Equation mass transfer

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 ... 

Simultaneous heat and mass transfer also occurs in drying processes, chemical reaction steps, evaporation, crystallisation, and distillation. In all of these operations transfer rates are usually fixed empirically. The process can be evaluated using either the heat- or mass-transfer equations. However, if the process mechanism is to be fully understood, both the heat and mass transfer must be described. Where that has been done, improvements in the engineering of the process usually result (see Process energy conservation). 

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 ... 

Over the years the original Evans diagrams have been modified by various workers who have replaced the linear E-I curves by curves that provide a more fundamental representation of the electrode kinetics of the anodic and cathodic processes constituting a corrosion reaction (see Fig. 1.26). This has been possible partly by the application of electrochemical theory and partly by the development of newer experimental techniques. Thus the cathodic curve is plotted so that it shows whether activation-controlled charge transfer (equation 1.70) or mass transfer (equation 1.74) is rate determining. In addition, the potentiostat (see Section 20.2) has provided... 

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).

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. 

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

A more recent review by Fahidy (FI) concerns the chemical engineering approach to electrochemical processes, such as fluidized-bed reactors, bipolar particulate reactors, pulsed electrochemical reactors, gas-phase electrochemical reactors, electrocrystallization and electrodissolution, and the enhancement of heat and mass transfer in electric fields. In this review, the author also discusses dimensionless mass-transfer equations applied in cell design. Such equations are reviewed in greater detail in Section VI. 

A unified gas hydrate kinetic model (developed at ARC) coupled with a thermal reservoir simulator (CMG STARS) was applied to simulate the dynamics of CH4 production and C02 sequestration processes in the Mallik geological zones. The kinetic model contains two mass transfer equations one equation transfers gas and water into hydrate, and a decomposition equation transfers hydrate into gas and water (Uddin etal. 2008a). 

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] ... 

Dimensionless blend time method, 16 688 Dimensionless groups, 15 685, 686t, 687t Dimensionless mass transfer equation,... 

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... 

Fig. 41. A comparison among mass transfer equations and the droplet evaporation data of Zhang and Davis (1987) and Taflin and Davis (1987).

Since the differential spreading pressure An will oppose the movement of the eddy at the interface, it will also oppose surface renewal and hence mass transfer. Equation (16) explains the form of the plot of l ig. 11. 

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... 

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