Steady states of a continuous-flow adiabatic chemical reactor

Flow reactors are widely used in the chemical industry for purposes of catalytic reactions [1,2]. Calculation of reactors of this type, even in one-dimemional approximation, is complicated and possible only with the use of numerical methods [1, 3]. Such calculations make it possible to find the steady-state distribution of temperature and concentration in the chemical reactor if one exists; in general, however, there may be other steady-state regimes which may be preferable from the standpoint of obtaining a different degree of conversion of the starting product, operating stability, etc.In this connection special interest attaches to the question of the existence and number of steady-state solutions of the system of equations describing the reactor process.This problem was previously considered in [4–7]. Thus, in [4, 5] it was pointed out that in certain special cases more than one steady-state regime may exist. In [6, 7] the question of sufficient conditions of uniqueness was investigated. In [7] it was shown that the steady-state regime is unique in the ease of short reactors or a dilute mixture of reactants. In [8] the problem of the existence and uniqueness of the steady-state regime was examined for a chain reaction model with direct application of the general theorems of functional analysis.The present paper includes an analysis of a very simple mathematical model of an adiabatic chemical reactor in which an exothermic or endothermie reaction takes place. It is established that in the case of an endothermic process a unique steady-state regime always exists. In the exothermic case the problem of the steady-state regime also always has a solution which, however, may be nonunique; the possibility of the existence of several steady-state regimes, associated with the form of the temperature dependence of the heat release rate, is substantiated.The authors thank G. I. Barenblatt, A. I. Leonov, L. M. Pis'men, and Yu. I. Kharkats for discussing and commenting on the work.     

Thermomechanical instability in steady operating regime of continuous-flow chemical reactor with fixed catalyzer bed

A continuous-flow chemical reactor with fixed bed is a vessel filled with a porous catalyzer through which a liquid or gaseous reagent mixture is filtered. The motion of the mixture in this system is maintained by the pressure differential which compensates for the hydraulic resistance.The complete system of equations describing the mass-and heat-transfer processes in the continuous-flow chemical reactor includes the mass, momentum, and energy conservation equations [1], Usually, in the study of the existence and stability of steady reactor operating regimes, very simple reactor models are analyzed, in which only the mass-conservation equation in the form of the diffusion equation and the energy-conservation equation in the form of the heat-conduction equation are taken into account. The equation of motion drops out of consideration, since the velocity of the reagent mixture with the reaction products through the reactor is considered given and unvarying for disturbances of the steady operating regime.The purpose of the present study is to indicate the possibility of instability in the steady-process regime of a chemical reactor, which is associated with the fact that any temperature (and also concentration) variation within the reactor with disturbances of the steady regime will, generally speaking, lead to variation of the mixture viscosity and, consequently, to variation of the hydraulic resistance and the filtration velocity. This interconnection is an additional factor which may have a significant effect on the behavior of disturbances of the steady regime.In contrast with thermal, kinetic, and thermokinetic instability [2], it is natural to refer to the instability being investigated as thermomechanical instability. Let us consider an example.     

Numerical study of steady magneto-convection around an adiabatic body inside a square enclosure in low Prandtl numbers

In this paper, the effects of Prandtl number on the steady magneto-convection around a centrally located adiabatic body inside a square enclosure are numerically investigated. Two-dimensional nonlinear governing equations are discretized using the control volume method and hybrid scheme. The equations are solved using SIMPLER algorithm. The results are displayed in the form of streamlines and isotherms when the Rayleigh number varies between 10³ and 10⁶, the Hartmann number changes between 0 and 100 and the Prandtl number ranges between 0.005 and 0.1. The ratio of the buoyancy force to the Lorentz force (Ra/Ha ²) is introduced as an index to compare the contribution of natural convection and magnetic field strength on heat transfer. The results obtained from numerical modeling show that the Prandtl number has not considerable effect on heat transfer at low Rayleigh numbers. The effect of magnetic field strength on convection is increased by increasing Prandtl number. The effect of Prandtl number on the average Nusselt number in the presence of a magnetic field is less than the case without a magnetic field.     

The stability of steady regimes in an isothermal chemical flow reactor

One of the fundamental problems in the theory of chemical reactors is the determination of the number of steady regimes and their stability. The problem of the number of steady regimes has been considered in many studies, for example, in [1–4]. The stability of a steady regime is usually established from an analysis of the behavior of small perturbations. The corresponding linear boundary-value problem for perturbations has been studied mainly in the limiting cases of ideal mixing and ideal displacement. When account was taken of longitudinal mixing, the only criteria obtained were ones which imposed fairly severe restrictions on the parameters [5]. In the present study numerical analysis is used in order to investigate the stability of steady concentration distributions in an isothermal chemical flow reactor with longitudinal mixing in the case of a single chemical reaction. The eigenvalues were obtained for the Sturm-Liouville problem, which fully characterize the stability for several laws of variation of the chemical reaction rate as a function of the concentration. A knowledge of the eigenvalues is essential, for example, in order to construct the stabilization system proposed in [6] for the unsteady regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 179–182, March–April, 1985.     

Stabilization of the unstable steady states of a conducting fluid in an electromagnetic field

The linear disturbance spectrum is determined for the solution with the fluid at rest and its unstable region identified. It is shown that an electric field has a destabilizing effect, while a magnetic field and viscosity affect only the rate of damping of the disturbances and not the actual existence of instability. An example of a feedback regulator that stabilizes two unstable modes is constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 138–142, November–December, 1993.     

Stability of steady nonisothermal flow in a vertical layer with permeable boundaries in the microconvection model

The problem of the stability of steady convective viscous incompressible fluid flow in a vertical layer with boundaries at different temperatures is considered in the presence of transverse blowing through the layer. The complete spectral problem is solved for a silicon melt. The neutral curve is constructed and the critical Grashof number is found. The numerical calculations show that the presence of transverse blowing significantly affects the flow stability. As compared with the Oberbeck-Boussinesq model, in the microconvection model the instability develops at lower wavenumbers.     

The existence and uniqueness of steady states for a class of chemical reaction networks

Communicated by the Editor     

Multiplicity of steady solutions in a two-sided lid-driven cavity with different aspect ratios

This study presents a continuation method to calculate flow bifurcation in a two-sided lid-driven cavity with different aspect ratios for anti-parallel motion. In anti-parallel motion, the top and bottom walls of the cavity move in opposite directions simultaneously, while the two walls both moving to the right give parallel motion at the same speed. Comprehensive bifurcation diagrams of the cavity flows with different aspect ratios of the cavities are derived via Keller’s continuation method, and linear- stability analysis is used to identify the nature of the various flow solutions. The Reynolds number (1 ≤ Re ≤ 1,200) is used as the continuation parameter to trace the solution curves. In anti-parallel motion, the evolution of the bifurcation diagrams in cases with different aspect ratios (1 ≤ AR ≤ 2.5) is illustrated. Two stable symmetric flows and one stable asymmetric flow are identified, and the existent regions of the stable flows in the aspect ratios and Reynolds numbers are distinguished. The newly found asymmetric flow state can be obtained at a high aspect ratio and a low Reynolds number.     

Oscillatory and steady convection in a dielectric viscoelastic layer subjected to a temperature gradient in the presence of an electric field

The linear stability, both stationary and oscillatory, of a horizontal layer of a dielectric Oldroyd fluid under the simultaneous action of a vertical a.c. field and a vertical temperature gradient is analysed. The stability regions in the space of the characteristic parameters are determined by means of a finite difference method. The physical mechanisms relevant to the problem are expounded using a heuristic argument.     

Nonequivalent similarity reductions of steady thermal boundary layer equations for an incompressible laminar flow over a continuous moving hot surface

This paper uses symmetry methods to obtain nonequivalent similarity reductions of the steady two-dimensional thermal boundary layer equations of an incompressible laminar flow. New similarity reductions are uncovered and results obtained through the classical dimensional analysis are recovered.     

Model of steady motion of the interface in a layer of a strongly superheated liquid

Steady propagation of the boundary of a vapor cavity in a layer of a metastable liquid along the heater surface is considered. The temperature and velocity of interface propagation are determined from the equations of conservation of mass, momentum, and energy in the neighborhood of the stagnation point of the vapor cavity and the condition of stability of steady motion of the interface. It is shown that a solution of these equations exists only if the liquid is heated above a threshold value. The calculated velocity of interface motion and the threshold value of temperature are in reasonable agreement with available experimental data for various liquids within wide ranges of saturation pressures and temperatures of the superheated liquid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 47–55, March–April, 2008.     

Magnetically stabilized bed reactor for selective hydrogenation of olefins in reformate with amorphous nickel alloy catalyst

A magnetically stabilized bed （MSB） reactor for selective hydrogenation of olefins in reformate was developed by combining the advantages of MSB and amorphous nickel alloy catalyst. The effects of operating conditions, such as temperature, pressure, liquid space velocity, hydrogen-to-oil ratio, and magnetic field intensity on the reaction were studied. A mathematical model of MSB reactor for hydrogenation of olefins in reformate was established. A reforming flow scheme with a post-hydrogenation MSB reactor was proposed. Finally, MSB hydrogenation was compared with clay treatment and conventional post-hydrogenation.     

Nonlinear behaviour of the instability of steady natural convection in a vertical air layer

In this paper we deal with the flow of natural convection between two vertical planes with horizontal temperature gradient. We show that periodic steady flow occur when Rayleigh number increases. Critical values are obtained numerically and a nonlinear analysis is presented according to Center manifold method.     

Bifurcation of the solution to the problem of steady traveling waves in a layer of viscous liquid on an inclined plane

Traveling waves in a viscous liquid flowing down an inclined plane can be described at small and moderate Reynolds numbers by an ordinary differential equation in the thickness of the layer [1, 2]. As the Reynolds number tends to zero, this equation goes over into an equation of third order with quadratic nonlinearity [3]. Periodic solutions of this last equation bifurcating from the plane-parallel solution have been investigated by Nepomnyashchii and Tsvelodub [3–6]. In the present paper, a study is made of the bifurcation of periodic solutions from periodic solutions, namely, an investigation is made of the values of the wave number for which a periodic solution is not unique; a bifurcation equation is derived, the number of bifurcating solutions is found, and their behavior near a bifurcation point is considered; and the bifurcating solutions are continued numerically with respect to a parameter (the wave number) from the neighborhoods of the bifurcation points.