Unsteady convective diffusion in capillary chromatography

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).

Unsteady convective diffusion in an annular catalytic reactor

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.

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.

Volume Contents and Author Index to Volume 39

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.

An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions

Hydrodynamic dispersion of reactive solute in a Hagen–Poiseuille flow of a layered liquid

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.

A dispersion model for initial consequence analysis based on diffusion equations

An optimized Schwarz waveform relaxation method for the unsteady convection diffusion equation in two dimensions

In protein threading, one is given a protein sequence, together with a database of protein core structures that may contain the natural structure of the sequence. The object of protein threading is to correctly identify the structure(s) corresponding to the sequence. Since the core structures are already associated with specific biological functions, threading has the potential to provide biologists with useful insights about the function of a newly discovered protein sequence. Statistical tests for threading results based on the theory of extreme values suggest several combinatorial problems. For example, what is the number of waysm′=#t{Li>xi}i=0n of choosing a sequence {Xi}i=1n from the set {1, 2, ...,t}, subject to the difference constraints {Li=Xi+1−Xi>xi}i=0n, whereX0=0,Xn+1=t+1, and {xi}i=0n is an arbitrary sequence of integers? The quantitym′ has many attractive combinatorial interpretations and reduces in special continuous limits to a probabilistic formula discovered by the Finetti. Just as many important probabilities can be derived from de Finetti's formula, many interesting combinatorial quantities can be derived fromm′. Empirical results presented here show that the combinatorial approach to threading statistics appears promising, but that structural periodicities in proteins and energetically unimportant structure elements probably introduce statistical correlations that must be better understood.

Suppose {Xn n ≥ 1} is a stationary sequence satisfying the D and D1 mixing conditions given by Adler [1]. Suppose further that g and h are functions on the positive integers such that h is positive, periodic with integer period p ≥ 1 and g satisfies a certain growth rate condition. By first exhibiting the weak convergence of a sequence of point processes related to , we derive the asymptotic distribution of . When {Xn} is in the maximal domain of attraction of A(x) = exp(-e -xx), {Mn} is attracted to (A(x))q/p where . If {Xn} is in the maximal domain of attraction of G(x) = exp(-x -α) then {Mn} is attracted to (G(x)) θ where . These asymptotic results lead to a new method for estimating the level of the k-year return period, a quantity of interest in many climatological studies. An analysis of the daily rainfall totals in Homestead, Florida from 1939-1979 is provided, illustrating this estimation procedure.

Many opportunities have been made available through the use of Information communication technologies (ICT) in organizations in general, and particularly in Small to medium-sized enterprises (SMEs) with the early adopters' successes well publicized. Thus, many organizations not taking advantages of the ICT are currently examining their options. The purpose of this study is to examine the factors that influence SMEs' decision in adopting ICT in business. The diffusion theory, the technology acceptance model and information richness will provide the theoretical foundation for this study. Currently in the data collection phase, the owner/manager or someone responsible for IT employed by the SMEs will be surveyed in this cross-industry study. Preliminary results from the study are reported as well as sample questions resulting from the study to be used in the final survey questionnaire.

The Orbiting Carbon Observatory-2 (OCO-2) collects infrared spectra from which atmospheric properties are retrieved. OCO-2 operational data processing uses optimal estimation (OE), a state-of-the-art approach to inference of atmospheric properties from satellite measurements. One of the main advantages of the OE approach is computational efficiency, but it only characterizes the first two moments of the posterior distribution of interest. Here we obtain samples from the posterior using a Markov Chain Monte Carlo (MCMC) algorithm and compare this empirical estimate of the true posterior to the OE results. We focus on 600 simulated soundings that represent the variability of physical conditions encountered by OCO-2 between November 2014 and January 2016. We treat the two retrieval methods as ensemble and density probabilistic forecasts, where the MCMC yields an ensemble from the posterior and the OE retrieval result provide the first two moments of a normal distribution. To compare these methods, we apply both univariate and multivariate diagnostic tools and proper scoring rules. The general impression from our study is that when compared to MCMC, the OE retrieval performs reasonably well for the main quantity of interest, the column-averaged $$\mathrm{{CO}}_{2}$$ concentration $$X_{\mathrm{{CO}}_{2}}$$ , but not for the full state vector $$\mathbf {X}$$ which includes a profile of $$\mathrm{{CO}}_{2}$$ concentrations over 20 pressure levels, as well as several other atmospheric properties.Supplementary materials accompanying this paper appear on-line.