Exact universal uniqueness criteria for the adiabatic tubular packed bed reactor

Exact, universal a priori bounds and regions of multiplicity for the entire tubular packed bed reactor are developed by application of a technique reported in Chang and Calo, Chem. Engng Sci. 1979 34 285 to a cascade two-phase cell model for an nth order chemical reaction with interphase resistance to mass and heat transport, Le ≠ 1 (or Le = 1) and either lumped parameter catalyst particles or with intraparticle concentration gradients with uniform temperature. Both the more common case of interphase heat transfer greater than interphase mass transfer rate and the inverse case of particle over-temperature have been considered. In all cases it has been shown that the reactor conservation equations can be decoupled at certain points along the bed determined by the Lewis number, and that questions of multiplicity and uniqueness reduce to consideration of a single algebraic equation which is actually a form of the two-phase adiabatic CSTR. Also as for the CSTR, the topology of the adiabatic packed bed reactor is shown to be the simple cusp catastrophe. The application of the resultant criteria is quite simple and represents a practical step in the design procedure for highly exothermic reactions in packed beds. A flow chart of a suggested procedure is included.     

Asymptotic solution to moving-bed exchange equations

Asymptotic solutions to countercurrent moving-bed exchange equations are discussed considering the effect of axial dispersion and interphase mass transfer. Existence conditions are derived by using perturbation theory in a properly defined moving coordinate system. The flow are ratio turns out to be a very important parameter and its effect on the exchange performance is carefully examined. In particular, the effect of interphase mass transfer disappears when the exchange zone becomes stationary. Applications are illustrated by using the Langmuir adsorption isotherm. Numerical solutions of the transient equations show that the asymptotic solution may be useful.     

Membrane contactor for CO2 absorption applying amino-acid salt solutions

A novel composite solution based on amino-acid salt as a CO₂ absorbent was proposed. Coupling process of membrane contactor and the composite solution was investigated. The performance of the coupling was experimentally compared between the single and composite solution. Overall mass transfer coefficients were determined. Effects of various factors such as flow rates and operation temperatures on mass transfer of membrane contactor were studied. Comparison of prediction for overall mass transfer coefficients using a resistance in series model with experimental values was performed. Results show that, performance of the composite solution is evidently better than that of the single solution. The overall mass transfer coefficient with the composite solution is much higher than that with the single solution. Higher operation temperature can enhance mass transfer of membrane contactor. Operation parameters such as flow rates can promote mass transfer, but the promotion is limited. Enhancement of mass transfer relies essentially on chemical reaction. Model values are in good agreement with experimental ones.     

RETARDATION OF DROPLET MOTION BY SURFACTANT. PART 1. THEORETICAL DEVELOPMENT AND ASYMPTOTIC SOLUTIONS

The equations describing the retardation by surfactant of an essentially spherical droplet in creeping flow through an unbounded Newtonian liquid are developed in a general way. The concentration distribution is determined by convection and the coupled mass transfer processes of liquid-phase diffusion, adsorption kinetics, and surface diffusion. Assuming that mass transfer occurs rapidly, an asymptotic solution is obtained showing uniform retardation, treating both phases and all mass transfer mechanisms simultaneously. For slow mass transfer, an asymptotic solution is obtained for stagnant cap behavior.     

RETARDATION OF DROPLET MOTION BY SURFACTANT. PART 1. THEORETICAL DEVELOPMENT AND ASYMPTOTIC SOLUTIONS

The equations describing the retardation by surfactant of an essentially spherical droplet in creeping flow through an unbounded Newtonian liquid are developed in a general way. The concentration distribution is determined by convection and the coupled mass transfer processes of liquid-phase diffusion, adsorption kinetics, and surface diffusion. Assuming that mass transfer occurs rapidly, an asymptotic solution is obtained showing uniform retardation, treating both phases and all mass transfer mechanisms simultaneously. For slow mass transfer, an asymptotic solution is obtained for stagnant cap behavior.     

MASS AND ENERGY TRANSFER IN MULTICOMPONENT SYSTEMS

In this paper, which is essentially an interpretative review, we develop the theory of multicomponent mass transfer using the Bird, Stewart and Lightfoot approach to binary mass transfer as a consistent basis. Methods of solving the coupled diffusion equations are discussed and procedures for estimating interphase mass transfer coefficients considered. The developed formalisms are applied to a few examples in the areas of distillation, condensation and extraction to demonstrate the importance of considering coupled diffusion effects in multicomponent transfers.     

MASS AND ENERGY TRANSFER IN MULTICOMPONENT SYSTEMS

In this paper, which is essentially an interpretative review, we develop the theory of multicomponent mass transfer using the Bird, Stewart and Lightfoot approach to binary mass transfer as a consistent basis. Methods of solving the coupled diffusion equations are discussed and procedures for estimating interphase mass transfer coefficients considered. The developed formalisms are applied to a few examples in the areas of distillation, condensation and extraction to demonstrate the importance of considering coupled diffusion effects in multicomponent transfers.     

Unsteady mass transfer around spheroidal drops in potential flow

Unsteady mass transfer in the continuous phase around spheroidal drops in potential flow and at high Peclet numbers has been theoretically studied. Analytical solutions for the concentration profile, the molar flux, the concentration boundary layer thickness, and the time to reach steady state are presented. The solution to the problem was obtained by the useful equations derived by Favelukis and Mudunuri for axisymmetric drops of revolution, with the only requirements being the shape of the drop and the tangential velocity at the surface of the drop. The solution suggests that, as the eccentricity increases, the total quantity of material transferred to or from the drop decreases (for prolate spheroids) and increases (for oblate spheroids). It was also determined that when the dimensionless time is greater than 2, then steady state is in practice obtained, with prolate drops attain steady-state conditions faster than oblate drops.     

Mass-transfer rates from single drops and drop swarms

In modelling the performance of liquid—liquid extractors the mass-transfer rates between drops and the surrounding liquid are assumed to be known. They are calculated using fundamental models from the literature which, however, are sometimes unreliable or too complicated. In this work, correlations based on numerous experimental data are developed for diffusional-type mass transfer, both for single drops and drop swarms. First, emprical equations are presented which characterize the continuous-side mass-transfer rate for rigid and internally circulating drops. An approach based on correlations of effective diffusivity is then used for mass transfer inside drops. Empirical equations correlating this parameter are developed both for single drops and for drop swarms.     

Mass transfer ambiguities in swirling pipe flows

A mathematical model of mass transfer in the boundary layer of swirl flow is developed. The equations obtained for Sherwood number and mass transfer coefficient generalize the equations obtained by Levich for the flat plate to the case of mass transfer in axisymmetrical swirl flow. For the first time it is shown that, in swirl flows, the mass transfer is not completely controlled by integral flow parameters (Reynolds number and the swirl number) but depends essentially on the type of vortex symmetry. The left-handed helical vortices generate wake-like swirl flows and increase mass transfer in comparison with axial flows. The right-handed helical vortices generate jet-like swirl flows, which can decrease mass transfer.     

带有矩形嵌件薄壁型腔内熔接过程动态模拟

为了准确模拟具有对称结构的带有矩形嵌件的薄壁型腔内熔接线的动态形成过程,采用Level Set/Ghost方法追踪充填阶段聚合物熔体前沿界面。引入具有高阶精度且数值稳定无振荡的5WENO（the fifth order weighted essentially nonoscillatory）格式对Level Set/Ghost方程进行数值求解,耦合求解物理量控制方程的一般差分格式实现熔接过程的动态模拟。数值算例对整个流场的压力、温度及速度进行了分析和讨论,并将熔接区域的压力、温度和非熔接区域的压力、温度进行了比较。数值结果与理论分析结果一致,且与前人数值结果相比有很好的精度。     

Exact solutions of the Newman,and of the handlos-baron,model equations for countercurrent flow extraction

The objective of this study was to develop numerical algorithms for the simultaneous solution of the mass transfer and hydrodynamic equations for swarms of liquid drops in a countercurrent flow liquidliquid extraction column. Two drop diffusion models were used—the Newman rigid drop, and the Handlos-Baron turbulent-circulating drop. Algorithms were developed which gave accurate and rapid solutions of the model equations. Mass transfer rates in a Pilot Plant Rotating Disc Contactor were predicted for comparison with experimental data and with rates predicted by approximate calculation methods. The latter were found to be 20–30% below the exact solution values.     

Rates of extraction of mercaptan sulfur from pentane by caustic solution

The interphase rate process for the extraction of mercaptan sulfur from n-pentane with a 10% caustic solution is examined. The system represents a case of mass transfer with an equilibrium chemical reaction occurring in one phase. Experimental studies were made with both a continuous flow stirred tank reactor (CFSTR) and an annular flow jet reactor (AFJR). The interphase volumetric mass transfer coefficient (K_La) was derived based on Olander's theory. The K_La data for each contactor was found to be a function of the power input per unit volume. The AFJR exhibited higher K_La values and equilibration of mercaptan sulfur was achieved in about three milliseconds.     

Algorithms for the solution of material balance equations for non-conventional multi-stage operations

Multicomponent distillation programs in general use are seldom capable of describing systems whose flow patterns differ from the conventional counter-current. However, these programs may be extended to include pumparounds or interphase mass transfer by algorithms which require neither additional internal iteration steps nor repetitive solutions. For a pumparound, a recursive procedure is developed to use the conventional Thomas Algorithm modified by an additional multiplier in the second step. To describe interphase mass transfer, the solution to the set of material balance equations requires a recursive technique in which the definitions for the algorithm alternate for successive stages. The use of recursional procedures is demonstrated on a multicomponent distillation column simulation.     

Numerical simulation of mass transport in a filter press type electrochemical reactor FM01-LC: Comparison of predicted and experimental mass transfer coefficient

This paper studies flow characteristics and their effect on local mass transfer rate to a flat plate electrode in a FM01-LC electrochemical reactor. 3D reactor simulations under limiting current and turbulent flow conditions were performed using potassium ferro-ferricyanide electrochemical system with sodium sulfate as supporting electrolyte. The model consists of mass-transport equations coupled to hydrodynamic solution obtained from Reynolds-averaged Navier–Stokes equations using standard k–? turbulence model, where the average velocity field, the turbulence level given by the eddy kinetic energy and the turbulent viscosity of the hydrodynamic calculation were used to evaluate the convection, turbulent diffusion and the concentration wall function. The turbulent mass diffusivity was evaluated by Kays–Crawford equation using heat and mass transfer analogies, while wall functions, for mass transport, were adapted from Launder–Spalding equations. Simulation results describe main flow properties, concentration profiles throughout the entire volume of the reactor and local diffusion flux over the electrode. Overall mass transfer coefficients estimated by simulation, without fitting parameters, agree closely with experimental coefficients determined from limiting current measurements (1.85% average error) for Re between 187 and 1407.     

Unsteady-State Wave Flows of a Gas–Solid Suspension with Gas-Flow-Rate Oscillations and Their Effect on the Heat and Mass Transfer

A set of equations of unsteady-state wave motion of the dispersed phase in a binary vapor–gas medium with allowance for the heat and mass transfer is obtained. By computational experiment, it is studied how gas pulsations at the channel inlet affect the flow of the gas–solid suspension and the intensity of heat and mass transfer under resonant conditions. In particular, it is shown that, in the parameter range studied, gas pulsations enhance the interphase heat and mass transfer.     

Modeling of a spiral-wound module and estimation of model parameters using numerical techniques

A model for a spiral-wound reverse osmosis system using the three-parameter membrane transport model by Spiegler-Kedem is presented. The pressure drops in the permeate channel, feed channel and also the variation of the mass transfer coefficient along the feed channel were taken into account. An analytical solution was not possible due to the large number of nonlinear model equations; therefore, a computer solution utilizing finite differences was employed. The data generated by the simulation of the proposed model clearly indicate that neglecting the variation in the mass transfer coefficient and pressure drop along the flow channels can lead to errors in permeate concentration, though the effect on permeate flow rate may not be significant. The significance of the reflection coefficient in the membrane transport model was also investigated. A method for estimation of the model parameters is also presented; previously reported experimental data were analyzed. Using this parameter-estimation program, a correlation for the mass transfer coefficient in the feed channel is proposed and compared with the correlation available in the literature.     

A MODEL FOR MASS TRANSFER IN OSCILLATING-CIRCULATING LIQUID DROPS†

Velocity vector computations based upon flow-visualization techniques have enabled the development of a detailed model for flow patterns in a droplet-continuum system typical of liquid-liquid solvent extraction. Experiments show that the streamlines for flow in and around circulating droplets which are forced into regular oscillation by a pulsing electric field can be modeled as the vectorial sum of low-Reynolds-number internal circulation (Hadamard flow) and an oscillatory streaming caused by the prolate-to-oblate ellipsoidal shape changes of the droplet (Dirichlet flow). The convective-diffusion equations have been derived and numerically solved for mass transfer from both circulating droplets (Hadamard flow) and oscillating-circulating droplets (mixed Hadamard-Dirichlet flow). The resulting rate of mass transfer from an oscillating-circulating droplet is significantly greater than that from the equivalent circulating droplet. The rate of mass transfer is dependent upon oscillation frequency and amplitude, droplet Peclet number, the interphase distribution coefficient of the transferring species, and physical properties of each fluid phase.     

Creeping flow mass transfer to a single active sphere in a random spherical inactive particle cloud at high schmidt numbers

Using the solution by Tam of Navier-Stokes equations for creeping flow around an active sphere surrounded by a random cloud of inactive spheres, an asymptotic solution of the convective diffusion equation is obtained for high Schmidt numbers. The Sherwood number for the overall mass transfer coefficient to the active sphere has been analytically related to the Peclet number as

It agrees very well with the experimental mass transfer data on single active spheres for σ = 0476, Re < 10 and large Sc. This analytical result becomes invalid as σ decreases to 0.33. Pfeffer's model for the same problem has excellent agreement with the mass transfer data on single active spheres for σ = 026, Re < 10 and Sc = 1600. Pfeffer's model seems to be quite satisfactory for the usual range of void volume fractions in packed beds. The present model seems to be more accurate at higher values of void volume fractions in packed and distended beds.     

Simulation of interfacial mass transfer by droplet dynamics using the level set method

A unique approach to simulate mass transfer across the moving droplet where mass transport equations and governing equations of the levels set method are solved separately is proposed in this work. Mass transfer coefficients of the chemical species can be computed by equating the diffusive flux and the mass transfer flux at the interface, which are found to be of the same order of magnitude as of those obtained using an empirical correlation. Simulations underestimate mass transfer coefficients by roughly 25% across the range of low Reynolds number studied systematically. The level set method is used to track the motion of the interface to study droplet dynamics and mass transfer across a moving droplet because of the ease in defining the local curvature of the interface and in capturing any topological changes. We perform various numerical simulations by varying the physical properties of the system, in order to analyze the influence of dimensionless numbers such as the Reynolds number (Re), the Eotvos number (Eo) and the Morton number (M) on the shape of a buoyancy-driven droplet and compare them with the various shape regimes of drops and bubbles reported by Clift et al. [1978. Bubbles, Drops and Particles. Academic Press, New York]. It is shown that larger deformation occurs for buoyancy-driven droplets when interfacial forces are considerably greater than viscous forces (M?1 and Eo>10) and the droplets are almost undeformed when viscous forces dominate interfacial forces (M>10³ and Eo>10).