The Mass-Transfer Equations

To apply the mass transfer equation for design, the interfacial area, a, and mass transfer coefficient kL must be calculated. The interfacial area is dependent upon the bubble size and gas hold-up in the mixing vessel as given by ... 

Experimental results for fixed packed beds are very sensitive to the structure of the bed which may be strongly influenced by its method of formation. GUPTA and Thodos157 have studied both heat transfer and mass transfer in fixed beds and have shown that the results for both processes may be correlated by similar equations based on. / -factors (see Section 10.8.1). Re-arrangement of the terms in the mass transfer equation, permits the results for the Sherwood number (Sh1) to be expressed as a function of the Reynolds (Re,) and Schmidt numbers (Sc) ... 

The mass transfer equations, Equations (ll.l)-(ll.lO), remain valid when A, replaces A,. Equations (11.27) and (11.28) contain one independent variable, 2, and two dependent variables, ai and Ug. There are also two auxiliary variables, the interfacial compositions a and a. They can be determined using Equations (11.5) and (11.6) (with A, replacing A). The general case regards K/f in Equation (11.4) as a function of composition. When Henry s law applies throughout the composition range, overall coefficients can be used instead of the individual film coefficients. This allows immediate elimination of the interface compositions ... 

The mass transfer equation applicable to the transport-limited extraction of a solute from an aqueous solution to an organic phase (sink conditions), was derived ... 

The basic theory of mass transfer to a RHSE is similar to that of a RDE. In laminar flow, the limiting current densities on both electrodes are proportional to the square-root of rotational speed they differ only in the numerical values of a proportional constant in the mass transfer equations. Thus, the methods of application of a RHSE for electrochemical studies are identical to those of the RDE. The basic procedure involves a potential sweep measurement to determine a series of current density vs. electrode potential curves at various rotational speeds. The portion of the curves in the limiting current regime where the current is independent of the potential, may be used to determine the diffusivity or concentration of a diffusing ion in the electrolyte. The current-potential curves below the limiting current potentials are used for evaluating kinetic information of the electrode reaction. 

The size, shape and charge of the solute, the size and shape of the organism, the position of the organism with respect to other cells (plankton, floes, biofilms), and the nature of the flow regime, are all important factors when describing solute fluxes in the presence of fluid motion. Unfortunately, the resolution of most hydrodynamics problems is extremely involved, and typically bioavailability problems under environmental conditions are in the range of problems for which analytical solutions are not available. For this reason, the mass transfer equation in the presence of fluid motion (equation (17), cf. equation (14)) is often simplified as [48] ... 

All of the above discussion of diffusion considers physical motion of particles excited by thermal energy of the system (because the system is not at 0 K), rather than by outside factors. Eddy diffusion is different. It is due to random disturbance in water by outside factors, such as fish swimming, wave motion, ship cruising, and turbulence in water. On a small length scale (similar to the length scale of disturbance), the disturbances are considered explicitly as convection or flow in the mass transfer equation (Equation 3-19). On a length scale much larger than the individual disturbances, the collective effect of all of the disturbances... 

FORMULATION OF THE MASS TRANSFER EQUATIONS AND THEIR SOLUTIONS IN LAPLACE FORM... 

When E, = Ej we see that kjk2 = sA,/sl2 — a ratio independent of temperature. Hence, if the mass-transfer equations are divided they may, for the case E, = E2, be integrated directly and the gradients evaluated at the slab surface, x = L. Thus ... 

Membrane Techniques in which the Mass Transfer Equations are Used... 

This last equation is analogous to the mass transfer equation 10.66 ... 

The number of moles transferred per unit volume can also he calculated from the mass transfer equation... 

The effectiveness factor is obtained by the resolution of the mass transfer equations [12] in a spherical pellet with simple assumptions ... 

The mass transfer equation is written in terms of the usual assumptions. However, it must be considered that because the concentration of the more abundant species in the flowing gas mixture (air), as well as its temperature, are constant, all the physical properties may be considered constant. The only species that changes its concentration along the reactor in measurable values is PCE. Therefore, the radial diffusion can be calculated as that of PCE in a more concentrated component, the air. This will be the governing mass transfer mechanism of PCE from the bulk of the gas stream to the catalytic boundaries and of the reaction products in the opposite direction. Since the concentrations of nitrogen and oxygen are in large excess they will not be subjected to mass transfer limitations. The reaction is assumed to occur at the catalytic wall with no contributions from the bulk of the system. Then the mass balance at any point of the reactor is... 

Prediction of C ji Equation (22-85) shows a semilog dependency of wall concentration on flux. Experimentally the dependence of flux on concentration usually deviates significantly from linearity well before the zero-flux intercept extrapolated from data in Eig. 22-66. Experimental data at very low flux are difficult to gather, but usually the flux is much higher than the values extrapolated assuming linearity. The mass-transfer equations predict a mass-transfer coefficient, k, without reference to flux, as they were formulated for nonmembrane systems. This k is used by assumption to predict the wall concentration up to the point at which flux becomes independent of pressure. 

Interpha e Gas Transfer. From the heavy reliance in this symposium (see Table II) on the mass transfer equations proposed by Davidson and Harrison (27) and by Kunii and Levenspiel (19), one might reasonably conclude that these approaches have been supported by at least the majority of experimental evidence. Nothing could be further from the truth. 

The flow rate of both phases, viscosity, density, surface tension, and size and shape of the packing determine the value of a . These same factors affect the value of the mass transfer coefficients Ky and Kx. Therefore, it is expedient to include a in the mass transfer equation and define two new quantities KyU and Kxa. These quantities would then be correlated with the solution parameters as functions of various chemical systems. If A is the absorption tower cross-sectional area, and z the packing height, then Az is the tower packing volume. Defining Ai as the total interfacial area ... 

The mass transfer equation formed with the mass flux then reads ... 

The mass transfer equations discussed above are now combined with a material balance on the transferred component to calculate the column or packing height required for a given separation. The column cross-sectional area A is assumed known at this point although in a complete column design A must be determined based on pressure drop considerations. The column, which is in countercurrent flow with only liquid feed and vapor product at the top, and vapor feed and liquid product at the bottom (absorber, stripper, column section), is deflned as follows ... 

The HTU is determined from experimental data or empirical correlations, and the NTU is related to equilibrium and operating data. Since both HTU and NTU depend on the particular definition of mass transfer coefficients and compositions in the mass transfer equations, it is important to use compatible pairs of HTU and NTU. For instance. Equations 15.24 and 15.25 are rewritten as... 

To illustrate this work, the mass transfer equation for sublimation from spheres at pressures near atmospheric, according to Ranz and Marshall (Rl), is... 

In the diffusion boundary layer d, in the mass transfer equation one can neglect molecular tangential diffusion transfer compared with the diffusion in the radial direction the convective terms are retained (but somewhat simplified by linearization near the interface). The concentration distribution in that region was obtained earlier in Section 4.6. 

In the rear stagnation region W(3), the mass transfer equation can be somewhat simplified. In doing so, one must take into account convective terms as well as the radial and tangential molecular diffusion components. 

When T (x, y) —> 0, we will have a zero value for 8A/8y and also for the current density. However, if we consider that the concentration at the point (x, y) = (0,0) is also changing with the same function in the mass transfer Equation 15.14, we can calculate for almost zero values of Tq(x,y) ... 

The mass transfer equation (5.2) remains the same and equation... 

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