Unsteady convective diffusion in an annular catalytic reactor

The effect of magnetic field on unsteady convective diffusion in a couple stress fluid (blood) is studied using a time dependent dispersion model. This model is used to calculate the mean concentration distribution of a solute, bounded by the porous layer and is expressed as a function of dimensionless axial distance and time. The magnetic field, arising as a body couple in the governing equations is shown to increase the axis dispersion coefficient. This is useful to the control of haemolysis caused by artificial organs implanted or extracorporeal. Dispersion coefficient and mean concentration are computed for different values of Hartmann number (M), Couple Stress Parameter (a) and Porous Parameter (σ).

The unsteady dispersion of a solute band by a steady pressure-driven flow in a semicircular microchannel is theoretically studied via the generalized dispersion model. Considering an irreversible first-order reaction at the curved wall while assuming a no-flux boundary condition at the flat wall, analytical solutions are obtained for the exchange, convection and dispersion coefficients as well as the dimensionless forms of solute concentration and mean solute concentration. The solutions are obtained assuming an initial solute band of arbitrary cross-sectional shape and axial distribution and the results are presented for both circular and semicircular shapes with the uniform distribution being a special case of the latter. Besides the general solutions, special solutions are also derived for uniform velocity and no-reaction cases. In the following, the influences of the initial concentration distribution and the Damköhler number, a measure of the reaction rate, on the transport coefficients and the concentration distribution are investigated in depth. It is demonstrated that the combination of the initial concentration distribution and the Damköhler number specifies the variations of the transport coefficients in the short term but the Damköhler number is the only parameter dominating the long-term values: the exchange and convection coefficients are increasing functions of the Damköhler number whereas the opposite is true for the dispersion coefficient. Moreover, we show that, provided the solute injection is appropriately positioned and shaped, liquid-phase transportation with little dispersion is possible in typical microchannels utilizing semicircular geometry.

Unsteady convective diffusion in capillary chromatography

The effects of the irreversible boundary reaction and the particle drag on mass transfer are studied analytically in concentric annulus flows. The solution of mathematical model, based on the generalized dispersion model brings out the mass transport following by the insertion of catheter on an artery in terms of the three effective transport coefficients, viz., the exchange, convection and diffusion coefficient. A general expression is derived which shows clearly the time dependent nature of the coefficients in the dispersive model. The complete time dependent expression for the exchange coefficient is obtained explicitly and independent of velocity distribution in the flow; however it does depend on the initial solute distribution. Because of the complexity of the problem only asymptotic large time evaluations are made for the convective and diffusion coefficients, but these are sufficient to give the physical insight into the nature of the problem of the effects of drag and absorption parameters. It is found that as absorption parameter increases exchange and convection coefficients will be enhanced, but diffusion coefficient will be reduced. After certain period of time exchange coefficient will be constant for different values annular gap. As the drag parameter increases convection and diffusion coefficients will be reduced. With the enhancement of catheter radius i.e., the annular gap will be reduced then the convection and diffusion coefficients will be decreased.

The unsteady dispersion of a solute has been discussed by the method of generalized dispersion technique in a blood-like liquid flowing through a pipe under the combined effects of finite yield stress and irreversible absorption into the wall.The solvent is enacted as a three-layered liquid by considering the center liquid as a Casson liquid (a core of red blood cell suspension) and a peripheral layer of plasma as a Newtonian liquid. An asymptotic representation for the convection and dispersion coefficients has been shown only for large values of time, which will not hamper the study of physical behavior of the system. The objective of the present study is to examine the nature of exchange coefficient, convective coefficient and in particular, dispersion coefficient together with mean concentration distribution under the effect of absorption parameter (​β)​, yield stress (​​τ​y​​​) (equivalently the plug radius (​​Rp​​​)) and peripheral layer variation (i.e., ratio of central core radius to normal artery radius (​​Ro​​​)). It is found that the presence of peripheral layer makes some important increment in dispersion coefficient compared to single phase Casson liquid for small absorption. Increase in both diffusivity (D*​​) and Peclet number (Pe) make a significant decrement in the magnitude of dispersion coefficient with respect to absorption rate. The decrease in peak of the mean concentration distribution with the increase in reaction rate is found irrespective of the nature of reaction.

In this paper, the streamline diffusion weak Galerkin finite element method is proposed and analyzed for solving unsteady time convection diffusion problem in two dimension. The v-elliptic property and the stability of this scheme are proved in terms of some conditions. We derive an error estimate in L2(μ) and H1(μ) norm. Numerical experiments have demonstrated the effectiveness of the method in solving convection propagation problems, and the theoretical analysis has been validated.

A systematic method is proposed to derive the dispersion equation in a cylindrical channel, based on the projection operator technique originated by Nakajima and Zwanzig. Under the general condition that the axial flow velocity and the diffusion coefficient are arbitrarily dependent on the radial distance, an analytical expression of the Taylor-type dispersion equation is obtained. In the classical Taylor problem, this expression coincides with the result of Gill and Sankarasubramanian. The sufficient condition, under which the approximate dispersion equation is valid, is explicitly presented. The transient term dependent on the radial concentration heterogeneity in the initial condition can be also accounted for. The present result can be applied to various problems of dispersion phenomena, such as unsteady mass or heat transfer in tubular mass or heat exchangers. Because the projection operator technique is a general method to treat dynamic systems with many degrees of freedom, this method can be applied systematically to various problems in transport phenomena.

Longitudinal dispersion coefficients in natural channels

The paper provides the statement of the problem, the development of a calculation method and an algorithm to solve the problems of propagation and absorption of non-axisymmetric natural waves in layered dissipative-inhomogeneous viscoelastic three-layer cylindrical bodies. A detailed analysis of well-known publications devoted to this problem is given. The paper posed the problem of non-axisymmetric natural wave propagation in three-layer cylindrical shells with fillers, taking into account the theory of hereditary viscoelasticity. Dispersion equations were obtained and the dependence of the phase velocity on the wavenumber for the three-layer shells was constructed. A calculation method was developed based on the Muller, Gauss, and orthogonal sweep methods. The program (in C++ language) was compiled based on the developed algorithm. Numerical results were obtained for the complex phase velocity depending on various wavenumbers and parameters of an axisymmetric cylindrical mechanical system for the Kirchhoff-Love and Timoshenko hypotheses. The change in the real and imaginary parts of the complex phase velocity under various parameters of the system was investigated for structurally homogeneous and inhomogeneous mechanical systems. The results obtained by the Kirchhoff-Love and Timoshenko hypotheses are in good agreement but differ in the wave absorbing abilities. It was found that a decrease in filler thickness at a rigid inner shell leads to a rapid increase in the real and imaginary parts of the vibration frequencies.

Preface to the Second Edition. Symbols. 1 Elementary Reactions in Ideal Reactors. 1.1 Material Balances. 1.2 Elementary Reactions. 1.3 Reaction Order and Mechanism. 1.4 Ideal, Isothermal Reactors. 1.5 Mixing Times and Scaleup. 1.6 Dimensionless Variables and Numbers. 1.7 Batch Versus Flow and Tank Versus Tube. 2 Multiple Reactions in Batch Reactors. 2.1 Multiple and Nonelementary Reactions. 2.2 Component Reaction Rates for Multiple Reactions. 2.3 Multiple Reactions in Batch Reactors. 2.4 Numerical Solutions to Sets of First-Order ODEs. 2.5 Analytically Tractable Examples. 2.6 Variable-Volume Batch Reactors. 2.7 Scaleup of Batch Reactions. 2.8 Stoichiometry and Reaction Coordinates. 3 Isothermal Piston Flow Reactors. 3.1 Piston Flow with Constant Mass Flow. 3.2 Scaleup Relationships for Tubular Reactors. 3.3 Scaleup Strategies for Tubular Reactors. 3.4 Scaling Down. 3.5 Transpired-Wall Reactors. 4 Stirred Tanks and Reactor Combinations. 4.1 Continuous Flow Stirred Tank Reactors. 4.2 Method of False Transients. 4.3 CSTRs with Variable Density. 4.4 Scaling Factors for Liquid Phase Stirred Tanks. 4.5 Combinations of Reactors. 4.6 Imperfect Mixing. 5 Thermal Effects and Energy Balances. 5.1 Temperature Dependence of Reaction Rates. 5.2 Energy Balance. 5.3 Scaleup of Nonisothermal Reactors. 6 Design and Optimization Studies. 6.1 Consecutive Reaction Sequence. 6.2 Competitive Reaction Sequence. 7 Fitting Rate Data and Using Thermodynamics. 7.1 Fitting Data to Models. 7.2 Thermodynamics of Chemical Reactions. 8 Real Tubular Reactors in Laminar Flow. 8.1 Flow in Tubes with Negligible Diffusion. 8.2 Tube Flows with Diffusion. 8.3 Method of Lines. 8.4 Effects of Variable Viscosity. 8.5 Comprehensive Models. 8.6 Performance Optimization. 8.7 Scaleup of Laminar Flow Reactors. 9 Packed Beds and Turbulent Tubes. 9.1 Packed-Bed Reactors. 9.2 Turbulence. 9.3 Axial Dispersion Model. 9.4 Scaleup and Modeling Considerations. 10 Heterogeneous Catalysis. 10.1 Overview of Transport and Reaction Steps. 10.2 Governing Equations for Transport and Reaction. 10.3 Intrinsic Kinetics. 10.4 Effectiveness Factors. 10.5 Experimental Determination of Intrinsic Kinetics. 10.6 Unsteady Operation and Surface Inventories. 11 Multiphase Reactors. 11.1 Gas-Liquid and Liquid-Liquid Reactors. 11.2 Three-Phase Reactors. 11.3 Moving-Solids Reactors. 11.4 Noncatalytic Fluid-Solid Reactions. 11.5 Scaleup of Multiphase Reactors. 12 Biochemical Reaction Engineering. 12.1 Enzyme Catalysis. 12.2 Cell Culture. 12.3 Combinatorial Chemistry. 13 Polymer Reaction Engineering. 13.1 Polymerization Reactions. 13.2 Molecular Weight Distributions. 13.3 Kinetics of Condensation Polymerizations. 13.4 Kinetics of Addition Polymerizations. 13.5 Polymerization Reactors. 13.6 Scaleup Considerations. 14 Unsteady Reactors. 14.1 Unsteady Stirred Tanks. 14.2 Unsteady Piston Flow. 14.3 Unsteady Convective Diffusion. 15 Residence Time Distributions. 15.1 Residence Time Theory. 15.2 Residence Time Models. 15.3 Reaction Yields. 15.4 Extensions of Residence Time Theory. 15.5 Scaleup Considerations. 16 Reactor Design at Meso-, Micro-, and Nanoscales. 16.1 Mesoscale Reactors. 16.2 Microscale Reactors. 16.3 Nanoscale Reactors. 16.4 Scaling, Up or Down. Suggested Further Readings. Problems. References. Index.

We investigate in this paper the simultaneous effects of electric field, couple stress, porous parameter and slip at the permeable surface on the generalized dispersion of an unsteady convective diffusion in a poorly conducting fluid in a channel bounded by porous layers. A two dimensional flow has been considered and the resulting partial differential equations have been solved analytically. The solutions are computed and the results show that the solute is dispersed relative to a plane moving with the mean speed of couple stress poorly conducting fluid with a relative unsteady dispersion coefficient. These relative unsteady dispersion coefficients are numerically computed and found that they increase with the increase in porous parameter and decrease with an increase in couple stress parameters. We have also estimated the contribution of diffusion and pure convection on the generalized dispersion coefficient. The effect of pure convection, neglecting diffusion terms on mean concentration is computed and the results show that the effect of pure convection decreases mean concentration compared to combined effect of convection and diffusion.

We construct three finite difference methods to solve a linearized Korteweg–de-Vries (KdV) equation with advective and dispersive terms and specified initial and boundary conditions. Two numerical experiments are considered; case 1 is when the coefficient of advection is greater than the coefficient of dispersion, while case 2 is when the coefficient of dispersion is greater than the coefficient of advection. The three finite difference methods constructed include classical, multisymplectic and a modified explicit scheme. We obtain the stability region and study the consistency and dispersion properties of the various finite difference methods for the two cases. This is one of the rare papers that analyse dispersive properties of methods for dispersive partial differential equations. The performance of the schemes are gauged over short and long propagation times. Absolute and relative errors are computed at a given time at the spatial nodes used.

Solute dispersion in Carreau–Yasuda fluid flow in a tube presented in Rana & Murthy (Proc. R. Soc. Lond.A, vol. 472, 2016, p. 20160294) was limited to a steady-state velocity profile due to the nonlinearity associated with the Yasuda parameter$a$with power-law exponent$n$. This limitation is overcome and the velocity profile is obtained for all values of the Yasuda parameter by using the Lagrange inversion theorem, which admits power series solution for the flow field. An analytical solution for the concentration distribution in the circular tube is obtained for the unsteady and pulsatile flow with$n\leq 1$and$\alpha <<1$and the numerical solution is presented for all values of$\alpha$and$n$. The solute dispersion is analysed analytically using the Sankarasubramanian–Gill generalized dispersion method and also using the Aris–Barton method of moments considering up to fourth-order moments. The solute dispersion is also simulated numerically by using a new class of computationally explicit Runge–Kutta method. The axial mean concentration of the solute is estimated by the exchange, convective and dispersion coefficients. The third- and fourth-order moments give rise to skewness and kurtosis revealing the deviation from the Gaussianity and reduction in the peak of the mean concentration profile at a small time of the solute injection. All time variations of these five moments against flow governing parameters are thoroughly investigated. The flow and dispersion regimes that are derived here for moments provide a good understanding of the solute dispersion in the tube. The increase in the Womersley frequency parameter led to a phase lag at each period. This work is the initiation of estimating the skewness and kurtosis in a non-yield stress fluid flow in a tube.

In describing various mass-transfer processes, wide use has been made of a diffusional model with an effective diffusion (dispersion) coefficient. The basic theory of dispersion was developed in [i, 2], where an equation with constant coefficients for the mean (over the cross section) particle concentration was proposed, together with a complex diffusion equation with a transfer rate depending on the coordinate transverse to the flow. Taylor calculated the dispersion coefficient and performed an experiment confirming the adequacy of the model. Subsequently, the dispersion theory was intensively developed, and several basic approaches to the derivation of the equivalent-diffusion equation and its generalization may be distinguished [3-5]. Here the dispersion equation is obtained by the smallperturbation method [6, 7], using Taylor considerations regarding the order of magnitude of terms in the initial diffusion equation. First, the impurity-dispersion equation is found with a weak dependence of the flow velocity on the impurity concentration. Then the problem is formulated for the first correction to the solution of the Taylor dispersion equation in the case of a parabolic profile of the flow velocity in the circular tube. Next, the analysis of the limiting solution of a characteristic nonsteady problem of impurity transfer around the wall of a prismatic channel of sufficiently general form is briefly considered.

Measurement of the axial dispersion of particles in a liquid fluidized bed applying a random walk method