Effective mass transfer coefficient

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) ... 

In the absence of viable cells in the bioreactor, an effective mass transfer coefficient can be obtained from... 

The expression for the enhancement factor E, eq. (35), has first been derived by van Krevelen and Hof-tijzer in 1948. These authors used Pick s law for the description of the mass transfer process and approximated the concentration profile of component B by a constant Xb, over the entire reaction zone. It seems worthwhile to investigate whether the same equation can be applied in case the Maxwell-Stefan theory is used to describe the mass transfer process. To evaluate the Hatta number, again an effective mass transfer coefficient given by eq. (34), is required. The... 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

From comparison of eqs (Cl) and (C2) it can be concluded that a better expression is obtained if Kas is replaced by an effective" mass transfer coefficient which takes into account the difference in binary mass transfer coefficients eq. (34). The more general expression for the effective mass transfer coefficient of component i is given by... 

From these results, the effective mass transfer coefficient ks [s" ] can be calculated and was found to vary from 8 10 m/s (10 rpm) to 1.2 10 m/s (40 rpm). Around 100 rpm, the system enters the kinetically limited regime (Ca=0.05). These results show that immobilized trypsin is very active and useful for determining mass transfer rates for liquid solid systems. 

In the case of two fluids, two films are developed, one for each fluid, and the corresponding mass-transfer coefficients are determined (Figure 3.2). In a fluid-solid system, there is only one film whereas the resistance within the solid phase is expressed by the solid-phase diffusion coefficient, however, in many cases an effective mass-transfer coefficient is used in the case of solids as well. Consider the irreversible catalytic reaction of the form... 

This problem considers the chemically reactive flow in a long, straight channel that represents a section of an idealized porous media (Fig. 4.32). Assume that the flow is incompressible and isothermal, but that it carries a trace compound A. The compound A may react homgeneously in the flow, and it may react heterogeneously at the pore walls. Overall, the objective of the problem is to characterize the chemically reacting flow problem, including the development of an effective mass-transfer coefficient as represented by a Sherwood number. 

For a resolution of question (3), either MASC or the simpler SSHTZ program was run under both isothermal and adiabatic conditions, with effective mass transfer coefficients chosen to simulate the stable portion of the sorption fronts. Fortunately, in most cases described below, the programs predicted that the steady-state MTZ lengths did not change by more than 10Z or so between the two extremes. Thus, an extensive analysis of the wall effects in the various columns was not required for proper interpretation of MTZ data. 

Using the computer programs discussed above, it is possible to extract from these breakthrough curves the effective local mass transfer coefficients as a function of CO2 concentration within the stable portion of the wave. These mass transfer coefficients are shown in Figure 15, along with the predicted values with and without the inclusion of the surface diffusion model. It is seen that without the surface diffusion model, very little change in the local mass transfer coefficient is predicted, whereas with surface diffusion effects included, a more than six-fold increase in diffusion rates is predicted over the concentrations measured and the predictions correspond very closely to those actually encountered in the breakthrough runs. Further, the experimentally derived results indicate that, for these runs, the assumption that micropore (intracrystalline) resistances are small relative to overall mass transfer resistance is justified, since the effective mass transfer coefficients for the two (1/8" and 1/4" pellets) runs scale approximately to the inverse of the square of the particle diameter, as would be expected when diffusive resistances in the particle macropores predominate. 

Table 3. Effective mass transfer coefficients obtained for Columns 1,2 and 3 under flowing and flow interrupt conditions.

In this study, the effects of cosolvent (EtOH) addition on the solubilization and recovery of PCE by a nonionic surfactant (Tween 80) was evaluated using a combination of batch, column and 2-D box studies. Batch results demonstrated that the addition of 5% and 10% EtOH increased the solubilization capacity of Tween 80 from 0.69 g PCE/g surfactant to 1.09 g PCE/g surfactant. For a 4% Tween 80 solution, this translates into a solubility enhancement of more than 50%, from 26,900 mg/L to 42,300. mg/L. When the surfactant formulations were flushed through soil columns containing residual PCE, effluent concentration data clearly showed that PCE solubilization was rate-limited, regardless of the EtOH concentration. Using analytical solutions to the 1-D ADR equation, effective mass transfer coefficients (Ke) were obtained from the effluent concentration data for both steady-state (A e ) and no flow conditions The addition of EtOH had... 

Equation 6.12 relates CP to flux (/) and the effective mass-transfer coefficient (kes) for retained solute. For a clean membrane keff = k, which is the boundary layer mass-transfer coefficient from Sherwood correlations [45, 46], However, for a membrane... 

When chemical reactions occur in an extraction process, the effective mass transfer coefficient may be higher or lower than that expected from purely physical considerations, e.g., Eqs. (10) and (11). For example, the slow interfacial reaction of Eq. (5) will tend to reduce the mass transfer rate. On the other hand, a rapid irreversible reaction can enhance the mass transfer rate. 

A practical approach is to use the venturi tube (see Fig. 10.13) with a swarm of representative droplets of a certain diameter to estimate an effective mass transfer coefficient, including diffusional and reactive resistances. The value obtained would be... 

Ranz and Marshall (13) carefully studied the evaporation of water droplets in still and moving air. They found that the steady-state temperature at the surface of a small evaporating drop was the wet bulb temperature of the surrounding air. Marshall considered the dependence of the effective mass transfer coefficient on air velocity to be determined largely by the boundary layer of less mobile air at the droplet surface, and he reported correlations which show that varied linearly with v1/2. Ranz and Marshall s data clearly show this relation, which we may write as Equation 5. 

In the absence of dissolved humic substances, the preceding equation reduces to the local mass transfer coefficient defined in Eq. (3). Furthermore, the corresponding average effective mass transfer coefficient, ke(t), applicable to the entire pool, can be expressed as... 

Goerke et al. [29] assumed that the actual thickness of the blood flow channels may be described by a truncated normal distribution. They then derived the following expression for the effective mass-transfer coefficient that incorporates the effects... 

The drawback of this approach compared with that described in Section 6.5.2 is that all errors are lumped into the isotherm parameters rather than the effective mass transfer coefficient, because either the wrong column or isotherm model is chosen. This approach is thus recommended to get a quick first idea of system behavior using only little amounts of sample, and not for a complete analysis, especially if binary mixtures with component interactions are investigated. The significance of the results decreases even further if some plant and packing parameters are only guessed or even neglected. 

If mass transfer in the film and diffusion inside the pores are taken into account the effective mass transfer coefficient is given as a series connection of the internal (1/kpore) and external (l/kf,im) mass transfer resistance (Eq. 6.138) ... 

The factor in front of Dm is an approximation of the internal tortuosity factor. For a porosity of 0.9 to 0.5, Dpore is 5 times smaller than Dm or even lower, so the contribution to the effective mass transfer coefficient Eq. 6.191 is... 

In chromatographic systems with relatively large effective mass transfer coefficients kefy (i.e. low mass transfer resistance) the influence of axial dispersion, especially eddy diffusion, dominates the concentration profile. HETPj and Nj is then independent of the interstitial velocity. 

Each particle in a bed of porous particles is surroimded by a laminar sublayer (Figure 5.4), through which mass transfer takes place only by molecular diffusion. On one side, this layer is exposed to the flowing mobile phase and is entirely accessible. On the other side, it wraps the particle wall and is accessible from the particle inside only at the pore openings. The thickness of this layer, hence the mass transfer coefficient, is determined by hydrodynamic conditions and depends on the flow velocity. The mass transfer rates can be correlated in terms of the effective mass transfer coefficient, fcy, defined according to a linear driving force equation ... 

With the composition derivatives obtained from Eq. 8.6.2 we may define an effective mass transfer coefficient by... 

Compute effective mass transfer coefficients, Compute [j8]. 

Pseudobinary (effective) mass transfer coefficient of component i in a mixture [m/s]... 

LI, RMAX simulation space limits m0 effective mass transfer coefficient... 

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