ON APPROXIMATE FIRE MODELING OF STEADY-STATE GAS COMBUSTION PROCESSES

Norkin N. N.

1. Introduction

A physicochemical process in general, and gas combustion in particular, is a combination of phenomena of chemical kinetics, hydrodynamics, heat transfer, etc.; therefore, the successful transfer of research results from conditions of one scale to conditions of another scale according to the scheme

\[ \text{Model }(M) \to \text{Production }(P) \tag{1} \]

is possible only under the condition of a comprehensive study of the process. The connecting link in this scheme is scientific design (pr); then the expanded representation of the problem of applying similarity theory to the investigation of processes may be expressed as

\[ M \to \text{pr} \to P \tag{2} \]

Consequently, one of the immediate tasks of scientific transfer is the development of a method for carrying out the intermediate link, i.e., design; the theoretical and experimental basis of design must be similarity theory and modeling.

In view of the complex connection between the principal and accompanying phenomena, a purely analytical determination of the functional dependence between the characteristic features of the phenomena is impossible; experimental studies are necessary. Similarity theory indicates the direction of experimental investigation and provides a method for the mathematical processing of the results of the investigation. Modeling, by making it possible to reveal experimentally the quantitative relationships of the phenomena being studied, makes it possible to establish generalized laws.

Thanks to the extensive and fruitful work of Academician M. V. Kirpichev and his school [1], similarity theory and the modeling method are successfully applied to the study of various physical phenomena, such as hydro-aerodynamic phenomena, heat transfer, etc. Recently, works have appeared in this field aiming to apply the general theory of similarity to the study of physicochemical phenomena, during which the physical and chemical nature of the initial substances changes; that is, the foundations of the theory of physicochemical similarity are beginning to be laid.

The most complete in this respect is the study by G. A. Dyakonov [2]. He analyzed the system of the basic equations and the uniqueness conditions of physicochemical processes, and also derived two new similarity criteria: the criterion of quasistaticity and the criterion of contact.

Having formulated the similarity conditions, G. A. Dyakonov comes to the conclusion that, in complex cases of transformation (a reversible process and a multicomponent ...

system) exact similarity of two phenomena can be achieved only when the same initial products are transformed, at identical process temperatures; but with certain, quite admissible simplifications, i.e., when approximate modeling is used, the boundaries of possible generalizations are expanded.

Some investigators [3] believe that transferring an experimental model process to industrial conditions while satisfying all the requirements imposed by similarity theory is meaningful only for a 2–5-fold increase in equipment productivity. As a consequence, at a higher increase in productivity, the model process should be transferred to a group of apparatuses operating in parallel. Such a general conclusion, not supported by experimental data, is undoubtedly not convincing.

At the same time, a conclusion is also drawn concerning the very limited possibilities of similarity theory in the study of chemical processes, when applying similarity analysis to systems with irreversible chemical transformations.

The theory and method of fire modeling of gas combustion have not been sufficiently developed. In studying turbulent-diffusion combustion of gases, G. P. Ivantsev [4], A. M. Petunin, and S. N. Syrkin [5] (TsKTI) carried out fire modeling only under the following conditions: a) identical temperatures in the prototype and the model, b) the same fuel, and c) an identical hydrodynamic regime. The above-mentioned investigators believe that, at high temperatures of the furnace process, the rate of chemical transformation is so great that it not only is not the limiting factor, but leads the phenomenon in the model and in the prototype, as it were, to an absence of difference (to self-similarity); therefore, in their opinion, the principal factor determining the combustion process is the turbulent mixing of fuel with air. As a result, the criteria of chemical transformation are omitted from the analysis, and only criteria of a hydrodynamic character are considered.

However, it seems to us that in modeling fire processes one cannot restrict the application of similarity theory solely to the domain of hydrodynamic phenomena and, in particular, solely to questions of turbulent mixing, since certain forms of gas combustion do not depend at all on mixing in the combustion process (for example, surface, flameless combustion); in this case the chemistry of the process plays the decisive role. The author of the present work has attempted to apply similarity theory to the modeling of laminar-diffusion combustion and surface catalytic (flameless) combustion, and has arrived at conclusions concerning the possibility of substantially broadening the conditions of modeling.

II. System of basic equations of steady-state gas combustion processes

The system of basic equations, which constitute, as it were, mathematical models of the classes of phenomena that determine the physicochemical process of combustion, includes the following equations:

I — the stoichiometric equation of the chemical reaction;

II — the kinetic equation of the chemical reaction;

III — the equation of material exchange;

IV — the equation of energy (heat) exchange;

V — the equation of proportionality between the amount of substances undergoing transformation and the heat of transformation;

VI — the equation of material connection between the processes of transformation and exchange;

VII — the equation of thermal coupling between the processes of transformation and exchange;
VIII — the equation of motion of material flows;
IX — the equations of material and energy exchange at the external boundaries of the reaction system.

The stoichiometric equation of a chemical reaction, which follows from the principle of conservation of matter, describes the scheme of chemical transformations, the nature and number of the reacting components

\[ \alpha_1 A_1 + \alpha_2 A_2 + \ldots \rightleftarrows \beta_1 B_1 + \beta_2 B_2 + \ldots \tag{3} \]

where \(A_1, A_2\) are the chemical components entering into the reaction;

\(\alpha_1, \alpha_2\) are the stoichiometric numbers of transformations of the chemical components entering into the reaction;

\(B_1, B_2\) are the chemical components formed as a result of the reaction;

\(\beta_1, \beta_2\) are the stoichiometric numbers of transformations of the chemical components formed as a result of the reaction.

The stoichiometric equation, which fixes the nature of the reacting and resulting substances, is identical for a class of similar phenomena and thus excludes the possibility of similarity of physicochemical processes between chemical reagents that differ in their nature. A different formulation of the question may lead to the investigation of a certain symbolic process instead of the real one, and the transfer of the results of studies of this symbolic process to the real one would be unlawful. From this point of view, such directions in work as modeling the combustion of coal by a solution of soda in water are hardly expedient. Of course, the possibility is not excluded of replacing the nature of the reagents and the scheme of chemical transformations when solving certain particular questions from the external environment accompanying the chemical transformation; this is sometimes done by the method of analogies, for example hydro-electrical analogies and thermo-electrical analogies.

The requirement that equation (3) be identical for a group of similar phenomena is not an obstacle, when modeling the combustion of gaseous mixtures, to replacing one kind of combustible gas mixture by another kind, for example, water gas by illuminating gas, etc., since for the majority of industrial combustible gases the number of chemical components is rather limited (hydrogen, carbon monoxide, methane), and from the standpoint of composition, but not of production technology, industrial combustible gases differ from one another only in the quantitative ratios among these three components.

The kinetic equation of a chemical reaction, which is a hidden expression of the principle of conservation of momentum, establishes that the instantaneous value of the rate of transformation is proportional, first, to the concentration of the substances undergoing transformation; second, to a power function of the temperature; and third, depends on the magnitude of the activation energy.

In general form, for a homogeneous reversible gas reaction, this equation is expressed as

\[ U_{\mathrm{n}}=\frac{DC_i}{d\tau}=K_1 f_1(C_{A1}\cdot C_{A2}\ldots)-K_2 f_2(C_{B1}\cdot C_{B2}\ldots) \tag{4} \]

Here \(U_{\mathrm{n}}\) — \(\left(\dfrac{\text{mol}}{\text{cm}^3\text{sec}}\right)\) — is the rate of chemical transformation;

\[ C_i \;-\;\left(\frac{\text{mol}}{\text{cm}^3}\right) \]

— is the concentration of component \(i\);

\(C_{A_1}, C_{A_2}\) — \(\left(\dfrac{\mathrm{mol}}{\mathrm{cm}^3}\right)\) — instantaneous value of the concentration of the components entering into the reaction;

\(C_{B_1}, C_{B_2}\) — \(\left(\dfrac{\mathrm{mol}}{\mathrm{cm}^3}\right)\) — the same, of the components formed in the course of the reaction;

\(f_1(C_{A_1} C_{A_2}\ldots)\) — a function of the concentration of the reacting substances; the form of the function depends on the mechanism of transformation (on the order of the reaction, etc.);

\(f_2(C_{B_1} C_{B_2}\ldots)\) — the same, of the substances obtained;

\(K_1\) — rate constant of the forward reaction;

\(K_2\) — rate constant of the reverse reaction.

In this case,

\[ \frac{DC_i}{d\tau} \]

is the substantial derivative, equal for a flow system to:

\[ \frac{DC_i}{d\tau}=\frac{\partial C_i}{\partial \tau}+w\frac{\partial C_i}{\partial x} \]

For steady-state processes occurring in apparatus with one characteristic dimension,

\[ \frac{DC_i}{d\tau}=w\frac{\partial C_i}{\partial x}. \]

Gas-combustion processes may be approximately assigned to the group of limiting nonstatic (irreversible) processes, since the rates of reverse reactions have practically a small significance only at temperatures above \(1700^\circ\mathrm{C}\).

Thus, according to Nernst and Wartenberg [6], the degree of dissociation of water vapor, i.e., the reaction reverse to the combustion of hydrogen, in the temperature interval \(2000^\circ\mathrm{K} \div 2500^\circ\mathrm{K}\) ranges from 0.582% to 4.21% (by weight). As a consequence of this, equation (4) is simplified by the disappearance of the term \(K_2 f_2(C_{B_1}C_{B_2}\ldots)=0\).

In addition, if one accepts the proposition of Academician N. N. Semenov that, for the majority of homogeneous gaseous combustion reactions, the reaction rate is proportional not only to the instantaneous amount of the substance being transformed, but also to the instantaneous amount of the substance already transformed, then the combustion-reaction mechanism and the value of the function \(f_1(C_{A_1} C_{A_2}\ldots)\) become more definite, i.e.,

\[ f_1(C_{A_1} C_{A_2}\ldots)=(C_{0A_1}-C_{\tau A_1})C_{\tau A_1}\cdot(C_{0A_2}-C_{\tau A_2})C_{\tau A_2}. \tag{5} \]

Here, in addition to the preceding notation, the following have been introduced:

\(C_{0A_1}\) — \(\left(\dfrac{\mathrm{mol}}{\mathrm{cm}^3}\right)\) — initial concentration of component \(A_1\);

\(C_{0A_2}\) — \(\left(\dfrac{\mathrm{mol}}{\mathrm{cm}^3}\right)\) — the same, of component \(A_2\);

\(C_{\tau A_1}\) — \(\left(\dfrac{\mathrm{mol}}{\mathrm{cm}^3}\right)\) — change in the concentration of component \(A_1\) by the given instant of time;

\(C_{\tau A_2}\) — the same, of component \(A_2\).

Thus, for a steady-state process of gas combustion, equation [4] can approximately be replaced by the following:

\[ U_{\mathrm{p}}=w\frac{\partial C_i}{\partial x} =K_1\,(C_{0A_1}-C_{\tau A_1})\,C_{\tau A_1}\cdot(C_{0A_2}-C_{\tau A_2})\,C_{\tau A_2} \tag{6} \]

Substituting the value of \(K\)—the constant of the chemical-reaction rate according to the Arrhenius law,

\[ K=Ae^{-\frac{E}{RT}}, \]

we obtain

\[ U_{\mathrm{p}}=w\frac{\partial C_i}{\partial x} =A\,(C_{0A_1}-C_{\tau A_1})\,C_{\tau A_1}\cdot(C_{0A_2}-C_{\tau A_2})\,C_{\tau A_2}e^{-\frac{E}{RT}} \tag{7} \]

The equation of material exchange, ensuring the supply of reacting substances to the system and the removal from the system of the reaction products formed, may be represented in the following form:

\[ U_0=-\sum_{\beta=x,y,z}\frac{\partial w C_i}{\partial \beta} +\sum_{\beta=x,y,z}D\frac{\partial}{\partial \beta}\left(\frac{\partial C_i}{\partial \beta}\right) \tag{8} \]

Here \(U_0\left(\dfrac{\mathrm{mol}}{\mathrm{cm}^3\,\mathrm{sec}}\right)\) is the rate of material exchange,

\[ w\left(\frac{\mathrm{cm}}{\mathrm{sec}}\right) \]

is the linear velocity of displacement of the material stream;

\[ D\left(\frac{\mathrm{cm}^2}{\mathrm{sec}}\right) \]

is the diffusion coefficient of the component;

the term

\[ \sum_{\beta=x,y,z}\frac{\partial w C_i}{\partial \beta} \]

characterizes the intensity of stream transfer;

the term

\[ \sum_{\beta=x,y,z}D\frac{\partial}{\partial \beta}\left(\frac{\partial C_i}{\partial \beta}\right) \]

characterizes the intensity of diffusion transfer.

The equation of energetic heat exchange within the system, expressing the relationship between the heat content of the system, the heat of stream (convective) transfer, the heat of conduction and radiation, has the form

\[ Q_0=\sum_{\beta=x,y,z}\frac{\partial}{\partial \beta}\left(\lambda\frac{\partial t}{\partial \beta}\right) +\sum_{\beta=x,y,z}(\rho\Delta t) +\sum_{\beta=x,y,z}\frac{\partial}{\partial \beta}\left[C_s\varphi\left(\frac{T}{100}\right)^4\right] +\sum_{\beta=x,y,z}C_p w\frac{\partial T}{\partial \beta}\gamma \tag{9} \]

Here \(Q_0\left(\dfrac{\mathrm{kcal}}{\mathrm{cm}^3\,\mathrm{sec}}\right)\) is the rate of heat exchange per unit time in a unit volume of the reaction system;

\[ C_p\left(\frac{\mathrm{kcal}}{\mathrm{g\,deg}}\right) \]

is the heat capacity of the flowing medium at constant pressure;

\[ \gamma\left(\frac{\mathrm{g}}{\mathrm{cm}^3}\right) \]

is the specific weight of the flowing medium,

\(t, T\) (deg) are the temperature of the flowing medium;

\[ \lambda\left(\frac{\mathrm{kcal}}{\mathrm{m\,sec\,deg}}\right) \]

is the thermal-conductivity coefficient of the flowing medium;

\(C_s\) is the radiation coefficient of an absolutely black body;

\(\varphi\) is the degree of blackness of the flowing medium;

The equation of proportionality between the heat of transformation and the quantity of transforming substances, as a consequence of applying the principle of conservation of energy under the conditions of a chemical process, has the form

\[ -q\left[\frac{\partial w C_i}{\partial \beta} +\sum_{\beta=x,y,z} D_i\frac{\partial}{\partial \beta} \left(\frac{dC_i}{d\beta}\right)\right] = Q_0+\sum_{\beta=x,y,z} C_p w_\gamma \frac{\partial T}{\partial \beta}. \tag{10} \]

Here, in addition to the notation indicated earlier,

\(q\) — \(\dfrac{\text{kcal}}{\text{mol}}\) — is the thermal effect of transformation of the component.

The connection between the processes of transformation and exchange is characterized by two equations: a) the equation of material connection and b) the equation of energetic (thermal) connection. Under the conditions of a steady-state transformation, when there must exist a dynamic equilibrium between the processes of transformation and exchange, it is obvious that the rates of transformation and exchange are equal to each other, i.e.,

\[ U_n=U_0 \tag{11} \]

When using this equation it is necessary to bear in mind that the exchange processes characterized by the terms on the right-hand side of equation (11) play a subordinate role with respect to the kinetic processes; this means that an arbitrary assignment of values of the quantities of the exchange process is inadmissible. In combustion processes, arbitrary assignment of exchange quantities does not entail regulation of the kinetics of the process, but leads to a violation of the stability of the process. Thus, if the exchange rates of the process are smaller than the rate of the chemical combustion reaction, this will lead to a violation of the continuity of the process; instead of continuous combustion there will occur repeated explosions and pops; if, on the other hand, the exchange rates of the process are greater than the rate of the chemical reaction, this will likewise lead to a violation of the continuity of the process, since combustion will cease as a result of the “blowing out” of the flame, or separation and dispersion of the flame.

The equation of energetic (thermal) connection in general form will be

\[ qU_n=Q_0 \tag{12} \]

The equation of motion, characterizing the interaction of the forces of inertia, gravity, external pressure, and internal friction, is the well-known Navier–Stokes equation of motion.

The equation of material exchange at the boundaries of the system characterizes the interaction of the moving reacting gas system with the surrounding medium. If the process of chemical transformation occurs throughout the entire volume of the reaction space bounded by enclosing surfaces, then material exchange at the boundaries of the system is absent, and each term of the equation characterizing material transport is separately equal to zero. If the process of chemical transformation does not extend over the entire volume of the reaction space, but has only a local significance, then the interaction of the moving reacting gas system with the surrounding medium may be expressed in the change in the concentration of the reacting substances as a result of the entrainment into the main moving stream of secondary air or inert gases (combustion products) from the surrounding medium.

In this case the conditions of material exchange at the boundary surface may be characterized by the equation of turbulent transfer of the quantity of motion (7)

\[ E=l^2\frac{dw}{dx}. \]

Here \(E\) \(\left(\dfrac{\mathrm{m}^2}{\mathrm{sec}}\right)\) is the kinematic exchange coefficient, having the dimension of a diffusion coefficient;

\(l\) (m) is the length of the mixing path in the direction perpendicular to the motion of the gas, or, otherwise, the width of the region covered by turbulent motion, measured from the flow axis;

\(w\) \(\left(\dfrac{\mathrm{m}}{\mathrm{sec}}\right)\) is the velocity of the main stream;

\(x\) (m) is the distance from the axis of the main stream.

The conditions of uniqueness that mark the individuality of phenomena belonging to a single class of physicochemical transformations (in the present case, to the class of gas combustion) are the following: (a) the composition of the reacting substances and the stoichiometric formula of the transformation; (b) the mechanism of the transformation; (c) the fields of concentrations, temperatures, physical constants, and velocities of motion of the initial substances at the initial moment of the transformation; (d) the laws governing changes in the physical constants during the transformation process; (e) the specificity of the geometrical conditions both within the reaction system and with respect to the external medium.

Some of these conditions in modeling must be identical, others similar. Thus, the composition of the reacting substances and the stoichiometric formula of the transformation under condition (a), and the mechanism of transformation under condition (b), must be identical for the phenomena being modeled; this requirement excludes the possibility of modeling phenomena of different chemical nature. However, if one takes into account, as was already noted above, that the diversity in the composition of the most widespread industrial combustible gases is due mainly only to different ratios among hydrogen \(\mathrm{H}_2\), carbon monoxide \(\mathrm{CO}\), and methane \(\mathrm{CH}_4\), and then considers that \(\mathrm{H}_2\) and \(\mathrm{CO}\) have one and the same stoichiometric formula of transformation and one and the same order and mechanism of combustion, then, because of these circumstances, no obstacles arise to carrying out approximate modeling of gas mixtures of different chemical composition. The remaining conditions of uniqueness must be similar, i.e., conditions (c), (d), and (e); but in the practical implementation of approximate modeling it may become necessary to abandon certain inessential dependences and relations, which will entail some distortion of exact similarity; the degree of distortion of similarity is established experimentally and is subsequently taken into account by introducing dimensionless simplexes additional to the similarity criteria.

III. Similarity Criteria for Approximate Fire Modeling

The similarity criteria whose numerical values must be identical for the processes being modeled are: the criteria of physicochemical transformation, the criteria of material and heat exchange, and hydrodynamic criteria. Under the conditions of a stationary physicochemical process, there is no need for independent consideration of the transformation equations and the exchange equations, since the dynamic equilibrium of the transformation and exchange processes requires equality of the rates of transformation and exchange, as a result of which the number of criteria is considerably reduced. It is further necessary to point out that the combustion of gas mixtures may be assigned to the category of extremely nonstatic processes with equilibrium shifted as far as possible toward the final products of the transformation; therefore the criterion of quasi-static-

... (as a measure of the ratio of the rates of the forward and reverse reactions and the relation between the concentrations of the initial and final products of transformation) and the contact criterion according to the system of G. K. Dyakonov (ENIN, Academy of Sciences of the USSR), naturally are not considered further for solving the problem posed.

The stoichiometric equation of the chemical reaction (3) does not yield similarity criteria, since the processes being modeled have one and the same stoichiometric formula of transformation.

The kinetic equation of the chemical reaction (7) and the material-exchange equation (8) are combined, for steady-state processes, into the equation of material relation between the processes of transformation and exchange:

\[ U_{\mathrm{n}}=-\sum_{\beta=x,y,z}\frac{\partial w C_i}{\partial \beta} +\sum_{\beta=x,y,z}D\frac{\partial}{\partial \beta}\left(\frac{\partial C_i}{\partial \beta}\right) \tag{14} \]

From this equation the exchange-kinetic criteria are obtained:

\[ K_{\mathrm{I}}=\frac{wC}{lU_{\mathrm{n}}};\quad K_{\mathrm{II}}=\frac{DC}{l^2U_{\mathrm{n}}} \quad \text{and the derivative} \quad K_{\mathrm{III}}=\frac{wl}{D}. \]

The quantity \(U_{\mathrm{n}}\) entering into these criteria—the rate of physicochemical transformation for combustion processes—may be determined according to the theory of thermal flame propagation (N. N. Semenov, Ya. B. Zel’dovich, D. A. Frank-Kamenetskii) from the equation:

\[ U_{\mathrm{n}}= \frac{\partial}{\partial \beta} \left[ \sqrt{ \frac{n\lambda A e^{-E/RT_1}\gamma C}{q(T_1-T_0)^m} \left(\frac{RT_1}{E}\right)^p } \right] \tag{15} \]

\[ \beta=x,y,z \]

where, for a first-order reaction, \(n=2;\ c=1;\ m=1;\ p=2\);

for a second-order reaction, \(n=4;\ c=c_0;\ m=2;\ p=3\).

The kinetic equation (7) and the heat-exchange equation (9) are combined, for steady-state processes, into the equation of thermal relation:

\[ qU_{\mathrm{n}}= \sum_{\beta=x,y,z}\left[ \frac{\partial}{\partial \beta} \left(\lambda \frac{\partial t}{\partial \beta}\right) \right] + \sum_{\beta=x,y,z}\frac{\partial}{\partial \beta}(\alpha \Delta t) + \]

\[ +\sum_{\beta=x,y,z}\frac{\partial}{\partial \beta} \left[ c_s \varphi \left(\frac{T}{100}\right)^4 \right] + \sum_{\beta=x,y,z}c_p w\frac{\partial T}{\partial \beta}\gamma \tag{16} \]

From this equation the exchange-kinetic criteria are obtained:

\[ K_{\mathrm{IV}}=\frac{\lambda t}{l^2qU_{\mathrm{n}}};\quad K_{\mathrm{V}}=\frac{\alpha t}{lqU_{\mathrm{n}}};\quad K_{\mathrm{VI}}=\frac{c_s\varphi T^4}{lqU_{\mathrm{n}}};\quad K_{\mathrm{VII}}=\frac{c_p\gamma wT}{lqU_{\mathrm{n}}}. \]

The criteria \(K_{\mathrm{IV}}, K_{\mathrm{V}}\), and \(K_{\mathrm{VI}}\) characterize the fraction of exchange heat by all modes of heat transfer relative to the heat of the chemical reaction, and \(K_{\mathrm{VII}}\)—the fraction of heat used to change the heat content of the gases.

Eliminating the quantity \(qU_{\mathrm{n}}\) gives the usual heat-exchange criteria \((Pe, B_0)\).

Combining the equation of material and thermal relation (14 and 16), we obtain the equation of proportionality between the quantity of substances undergoing transformation and the heat of transformation:

\[ -q\left[ \frac{\partial wC_i}{\partial \beta} + \sum_{\beta=x,y,z}D_i\frac{\partial}{\partial \beta} \left(\frac{\partial C_i}{\partial \beta}\right) \right] = Q_0+ \sum_{\beta=x,y,z}c_p w\gamma \frac{\partial T}{\partial \beta} \tag{17} \]

The criteria obtained from this equation are the following:

\[ K_{\mathrm{VIII}}=\frac{qC_i}{\gamma c_pT};\quad K_{\mathrm{IX}}=\frac{qDC}{\gamma lc_pwT} \quad \text{and the derivative} \quad K_{\mathrm{III}}=\frac{wl}{D}. \]

The criterion \(K_{\mathrm{VIII}}\) characterizes the proportionality of concentration and temperature fields in the directions of motion of the flame gases; \(K_{\mathrm{IX}}\), the same in the direction of diffusion, and the derivative \(K_{\mathrm{III}}\), the ratio between convective and diffusion transfer in the material system.

From the equation of motion of the material fluxes, the usual criteria are obtained:

\[ Re=\frac{wl}{\nu};\quad Ar=\frac{l^{3}}{\nu^{2}}\cdot\frac{\Delta\rho}{\rho}\,g\quad \text{and}\quad Eu=\frac{\Delta P}{\rho\omega^{2}} \]

From the equations of material and energy exchange at the external boundaries of the reaction system (13), the criteria are obtained: for material exchange, from the equation of turbulent transfer of momentum,

\[ K_{\mathrm{X}}=\frac{E}{lw} \]

and for heat exchange, analogously to the criteria \(Pe\), \(Nu\), \(Bo\).

The criterion of turbulent momentum transfer is appropriately used to characterize phenomena connected with turbulent mixing at the boundaries of the system, for example, in the recirculation of furnace gases with a flame front, in combustion in an unclosed system, etc. The criteria of heat exchange at the boundaries of a closed system can sometimes be replaced by the efficiency of the installation, taking into account heat losses to the external medium.

The general integral of the system of basic equations can be solved in the form of a functional relation between the determining and the determined criteria. The choice of one or another of the determined criteria is arbitrary, depending on the objective of the planned experimental study and on the form of gas combustion. In what follows, analysis of the relations between criteria is carried out for two forms of gas combustion: for laminar-diffusion combustion and for surface catalytic (flameless) combustion.

In carrying out approximate fire modeling of flame laminar-diffusion combustion, taking into account the characteristic features of this form of combustion, the following criteria may be excluded from consideration:

\[ K_{1}=\frac{wC}{lU_{\mathrm{n}}};\quad K_{\mathrm{III}}=\frac{wl}{D};\quad K_{\mathrm{IV}}=\frac{\lambda t}{l^{2}qU_{\mathrm{n}}};\quad K_{\mathrm{V}}=\frac{at}{lqU_{\mathrm{n}}} \]

and all criteria of motion, owing to the following:

a) the criteria \(K_{1}\) and \(K_{\mathrm{III}}\) are associated with convective transfer, but laminar-diffusion combustion depends predominantly on diffusion transfer of matter and heat;

b) on the same basis, the criteria \(K_{\mathrm{IV}}\) and \(K_{\mathrm{V}}\) need not be considered, since the predominant type of heat transfer is radiation; as for heat transfer within the flame by conduction, the role of conduction is reflected in the kinetic formula by means of the quantity \(U_{\mathrm{n}}\), calculated according to the theory of thermal propagation of flame;

c) owing to the self-similarity of laminar flows at any value of \(Re\) below the critical one, the requirement \(Re=\mathrm{idem}\) is not obligatory; however, the geometric similarity of the modeled phenomena must ensure the obligatory condition \(Re<2320\). It should then be noted that, in the free diffusive development of the flame, the value of the criterion \(Eu\) is not significant. The phenomenon of curvature of the flame axis in the vertical plane may be of interest; in this case the criterion \(Ar\) is of substantial importance.

Consequently, the phenomenon of laminar-diffusion combustion is characterized in its main features by a functional relation among the criteria:

\[ K_{\mathrm{II}}=\frac{DC}{l^2U_{\mathrm{n}}};\quad K_{\mathrm{VI}}=\frac{C_s\varphi T^4}{lqU_{\mathrm{n}}};\quad K_{\mathrm{VII}}=\frac{C_p\gamma wT}{lqU_{\mathrm{n}}};\quad K_{\mathrm{VIII}}=\frac{qC}{\gamma C_pT};\quad K_{\mathrm{IX}}=\frac{qDC}{\gamma l C_p wT} \]

and the condition \(Re<2320\);

but since

\[ \frac{K_{\mathrm{VIII}}}{K_{\mathrm{IX}}}\cdot\frac{\nu}{Re.D}=Pr' \quad \text{and} \quad \frac{K_{\mathrm{VII}}}{K_{\mathrm{VI}}}=\frac{C_p\gamma w}{C_s\varphi T^3}=Bo, \]

the similarity equation for characterizing the main features of laminar-diffusion combustion takes the form

\[ K_{\mathrm{II}}=f(Pr',Bo)\quad \text{for } Re<2320 \]

or

\[ \frac{DC}{l^2U_{\mathrm{n}}}=f(Pr',Bo)\quad \text{for } Re<2320 \tag{18} \]

The linear dimension \(l\) should be taken in the direction of diffusion, i.e., as the thickness of the diffusing layers.

Modeling for finding the functional relation in the above form of the similarity equation makes it possible to establish the law governing the change in the relative flame length in the direction of motion, \(\frac{L_{\mathrm{m}}}{L_{\mathrm{p}}}\), as a function of the diffusion-kinetic conditions, the properties of the material system \((Pr')\), heat transfer by radiation \((Bo)\), and the thickness of the diffusing gas layers \((l)\), i.e., to solve the basic question that is of substantial practical importance in diffusion combustion.

The phenomenon of surface catalytic (flameless) combustion is characterized in its main features by a mutual relation among the criteria:

\[ K_{\mathrm{I}}=\frac{wC}{lU_{\mathrm{n}}};\quad K_{\mathrm{V}}=\frac{\alpha t\lambda}{lqU_{\mathrm{n}}};\quad K_{\mathrm{VI}}=\frac{C_s\varphi T^4}{lqU_{\mathrm{n}}};\quad K_{\mathrm{VII}}=\frac{C_p\gamma wT}{lqU_{\mathrm{n}}}; \]

\[ K_{\mathrm{VIII}}=\frac{qC}{\gamma C_pT};\quad Re=\frac{wd\rho}{\mu}\quad \text{and}\quad Eu=\frac{\Delta P}{\rho w^2} \]

The remaining criteria may be excluded from consideration because of their insignificant influence on the essential features of the phenomenon when carrying out approximate fire modeling.

The number of criteria subject to consideration can be reduced by transforming them;

\[ \frac{K_{\mathrm{V}}}{K_{\mathrm{VI}}}=\frac{\alpha}{C_s\varphi T^3}=Bo' \]

shows the relation between the convective and radiative components of external heat exchange.

The criterion

\[ \frac{K_{\mathrm{I}}}{K_{\mathrm{VIII}}}=K_{\mathrm{VII}}=\frac{wC_p\gamma T}{lqU_{\mathrm{n}}} \]

establishes the relation between the heat content of the gases (combustion products) and the heat of combustion. The ratio \(\frac{C_p\gamma T}{q}\) represents a pi-

pyrometric combustion coefficient $\eta$; therefore, instead of the criteria $K_{\mathrm{I}}$, $K_{\mathrm{VII}}$ and $K_{\mathrm{VIII}}$, one new criterion is introduced,

\[ K_{\mathrm{XI}}=\eta\cdot\frac{w}{lU_{\mathrm{n}}}, \]

which shows the relation between the pyrometric combustion coefficient and the flame-propagation velocity.

The hydrodynamic criteria $Re$ and $Eu$ are of substantial importance. The value of the criterion $Re$ determines the range of velocities and the deformation of the flame front (a conical flame front at large values of $Re$ becomes almost plane). The value of the criterion $Eu$ determines the resistance of the catalytic system and the required pressure.

Thus, the basic features of surface catalytic combustion are characterized by the dependence between the criteria:

\[ Bo';\quad \eta\,\frac{w}{lU_{\mathrm{n}}};\quad Re\ \text{and}\ Eu. \]

The linear dimension $l$ entering into the criterion characterizes the length of the channel in a channel burner, on whose surface combustion takes place; for burners made of a granular bed, $l$ is a quantity inverse to the specific surface of the bed, i.e.

\[ |l|=\left|\frac{1}{s}\right|. \]

The principal question whose solution is of direct practical importance in catalytic surface combustion is the question of the heat intensity of combustion

\[ \left(\frac{\mathrm{kcal}}{\mathrm{m}^{3}\,\mathrm{h}}\ \text{or}\ \frac{\mathrm{kcal}}{\mathrm{m}^{2}\,\mathrm{h}}\right) \]

and its dependence on the catalytic influence of the material of the combustion bed, the specific surface, and the blowing velocity of the combustible mixture.

In accordance with this, it is expedient to carry out the experimental study by means of approximate fire modeling and to process the results in the form of a relation according to the similarity equation

\[ K_{\mathrm{XI}}=f(Bo',\,Re,\,Eu). \]

The desired value of the heat intensity of combustion is directly related to the pyrometric combustion coefficient.

IV. Conclusion

1. On the basis of an analysis of the literature of other investigators, and also in development of the author’s own work, a system of equations has been described which characterize the physicochemical process of gas combustion under steady-state conditions.

2. Criteria following from the basic equations of gas combustion have been identified. Discussion of the specific conditions for the development of steady-state gas-combustion processes made it possible to reduce a large number of criteria to a considerably smaller number, without impairing the completeness with which the basic properties of the phenomena under consideration are represented.

3. Criterion equations have been compiled for such forms of gas combustion as: a) laminar-diffusion combustion and b) surface catalytic (flameless) combustion.

These criterion equations constitute the basic content of the proposed hypothesis of approximate fire modeling of steady-state gas-combustion processes.

References Cited

1. M. V. Kirpichev and M. A. Mikheev. Modeling of Thermal Devices, Publishing House of the Academy of Sciences of the USSR, 1936.

2. G. A. D’yakonov. Reports of the Academy of Sciences of the USSR, New Series, XXXIX, 4, 1943, and Proceedings of the Department of Technical Sciences of the Academy of Sciences of the USSR, 7–8, 1944.

3. Damköhler. Advances in Chemistry, VII, 5, 1938.

4. G. P. Ivantsov. In the collection of works of Stal’proekt, Problems of Gas Motion in Furnaces, edited by G. P. Ivantsov, 1936.

5. A. M. Petunin, S. N. Syrkin. Soviet Boiler-and-Turbine Construction, 8, 1937.

6. Aufhäuser. Fuel and Its Combustion, 1933.

7. Cited from the collection of translated articles Problems of Turbulence, 1936.