A dynamic forward mixing model for evaluating the mass transfer performances of an extraction column

It is well known that the droplet behavior of the dispersed phase in extraction equipments has a strong influence on the mass transfer performances. It is, and will continuously be a key project for design and scaling up of extraction columns. In this work, a dynamic mass transfer model, considering the effect of forward mixing led by the drop size distribution and the axial mixing of the continuous phase, has been developed, by which the axial mixing characteristic can be easily evaluated when a stimulus-response dynamic curve is obtained. In order to test the mass transfer model and to study in the effect of droplet coalescence on mass transfer performance, a typical experimental system of 30% tributyl phosphate (in kerosene)-nitric acid-water with interface intension of 0.00995 N/m was chosen to investigate the mass transfer in a coalescence-dispersion pulsed-sieve-plate extraction column (CDPSEC) with 150 mm in diameter. The two-point dynamic method was applied to get the stimulus-response curves. With these results the axial mixing of the CDPSEC were evaluated. The calculated results showed that the response curves could be predicted with the new mass transfer model very well. The model has marked advantages over the traditional diffusion model. It is closer to the practice, easier to solve for the mathematical equations and boundary conditions, and has only one parameter to be optimized. The calculated results also showed that the influence of local coalescence of droplets on mass transfer performances is obvious.     

A new equation for estimating the height of the mass transfer unit in extraction columns

In this study, a new equation for estimating the height of the mass transfer unit, H_oxp, for mutually insoluble extraction systems was developed. The equation's accuracy and robustness were tested by comparing the predicted results with experimental data in the literature. With the same apparent interfacial tension, the mass transfer unit height decreased hyperbolically with the increase of the overall interphase volume mass transfer coefficient, despite the differences of column dimensions, experimental systems and operating conditions.     

Estimation of axial dispersion in pulsed-plate extraction columns

In this study, a two-region physical model was used to quantitatively estimate axial dispersion in pulsed-plate extraction columns, considering the operating conditions, geometry parameters and the physical properties of the experimental system. The calculated results fit the experimental axial dispersion coefficients well in nine geometries of pulsed-plate columns. This model is of importance to the design and scale-up of pulsed-plate extraction columns.     

New predictive correlation for mass transfer coefficient in structured packed extraction columns

Developments in the area of packed columns, particularly structured packed columns, are ongoing, specifically in the area of liquid–liquid extractions in different industries. In the present study, mass transfer coefficients have been obtained experimentally in a structured packed extraction column to develop a new correlation for prediction of continuous phase Sherwood number. The experiments were carried out for toluene/acetic acid/water and n-butyl acetate/acetic acid/water systems with counter current flow in different heights of column. A new dimensionless parameter, d₃₂/h, is introduced in proposed equation. This number considers the effect of column height (h) and mean drop diameter (d₃₂) jointly. The main advantage of this approach is that the principal effect of column height is considered in correlation without which the experimental data could not be fitted with a acceptable accuracy.     

Application of surface-renewal-stretch model for interface mass transfer

A new surface-renewal-stretch (SRS) model was developed to correlate experimental data for the time-average overall mass transfer coefficient, K_L,av, in liquid-liquid and gas-liquid mass transfer systems. The model is based on the equation of continuity, which includes both turbulent and convective mass transfer at the liquid-liquid and gas-liquid interfaces. The model incorporates Dankwerts surface-renewal model with the penetration theory for surface stretch proposed by Angelo et al. [Angelo, J.B., Lightfoot, E.N., Howard, D.W., 1996. Generalization of the penetration theory for surface stretch: application to forming and oscillation drops. A.I.Ch.E. Journal 12 (4) 751-760]. We used our new SRS mass transfer model to correlate successfully the existing interface mass transfer experimental data from published literature. As a result, the experimental mass transfer coefficient data was predicted with a high degree of accuracy.     

Numerical simulations of gas-liquid mass transfer in bubble columns with a CFD-PBM coupled model

Gas-liquid mass transfer in a bubble column in both the homogeneous and heterogeneous flow regimes was studied by numerical simulations with a CFD-PBM (computation fluid dynamics-population balance model) coupled model and a gas-liquid mass transfer model. In the CFD-PBM coupled model, the gas-liquid interfacial area a is calculated from the gas holdup and bubble size distribution. In this work, multiple mechanisms for bubble coalescence, including coalescence due to turbulent eddies, different bubble rise velocities and bubble wake entrainment, and for bubble breakup due to eddy collision and instability of large bubbles were considered. Previous studies show that these considerations are crucial for proper predictions of both the homogenous and the heterogeneous flow regimes. Many parameters may affect the mass transfer coefficient, including the bubble size distribution, bubble slip velocity, turbulent energy dissipation rate and bubble coalescence and breakup. These complex factors were quantitatively counted in the CFD-PBM coupled model. For the mass transfer coefficient k_l, two typical models were compared, namely the eddy cell model in which k_l depends on the turbulent energy dissipation rate, and the slip penetration model in which k_l depends on the bubble size and bubble slip velocity. Reasonable predictions of k_la were obtained with both models in a wide range of superficial gas velocity, with only a slight modification of the model constants. The simulation results show that CFD-PBM coupled model is an efficient method for predicting the hydrodynamics, bubble size distribution, interfacial area and gas-liquid mass transfer rate in a bubble column.     

Numerical simulation of unsteady mass transfer by the level set method

Unsteady mass transfer to/from a single drop in the continuous phase is formulated and numerically simulated in a moving reference coordinate system by solving the motion and mass transfer equations of an accelerating drop coupled with a level set equation for capturing the interface. Numerical simulation demonstrates the evolution of mass transfer rate and average drop concentration. Numerical simulation of the flow field and the concentration field simultaneously in each time step is compared with experimental data on single drop motion and mass transfer in two typical solvent extraction systems. The numerical predictions are found in good accord with the experimental measurements. The present numerical procedure in which the flow field is solved in a coupled way with the concentration field gives more accurate prediction than the previous decoupling algorithm by the authors.     

Transient convective mass transfer for a fluid sphere dissolution in an alternating electric field

In this study, we considered mass transfer in a binary system comprising a stationary fluid dielectric sphere embedded into an immiscible dielectric liquid under the influence of an alternating electric field. Fluid sphere is assumed to be solvent-saturated so that an internal resistance to mass transfer can be neglected. Mass flux is directed from a fluid sphere to a host medium, and the applied electric field causes a creeping flow around the sphere. Droplet deformation under the influence of the electric field is neglected. The problem is solved in the approximations of a thin concentration boundary layer and finite dilution of a solute in the solvent. The thermodynamic parameters of a system are assumed constant. The nonlinear partial parabolic differential equation of convective diffusion is solved by means of a generalized similarity transformation, and the solution is obtained in a closed analytical form for all frequencies of the applied electric field. The rates of mass transfer are calculated for both directions of fluid motion—from the poles to equator and from the equator to the poles. Numerical calculations show essential (by a factor of 2/3) enhancement of the rate of mass transfer in water droplet-benzonitrile and droplet of carbontetrachloride-glycerol systems under the influence of electric field for a stagnant droplet. The asymptotics of the obtained solutions are discussed.     

Numerical simulation of interphase mass transfer with the level set approach

A level set approach is applied for simulating the interphase mass transfer of single drops in immiscible liquid with resistance in both phases. The control volume formulation with the SIMPLEC (semi-implicit method for pressure-linked equations consistent) algorithm incorporated is used to solve the governing equations of incompressible two-phase flow with deformable free interface on a staggered Eulerian grid. The solution of convective diffusion equation for interphase mass transfer is decoupled with the momentum equations. Different spatial discretization schemes including the fifth-order WENO (weighted essentially nonoscillatory), second-order ENO (essentially nonoscillatory) and power-law schemes, are tested for the solution of mass transfer to or from single drops. The conjugate cases with different equilibrium distribution coefficients are simulated successfully with the transformation of concentrations, molecular diffusivities, mass transfer time and velocities. The predicted drop concentration, overall mass transfer coefficient and flow structure are compared with the reported experimental data of a typical extraction system, i.e., n-butanol-succinic acid-water, and good agreement is observed.     

A tool for mass transfer evaluation in packed-bed columns for chemical engineering students

This paper presents a Microsoft Excel tool to calculate liquid-gas mass transfer coefficients in packed towers to support numerical design activities in the courses of Unit Operations for Industrial Process and Sustainable Process Design for the Master’s degree in Chemical Engineering of the University of Naples Federico II (Italy).The Mass Transfer Solver Tool (MT Solver Tool) uses several available models to estimate, separately, the values of liquid and gas mass-transfer coefficients and the wet surface area for 144 random and structured packings of interest for absorption/stripping and distillation processes. In addition, a separate spreadsheet can be used in a user-defined mode, to evaluate the mass transfer coefficients with new packing types or to interpret experimental data when the geometrical and physical characteristics of the packing are known. Eventually, the tool is supplied with a data library, where packing geometry and model fitting parameters can be retrieved.The software is aimed to support students and educators in the Unit Operations for Industrial Process and Sustainable Process Design courses. In particular, this is meant to be an example on how the accuracy of design algorithms adopted in unit operation processes is affected by the use of the underpinning correlations for mass transfer rate or pressure drops. Besides, this is aimed to encourage comparison of different correlations when exact field data are not available. Besides, chemical engineers and researchers interested in packed columns design and modelling data may also benefit from the utilization of the software. The MT Solver Tool was introduced to students in a dedicated tutorial lesson after lecturers on packed column design algorithms for distillation, absorption and stripping. Most of the students of the course participated to a group training aimed to simulate the design of an absorption column supported by the MT Solver Tool providing feedback on its application.After the training, an anonymous survey was proposed to the students to monitor the approval rating of the proposed activity and the use of the MT Solver Tool software to support numerical calculations.     

Transient rise velocity and mass transfer of a single drop with interfacial instabilities—Numerical investigations

Full 3D-simulations of transient interfacial mass transfer accompanied by Marangoni convection at a single spherical droplet in a quiescent liquid were performed in a moving reference coordinate system. The flow and concentration field are solved simultaneously, coupled via the additional Marangoni stress generated by concentration gradients at the interface. Fluid dynamics and mass transfer are investigated in the Marangoni convection dominated toluene/acetone/water system. The numerical results are qualitatively and quantitatively compared with own experimental results. The simulation results reveal that mass transfer is always enhanced—compared to calculations where no Marangoni convection appears—independently from the initial solute concentration. The enhancement factor of mass transfer ranges between 2 and 3.     

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).     

Drop size correlation and population balance model for an agitated-pulsed solvent extraction column

Agitated-pulsed column (APC) is a newly designed extraction column with excellent mass transfer performance. In this work, Sauter mean drop diameter d₃₂ and drop size distribution was investigated under different operation conditions in a 25 mm diameter APC. The results show that with an increase in pulsation intensity and agitation speed the drop size distribution is narrowed and d₃₂ is decreased significantly. With increasing dispersed-phase velocity, d₃₂ increased and drop size distribution become narrow, while there was no noticeable change with continuous velocity. The cumulative size distribution was found to be predicted well using the Inverse Gaussian function. A new correlation was proposed to predict the experimental d₃₂ data of the APC column used in this study. Furthermore, population balance model was applied to predict the drop size distribution with refitted parameters in the breakage, coalescence kernels functions.     

Drop size distribution and mean drop size in a pulsed packed extraction column

Drop size distribution and mean drop size are used for calculation of interfacial area available for mass transfer. In this study, the drop size distribution and Sauter mean drop diameter (d₃₂) have been investigated using three different liquid systems in the absence of mass transfer in a pilot plant pulsed packed column. The drop size was measured at four different points along the active column height. Three operating variables have been studied including the pulse intensity (af) and flow rates of both liquid phases. The effect of liquid properties and height of the active column were also investigated. A combination of the pulse intensity and interfacial tension had the largest effect on the drop size distribution while none of the flow rates were of significance. The height of the column played an important role at the bottom of the active column, but the associated effect was reduced with increase of the height. Finally, a normal probability function of number density was proposed for prediction of the drop size distribution with an Average Absolute Relative Error (AARE) of 8.8% for their optimized constant. Furthermore, two correlations were presented involving height or flow rates of the two phases along with operating variables and physical properties of the liquids. These correlations had AARE values of about 8.5 and 7.8%, respectively.     

Unsteady mass transfer in the continuous phase around axisymmetric drops of revolution

Unsteady mass transfer in the continuous phase around any axisymmetric drop of revolution at high Peclet numbers has been theoretically studied. General equations for the concentration profile, the molar flux, the concentration boundary layer thickness, and the time to reach steady state have been obtained using a similarity transformation and by the method of characteristics. Solutions for large number of problems can be immediately obtained, with the only requirements being the shape of the drop and the tangential velocity at the surface of the drop.     

Numerical solution of the bivariate population balance equation for the interacting hydrodynamics and mass transfer in liquid-liquid extraction columns

A comprehensive model for predicting the interacting hydrodynamics and mass transfer is formulated on the basis of a spatially distributed population balance equation in terms of the bivariate number density function with respect to droplet diameter and solute concentration. The two macro- (droplet breakage and coalescence) and micro- (interphase mass transfer) droplet phenomena are allowed to interact through the dispersion interfacial tension. The resulting model equations are composed of a system of partial and algebraic equations that are dominated by convection, and hence it calls for a specialized discretization approach. The model equations are applied to a laboratory segment of an RDC column using an experimentally validated droplet transport and interaction functions. Aside from the model spatial discretization, two methods for the discretization of the droplet diameter are extended to include the droplet solute concentration. These methods are the generalized fixed-pivot technique (GFP) and the quadrature method of moments (QMOM). The numerical results obtained from the two extended methods are almost identical, and the CPU time of both methods is found acceptable so that the two methods are being extended to simulate a full-scale liquid-liquid extraction column.     

Correlation of the Drop Size in a Modified Scheibel Extraction Column

Experiments were designed to ascertain the main factors for the Sauter mean drop size (d₃₂) of the dispersed phase in a three‐stage modified Scheibel extraction column with no mass transfer. A precise correlation applied to the liquid‐liquid systems with low interfacial surface tension was proposed for calculating d₃₂. The maximum relative error for all data was 16.0 % and the mean relative error ±4.6 %.     

Simultaneous measurement of flow pattern and mass transfer coefficient in bubble columns

Mass transfer studies were carried out in a bubble column using the chemical method. Catalytic oxidation of sodium sulfite was chosen for the studies and the corresponding specific rates of oxidation were obtained using a stirred cell. Laser Doppler anemometer (LDA) was used to measure the instantaneous velocities in the same stirred cell as well as in bubble columns (100 and i.d.). An efficient algorithm based on the multiresolution analysis of the velocity-time data using wavelets was used for the isolation of data belonging to the gas and liquid phases. Eddy isolation model was used for the characterization of the eddy motion including the estimation of the energy dissipation rate. Using the knowledge of eddy motion, a methodology was developed for the prediction of true mass transfer coefficient (k_L) in a stirred cell as well as in bubble columns. The predicted values of k_L have been compared with the experimental values obtained by the chemical method.