Mass transfer coefficients variation with flow

These fluid-solid dimensionless correlations can conceal how the mass transfer coefficient varies with fluid flow v and diffusion coefficient D, just as those for fluid-fluid interfaces obscured these variations. Often k varies with the square root of v . The variation is lower for some laminar flows and higher for some turbulent flows. Usually, k is said to vary with though this variation is rarely checked carefully by those who develop the correlations. Variation of k with does have some theoretical basis, a point explored further in Chapter 9. 

Solution The answer in this case depends on how the mass transfer coefficient varies with fluid flow, or, in more general terms, on how the Sherwood number varies with the Reynolds number. This variation depends on the specific experimental situation. For the spinning disc described in Section 3.4,... 

For simplicity, this section discusses only the mass transfer of one component in a liquid-liquid system with negligible miscibility of both liquids and with one transitional component. On the other hand, calculations must consider mass transfer rates of several components and more or less strong variation in the mass flows along the column, where both complicate the equation considerably [21-23]. Chemical reactions may cause further complications. Their kinetics can enhance the mass transfer coefficients and, therefore, the reaction equations have to be part of the mathematical model of the extractor [24,25]. 

The Sherwood number can be determined from the solution of the nondimensional problem by evaluating the nondimensional mass-fraction gradients at the channel wall and the mean mass fraction, both of which vary along the channel wall. With the Sherwood number, as well as specific values of the mass flow rate, fluid properties, and the channel geometry, the mass transfer coefficient hk can be determined. This mass-transfer coefficient could be used to predict, for example, the variation in the mean mass fraction along the length of some particular channel flow. 

A fundamental shortcoming of the Chilton-Colburn approach for multicomponent mass transfer calculations is that the assumed dependence of [/ ] on [Sc] takes no account of the variations in the level of turbulence, embodied by r turb/, with variations in the flow conditions. The reduced distance y is a function of the Reynolds number y = (y/R )(//8) / Re consequently. Re affects the reduced mixing length defined by Eq. 10.2.21. An increase in the turbulence intensity should be reflected in a relative decrease in the influence of the molecular transport processes. So, for a given multicomponent mixture the increase in the Reynolds number should have the direct effect of reducing the effect of the phenomena of molecular diffusional coupling. That is, the ratios of mass transfer coefficients 21/ 22 should decrease as Re increases. 

For SLM configurations Now let us consider the SLM module design. The same as with the BLM systems, individual mass-transfer coefficients of solute species in the feed, and strip interfacial boundary layers are determined experimentally by feed and strip flow rate variations, using relations (51) and (53). 

In most studies in which attempts have been made to investigate the relationship between hydrodynamics and thrombus formation, the flow rate of blood through the test device was changed to alter the hydrodynamics. Unfortunately, for many configurations [for example, a tube (9)] a change in flow rate also changed the residence time of blood in the device, the mass transfer coefficients, as well as shear rates at the biomaterial-blood interface. This simultaneous variation in hemodynamic parameters makes it difficult to assign specific cause and effect relationships to the results obtained with these methods. 

The effect of reactor size on behavior was identified in the discussion of fluid dynamics and mass transfer in Chapter 2. Variation of velocity with position and the formation of boundary layers cause changes in mass transfer coefficients with electrode coordinates. We saw that for developing laminar flow, mass transfer coefficients and hence limiting currents decrease with increasing electrode length. Reactor operation should be at current densities... 

A reactor channel with a square cross section produces longitudinal variation of mass transfer and a spanwise distribution of the mass transfer coefScient (Fig. 5.15b). In the recirculating flow region mass transfer coefficients are highest at the center of the electrode and decrease toward... 

A falling film reactor is essentially a vertical cylinder, where liquid flows downward in a thin film along the wall, and gas flows in the core. Relatively high mass transfer coefficients are obtained both in the liquid and in the gas phases. The liquid phase can be cooled effectively via the wall. Therefore this type of reactor is preferred for very rapid exothermic gas liquid reactions. There are two variations, one consists of a tube bundle, the other consists of one cylinder with a rotor. 

Overall mass transfer coefficients show similar variations with gas and liquid flow rates as in absorption (see Chapter 3). The effects of temperature and pressure are expected to be the same for the gas-film and liquid-film mass transfer coefficients as in absorption operations. Because liquid-film mass transfer coefficients, as well as the Henry s Law constant (H) and the equilibrium ratio (K), increase with rising temperature, stripping is facilitated by heating the liquid phase. 

In many situations the concentrations of solute in the bulk fluid, and even at the fluid interface, may vaiy in the direction of flow. Further, the mass-transfer coefficients depend upon fluid properties and rate of flow, and if these vary in the direction of flow, the coefficients will also. The flux of Eqs. (3.1) and (3.3) to (3.6) is therefore a local flux and will generally vary with distance in the direction of flow. This problem was dealt with, in part, in the development leading to Illustration 3.1. In solving problems where something other than the local flux is required, allowance must be made for these variations, ITiis normally requires some considerations of material balances, but there is no standard procedure. An example is offered below, but it must be emphasized that generally some sort of improvisation for the circumstances at hand will be required. 

Another term used to characterize the transport properties of dialysis membranes is the so-called mass transfer area coefficient (MTAC), which is the product of the mass transfer coefficient (Ko) times the membrane surface area (A), or KoA. Usually, the terms MTAC and KoA reported are those for urea. While Ko should equal the maximum clearance obtained at high blood and dialysate flow rates, reports in the dialysis Hterature (Leypoldt et al., 1997) discuss the variation of KoA with dialysate flow rate. Such reports reflect the manufacturers or others inappropriate extrapolation of KoA from data obtained at typically clinically relevant flow rates, which arc not high enough to minimize boundary layer resistance. While the measurement of in vivo rather than in vitro characteristics of dialyzers is meant to provide more accurate or realistic information, it can be misleading in this context. 

This simplest theory says that the mass transfer coefficient k is proportional to the diffusion coefficient D and independent of the fluid velocity v. Doubling diffusion doubles mass transfer doubling flow has no effect. This is not at all what we set out to predict, given in Equation 9.0-1. Of course, the variation of k with v has been lumped into the unknown fllrn thickness /. This thickness is almost never known a priori, but must be found from measurements of k and D. But if we cannot predict k from the film theory, what value has this theory ... 

The mass transfer theories developed in the previous sections of this chapter are not especially successful. To be sure, the penetration and surface-renewal theories do predict that mass transfer does vary with the square root of the diffusion coefficient, consistent with many correlations. However, neither the film theory nor the surface-renewal theory predicts how mass transfer varies with flow. The penetration theory predicts variation with the square root of flow, less than that indicated by most correlations. This failure to predict the variation of mass transfer with flow is especially disquieting the film and penetration theories should bracket all behavior because a thin film and a semi-infinite slab bracket all possible geometries. 

Example 9.3-1 Apparent mass transfer coefficients caused by bypassing. This example illustrates how uneven flow can give apparent variations of mass transfer coefficients with flow which are larger than those expected from the theories described earlier. These flow variations are close to those reported in popular correlations. 

B Shrinking particle External diffusion iC oc (cjt) (small particles) a (cr) (larger particles) Weak temperature variation Independent for small partieles only The exact variation with flow depends on the mass transfer coefficient... 

Although this result has the same variation with time as do the cases where surface reaction controls, it shows a square-root dependence on flow. It also shows a smaller variation with temperature, for mass transfer coefficients vary much less with temperature than reaction-rate constants. 

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