Mass Transfer Equation. Laminar Flows

Let us write out the main equations and boundary conditions used in the mathematical statement of physical and chemical hydrodynamic problems. More detailed derivations of these equations and boundary conditions, analysis of their scope, various physical models of numerous related problems, solution methods, as well as applications of the results, can be found in the books [35, 121, 159, 185, 199, 406], 

We assume that the medium density and viscosity are independent of concentration and temperature, and hence, the concentration and temperature distributions do not affect the flow field. This allows one to analyze the hydrodynamic problem about the fluid motion and the diffusion-heat problem of finding the concentration and temperature fields independently. (More complicated problems in which the flow field substantially depends on diffusion-heat factors will be considered later in Chapter 5.) It is assumed that the information about the fluid velocity field necessary for the solution of the diffusion-heat problem is known. We also assume that the diffusion and thermal conductivity coefficients are independent of concentration and temperature. For simplicity, we restrict our consideration to the case of two-component solutions. 

In Cartesian coordinates X,Y,Z, solute transfer in absence of homogeneous transformations is described by the equation 

Equation (3.1.1) reflects the fact that the transfer of a substance in a moving medium is due to two distinct physical mechanisms. First, there is molecular diffusion due to concentration difference in a liquid or gas, which tends to equalize the concentrations. Second, the solute is carried along by the moving medium. The combination of these two processes is usually called convective diffusion [133, 270], 

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The basic theory of mass transfer to a RHSE is similar to that of a RDE. In laminar flow, the limiting current densities on both electrodes are proportional to the square-root of rotational speed they differ only in the numerical values of a proportional constant in the mass transfer equations. Thus, the methods of application of a RHSE for electrochemical studies are identical to those of the RDE. The basic procedure involves a potential sweep measurement to determine a series of current density vs. electrode potential curves at various rotational speeds. The portion of the curves in the limiting current regime where the current is independent of the potential, may be used to determine the diffusivity or concentration of a diffusing ion in the electrolyte. The current-potential curves below the limiting current potentials are used for evaluating kinetic information of the electrode reaction. 

An exclusively analytical treatment of heat and mass transfer in turbulent flow in pipes fails because to date the turbulent shear stress Tl j = —Qw w p heat flux q = —Qcpw, T and also the turbulent diffusional flux j Ai = —gwcannot be investigated in a purely theoretical manner. Rather, we have to rely on experiments. In contrast to laminar flow, turbulent flow in pipes is both hydrodynamically and thermally fully developed after only a short distance x/d > 10 to 60, due to the intensive momentum exchange. This simplifies the representation of the heat and mass transfer coefficients by equations. Simple correlations, which are sufficiently accurate for the description of fully developed turbulent flow, can be found by... 

What important dimensionless number(s) appear in the dimensionless partial differential mass transfer equation for laminar flow through a blood capillary when the important rate processes are axial convection and radial diffusion ... 

The solution to this laminar boundary layer problem must satisfy conservation of species mass via the mass transfer equation and conservation of overall mass via the equation of continuity. The two equations have been simplified for (1) two-dimensional axisymmetric flow in spherical coordinates, (2) negligible tangential diffusion at high-mass-transfer Peclet numbers, and (3) negligible curvature for mass flux in the radial direction at high Schmidt numbers, where the mass transfer... 

Axial dispersion in packed beds, and Taylor dispersion of a tracer in a capillary tube, are described by the same form of the mass transfer equation. The Taylor dispersion problem, which was formulated in the early 1950s, corresponds to unsteady-state one-dimensional convection and two-dimensional diffusion of a tracer in a straight tube with circular cross section in the laminar flow regime. The microscopic form of the generalized mass transfer equation without chemical reaction is... 

There are more data for heat transfer in laminar flow than for mass transfer, and the correlations should be similar, with Pr and Nu replacing Sc and Sh. An empirical equation for heat transfer at Graetz numbers greater than 20 is [15]... 

The heat and mass transfer and fluid flow phenomena in the planar micro-SOFC is described by the CFD model. Due to the low gas velocity and small size of the SOFC, the Reynolds number in the micro-channel is usually much lower than 100 (Yuan et al., 2003). Thus, the gas flow in an SOFC is typically laminar. From a heat transfer analysis, it is found that the local thermal equilibrium assumption is valid for the porous electrodes of an SOFC (Zheng et al., 2013). The governing equations for the CFD model include mass conservation, momentum conservation, energy conservation, and species conservation (Wang, 2004) ... 

A model of a reaction process is a set of data and equations that is believed to represent the performance of a specific vessel configuration (mixed, plug flow, laminar, dispersed, and so on). The equations include the stoichiometric relations, rate equations, heat and material balances, and auxihaiy relations such as those of mass transfer, pressure variation, contac ting efficiency, residence time distribution, and so on. The data describe physical and thermodynamic properties and, in the ultimate analysis, economic factors. 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

In turbulent flow, the edge effect due to the shape of the support rod is quite significant as shown in Fig. 6. The data obtained with a support rod of equal radius agree with the theoretical prediction of Eq. (52). The point of transition with this geometry occurs at Re = 40000. However, the use of a larger radius support rod arbitrarily introduces an outflowing radial stream at the equator. The radial stream reduces the stability of the boundary layer, and the transition from laminar to turbulent flow occurs earlier at Re = 15000. Thus, the turbulent mass transfer data with the larger radius support rod deviate considerably from the theoretical prediction of Eq. (52) a least square fit of the data results in a 0.092 Re0 67 dependence for... 

Shah and Nelson [33] introduced a convective mass transport device in which fluid is introduced through one portal and creates shear over the dissolving surface as it travels in laminar flow to the exit portal. They demonstrated that this device produces expected fluid flow characteristics and yields mass transfer data for pharmaceutical solids which conform to convective diffusion equations. 

Considerable interest has been generated in turbulence promoters for both RO and UF. Equations 4 and 5 show considerable improvements in the mass-transfer coefficient when operating UF in turbulent flow. Of course the penalty in pressure drop incurred in a turbulent flow system is much higher than in laminar flow. Another way to increase the mass-transfer is by introducing turbulence promoters in laminar flow. This procedure is practiced extensively in enhanced heat-exchanger design and is now exploited in membrane hardware design. 

In Sect. 3.2, the development of the design equation for the tubular reactor with plug flow was based on the assumption that velocity and concentration gradients do not exist in the direction perpendiculeir to fluid flow. In industrial tubular reactors, turbulent flow is usually desirable since it is accompanied by effective heat and mass transfer and when turbulent flow takes place, the deviation from true plug flow is not great. However, especially in dealing with liquids of high viscosity, it may not be possible to achieve turbulent flow with a reasonable pressure drop and laminar flow must then be tolerated. 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

In the case where the fluid flow outside the tubes is parallel to the tubes and laminar (as in some membrane devices), the film coefficient of mass transfer on the outer tube surface can be estimated using Equation 6.26a and the equivalent diameter as calculated with Equation 5.10. 

Because of the very small fluid channels (Re is very small), the flows in microreactor systems are always laminar. Thus, mass and heat transfers occur solely by molecular diffusion and conduction, respectively. However, due to the very small transfer distances, the coefficients of mass and heat transfer are large. Usually, film coefficients of heat and mass transfer can be estimated using Equations 5.9b and 6.26b, respectively. 

For packed beds of naphthalene and caffeine, Lim et al. [28] took into consideration mass--transfer by both types of convection in opposite flows. Their equation for laminar flow in packed beds (10 to 203 bar and 35 to 45°C) is as follows ... 

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