Mass transfer coefficients limits

Inspection of Eqs. (14-71) and (14-78) reveals that for fast chemical reactions which are liquid-phase mass-transfer limited the only unknown quantity is the mass-transfer coefficient /cl. The problem of rigorous absorber design therefore is reduced to one of defining the influence of chemical reactions upon k. Since the physical mass-transfer coefficient /c is already known for many tower packings, it... 

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. 

Ga.s-Lic(uid Ma.s.s Tran.sfer Gas-liqiiid mass transfer norrnallv is correlated bv means of the mass-transfer coefficient K a ersiis powder le el at arioiis superficial gas elocities. The superficial gas clocitv is the ohirne of gas at the a erage temperature and pressure at the midpoint in the tank di ided bv the area of the essel. In order to obtain the partial-pressure dri ing force, an assumption must be made of the partial pressure in equilibrium wdth the concentration of gas in the liquid, Manv times this must be assumed, but if Fig, 18-26 is obtained in the pilot plant and the same assumption principle is used in e ahiating the mixer in the full-scale tank, the error from the assumption is limited. 

In order to allow integration of countercurrent relations like Eq. (23-93), point values of the mass-transfer coefficients and eqiiilibrium data are needed, over ranges of partial pressure and liquid-phase compositions. The same data are needed for the design of stirred tank performance. Then the conditions vary with time instead of position. Because of limited solubihty, gas/liquid reactions in stirred tanks usually are operated in semibatch fashion, with the liquid phase charged at once, then the gas phase introduced gradually over a period of time. CSTR operation rarely is feasible with such systems. 

Pore diffusion limitation was studied on a very porous catalyst at the operating conditions of a commercial reactor. The aim of the experiments was to measure the effective diffiisivity in the porous catalyst and the mass transfer coefficient at operating conditions. Few experimental results were published before 1970, but some important mathematical analyses had already been presented. Publications of Clements and Schnelle (1963) and Turner (1967) gave some advice. 

This equation is that of a first-order reaction process, and thus the fraction of material electrolysed at any instant is independent of the initial concentration. It follows that if the limit of accuracy of the determination is set at C, = 0.001 C0, the time t required to achieve this result will be independent of the initial concentration. The constant k in the above equation can be shown to be equal to Am/ V, where A is the area of the pertinent electrode, V the volume of the solution and m the mass transfer coefficient of the electrolyte.20 It follows that to make t small A and m must be large, and V small, and this leads to the... 

Checking the absence of external mass transfer limitations is a rather easy procedure. One has simply to vary the total volumetric flowrate while keeping constant the partial pressures of the reactants. In the absence of external mass transfer limitations the rate of consumption of reactants does not change with varying flowrate. As kinetic rate constants increase exponentially with increasing temperature while the dependence of mass transfer coefficient on temperature is weak ( T in the worst case), absence... 

GP 9] [R 16[ The extent of external transport limits was made in an approximate manner as for the internal transport limits (see above), as literature data on heat and mass transfer coefficients at low Peclet numbers are lacking [78]. Using a Pick s law analysis, negligible concentration differences from the bulk to the catalyst sur-... 

For a non-linear equilibrium relationship, in which the slope of the equilibrium curve varies with concentration, the magnitudes of the overall mass transfer coefficients will also vary with concentration, even when the film coefficients themselves remain constant. The use of overall mass transfer coefficients in mass transfer rate equations should therefore be limited to the case of linear equilibria or to situations in which the mass transfer coefficient is known to be... 

It was shown later that a mass transfer rate sufficiently high to measure the rate constant of potassium transfer [reaction (10a)] under steady-state conditions can be obtained using nanometer-sized pipettes (r < 250 nm) [8a]. Assuming uniform accessibility of the ITIES, the standard rate constant (k°) and transfer coefficient (a) were found by fitting the experimental data to Eq. (7) (Fig. 8). (Alternatively, the kinetic parameters of the interfacial reaction can be evaluated by the three-point method, i.e., the half-wave potential, iii/2, and two quartile potentials, and ii3/4 [8a,27].) A number of voltam-mograms obtained at 5-250 nm pipettes yielded similar values of kinetic parameters, = 1.3 0.6 cm/s, and a = 0.4 0.1. Importantly, no apparent correlation was found between the measured rate constant and the pipette size. The mass transfer coefficient for a 10 nm-radius pipette is > 10 cm/s (assuming D = 10 cm /s). Thus the upper limit for the determinable heterogeneous rate constant is at least 50 cm/s. 

Reactors which generate vortex flows (VFs) are common in both planktonic cellular and biofilm reactor applications due to the mixing provided by the VF. The generation of Taylor vortices in Couette cells has been studied by MRM to characterize the dynamics of hydrodynamic instabilities [56], The presence of the coherent flow structures renders the mass transfer coefficient approaches of limited utility, as in the biofilm capillary reactor, due to the inability to incorporate microscale details of the advection field into the mass transfer coefficient model. 

The constants characterizing the electrode reaction can be found from this type of polarization curve in the following manner. The quantity k"e is determined directly from the half-wave potential value (Eq. 5.4.27) if E0r is known and the mass transfer coefficient kQx is determined from the limiting current density (Eq. 5.4.20). The charge transfer coefficient oc is determined from the slope of the dependence of In [(yd —/)//] on E. 

The RHSE has the same limitation as the rotating disk that it cannot be used to study very fast electrochemical reactions. Since the evaluation of kinetic data with a RHSE requires a potential sweep to gradually change the reaction rate from the state of charge-transfer control to the state of mass transport control, the reaction rate constant thus determined can never exceed the rate of mass transfer to the electrode surface. An upper limit can be estimated by using Eq. (44). If one uses a typical Schmidt number of Sc 1000, a diffusivity D 10 5 cm/s, a nominal hemisphere radius a 0.3 cm, and a practically achievable rotational speed of 10000 rpm (Re 104), the mass transfer coefficient in laminar flow may be estimated to be ... 

This simple experiment illustrates the basic features of the limiting-current method. A particular electrode reaction proceeds at the highest possible rate, indicated by a current plateau. From the limiting current thus recorded the mass-transfer rate and the mass-transfer coefficient at the electrode in question may be determined. 

In the limiting-current method of measuring mass-transfer coefficients, the reacting-ion concentration at the electrode is made vanishingly small by applying a sufficiently large potential. In this case, therefore, only the current density and the bulk concentration need to be known. 

However, flow generated by a cylinder rotating at high speed was subsequently used by others, and in particular by King and co-workers (K3, K4a), to demonstrate that dissolution and electrochemical corrosion may both be transport limited. The dependence of the mass-transfer coefficient on the rotation rate and on the diffusivity of the dissolving species was established by correlation of experimental data (see Table VII, System 43). 

Finally, when calculating mass-transfer coefficients from limiting-current values, the following factors should be considered ... 

Steps 1 and 7 are highly dependent on the fluid flow characteristics of the system. The mass velocity of the fluid stream, the particle size, and the diffusional characteristics of the various molecular species are the pertinent parameters on which the rates of these steps depend. These steps limit the observed rate only when the catalytic reaction is very rapid and the mass transfer is slow. Anything that tends to increase mass transfer coefficients will enhance the rates of these processes. Since the rates of these steps are only slightly influenced by temperature, the influence of these processes... 

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