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.     

Stationary operating regimes of chemical reactors

We investigate the existence and also the number of possible stationary operating regimes of continuous reactors of finite length with a fixed fine-grained catalyzer layer. It is assumed that the effective chemical reaction rate is expressed by a single-valued function of very general form of the temperature and concentration of the primary component in the stream. It is shown in section 1 that in an adiabatic reactor the solution of the direct and inverse problems of finding the stationary regimes always exists and for the inverse problem the solution is unique. In section 2 we establish some sufficient conditions for the uniqueness of the direct problem for the case in which the effective thermal conduction and diffusion coefficients are equal. In section 3 we examine a very simple diffusion model of a reactor with heat removal. An attempt is made to determine the region of variation of the parameters (characterizing the temperature of the supplied mixture, its input rate, heat removal and reactor length) in which the various stationary regimes exist, in particular the low and high temperature regimes and also both of these regimes together.     

Uniqueness and stability of steady states of continuous chemical reactors

We examine the uniqueness and stability of the solutions to the problem of steady-state operation of a continuous chemical reactor in which longitudinal diffusion and heat conduction are taken into account. We investigate an adiabatic reactor in which the concentration and temperature distributions are similar (the thermal diffusivity and diffusion coeffecient are equal) and an isothermic reactor. These two cases are considered together because the mathematical formulations of the problem are equivalent.The question of the existence and number of steady states was analyzed in [1, 2], where references were made to earlier investigations. The results obtained in [1, 2] are now extended. The stability of the steady states is investigated by the small-perturbation method.     

On analysis of the steady flow in an irrectangular parallel-plate flow chamber

I.IntroductionForthestudyofrelationshipbetweencell'sgrowthandshearstress,thesubjectthatshouldbeworkedoutatfirstishowtosimulateshearstressenvironnlelltwhichsuitcell'sgrowthandofwhichtheshearstresscanbecalculatedconvelliently.NowthePPFCisoneofthemainapparatusesthatsimulatetheshearstressenvironmelltforcell'sgrowthinvitro.Forexample,tilerectangularPPFCisusedforthestudyoftherelationshipbetweentheendotheliumcells'injuryinartery,orplatelet'sadhesionordeformationalldatherosclerosisl3].Apartfromthe…     

On the measurement of normal stress differences in steady shear flow

Summary Compared to the similar pressure-distribution cone-and-plate apparatus of Adams and Lodge (4), the new apparatus' improvements include: temperature control of the cone (as well as the plate); increased stiffening of the frame; four (not three) pressuremeasuring holes in the cone/plate region; inclusion of a pressure-measuring hole on the axis under the cone truncation; exclusive use of a vertical free liquid boundary at the cone rim (without a sea of liquid). Temperature control of the rotating cone and of the fixed plate leads to acceptable temperature uniformity in the test liquid for working temperatures within 10°C or 20°C of ambient; the corresponding interval is about 1°C if the cone temperature control is abandoned. Pressure gradients measured using a Newtonian liquid agree with those calculated using Walters' eq. (3). For a viscoelastic liquid, after subtracting inertial contributions, pressure distributions measured at a given shear rate in the cone/plate region do not change when the gap angle is changed from 2° to 3°, showing that the effects of secondary flow are negligible. Values ofN ₃ =N ₁ + 2N ₂ obtained from the gradients of these distributions are believed to be in error by not more than ±1 Pa, in favorable cases. The present most useful ranges are: 10 to 5000 Pa forN ₃; 0.1 to 200 sec^–1 for shear rate; up to 5 Pa s for viscosity; and 5 to 40°C for temperature. As an application, it is shown that adding 0.1% of a high molecular weight polyisobutylene to a 2% polyisobutylene solution doublesN ₃ and has no detectable effect on the viscosity measured at low shear rates with a Ferranti-Shirley viscometer.

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Zusammenfassung Im Vergleich zu dem ähnlichen Kegel-Platte-Gerät von Adams und Lodge (4) zur Messung der Druckverteilung wurden an dem neuen Gerät die folgenden Verbesserungen vorgenommen: Temperaturregelung an Kegel und Platte, Versteifung des Rahmens, vier (anstatt drei) Druckmeßlöcher im Kegel-Platte-Bereich, ein zusätzliches Druckmeßloch auf der Achse unter der Kegelstumpf-Deckfläche, ausschließliche Verwendung einer vertikalen freien Grenzfläche der Flüssigkeit am Kegelrand (ohne umgebenden Flüssigkeitssee). Die Temperaturregelung des rotierenden Kegels und der festen Platte führt zu einer ausreichenden Temperaturgleichförmigkeit in der Testflüssigkeit für Betriebstemperaturen, die höchstens um 10–20°C von der Umgebungstemperatur abweichen. Dieses Intervall beträgt dagegen nur etwa 1°C, wenn auf die Temperaturregelung am Kegel verzichtet wird. Für newtonsche Flüssigkeiten entsprechen die gemessenen Druckgradienten den mittels der Gleichung von Walters (3) berechneten. Für viskoelastische Flüssigkeiten zeigen sich bei der Änderung des Spaltwinkels von 2° auf 3° nach Abzug der Trägheitsbeiträge keine Änderungen der bei einer bestimmten Schergeschwindigkeit gemessenen Druckverteilung. Dies zeigt, daß Sekundärströmungseffekte vernachlässigbar sind. Es darf angenommen werden, daß die Werte vonN ₃ =N ₁ + 2N ₂, die man aus den Gradienten dieser Verteilungen erhält, unter günstigen Umständen mit einem Fehler von nicht mehr als ±1 Pa behaftet sind. Gegenwärtig liegen die günstigsten Bereiche bei 10 bis 5000 Pa fürN ₃, 0,1 bis 200 s^–1 für die Schergeschwindigkeit, unterhalb von 5 Pa s für die Viskosität und 5 bis 40°C für die Temperatur. Als Anwendung wird gezeigt, daß ein Zusatz von 0,1% hochmolekularen Polyisobutylens zu einer 2%igen Polyisobutylenlösung den Wert vonN ₃ verdoppelt, aber keinen erkennbaren Einfluß auf die (bei geringen Schergeschwindigkeiten mit einem Ferranti-Shirley-Viskosimeter gemessen) Viskosität hat.

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udsf unidirectional shear flow - TCP truncated-cone and plate - N ₁,N ₂ 1st and 2nd normal stress differences in udsf - N ₃ N ₁ + 2N ₂ - : = A is defined by the equationA := B - P ^* hole pressurePw – Pm; Pw, Pm = pressures measured by flush transducer and by hole-mounted transducer - t time - , strain rate, shear rate - (P,t) covariant body metric tensor at particleP and timet - ⁱ , _i covariant and contravariant udsf body base vectors (i = 1, 2, 3) - ^–1 inverse of - R, plate radius, cone/plate gap angle - r ₀,h ₀ radius and height of cone truncation - r,, spherical polar coordinates; cone axis = 0; plate surface = /2 - physical components of stress; for a tensile component - cone angular velocity - p on the plate = /2 - ,T, density, absolute temperature, viscosity - P 0.15 ²(r ² –R ²) (inertial contribution) [2.7] - P _ve contribution [2.8] from flow perturbations of viscoelastic origin - r _i i = 1,2,3,4; values ofr at centers of holes in cone/plate region - P _i () pressure change recorded by transducerTi when cone angular velocity goes from zero to - 1/2 {P _i ()+ P _i (–)} (average for 2 senses of rotation) - rim pressure, from least-squares line through four points - Re Reynolds' number:R ²/ - (P,t)/t With 11 figures and 2 tables     

Selection of steady regimes of a one-parameter family in the problem of plane convective flow through a porous medium

Convective flow through a porous medium in a rectangular vessel with a linear temperature profile steadily maintained on the boundary is investigated. On the basis of numerical experiments the realization of steady regimes that belong to a globally stable, one-parameter family is studied at different vessel dimensions and initial temperature distributions. The regime selection is shown to strongly depend on the initial fluid temperature: a vicinity of two regimes is realized from initial data similar with the state of rest; at high initial fluid heating the regimes are selected from a vicinity of two other regimes; and in intermediate situations any of infinite number of steady regimes can be attained.     

Influence of transport processes on steady regimes of a chemical flow reactor with reaction of the type a → b,a+b→2c

An approximate analytic solution is obtained to the problem of the concentration distribution in a one-dimensional chemical flow reactor for a reaction of the type A B, A + B 2C in the approximation of a weak reaction. The method of small perturbations up to terms of third order is used. A study is made of the influence of the intensity of longitudinal mixing on the degree of transformation, the selectivity, and the yield. It is found that the optimal level of longitudinal mixing in the weak reaction approxmation is determined by a dimensionless parameter that is a combination of the dimensionless rates of the chemical reactions. The dependence of the optimal Péclet number on this parameter is found. Characteristic regions are determined on the plane of the determining parameters in which the interaction of the physical and chemical processes in the reactor leads to different results, which must be taken into account when choosing the type and the parameters of a chemical reactor.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 106–113, November–December, 1982.     

On the steady laminar flow in a curved pipe of varying elliptic cross-section

The fully developed steady flow of a fluid through a curved tube with elliptic cross-section is studied, the cross-sectional area varying slowly with longitudinal distance. Using a perturbation scheme in terms of two small parameters (geometric parameter and curvature parameter), complete analytical solutions are obtained to the first order. The effects of constriction combined with constant curvature of the centre line are discussed. The phenomenon of secondary flow, shear stress and the increased impedance (due to the constriction) are calculated. The possibility of the application of the results to various engineering problems and physiological flows is indicated.     

Secondary regimes in couette flow

We consider the asymptotic solutions of secondary steady flows in a fluid contained between cylinders rotating in the same direction for large Reynolds numbers.The existence of secondary axisymmetric steady flows in a fluid contained between cylinders rotating in the same direction was shown in [1, 2]. In the following we present the asymptotic behavior of such solutions for the case of large Reynolds numbers. The construction follows the scheme suggested in [3].     

On chemical reactions in internal flow systems

Summary The basic equations for a non-isothermal homogeneous reaction in an internal flow system with a fully established velocity field are presented. It is shown that the specie concentrations can be found from a single function, i.e. the degree of advancement of the reaction, while the temperature is determined through a generalized enthalpy function which takes into account the thermal exchange with the exterior. Under mildly non-isothermal conditions, to which the analysis applies primarily, the determination of the two basic functions reduces to the solution of two linear parabolic equations and their associated Sturm-Liouville systems. An illustration of the analysis is given and a number of extensions are indicated. This research was supported in part by the Office of Naval Research under Contract Nonr-222(04).     

On the steady solution of linear advection problems in steady recirculating flows

Numerical modelling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic behaviour, and the fluid constitutive equation, which is an advection equation linked to the fluid history. In this paper we prove that linear steady advection problems in steady recirculating flows have only one solution when the kinematics differs from a rigid motion. We also give a numerical procedure to determine this steady solution. We will describe this numerical procedure for two linear models the first will be the SFRT flow model and the second will be a simplified linear formulation of the Pom–Pom viscoelastic model.     

Magnetohydrodynamic steady flow computations in three dimensions

A complete three-dimensional mathematical model has been developed governing the steady, laminar flow of an incompressible fluid subjected to a magnetic field and including internal heating due to the Joule effect, heat transfer due to conduction, and thermally induced buoyancy forces. The thermally induced buoyancy was accounted for via the Boussinesq approximation. The entire system of eight partial differential equations was solved by integrating intermittently a system of five fluid flow equations and a system of three magnetic field equations and transferring the information through source-like terms. An explicit Runge-Kutta time-stepping algorithm and a finite difference scheme with artificial compressibility were used in the general non-orthogonal curvilinear boundary-conforming co-ordinate system. Comparison of computational results and known analytical solutions in two and three dimensions demonstrates high accuracy and smooth monotone convergence of the iterative algorithm. Results of test cases with thermally induced buoyancy demonstrate the stabilizing effect of the magnetic field on the recirculating flows.     

Determination of normal-stress differences in steady shear flow

Rheologica Acta - A new wide-gap concentric cylinder apparatus has been constructed for the measurement of pressures generated in Couette flow; diaphragm-capacitance pressure transducers in the...