Transfer, mass coefficient

The mass flux, the rate of mass transfer qG per unit area, is proportional to a concentration difference. If a solute transfers from the gas to the liquid phase, its mass flux from the gas phase to the interface Nc is 

Similarly, the liquid-side phase mass flux Nt is 

3 Concentration profile near a gas-liquid interface and an equilibrium curve. 

Since the amount of solute transferred from the gas phase to the interface must equal that from the interface to the liquid phase, 

It is hard to determine the mass-transfer coefficient according to Eq. (9.6) or Eq. (9.7) because we cannot measure the interfacial concentrations, CL, or CG. Therefore, it is convenient to define the overall mass-transfer coefficient as follows  

In view of what has been said so far regarding the mixing of highly viscous Newtonian and non-Newtonian systems, a reduction in the mass transfer coefficient, kia is inevitable. This is indeed borne out well by the few studies 

Dimensionless empirical correlations relating k a to the system geometry, and kinematic and physical variables are available in the literature [Tatterson, 1991], For example, the following correlation due to Kawase and Moo-Yormg [1988] is one which embraces a very wide range of power-law parameters  

Equation (8.18) is not dimensionally consistent and all quantities must be expressed in S.I. units that is, k a, the volumetric mass transfer coefficient (s ) p, the density of liquid (kg/m ) (PAO, the power input per rmit volmne of dispersion (W/m ) a, the interfacial tension (N/m) Vg, the superficial velocity of gas (m/s) Vt, the terminal velocity of a single bubble in a quiescent medium (m/s), (Kawase and Moo-Yoimg recommended a constant value of 0.25 m/s) (Xes, the effective viscosity estimated using equation (8.8) (Pa-s) /Ltw, the viscosity of water (Pa s) Dl, the diffusivity of gas into liquid (m /s) and m is the power-law consistency coefficient (Pa s ). Equation (8.18) applies over the following ranges of conditions 0.59 n 0.95 0.0036 m 10.8 Pa-s and 0.15 Dt 0.6 m. 

A comprehensive discussion of other contemporary work in this field is available elsewhere [Nienow and Ulbrecht, 1985 Hamby et al., 1992 Herbst et al, 1992]. 

Consider a binary mixture consisting of components A and B. If component A moves with a velocity of v and the component B with a velocity of v there is a force against the motion of component A that is proportional to the velocity difference (5 a. Vfj). This is the physical content of Pick s law in the steady-state condition. 

Note Equation (4.241) characterizes diffusion when the mixture element is in steady state with no turbulence. Diffusion in a pipe can be represented by Eq. (4.241) in convective mass transfer the flow and turbulence are important. 

The net flow of component A with Stefan flow taken into consideration is 

Component A diffuses due to the concentration gradient -dc /d. Component B diffuses due to the mean molar velocity v, v = (+ CgVg )lc, like a fish swimming upstream with the same velocity as the flowing water, /y = 0, with regard to a fixed point. 

In a distillation process the diffusion is nearer to the case = -/g = constant or component B absorbs in place of the vaporizing component A, and now g 0. If = -/g, the concentrations are similar to those presented in Fig. 4.35. 

External Diffusion Effects on Heterogeneous Reactions Chap. 11 CAb 

For either EMCD or dilute concentrations, the solution was shown to be in the form 

The ratio of the diffusivity to Ae film-thickness is the mass transfer coefficient, kc. The tildas in the terms kc and 6 denote that they are, respectively, the local transfer coefficient and the boimdary layer thickness at a particular point P on the sphere, 

The average mass transfer coefficient over the surface of area A is 

The average molar flux from the bulk fluid to the surface is 

---

Kovats retention indices RI Mass transfer coefficient h... 

The relationship of the overall gas-phase mass transfer coefficient to the individual film coefficients maybe found from equations 4 and 5, assuming a straight equiHbrium line ... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Equations 11 and 12 caimot be used to predict the mass transfer coefficients directly, because is usually not known. The theory, however, predicts a linear dependence of the mass transfer coefficient on diffusivity. 

Rate Equations with Concentration-Independent Mass Transfer Coefficients. Except for equimolar counterdiffusion, the mass transfer coefficients appHcable to the various situations apparently depend on concentration through thej/g and factors. Instead of the classical rate equations 4 and 5, containing variable mass transfer coefficients, the rate of mass transfer can be expressed in terms of the constant coefficients for equimolar counterdiffusion using the relationships... 

This leads to rate equations with constant mass transfer coefficients, whereas the effect of net transport through the film is reflected separately in thej/gj and Y factors. For unidirectional mass transfer through a stagnant gas the rate equation becomes... 

Equation 28 and its liquid-phase equivalent are very general and valid in all situations. Similarly, the overall mass transfer coefficients may be made independent of the effect of bulk fiux through the films and thus nearly concentration independent for straight equilibrium lines ... 

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

Neither the penetration nor the surface renewal theory can be used to predict mass transfer coefficients directiy because T and s are not normally known. Each suggests, however, that mass transfer coefficients should vary as the square root of the molecular diffusivity, as opposed to the first power suggested by the film theory. 

To use all of these equations, the heights of the transfer units or the mass transfer coefficients and must be known. Transfer data for packed columns are often measured and reported direcdy in terms of and and correlated in this form against and... 

Experimental Mass Transfer Coefficients. Hundreds of papers have been pubHshed reporting mass transfer coefficients in packed columns. For some simple systems which have been studied quite extensively, mass transfer data may be obtained directiy from the Hterature (6). The situation with respect to the prediction of mass transfer coefficients for new systems is stiU poor. Despite the wealth of experimental and theoretical studies, no comprehensive theory has been developed, and most generalizations are based on empirical or semiempitical equations. 

Other correlations based partially on theoretical considerations but made to fit existing data also exist (71—75). A number of researchers have also attempted to separate from a by measuring the latter, sometimes in terms of the wetted area (76—78). Finally, a number of correlations for the mass transfer coefficient itself exist. These ate based on a mote fundamental theory of mass transfer in packed columns (79—82). Although certain predictions were verified by experimental evidence, these models often cannot serve as design basis because the equations contain the interfacial area as an independent variable. 

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

The rate of mass transfer,/, is then assumed to be proportional to the concentration differences existing within each phase, the surface area between the phases,, and a coefficient (the gas or Hquid film mass transfer coefficient, k or respectively) which relates the three. Thus... 

The Mass Transfer Coefficient, Because of the interaction between and a when air is dispersed in the media, the two have rarely... 

The value of the saturation concentration,, is the spatial average of the value determined from a clean water performance test and is not corrected for gas-side oxygen depletion therefore K ji is an apparent value because it is determined on the basis of an uncorrected. A tme volumetric mass transfer coefficient can be evaluated by correcting for the gas-side oxygen depletion. However, for design purposes, can be estimated from the surface saturation concentration and effective saturation depth by... 

Interfacial Mass-Transfer Coefficients. Whereas equiHbrium relationships are important in determining the ultimate degree of extraction attainable, in practice the rate of extraction is of equal importance. EquiHbrium is approached asymptotically with increasing contact time in a batch extraction. In continuous extractors the approach to equiHbrium is determined primarily by the residence time, defined as the volume of the phase contact region divided by the volume flow rate of the phases. 

The mass-transfer coefficients are typically between 10 to 100 p.m/s, depending on hydrodynamic conditions and the values of D. 

Values of the mass-transfer coefficient k have been obtained for single drops rising (or falling) through a continuous immiscible Hquid phase. Extensive Hterature data have been summarized (40,42). The mass-transfer coefficient is often expressed in dimensionless form as the Sherwood number ... 

The values of k and hence Sb depend on whether the phase under consideration is the continuous phase, c, surrounding the drop, or the dispersed phase, d, comprising the drop. The notations and Sh are used for the respective mass-transfer coefficients and Sherwood numbers. 

Although the adsorption of surfactants tends to reduce mass-transfer coefficients by suppressing drop circulation, a sharp increase in mass transfer... 

Mass-Transfer Coefficients with Chemical Reaction. Chemical reaction can occur ia any of the five regions shown ia Figure 3, ie, the bulk of each phase, the film ia each phase adjacent to the iaterface, and at the iaterface itself. Irreversible homogeneous reaction between the consolute component C and a reactant D ia phase B can be described as... 

The enhanced rate expressions for regimes 3 and 4 have been presented (48) and can be appHed (49,50) when one phase consists of a pure reactant, for example in the saponification of an ester. However, it should be noted that in the more general case where component C in equation 19 is transferred from one inert solvent (A) to another (B), an enhancement of the mass-transfer coefficient in the B-rich phase has the effect of moving the controlling mass-transfer resistance to the A-rich phase, in accordance with equation 17. Resistance in both Hquid phases is taken into account in a detailed model (51) which is apphcable to the reversible reactions involved in metal extraction. This model, which can accommodate the case of interfacial reaction, has been successfully compared with rate data from the Hterature (51). 

---