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Distribution of Active Reaction Centers in the Combustion Zone
V. N. Kondrat’ev
The course of chemical reactions in flames is determined by various physicochemical and aerodynamic factors that characterize the conditions of flame combustion. One of the most important chemical factors is the concentration of various substances in particular parts of the flame. From the point of view of the kinetics and mechanism of combustion reactions, the concentration of active substances—free atoms and radicals that determine the chain development of the reaction—is of special interest. Therefore, the study of the distribution of active substances in the flame zone should be one of the most powerful methods for investigating combustion processes.
The concentration of a given active substance in a given elementary volume of the combustion zone is a direct indicator of those chemical processes that occur in this volume or in its immediate vicinity. This applies especially to ordinary flames, i.e., to flames burning at atmospheric pressure and at a temperature of the order of 2000°. Indeed, if one calculates the length of the diffusional displacement \(x\) of a molecule of an active substance during the time \(\tau\), equal to its mean lifetime under flame conditions, it is not difficult to show that this length, in the case of ordinary flames, does not exceed, in order of magnitude, one tenth of a millimeter. Namely, substituting into the well-known formula
\[ x^2 = \frac{4}{3\pi}\frac{v^2}{z}\tau \tag{1} \]
(\(v\) is the velocity of the molecule, \(z\) is the number of gas-kinetic collisions it experiences per second) the time \(\tau\), calculated on the assumption that the destruction of the active substance is associated with its bimolecular reaction with one of the initial substances, and equal to
\[ \tau = \frac{1}{az} e^{\frac{A}{RT}} \tag{2} \]
($\alpha$ is the ratio of the partial pressure of the initial substance in the given element of the flame volume to the total pressure, $\varepsilon$ is the steric factor, and $A$ is the activation energy), we obtain
\[ x \approx \frac{T}{1000} \frac{e^{\frac{A}{2RT}}}{\sqrt{\alpha \varepsilon p_A}}\,10^{-3}\ \text{mm} \tag{3} \]
($p_A$ is the total pressure in atmospheres). From the last formula, for $T = 2000^\circ$ abs., $A = 5000 — 10000$ cal per mole, $\alpha = \frac{1}{2}$, $\varepsilon = 0.1$ and $p_A = 1$ atm, we find $x \approx 0.02 — 0.1$ mm. For the same values of $\alpha$, $\varepsilon$, and $A$, but with $T = 1000^\circ$ abs. and $p_A = 0.01$ atm, corresponding to the conditions of a rarefied flame, we find $x \approx 1 — 5$ mm—in agreement with the diffuse character of these flames.
The difficulty of the problem of measuring the distribution of active substances in a flame, caused by their extreme lability, requires the use of refined methods of investigation for its solution. This condition is satisfied by modern physical methods and, above all, by the spectroscopic method. As is known, the latter is used in two forms: as an emission method, based on the study of emission spectra, and as an absorption method, consisting in the study of absorption spectra.
The first of these methods, which has a history of more than half a century, when applied to various flames made it possible to establish the presence in these flames of free radicals of the type OH, CH, CS, SO, PO, C₂, HCO, etc. On the basis of data obtained with the aid of this method, various authors have drawn a number of conclusions concerning the nature of the chemical reactions occurring in flames under one or another set of combustion conditions. Thus, for example, it was established that in the outer cone of the Bunsen flame the hydroxyl bands are especially intense; the presence of the latter must therefore be connected with the oxidizing properties of the flame. Conversely, the great intensity of the C₂ bands in the inner cone, together with the HCO bands observed in the spectra of a number of flames, is a characteristic sign of an excess of fuel. In a number of cases it is possible to trace how, with enrichment of the fuel mixture, the increase in the intensity of the C₂ bands proceeds in parallel with the weakening of the hydroxyl bands.
The emission method, however, gives only a partial and not always unambiguous solution of the problem of the distribution of active centers in flames, since the intensity of radiation ($I$) is far from always proportional to the concentration of the radiating substance ($c$). This proportionality holds only in the case of a purely thermal origin of the radiation (temperature glow), when the radiation intensity is determined by the Boltzmann factor
\[ I = A e^{-\frac{E}{RT}} \cdot c \tag{4} \]
($A$—the probability of emission, $E$—the excitation energy). However, in this case as well, the distribution of concentrations can be obtained from measurements of the distribution of radiation intensity only if the distribution of temperature in the flame is known. True, the spectroscopic method gives, in principle, the possibility of measuring the temperature of individual regions of the flame; however, from a methodological standpoint this is by no means an easy task.
In those cases where the radiation of the flame is chemiluminescence, i.e., nonequilibrium radiation, the relation between intensity and concentration is a complex function of physical and kinetic factors, by virtue of which even an approximate proportionality between the radiation intensity and the concentration of the emitting substance may fail to hold. Therefore conclusions about the distribution of the active substance in the flame, drawn on the basis of studying the distribution of the intensity of its radiation, may prove erroneous.
Free from a number of the shortcomings mentioned is the second spectroscopic method—the absorption method. However, the low sensitivity of this method, in its usual form considerably inferior to the sensitivity of the emission method, makes it practically applicable only in very rare cases. There are two possibilities for increasing the sensitivity of this method, which significantly broaden the limits of its applicability. The first of them consists in using instruments of high resolving power, for example, a diffraction grating, whose sensitivity is 1–2 orders of magnitude greater than the sensitivity of ordinary spectrographs. The low sensitivity of the latter is due to the circumstance that the width of the spectral interval $\Delta\nu$ resolved by them, as a rule, many times exceeds the width of the absorption lines $\nu'$, as a consequence of which the average weakening of intensity (in the interval $\Delta\nu$), which is a measure of the absorption intensity, turns out to be negligibly small. In the case of instruments of high resolving power, the magnitude $\Delta\nu$ is comparable with $\nu'$, which is the reason for their comparatively great sensitivity. However, these instruments possess very low luminosity, which makes their use for solving the problem of interest to us almost impossible.
Another possibility for increasing the sensitivity of the absorption method consists in applying line absorption, the idea of which is that, instead of a light source with a continuous spectrum, a source of line radiation is used, with lines coinciding with the absorption lines. As such a source it is natural to use an electric discharge in a gas giving a spectrum coinciding with the absorption spectrum being studied; in some cases it is also possible to use the multiline arc spectrum of metals—provided coincident lines are present. Under this condition, owing to the absence of an unabsorbed background near the absorption lines, what is measured is no longer the average but the true weakening of inten-
sensitivity than practically the same sensitivity that is achieved also with instruments of high resolving power. At the same time, the sufficiently large luminosity of medium-dispersion spectrographs provides the practical possibility of applying the line method[^1] of absorption.
It was precisely by means of this method that, at the Institute of Chemical Physics of the Academy of Sciences of the USSR, it proved possible not only to detect certain radicals in various flames, but also to measure their distribution in the flame. To illustrate the possibilities of the spectroscopic absorption method, we shall indicate some results obtained by this method and concerning the question of the distribution of active substances in the flame itself or in the space surrounding it.
Fig. 1.
In Fig. 1 are given the distribution curves of the CS radical along the flame zone of carbon disulfide, obtained for different oxygen contents in the combustible mixture (the order of the curves I, II, III, and IV corresponds to an increase in the oxygen content)[^2]. From these curves it follows that the concentration of CS falls toward the end of the zone (on the right), the more sharply the poorer the mixture, which is naturally connected with the reaction of CS with oxygen. From the presence of parallelism between the concentration of COS in the reaction products and the concentration of CS in the flame, it may be assumed that the disappearance of CS is associated with its conversion into COS.
As is seen from this example, the study of the distribution of an active substance in the combustion zone leads to definite conclusions regarding the chemical mechanism of the reaction. In Fig. 1, attention is also drawn to the significant concentration of the CS radical at the very end of the combustion zone (equal to \(\sim 10\) cm), observed in the case of richer mixtures. The result of this is the removal of CS from the flame zone, which was also established experimentally[^3].
Fig. 2.
Along with the study of the distribution of active substances along the flame zone, the study of their radial distribution (from the flame axis) is also of great interest.
An attempt at such an investigation was made by Avramenko, who studied the radial distribution of hydroxyl near a hydrogen flame.
In this case, a narrow beam of light from a line source passed at various distances \(x\) from the flame axis (see Fig. 2), burning in a quartz vessel with flat parallel windows. Denoting
the concentration of hydroxyl at a distance \(r\) from the flame axis by \(n(r)\), and the length element along the ray by \(dl\), then, for the quantity \(\lg \left(\dfrac{I_0}{I}\right)_x\) measured in Avramenko’s experiments (\(I_0\) and \(I\) are respectively the intensities of the incident and transmitted rays), we shall have
\[ N(x)=\lg \left(\frac{I_0}{I}\right)_x =2B\int_0^\infty \varphi(T)n(r)\,dl \tag{5} \]
or, since
\[ dl=\frac{r\,dr}{\sqrt{r^2-x^2}}, \]
\[ N(x)=2B\int_x^\infty \frac{\varphi n r\,dr}{\sqrt{r^2-x^2}} . \tag{6} \]
Here \(B\) is a constant proportional to the probability of absorption, and
\[ \varphi(T)= \frac{e^{-\frac{hF(J'')}{kT}}}{1+e^{-\frac{h\omega}{kT}}}\, \frac{1}{\nu' Z}, \tag{7} \]
where \(F(J'')\) is the rotational term and \(h\omega\) is the vibrational quantum of hydroxyl, \(\nu'\) is the width of the absorption line, and \(Z\) is the rotational sum of states. The problem consists in finding, on the basis of the measured values of the quantity \(N(x)\), the radial distribution of hydroxyl, i.e. the function \(n(r)\). This problem can be solved in the following way.
Taking the integral
\[ \frac{1}{B}\int_z^\infty \frac{N(x)x\,dx}{\sqrt{x^2-z^2}} \]
and substituting \(N(x)\) into it, we obtain
\[ 2\int_z^\infty \frac{x\,dx}{\sqrt{x^2-z^2}} \int_x^\infty \frac{\varphi n r\,dr}{\sqrt{r^2-x^2}} . \]
A transformation based on Dirichlet’s formula then gives:
\[ \frac{1}{B}\int_z^\infty \frac{N(x)x\,dx}{\sqrt{x^2-z^2}} = 2\int_z^\infty \varphi n r\,dr \int_z^r \frac{x\,dx}{\sqrt{(r^2-x^2)(x^2-z^2)}} = \pi\int_z^\infty \varphi n r\,dr, \]
whence it follows that:
\[ \varphi n=-\frac{1}{\pi rB}\frac{d}{dz}\int_z^\infty \frac{N(x)\,x\,dx}{\sqrt{x^2-z^2}} =-\frac{1}{\pi rB}\frac{d}{dr}\int_r^\infty \frac{N(x)\,x\,dx}{\sqrt{x^2-r^2}} = \]
\[ =-\frac{1}{\pi B}\int_r^\infty \frac{N'(x)\,dx}{\sqrt{x^2-r^2}}, \tag{8} \]
where \(N'(x)=\dfrac{dN(x)}{dx}\).
The values of the quantity \(N(x)\) measured by Avramenko for two pressures of the explosive mixture are shown in Fig. 3 (the hatched part of the drawing denotes the visible zone of the flame). The solid curve for a pressure of \(50\ \mathrm{mm}\ \mathrm{Hg}\) is constructed according to the formula \(N(x)=0.71(1-0.0075x^2)\), and the straight line for a pressure of \(40\ \mathrm{mm}\ \mathrm{Hg}^{*}\) according to the formula \(N(x)=0.7\left(1-\dfrac{x}{9}\right)\).
Fig. 3.
From these formulas it follows:
for a pressure of \(50\ \mathrm{mm}\ \mathrm{Hg}\)
\[ N'(x)\sim -x, \]
for a pressure of \(40\ \mathrm{mm}\ \mathrm{Hg}\)
\[ N'(x)=\mathrm{const.}<0. \]
Substituting these values of \(N'(x)\) into the formula for \(\varphi n\), as a result of integration [replacing the upper limit of integration by the finite value \(x_0\), satisfying the condition \(N(x_0)=0\)], we find:
\[ (\varphi n)_{50\ \mathrm{mm}}\sim \sqrt{x_0^2-r^2} \]
and
\[ (\varphi n)_{40\ \mathrm{mm}}\sim \ln \frac{x_0+\sqrt{x_0^2-r^2}} {x_0-\sqrt{x_0^2-r^2}}. \tag{9} \]
To find the function \(n(r)\) it is necessary to know \(\varphi(r)\), i.e. the radial temperature distribution \(T(r)\), which can be obtained from the equation
\[ \frac{d}{dr}\left(xr\frac{dT}{dr}\right)=0, \tag{10} \]
where \(x\) is the coefficient of thermal conductivity. Taking the latter to be proportional to \(T^2\), we find
\[ T^3=T_0^3-\left(T_0^3-T_R^3\right)\frac{\ln r/r_0}{\ln R/r_0} \tag{11} \]
\[ \text{*) Pressure in the zone of the rarefied flame.} \]
(\(T_0\) is the flame temperature and \(T_R\) is the wall temperature) or, in view of \(T_0^3 \gg T_R^3\):
\[ T = T_0 \sqrt[3]{\frac{\lg \dfrac{R}{r}}{\lg \dfrac{R}{r_0}}}, \tag{12} \]
where \(r_0\) is the radius of the visible zone (\(r_0 = 2\) mm) and \(R\) is the radius of the reaction vessel (\(R = 13\) mm). The temperature distribution obtained from the last expression is shown in Fig. 4 (upper curve, \(T_0 = 2000^\circ\) abs., \(T_R = 400^\circ\) abs.).
Knowing \(T(r)\), on the basis of formula (7) we find \(\varphi(r)\). The values of the quantity \(\dfrac{1}{\varphi(r)}\) calculated in this way (in arbitrary units) are also shown in Fig. 4 (lower curve).
The functions \(n(r)\), calculated from \(\varphi(r)\) by formulas (9) for pressures of 40 and 50 mm Hg, are graphically represented in Fig. 5 (in arbitrary units). The same figure gives the curve \(n(r)\), calculated from the diffusion equation
\[ \frac{d}{dr}\left(Dr\frac{dn}{dr}\right)=0 \tag{13} \]
(\(D\) is the diffusion coefficient), which, together with the heat-conduction equation, gives
\[ n(r)\sim T(r)-T_R. \tag{14} \]
As is evident from Fig. 5, the concentration of hydroxyl near the visible flame zone decreases more rapidly than follows from the diffusion equation.
Fig. 4.
Visible graph labels: \(T/T_0\), \(\dfrac{1}{\varphi(r)}\), \(x\) mm.
Fig. 5.
Visible graph labels: \(n(r)\), \(r\) mm, 40 mm, 50 mm.
The reason for this, apparently, is a chemical reaction that does not completely cease beyond the limits of the visible zone. The correctness of this conclusion is evident from the fact that photographing a flame with quartz optics gives flame dimensions exceeding the dimensions of the visible zone (Avramenko). This testifies to the presence of excited hydroxyl beyond the latter and, consequently, to a reaction proceeding outside the visible zone of the flame.
The examples given indicate new experimental possibilities for investigating chemical processes in flames on the basis of the use of the absorption spectroscopic method.
References
- V. Kondrat'ev, Free Hydroxyl, GONTI, Moscow, 1939.
- V. Kondrat'ev, Journal of Physical Chemistry 14, 287 (1940).
- V. Kondrat'ev, Journal of Physical Chemistry 13, 1260 (1939).