Rate Equation Under Mass Transfer Control

The electrochemicar rate, equation presented in Sections 4,2.2 and 4.2.3 is applicable when the electrochemicat reactions are under activation or kinetic control. However, when mass transfer of the el ectroacfive species becomes rate controlling, the rate equation requires modification, as shown below. 

The mass transfer of an ionic species M+ from the bulk of the solution to the cathode surface during its reduction, is controlled by two factors, migration and diffusion [16]. 

= equivalent conductance (mhocm -i K — conductivity of the solution (mho cm ) m and d (in the subscripts) = migration and diffusion, respectively. 

When the solution contains supporting electrolyte, the contribution of migration to mass transfer is small. Hence, the flux, v, or the current density, /, is a result of diffusion, since the charge-transfer step is considered to be fast. The concentration at the cathode surface. Cs, decreases under the.se conditions as shown in Fig. 4.2.5. Assuming a linear distribution of concentration in the diffusion layer thickness 5. we have, at steady. state  

Equation (41) shows that the reaction rate is highest when Cs approaches zero. Since i = Fv, Eq. (41) can be rewritten as  

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This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

Finally, when the corrosion rate is under complete mass transfer control, the corrosion rate can be derived from an equation of the form ... 

In contrast, physical adsorption is a very rapid process, so the rate is always controlled by mass transfer resistance rather than by the intrinsic adsorption kinetics. However, under certain conditions the combination of a diffiision-controUed process with an adsorption equiUbrium constant that varies according to equation 1 can give the appearance of activated adsorption. 

Direct Chlorination of Ethylene. Direct chlorination of ethylene is generally conducted in Hquid EDC in a bubble column reactor. Ethylene and chlorine dissolve in the Hquid phase and combine in a homogeneous catalytic reaction to form EDC. Under typical process conditions, the reaction rate is controlled by mass transfer, with absorption of ethylene as the limiting factor (77). Ferric chloride is a highly selective and efficient catalyst for this reaction, and is widely used commercially (78). Ferric chloride and sodium chloride [7647-14-5] mixtures have also been utilized for the catalyst (79), as have tetrachloroferrate compounds, eg, ammonium tetrachloroferrate [24411-12-9] NH FeCl (80). The reaction most likely proceeds through an electrophilic addition mechanism, in which the catalyst first polarizes chlorine, as shown in equation 5. The polarized chlorine molecule then acts as an electrophilic reagent to attack the double bond of ethylene, thereby faciHtating chlorine addition (eq. 6) ... 

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

Transport Processes. The velocity of electrode reactions is controlled by the charge-transfer rate of the electrode process, or by the velocity of the approach of the reactants, to the reaction site. The movement or trausport of reactants to and from the reaction site at the electrode interface is a common feature of all electrode reactions. Transport of reactants and products occurs by diffusion, by migration under a potential field, and by convection. The complete description of transport requires a solution to the transport equations. A full account is given in texts and discussions on hydrodynamic flow. Molecular diffusion in electrolytes is relatively slow. Although the process can be accelerated by stirring, enhanced mass transfer... 

No study has been made to discover which of the several resistances is important, but a simple rate equation can be written which states that the rate of the over-all process is some function of the extent of departure from equilibrium. The function is likely to be approximately linear in the departure, unless the intrinsic crystal growth rate or the nucleation rate is controlling, because the mass and heat transfer rates are usually linear over small ranges of temperature or pressure. The departure from equilibrium is the driving force and can be measured by either a temperature or a pressure difference. The temperature difference between that of the bulk slurry and the equilibrium vapor temperature is measured experimentally to 0.2° F. and lies in the range of 0.5° to 2° F. under normal operating conditions. 

It is assumed that all electrons transfers from the particle conduction band and surface states to the electrode take place under conditions where the current is mass transport controlled. The first order rate constant kg describes electron promotion either by thermal or photonic processes, and the rate constant k describes the loss of the electrons from the conduction band or surface states by a process which is first order in electron concentration. The validity of this assumption will be discussed later. There will be an equation similar to equation (71) for each value of m. If each equation is multiplied by its value of m and the engendered set of equations summed, it is possible to obtain the simple result that ... 

Here tm is the mass-transfer time. Only under slow reaction kinetic control regime can intrinsic kinetics be derived directly from lab data. Otherwise the intrinsic kinetics have to be extracted from the observed rate by using the mass-transfer and diffusion-reaction equations, in a manner similar to those defined for catalytic gas-solid reactions. For instance, in the slow reaction regime,... 

Several of the factors of Figure 3 controlling the activity and selectivity of the biphasic selective hydrogenation of ,/ -unsaturated aldehydes to allylic alcohols, for instance, 3-methyl-2-butenaldehyde to 3-methyl-2-buten-l-ol (Eq. 11) with rutheni-um-sulfonated phosphine catalysts were investigated [11], such as the effect of agitation speed and the influence of aldehyde, ligand, and metal concentrations. Under optimized reaction conditions, where gas-liquid mass transfer was not rate-determining, the kinetic equation (Eq. 12) was found to apply. A zero-order dependence with respect to the concentration of the ,/i-unsaturated aldehyde was found. 

Under such circumstances, even fairly large percentage changes in kx will not significantly affect K, and efforts to increase the rate of mass transfer would best be directed toward decreasing the gas-phase resistance. Conversely, when m is very large (solute A relatively insoluble in the liquid), with kx and k nearly equal, the first term on the right of equation (3-23) becomes minor and the major resistance to mass transfer resides within the liquid, which is then said to control the rate. Ultimately, this becomes... 

Under steady-state conditions in a plug-flow tubular reactor, the onedimensional mass transfer equation for reactant A can be integrated rather easily to predict reactor performance. Equation (22-1) was derived for a control volume that is differentially thick in all coordinate directions. Consequently, mass transfer rate processes due to convection and diffusion occur, at most, in three coordinate directions and the mass balance is described by a partial differential equation. Current research in computational fluid dynamics applied to fixed-bed reactors seeks a better understanding of the flow phenomena by modeling the catalytic pellets where they are, instead of averaging or homogenizing... 

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