Other Definitions of Mass Transfer Coefficients

Effect Basic equation Rate Force Coefficient 

Mass transfer = kAci Flux per area relative to an interface Difference of concentration The mass transfer coefficient k ( = Ljt) is a function of flow 

Diffusion -ji = DVci Flux per area relative to the volume average velocity Gradient of concentration The diffusion coefficient D ([=]Z, /0 is a physical property independent of flow 

Dispersion — ci v] = EVc Flux per area relative to the mass average velocity Gradient of time averaged concentration The dispersion coefficient E ( = L ji) depends on the flow 

Homogeneous chemical reaction r = CiCi Rate per volume Concentration The rate constant ([=]l/0 is a chemical property independent of flow 

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Since concentrations may be defined in a number of equivalent ways, other definitions of mass-transfer coefficients for this case (NB - 0 dilute solutions) are frequently used, such as... 

This chapter discusses mass transfer coefficients for dilute solutions extensions to concentrated solutions are deferred to Section 9.5. In Section 8.1, we give a basic definition for a mass transfer coefficient and show how this coefficient can be used experimentally. In Section 8.2, we present other common definitions that represent a thicket of prickly alternatives rivaled only by standard states for chemical potentials. These various definitions are why mass transfer often has a reputation with students of being a difficult subject. In Section 8.3, we list existing correlations of mass transfer coefficients and in Section 8.4, we explain how these correlations can be developed with dimensional analysis. Finally, in Section 8.5, we discuss processes involving diffusion across interfaces, a topic that leads to overall mass transfer coefficients found as averages of more local processes. This last idea is commonly called mass transfer resistances in series. 

The mass-transfer coefficients, by definition, are equal to the ratios of the molar mass flux to the concentration driving forces. The mass-transfer coefficients are related to each other as follows ... 

When transport occurs between phases, we usually express the flux as a function only of the concentration difference of the diffusing gas. The proportionality constant here is known as the mass transfer coefficient (with units of distance/time). More puristically, it is referred to as the phenomenological mass transfer coefficient, to distinguish it from other, often more useful, definitions. This is usually the chemical engineer s... 

As the resistance to mass transfer in the gas phase may be neglected, the diffusive flux of evaporation Ny, in moles per unit area and unit time, will be obtained with the help of a mass transfer coefficient kfy.. There are two equivalent alternative definitions [Eq. (28)], one using a driving force in terms of mole concentrations, the other in terms of activities (asterisks mark a concentration or activity value at the interface, considered to lie at y = 0). 

In equimolar counterdiffusion, knowledge of the mass-transfer coefficient (or fc or and the values of partial pressures (or concentrations or mole fractions) at the two ends of the diffusion path (z = 0 and z = Sg) were sufficient to determine Naz (or Ngz). In all other cases of molecular diffusion, the flux ratio Nr needs to be known [Pg.104]

Heat and mass transfer coefficients are usually reported as correlations in terms of dimensionless numbers. The exact definition of these dimensionless numbers implies a specific physical system. These numbers are expressed in terms of the characteristic scales. Correlations for mass transfer are conveniently divided into those for fluid-fluid interfaces and those for fluid-solid interfaces. Many of the correlations have the same general form. That is, the Sherwood or Stanton numbers containing the mass transfer coefficient are often expressed as a power function of the Schmidt number, the Reynolds number, and the Grashof number. The formulation of the correlations can be based on dimensional analysis and/or theoretical reasoning. In most cases, however, pure curve fitting of experimental data is used. The correlations are therefore usually problem dependent and can not be used for other systems than the one for which the curve fitting has been performed without validation. A large list of mass transfer correlations with references is presented by Perry [95]. 

Other values often used instead of the mass transfer coefficient and die average driving force are the height of a mass transfer unit and the number of mass transfer units. To definite them, let us proceed fitan equations (194) and (198) and fi om the equation of the mass balance of the apparatus. From these equations it follows ... 

Most of the variables listed m be determined readily by experiment or else studied in the thermodynamic or chemistry literature. However some of the variables such as surface radiative emissivity and convective heat transfer coefficient ean be more difficult to ascertain. More difficult still is a definitive value for the mass transfer coefficient of volatiles, because of the complexity of measuring gas mass transfer through a burning thermoset composite laminate. Incorrect values for these parameters taken together can contribute to a substantial error in the predictive capacity of a model, even where the balance of the other properties have been correctly evaluated and deployed. 

In fact, the mass transfer coefficient is often an ambiguous concept, reflecting nuances of its basic definition. To begin our discussion of these nuances, we first compare the mass transfer coefficient with the other rate constants given in Table 8.2-1. The mass transfer coefficient seems a curious contrast, a combination of diffusion and dispersion. Because it involves a concentration difference, it has different dimensions than the diffusion and dispersion coefficients. It is a rate constant for an interfacial physical reaction, most similar to the rate constant of an interfacial chemical reaction. 

Unfortunately, the definition of the mass transfer coefficient in Table 8.2-1 is not so well accepted that the coefficient s dimensions are always the same. This is not true for the other processes in this table. For example, the dimensions of the diffusion coefficient are always taken as ifjt. If the concentration is expressed in terms of mole fraction or... 

In this book, we always choose to use the local concentration difference at a particular position in the column. Such a choice implies a local mass transfer coefficient to distinguish it from an average mass transfer coefficient. Use of a local coefficient means that we often must make a few extra mathematical calculations. However, the local coefficient is more nearly constant, a smooth function of changes in other process variables. This definition was implicitly used in Examples 8.1-1, 8.1-3, and 8.1-4 in the previous section. It was used in parallel with a type of average coefficient in Example 8.1-2. 

Finally, mass transfer coefficients can be complicated by diffusion-induced convection normal to the interface. This complication does not exist in dilute solution, just as it does not exist for the dilute diffusion described in Chapter 2. For concentrated solutions, there may be a larger convective flux normal to the interface that disrupts the concentration profiles near the interface. The consequence of this convection, which is like the concentrated diffusion problems in Section 3.3, is that the flux may not double when the concentration difference is doubled. This diffusion-induced convection is the motivation for the last definition in Table 8.2-2, where the interfacial velocity is explicitly included. Fortunately, many transfer-in processes like distillation often approximate equimolar counterdiffusion, so there is little diffusion-induced convection. Also fortunately, many other solutions are dilute, so diffusion induced convection is minor. We will discuss the few cases where it is not minor in Section 9.5. 

We now turn to mass transfer across interfaces, from one fluid phase to the other. This is a tricky subject, one of the main reasons that mass transfer is felt to be a difficult subject. In the previous sections, we used mass transfer coefficients as an easy way of describing diffusion occurring from an interface into a relatively homogeneous solution. These coefficients involved approximations and sparked the explosion of definitions exemplified by Table 8.2-2. Still, they are an easy way to correlate experimental results or to make estimates using the published relations summarized in Tables 8.3-2 and 8.3-3. 

Other definitions of HTU and NTU can be based on other forms of the overall mass transfer coefficients. The use of transfer units is a rough parallel with the use of stages in distillation or the term theoretical plates in chromatography. As such, it is a historical genuflection by the more recent absorption analyses in the direction of the older equilibrium stage separation analysis. 

These definitions, formally different with respect to those reported in eqn (14.10), approach each other in the case of negligible inhibition. This approach is very useful because, with this choice, CPC is directly related to the flux reduction due only to external mass transfer resistance and not to the inhibition phenomenon. Moreover, the polarization and inhibition effect are able to be separately identified and split into their own different contributions, which can thus be analyzed to provide a better understanding of the coupled influence of these two phenomena. This is done by defining another coefficient, i.e. the inhibition coefficient, IC, which is on the other hand a quantitative indicator of the inhibition phenomenon only (see next section). 

Other driving force units and corresponding coefficients can be used in analogous definitions. Equation (2.4-2) explicitly allows for the bulk flow contribution to the molar flux relative to stationary coordinates. The coefficient or its analogues is thought to exhibit a somewhat less complex relationship to composition, flow conditions, and geometry. Further discussion of the relationships among those coefficients is provided in Section 2.4-1 in terms of particular models for turbulent mass transfer. 

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