Mass and heat transfer during solid sphere dissolution in a non-uniform fluid flow

We studied a nonisothermal dissolution of a solvable solid spherical particle in an axisymmetric non-uniform fluid flow when the concentration level of the solute in the solvent is finite (finite dilution of solute approximation). It is shown that simultaneous heat and mass transfer during solid sphere dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with a parabolic velocity profile can be described by a system of generalized equations of convective diffusion and energy. Solutions of diffusion and energy equations are obtained in an exact analytical form. Using a general solution the asymptotic solutions for heat and mass transfer problem during spherical solid particle dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with parabolic velocity profile are derived. Theoretical results are in compliance with the available experimental data on falling urea particles dissolution in water and for solid sphere dissolution in a shear flow.     

Equivalent Diffusion in a Thin Film Flow with a Non-One-Dimensional Velocity Field and Anisotropic Diffusion Tensor

For describing the mass transfer processes in channels, Taylor's dispersion theory is widely used. This theory makes it possible, with asymptotic rigor, to replace the complete diffusion (heat conduction) equation with a convective term that depends on the coordinate transverse to the flow by an effective diffusion (dispersion) equation with constant coefficients, averaged over the channel cross-section. In numerous subsequent studies, Taylor's theory was generalized to include more complex situations, and novel algorithms for constructing the dispersion equations were proposed. For thin film flows a theory similar to Taylor's leads to a matrix of dispersion coefficients.In this study, Taylor's theory is extended to film flows with a non-one-dimensional velocity field and anisotropic diffusion tensor. These characteristics also depend to a considerable extent on the spatial coordinates and time. The dispersion equations obtained can be simplified in regions in which the effective diffusion coefficient tensor changes sharply.     

Numerische Berechnung des Partikeltransportes zu einer laminar angeströmten Kreisscheibe

Increasingly process steps become important, in which particles as product particles or contaminants are deposited on substrates out of the gas phase. In this paper the particles transport processes are investigated close to the surface of a circular plate surrounded by a laminar flow. The analogy between the governing equations of momentum, energy and mass is applied to the extended diffusion equation. In the nondimensional form the results of the numerical calculations give informations about velocity, temperature and particle concentration boundary layer thickness as well as their distributions. Especially the impact of external forces on particle concentration boundary layer thickness and profile is discussed. The transport of submicron particles to the surface due to convection, diffusion, gravity and thermophoretic forces acting independently is investigated. In the used normalized form the different forces are acting as one resulting force independently of their origin. Their resulting effect in comparison to the effect due to convective diffusive transport is important for particle deposition.     

Soret and dufour effects on MHD free convective heat and mass transfer with thermophoresis and chemical reaction over a porous stretching surface: Group theory transformation

The group theoretic method is applied for solving the problem of the combined influence of the thermal diffusion and diffusion thermoeffect on magnetohydrodynamic free convective heat and mass transfer over a porous stretching surface in the presence of thermophoresis particle deposition with variable stream conditions. The application of one-parameter groups reduces the number of independent variables by one; consequently, the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The equations along with the boundary conditions are solved numerically by using the Runge-Kutta-Gill integration scheme with the shooting technique. The impact of the Soret and Dufour effects in the presence of thermophoresis particle deposition with a chemical reaction plays an important role on the flow field.     

Asymptotic form of the admixture transport equation for a channel flow with an anisotropic diffusion tensor

The problem of the asymptotically correct reduction of a 3-D mass (heat) transfer equation to a 1-D equation in a flow with anisotropic diffusion properties is considered. The convective mass (heat) transfer domain is a cylindrical channel of arbitrary cross section. The diffusion coefficient matrix is assumed to be independent of the spatial coordinates. In the equivalent diffusion equation constructed, a certain effective diffusion (dispersion [1]) coefficient is introduced. Formulas for this coefficient are obtained. A relation between the effective diffusion coefficient calculations and the problem of minimization of a certain functional is established, i. e. the possibility of calculations based on variational methods is noted. An example of an exact calculation of the effective diffusion coefficient is considered. The possibility of a generalization of the problem, in which the effective diffusion (heat conduction) equation is essentially a nonlinear equation of general form for the one-dimensional case, is indicated. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 110–123, March–April, 2000.     

Forced Convection Heat Transfer of Nanofluids in a Porous Channel

This article is concerned with the effects of flow and migration of nanoparticles on heat transfer in a straight channel occupied with a porous medium. Investigation of force convective heat transfer of nanofluids in a porous channel has not been considered completely in the literature and this challenge is generally considered to be an open research topic that may require more study. The fully developed flow and steady Darcy?CBrinkman?CForchheimer equation is employed in porous channel. The thermal equilibrium model is assumed between nanofluid and solid phases. It is assumed that the nanoparticles are distributed non-uniformly inside the channel. As a result the volume fraction distribution equation is also coupled with governing equations. The effects of parameters such as Lewis number, Schmidt number, Brownian diffusion, and thermophoresis on the heat transfer are completely studied. The results show that the local Nusselt number is decreased when the Lewis number is increased. It is observed that as the Schmidt number is increased, the wall temperature gradient is decreased and as a consequence the local Nusselt number is decreased. The effects of Lewis number, Schmidt number, and modified diffusivity ratio on the volume fraction distribution are also studied and discussed.     

Motion and heat and mass transfer in a disperse admixture in turbulent nonisothermal jets of a gas and a low-temperature plasma

The motion and heat and mass transfer of particles of a disperse admixture in nonisothermal jets of a gas and a low-temperature plasma are simulated with allowance for the migration mechanism of particle motion actuated by the turbophoresis force and the influence of turbulent fluctuations of the jet flow velocity on heat and mass transfer of the particle. The temperature distribution inside the particle at each time step is found by solving the equation of unsteady heat conduction. The laws of scattering of the admixture and the laws of melting and evaporation of an individual particle are studied, depending on the injection velocity and on the method of particle insertion into the jet flow. The calculated results are compared with data obtained with ignored influence of turbulent fluctuations on the motion and heat and mass transfer of the disperse phase. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 95–108, May–June, 2008.     

Numerical simulation of particle‐laden flows by the residual‐based variational multiscale method

We present a finite element residual‐based variational multiscale formulation applied to the numerical simulation of particle‐laden flows. We employ a Eulerian–Eulerian framework to describe the flows in which the mathematical model results from the incompressible Navier–Stokes equation combined with an advection–diffusion transport equation. Special boundary conditions at the bottom are introduced to take into account sediments deposition. Computational experiments are organized in two examples. The first example deals with the well‐known gravity current benchmark, the lock‐exchange configuration. The second also employs for the current initiation the lock configuration, but the sediment particles are endowed with a deposition velocity and are allowed to leave the domain in the moment they reach the bottom. This is intended to mimic, partially, as the bed morphology is not allowed to change, the deposition process, in which sediment deposits are no longer carried by the flow. The spatial pattern of the deposition and its correlation with flow structures are the main focus of this analysis. Numerical experiments have shown that the present formulation captures most of the relevant turbulent flow features with reasonable accuracy, when compared with highly resolved numerical simulations and experimental data. Copyright © 2013 John Wiley & Sons, Ltd.     

振荡潜流带沉积层?水界面污染物输运的研究

潜流带中污染物质交换与输运特性是影响水资源环境的重要问题之一. 本文对底部为高渗透沉积层的三维槽道振荡流高Schmidt数传质问题进行了大涡模拟研究. 采用动力学亚格子模型来封闭滤波后的三维不可压缩Navier-Stokes方程以及污染物输运方程, 同时采用修正的Darcy-Brinkman-Forcheimer模型来描述沉淀有锌离子污染溶质的可渗透沉积层. 通过对沉积层内外流场和浓度场的统计特性以及瞬态结构的分析, 探究了上覆水体中振荡流驱动作用对污染物输运的动力学影响以及扩散率随振荡周期和振荡角的变化规律. 研究结果表明, 浓度通量中的湍流浓度分量在垂向物质交换中起主导作用, 流向、展向速度, 湍流强度和污染物浓度的波动跟随振荡驱动力呈现准周期变化, 同时发现沉积层?水交界面处的湍流浓度通量与法向湍流强度两者之间的变化具有明确的相关性. 并且在较大振荡角和低频振荡的情况下, 沉积层?水交界面处的有效扩散率增大, 这主要是来自于沉积层?水交界面处流体的猝发行为有效促进了湍流混合和物质交换, 进而增强了污染物的垂向输运.      

Experimental study of critical heat flux enhancement during forced convective flow boiling of nanofluid on a short heated surface

Enhancements of nucleate boiling critical heat flux (CHF) using nanofluids in a pool boiling are well-known. Considering importance of flow boiling heat transfer in various practical applications, an experimental study on CHF enhancements of nanofluids under convective flow conditions was performed. A rectangular flow channel with 10-mm width and 5-mm height was used. A 10 mm-diameter disk-type copper surface, heated by conduction heat transfer, was placed at the bottom surface of the flow channel as a test heater. Aqueous nanofluids with alumina nanoparticles at the concentration of 0.01% by volume were investigated. The experimental results showed that the nanofluid flow boiling CHF was distinctly enhanced under the forced convective flow conditions compared to that in pure water. Subsequent to the boiling experiments, the heater surfaces were examined with scanning electron microscope and by measuring contact angle. The surface characterization results suggested that the flow boiling CHF enhancement in nanofluids is mostly caused by the nanoparticles deposition of the heater surface during vigorous boiling of nanofluids and the subsequent wettability enhancements.     

Mass transfer enhancement by longitudinal vortices

From numerical and experimental investigations it has been recently established that convective heat transfer can be dramatically enhanced by the generation of longitudinal vortices in the flow. The phenomenological similarity between heat and mass transfer suggests that longitudinal vortices should increase also mass transfer. The mixing between two parallel streams of two components in a rectangular channel with and without a pair of rectangular winglets as vortex generators has been numerically investigated. The results show that one pair of vortex generators can increase the global mixing by more than 50 for laminar flow. This global mixing has been defined as the sum of the square of the differences of concentrations.     

Effect of a Dispersed Admixture on the Flow Pattern and Heat Transfer in a Supersonic Dusty-Gas Flow Around a Cylinder

A mathematical model of two-phase (gas-solid particle) flow which takes into account particle-particle collisions and the feedback effect of the admixture on the gas parameters is proposed. The dispersed phase is described by a kinetic equation of the Boltzmann type and the carrier gas by modified Navier-Stokes equations. Using this model, a supersonic uniform dusty-gas flow past a cylinder is calculated. The fields of the macroparameters of the admixture and the carrier medium are obtained. The dependence of the heat transfer at the stagnation point on the relative particle size and the free-stream admixture concentration is studied in detail. The ranges of these parameters on which particle collisions and the feedback effect of the admixture on the carrier-gas flow are important are found.     

A generalized mass transfer law unifying various particle transport mechanisms in dilute dispersions

A generalized mass transfer law for dilute dispersion of particles (or droplets) of any sizes suspended in a fluid has been described, which can be applied to turbulent or laminar flow. The generalized law reduces to the Fick’s law of diffusion in the limit of very small particles. Thus the study shows how the well-known and much-used Fick’s law of diffusion fits into the broader context of particle transport. The general expression for particle flux comprises a diffusive flux due to Brownian motion and turbulent fluctuation, a diffusive flux due to temperature gradient (thermophoresis plus stressphoresis) and a convective flux that arises primarily due to the interaction of particle inertia and the inhomogeneity of the fluid turbulence field (turbophoresis). Shear-induced lift force, electrical force, gravity, etc. also contribute to the convective flux. The present study includes the effects of surface roughness, and the calculations show that the presence of small surface roughness even in the hydraulically smooth regime significantly enhances deposition especially of small particles. Thermophoresis can have equally strong effects, even with a modest temperature difference between the wall and the bulk fluid. For particles of the intermediate size range, turbophoresis, thermophoresis and roughness are all important contributors to the overall deposition rate. The paper includes a parametric study of the effects of electrostatic forces due to mirror charging. The present work provides a unified framework to determine the combined effect of various particle transport mechanisms on mass transfer rate and the inclusion of other mechanisms not considered in this paper is possible.     

Thermophoresis particle deposition effects in a free convective doubly stratified medium over a vertical plate

An analysis is performed to study the thermophoresis effects in a transient free convective flow of a viscous, incompressible fluid past an isothermal vertical plate in a doubly stratified medium. The governing boundary layer equations are solved numerically using an implicit finite difference scheme of Crank-Nicolson type. The influence of thermophoresis on particle deposition velocity and particle concentration in a doubly stratified medium are analyzed and illustrated graphically. As well the influence of thermal and mass stratification on velocity, temperature and concentration are also investigated and presented. The influence of the parameters on local as well as average skin-friction, the rate of heat and mass transfer are presented graphically and discussed. The results are compared with particular solutions available in the literature and are found to be in good agreement.     

Discretization of an admixture flux within the framework of the fractal model of anomalous diffusion

Within the framework of the fractal mobile-immobile medium model describing non-Fickian effects occurring in admixture seepage due to particle adhesion to the solid matrix, an expression for the admixture flux is derived. Flow discretization intended for finite-difference calculations is proposed and used as a basis for a conservation-law scheme for solving the model equations with account for admixture sources. Several one-dimensional test problems of admixture propagation in an imposed seepage flow are solved using the approach developed.     

Convective diffusion to a drop under arbitrary conditions of absorption. Diffusion boundary layer approximation

The diffusion boundary layer approximation is used to investigate the stationary convective diffusion of substance dissolved in a flow to a spherical drop under arbitrary conditions of absorption on its surface, in particular when a chemical reaction of arbitrary order takes place on the surface. An integral equation is obtained for the local diffusion flux to the surface of the drop. It is shown that 1) the total Sherwood number increases with increasing rate of the reaction and decreases with increasing exponent of the reaction rate; 2) with increasing Péclet number, saturation occurs (i.e., the total diffusion flux to the surface of the drop tends to a limiting value, which depends only on the reaction kinetics). The case of total absorption of diffusing, substance on the surface of reacting solid and liquid particles in a homogeneous Stokes flow at large Péclet numbers was investigated in [1]. The problem of convective diffusion to the surface of a solid spherical particle in the case of mixed kinetics was considered in [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 64–69, November–December, 1979.     

Asymptotic solutions of the problem of heat and mass transfer of particles in a fluid

Some questions related to asymptotic analysis (as P , where P is the Péclet number) of problems involving heat and mass transfer of particles in a fluid are considered. The first part of the paper investigates the stationary convective diffusion of a solute to a particle near its front critical point (incidence point). An explicit expression is obtained for the concentration in the region of the front critical point of a solid or liquid particle around which a Stokes flow occurs. In the second part of the paper, a unified formula is obtained for the concentration distribution behind the particle. Appropriate limits in this formula determine the concentration in the mixing region and the inner and convective boundary-layer regions of the diffusion wake. In the final part of the paper, a study is made of the diffusion to a chain of absorbing solid spheres of equal radius a at distances1, 1 1/a P^1/3, from each other on the axis of an oncoming Stokes flow; an integral equation is obtained for the local diffusion flux when a chemical reaction with arbitrary kinetics takes place on the surfaces of the spheres. A certain heterogeneity of the material in the paper is due to the investigation in it of various questions that arise in the solution of more general problems (see, for example, [1–14]) which have not been considered hitherto.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 1, pp. 89–96, January–February, 1979.     

A finite difference self-adaptive mesh solution of flow in a sedimentation tank

A new numerical model has been developed to evaluate the removal efficiency of primary sedimentation clarifiers operating at neutral density condition. The velocity and concentration fields as well as the development in time and space of the settled particle bed thickness are simulated. The main difficulties in simulation of velocity and concentration fields are related to (1) numerical instabilities produced by the prevalence of convective terms in the unknown variable high-gradient regions and (2) turbulence effects on the suspension of solid particles from the settled bed. The need to overcome the numerical instabilities without the upwind difference approximation, which introduces high numerical viscosity, suggests the use of non-uniform grids of calculation. The velocity field is obtained by solving the motion equations in the vorticity and streamfunction formulation by means of a new numerical method based upon a dynamically self-adjusting calculation grid. These grids allow for a finer mesh following the evolution of the unknown quantities. A k–? model is used to simulate turbulence phenomena. The sedimentation field is found by solving the diffusion and transport equation of the solid particle concentration. Boundary conditions on the bottom line are imposed relating the amount of turbulence flux and sedimentation flux to the actual concentration and the reference concentration. Such an approach makes it possible to represent the solid particle suspension from the bottom, taking into account its dependence on (1) the characteristics and the evolution in time of the settled bed, (2) the velocity component parallel to the bottom line and (3) the turbulence structure.