dashed line, indicated for the oxygen curve, corresponds to the calculated values obtained from A. S. Predvoditelev’s refined solution, but without taking into account
Fig. 3. Successive photographs of the burning-out of a spherical particle in an air stream at a low flow velocity. (Experiments of Tsukhanova and Kolodkina.)
secondary processes. It is seen that the theoretical calculation agrees with the experimental data only up to a temperature of \(900^\circ\text{C}\), after which the curve forms a characteristic kink, and the actual combustion rate considerably exceeds the calculated one,
§ 3. THE CONCEPT OF THE REACTION GAS-EXCHANGE COEFFICIENT. THE PROCESS OF INTERNAL REACTION
If internal reaction takes place and if secondary processes are taken into account, then, from the point of view of the mathematical problem, the combustion of carbon contains 2 systems of differential equations for three reaction components, namely: for \(O_2\), \(CO\), and \(CO_2\).
Fig. 4. Successive motion-picture frames of the burning out of a spherical particle in an air stream at a high flow velocity.
Fig. 5. Temperature dependence of the surface combustion rate of a spherical particle of electrode carbon in an air stream.
Fig. 6. Gas composition at the end of the coal channel as a function of the temperature of the channel walls. Air flow rate 10 l/min. Channel diameter 12 mm, length 950 mm.
One system corresponds to the diffusion equation in the gas phase, while the other describes the process of diffusion and reaction inside porous carbon. The boundary conditions are formulated on the basis of the conditions existing at the phase interface, which is taken to be the geometrically visible surface of the carbon.
In general form the equations and boundary conditions were compiled by L. N. Khitrin and then analyzed by him for the specific case of combustion of a spherical carbon particle. For brevity of exposition, the form of these equations is not given. They are extremely complicated, and in their complete form their application to the solution of practical problems is hopeless. The complete mathematical formulation and subsequent analysis in accordance with the experimental material pursued the aim of establishing the physicochemical meaning of the process characteristics obtained from experiment and of constructing a rational scheme for its calculation. An essential conclusion was drawn concerning the significance of the quantity \(k\) entering equation (3), which is customarily understood as the rate of the chemical reaction and is usually calculated from experimental data by formula (4). In reality, from the experimental data one calculates not \(k\), but \(\alpha\), determined by the following equation:
\[ \alpha = k + \frac{D_i \left(\dfrac{\partial c_i}{\partial n}\right)_F}{f(c_F)} . \tag{5} \]
In expression (5), \(k\)—the surface-reaction constant—enters as a term; the second term depends on the physical properties of the carbon, the distribution of fluxes, and the distribution of concentration over the visible surface of the carbon. For carbon dioxide and carbon monoxide the analogous relations are still more complicated.
In the case of a first-order reaction, equation (5) reduces to the following form:
\[ \alpha = k + D_i \left[\frac{\partial}{\partial n}\left(\ln \frac{c_i}{c_0}\right)\right], \tag{6} \]
i.e., \(\alpha\) is a constant with respect to \(f(c)\).
Consequently, in this case one may use boundary condition (3) to calculate the combustion process, but it must only be remembered that \(\alpha\) is not a surface-reaction constant, but accounts for the complex exchange process at the phase interface caused by the chemical process inside the carbon and on its surface. It is therefore expedient to call this quantity and the analogous quantities for \(\mathrm{CO_2}\) and \(\mathrm{CO}\) by the more general term—coefficient of reactive gas exchange. The manner of introducing it somewhat resembles the manner of introducing heat-transfer coefficients.
From the standpoint of the possibility of practical use for calculations of the quantities \(\alpha\) computed from experimental data, the question of the properties of these quantities is of great importance. In Fig. 7 the dependence of \(\lg \alpha\) on \(\frac{1}{T}\) is presented, making it possible easily to reveal the chemical properties of the quantity \(\alpha\). In the region of low temperatures, up to values of the criterion \(\lambda R=\sqrt{\frac{k_i}{D_i}}R<0.55\), the course of the dependence is unambiguous and rectilinear. The slope characterizes the temperature coefficient, equal to \(E\). This region corresponds to the condition when the influence of external diffusion processes is absent. In the region of values
\[ 0.55<\sqrt{\frac{k_i}{D_i}}R<40 \]
the quantity \(\alpha\) proves to depend on the external diffusion environment, i.e., it depends on particle size, concentration, and other circumstances. If investigations are carried out in this region and an attempt is made to generalize the results by means of relations of the type (3) or (4), then it is clear that this will have no great scientific meaning. The most varied dependences on concentration, size, and temperature may be obtained, not reflecting the nature of the phenomenon. Beyond this comparatively narrow transition region follows the region
\[ \sqrt{\frac{k_i}{D_i}}R>40, \]
in which processes of external diffusion play a major role.
Fig. 7. Theoretical dependence of the coefficient \(\alpha\) for a sphere on temperature.
Usually of practical interest are values of \(\sqrt{\frac{k_i}{D_i}}R\) up to 1000. The coefficient \(\alpha\) here again assumes an unambiguous and stable value and again acquires a temperature coefficient equal to \(E\). Consequently, in the principal regions the coefficient of reaction gas exchange is a function only of the temperature of the carbon, like a purely chemical constant.
Experimental investigations of the oxidation process of various coals corresponding to the region \(\sqrt{\frac{k_i}{D_i}}R<0.55\) (L. N. Khitrin, Z. F. Chukhanov, E. S. Golovina, S. E. Khaikina, and others) show that the combustion rate for small particles pro-
proportional to their volume. For larger particles, when the retarding effect of internal diffusion begins to make itself felt, the combustion rate can be calculated on the basis of relation (6). Internal reaction was directly investigated by A. S. Predvoditelev and S. E. Khaikina, and later by E. S. Golovina. Values were obtained for the coefficients of internal diffusion \(D_i\) and other characteristics of this process.
When the coefficients of reactive gas exchange are used as the basic calculated characteristics of combustion, the following circumstance becomes important. These coefficients are obtained from experimental data by calculation, and not by direct measurements. The calculations are based, first, on the assumption that the chemical-reaction order is first order for the reactions \(C + O_2\) and \(CO_2 + C\). It is known, however, that the order of these reactions may differ greatly from first order. We have followed the path of approximating by first-order laws, and our experiments, carried out over wide ranges of variation of the individual parameters, show that this path does not lead to any great distortion of reality, since the distortion of the law with respect to the effect of concentration proves to be of little significance for obtaining the principal dependence of \(\alpha\) on temperature. Nevertheless, this assumption may prove to be significant in certain special cases.
Second, in determining these coefficients it is necessary to calculate the diffusion of the reacting gases to the surface of the coal. These calculations are again connected with certain assumptions and suppositions that affect the magnitude of \(\alpha\). In particular, for example, the form of the velocity-distribution function at the surface, the value of the diffusion coefficient, and its dependence on temperature are significant. The effect of the form of the velocity-distribution function was clarified by A. S. Predvoditelev and O. A. Tsukhanova from the results of studies of the combustion of a carbon channel. A. S. Predvoditelev considered and solved problems of combustion of a carbon channel under the assumption of a plane and a parabolic profile of the velocity distribution over the channel cross section. Application of these solutions to experimental data gives somewhat different absolute values for \(\alpha\), especially in the region of high temperatures; but in either case \(\alpha\) remains a function only of temperature.
The choice of the temperature dependence of the diffusion coefficient affects the result in the same direction. This circumstance opens up great possibilities for constructing simplified calculation schemes of the process, as discussed below, in § 5.
Consequently, it may be considered proven that the values of \(\alpha\) can be taken as practical reaction characteristics of the fuel and used as the basis for the calculation and analysis of the processes of coal combustion.
This is shown especially convincingly by Fig. 8, which gives values for the coefficient of the reaction \(C+O_2\) for electrode carbon (denoted \(\alpha^c\)) according to Khitrin’s data for a spherical particle (solid lines) and according to Tsukhanova’s data for a straight channel (dashed line). These values agree well, although they were obtained under quite different diffusion conditions. The same also applies to the values of the coefficient for the reaction \(C+CO_2\) (denoted \(\alpha^c_{21}\)).
Fig. 8. Results of the calculation of \(\alpha\) for electrode carbon.
The generalization of the calculation of the coefficient of reaction gas exchange makes it possible to use the data obtained in considering the combustion of coal dust.
In the combustion of coal dust, practically the entire process takes place on the visible surface. From this standpoint, Khitrin processed P. A. Serebryakov’s experiments on the combustion of anthracite dust and compared them with his data on the combustion of anthracite particles. The experimental data correspond completely to the calculated values.
§ 4. BASIC REGULARITIES OF THE PROCESS. THE ROLE OF SECONDARY REACTIONS. THE INFLUENCE OF ADDITIVES ON THE GASIFICATION PROCESS
The coefficient of reaction gas exchange adopted as the determining quantity makes it possible to establish the basic regularity of the combustion process in the form of the dependence of the criterion \(\dfrac{\alpha R}{D}\) on the criteria characterizing diffusive transfer. It has been established that in the case of the external problem the combustion process is determined by the cri-
... the criterion \(\dfrac{wd}{D}\), and in the case of the internal problem the criterion \(P_{l\text{diff}}\dfrac{d}{z}=\dfrac{wd^2}{Dz}\).
L. N. Khitrin, applying the method of the reduced film, considered the combustion of a carbon particle accompanied by the afterburning of carbon monoxide and the reduction of carbon dioxide. In Fig. 9 the dependence of the quantities \(\dfrac{K_s d}{c_0D}\) on \(\dfrac{wd}{D}\) is presented in logarithmic coordinates.
Fig. 9. Dependence of \(\dfrac{K_s d}{c_0D}\) on \(\dfrac{wd}{D}\) for a spherical particle.
Along the abscissa axis the value \(\lg \dfrac{wd}{D}\) is plotted, and along the ordinate axis \(\lg \dfrac{K_s d}{c_0D}\).
The solid curve corresponds to the theoretical dependence obtained by Khitrin:
\[ K_s=\frac{12}{32}\,\frac{\delta}{\sqrt{\pi}}\,(1+\beta)c_0\sqrt{\frac{wD}{d}}. \]
The horizontal dashed straight line characterizes the limiting state corresponding to natural diffusion. The degree of participation of carbon monoxide combustion in the process of particle combustion in the latter case is determined by the quantity \(\dfrac{\nu R^2}{D}\), and, consequently, the role of carbon monoxide combustion is the smaller, the smaller the particle.
In combustion under the conditions of the internal problem, the criterion for afterburning of carbon monoxide is the quantity \(\dfrac{\chi z}{w}\), i.e., all other conditions being equal, the yield of carbon monoxide will increase as the residence time of the gas in the channel decreases. The intertwining of the main combustion reaction with secondary reactions leads to the fact that the gas composition near the coal surface depends strongly on the temperature and by no means corresponds to the so-called “primary” composition of the products of the reaction \(\mathrm{C}+\mathrm{O}_2\). Owing to the afterburning reaction of carbon monoxide, within a certain temperature range the process may proceed as if only carbon dioxide were formed, independently of the composition of the primary oxides. Conversely, at high temperature, owing to the formation of \(\mathrm{CO}_2\), combustion proceeds as if
Fig. 10. Example of the distribution of concentrations of reacting gases near the coal surface
only carbon monoxide were formed. In the case of maximally high temperatures, oxygen does not reach the coal surface at all, and the whole process occurs through the decomposition of carbon dioxide followed by combustion of carbon monoxide at some distance from the surface. Fig. 10 shows how the concentration field can vary near a burning spherical coal particle. It has not yet been possible to determine exactly the composition of the primary oxides, but one may say that the ratio of concentrations
\[ \frac{\mathrm{CO}}{\mathrm{CO}_2} \simeq 1 \]
is justified for the temperature range from \(800\) to \(1100^\circ\mathrm{C}\).
COMBUSTION OF CARBON
Phlegmatizing admixtures can exert a strong influence on the afterburning of carbon monoxide. In studying the influence of halide vapors, in particular iodine, on the gasification process, Tsukhanova established that the addition of iodine vapors to the blown-in gas causes a sharp increase in the yield of CO. Correspondingly, the rate of combustion of carbon increases. Fig. 11 gives the composition of the combustion products at the end of the carbon channel as a function of the temperature of the coal. The dashed lines are plotted from data obtained under the same conditions, but in the absence of iodine. Iodine vapors inhibit the reaction \(CO+O_2\) to a certain extent. Owing to this, oxygen obtains access to the carbon surface and is not consumed in the afterburning of carbon monoxide. It is interesting that the amount of oxygen burned is the same in both cases. This circumstance indicates that the afterburning of carbon monoxide takes place near the carbon surface. Intensification of the process and an increase in the yield of carbon monoxide upon the addition of halides may find practical application.
Fig. 11. Gas composition at the end of the carbon channel upon addition of iodine vapors to the blown-in gas. Flow rate 5 l/min, \(d=5\) mm, \(l=230\) mm.
§ 5. APPROXIMATE METHODS OF CALCULATION
The calculation relations that were obtained for the processing of experimental data usually have a rather complex and cumbersome form. The application of exact relations that take into account all the features of the process had methodological significance. Naturally, if one regards the calculation part of the work not merely as a means of investigation, but has practical applications in mind, then the calculation relations must be and can be simplified.
Ways of such simplification were indicated by A. S. Predvoditelev in 1939, when he gave a fundamental solution of the problem of combustion of a layer. The idea of A. S. Predvoditelev’s method consists in introducing generalized, total coefficients of reaction gas exchange, for which the basic dependences are established.
from appropriately arranged experiments, and are then generalized to the real process by methods of modeling.
The meaning of such aggregate coefficients can be understood from consideration of the approximate solution of the problem of combustion in a channel with allowance for the combustion of carbon monoxide, carried out by Tsukhanova. The mathematical formulation of the problem of combustion in a coal channel reduces to a system of diffusion equations in second-order partial derivatives with a source term. The boundary conditions express the equality between the amount of oxygen reacting at the surface and the amount of oxygen arriving at this surface by diffusion, and the law of diffusion of carbon monoxide from the surface. An exact solution of such a problem is very difficult. Let us introduce a simplification based on the fact that the form of the distribution function over the cross section, within known limits, does not affect the character of the functional relation between the mean value of the unknown function and the principal criteria, but is reflected only in the magnitude of the numerical constants.
Suppose that the concentration distribution over the cross section has the form
\[ c=A\left[1+\lambda \frac{r^2}{R^2}\right], \]
where \(A\) and \(\lambda\) are unknown functions of \(z\). We shall now form an equation for the cross-section-mean values of the concentrations, by averaging the differential equations with respect to \(r\) and using the boundary conditions to eliminate the unknowns \(A\) and \(\lambda\). Then we obtain a system of ordinary differential equations of the following form:
\[ w\frac{dc_1}{dz}=-pc_1-qc_2, \]
\[ w\frac{dc_2}{dz}=p\beta c_1-\alpha q c_2, \]
and the boundary condition of the form: \(c_1=c_0,\ c_2=0\) at \(z=0\).
The solution of such an equation presents no difficulty and gives the value of the cross-section-mean concentrations of oxygen and carbon monoxide as functions of the parameters \(\frac{pz}{w}\) and \(\frac{qz}{w}\). The quantities \(p\) and \(q\), in turn, are functions of \(\alpha, R, D, \chi\) and numerical constants. The values of the numerical constants depend on the law of distribution over the cross section. The expression for the parameter \(p\) has the following form:
\[ p=-\frac{4D}{R^2}\frac{1}{\dfrac{2D}{\alpha R}+a_1}. \]
Here \(a_1\) is a numerical constant whose value, in the case of a parabolic velocity profile, is equal to \(1/3\), and in the case of a flat ...
profile \(1/2\), and in the case of combustion under turbulent conditions \(2\). The combustion of a carbon channel under turbulent conditions was studied by S. A. Goldenberg. In this case the quantity \(D\) should be replaced by the value of the turbulent diffusion coefficient, while the values of \(\alpha\) coincide with those obtained from experiments on the combustion of spherical particles and a channel. The composition of the combustion products for a carbon channel is calculated by means of the approximate formulas with accuracy sufficient for practice.
In many practically important cases the form of the distribution function is unknown, and averaged values of the parameters are used. For example, when considering the flow of gas through a layer of loose material, the flow velocity is determined by the value of the so-called filtration velocity. In calculating the combustion of a fuel bed and of a suspension, it is hopeless to try to solve the problem exactly. On the basis of the functional relation between the criteria, one should, from appropriately formulated experiments, find the values of the total coefficients analogous to the coefficients \(p\) and \(q\). For the case of combustion of a coal bed, Kh. I. Kolodtsev and M. K. Grodzovskii carried out experimental investigations with subsequent processing of the data by the method of A. S. Predvoditelev.
Research in this field is continuing at the present time. Part of the results is contained in the monograph Combustion of Carbon. They are of great technical importance and could constitute the content of a separate article.
In the present article we have touched only upon the question of the physicochemical nature of the process of carbon combustion, which is the most complete and finished part of the study.
It should be noted that the developed methods of investigation and analysis of the process of carbon combustion have a more general significance and can be applied not only to carbon, but in general to heterogeneous reactions. Likewise, the approximate method of solving differential equations by averaging can be successfully applied in other fields, in particular in solving problems of heat exchange.
NOTATION
\(K_s\left[\dfrac{g}{\mathrm{cm}^2\,\mathrm{sec}}\right]\)—combustion rate.
\(k\)—rate constant of the chemical reaction of carbon with oxygen,
\(\chi\)— » » » » carbon monoxide with oxygen,
\(T\)—absolute temperature,
\(E\)—activation energy,
\(R\)—gas constant,
\(D\)—coefficient of molecular diffusion,
\(D_i\)—coefficient of diffusion inside the particle,
\(c\)—concentration,
$c_0$ — initial concentration,
$c_i$ — concentration inside the particle,
$c_F$ — concentration at the particle surface,
$c_1$ — concentration of oxygen,
$c_2$ — concentration of carbon monoxide,
$c_3$ — concentration of carbon dioxide,
$R$ — radius of the particle or channel.
$d = 2R$.
$n$ — direction of the normal,
$r, z$ — coordinates.
$a$ — coefficient of reaction gas exchange
$a^{c}$ — coefficient of the reaction gas exchange of $C + O_2$,
$a^{c}_{21}$ — coefficient of the reaction gas exchange of $CO_2 + C$,
$S_i$ — internal surface area per unit volume of coal,
$\delta$ — thickness of the film of burning carbon monoxide,
$u_{\mathrm{diff}}$ — diffusion rate,
$w$ — gas velocity,
$\alpha, \beta$ — stoichiometric factors.