Mass transfer equation for

This process has been used for various situations (1—14). Eor the condensation of a single component from a binary gas mixture, the gas-stream sensible heat and mass-transfer equations for a differential condenser section take the following forms ... 

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 ... 

Fig. 10. Numerical solutions of the forced-convection mass-transfer equation for the case of irreversible first-order chemical reaction [after Johnson et al. (J4)] (Solid lines— rigid spheres dashed lines—circulating gas bubbles).

An analytical solution of these mass-transfer equations for linear equilibrium was found by Thomas [36] for fixed bed operations. The Thomas solution can be further simplified if one assumes an infinitely small feed pulse (or feed arc in case of annular chromatography), and if the number of transfer units (n = k0azlu) is greater then five. The resulting approximate expression (Sherwood et al. [37]) is... 

Mass transfer equations for oxygen and carbon dioxide ... 

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

The sharp concentration fronts shown in Fig. [4.1-2 and 14.1-3 never occur in practice. The zones are always diluted and broadened by mass transfer and dispersion for both linear and nonlinenr isotherms. The complete solution of (he equilibrium equations, mass balances, and mass transfer equations for nonlinear systems is a formidable task requiring numerical solutions. For I incur systems the task is much easier and very useful solutions have been developed. Even thongh large-scale chromatography often is operated in the u out inear range, the linear analyses are valnable siace they can provide a qualitative feel (quantitative for linear systems) for band broadening effects. 

RATE OF MASS TRANSFER. Equations for mass transfer in fixed-bed adsorption are obtained by making a solute material balance for a section dL of the bed, as shown in Fig. 25.8. The rate of accumulation in the fluid and in the solid is the difference between input and output flows. The. change in superficial velocity is neglected ... 

As with LSV or CV, the mass-transfer equations for the various species must be changed to take account of loss or production of material because of the coupled reactions. Thus for the CE mechanism [(12.3.1) and (12.3.2)] at the RDE, (9.3.13) becomes... 

The solution of this equation would then follow as described in Section 9.4. The modifications of the mass-transfer equations for the different cases generally follow those used in voltammetric methods as shown in Table 12.2.1. Appropriate dimensionless parameters are listed in Table 12.3.1. It is usually not possible to solve these equations analytically, so various approximations (e.g., the reaction layer approach, as described in Section 1.5.2), digital simulations, or other numerical methods must be employed. The behavior of systems at the RDE can be analyzed by means of the zone diagrams employed for voltammetry (Section 12.3) by redefining the parameter A. This is accomplished by re-... 

When cloud and fog droplets have diameters significantly larger than 1 pm, mass transfer of water to a droplet can be expressed by the mass transfer equation for the continuum regime (see Chapter 12)... 

The subject has recently been reviewed by Carta and Cincotti (1998), who numerically integrated the mass transfer equations for intra-particle transport, to obtain the exact concentration profiles within the solid particle in different conditions. In this way they were able to evaluate the relative merits of the different approximations proposed and, from the results of their analysis, suggested an expression for the intra-particle mass transfer rate. Their expression explicitly takes into account both the form of the ion exchange isotherm and the value of the partition ratio A=psQlsN, in which Q is the ion exchange capacity of the solid considered, ps is its density, e is the void fraction and N is the total normality of the solution. 

MASS TRANSFER EQUATION FOR PSEUDO-BINARY MIXTURES... 

SIMPLIFICATION OF THE MASS TRANSFER EQUATION FOR PSEUDO-BINARY INCOMPRESSIBLE MIXTURES WITH CONSTANT PHYSICAL PROPERTIES... 

The product rule for the divergence operator is applied to both terms on the right-hand side of equation (9-27). In any coordinate system, the divergence of the product of a scalar and a vector is expanded as a product of the scalar and the divergence of the vector plus the scalar (i.e., dot) product of the vector and the gradient of the scalar. This vector identity was employed in equation (9-14). The pseudo-binary mass transfer equation for component i is... 

Hence, one term each on the left- and right-hand sides of equation (9-29) is zero. Since the mass density of component i, p, and the molar density of component i, Q, are related by molecular weight, division by MW, produces the final form of the mass transfer equation for incompressible pseudo-binary mixtures with constant physical properties ... 

The objective of this section is to identify the dimensionless transport numbers that appear in the mass transfer equation for component i. Order-of-magnitude estimates of the importance of one mass transfer rate process relative to another... 

The product of the Reynolds and Schmidt numbers, which counts as one dimensionless number, is equivalent to the Peclet number for mass transfer, PeMx- The Peclet number represents the ratio of the convective mass transfer rate process to the diffusion rate process of component, and it appears on the left-hand side of the dimensionless mass transfer equation for component i. The remaining r dimensionless transport numbers can be treated simultaneously because they represent ratios of scaling factors for the reactant-product conversion rate due to the jth independent chemical reaction relative to the rate of diffusion of component I. Hence,... 

For unsteady-state diffusion into a quiescent medium with no chemical reaction, the mass transfer Peclet number does not appear in the dimensionless mass transfer equation for species i because it is not appropriate to make variable time t dimensionless via division by L/ v) if there is no bulk fluid flow (i.e., (d) = 0). In this case, the first term on each side of equation (10-11) survives, which corresponds to the unsteady-state diffusion equation. However, the characteristic time for diffusion of species i over a length scale L, given by L /50i,mix. replaces L/ v) to make variable time t dimensionless. Now, the accumulation and diffusional rate processes scale as CAo i.mix/A, with dimensions of moles per volume per time. Since both surviving mass transfer rate processes exhibit the same dimensional scaling factor, there are no dimensionless numbers in the mass transfer equation which describes unsteady-state diffusion for species i in nonreactive systems. 

What important dimensionless number(s) appear in the dimensionless partial differential mass transfer equation for laminar flow through a blood capillary when the important rate processes are axial convection and radial diffusion ... 

Locally Flat Description. Analogous to the discussion on pages 279-280, one invokes the thin boundary layer approximation for either short contact times or small diffusivities and arrives at a locally flat description of the mass transfer equation for Ca(t, f) ... 

Dimensionless Molar Density. The final form of the mass transfer equation for Cp, y, t), which will be used to calculate the concentration profile and boundary layer thickness of species A in the liquid phase, is... 

Upon substitution into the mass transfer equation for P y, t), given by (11-191), with y = I Sc, one obtains... 

The mass transfer equation for the dimensionless molar density profile of mobile component A is... 

Answer Begin with the steady-state mass transfer equation for species A in an incompressible fluid with no chemical reaction. In vector form,... 

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