Heat transfer to submerged surfaces in coarse-grained pressurized fluidized beds and fixed beds

This contribution reports on the theory underlying a uniform representation of heat transfer to submerged surfaces in fixed bed reactors and of gas convective part of heat transfer in fluidized beds with coarse-grained bulk solids and/or at elevated pressure. Based on an analysis of the pressure drop behaviour of fixed bed percolation at different gas pressures and with different bulk solids, a new dimensionless pressure drop parameter was developed. Fixed bed heat transfer data are very well correlated by this new dimensionless number. As soon as fluid throughput is in excess of minimum fluidization velocity, the pressure drop parameter transforms into the well-known Archimedes number. These two dimensionless numbers are connected by the condition of equilibrium for pressure drop and mass of practices in a fluidized bed. This equilibrium is fulfilled as soon as fluidization commences. Up to now, the Archimedes number has been generally accepted as the significant parameter, determining the gas convective part of heat transfer in fluidized beds; however, without any physical interpretation of this parameter. Introduction of the pressure drop number, which is consistent with the Archimedes number, reduces the heat transfer behaviour to pressure drop characteristics. The usefulness of this concept is proven by the comparison of experimental results and prediction.     

Development of design charts for fixed-bed adsorption

A mathematical model of the dynamic behaviour of non-isothermal fixed-bed adsorbers has been developed which takes into account the various mass and heat transfer resistances. Comparison of experimental and simulated results confirms that the model can predict the adsorption and desorption breakthrough curves of an adiabatically operated column, using only equilibrium data and tortuosity factors obtained from single pellet experiments. A simplified model with a reduced number of parameters was derived by investigation of the dimensionless transfer parameters under industrial conditions. It becomes evident that the main transfer mechanisms are convective heat and mass transfer in the bulk flow and diffusion within the pores of the particle. Dimensionless effluent concentration is expresses in terms of dimensionless time, a transport parameter, a non-isothermal parameter, the adsorption equilibrium and the inlet and initial concentrations and temperatures in the simplified model. For a chosen system of adsorbate and adsorbent, design charts can be developed by computer simulation, to determine graphically the breakthrough time as a function of significant process parameters, i.e. the dimensionless transfer parameter and the feed concentration.     

Relationship of system properties to performance in facilitated transport systems

A modelling procedure for the prediction of the performance of facilitated transport membranes is presented and illustrated with examples. Measurable system properties are used throughout. There are no empirical or adjustable parameters.The optimal range of the equilibrium constant for the reversible reaction between the gas solute and the carrier has previously been determined as well as the corresponding maximum facilitation factor. These results can be used to determine the optimal property set, to screen potential carrier molecules, as the basis for carrier structural modifications, to indicate the optimal range for the solute feed concentration, and to compare actual to optimal performance.Several uses are given for an analytical solution for the facilitation factor for gas transport. One can estimate the facilitation factor, calculate a single dimensionless number, tanh λ/λ, which can give the degree of facilitation, and use this result as the basis for a data plot to estimate the carrier gas complex diffusion coefficient and the external mass transfer resistance.     

UNSTEADY MHD BOUNDARY LAYER FLOW WITH DIFFUSION AND FIRST-ORDER CHEMICAL REACTION OVER A PERMEABLE STRETCHING SHEET WITH SUCTION OR BLOWING

In this study, unsteady MHD boundary layer flow with diffusion of chemically reactive species undergoing first-order chemical reaction over a permeable stretching sheet with suction or blowing and also with power-law variation in wall concentration is investigated. Using similarity transformation, the governing partial differential equations are converted into nonlinear self-similar ordinary differential equations. The transformed equations are then solved by the finite difference method using the quasi-linearization technique. Due to the increase in the unsteadiness parameter, the velocity initially decreases, but after a certain point it increases. A similar effect is also observed in case of concentration distribution. The increase in magnetic parameter causes a decrease in velocity and an increase in concentration. For increasing strength of applied suction both momentum and concentration boundary layer thicknesses decrease. On the other hand, applied blowing has reverse effects. Moreover, the mass transfer from the sheet is enhanced with increasing values of Schmidt number, reaction rate parameter, and also power-law exponent (related to wall concentration distribution). For high negative values of the power-law exponent, mass absorption at the sheet occurs. Moreover, due to increase of unsteadiness, this mass absorption is prevented.     

MHD flow and heat transfer of a micropolar fluid over a nonlinear stretching surface with variable surface heat flux and heat generation

An analysis has been carried out to study magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting micropolar fluid over a nonlinear stretching surface with variable wall heat flux in the presence of heat generation/absorption and a non‐uniform transverse magnetic field. The governing system of partial differential equations is first transformed into a system of ordinary differential equations using similarity transformation. The transformed equations are solved numerically. Results for the dimensionless velocity, micro‐rotation, and temperature profiles are displayed graphically delineating the effects of various parameters characterising the flow. The results show that the velocity profile decreases as the magnetic parameter and the velocity exponent increase, while it increases as the material parameter increases. The results show also that the temperature profile increases as the magnetic parameter, the velocity exponent, and the heat generation parameter increase. Furthermore, the temperature profile decreases as the material parameter, the heat absorption parameter, and the Prandtl number increase.     

Analytical approximate solution of competitive facilitated transport of acid gases through liquid membranes

An analytical approximate solution for the competitive facilitation factor of components A and E across a liquid membrane is developed in the case of instantaneous reactions inside the liquid membranes. This analytical solution solves the dimensionless, nonlinear diffusion-reaction transport problem governing the competitive facilitated transport of two gaseous components through liquid membranes. Prediction of the facilitation factors has been obtained for the equilibrium chemical reaction regime, considering the unequal complexes diffusivities and cases of zero and nonzero permeate side solute concentrations. This mathematical solution leads to analytical expressions for the concentration profiles of the species across the liquid membrane. In comparison with the present numerical solution and also numerical calculations and experimental data from the open literature, the difference between the analytical predictions and those obtained from the numerical solution were found to be in well agreement.     

MARANGONI AND ELECTRIC FIELD EFFECTS UPON MASS TRANSFER TO TRANSLATING DROPS

Interfacial stresses arising from the transfer of an interfacial tension-lowering solute (Marangoni effects) or the presence of an electric field may significantly alter the interfacial velocity of a moving droplet with consequent effects upon the rate of convective mass exchange between the drop and the external medium. The present theoretical study considers separately the Marangoni and electric field effects and delineates the conditions under which mass transfer enhancement should be observed. For Peclet numbers in the range of 10-100, enchancement is predicted for a Marangoni to Peclei number ratio of 100 or for a dimensionless electric velocity of 0.1, As an example, numerical results for a Peclet number of 50 indicate an increase of approximately 20% in the mass transfer coefficient for either a Marangoni number of 5000 or a dimensionless electric velocity of 0.5 for phases of comparable transport properties and solute solubilities.     

UNSTEADY FLUID AND HEAT FLOW INDUCED BY A STRETCHING SHEET WITH MASS TRANSFER AND CHEMICAL REACTION

The effect of chemical reaction on the flow, heat, and mass transfer within a viscous fluid on an unsteady stretching sheet is examined. The stretching rate, temperature and concentration of the sheet, and the chemical reaction rate are assumed to vary with time. The time-dependent boundary layer equations governing the flow are reduced through a convenient similarity transformation to a set of ordinary differential equations, which are numerically solved by applying the fourth-order Runge-Kutta-Fehlberg scheme with the shooting technique. Results for the velocity, temperature, and concentration distributions as well as the wall temperature and concentration gradients are presented graphically for various values of the unsteadiness parameter A, Prandtl number Pr, Schmidt number Sc, and chemical reaction parameter γ.     

Solid–liquid mass transfer in a gas sparging contactor equipped with a tube of circular fins

The solid–liquid mass transfer rate at a stack of circular fin surfaces in a gas sparging contactor was investigated. A diffusion-controlled dissolution technique of copper in an acidified chromate solution was employed. Variables studied included the number of actively exposed fins ranging from 5 to 20, pertinent physical properties of the solution, and air superficial velocity. Experimental data showed that the rate of the diffusion-controlled mass transfer increases with increasing superficial air velocity and decreases with increasing chromate solution acid concentration. Moreover, at relatively low superficial air velocity, increasing the number of actively exposed fins results into a continuous increase in the mass transfer coefficient. At relatively higher superficial air velocity, however, the mass transfer coefficient decreases in the 5–10 range of actively exposed fins and then reverts to increase in the 15–20 range. An empirical correlation relating the mass transfer j factor to Re_g, Fr, and a dimensionless height defined as the ratio of the height of actively exposed fins to the column equivalent diameter was developed based on the data generated in this study, with ±6.45% average deviation.     

Combined effects of internal heat generation and higher order chemical reaction on the non‐darcian forced convective flow of a viscous incompressible fluid with variable viscosity and thermal conductivity over a stretching surface embedded in a porous medium

In this paper, we study the combined effects of internal heat generation and higher order chemical reaction on a steady two‐dimensional non‐Darcian forced convective flow of a viscous incompressible fluid with variable dynamic viscosity and thermal conductivity in a fluid saturated porous medium passing over a linear stretching sheet. Using similarity transformations, the governing nonlinear‐coupled partial differential equations are made dimensionless and solved numerically for similarity solutions using very robust computer algebra software Maple 8. The non‐dimensional velocity, temperature and concentration distributions are presented graphically for various pertinent parameters such as relative temperature difference parameter, Darcy number, porosity parameter, reaction rate parameter and the order of the chemical reaction. The variations of Prandtl number and Schmidt number within the boundary layer are also displayed graphically when the fluid dynamic viscosity and thermal conductivity are temperature dependent. From the present numerical computations it is found that Prandtl number as well as Schmidt number must be taken as variables within the flow domain when the fluid's dynamic viscosity and thermal conductivity are variable. In the presence of internal heat generation, dynamic viscosity and thermal conductivity of the fluid are found to be higher than when it is absent. Increasing Darcy number reduces dynamic viscosity as well as thermal conductivity whereas increasing pore size reduces the Schmidt number and increases the Prandtl number within the boundary layer. For higher order reaction the rate of increase in mass transfer function is less compared to the rate of increase for the lower order reaction. © 2011 Canadian Society for Chemical Engineering     

Performance characteristics of a cstr containing immobilised enzyme particles

The effects of internal and external substrate diffusion resistances on the performance of a continuous stirred tank reactor are analysed in this work. Both immobilised enzymatic reactions with and without substrate inhibitions are considered. The substrate conversion for the reaction without substrate inhibition is dependent on four dimensionless parameters: the Thiele modulus, the dimensionless Michaelis constant, the mass transfer Nusselt number and β, which represents a combination of particle hold-up, maximum reaction rate, input substrate concentration and substrate residence time in the continuous stirred tank reactor. For the corresponding reaction with substrate inhibition, the effect of the additional dimensionless inhibition constant on the substrate conversion is also very significant. The substrate conversion generally decreases with decrease in dimensionless parameter β, increase in Thiele modulus and decrease in mass transfer Nusselt number. For the reaction with small Thiele modulus, β and strong substrate diffusion resistances, a multireactor system may be needed if a certain desired substrate conversion is required. The single CSTR model can be extended to describe the multireactor system and the effect of the number of reactors on the substrate conversion for a two- or more reactor system is also examined.     

Two-phase flow—IV. Gas and liquid side mass transfer coefficients

Mass transfer coefficients in gas and liquid have been obtained for the case of cocurrent gas—liquid flow through a vertical tube 6 mm i.d. by absorbing sulphur dioxide into sodium hydroxide solution and carbon dioxide into sodium carbonate—sodium bicarbonate solution respectively. The liquid side mass transfer coefficient was found to increase with the gas velocity but showed a maximum when plotted against the liquid velocity. A model based on the analogy between momentum and mass transfer has been proposed for the rate of mass transfer in the liquid phase. The mass transfer coefficient in the gas phase increases with the gas velocity but the liquid velocity has an opposite effect. A correlation in terms of dimensionless groups is presented for the gas side mass transfer coefficient.     

Effects of higher order chemical reaction on micropolar fluid flow on a power law permeable stretched sheet with variable concentration in a porous medium

An analysis has been carried out to obtain the effects of higher order chemical reaction on flow and mass transfer characteristics of micropolar fluids past a nonlinear permeable stretching sheet immersed in a porous medium with variable concentration of the reactant. The local similarity solutions for the flow, microrotation and mass transfer are obtained numerically and are illustrated graphically for various material parameters. Comparisons carried out with results from previously published work present excellent agreement. The results show that rate of mass transfer from the sheet to the surrounding fluid decreases significantly with the increase of nonlinear stretching parameter and increases with the increase of concentration parameter. The mass transfer of the reactive species strongly depends on the reaction rate parameter as well as order of the chemical reaction. It is stronger for the first‐order reaction than that for the higher order reaction. The rate of surface mass transfer decreases with the increase of the Darcy parameter. The results also show that the effect of these parameters on the micropolar fluids are less compared to the Newtonian fluids.     

Intensification of mass transfer in liquid fluidized beds with inert particles

The influence of inert particles on liquid/solid mass transfer is studied in fluidized beds by using a binary-mixture of solids of differing size and density. The addition of inert particles of higher density and smaller diameter, e.g. glass beads, exerts remarkable effects on mass transfer coefficients in comparison to that of mono-component active particles at the same liquid velocity. The extent of the effect on liquid–solid mass transfer coefficients increases with an increasing fraction of the small inert particles in the mixture. The liquid–solid mass transfer coefficients for binary-mixtures are well correlated in terms of dimensionless groups and the voidage parameter.     

Mixed convection in a doubly stratified micropolar fluid saturated non‐Darcy porous medium

The effects of thermal and solutal stratification on mixed convection along a vertical plate embedded in a micropolar fluid saturated non‐Darcy porous medium are analysed. The nonlinear governing equations and their associated boundary conditions are initially cast into dimensionless forms by pseudo‐similarity variables. The resulting system of equations is then solved numerically using the Keller‐box method. The numerical results are compared and found to be in good agreement with previously published results as special cases of the present investigation. The velocity, microrotation, temperature and concentration profiles are shown for different values of the coupling number, non‐Darcy parameter, mixed convection parameter, thermal and solutal stratification parameters. The numerical values of the skin friction, wall couple stress, heat and mass transfer rates for different values of governing parameters are also tabulated. © 2011 Canadian Society for Chemical Engineering     

Test of flow field on the annular meridian plane in a tubular membrane separator with rotary tangential flow

Enhancement of membrane microfiltration by rotary tangential flow is a new technique, which is based on the hydrocyclone mechanism. It improved the structure of the general membrane separator and the form of the liquid suspension flowing into the separator, so as to increase membrane fluxes and decrease membrane fouling. In our research, a tubular membrane separator with rotary tangential flow was designed for the first time. The flow field characteristics of polypropylene tubular membrane microfiltration in this tubular separator were studied systematically by means of the Particle Image Velocimetry (PIV) test. Streamlines and velocity distributions of the meridian plane of the separator under different operating parameters were obtained. The velocity distribution characteristics of rotary circular tangential flow were analyzed quantitatively with the following conclusions being obtained:

• (1) In the non‐vortex area, no matter how the operating parameters (flux, entry pressure) change, the velocity near the rotary tangential flow entrance is higher than the velocity far from the entrance at the same radial coordinates. In the vortex area, generally the flow velocity of the inner vortex is lower than that of the outer vortex. At the vortex center, the velocity is the lowest, the radial velocity being generally equal to zero. In the vortex zone, the radial velocity is less than the axial velocity.
• (2) Under test conditions, the radial velocity and the axial velocity of the vortexes' borders are 1–2 times the average axial velocity in the annular gap of the membrane module. The maximum radial velocity and axial velocity of Taylor vortexes are 2–5 times the average axial velocity in the annular gap of the membrane module.
• (3) In the vortexes that formed on the meridian plane, it was found that mass transfer occurred between the inner and outer parts of the fluid. Much fluid moved from the outer vortexes into the inner ones, which was able to prevent particles blocking the membrane tube.

Copyright © 2004 Society of Chemical Industry     

Multicomponent ion exchange column dynamics

Simple model equations which consider different rate control mechanisms are formulated for fixed bed multicomponent ion exchange processes. Efficient and accurate numerical methods are developed for solving these equations for liquid phase, solid phase or combined phase control. The algorithms are applicable to both ion exchange and liquid adsorption and are shown to be extendable to a general form of isotherms. The accuracy of the numerical algorithm is evidenced by the fact that the asymptotic limiting equilibrium solutions are closely approached as the dimensionless length parameter is increased, regardless of the rate control mechanism. Numerical examples for ion exchange applications are presented. These examples included multicomponent elution and purification problems. The effect of mass transfer resistance was also examined. Other examples examine the validity of a model reduction assumption and the comparison of equilibrium theory with the results obtained using the present finite mass transfer rate model.     

Heat Transfer Effects in Facilitated Transport Liquid Membranes

Abstract

Various mathematical models have been developed to describe facilitated transport. There are two limiting regimes where steady-state analytical solutions are available; diffusion-limited (reaction equilibrium) and reaction-limited (frozen condition). For intermediate cases, numerical solutions are available. All of these models are valid for isothermal conditions. It is possible in practice that the system may not be isothermal. The gas streams on each side of the membrane may be at different temperatures and/or there can be heat of reaction effects. These effects can cause the total facilitated flux to deviate from the isothermal case.

The results of including temperature effects are incorporated in a dimensionless factor θ. θ is the facilitation factor for the non-isothermal case divided by the facilitation factor for the isothermal case. Results show that the imposition of a temperature gradient can cause a significant increase or decrease in the facilitated flux. In extreme cases, there can even be a reversal in the direction of the facilitated flux. The heat of reaction had no noticeable effect for the conditions studied.     

A comparative analysis of unsteady and steady Buongiorno's Williamson nanoliquid flow over a wedge with slip effects

Comparison between unsteady and steady MHD Buongiorno's model Williamson nanoliquid flow through a wedge with slip effects, chemical reaction and radiation is made in this analysis. Thermophoresis and Brownian motion are also considered in this study. Appropriate similarity variables are presented to transmute the governing PDEs into the set of non-linear ODEs. The most widely authenticated finite element method is implemented to analyze these set of ODEs numerically. The behavior of concentration, temperature and velocity sketches for varied values of relevant parameters is numerically calculated and the outcomes are plotted through graphs. The numerical values of dimensionless rates of mass transfer, heat and velocity are also evaluated and depicted through tables. It is noted that with upsurging values of angle of wedge parameter, the distributions of temperature of the liquid intensify in both steady and unsteady cases.     

Sherwood number in flow through parallel porous plates (Microchannel) due to pressure and electroosmotic flow

An expression for Sherwood number is developed from first principles for combined pressure‐driven and electroosmotic flow in a porous rectangular microchannel. This quantifies the mass transfer of an electrically neutral solute in the microchannel and is useful for designing microfluidic devices and porous media flows. The convective‐diffusive species balance equation, coupled with the velocity field, is solved within the mass transfer boundary layer utilizing similarity method. From the simulations, it is observed that the Sherwood number increases as the electric double layer near the channel wall becomes more compact (as manifested through a decrease in the Debye length), and it reaches a constant value around the scaled Debye length of 40. The Sherwood number becomes constant at higher Debye lengths as electrokinetic effects become negligible. A detailed analysis of dependence of Reynolds number, dimensionless permeation velocity, ratio of driving force and scaled Debye length on Sherwood number is presented. © 2011 American Institute of Chemical Engineers AIChE J, 58: 1693–1703, 2012