PHYSICAL CHEMISTRY

V. N. VILYUNOV

ON THE MATHEMATICAL THEORY OF THE STEADY BURNING RATE OF A CONDENSED SUBSTANCE

(Presented by Academician V. N. Kondrat’ev, 2 VII 1960)

The mathematical theory of powder combustion was proposed by Ya. B. Zel’dovich \((^1)\). This theory is applicable to the region of very high pressures, when the heat release in the condensed phase (\(k\)-phase) may be neglected in comparison with the heat coming from the gas phase (\(g\)-phase), and when the combustion process is controlled by reactions of the \(g\)-phase. In the present note a generalization of the theory is given for the case of combustion of a condensed substance at low pressures.

A three-stage model of the combustion process is adopted \((^{2,3})\) (\(\alpha\)-, \(\beta\)-, \(\gamma\)-stages). A coordinate system is considered in which the flame is stationary. The steady process of diffusion-thermal propagation of the flame of the \(i\)-th stage is described by the system of equations

\[ \lambda_i T'' - c_i m T' + Q_i f_i(n,T) = 0, \tag{1} \]

\[ \rho_i D_i n'' - m n' - f_i(n,T) = 0, \tag{2} \]

where \(T(y)\) is the temperature; \(n(y)\) is the ratio of the dimensionless concentrations of the reacting substance; \(\lambda_i, D_i\) are the coefficients of thermal conductivity and diffusion; \(c_i\) is the heat capacity; \(Q_i\) is the thermal effect of the reaction; \(f_i(n,T)\) is the total rate of the chemical reaction; \(m\) is the mass burning rate of the powder; \(\rho_i\) is the density.

\(\alpha\)-Stage (\(k\)-phase). It is assumed that \(D_\alpha = 0\) and that the rate of the total reaction does not depend on pressure. Equations (1)—(2) admit a first integral

\[ \lambda_\alpha T'_s = c_\alpha m (T_s - T_0) - mQ_\alpha \quad \text{or} \quad \lambda_\alpha T'_s = c_\alpha m (T_s - T_{s1}), \tag{3} \]

where \(T_s\) is the surface temperature of the \(k\)-phase, depending parametrically on the pressure \(p\) and the initial temperature \(T_0\); \(Q_\alpha = c_\alpha (T_{s1} - T_0)\), where \(T_{s1}\) is the surface temperature in flameless combustion \((^2)\), when the heat inflow from the \(g\)-phase is negligibly small.

The mass burning rate is found by integrating the heat-conduction equation (1), \(i=\alpha\), under the boundary conditions

\[ y=-\infty,\quad T=T_0 \quad (T'(-\infty)=0); \]

\[ y=0,\quad T=T_s,\quad \lambda_\alpha T'_s = c_\alpha m (T_s - T_{s1}). \tag{4} \]

In particular, using the approximate method of Ya. B. Zel’dovich and D. A. Frank-Kamenetskii \((^4)\), we find the dependence of the mass burning rate on the physicochemical characteristics of the \(k\)-phase:

\[ m^2 = 2\delta k_\alpha \varkappa_\alpha \frac{RT_s^2}{E_\alpha(2T_s - T_{s1} - T_0)} e^{-E_\alpha/RT_s}, \tag{5} \]

where \(\delta\) is the powder density, \(k_\alpha\) is the pre-exponential factor, \(\chi_\alpha\) is the thermal diffusivity of the \(\kappa\)-phase.

Equation (5) coincides with the formula of A. G. Merzhanov and F. I. Dubovitskii \((^5)\). The width of the preheating zone of the \(\kappa\)-phase is approximately determined by the expression

\[ y_\alpha=\frac{\lambda_\alpha}{c_\alpha m}\ln\frac{0.05T_0}{T_s-T_0}, \tag{6} \]

Equation (5) still does not make it possible to determine the dependence of the linear burning rate on pressure, since it contains the surface temperature \(T_s\)—as yet an unknown function of pressure; in order to determine \(m=m(p)\) and \(T_s=T_s(p)\), we shall relate the mass burning rate to the physicochemical characteristics of the \(\beta\)-stage.

\(\beta\)-Stage. It is assumed that in the \(\beta\)-stage a net exothermic reaction takes place, leading to products of incomplete combustion (NO, CO). At low pressures this stage is the final stage of the combustion process. The maximum temperature reached in this stage is \(T_{11}\). At high pressures the processes occurring in the \(\beta\)-stage begin to be affected by the flame stage (\(\gamma\)-stage); in this case, at the boundary between the \(\beta\)- and \(\gamma\)-stages a temperature \(T_1>T_{11}\) is reached. It is assumed that \(T_1\), like \(T_s\), depends parametrically on \(p\) and \(T_0\). Using the usually adopted assumption \(D_\beta=\lambda_\beta/c_\beta\rho_\beta\), we find the first integral of the system (1), (2), \(i=\beta\):

\[ n=\frac{a}{a_{s1}}=\frac{T_1-T}{T_1-T_s} \tag{7} \]

(the integral of Ya. B. Zeldovich).

The expression for the mass velocity is found by integrating equations (1), \(i=\beta\), under the boundary conditions

\[ \begin{gathered} y=0,\quad T=T_s,\quad \lambda_\alpha T'_s=c_\alpha m\,(T_s-T_{s1});\\ y=y_1,\quad T=T_1,\lambda_\beta T'_1=c_\beta m\,(T_1-T_{11}). \end{gathered} \tag{8} \]

Following the method \((^4)\), we obtain:

\[ m^2= \frac{ 2\lambda_\beta Z_\beta(\nu_\beta)!\left(\dfrac{RT_1^2}{E_\beta}\right)^{\nu_\beta+1} }{ c_\beta(2T_1-T_{11}-T_{s1})(T_1-T_s)^{\nu_\beta} } e^{-E_\beta/RT_1}, \qquad Z_\beta=k_\beta\mu_\beta a_{s1}^{\nu_\beta-1} \left(\frac{p}{RT_1}\right)^{\nu_\beta}. \tag{9} \]

Formula (9) gives the relation between the mass burning rate and the physicochemical characteristics of the \(\beta\)-stage. In the integration, the usual expression for the net rate of the chemical reaction was adopted,

\[ f_\beta(T)=k_\beta\frac{\mu_\beta}{a_{s1}} \left(\frac{a_sp}{RT}\right)^{\nu_\beta} \left(\frac{T_1-T}{T_1-T_s}\right)^{\nu_\beta} e^{-E_\beta/RT}, \tag{10} \]

where \(\mu_\beta\) is the mean molecular weight; \(\nu_\beta\) is the overall reaction order; \(E_\beta\) is the effective value of the activation energy; \(a_{s1}\) is the relative concentration of the reacting substance at the interface between the \(\alpha\)- and \(\beta\)-stages in flameless combustion; \(a_s\) is the relative concentration when the processes of the \(\kappa\)-phase are affected by the \(\varepsilon\)-phase.

Formula (9) was obtained under the assumption \(a_s\equiv a_{s1}\).

The width of the preheating zone of the \(\beta\)-stage is determined by the formula

\[ y_1\simeq\frac{\lambda_\beta}{c_\beta m}\ln\frac{T_1-T_{s1}}{T_s-T_{s1}}. \tag{11} \]

\(\gamma\)-Stage (flame stage). In this stage an exothermic reaction of interaction takes place between the products of the reaction of the \(\beta\)-stage, chiefly...

way between NO and CO. Taking \(D_\gamma=\lambda_\gamma/c_\gamma \rho_\gamma\), we find \(n=\dfrac{T_{21}-T}{T_{21}-T_1}\). Integrating equations (1), \(i=\gamma\), under the boundary conditions:

\[ y=y_1,\quad T=T_1,\quad \lambda_\beta T_1'=c_\beta m\,(T_1-T_{11}); \]

\[ y=+\infty,\quad T=T_{21}\,(T'(+\infty)=0), \tag{12} \]

we obtain

\[ m^2=\frac{2\lambda_\gamma Z_\gamma(\nu_\gamma)!\,(RT_{21}/E_\gamma)^{\nu_\gamma+1}} {c_\gamma (T_{21}-T_{11})(T_{21}-T_1)\nu_\gamma} e^{-E_\gamma/RT_{21}},\quad Z_\gamma=k_\gamma \mu_\gamma a_1^{\nu_\gamma-1}\left(\frac{p}{RT_{21}}\right)^{\nu_\gamma}. \tag{13} \]

The width of the preheating zone of the flame stage is found from the formula

\[ y_2-y_1=\frac{\lambda_\gamma}{c_\gamma m}\ln\frac{T_{21}-T_{11}}{T_1-T_{11}}. \tag{14} \]

As the pressure increases, \(T_1\) and \(m\) increase, and consequently the preheating width decreases. In the limiting case, when \(T_1\to T_{21}\), combustion will take place in two stages. This limiting case will probably be observed in the combustion of powder at very high pressures, when the flame is very close to the surface of the \(k\)-phase.

Table 1

Formulas (5), (9), (13) solve the problem in principle, i.e., they make it possible to calculate the dependences \(m=m(p,T_0)\), \(T_s=T_s(p,T_0)\), \(T_1=T_1(p,T_0)\). The calculations are considerably simplified if one considers powder combustion at low pressures, when the influence of the flame stage may be neglected \((T_1\equiv T_{11})\). For this particular case the pressure coefficient \(k_p\) and the temperature coefficient \(k_T\) are expressed in the form:

\[ k_p=\left(\frac{\partial\ln m}{\partial\ln p}\right)_{T_0} \simeq \frac{\nu_\beta}{2} \left[ 1-\frac{\nu_\beta}{T_{11}-T_s}\frac{RT_s^2}{E_\alpha} \right]^{-1}, \tag{15} \]

\[ k_T=\left(\frac{\partial\ln m}{\partial T_0}\right)_p \simeq \frac{k_p}{\nu_\beta} \left[ \frac{E_\beta}{RT_{11}^2} -\frac{\nu_\beta}{T_{11}-T_s} +\frac{\nu_\beta+2}{T_{11}} \right]. \tag{16} \]

Analysis of formulas (15) and (16) shows that with increasing \(T_s\) (or pressure) \(k_p\) increases; the temperature coefficient \(k_T\), for \(A>1\), increases with increasing \(T_s\), while for \(A<1\) it decreases with increasing \(T_s\). The parameter \(A\) is equal to

\[ A=\frac{E_\alpha}{E_\beta}\left(\frac{T_{11}}{T_s}\right)^2 \left(1+\frac{\nu_\beta+2}{E_\beta}RT_{11}\right)^{-1}. \tag{17} \]

Using the formulas obtained, a sample calculation was carried out for nitroglycerin powder. The composition and experimental data for this powder are given in work (6). The data used for the calculation are: \(T_{s1}=700^\circ\mathrm{K}\),

\[ E_{\alpha}^{*}=32000\ \text{cal/mole},\quad \delta=1.6\ \text{g/cm}^{3},\quad c_{\alpha}=c_{\beta}=0.35\ \text{cal/g·deg},\quad T_{11}=1085^\circ\text{K}, \]
\[ E_{\beta}=21000\ \text{cal/mole},\quad \lambda_{\alpha}=\lambda_{\beta}=4\cdot10^{-4}\ \text{cal/cm·sec·deg},\quad \mu_{\beta}=27\ \text{g/mole}, \]
\[ k_{\beta}=0.93\cdot10^{10}\ \text{sec}^{-1},\quad \nu_{\beta}=1. \]

To simplify the calculations, the exact formula (5) was replaced by the approximate one

\[ V=B_{\alpha}e^{-E_{\alpha}^{*}/2RT_{s}},\quad \text{where } B_{\alpha}=0.875\cdot10^{4}\ \frac{\text{cm}}{\text{sec}}. \]

The results of the calculation are summarized in Table 1.

Siberian Physicotechnical Institute
at Tomsk State University
named after V. V. Kuibyshev

Received
23 VI 1960

REFERENCES

1. Ya. B. Zel’dovich, ZhETF, 12, 498 (1942).
2. P. F. Pokhil, Collected Papers: Physics of Explosion, 2, 1953.
3. K. K. Andreev, Thermal Decomposition and Combustion of Explosives, Moscow, 1957.
4. Ya. B. Zel’dovich, D. A. Frank-Kamenetskii, ZhFKh, 12, 100 (1938).
5. A. G. Merzhanov, F. I. Dubovitskii, DAN, 129, No. 1 (1959).
6. R. D. Geckler, Selected Combustion Problems, London, 1954, p. 289.