Laminar flow in channels with wall suction or injection: a new model to study multi-channel filtration systems

A one-dimensional model to determine the laminar flow of a fluid in a porous channel with wall suction or injection is proposed. The approach is based on the integration of the Navier-Stokes equations using the analytical solutions for the two-dimensional local velocity and pressure fields obtained from the asymptotic developments at low filtration Reynolds number proposed by Berman (J. Appl. Phys. 24 (1953) 1232) and Yuan and Finkelstein (Trans. ASME 74 (1956) 719). It is noticeable that the resulting one-dimensional model preserves the whole flow properties, in particular the inertial terms which can affect the wall suction conditions and the spatial distribution of the growing particle cake layer at the wall encountered in filtration processes. The model is validated in the case of a single porous channel of rectangular or circular cross-section with uniform or variable wall suction. Then the model is applied to a two-dimensional multi-channel system which consists of a great number of adjacent entrance and exit channels connected by a filter porous medium. It is shown that the effect of non-uniform boundary conditions and the influence of heterogeneous geometrical characteristics on the heterogeneity of the fluid flow structure can be studied using such a model.     

Finite element modelling of concentration profiles in flow domains with curved porous boundaries

Domains containing porous boundaries are a feature of important processes such as crossflow filtration commonly used in a variety of industries ranging from medical to oil and gas to water treatment. Prediction of concentration profiles of solids being carried by the fluid stream in these types of flow domains is a necessary step in the design of efficient systems used in such operations. The novel approach of the present paper is shown to provide a very convenient technique for simulating concentration profiles within geometrically complex domains. The main feature of this approach is the construction of a very simple technique for the handling of concentration boundary conditions along the porous walls. Therefore this model provides a necessary and useful step towards the development of a fast engineering technique for the quantitative analysis of crossflow filters. The developed algorithm is based on the Streamline Upwind Petrov Galerkin finite element scheme (SUPG) for the solution of flow and convective dispersion equations in two dimensional domains with solid and porous walls. Using iterative algorithms the model can take into account process dependent changes of physical parameters that inevitably occur as a portion of the fluid seeps through the porous wall in these domains. The numerical tests prove the stability of the developed scheme and show that the model can produce results that are consistent with theoretical considerations.     

CHEMICAL REACTION ON NON-LINEAR BOUNDARY LAYER FLOW OVER A POROUS WEDGE WITH VARIABLE STREAM CONDITIONS

An analysis is presented to investigate the effects of variable viscosity and thermal stratification on non-Darcy MHD mixed convective heat and mass transfer of a viscous, incompressible, and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform non-Darcian porous medium in order to allow for possible fluid wall suction or injection. The governing partial differential equations of the problem, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary layer analysis. Numerical calculations up to third-order level of truncation are carried out for different values of dimensionless parameters. The results are presented graphically, and the conclusion is drawn that the flow field and other quantities of physical interest are significantly influenced by these parameters. The results are compared with those known from the literature, and excellent agreement between the results is obtained.     

Thermophoresis and chemical reaction effects on non-Darcy MHD mixed convective heat and mass transfer past a porous wedge in the presence of variable stream condition

An analysis is presented to investigate the effect of thermophoresis particle deposition and variable viscosity on non-Darcy MHD mixed convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform non-Darcian porous medium in order to allow for possible fluid wall suction or injection. The governing partial differential equations of the problem, subjected to their boundary conditions, are solved numerically by applying an efficient solution scheme for local nonsimilarity boundary layer analysis. Numerical calculations are carried out for different values of dimensionless parameter in the problem and an analysis of the results obtained show that the flow field is influenced appreciably by the applied magnetic field. The results are compared with those known from the literature and excellent agreement between the results is obtained.     

FLUID DYNAMICS IN A TUBULAR MEMBRANE: THEORY AND EXPERIMENT

Measurements of trans-cartridge [axial] pressure drops for pure liquid flowing in a porous tube under all regimes of flow as a function of wall suction and axial flow rate are reported. At very low axial flow rates [Re_F≤ 1,000], low values of wall suction [Re_W≥ 0.25] have a minimal effect on the non-dimensional axial pressure drop. At very high axial flow rates [Re_F≤20,000], however, all values of wall suction have a minimal effect on the axial pressure drop, Wall suction, on the other hand, has its maximum effect on the axial pressure drop at intermediate axial flow rates [1,000 Re_F≥15,000]. It is in this range that most commercial membrane modules operate.

Starting with the equations of continuity and Navier-Stokes we have developed two relatively simple approximate analytical solutions of this problem. The first approach assumes an average constant wall flux and includes the effect of the inertial terms while the second approach accounts for axial pressure-dependent flux but neglects the inertial terms. Both analytical models are useful qualitatively. However, neither was able to predict performance accurately. The model with constant wall flux and inertia predicts quite well the pressure drop as a function of ReF for low suction (Rew = 0.25)Deviations between model and experiment increased with increasing Re_Eand Re_F. Results from the model with pressure-dependent wall flux but without inertia fluctuated widely about the measurements for increasing Rew. This model is useful only for very small Rew. When the experimental conditions were such that the parabolic profile was distorted [i.e., for Re_W≤ 0.5] the theories were invalid and were unable to predict the measurements accurately.     

A NEW CLASS OF SIMILARITY SOLUTIONS OF AN UNSTEADY ELECTRICALLY CONDUCTING FREE-FORCED CONVECTIVE FLOW IN A VERTICAL POROUS SURFACE WITH DUFOUR AND SORET EFFECTS

The 2-D unsteady magnetohydrodynamic free-forced convective boundary layer flow of a viscous incompressible fluid is studied numerically taking into account heat and mass transfer. The fluid is subjected to uniform heat and mass fluxes embedded in a porous medium by the presence of coupled Dufour and Soret effects. A new class of similarity equations has been obtained by introducing a time-dependent length scale and a corresponding similarity variable. The resulting equations are then integrated numerically using the Nachtsheim-Swigert shooting iteration technique along with the sixth-order Runge-Kutta integration scheme. By developing locally similar solutions of the fluid flow, the behavior of the velocity, temperature, and concentration fields as well as the rate of heat transfer, wall temperature gradient, rate of mass transfer, and skin friction coefficient have been investigated. The effects of Grashof number (Gr), modified Grashof number (Gm), combined effects of the porous and magnetic parameter (S), suction/injection parameter Fw, Brinkman number (Br), Soret number (Sr), and Dufour number (D_f) have been observed on the flow field and discussed.     

Numerical Simulation of the Lock-In Effect of a Fixed-Fixed Tube in Cross Flow

This paper deals with the development of a numerical calculation code that is able to simulate the three-dimensional flow through a heat exchanger tube bundle and therefore allows a coupled calculation of fluid-structure interaction between the flow and the tube bundle. The incompressible flow field is calculated by a Navier-Stokes solver using a first-order power law scheme, a SIMPLEC algorithm to calculate the pressure and velocity correction fields, and a line-by-line Gauss-Seidl tridiagonal algorithm to solve the linearised system of equations. The transient parts of the Navier-Stokes equations are discretised by a second-order forward finite differencing scheme. The turbulence is examined with the aid of a large-eddy turbulence model. The transient fluid forces acting on the tubes are calculated by integration of all local flow pressure values on the surfaces of the tubes. As an example a single fixed-fixed cylinder in a flow channel is considered using the structural calculation part already developed as well as the new flow field and flow forces subroutines. The time series of the tube's motion and the fluid forces acting on the tube are analysed by Fourier's transformation. The lock-in effect occurring when the vortex shedding frequency approaches the first natural frequency of the tube can be excellently demonstrated by varying the inflow velocity over a wide range of Reynolds numbers.     

湍流促进器强化微滤过程的实验与CFD模拟研究(英文)

This paper reports experimental and computational fluid dynamics(CFD) studies on the performance of microfiltration enhanced by a helical screw insert.The experimental results show that the use of turbulence pro-moter can improve the permeate flux of membrane in the crossflow microfiltration of calcium carbonate suspension,and flux improvement efficiency is strongly influenced by operation conditions.The energy consumption analysis indicates that the enhanced membrane system is more energy saving at higher feed concentrations.To explore the intrinsic mechanism of flux enhancement by a helical screw insert,three-dimensional CFD simulation of fluid flow was implemented.It reveals that hydrodynamic characteristics of fluid flow inside the channel are entirely changed by the turbulence promoter.The rotational flow pattern increases the scouring effect on the tube wall,reducing the particle deposition on the membrane surface.The absence of stagnant regions and high wall shear stress are respon-sible for the enhanced filtration performance.No secondary flow is generated in the channel,owing to the streamline shape of helical screw insert,so that the enhanced performance is achieved at relatively low energy consumption.     

Viscous flow in a channel partially filled with a porous medium and with wall suction

In this paper, we consider a coupled two-dimensional flow of a Newtonian fluid, both above and through a porous medium. In the fluid-only region, the two-dimensional flow field is governed by the Navier-Stokes equation. We consider the Brinkman-extended Darcy law relationship in the porous medium. Inertial terms are retained in the formulation and the interface conditions between the two domains are those as outlined by Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). It should be noted that these interface conditions are formulated with an empirical constant β that is unknown a priori. The model equations were solved using two independent methods. In the first method, we pose a similarity variable and reduce the governing equations to two, coupled, non-linear ordinary differential equations. In the second approach, the governing equations were re-posed as a one-domain problem, using the procedure outlined by Basu and Khalili (Phys. Fluids 11 (1999) 1031), so that the conditions at the interface need not be considered. The resulting equation was solved directly, in primitive variable form, using a finite volume formulation. This enabled us to determine β by comparing the resulting solutions.     

CFD Simulation and Modeling of Membrane‐Assisted Separation of Organic Compounds from Wastewater

The extraction of volatile organic compounds (VOC) from water in porous hollow fibers was simulated with toluene, a hazardous material. The system to be simulated included a VOC stream and air as stripping gas, which were contacted using a porous hollow‐fiber membrane contactor. To model the process, the contactor was considered as three compartments, including shell side, porous membrane, and tube side. The model equations were derived and solved using computational fluid dynamics of momentum and mass transfer in all zones of the contactor. The profiles of concentration and pressure were obtained for the VOC in the hollow fibers.     

动态膜的形成机理及其水处理性能研究

以陶瓷管为载体,对高岭土动态膜的形成及其污水处理性能进行了详细的研究.实验中采用错流过滤方式涂膜,考查了跨膜压差、涂膜液浓度、错流速度及涂制时间等对动态膜形成的影响.通过对实验数据分析可知,动态膜形成初期10～13 min,膜的形成过程可用标准过滤模型描述,在此期间颗粒堵塞载体膜管孔道,致使渗透液通量急剧减小;之后,膜的形成过程符合滤饼过滤模型,这一阶段颗粒主要在载体膜管内壁面沉积,渗透液通量缓慢下降直至基本稳定.制备的动态膜可用于处理城市污水厂二级出水,动态膜对浊度去除率基本上为100%,对COD也有一定的去除作用.     

Flow Characteristics of Gas Permeation through a PVF Porous Tube

PVF (polyvinyl formal) porous materials have attractive properties, such as noise attenuation, good structural integrity, thermal and chemical stability, high permeability and large specific surface area, for many flow‐through applications. Several characteristics of the porous material will have an impact on the permeability, and gas flow and diffusion. However, the shape and the design of the device may also have significant impact on the gas flow. A porous media model and Darcy‐Forchheimer principle were used as the basic theoretical frame. The unified governing equations were used to describe the compressible flow in and out of a PVF porous tube. A robust NND numerical scheme was used to discretize the equations and the TDBC (time‐dependent boundary conditions) were used to treat the nonreflective boundaries. Numerical simulations of an interior and exterior flow field of a PVF porous tube were completed. The detailed flow characteristics of the inner and outer flow fields of the tube were obtained. The velocity distribution of the outside of the tube compare very well with the experimental data.     

CFD-based optimization and design of multi-channel inorganic membrane tubes☆

As a major configuration of membrane elements,multi-channel porous inorganic membrane tubes were studied by means of theoretical analysis and simulation.Configuration optimization of a cylindrical 37-channel porous inorganic membrane tube was studied by increasing membrane filtration area and increasing permeation efficiency of inner channels.An optimal ratio of the channel diameter to the inter-channel distance was proposed so as to increase the total membrane filtration area of the membrane tube.The three-dimensional computational fluid dynamics(CFD) simulation was conducted to study the cross-flow permeation flow of pure water in the 37-channel ceramic membrane tube.A model combining Navier–Stokes equation with Darcy's law and the porous jump boundary conditions was applied.The relationship between permeation efficiency and channel locations,and the method for increasing the permeation efficiency of inner channels were proposed.Some novel multichannel membrane configurations with more permeate side channels were put forward and evaluated.     

Laminar flow with injection through a long dead‐end cylindrical porous tube: Application to a hollow fiber membrane

Effects of membrane length and hydraulic resistance on the steady‐state laminar flow of a fluid with injection in a dead‐end cylindrical porous tube have been investigated using the perturbation approach of the Navier‐Stokes and continuity equations. Analytic solutions of the dynamic equations, reduced to non linear differential equations, were obtained and applied to submerged, dead‐end hollow fiber membranes of large resistances and small wall Reynolds numbers. The velocity and pressure profiles were solved using the method of separation of variables as with physical properties of the membrane and fluid and the wall velocity at the dead end. The constant flux approximation was found to be valid only for a short membrane with a large hydraulic resistance. The constant permeability approximation used in this study is universal for a membrane of an arbitrary length, through which the fluid velocities and pressure increase exponentially from the dead end to the open end. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

甲烷水蒸气重整制氢反应及其影响因素的数值分析

本文通过建立包含动量、能量、质量以及化学反应的多物理场耦合数值模型, 以多孔介质模型表征催化剂层, 对工业转化炉管中的甲烷水蒸气重整制氢过程进行了详细分析。计算得到了转化炉管内甲烷重整过程反应物及产物气体的速度、温度及浓度场分布, 以此分析了甲烷重整制氢过程的反应特性, 并阐明了转化炉管的壁面温度、原料气入口水碳比以及入口速度对甲烷转化率的影响。结果表明:水蒸气重整在转化炉管的入口区域反应迅速, 沿着气体流动方向, 反应速率由于反应物浓度的不断降低而减小, 导致混合气体流动速度和温度也逐渐趋于稳定;水碳比和转化管壁面温度的增加以及原料气体入口流速的降低, 都会提高甲烷的转化率。本文所得到的结论对于优化实际生产中甲烷水蒸气重整制氢反应的工况条件具有一定的参考和借鉴意义。     

考虑局部非热平衡的流体层流横掠多孔介质中恒热流平板的传热分析

对流体层流横掠多孔介质中恒热流加热的平板,应用Brinkman-Forchheime-extended Darcy流动模型和流体与多孔介质之间局部非热平衡理论建立守恒方程组,应用数量级分析和积分法,得出了速度边界层厚度、热边界层厚度、壁面黏性摩擦系数和对流传热系数、流体与多孔介质之间局部温差的计算公式。结果表明,速度边界层与光板时明显不同,其在平板前端迅速增长,之后越来越平坦,趋于一个恒定值;而热边界层则沿着流动方向不断增长,类似于光板时的情况;局部的表面对流传热系数在平板前端达最大值,之后逐渐减小,也类似于光板时的情况;多孔介质与流体间的局部温差在平板前端达最大值,之后呈现沿着流动方向逐渐减小的变化趋势。     

Dynamics of the flow of a dense fluid in narrow pores

A new approach was employed for the theoretical study of a dense fluid flow (gas and liquid) in narrow pores. Calculations were made on the basis of the Navier-Stokes equations. The transfer coefficients and the equations of state were derived within the framework of the simplest molecular model, namely, the lattice gas model. This model takes into account the volume of molecules and their interaction with one another. The transfer coefficients and the equations of state prove to be dependent on the local values of the fluid density and temperature. A study was performed into the dynamic modes of flow of a monoatomic gas (argon) in slitlike pores of various widths at a given pressure drop across the pore ends. It was shown that, at low densities, there is a high anisotropy of the flow owing to the attraction of argon atoms to the pore walls, which gives rise to a strong dependence of the local viscosity on the distance from the pore wall and to the film flow. An investigation was carried out into the dynamics of the interaction between the gas and liquid zones, which is accompanied by the breakdown of the interface.     

A spatially resolved model for pressure filtration of edible fat slurries

A spatially resolved one dimensional pressure filtration model was developed for a slurry of edible fat crystals. The model focuses on the expression step in which a cake is compressed to force the liquid through a filter cloth. The model describes the local oil flow in the shrinking cake modeled as a porous nonlinear elastic medium existing of two phases, viz. porous aggregates and interaggregate liquid. Conservation equations lead to a set of two differential equations (vs. time and vs. a material coordinate ω) for two void ratios, which are solved numerically by exploiting a finite-difference scheme. A simulation with this model results in a spatially resolved cake composition and in the outflow velocity, both as a function of time, as well as the final solid fat contents of the cake. Simulation results for various filtration conditions are compared with experimental data collected in a pilot-plant scale filter press.     

Effect of Sub-Grid Scales on Large Eddy Simulation of Particle Deposition in a Turbulent Channel Flow

Large-eddy simulations (LES) of particle transport and deposition in turbulent channel flow were presented. Particular attention was given to the effect of subgrid scales on particle dispersion and deposition processes. A computational scheme for simulating the effect of subgrid scales (SGS) turbulence fluctuation on particle motion was developed and tested. Large-eddy simulation of Navier-Stokes equations using a finite volume method was used for finding instantaneous filtered fluid velocity fields of the continuous phase in the channel. Selective structure function model was used to account for the subgrid-scale Reynolds stresses. It was shown that the LES was capable of capturing the turbulence near wall coherent eddy structures.

The Lagrangian particle tracking approach was used and the transport and deposition of particles in the channel were analyzed. The drag, lift, Brownian, and gravity forces were included in the particle equation of motion. The Brownian force was simulated using a white noise stochastic process model. Effects of SGS of turbulence fluctuations on deposition rate of different size particles were studied. It was shown that the inclusion of the SGS turbulence fluctuations improves the model predictions for particle deposition rate especially for small particles. Effect of gravity on particle deposition was also investigated and it was shown that the gravity force in the stream wise direction increases the deposition rate of large particles.