Fluid systems convective mass transfer

One of the simplest models for convective mass transfer is the stirred tank model, also called the continuously stirred tank reactor (CSTR) or the mixing tank. The model is shown schematically in Figure 2. As shown in the figure, a fluid stream enters a filled vessel that is stirred with an impeller, then exits the vessel through an outlet port. The stirred tank represents an idealization of mixing behavior in convective systems, in which incoming fluid streams are instantly and completely mixed with the system contents. To illustrate this, consider the case in which the inlet stream contains a water-miscible blue dye and the tank is initially filled with pure water. At time zero, the inlet valve is opened, allowing the dye to enter the... 

Figure 3 The plug flow model for convective mass transfer. The drawing shows a tube of diameter D and length L. Discrete fluid plugs (shaded rectangles) move down the tube fluid within each plug is completely mixed, while there is no mixing between adjacent plugs. The system is a cylindrical section of the tube between z + z and Az and is fixed in space. Fluid enters the system at z with density p(z) and volumetric flow rate q(z) fluid exits the system at z + Az with density p(z + Az) and volumetric flow rate q(z, + Az).

In this text, the conversion rate is used in relevant equations to avoid difficulties in applying the correct sign to the reaction rate in material balances. Note that the chemical conversion rate is not identical to the chemical reaction rate. The chemical reaction rate only reflects the chemical kinetics of the system, that is, the conversion rate measured under such conditions that it is not influenced by physical transport (diffusion and convective mass transfer) of reactants toward the reaction site or of product away from it. The reaction rate generally depends only on the composition of the reaction mixture, its temperature and pressure, and the properties of the catalyst. The conversion rate, in addition, can be influenced by the conditions of flow, mixing, and mass and heat transfer in the reaction system. For homogeneous reactions that proceed slowly with respect to potential physical transport, the conversion rate approximates the reaction rate. In contrast, for homogeneous reactions in poorly mixed fluids and for relatively rapid heterogeneous reactions, physical transport phenomena may reduce the conversion rate. In this case, the conversion rate is lower than the reaction rate. 

Next, a mathematical model that allows description of the separation and concentration of the components of a metallic mixture will be detailed the principal assumptions of the model are (1) convective mass transfer dominates diffusive mass transfer in the fluid flowing inside the HFs, (2) the resistance in the membrane dominates the overall mass transport resistance, therefore the overall mass transfer coefficient was set equal to the mass transfer coefficient across the membrane, and (3) chemical reactions between ionic species are sufficiently fast to ignore the contribution of the chemical reaction rates. Thus, the reacting species are present in equilibrium concentration at the interface everywhere [31,32,58,59]. For systems working under nonsteady state, it is also necessary to describe the change in the solute concentration with time both in the modules and in the reservoir tanks. The reservoir tanks will be modeled as ideal stirred tanks. 

We saw above that the concentration gradient at an electrode will be linear with respect to the spatial coordinate perpendicular to the electrode surface if the anode/cathode cell were operated at a constant current density and if the fluid velocity were zero. In actuality, there will always be some bulk liquid electrolyte stirring during current flow, either an imposed forced convection velocity or a natural convection fluid motion due to changes in the reacting species concentration and fluid density near the electrode surface. In electrochemical systems with fluid flow, the mass transfer and hydrodynamic fluid flow equations are coupled and the solution of the relevant differential equations is often a formidable task, involving complex mathematical and/or numerical solution techniques. The concept of a stagnant diffusion layer or Nemst layer parallel and adjacent to the electrode surface is often used to simplify the analysis of convective mass transfer in... 

At low Re, the viscous effects dominate inertial effects and a completely laminar flow occurs. In the laminar flow system, fluid streams flow parallel to each other and the velocity at any location within the fluid stream is invariant with time when boundary conditions are constant. This implies that convective mass transfer occurs only in the direction of the fluid flow, and mixing can be achieved only by molecular diffusion [37]. By contrast, at high Re the opposite is true. The flow is dominated by inertial forces and characterized by a turbulent flow. In a turbulent flow, the fluid exhibits motion that is random in both space and time, and there are convective mass transports in all directions [38]. 

Ordinary diffusion involves molecular mixing caused by the random motion of molecules. It is much more pronounced in gases and Hquids than in soHds. The effects of diffusion in fluids are also greatly affected by convection or turbulence. These phenomena are involved in mass-transfer processes, and therefore in separation processes (see Mass transfer Separation systems synthesis). In chemical engineering, the term diffusional unit operations normally refers to the separation processes in which mass is transferred from one phase to another, often across a fluid interface, and in which diffusion is considered to be the rate-controlling mechanism. Thus, the standard unit operations such as distillation (qv), drying (qv), and the sorption processes, as well as the less conventional separation processes, are usually classified under this heading (see Absorption Adsorption Adsorption, gas separation Adsorption, liquid separation). 

This chapter will focus on three types of membrane extracorporeal devices, hemodialyzers, plasma filters for fractionating blood components, and artificial liver systems. These applications share the same physical principles of mass transfer by diffusion and convection across a microfiltration or ultrafiltration membrane (Figure 18.1). A considerable amount of research and development has been undertaken by membrane and modules manufacturers for producing more biocompatible and permeable membranes, while improving modules performance by optimizing their internal fluid mechanics and their geometry. 

The hydrodynamic conditions influence the concentration distribution explicity through the velocity term present in the convective diffusion equation. For certain well-defined systems the fluid flow equations have been solved, but for many systems, especially those with turbulent flow, explicit solutions have not been obtained. Consequently, approximate techniques must frequently be used in treating mass transfer. 

The plug-flow model indicates that the fluid velocity profile is plug shaped, that is, is uniform at all radial positions, fact which normally involves turbulent flow conditions, such that the fluid constituents are well-mixed [99], Additionally, it is considered that the fixed-bed adsorption reactor is packed randomly with adsorbent particles that are fresh or have just been regenerated [103], Moreover, in this adsorption separation process, a rate process and a thermodynamic equilibrium take place, where individual parts of the system react so fast that for practical purposes local equilibrium can be assumed [99], Clearly, the adsorption process is supposed to be very fast relative to the convection and diffusion effects consequently, local equilibrium will exist close to the adsorbent beads [2,103], Further assumptions are that no chemical reactions takes place in the column and that only mass transfer by convection is important. 

Mass transfer can result from several different phenomena. There is a mass transfer associated with convection in that mass is transported from one place to another in the flow system. This type of mass transfer occurs on a macroscopic level and is usually treated in the subject of fluid mechanics. When a mixture of gases or liquids is contained such that there exists a concentration gradient of one or more of the constituents across the system, there will be a mass transfer on a microscopic level as the result of diffusion from regions of high concentration to regions of low concentration. In this chapter we are primarily concerned with some of the simple relations which may be used to calculate mass diffusion and their relation to heat transfer. Nevertheless, one must remember that the general subject of mass transfer encompasses both mass diffusion on a molecular scale and the bulk mass transport, which may result from a convection process. 

The mass transfer coefficient (3C with SI units of m/s or m3/(sm2) is defined using these equations. It is a measure of the volumetric flow transferred per area. The concentration difference Aca defines the mass transfer coefficient. A useful choice of the decisive concentration difference for mass transfer has to be made. A good example of this is for mass transfer in a liquid film, see Fig. 1.41 where the concentration difference cA0 — cM between the wall and the surface of the film would be a a suitable choice. The mass transfer coefficient is generally dependent on the type of flow, whether it is laminar or turbulent, the physical properties of the material, the geometry of the system and also fairly often the concentration difference Aca. When a fluid flows over a quiescent surface, with which a substance will be exchanged, a thin layer develops close to the surface. In this layer the flow velocity is small and drops to zero at the surface. Therefore close to the surface the convective part of mass transfer is very low and the diffusive part, which is often decisive in mass transfer, dominates. 

Mass transfer to a particle in a translational flow, considered in Section 4.4, is a good model for many actual processes in disperse systems in which the velocity of the translational motion of particles relative to fluid plays the main role in convective transfer and the gradient of the nonperturbed velocity field can be neglected. 

The primary goal in using perfusion based microfluidic cell culture chips is to achieve a high degree of control over the microenvironment that the cultured cells are exposed to, i.e. to create a microenvironment that resembles the one cells are exposed to in vivo. In order to illustrate the difference between traditional culture vessels and perfusion based microfluidic cell culture chips, an effective culture volume (ECV) has been proposed as a descriptive concept [4]. ECV is a combined function of the characteristic mode of mass transfer (convection or diffusion), the magnitude of the mass transfer in all directions (x, y, z) in space (Fig. 2) as well as the extent of protein adsorption to the surfaces in a system. In vivo systems, e.g. tissue as part of an organ, are characterized by a fluid volume that is comparable to the volume of the cells in the tissues as well as by diffusive mass transfer. Based on these features the ECV of in vivo systems is small. 

The transfer of heat in a fluid may be brought about by conduction, convection, diffusion, and radiation. In this section we shall consider the transfer of heat in fluids by conduction alone. The transfer of heat by convection does not give rise to any new transport property. It is discussed in Section 3.2 in connection with the equations of change and, in particular, in connection with the energy transport in a system resulting from work and heat added to the fluid system. Heat transfer can also take place because of the interdiffusion of various species. As with convection this phenomenon does not introduce any new transport property. It is present only in mixtures of fluids and is therefore properly discussed in connection with mass diffusion in multicomponent mixtures. The transport of heat by radiation may be ascribed to a photon gas, and a close analogy exists between such radiative transfer processes and molecular transport of heat, particularly in optically dense media. However, our primary concern is with liquid flows, so we do not consider radiative transfer because of its limited role in such systems. 

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