Mass transfer coefficients geometries

Correlations for the mass-transfer coefficient, as the Sherwood number for various membrane geometries have been reviewed (39). 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

With complicated geometries, the product of the interfacial area per volume and the mass-transfer coefficient is required. Correlations of kop or of HTU are more accurate than individual correlations of k and since the measurements are simpler to determine the produc t kop or HTU. 

To determine the mass-transfer rate, one needs the interfacial area in addition to the mass-transfer coefficient. For the simpler geometries, determining the interfacial area is straightforward. For packed beds of particles a, the interfacial area per volume can be estimated as shown in Table 5-27-A. For packed beds in distillation, absorption, and so on in Table 5-28, the interfacial area per volume is included with the mass-transfer coefficient in the correlations for HTU. For agitated liquid-liquid systems, the interfacial area can be estimated... 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

The principal applications of mass transfer are in the fields of distillation, gas absorption and the other separation processes involving mass transfer which are discussed in Volume 2, In particular, mass transfer coefficients and heights of transfer units in distillation, and in gas absorption are discussed in Volume 2,. In this section an account is given of some of the experimental studies of mass transfer in equipment of simple geometry, in order to provide a historical perspective. 

The determination of the external mass transfer coefficient of CO, k, (see Equation 3) deserves brief comments. Since the complex geometry and flow characteristics in the reactor cell precluded a reliable estimation of k based on correlations given in the literature, the CO oxidation activities of the catalyst... 

Due to the difficulties in having rigorous analytical expressions for the flux at any given geometry and flow conditions, in many instances it is convenient to include all the characteristics of the supply in the mass transfer coefficient ms and use expression (50). It must be pointed out that expression (50), stating a linearity between the flux and the difference between bulk and surface concentrations, cannot be - in general valid for nonlinear processes, such as coupled complexation of the species i with any other species (see Chapters 4 and 10 for a more detailed discussion). 

It should be obvious from the above that fluid-management techniques which Improve the mass-transfer coefficient (k) with minimum power consumption are most desirable. However, in some cases, low-cost membrane configurations with inefficient fluid management may be more cost effective. In any case, it is important to understand quantitatively how tangential velocity and mem-brane/hardware geometry affects the mass-transfer coefficient. 

The mass and heat transfer analogies make possible an evaluation of the mass-transfer coefficient (k) and provide insight into how membrane geometry and fluid-flow conditions can be specified to optimize flux (4). For laminar flow ... 

The basic membrane/hardware geometries-tubes, plate and frame, and hollow fibers can of course all be operated with fluid management techniques which are relatively efficient or non-efficient. The remainder of this paper will adress itself to novel fluid-management techniques which utilize the basic membrane/hardware geometries but which seek to augment the mass-transfer coefficient even further. 

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. 

Many parameters affect the mass transfer between two phases. As we discussed above, the concentration gradient between the two phases is the driving force for the transfer and this, together with the over-all mass transfer coefficient, determines the mass transfer rate. The influence of process parameters (e. g. flow rates, energy input) and physical parameters (e. g. density, viscosity, surface tension) as well as reactor geometry are summed up in the mass transfer coefficient. The important parameters for Kta in stirred tank reactors are ... 

The shrinking-core model (SCM) is used in some cases to describe the kinetics of solid and semi-solids-extraction with a supercritical fluid [22,49,53] despite the facts that the seed geometry may be quite irregular, and that internal walls may strongly affect the diffusion. As will be seen with the SCM, the extraction depends on a few parameters. For plug-flow, the transport parameters are the solid-to-fluid mass-transfer coefficient and the intra-particle diffusivity. A third parameter appears when disperse-plug-flow is considered [39,53],... 

Packed height is determined from the relationships in Section III. Application of these relationships requires knowledge of the liquid and gas mass transfer coefficients. It is best to obtain these from experimental data on the system if available, but caution is required when extending such data to column design, because mass transfer coefficients depend on packing geometry, liquid and gas distribution, physical properties, and gas and liquid loads, and these may vary from one contactor to another. 

Correlations for mass transfer coefficients for a wide range of geometries and flow conditions are available in the literature.7... 

In extraction columns, it is possible to find droplet swarms where the local velocities near the droplet surface are higher, this being due to the lower free area available for the countercurrent flowing continuous phase. Wake and Marangoni influences make the prediction of a physical mass transfer coefficients difficult. With reactive extraction the influence of interfacial kinetics on overall mass transfer is generally not negligible. In any case, a combination of reactive kinetics with any eddy mass transfer model is recommended, whereas the latter could rely on correlations derived for specific column geometries. 

The hydrodynamic parameters that are required for stirred tank design and analysis include phase holdups (gas, liquid, and solid) volumetric gas-liquid mass-transfer coefficient liquid-solid mass-transfer coefficient liquid, gas, and solid mixing and heat-transfer coefficients. The hydrodynamics are driven primarily by the stirrer power input and the stirrer geometry/type, and not by the gas flow. Hence, additional parameters include the power input of the stirrer and the pumping flow rate of the stirrer. 

Both criteria for extraparticle gradients contain observables and can be calculated based on experimental observations of reaction rates. For heat and mass transfer coefficients in packed beds various correlations exist in terms of dimensionless numbers. In Table 1 the most appropriate ones for laboratory reactors are given [5, 7, 30, 31]. Values of k( and h for gases in laboratory systems range between O.l-lOms-1 and 100-1000JK-1s-1 m-2, respectively. In the case of monoliths, other correlations should be used because of the different geometry [32-34],... 

When two liquids are immiscible, the design parameters include droplet size distribution of the disperse phase, coalescence rate, power consumption for complete dispersion, and the mass-transfer coefficient at the liquid-liquid interface. The Sauter mean diameter, dsy, of the dispersed phase depends on the Reynolds, Froudes and Weber numbers, the ratios of density and viscosity of the dispersed and continuous phases, and the volume fraction of the dispersed phase. The most important parameters are the Weber number and the volume fraction of the dispersed phase. Specifically, dsy oc We 06(l + hip ), where b is a constant that depends on the stirrer and vessel geometry and the physical properties of the system. Both dsy and the interfacial area aL remain unaltered, if the same power per unit volume (P/V) is used in the scale-up. 

Silverman has defined a number of useful expressions that allow one to utilize the rotating cylinder method with a variety of practical geometries (12,15). Both shear stresses and mass transfer coefficients are included in the derivations described (12). Table 1 in NACE standard TM-0270-72 summarized the various features of experimental systems for studying flow induced corrosion (22). 

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