Rate of transfer

The right hand side represents the rate of momentum transfer from species r by mechanism (i) and, combining this with the rate of transfer by mechanism (ii) as given by equation (2.IS), we obtain... 

From equation 23, it can be seen that the higher the power input per unit volume, the lower the oxygen transfer efficiency. Therefore, devices should be compared at equal transfer rates. AH devices become less energy efficient as rates of transfer increase (3). 

Any one of the five basic processes may be responsible for limiting the extraction rate. The rate of transfer of solvent from the bulk solution to the soHd surface and the rate into the soHd are usually rapid and are not rate-limiting steps, and the dissolution is usually so rapid that it has only a small effect on the overall rate. However, knowledge of dissolution rates is sparse and the mechanism may be different in each soHd (1). 

Agitation of the Fluid. Agitation of the solvent increases local turbulence and the rate of transfer of material from the surface of the particles to the bulk of the solution. Agitation should prevent settling of the soHds, to enable most effective use of the interfacial area. 

Performance assessments are predictions of radioactivity releases, the rate of transfer of contaminants through various media, and the potential for hazard to the pubHc. These are based on a combination of experimental data obtained in the process called site characterization and detaded computations about radionuchdes and their effects. The progressive attack on the metal or ceramic waste container, the diffusion of water into the waste form, the leaching of the radioactive compounds, diffusion out, and washing away of radionuchdes are all considered. 

Removing an analyte from a matrix using supercritical fluid extraction (SEE) requires knowledge about the solubiUty of the solute, the rate of transfer of the solute from the soHd to the solvent phase, and interaction of the solvent phase with the matrix (36). These factors collectively control the effectiveness of the SEE process, if not of the extraction process in general. The range of samples for which SEE has been appHed continues to broaden. Apphcations have been in the environment, food, and polymers (37). 

A simplified diagram representing the various reservoirs and transport mechanisms and pathways involved in the cycles of nutrient elements at and above the surface of the Earth is given in Eigure 1. The processes are those considered to be the most important in the context of this article, but others of lesser significance can be postulated. Eor some of the elements, notably carbon, sulfur, chlorine, and nitrogen, considerable research has been done to evaluate (quantitatively) the amount of the various elements in the reservoirs and the rates of transfer. 

Under equiUbrium or near-equiUbrium conditions, the distribution of volatile species between gas and water phases can be described in terms of Henry s law. The rate of transfer of a compound across the water-gas phase boundary can be characterized by a mass-transfer coefficient and the activity gradient at the air—water interface. In addition, these substance-specific coefficients depend on the turbulence, interfacial area, and other conditions of the aquatic systems. They may be related to the exchange constant of oxygen as a reference substance for a system-independent parameter reaeration coefficients are often known for individual rivers and lakes. 

Transfer of Disperse Dye on Polyester. A specimen of dyed polyester is placed in a standard dyebath with an equal weight of undyed polyester and the dyeing cycle completed. The rate of transfer from dyed to undyed fabric is compared to that obtained with a range of five standard dyes and the dye under test is given the same number as the dye it most closely resembles. 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

For systems in which the solute concentrations in the gas and hquid phases are dilute, the rate of transfer may be expressed by equations which predic t that the rate of mass transfer is proportional to the difference between the bulk concentration and the concentration at the gas-liquid interface. Thus... 

Tbe mass-transfer coefficients k c and /cf by definition are equal to tbe ratios of tbe molal mass flux Na to tbe concentration driving forces p — Pi) and (Ci — c) respectively. An alternative expression for tbe rate of transfer in dilute systems is given by... 

The interfacial mole fractions yj and Xj can be determined by solving Eq. (5-252) simultaneously with the equilibrium relation = F(x,) to obtain y and Xj. The rate of transfer may then be calculated from... 

If the equilibrium relation y° = F Xi) is sufficiently simple, e.g., if a plot of yfversus Xi is a straight hne, not necessarily through the origin, the rate of transfer is proportional to the difference between the bulk concentration in one phase and the concentration (in that same phase) which would be in equilibrium with the bulk concentration in the second phase. One such difference isy — y°, and another is x° — x. In this case, there is no need to solve for the interfacial compositions, as may be seen from the following derivation. 

If it is desired to calculate the rate of transfer from the overall concentration difference based on bulk-hquid compositions (x° — x), the appropriate overall coefficient Kl is related to the individual coefficients by the equation... 

When the equilibrium curve is not straight, there is no strictly logical basis for the use of an overall transfer coefficient, since the value of m will be a function of position in the apparatus, as can be seen from Fig. 5-27. In such cases the rate of transfer must be calculated by solving for the interfacial compositions as described above. 

Experimentally observed rates of mass transfer often are expressed in terms of overall transfer coefficients even when the eqmlibrium lines are curved. This procedure is empirical, since the theory indicates that in such cases the rates of transfer may not vary in direct proportion to the overall bulk concentration differences y — y°) and (x° — x) at all concentration levels even though the rates may be proportional to the concentration difference in each phase taken separately, i.e., Xi — x) and y — y ). 

When liqiiid-phase chemical reactions are extremely slow, the gas-phase resistance can be neglected and one can assume that the rate of reaction has a predominant effect upon the rate of absorption. In this case the differential rate of transfer is given by the equation... 

Batch Stirred Tanks Tanks agitated by coaxial impellers (turbines, paddles, or propellers) are commonly used for batch dissolution of solids in liquids and may be used for leaching fine solids. Insofar as the controlhng rate in the mass transfer is the rate of transfer of mate-... 

Dispersion The movement of aggregates of molecules under the influence of a gradient of concentration, temperature, and so on. The effect is represented hy Tick s law with a dispersion coefficient substituted for molecular diffusivity. Thus, rate of transfer = —Dj3C/3p). 

The numerical solution of these equations is shown in Fig. 23-28. This is a plot of the enhancement fac tor E against the Hatta number, with several other parameters. The factor E represents an enhancement of the rate of transfer of A caused by the reaction compared with physical absorption with zero concentration of A in the liquid. The uppermost line on the upper right represents the pseudo-first-order reaction, for which E = P coth p. 

Liquid/Solid Mass Transfer The dissolved gas and the solvent react in contact with the surface of the catalyst. For studying the rate of transfer to the surface, an often-used system was benzoic acid or naphthalene in contact with water. A correlation of Dharwadkar and Sylvester (AJChE Journal, 23, 376 [1977]) that agrees well with some others is... 

Unloading or loading liquefied gas escapes should be limited to those vented via safety devices or authorized by the harbour master. Rate of transfer should ensure pipelines cool gradually lines should be vented safely at the end of the transfer. 

This form is partieularly appropriate when the gas is of low solubility in the liquid and "liquid film resistanee" eontrols the rate of transfer. More eomplex forms whieh use an overall mass transfer eoeffieient whieh ineludes the effeets of gas film resistanee must be used otherwise. Also, if ehemieal reaetions are involved, they are not rate limiting. The approaeh given here, however, illustrates the required ealeulation steps. The nature of the mixing or agitation primarily affeets the interfaeial area per unit volume, a. The liquid phase mass transfer eoeffieient, kL, is primarily a funetion of the physieal properties of the fluid. The interfaeial area is determined by the size of the gas bubbles formed and how long they remain in the mixing vessel. The size of the bubbles is normally expressed in terms of their Sauter mean diameter, dj, whieh is defined below. How long the bubbles remain is expressed in terms of gas hold-up, H, the fraetion of the total fluid volume (gas plus liquid) whieh is oeeupied by gas bubbles. 

The performance equation of a mixer relates mixer size or mixing time to the input and output of the mixing device. The rate of transfer, r, incorporates the mass transfer coefficient, kL, and interfacial area, a, as calculated above. This rate can be used in conjunction with a material balance to relate concentrations of interest to time or size variables. 

Rates of transfer of mass, energy, individual eomponents, or speeies aeross the boundaries of the region. 

The diffusion coefficient, or diffusivity (D), is a measure of tlie rate of transfer of one substance tlirough another. The diffusivity for component A migrating through a solute B, for example, is given by the ratio of the flux Ja (mass... 

---