Mass transfer equilibrium conditions

It should be noted that the highest possible absorption rates will occur under conditions in which the hquid-phase resistance is negligible and the equilibrium back pressure of the gas over the solvent is zero. Such situations would exist, for instance, for NH3 absorption into an acid solution, for SO9 absorption into an alkali solution, for vaporization of water into air, and for H9S absorption from a dilute-gas stream into a strong alkali solution, provided there is a large excess of reagent in solution to consume all the dissolved gas. This is known as the gas-phase mass-transfer limited condition, wrien both the hquid-phase resistance and the back pressure of the gas equal zero. Even when the reaction is sufficiently reversible to allow a small back pres-... 

Assume that a simple film model exists for the mass transfer, equilibrium is established at the gas-liquid interface, and the diffusion occurs at isobaric and isothermal conditions. Also assume that neither helium nor argon is absorbed so that N2=N3 = 0. Then, the Maxwell-Stefan equations for the diffusion of argon and helium are... 

These models actually study kinetics of the mass transfer in conditions of the geological and hydro-geodynamic submodels. Thereby they accept chemical equilibrium only for homogenous reactions in water and ion exchange. Mass transfer between media is stretched in time, and values... 

For liquid/liquid extraction, data on mass transfer rate of the system at typical operating conditions are required. Also required are an applicable liquid/liquid equilibrium curve and data on chemical reactions occurring after mass transfer in the mixer. 

Equilibrium Considerations - Most of the adsorption data available from the literature are equilibrium data. Equilibrium data are useful in determining the maximum adsorbent loading which can be obtained for a specific adsorbate-adsorbent system under given operating conditions. However, equilibrium data by themselves are insufficient for design of an adsorption system. Overall mass transfer rate data are also necessary. 

Under certain conditions, it will be impossible for the metal and the melt to come to equilibrium and continuous corrosion will occur (case 2) this is often the case when metals are in contact with molten salts in practice. There are two main possibilities first, the redox potential of the melt may be prevented from falling, either because it is in contact with an external oxidising environment (such as an air atmosphere) or because the conditions cause the products of its reduction to be continually removed (e.g. distillation of metallic sodium and condensation on to a colder part of the system) second, the electrode potential of the metal may be prevented from rising (for instance, if the corrosion product of the metal is volatile). In addition, equilibrium may not be possible when there is a temperature gradient in the system or when alloys are involved, but these cases will be considered in detail later. Rates of corrosion under conditions where equilibrium cannot be reached are controlled by diffusion and interphase mass transfer of oxidising species and/or corrosion products geometry of the system will be a determining factor. 

The penetration theory has been used to calculate the rate of mass transfer across an interface for conditions where the concentration CAi of solute A in the interfacial layers (y = 0) remained constant throughout the process. When there is no resistance to mass transfer in the other phase, for instance when this consists of pure solute A, there will be no concentration gradient in that phase and the composition at the interface will therefore at all Limes lie the same as the bulk composition. Since the composition of the interfacial layers of the penetration phase is determined by the phase equilibrium relationship, it, too. will remain constant anil the conditions necessary for the penetration theory to apply will hold. If, however, the other phase offers a significant resistance to transfer this condition will not, in general, be fulfilled. 

However, under working conditions, with a current density j, the cell voltage E(j) decreases greatly as the result of three limiting factors the charge transfer overpotentials r]a,act and Pc,act at the two electrodes due to slow kinetics of the electrochemical processes (p, is defined as the difference between the working electrode potential ( j), and the equilibrium potential eq,i). the ohmic drop Rf. j, with the ohmic resistance of the electrolyte and interface, and the mass transfer limitations for reactants and products. The cell voltage can thus be expressed as... 

An alternative approach to the solution of the system dynamic equations, is by the natural cause and effect mass transfer process as formulated, within the individual phase balance equations. This follows the general approach, favoured by Franks (1967), since the extractor is now no longer constrained to operate at equilibrium conditions, but achieves this eventual state as a natural consequence of the relative effects of solute accumulation, solute flow in, solute flow out and mass transfer dynamics. 

The archetypal, stagewise extraction device is the mixer-settler. This consists essentially of a well-mixed agitated vessel, in which the two liquid phases are mixed and brought into intimate contact to form a two phase dispersion, which then flows into the settler for the mechanical separation of the two liquid phases by continuous decantation. The settler, in its most basic form, consists of a large empty tank, provided with weirs to allow the separated phases to discharge. The dispersion entering the settler from the mixer forms an emulsion band, from which the dispersed phase droplets coalesce into the two separate liquid phases. The mixer must adequately disperse the two phases, and the hydrodynamic conditions within the mixer are usually such that a close approach to equilibrium is obtained within the mixer. The settler therefore contributes little mass transfer function to the overall extraction device. 

Owing to the intensive agitation conditions and intimate phase dispersion, obtained within the mixing compartment, the mixer can usually be modelled as a single, perfectly mixed stage in which the rate of mass transfer is sufficient to attain equilibrium. As derived previously in Sec. 3.3.1.3, the component balance equations for the mixer, based on the two combined liquid phases, is thus given by... 

The mass transfer coefficient can be increased to obtain an approach to equilibrium conditions. 

Mass transfer in either the stationary or mobile phase is not instantaneous and, consequently, complete equilibrltui is not established tinder normal separation conditions. The result is that the solute concentration profile in the stationary phase is always displaced slightly behind the equilibrluM position and the mobile I se profile is similarly slightly in advance of the equilibrium position. The combined peak observed at the column outlet is broadened about its band center, which is located where it would have been for instantaneous equilibrium, provided the degree of nonequllibrluM is small. The stationary phase contribution to Mass transfer is given by equation (1.25)... 

The assumptions inherent in the derivation of the Hertz-Knudsen equation are (1) the vapor phase does not have a net motion (2) the bulk liquid temperature and corresponding vapor pressure determine the absolute rate of vaporization (3) the bulk vapor phase temperature and pressure determine the absolute rate of condensation (4) the gas-liquid interface is stationary and (5) the vapor phase acts as an ideal gas. The first assumption is rigorously valid only at equilibrium. For nonequilibrium conditions there will be a net motion of the vapor phase due to mass transfer across the vapor-liquid interface. The derivation of the expression for the absolute rate of condensation has been modified by Schrage (S2) to account for net motion in the vapor phase. The modified expression is... 

The linear driving force (LDF) approximation is obtained when the driving force is expressed as a concentration difference. It was originally developed to describe packed-bed dynamics under linear equilibrium conditions [Glueckauf, Trans. Far. Soc., 51, 1540 (1955)]. This form is exact for a nonlinear isotherm only when external mass transfer is controlling. However, it can also be used for nonlinear systems with pore or solid diffusion mechanisms as an approximation, since it provides qualitatively correct results. 

The proposed catalyst loading, that is the ratio by volume of catalyst to aniline, is to be 0.03. Under the conditions of agitation to be used, it is estimated that the gas volume fraction in the three-phase system will be 0.15 and that the volumetric gas-liquid mass transfer coefficient (also with respect to unit volume of the whole three-phase system) kLa, 0.20 s-1. The liquid-solid mass transfer coefficient is estimated to be 2.2 x 10-3 m/s and the Henry s law coefficient M = PA/CA for hydrogen in aniline at 403 K (130°C) = 2240 barm3/kmol where PA is the partial pressure in the gas phase and CA is the equilibrium concentration in the liquid. 

In industrial operations, adsorption is accomplished primarily on the surfaces of internal passages within small porous particles. Three basic mass transfer processes occur in series (1) mass transfer from the bulk gas to the particle surface, (2) diffusion through the passages within the particle, and (3) adsorption on the internal particle surfaces. Each of the processes depends on the system operating conditions and the physical and chemical characteristics of the gas stream and the solid adsorbent. Often, one of the transfer processes will be significantly slower than the other two and will control the overall transfer rate. The other process will operate nearly at equilibrium. 

The boundary conditions require knowledge of the interface concentration of hydrogen ChjL to compute E (see below). For hydrogenations, the equilibrium concentration (ChjL= CfJ L) can be used, albeit with the assumption of no mass transfer resistance on the gas side. Otherwise, it must be determined using Eq. (4). The boundary conditions for the substrate S state that it is not transferred to the gas phase - that is, S is not vaporized. This assumption is most often... 

Gas production and subsequent pressure-time histories can be investigated successfully only in pressure vessels such as the VSP. If the gaseous product dissolves partly in the reaction mixture (i.e., the vapor-liquid equilibrium is changed), careful investigations of the pressure effect within the possible variations of the operating conditions are necessary. Pressurized vessels are also useful to investigate any mass transfer improvement for gas-liquid or gas-dissolved (suspended) solid reactions. 

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