Mass transfer diffusion-limited

As the temperature is varied in a reactor, we should expect to see the rate-controlling step vary. At sufficiently low temperature the reaction rate coefficient is small and the overall rate is reaction limited. As the temperature increases, pore diffusion next becomes controlling (Da is nearly independent of temperature), and at sufficiency high temperature external mass transfer might limit the overall process. Thus a plot of log rate versus 1 / T might look as shown in Figure 7-15. 

Limitation of a reaction by translational diffusion in solution is a rather rare case. Much more frequently the limitation of the observed overall reaction rate is by external mass transfer (through a laminar film around a solid macroscopic carrier) (Chapter 5, Section 5.5.1) or internal mass transfer (diffusion of substrate or product through the pores of a solid carrier or a gel network to an enzyme molecule in the interior of the carrier) (Chapter 5, Section 5.5.2). 

Enhanced Mass Transfer, Diffusivity Supercritical fluids share many of the advantages of gases, including lower viscosities and higher diffusivities relative to liquid solvents, thereby potentially providing the opportunity for faster rates, particularly for diffusion-limited reactions. 

This same reaction sequence can be used to describe the thermal decomposition of polymers under reducing conditions. In this case, the value of n is equal to 0 and h is usually set equal to 1.0 in the generalized reaction. Under these conditions, the mass transfer is limited to the removal of the volatiles from the porous green body. This mass transfer can be limited by the pore diffusion or the boundary layer. We still must consider that the surface reaction or the steps of heat transfer in the boundary layer or heat conduction in the porous body could also be rate controlling in this case of the thermal decomposition of polymers under reducing conditions. 

Diffusion coefficients are needed in the design of mass transfer processes, such as gas absorption, distillation, and liquid-liquid extraction, as well as in catalytic reactions where mass transfer can limit the rate of reaction. 

A measure of the absence of internal (pore diffusion) mass transfer limitations is provided by the internal effectiveness factor, t, which is defined as the ratio of the actual overall rate of reaction to the rate that would be observed if the entire interior surface were exposed to the reactant concentration and temperature existing at the exterior of the catalyst pellet. A value of 1 for rj implies that all of the sites are being utilized to their potential, while a value below, say, 0.5, signals that mass transfer is limiting performance. The value of rj can be related to that of the Thiele modulus, 4>, which is an important dimensionless parameter that roughly expresses a ratio of surface reaction rate to diffusion rate. For the specific case of an nth order irreversible reaction occurring in a porous sphere,... 

The above mass transfer equations, although based on sound molecular diffusion principles, are limited in their applicability in a number of ways. A basic condition for their validity is the assumption of equimolar or dilute unimolar mass transfer. This limits the NTU and HTU approach to processes that are essentially either binary (distillation) or ternary (absorption or stripping) with only one component crossing the phase boundary. Another shortcoming of the transfer units technique is its exclusion of energy balances or temperature calculations. 

The results obtained in equations (8-136) to (8-142) assume constant B, i.e., the reaction is pseudo-first-order in A. Another limiting case that yields to analytical solution is that in which the rate of reaction is very rapid and the reaction occurs wholly within the film. Here we consider the reaction A -I- P to occur very rapidly compared to mass-transfer/diffusion rates. The profiles look as in Figure 7.17b, and the overall flux and enhancement factor are given by... 

In a porous medium, assuming surface diffusion is negligible, the mass transfer is limited by viscous resistance, resulting from the momentum transferred to the membrane, Knudsen diffusion resistance due to molecule-membrane collisions and ordinary diffusion due to collisions between molecules. Predominance, coexistence, or transitions between these different mechanisms are estimated by the dimensionless Knudsen number (Kn) that compares the mean free path i of diffusing molecules... 

In the case of a fuel cell, the situation is more complex because several phenomena occur with a relative importance depending on the fuel cell and experimental conditions ohmic drop due to the electrolyte and charge transfer resistances, anode and cathode overpotentials due to charge transfer and/or mass transfer (essentially limited by diffusion). Therefore, anode overpotential is the sum of charge transfer and diffusion transfer (77 ) at the anode = ri, +... 

Figure 3 shows the corresponding semi-integrals of the current, the hysteresis of which proves that the reduction step is not fast. But the cathodic limiting plateau proves that mass transfer is limited by diffusion. From the limiting value of the semi-integral, the value of the diffiision coefficient of HfCU . could be estimated as 2.4 10 cm s The... 

Figure 6 shows that at 700°C a single reduction step for which mass transfer is limited by diffusion (see semi-integral curve on figure 7). Since the peak intensity is proportional to the square root of the sweep rate, the reaction can be either reversible or irreversible, but by no means quasireversible. A complete study shows that this reaction can be considered as reversible. 

These results indicate that, as n increases, (yij) , the bottom product composition, tends to zero (relation (7.1.86a)). Although as n increases (yij-) increases, relation (7.1.85a) suggests that (yij-) = yiT)n-i- It system, the ratio (yi7-) /(yi does not tend to infinity, as suggested by (7.1.87) for large n-, instead, mass-transfer rate limitations between the phases and axial diffusion limit the possible enrichment (Pigford et ah, 1969a). Figure 7.1.19 compares the predictions from this equilibrium theory with the experimental data on an n-heptane-toluene separation... 

To increase the number of theoretical plates without increasing the length of the column, it is necessary to decrease one or more of the terms in equation 12.27 or equation 12.28. The easiest way to accomplish this is by adjusting the velocity of the mobile phase. At a low mobile-phase velocity, column efficiency is limited by longitudinal diffusion, whereas at higher velocities efficiency is limited by the two mass transfer terms. As shown in Figure 12.15 (which is interpreted in terms of equation 12.28), the optimum mobile-phase velocity corresponds to a minimum in a plot of H as a function of u. 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

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