On the distribution of oxidizer concentration in a burning fuel particle
Abstract Generated abstract
This paper examines oxygen diffusion and consumption inside a burning spherical fuel particle, emphasizing the effect of a time-varying internal reaction coefficient during burnout. It formulates a nonstationary diffusion problem with a moving particle boundary, introduces an approximate reacting-layer representation to simplify the boundary balance, and assesses the error of the linearized approximation as acceptable for kinetic calculations. A dimensionless system is then solved numerically for different values of the internal-contact criterion to obtain oxidizer concentration distributions and the degree of internal surface utilization. The results indicate a progressive shift of the reaction regime toward the internal kinetic region as burnout proceeds, providing guidance on when furnace intensification should emphasize improved oxidizer supply or increased temperature.
Full Text
On the Distribution of Oxidizer Concentration in a Burning Fuel Particle
E. A. Kon’kov
(Presented by the Department of Boiler Engineering and Boiler Installations)
The determination of specific practical measures for optimizing the operating process of a pulverized-coal furnace is closely connected with the need to know reliably the regime of fuel reaction. In the general case, the establishment of this regime can be carried out on the basis of solving the problem of oxidizer diffusion into the fuel particle in comparison with the intensity of oxygen consumption within the volume of the solid fuel phase. The classical solution of such a problem by means of its quasi-stationary representation was given by L. N. Khitrin [1]; however, a number of simplifying assumptions limit the range of practical applicability of this solution. One such simplification is the use of the constant of internal fuel reaction as a quantity essentially constant in time. For a real fuel, the assumption that this quantity has a variable character in the course of the burnout process appears more probable [2]. By using specific regularities of its change with burnout, one may pose the problem with a closer approximation to the actual combustion conditions.
The indicated complication will limit the possibility of obtaining a regular analytical solution of the problem; however, such a formulation may be considered expedient if one is oriented toward the application of numerical methods of solution using electronic-computing equipment. The latter circumstance also makes it possible to take into account the condition of mobility of the geometric boundary of the reacting fuel particle during burnout, i.e., to pose the problem in its greatest real completeness. The initial system of equations in this case may be taken as follows:
\[ D_i\left[\frac{\partial^2 c}{\partial x^2}+\frac{2}{x}\frac{\partial c}{\partial x}\right]-\alpha(\tau)c=\frac{\partial c}{\partial \tau} \tag{1} \]
\[ 4\pi \ell^2 D_i \frac{\partial c}{\partial x} = \int_{0}^{\ell} c(x)\,4\pi \rho^2\,d\rho\,\alpha(\tau), \tag{2} \]
in which equation (1) describes the process of nonstationary diffusion of oxygen into the fuel sphere, while relation (2) determines the oxygen balance at the outer boundary. In equations (1)—(2) the following notation is adopted:
\(\rho\)—the outer radius of the burning fuel particle, \(\rho=\rho(\tau)\);
\(x\)—an arbitrary radius inside the fuel sphere \((x \leqslant \rho)\);
\(\alpha(\tau)\)—the time-variable coefficient of internal reaction;
\(c\)—oxygen concentration;
\(D_i\)—coefficient of internal diffusion;
\(\tau\)—time.
Since the solution of the integro-differential system (1)—(2) is associated with considerable technical difficulties even when using machine computation, owing to the necessity of employing some cumbersome approximate method (for example, the method of successive iterations), expression (2) can be simplified as follows. Let us introduce into consideration the integral analogue of the thickness of the reacting layer [3]
\[ \varepsilon_0=\int_0^\rho \alpha(\tau)c(x)\,dx=\sqrt{\frac{D_i}{\alpha}}. \tag{3} \]
Then the effective reacting volume of the fuel particle can be expressed as
\[ V=\frac{4}{3}\pi\left[\rho^3-(\rho-\varepsilon_0)^3\right], \tag{4} \]
and equation (2) can be replaced by the equivalent algebraic equation
\[ 4\pi\rho^2D_i\frac{\partial c}{\partial x} = \frac{4}{3}\pi\left[\rho^3-(\rho-\varepsilon_0)^3\right]\alpha(t)c \tag{5} \]
where \(c_0\) is the oxygen concentration at the outer boundary of the fuel sphere (we solve the problem under boundary conditions of the first kind and assuming symmetry of the outer boundary with respect to the oxidizer flux).
If, further, expression \(\left[\rho^3-(\rho-\varepsilon_0)^3\right]\) is linearized with respect to \(\varepsilon_0\), i.e., set
\[ V=\pi\rho^2 A\varepsilon_0, \tag{6} \]
then the solution of the problem is substantially facilitated.
In doing so, the following should be borne in mind. As a detailed analysis of function (4) shows, it is characterized by rapid growth as a function of \(\varepsilon_0\) and already at \(\varepsilon_0=0.5\rho\) reaches a value of \(\sim 0.9\) of its maximum value. On the other hand, it is known that the boundaries of the regime regions of reaction are determined with the same accuracy (10%) [4], i.e., it may be assumed that the purely kinetic reaction regime already begins at \(\rho=0.5\). Taking this into account, linearization of expression (4) is required within the limits \(0 \leqslant \varepsilon_0 \leqslant 0.5\rho\). We shall see that in the indicated range of values of \(\varepsilon_0\) function (4) is linearized with acceptable accuracy. Taking (6) and equating the right-hand sides of (4) and (6), after identical transformations we obtain
\[ \frac{1}{3}\left[3\varphi-3\varphi^2+\varphi^3\right]=A\varphi, \tag{7} \]
where we have set \(\varphi=\dfrac{\varepsilon_0}{\rho}\).
Solving the cubic equation (7) graphically, we find that
\[ A=\frac{2}{3}. \]
Let us check the accuracy of the adopted approximation, bearing in mind the limits of its applicability \(0 \leq \varphi \leq 0.5\). Denote:
\[ \frac{1}{3}\left[\varphi^3-3\varphi^2+3\varphi\right]-A\varphi = \frac{1}{3}\left[\varphi^3-3\varphi^2+3\varphi\right]-\frac{2}{3}\varphi =F(\varphi). \tag{8} \]
Equating the first derivative of (8) to zero, we find the values of \(\varphi\) corresponding to the extrema of this function:
\[ \frac{dF}{d\varphi}=3\varphi^2-6\varphi+1=0. \tag{9} \]
Equation (9) has only two real roots, \(\varphi_1=0.186\) and \(\varphi_2=1.814\). We discard the latter root as lying outside the adopted approximation region, i.e. \(\varphi_2=1.814>\varphi_k=0.5\). The maximum relative error of the approximation corresponding to the value \(\varphi_1\) is
\[ \delta_{\varphi_1} = \frac{\left[\varphi_1^3-3\varphi_1^2+3\varphi_1\right]-2\varphi_1} {\varphi_1^3-3\varphi_1^2+3\varphi_1} = \]
\[ = \frac{0.186^3-3\cdot 0.186^2+3\cdot 0.186-2\cdot 0.186} {0.186^3-3\cdot 0.186^2+3\cdot 0.186} \cdot 100\% = 19.3\%. \]
Let us also check the approximation error at the boundary of the domain, i.e. at \(\varphi_k=0.5\):
\[ \delta_{\varphi_k} = \frac{\left[\varphi_k^3-3\varphi_k^2+3\varphi_k\right]-2\varphi_k} {\varphi_k^3-3\varphi_k^2+3\varphi_k} = \]
\[ = \frac{0.5^3-3\cdot 0.5^2+3\cdot 0.5-2\cdot 0.5} {0.5^3-3\cdot 0.5^2+3\cdot 0.5} \cdot 100\% = 14.3\%. \]
Thus, the maximum approximation error corresponds to the value \(\varphi=0.186\) and does not exceed \(20\%\), which, considering the generally rather low accuracy of kinetic calculations [5], may be regarded as entirely acceptable.
Using the empirical law for the variation of the internal-reaction coefficient in the form [2]: \(\alpha(\tau)=\alpha_{\mu}\cdot e^{-\mathrm{Fo}}\), after reducing the original system of equations to dimensionless form, it may be represented as:
\[ \left\{ \begin{aligned} \frac{\partial \Theta}{\partial \mathrm{Fo}} &= \frac{1}{4W_0} \left[ \frac{\partial^2 \Theta}{\partial \xi^2} + \frac{2}{\xi}\frac{\partial \Theta}{\partial \xi} \right] - W_0 e^{-\mathrm{Fo}} \\ \psi &= e^{-\frac{\mathrm{Fo}}{2}} \end{aligned} \right. \tag{10} \]
\[ \tag{11} \]
under the following system of uniqueness conditions
\[ \Theta(\psi,\mathrm{Fo})=1 \tag{12} \]
\[ \frac{\partial \Theta(0,\mathrm{Fo})}{\partial \xi}=0 \tag{13} \]
and under the following ranges of variation of the independent variables
\[ \begin{gathered} 0 \leq \xi \leq \psi,\\ 0 \leq \mathrm{Fo} \leq \infty . \end{gathered} \tag{14} \]
In equations (10)—(14):
\[ \Theta=\frac{c}{c_0} \]
is the dimensionless oxygen concentration;
\[ \mathrm{Fo}=\frac{\tau \cdot D_i}{\rho^2} \]
is the diffusional analogue of the Fourier criterion;
\[ W_0=\frac{\rho_0^2 \alpha_k}{D_i} \]
is the integral analogue of the internal-contact criterion;
\[ \psi=\frac{\rho}{\rho_0} \]
is the dimensionless outer radius of the fuel particle at an arbitrary instant of time;
\[ \xi=\frac{x}{\rho_0} \]
is the dimensionless radius of an arbitrary sphere inside the fuel particle.
By a numerical solution of the system (10)—(14) on the Minsk-1 electronic computer, there was obtained (for each preliminarily adopted value of \(W_0\)) the distribution of the oxidizer over the volume (radius) of the burning fuel particle for the corresponding values of \(\mathrm{Fo}\), in the form of a tabulated function \(c=c(\rho)\). These data were then used to calculate the degree of participation of the internal surface of the fuel at various instants of the combustion process according to the relation
Fig. 1. Use of the internal surface of the fuel as a function of the degree of burnout. 1—for \(W_0=10\); 2—for \(W_0=4.0\); 3—for \(W_0=1.0\); 4—for \(W_0=0.7\).
\[ \gamma-\frac{\displaystyle \int_0^\rho c(x)x^2\,dx}{\displaystyle \frac{4}{3}\cdot c_0 \rho^3}. \]
The values of $\gamma$, found both as a function of combustion time and as a function of the degree of burnout (Fig. 1), make it possible, from the available characteristics of the fuel being burned, to determine the regime of its reaction at all stages of the burnout process and, accordingly, to determine the limits of the practical effectiveness of particular measures for intensifying the furnace process. Thus, from Fig. 1 it is seen that at $w_0 = 1.0$ the combustion regime passes into the internal kinetic region at a degree of burnout $\Omega \approx 0.9$. This means that, upon reaching such a degree of burnout, practical measures for intensifying the process should not be sought on the basis of turbulizing the flame, which increases the supply of oxygen to the reacting surface; rather, measures should be selected that raise the temperature level of the process. At earlier stages of burnout, on the contrary, means that accelerate the supply of oxidizer to the reaction surface will be more effective, since in this case the combustion regime proceeds in the diffusion region. At intermediate stages of burnout, the effectiveness of the considered means of controlling the combustion process will vary, and the choice of one or another of them, or of a combination of them, must be made depending on the specific conditions.
Conclusions
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An approximate method is considered for the numerical calculation of the degree of utilization of the internal surface of a burning fuel under conditions of an internal transition regime, taking into account the internal-reaction constant that changes during the combustion process.
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The dynamics of the shift of the reaction toward the internal kinetic region as the fuel burns out is shown.
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Considerations are given regarding the choice of practical measures for intensifying the flame process.
REFERENCES
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L. N. Khitrin. Physics of Combustion and Explosion. Moscow State University Press, 1957.
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E. A. Konkov, I. K. Lebedev. On determining the reaction properties of solid fuels in a pulverized stream at different stages of burnout. Present collection.
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B. V. Kantorovich. Fundamentals of the Theory of Combustion and Gasification of Solid Fuel. Publishing House of the Academy of Sciences of the USSR, 1958.
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Z. F. Chukhanov. Some Problems of Fuel and Power Engineering. Publishing House of the Academy of Sciences of the USSR, Moscow, 1961.
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L. N. Khitrin. The equation of combustion of a stream of pulverized fuel (coke) in the presence of oxygen chemisorption. IFZh, No. 7, 1962.