Abstract Generated abstract
This paper examines whether an upper pressure limit for thermal self-ignition can occur in a chain reaction without branching, contrary to the usual expectation that such limits require chain branching. A hypothetical one-center gas reaction in a spherical reactor is analyzed, with radical initiation at the wall, chain propagation in the gas, and third-body termination on an impurity, using diffusion and heat-conduction equations with an approximate temperature dependence of the rate constant. The analysis reduces the ignition condition to a Bessel-function criterion and then to an approximate inequality involving the radical diffusion length. It shows that because the effective reaction-zone thickness decreases strongly with pressure, the self-ignition condition can cease to be satisfied at high pressure, producing an upper ignition limit in an unbranched chain reaction.
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PHYSICAL CHEMISTRY
B. V. Pavlov
ON THE UPPER LIMIT OF SELF-IGNITION IN A CHAIN UNBRANCHED REACTION
(Presented by Academician V. N. Kondrat’ev, November 28, 1964)
The condition of thermal self-ignition for a homogeneous gas reaction is determined by the relation (\(^1,{}^2\))
\[ wq \frac{E}{RT_0^2} \sim \frac{a}{r_0^2}, \]
where \(T_0\) is the temperature of the reactor wall; \(w\) is the reaction rate at \(T=T_0\); \(q\) and \(E\) are the heat and activation energy of the reaction; \(r_0\) is the linear dimension of the reactor, and \(a\) is a constant. It follows from this that, with increasing pressure, the conditions for ignition can only improve. Therefore it is considered that the upper limit of self-ignition can arise only in a chain branched reaction. However, one can simply indicate a concrete scheme of a chain unbranched reaction in which the existence of an upper limit is possible.
- Let us consider the following hypothetical one-center scheme.
\[ \begin{array}{lll} \mathrm{A}+\text{wall}\to \mathrm{R}, & & \text{initiation on the surface of the reactor.}\\ \mathrm{R}+\mathrm{A}\to \text{product}+\mathrm{R}, & & k,\ E,\ \text{chain propagation.}\\ \mathrm{R}+\mathrm{B}+\mathrm{M}\to \mathrm{RB}+\mathrm{M}, & & k_p,\ \text{termination on impurity B with participation of a third}\\ & & \text{particle M.} \end{array} \]
The distribution of radicals over the radius in a spherical reactor is determined by the diffusion equation
\[ D\nabla_r^2[\mathrm{R}]-k_p[\mathrm{R}][\mathrm{B}][\mathrm{M}]=0, \qquad [\mathrm{R}]=[\mathrm{R}]_{r_0}\frac{r_0}{r}\frac{\operatorname{sh} r/\rho}{\operatorname{sh} r_0/\rho}, \tag{1} \]
where \(D\) is the diffusion coefficient of the radicals; \(\rho=1/\sqrt{k_p[\mathrm{B}][\mathrm{M}]/D}\), and \(r_0\) is the reactor radius.
In the case when \(r_0/\rho \gg 1\), \([\mathrm{R}]\) at the wall is well approximated by the exponential
\[ [\mathrm{R}] \simeq [\mathrm{R}]_{r_0}e^{(r-r_0)/\rho}, \tag{2} \]
and the reaction rate near the reactor surface will be
\[ w=k[\mathrm{R}]_{r_0}[\mathrm{A}]e^{(r-r_0)/\rho}. \tag{3} \]
Near the center (3) is certainly incorrect, but for solving the heat-conduction equation this is immaterial, provided only that the approximating function at \(r\sim 0\) is sufficiently small; (3) satisfies this requirement, and it may be used in solving the heat-conduction equation.
- The temperature distribution over the reactor is given by the heat-conduction equation, in which, to simplify the calculation, we approximate the dependence of \(k\) on \(T\) by the first two terms of the expansion of \(k\) in powers of \((T-T_0)\):
\[ \lambda \nabla_r^2 T + qk_{T_0} \left( 1+\frac{E(T-T_0)}{RT_0^2} \right) [\mathrm{R}]_{r_0}[\mathrm{A}]e^{(r-r_0)/\rho} =0; \tag{4} \]
\(\lambda\) is the coefficient of thermal conductivity; \(q\) is the thermal effect of the reaction; \(T_0\) is the temperature of the reactor wall; \(k_{T_0}\) is the value of \(k\) at \(T-T_0\).
The solution of (4) must be made subject to the boundary conditions
\[ \text{a) } T=T_0 \quad \text{at } r=r_0; \qquad \text{b) } T \text{ is bounded at } r=0. \tag{5} \]
By the change of variables \(T=T_0+u/r-RT_0^2/E,\quad x^2=4a\rho^2e^{r/\rho}\), where
\[ a=q\frac{k_{T_0}}{\lambda}\frac{E}{RT_0^2}[A][R]_{r_0}e^{-r_0/\rho}, \tag{4} \]
it is reduced to Bessel’s equation
\[ x^2u''+xu'+x^2u=0. \]
The solution of (4) with conditions (5) is the function
\[ T=T_0+\frac{RT_0^2}{E}\left\{ \frac{r_0}{r} \frac{ J_0(2\rho\sqrt{a}\,e^{r/2\rho})-Y_0(2\rho\sqrt{a}\,e^{r/2\rho})J_0(2\rho\sqrt{a})/Y_0(2\rho\sqrt{a}) }{ J_0(2\rho\sqrt{a}\,e^{r/2\rho})-Y_0(2\rho\sqrt{a}\,e^{r/2\rho})J_0(2\rho\sqrt{a})/Y_0(2\rho\sqrt{a}) } -1\right\}, \]
where \(J_0\) and \(Y_0\) are Bessel functions of the first and second kinds of zero order. For
\[ J_0(2\rho\sqrt{a}\,e^{r_0/2\rho}) - Y_0(2\rho\sqrt{a}\,e^{r_0/2\rho})J_0(2\rho\sqrt{a})/Y_0(2\rho\sqrt{a})=0 \tag{6} \]
the temperature tends to \(\infty\), and the relation between the parameters for which (6) is satisfied gives the self-ignition condition.
Owing to the smallness of \(2\rho\sqrt{a}\), \(Y_0(2\rho\sqrt{a})\) is a large negative quantity, so that the second term in (6) is small. Therefore one may use the approximate equation
\[ J_0(2\rho\sqrt{a}\,e^{r_0/2\rho})=0, \tag{7} \]
whence we obtain the self-ignition condition in the form
\[ \frac{D}{k_p[B][M]}\,q\,\frac{k_{T_0}}{\lambda}\frac{E}{RT_0^2}[A][R]_{r_0}\geq \frac{\mu_1^2}{4} \tag{8} \]
(\(\mu_1\) is the first zero of \(J_0\)).
- The value of the concentration \([R]_{r_0}\) near the wall is determined by the boundary condition
\[ -D\,d[R]/dr\big|_{r=r_0}=\chi_2[R]-\chi_1[A], \tag{9} \]
where \(\chi_1[A]\) is the number of impacts of molecules \(A\) on \(1\ \mathrm{cm}^2\) of the reactor surface leading to the formation of radicals \(R\); \(\chi_2[R]\) is the number of impacts on \(1\ \mathrm{cm}^2\) of the surface in which radicals \(R\) are destroyed.
Using (2), from (9) we obtain \([R]_{r_0}=\chi_1[A]/(\chi_2+\sqrt{k_p[B][M]D})\). Since \([A]\sim p,\ [B]\sim p,\ [M]\sim p,\ D\sim 1/p\), then
\[ [R]_{r_0}\sim p/(a+p^{1/2})\quad (a\text{ is a constant}). \tag{10} \]
Substituting (10) into (8), it is easy to see that the left-hand side decreases with increasing pressure, so that the point of equality in (8) determines the upper limiting value of the pressure at which self-ignition of the reacting mixture still occurs.
- Condition (8) is easily transformed to the form
\[ w_{r_0}q\frac{E}{RT_0^2}\geq \frac{\mu_1^2}{4}\frac{\lambda}{\rho^2}, \]
where \(w_{r_0}\) is the reaction rate near the wall.
In this form the self-ignition condition has the usual form, but instead of the reactor radius \(r_0\) there appears the thickness of the reaction zone \(\rho\), which depends strongly, as \(p^{-3/2}\), on the pressure. It is precisely this latter circumstance that is the physical cause of the appearance of the upper limit.
Institute of Chemical Physics
Academy of Sciences of the USSR
Received
23 XI 1964
REFERENCES
¹ N. N. Semenov, Chain Reactions, Leningrad, 1934.
² D. A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Moscow–Leningrad, 1947.