Systems with Finite Mass Transfer Resistance

ISOTHERMAL SYSTEMS WITH FINITE MASS TRANSFER 

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Isothermal Systems with Finite Mass Transfer Resistance 291... 

ADIABATIC AND NEAR ADUBATIC SYSTEMS WITH FINITE MASS TRANSFER RESISTANCE... 

Tertiary current distribution. This method of analysis applies to those systems where there is significant mass transport and electrode polarization effects. Electrode kinetics is considered, with electrode surface concentrations of reactant and/or products that are no longer equal to those in the bulk electrolyte due to finite mass transfer resistance. The analysis of tertiary current distributions is complex, involving the solution of coupled... 

Broadening of he chromatographic response can also be caused by other effects such as finite mass transfer resistance, and this is the basis of the chromatographic ipethod for measuring mass transfer coefficients. Clearly in any application of thii method it is essential to confirm that the system is indeed linear and he peak broadening results from mass transfer resistance rather than from nonlinearity of the isotherm. With strongly adsorbed species this may require jworking at extremely low concentration levels at which accurate measurernents may be difficult. 

One important result which follows from the equilibrium theory analysis is that as long as the dimensionless bed length (/ -/o) exceeds a critical minimum value a pure raffinate product will be obtained. However, the results of an experimental study carried out by Mitchell and Shendalman with the COj-He-silica gel system showed considerable deviations from the predictions of simple equilibrium theory, indicating that kinetic or dispersive effects are important. A modified theory which included finite mass transfer resistance was developed but proved only marginally more successful in accounting for the observed behavior of the system. 

Current and potential distributions are affected by the geometry of the system and by mass transfer, both of which have been discussed. They are also affected by the electrode kinetics, which will tend to make the current distribution uniform, if it is not so already. Finally, in solutions with a finite resistance, there is an ohmic potential drop (the iR drop) which we minimise by addition of an excess of inert electrolyte. The electrolyte also concentrates the potential difference between the electrode and the solution in the Helmholtz layer, which is important for electrode kinetic studies. Nevertheless, it is not always possible to increase the solution conductivity sufficiently, for example in corrosion studies. It is therefore useful to know how much electrolyte is necessary to be excess and how the double layer affects the electrode kinetics. Additionally, in non-steady-state techniques, the instantaneous current can be large, causing the iR term to be significant. An excellent overview of the problem may be found in Newman s monograph [87]. 

In the frequency response method, first applied to the study of zeolitic diffusion by Yasuda [29] and further developed by Rees and coworkers [2,30-33], the volume of a system containing a widely dispersed sample of adsorbent, under a known pressure of sorbate, is subjected to a periodic (usually sinusoidal) perturbation. If there is no mass transfer or if mass transfer is infinitely rapid so that gas-solid mass-transfer equilibrium is always maintained, the pressure in the system should follow the volume perturbation with no phase difference. The effect of a finite resistance to mass transfer is to cause a phase shift so that the pressure response lags behind the volume perturbation. Measuring the in-phase and out-of-phase responses over a range of frequencies yields the characteristic frequency response spectrum, which may be matched to the spectrum derived from the theoretical model in order to determine the time constant of the mass-transfer process. As with other methods the response may be influenced by heat-transfer resistance, so to obtain reliable results, it is essential to carry out sufficient experimental checks to eliminate such effects or to allow for them in the theoretical model. The form of the frequency response spectrum depends on the nature of the dominant mass-transfer resistance and can therefore be helpful in distinguishing between diffusion-controlled and surface-resistance-controlled processes. 

In case of packed columns, a qualitatively different behavior can be found for finite and infinite intra-partide mass transfer resistance. For vanishing mass transfer resistance inside the catalyst a small number of solutions, typically three, can be observed. Note, that this is consistent with the TAME case discussed above. Instead, for finite transport inside the catalyst a very large number of solutions can be observed. An example is shown in Fig. 10.17, right. It was conjectured by Mohl et al. [74], that this behavior is caused by isothermal multiplidty of the single catalyst pellet and is therefore similar to the well-known fixed-bed reactor [38, 77]. However, further research is required to verify this hypothesis. Further, it was shown by Mohl et al. [74] that in both cases the number of solutions may crucially depend on the discretization of the underlying continuously distributed parameter system. A detailed discussion is given by Mohl et al. [74]. 

In an adiabatic adsorption column the temperature front generally travels at a velocity which is different from the velocity of the primary mass transfer front and, since adsorption equilibrium is temperature dependent, a secondary mass transfer zone is established coincident with the thermal front. In a system with finite heat loss from the column wall one may approach either the isothermal situation with a single mass transfer zone or the adiabatic situation with two mass transfer zones, depending on the relative rates of heat generation and dissipation from the column wall. In the former case the effect of finite heat transfer resistance is to widen the mass transfer zone relative to an isothermal system. 

Constant Pattern Behavior. In a real system the finite resistance to mass transfer and axial mixing in the column lead to departures from the idealized response predicted by equilibrium theory. In the case of a favorable isotherm the shock wave solution is replaced by a constant pattern solution. The concentration profile spreads in the initial region until a stable situation is reached in which the mass transfer rate is the same at all points along the wave front and exactly matches the shock velocity. In this situation the fluid-phase and adsorbed-pliase profiles become coincident, as illustrated in Figure 13. This represents a stable situation and the profile propagates without further change in shape—lienee the term constant pattern. The form of the concentration profile under constant pattern conditions may be easily deduced by integrating the mass transfer rate expression subject to the condition c/c0 = q/qQy where qfj is the adsorbed phase concentration in equilibrium with c(y... 

Figure 10.6 shows that the overall impedance of the system decreases after addition of plasticizer. The data are in agreement with the increase observed in ionic conductivity. From the parameters obtained by fitting the experimental data shown in Fig. 10.6, the apparent diffusion coefficient can be estimated using equation 10.7,where 4 is the thickness of the electrolyte film and 5 is a parameter related to the element O in the equivalent circuit proposed, which accounts for a finite-length Warburg diffusion (Zd), which represents a kind of resistance to mass transfer. 

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