Thin Concentration Boundary Layer Research Articles

A Simulation Framework for Electrochemical Processes with Electrolyte Flow

Software tools for simulating electrochemical processes (e.g. COMSOL Multiphysics, ELSYCA) are mostly of commercial type. Besides, three-dimensional simulations in complex cell geometries are known to become resource-expensive, as typically thin concentration boundary layers need to be resolved. This work presents a simulation framework for electrochemical processes based on the open source platform OpenFOAM. The finite volume method used and combined with domain decomposition is able to efficiently benefit from multi-core computer architectures. Our framework takes into account electrolyte flow, which is well known to affect mass transfer, and allows to consider multi-species electrolytes and forcing of the electrolyte. The stability and fast convergence of the method presented is found to rely on the linearization of the Butler-Volmer condition in the iterative solver. The framework is validated against an analytical solution valid for simplified conditions and an electrodeposition process at a conically shaped electrode in an external magnetic field. The latter exhibits transient departure of the concentration boundary layer from the cathode, and excellent agreement with COMSOL simulation results is found.

Double-diffusive convection in viscous flow over a rotating porous and moving disk

This paper presents the double-diffusive convection in the flow of viscous fluid over a heated, rotating, porous and moving disk. All the field quantities, defined at the surface of such disk, are nonuniform and nonlinear, therefore, the boundary conditions of the simulated problem are nonhomogeneous and variable. A set of new variables is found and it has simplified the system of six PDE's along with the boundary conditions into a system of coupled ODE's of boundary value type. The system of nonlinear, coupled and boundary value ODE's has some nine dimensionless parameters, whereas, effects of these numbers have been seen on all field variables, tangential and surface shear stresses and rates of heat and mass transfer. The present simulations have generalized the concept of rotating disk flows in the sense that the three components of velocity, associated with the disk, are taken nonuniform and nonlinear, whereas, the other thermodynamic properties, specified at the surface of the disk, also have the nonlinear nature. In this investigation, we focused on weak and strong assisting and opposing flows of liquids and gases over a rotating disk in the presence of stretching, shrinking, injection and suction velocities. Each component of velocity is effectively (slightly) enhanced with the variation of and in case of assisting (opposing) flow in the presence of stretching/shrinking and injection/suction. Moreover, similar behavior of the other field variables and quantities of physical interests has been noticed. Thin momentum, thermal and concentration boundary layers are appeared in some cases, which are clearly demonstrated in graphs in special situations.

Impact of large-scale effects on mass transfer and concentration polarization in Reverse Osmosis membrane systems

We present well-resolved computational fluid dynamics simulations of a large-scale reverse osmosis membrane-spacer configuration (∼ 1 m). Our computational model solves the flow and transport equations with variable solute-dependent properties. We utilize a high resolution computational mesh to resolve all relevant length scales associated with spacer-induced mixing and thin concentration boundary layers. An important contribution of this work is the development of a modified mass-transfer correlation that accounts for the development of the concentration boundary layer along the channel. A set of 2D axisymmetric simulations were performed for a spiral wound module layer with varying cross-flow conditions and spacer diameters which indicate a significant entrance length effect for concentration profile development at lower flow rates while mixing effects dominate at higher flow rates. The mass-transfer correlations at higher flow rates compare well with published correlations while a surrogate model for Sherwood number was obtained that depends on an additional similarity variable that accounted for entrance length effects at lower flow rates. Finally, a large-scale membrane-spacer design relevant to high-pressure reverse osmosis is studied with a non-uniform arrangement of spacers, which indicate a substantial saving in pressure drop (∼ 40%) compared to traditional uniformly spaced pattern with minor variations (∼ 2%) in concentration polarization, product water quality (∼ 1%) and water recovery (∼ 7%) compared to a uniform spacer pattern.

A PLIC-based method for species mass transfer at free fluid interfaces

A finite volume based method to govern mass transfer of a diluted transition component across moving fluid–fluid interfaces is presented. The concentration jump at the interface is described by a constant distribution coefficient, while Marangoni effects and volume changes due to mass transfer are neglected. The interface position is captured by means of a level set function and subsequently reconstructed using a PLIC (piecewise linear interface construction) algorithm. For the discretisation of the species transport equation, cells containing the interface are split at the reconstructed planes. The concentration of the transition component is represented by two variables, each being valid in one phase only. The cell splitting and the definition of two concentration fields is essential to avoid artificial mass transfer due to discretisation errors. The interfacial boundary conditions are the concentration jump and the continuity of species fluxes. They are implemented in a robust and fast converging algorithm.The method has been validated for two liquid–liquid systems and one gas-liquid system by simulating a single moving droplet/bubble. The results are in good agreement with reference data, even for the gas–liquid system with thin liquid-side concentration boundary layers. In contrast to one-field formulations, this method is suitable for convection-dominated mass transport and significant movement of the interface due to the prevention of non-physical effects.

Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles

This paper presents a novel subgrid scale (SGS) model for simulating convection-dominated species transport at deformable fluid interfaces. One possible application is the Direct Numerical Simulation (DNS) of mass transfer from rising bubbles. The transport of a dissolving gas along the bubble–liquid interface is determined by two transport phenomena: convection in streamwise direction and diffusion in interface normal direction. The convective transport for technical bubble sizes is several orders of magnitude higher, leading to a thin concentration boundary layer around the bubble. A true DNS, fully resolving hydrodynamic and mass transfer length scales results in infeasible computational costs. Our approach is therefore a DNS of the flow field combined with a SGS model to compute the mass transfer between bubble and liquid. An appropriate model-function is used to compute the numerical fluxes on all cell faces of an interface cell. This allows to predict the mass transfer correctly even if the concentration boundary layer is fully contained in a single cell layer around the interface. We show that the SGS-model reduces the resolution requirements at the interface by a factor of ten and more. The integral flux correction is also applicable to other thin boundary layer problems. Two flow regimes are investigated to validate the model. A semi-analytical solution for creeping flow is used to assess local and global mass transfer quantities. For higher Reynolds numbers ranging from Re=100 to Re=460 and Péclet numbers between Pe=104 and Pe=4⋅106 we compare the global Sherwood number against correlations from literature. In terms of accuracy, the predicted mass transfer never deviates more than 4% from the reference values.

Explicit Radial‐Basis‐Function‐Based Finite‐Difference Method for Interfacial Mass‐Transfer Problems

AbstractThe development of an efficient and accurate numerical method, able to capture the very thin concentration boundary layer in cases of interfacial mass transfer at fluid interfaces in two‐phase systems, is a challenging task. In this study, a mesh‐less finite‐difference method based on radial basis functions has been adopted. The use of radial basis functions has proven to be helpful in multidimensional problems with complex geometries, in which the use of scattered sets of nodes is preferable. Unlike most of the applications reported in literature, the proposed method has been employed in two‐phase flows, with two regions connected by a surface, i.e., the fluid interface, on which additional boundary conditions have to be fulfilled. Two validation test cases with increasing complexity are presented.

Boundary layers at a dynamic interface: Air‐sea exchange of heat and mass

AbstractExchange of mass or heat across a turbulent liquid‐gas interface is a problem of critical interest, especially in air‐sea transfer of natural and anthropogenic gases involved in the study of climate. The goal in this research area is to determine the gas flux from air to sea or vice versa. For sparingly soluble nonreactive gases, this is controlled by liquid phase turbulent velocity fluctuations that act on the thin species concentration boundary layer on the liquid side of the interface. If the fluctuations in surface‐normal velocity and gas concentration are known, then it is possible to determine the turbulent contribution to the gas flux. However, there is no suitable fundamental direct approach in the general case where neither nor can be easily measured. A new approach is presented to deduce key aspects about the near‐surface turbulent motions from measurements that can be taken by an infrared (IR) camera. An equation is derived with inputs being the surface temperature and heat flux, and a solution method developed for the surface‐normal strain experienced over time by boundary layers at the interface. Because the thermal and concentration boundary layers experience the same near‐surface fluid motions, the solution for the surface‐normal strain determines the gas flux or gas transfer velocity. Examples illustrate the approach in the cases of complete surface renewal, partial surface renewal, and insolation. The prospects for use of the approach in flows characterized by sheared interfaces or rapid boundary layer straining are explored.

Effects of solute concentration in liquid on pore shape in solid

The effects of initial solute gas concentration in the liquid on the shape of a pore, resulting from a nucleated bubble entrapped by a solidification front, are predicted in this work. Solute concentration in liquid is responsible for solute transfer across the bubble cap, gas pressure in the pore, and shapes of the cap and pore in solid. Distributions and shapes of pores in solids influence not only microstructure of materials, but also contemporary issues of biology, engineering, foods, geophysics and climate change, etc. In this work, the relevant pore shape delineated by tracing contact angle of the cap to a first approximation is determined by accounting for mass and momentum transport across a self-consistent shape of the cap whose surface is satisfied by physico-chemical equilibrium, as proposed previously. This work finds that there exist three different mechanisms for pore formation, depending on directions and magnitude of solute transfer across the cap. Case 1 is subject to solute transport from the pore into surrounding liquid as a result of the cap emerged from a thin concentration boundary layer on the solidification front. An increase in initial solute concentration in liquid decreases pore radius and times for bubble entrapment. Opposite directions of solute transport across the cap submerged into a thick concentration boundary layer along the solidification front, however, cannot result in bubble entrapment, because solute concentration at the cap increases and decreases rapidly in late stage in Cases 2a and 2b, respectively. The predicted pore shape in solid agrees with experimental data. The pore shape therefore can be controlled by initial solute concentration to change directions and magnitudes of solute transport across the cap.

Lagrangian wall shear stress structures and near-wall transport in high-Schmidt-number aneurysmal flows

The wall shear stress (WSS) vector field provides a signature for near-wall convective transport, and can be scaled to obtain a first-order approximation of the near-wall fluid velocity. The near-wall flow field governs mass transfer problems in convection-dominated open flows with high Schmidt number, in which case a flux at the wall will lead to a thin concentration boundary layer. Such near-wall transport is of particular interest in cardiovascular flows whereby haemodynamics can initiate and progress biological events at the vessel wall. In this study we consider mass transfer processes in pulsatile blood flow of abdominal aortic aneurysms resulting from complex WSS patterns. Specifically, the Lagrangian surface transport of a species released at the vessel wall was advected in forward and backward time based on the near-wall velocity field. Exposure time and residence time measures were defined to quantify accumulation of trajectories, as well as the time required to escape the near-wall domain. The effect of diffusion and normal velocity was investigated. The trajectories induced by the WSS vector field were observed to form attracting and repelling coherent structures that delineated species distribution inside the boundary layer consistent with exposure and residence time measures. The results indicate that Lagrangian WSS structures can provide a template for near-wall transport.

A new analysis of fine aerosol capture by raindrops at terminal velocities

The diffusional capture of submicron aerosols by a falling raindrop at its terminal velocity is theoretically analyzed by employing the thin concentration boundary layer theory. The flow fields around the raindrop are assumed to be viscous and potential flows for small and large drops, respectively. The mass transfer rate of submicron aerosols at the front stagnation point of the raindrop is determined by using the boundary layer expansion technique in the thin concentration boundary layer. Comparing the present results with experimental data and analytical results in previous studies, new empirical correlations are regressed and recommended. The collection efficiency of submicron aerosols by either a large or a small raindrop is proportional to Pe−1/2, and interception effect is negligible. The expressions are different from the earlier results for the same or similar problems, e.g., deposition on a hard spherical collector and the below-cloud scavenging process. In addition, the results show that the velocity components of both viscous and potential flows can be expressed as Taylor series with acceptable accuracies in the concentration boundary layer near the front stagnation point of the raindrop.

A calibrated hydrogen-peroxide direct-borohydride fuel cell model

A numerical model with global reaction rates is calibrated to measurements from a simple hydrogen-peroxide direct-borohydride fuel cell (H2O2-DBFC), and then used to unravel complex electrochemical and competing parasitic reactions. In this H2O2-DBFC, fuel (1–50 mM NaBH4/2 M NaOH) is oxidized at a Au anode and oxidizer (10–40 mM H2O2/1 M H2SO4) is reduced at a Pd:Ir cathode. Polarization curves and electrode potentials, as functions of fuel and oxidizer feeds support global reaction rate parameter fitting. The measurements and calibrated model showed H2O2 decomposition at the cathode depresses open circuit voltage from 3.01 V theoretical to 1.65 V, and when H2O2 supply is limited, cathode potentials are sufficiently negative to make H+ reduction to H2 thermodynamically favorable. Calibrated model results show that thin concentration boundary layers limit reactant utilization and current density. Decreasing the inlet concentrations, flow rates, and cell voltage slow parasitic reactions and favor desirable charge transfer reactions. Peak conversion efficiency and peak power density coincide because thermodynamic efficiency and parasitic reaction rates decrease (relative to charge transfer reaction rates) with increasing current density. We conclude that the performance of a fuel cell with parasitic side reactions can be predicted through numerical modeling.

Simulations of solid–liquid mass transfer in fixed and fluidized beds

Abstract Solid-to-liquid mass transfer as a result of liquid flow through static and dynamic assemblies of mono-sized solid spheres has been simulated. The three-dimensional, transient simulations fully resolve the flow of liquid in the interstitial space between the spheres by means of a lattice-Boltzmann scheme. For the mass transfer process, a finite volume method on Coupled Overlapping Domains (CODs) is used. Spherical grids around the particles allow for resolving the thin scalar concentration boundary layers that are the result of high Schmidt numbers (Sc up to 1000). The spherical grids – one for each sphere – are coupled to an outer cubic grid. Particle-based Reynolds numbers are in the range 1–10; solids volume fractions are 0.15–0.40. The results demonstrate the validity of the COD approach for multi-particle systems, and provide insight in the dependencies of the Sherwood number on the solid–liquid system characteristics. It is shown that for moderate solid-over-liquid density ratios, fluidized particle assemblies have lower Sherwood numbers than fixed beds.

Effects of shape oscillation on mass transfer from a Taylor bubble

Interface tracking simulations of mass transfer from Taylor bubbles were carried out to investigate effects of shape oscillation on the mass transfer. Mass transfer from carbon dioxide Taylor bubbles in a glycerol-water solution was also measured to obtain experimental data for the validation of the numerical method. A high spatial resolution was used to resolve thin concentration boundary layers on the bubble interface, which enabled us to capture the agitation of boundary layer due to interfacial waves. The predicted mass transfer coefficients were in good agreements with the experimental data, provided that the spatial resolution was high enough to capture thin concentration boundary layers of high Schmidt number Taylor bubbles. The simulations have made it clear that the effect of agitation of concentration boundary layer due to interfacial waves on the total mass transfer rate is small and a fluctuation of the Sherwood number is caused by a fluctuation of the bubble surface area. This result implies that interface tracking methods using boundary layer approximations can give accurate predictions for mass transfer from bubbles even when the flow field is not fully resolved when the wave effect on the net mass transfer is negligible.

Concentration boundary layer visualization in nanofiltration by holographic interferometry with light deflection correction

Light deflection effects in thin concentration boundary layers adjacent to nanofiltration (NF) or reverse osmosis (RO) membranes are an important source of error associated to the measurement of concentration profiles by holographic interferometry (HI). In this work, a numerical method was developed that corrects the light deflection and minimizes this error. A classical HI setup was used to visualize the concentration boundary layer in a NF cell with a rectangular narrow channel (2mm height and 15mm width), in laminar flow using a He-Ne laser. Experiments were performed with binary aqueous solutions of K2SO4, glucose and sucrose, for two different concentrations (2 and 4kg/L) and pressures (5 and 8bar) and with average velocities ranging from 0.15 to 2.5cm/s. A ray tracing numerical method was used to simulate the interferogram and the concentration profile was estimated by minimizing the difference between the predicted and experimental interferograms. The concentration profile estimated for each experimental condition was compared with the theoretical one obtained by computational fluid dynamics. The results indicate that light deflection can be neglected if the ratio of the difference of the refractive index between the membrane surface and the bulk of the aqueous solution, Δnbl, to the boundary layer thickness, δ is below 1m−1. For Δnbl/δ>2m−1, light deflection correction is required for quantification purposes. HI with light deflection correction can measure concentration profiles in 2D concentration boundary layers with a thickness of the order of 100μm or higher.

Interface Tracking Simulation of Mass Transfer from a Dissolving Bubble

An interface tracking method for predicting bubble dissolution process is proposed. A non-diffusive scheme for advecting species concentrations is adopted to accurately compute the volume change due to mass transfer. The applicability of the proposed method is examined through several benchmark tests, i.e. mass transfer from stationary bubbles and that from free rising bubbles. Dissolution of single carbon dioxide bubbles rising through a vertical pipe filled with water is also simulated to examine its applicability to high bubble Reynolds number and high Schmidt number conditions. The bubble initially consists of carbon dioxide only, whereas nitrogen and oxygen are dissolved in water. The volume change due to dissolution of carbon dioxide from the bubble and evaporation of nitrogen and oxygen from water are also accounted for. A high spatial resolution is used in this simulation to capture a thin concentration boundary layer in the vicinity of the bubble surface. As a result the following conclusions are obtained: (1) the proposed two-variable method yields no artificial diffusion of species concentrations through the interface and well conserves the species moles, (2) the method gives accurate predictions for the concentration profiles about the stationary dissolving bubbles, (3) the mass transfer rate from bubbles at intermediate Reynolds numbers and a low Schmidt number is well predicted, and (4) the method gives good predictions for the mass transfer from the high Schmidt and high Reynolds number bubbles containing multiple gaseous species and for the volume change due to the mass transfer, provided that the spatial resolution is high enough to capture thin concentration boundary layers.

Uptake of soluble gaseous pollutants by rain droplets in the atmosphere with nocturnal temperature profile

We analyze the uptake of gaseous pollutants by the rain droplets falling in the atmosphere with nocturnal temperature inversion. The rate of uptake of soluble trace gases by falling rain droplets is determined by solving energy and mass conservation equations. In the analysis we accounted for the accumulation of the soluble gas and energy in the bulk of the falling rain droplet. The problem is solved in the approximation of a thin concentration and temperature boundary layers in the vicinity of the droplet surface. It is assumed that the bulk of a droplet, beyond the diffusion boundary layer, is completely mixed and distributions of concentration of the absorbate and temperature are homogeneous and time-dependent in the bulk. The problem is reduced to a system of linear-convolution Volterra integral equations of the second kind which is solved numerically. Calculations are performed using available experimental data on nocturnal temperature profiles in the atmosphere. It is shown than if the concentration of gaseous pollutants in the atmosphere is homogeneous and the absolute temperature in the atmosphere increases with altitude, a droplet absorbs gas during all the period of its fall. In the case when the temperature–altitude curve comprises the nocturnal inversion and temperature fall segments, gas absorption by a falling rain droplet can be replaced by desorption and vice versa. Neglecting temperature inhomogeneity in the atmosphere caused by nocturnal temperature profile leads to significant underestimation of the concentration of gaseous pollutants inside a droplet on the ground. The calculations performed using temperature profiles measured by Corsmeier et al. (1997) showed that the underestimation of the concentration of gaseous pollutants in rain droplets at the ground can exceed 20%.

Scavenging of soluble trace gases by falling rain droplets in inhomogeneous atmosphere

We analyze non-isothermal absorption of trace gases by the rain droplets with internal circulation which is caused by interfacial shear stresses. It is assumed that the concentration of soluble trace gases and temperature in the atmosphere varies in a vertical direction. The rate of scavenging of soluble trace gases by falling rain droplets is determined by solving heat and mass transfer equations. In the analysis we accounted for the accumulation of the absorbate in the bulk of the falling rain droplet. The problem is solved in the approximation of a thin concentration and temperature boundary layers in the droplet and in the surrounding air. We assumed that the bulk of a droplet, beyond the diffusion boundary layer, is completely mixed and concentration of the absorbate and temperature are homogeneous and time-dependent in the bulk. By combining the generalized similarity transformation method with Duhamel's theorem, the system of transient conjugate equations of convective diffusion and energy conservation for absorbate transport in liquid and gaseous phases with time-dependent boundary conditions is reduced to a system of linear convolution Volterra integral equations of the second kind which is solved numerically. Calculations are performed using available experimental data on concentration and temperature profiles in the atmosphere. It is shown than if concentration of a trace gas in the atmosphere is homogeneous and temperature in the atmosphere decreases with height, beginning from some altitude gas absorption is replaced by gas desorption. Neglecting temperature inhomogenity in the atmosphere described by adiabatic lapse rate leads to essential overestimation of the trace gas concentration in a droplet on the ground.

Effect of Altitude Concentration Gradient of Soluble Gaseous Pollutants on Their Scavenging by Falling Rain Droplets

Abstract This paper analyzes absorption of soluble atmospheric trace gases by falling rain droplets with internal circulation, which is caused by interfacial shear stresses. It is assumed that the concentration of soluble trace gases in the atmosphere varies in a vertical direction. In the analysis the accumulation of the absorbate in the bulk of the falling rain droplet was accounted for. The problem is solved in the approximation of a thin concentration boundary layer in the droplet and in the surrounding air. It was assumed that the bulk of a droplet, beyond the diffusion boundary layer, is completely mixed and that concentration of the absorbate is homogeneous and time dependent in the bulk. By combining the generalized similarity transformation method with Duhamel’s theorem, the system of transient conjugate equations of convective diffusion for absorbate transport in liquid and gaseous phases with time-dependent boundary conditions is reduced to a linear-convolution Volterra integral equation of the second kind, which is solved numerically. It is shown that the nonuniform vertical distribution of absorbate in a gaseous phase strongly affects mass transfer during gas absorption by a falling droplet.

Mass transfer and concentration contours between an oblate spheroid buried in granular beds and a flowing fluid

The present work describes the mass transfer process between a moving fluid and a soluble oblate spheroid buried in a packed bed of small inert particles with uniform voidage. A general solution of this problem is not available analytically and in the present work we only consider the limiting cases of mass transfer with a thin concentration boundary layer, to which corresponds high values of Peclet number. The solution of this problem is useful in the analysis of a variety of physical situations, such as the analytical models of continuous injection of solute at a point source, in a uniform stream, to estimate the distance from the “contaminant source” beyond which the levels of contaminant are expected to fall below some safe limit. The obtained analytical solution, Eq. (20), describes the mass transfer between a large soluble oblate spheroid and the fluid flowing around it in the interstices of a packed bed, for high values of Peclet number. This expression was also used to obtain the concentration contour surfaces using the well-known analytical solution of continuous injection of solute at a point source.

Rotating Disk Flow in Electrochemical Cells: A Coupled Solution for Hydrodynamic and Mass Equations

This work deals with the steady-state solution of a rotating disk flow, coupled, through the fluid viscosity, to the mass-concentration field of chemical species. The configuration refers to electrochemical cells where the working electrode consists of an iron rotating rod which is dissolved into the electrolyte, a sulfuric acid solution. Dissolution of the electrode gives rise to a thin concentration boundary layer, which, together with the potential applied to the electrode, results in an increase in the fluid viscosity and a decrease in the diffusion coefficient close to the electrode surface, both affecting the current. A phenomenological law is assumed, relating the fluid viscosity to the concentration of relevant chemical species. Parameters appearing in this law are evaluated based on experimental electrochemical data. The steady-state solution is obtained by solving the coupled hydrodynamic and mass-concentration equations.