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The influence of the flow of the reacting gas on the conditions for a thermal?explosion

Contributed by A. J. Chorin
Abstract
The classical problem of thermal explosion is modified so that the chemically active gas is not at rest but is flowing in a long cylindrical pipe. Up to a certain section the heatconducting walls of the pipe are held at low temperature so that the reaction rate is small and there is no heat release; at that section the ambient temperature is increased and an exothermic reaction begins. The question is whether a slow reaction regime will be established or a thermal explosion will occur. The mathematical formulation of the problem is presented. It is shown that when the pipe radius is larger than a critical value, the solution of the new problem exists only up to a certain distance along the axis. The critical radius is determined by conditions in a problem with a uniform axial temperature. The loss of existence is interpreted as a thermal explosion; the critical distance is the safe reactor’s length. Both laminar and developed turbulent flow regimes are considered. In a computational experiment the loss of the existence appears as a divergence of a numerical procedure; numerical calculations reveal asymptotic scaling laws with simple powers for the critical distance.
1. Introduction
The classical problem of a thermal explosion in a long cylindrical vessel containing a reacting gas at rest was first formulated and solved by D. A. FrankKamenetsky in 1939 (see ref. 1). The basic result is that there is no spontaneous thermal explosion if the radius of the vessel r_{0} is less than a critical value r_{?} determined by the gas properties and the properties of the exothermal chemical reaction. If r_{0} > r_{?}, a quiet evolution of the chemical reaction is impossible and a thermal explosion occurs.
In the present paper the following question is asked: what will happen if the long cylindrical vessel is in fact a pipe where the chemically active gas is flowing? At a certain section of the pipe (see Fig. 1) we suddenly increase the temperature of the walls and the rate of the exothermic reaction increases. How will the critical conditions for thermal explosion be affected by the gas flow? In the present paper we provide a mathematical formulation of the problem. Our first result is rather obvious: if the radius of the pipe is less than critical at the increased wall temperature, nothing will happen; the reactor will be safe for arbitrary length. The second result seems to be deeper: if the radius of the pipe is larger than critical, there exists a distance along the pipe up to which the solution of the mathematical problem exists, and after which it ceases to exist. This critical length depends on the flow conditions and determines the “safe” length of the reactor, up to which the thermal explosion does not happen.
We want to mention here the remarkable paper (2) in which the influence of a flow on the critical conditions for thermal explosion was first investigated. The flow under consideration in ref. 2 was different—free convection—and therefore the mathematical problem was also different from ours.
2. Problem Formulation: Laminar Flow
We make the same basic assumptions concerning the exothermic chemical reaction as in the classical theory of thermal explosion (see ref. 1): Arrhenius dependence of the reaction rate on the temperature, large activation energy, and neglect of the longitudinal heat transfer in comparison with the contribution of the advection. Then the equation of the energy balance takes the following form: 1 Here T is the absolute temperature of the gas, c is the heat capacity, z is the longitudinal coordinate reckoned from the section where the temperature was elevated, (r) is the longitudinal fluid velocity, r is the current radius, k is the heat conductivity corresponding to the flow regime, Q is the thermal effect of the reaction, σ(T) is the preexponential factor in the Arrhenius reaction rate (a slow function of the temperature), E is the activation energy, and R is the universal gas constant. Applying the Frank–Kamenetsky high activation energy approximation (see ref. 1), and going to dimensionless variables, we reduce Eq. 1 to the form 2 Here ρ = r/r_{0} is the dimensionless radius, κ is the temperature diffusivity appropriate for the flow regime under consideration, κ_{0} is the molecular temperature diffusivity, V(ρ) = (r)/v?, v? = G/πr_{0}^{2}, is the average velocity, G is the flow discharge rate, ζ = z/(G/πν_{0})Pr is the dimensionless longitudinal coordinate, ν_{0} is the molecular kinematic viscosity, and Pr = ν_{0}/κ_{0} is the Prandtl number, a characteristic gas constant. Furthermore, λ = r_{0}/?, where ? = (e^{φ}κ_{0}T_{0}c/Qφσ(T_{0}))?, φ = E/RT_{0}, u = E(T ? T_{0})/RT_{0}^{2}, T_{0} being the wall temperature at z > 0. For a gas temperature uniformly distributed along the z axis in a laminar flow ?_{z}T = 0, κ = κ_{0} and we return to the classical problem formulation (see ref. 1): 3 In the case of laminar motion it is easy to show that V(ρ) = 2(1 ? ρ^{2}); therefore, Eq. 2 assumes the form 4 (we redefined ζ : ζ = z/(2G/πν_{0})Pr). The boundary conditions for the solution u(ρ, ζ) take the form 5 The first condition reflects the assumption that the temperature of the flowing gas in the whole inlet section z = 0 is equal to the temperature of the wall for z > 0. This condition can be replaced by an arbitrary inlet condition u(ρ, 0) = u_{0}(ρ). The second and third conditions are the same as those in the classical problem formulation. It is clear that ?_{ζ}u is positive; therefore, for λ < λ_{cr} = , the solution of the problem does exist for 0 ≤ ζ < ∞. For λ > λ_{cr} the solution to the problem 4–5 does exist only at 0 ≤ ζ ≤ ζ_{0}, where ζ_{0} is a certain finite positive constant that depends on λ. The critical dimensional length is 6 In section 4 the numerical solution of the problem 4–5 will be presented.
3. Problem Formulation: Developed Turbulent Flow
In this case we use the power law proposed earlier (see ref. 3), according to which the longitudinal velocity distribution takes the form 7 Here y = r_{0} ? r is the distance from the wall, Re = 2v?r_{0}/ν_{0} is the Reynolds number, and _{?} is the “friction” velocity.
Eq. 2 in this case is reduced to the form 8 The first two boundary conditions (5) preserve their form in the turbulent case; however, the third condition—the condition of zero influx at the axis—takes the form 9 To obtain 8,9 an expression for the turbulent temperature diffusivity κ is needed. This quantity is determined by a relation κ = A(Re, Pr)ν_{T}, where ν_{T} is the turbulent kinematic viscosity and A(Re, Pr) is a dimensionless factor that we do not necessarily assume to be equal to unity. By definition ν_{T} = _{?}^{2}ρ/?_{y}, so that 10 The quantity (_{?}d/ν)^{1?α} is a function of α, or, which is the same, of the Reynolds number Re (see ref. 3). Substituting this expression into 2, and redefining ζ and ? by 11 we obtain the basic equation 8. Condition 9 is obtained from 10 and the from the condition of zero flux at the axis ρ = 0.
The critical dimensional length z_{0} is therefore determined by the relation 12 To demonstrate the difference between condition 9 and the corresponding condition for the laminar case, consider the equation 13 which, in the subcritical case, describes the temperature distribution at large distances from the inlet section, where the longitudinal temperature variation is negligible. If condition 9 holds, u(0) is finite, so that integrating 13 we obtain 14 Thus the integral curves of 13 that satisfy condition 9 form a oneparameter family having u(0) as parameter. This family possesses an envelope, and the value of the parameter λ that corresponds to the intersection of the envelope with the ρ axis at ρ = 1 is the critical value.
In the following section the value of λ_{cr} will be determined numerically, in the case of turbulent flow, as a function of the Reynolds number, and for values λ > λ_{cr} the critical dimensionless length ζ_{0} will also be determined.
4. Numerical Solutions
It is convenient to use the coordinates ρ,ξ = ζ/ζ_{0} in the numerical solution of the problems 4–5 and 8,5,9 so that the domain of integration is fixed: 0 ≤ ρ ≤ 1, 0 ≤ ξ ≤ 1; in these coordinates ζ_{0} appears as a parameter in the differential equations. The resulting equations are discretized and yield a system of algebraic equations for u at grid points. This system is solved by Newton’s method. If Newton’s method converges for a particular pair λ,ζ_{0} we decide that the pair is inside the existence region in the λ,ζ_{0}plane. If the method does not converge, the pair is outside.
Newton’s method is iterative and an initial condition for iteration must be chosen. If this condition is close to an actual solution then the method is rapidly convergent. If the initial condition is not close to the actual solution, then the method can fail to converge even when the solution does exist. To pick the initial conditions, we first assume that for any λ, a solution can be found if the value of parameter ζ_{0} is sufficiently small. We pick λ, compute a solution for some ζ_{0} sufficiently small so that the solver converges, and solve again starting from the initial conditions given by the preceding solution. When ζ_{0} becomes so large that the solver fails to converge, we say that we have crossed the existence boundary. In the laminar case, whenever 0 < λ < , our method identifies convergent solutions for all values of ζ_{0} tested. When λ > , our method identifies convergent solutions for all values of ζ_{0} less than a critical value ζ_{0}(λ).
Fig. 2 shows the graph of ζ_{0}(λ) for the laminar case. The salient feature is the divergence of ζ_{0}(λ) at the point λ = : ζ_{0}() = ∞. Similar calculations were also performed for the turbulent case. Fig. 3 shows the graph of λ_{cr}^{2} as a function of the Reynolds number. Fig. 4 shows the graph of the critical values ζ_{0}(λ).
Conclusion
The formulation of a new mathematical problem for chemically active gas flow in a pipe is presented. It is shown that for a supercritical regime, where the radius of the pipe is larger than a critical value, the solution of this problem exists only for an interval of bounded length along the pipe axis. The loss of the existence of the solution is naturally interpreted as a thermal explosion. The dimensional critical length ζ_{0} is determined both for laminar and for developed turbulent flows.
In the case of laminar flow the results of the numerical computation of ζ_{0}(λ) (λ is the parameter proportional to the radius of the pipe) are presented in Fig. 2. The asymptotic branch of the curve ζ_{0}(λ) at large λ corresponds to the scaling law ζ_{0}(λ) = 1.77λ^{?11/4}. The results of the numerical computation of λ_{cr} as a function of the Reynolds number for developed turbulent flow are presented in Fig. 3, and the function ζ_{0}(λ) for developed turbulent flow at various values of the Reynolds number is presented in Fig. 4. An interesting property of this graph is the collapse of the curves corresponding to different values of the Reynolds number in a very large (3 orders of magnitude) range of Re starting from rather moderate values of λ. The asymptotic branch of the curve ζ_{0}(λ) at large λ corresponds to the scaling law ζ_{0}(λ) = 0.78λ^{?9/4}. The origin of the observed scaling laws remains unknown to us.
Acknowledgments
This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract DEAC0376SF00098, and in part by the National Science Foundation under Grant DMS9414631.
 Accepted September 30, 1997.
 Copyright ? 1997, The National Academy of Sciences of the USA
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