On the Calculation of Kinetic Constants of an Isothermal Pulverized-Coal Flame at Various Stages of Burnout
Abstract Generated abstract
This paper develops a calculation method for determining kinetic constants of an isothermal pulverized-coal flame at successive stages of burnout. Under assumptions of one-dimensional laminar flow, constant temperature, no particle slip, spherical particles, and reaction over the external geometric surface, the flame is treated as a homogeneous system and a zonal integration of the burnout equation is derived. The resulting relations connect segmental burnout, residence time, reaction surface, and temperature, allowing apparent pre-exponential factors and activation energies to be obtained graphically for individual burnout intervals. The paper also indicates how the formulation changes when the excess-air coefficient differs from unity and discusses estimation of the reaction surface from particle size and degree of burnout.
Full Text
ON THE CALCULATION OF KINETIC CONSTANTS OF AN ISOTHERMAL PULVERIZED-COAL FLAME AT VARIOUS STAGES OF BURNOUT
E. A. KONKOV
(Presented by the scientific seminar of the Department of Boiler Engineering and Boiler Installations)
Let us restrict the burning dust–air flow moving in a cylindrical tube of radius \(R\) by the following conditions:
1) the flow is one-dimensional and laminar \((\operatorname{Re} \ll 2000)\);
2) the motion of the flow is uniform \((T = \mathrm{const})\);
3) the velocity of motion of the particles relative to the flow is zero.
Under these conditions the pulverized-coal flame may be regarded as a homogeneous system, and the method of comprehensive investigation of burnout [1] may be applied to it. In addition, we shall consider the fuel particles to be spherical, and their burnout to occur over the external geometric surface.
The differential equation of burnout of the flow under consideration is
\[ \frac{dG_{\mathrm{t}}}{dx} = M \alpha c s, \tag{1} \]
where \(G_{\mathrm{t}}\) is the consumption of unburned fuel in the section of the combustion chamber
(per unit cross-sectional area of the chamber);
\(M\) is the stoichiometric coefficient;
\(c\) is the oxygen concentration in the section under consideration;
\(\alpha\) is the apparent reaction-rate constant;
\(s\) is the reaction surface per unit volume of the solid phase.
It is obvious that, with an excess-air coefficient equal to unity and constant along the length of the reaction tube, the amount of unburned fuel is proportional to the amount of unused oxidizer, i.e.,
\[ G_{\mathrm{t}} = \omega c M, \tag{2} \]
where \(\omega\) is the flow velocity.
Substituting the value of \(G_{\mathrm{t}}\) from (2) into (1), we obtain
\[ \omega \frac{dc}{dx} = \alpha c s. \tag{3} \]
Integrating equation (3) with the initial condition
\[ x = 0;\quad c = c_0, \tag{4} \]
after exponentiation we obtain
\[ c=c_0 \exp - \int_0^x \frac{\alpha s}{\omega}\,dx, \tag{5} \]
where \(c_0\) is the concentration of the oxidizer at the inlet to the reaction tube. The quantity
\[ z=\int_0^x \frac{\alpha s}{\omega}\,dx \tag{6} \]
is, in the general case, a function of a large number of variables (kinetic, regime, etc.), as a consequence of which integral (6) cannot be evaluated in quadratures.
However, if for each of the narrow limits of burnout one assumes
\[ \alpha=\bar{\alpha}=\mathrm{const}, \]
\[ s=\bar{s}=\mathrm{const}, \tag{7} \]
where \(\bar{\alpha}\) and \(\bar{s}\) are the mean values of \(\alpha\) and \(s\) over the segment under consideration, then zonal integration can be carried out:
\[ z=\sum_{i=1}^{n}\int \frac{\bar{\alpha}_i s_i}{\omega}\,dx =\sum_{i=1}^{n}\frac{\bar{\alpha}_i s_i}{\omega}(x_i-x_{i-1}). \tag{8} \]
Thus,
\[ c_x=\frac{c}{c_0} =\exp -\sum_{i=1}^{n}\frac{\alpha_i s_i}{\omega}(x_i-x_{i-1}). \tag{9} \]
In expressions (7)—(9) and those that follow, \(i\) denotes the number of the section at the end of the investigated burnout segments, of which there are \(n\) in all.
According to the \(\alpha\)-formula [2], for \(\beta=1\),
\[ \Omega=(1-c_x), \tag{10} \]
where \(\Omega\) is the degree of flame burnout,
\(\beta\) is the excess-air coefficient.
From (9) and (10) we obtain for the \(i\)-th burnout segment
\[ \ln(1-\Omega_i)=-\bar{\alpha}_i s_i \tau_i =a_{0i}\exp -\frac{E_i}{RT}\,s_i\tau_i, \tag{11} \]
where
\[ \tau_i=\frac{x_i-x_{i-1}}{\omega} \]
is the residence time of the particles in the \(i\)-th segment;
\[ (1-\Omega_i)=1-\frac{c_{i-1}-c_i}{c_{i-1}} \]
is the degree of underburning of the flame in this segment.
Taking logarithms of equation (11) once again, we obtain the final expression for determining the kinetic constants for the \(i\)-th segment:
\[ \ln\left[\frac{-(\ln 1-\Omega_i)}{s_i\tau_i}\right] =\ln a_{0i}-\frac{E_i}{RT}, \tag{12} \]
which is easily solved by the usual graphical method [3].
By a procedure analogous to that considered, one can obtain calculation relations also for the case \(\beta\ne 1\). It is not difficult, for example, to show that in this case the expression for the oxidizer concentration in an arbitrary
the reactor cross section, in comparison with equation (5), is modified as follows
\[ c=\frac{c_0}{\beta}\exp-\int_0^x \frac{\alpha s\beta}{\omega}\,dx+\frac{c_0}{\beta}(\beta-1). \tag{5′} \]
The main complication of the method in the present case is connected with the fact that the excess-air coefficient is then variable along the length of the reaction tube. This imposes an additional condition on the integrability of expression (6)
\[ \beta_i=\bar{\beta}_i=\text{const} \tag{7′} \]
and is practically connected with difficulties in estimating the quantity \(\Omega^i\) for different sections.
The value of the reaction surface entering equation (12) is easily estimated only when the coefficient of reaction gas exchange is used as the criterion of reactivity, since in this case it is conventionally taken equal to the external geometric surface of the particles. In this case the average value of the reaction-surface quantity in each of the burnout sections can be found by linear interpolation of its initial and final values for the corresponding section, since the dependence of the surface on the degree of burnout has the form of a semicubical parabola
\[ s_z=s_0\sqrt[3]{(1-\Omega)^2}, \tag{13} \]
which is approximated quite accurately by a linear dependence up to very deep degrees of burnout (Fig. 1).
The value \(S_0\) is calculated from the initial particle size.
Fig. 1. Dependence of the external surface of fuel particles on the degree of burnout
\[
\left(s=\frac{s_r}{s_0}=\sqrt[3]{(1-\Omega)^2}\right)
\]
Conclusions
- A method is considered for calculating kinetic constants from the data of a study of a one-dimensional isothermal flame at different stages of burnout.
References
-
B. V. Kantorovich. Fundamentals of the Theory of Combustion and Gasification of Solid Fuel. Publishing House of the Academy of Sciences of the USSR, Moscow, 1958.
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A. B. Reznyakov. Combustion of a Pulverized-Coal Flame as a Polydisperse System. Alma-Ata, 1958.
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G. M. Panchenkov and V. P. Lebedev. Chemical Kinetics and Catalysis. Moscow State University Press, 1961.