Equilibrium, mass transfer

Approximately 7.9 and 7.6 pore volumes (8.7 L and 8.3 L) of surfactant solution were flushed through Box A and Box B, respectively. The effects of non-equilibrium mass transfer on PCE recovery were assessed through a series of flow interruptions, lasting from 12 to 17 hours. PCE remaining in each box after the surfactant flushing procedure was extracted with isopropanol and analyzed by GC. Total PCE mass balances of greater than 94% PCE was achieved in both box studies. 

A number of industrially important processes, such as distillation, absorption, and extraction, bring two phases into contact. When the phases are not in equilibrium, mass transfer occurs between the phases. The rate of transfer of each species depends on the departure of the system from equilibrium. Quantitative treatment of mass-transfer rates requires knowledge of the equilibrium states (T, P, and compositions) of the system. 

MINTEQ ( was the equilibrium mass-transfer code chosen to model the geochemical interaction between leachate and sandstone. The model was selected to perform three functions for which it is well-suited ... 

At the microscopic, molecular level, very complex theoretical equations are required to describe the chromatographic process. These include expressions for laminar or turbulent fluid flow random walk, diffusional broadening of analyte bands in both the mobile and stationary phases and the kinetics of near-equilibrium mass transfer between the phases. Such discussions are beyond the scope of this text. 

At equilibrium, mass transfer between the phases of a system is reversible and a material balance requires... 

A fully description of the non-equilibrium mass transfer phenomenon embraces simultaneously a Maxwell-Stefan diffusion model and a flow model. The most commonly used approach is the film model, which is addressed in more detail in the explanatory note 2.1. 

Author Equilibrium Mass Transfer Heat Transfer Experimental System Remarks... 

A widely used method for adsorption of solutes from liquid or gases employs a fixed bed of granular particles. The fluid to be treated is usually passed down through the packed bed at a constant flow rate. The situation is more complex than that for a simple stirred-tank batch process which reaches equilibrium. Mass-transfer resistances are important in the fixed-bed process and the process is unsteady state. The overall dynjimics of the system determines the efliciency of the process rather than just the equilibrium considerations. 

For proper design and simulation of HDT reactors, kinetic and reactor modeling are aspects that need to be deeply studied however, this is not a trivial task due to the numerous physical and chemical processes that occur simultaneously in the reactor phase equilibrium, mass transfer of reactants and products between the gas-liquid-solid phases, diffusion inside the catalyst particle, a complex reaction network, and catalyst deactivation. Ideally, the contribution of all these events must be coupled into a robust reactor performance model. The level of sophistication of the model is generally defined based upon the pursued objectives and prediction capability [4]. 

Equation 28 and its liquid-phase equivalent are very general and valid in all situations. Similarly, the overall mass transfer coefficients may be made independent of the effect of bulk fiux through the films and thus nearly concentration independent for straight equilibrium lines ... 

Design Methods. Improvements ia the ability to predict multicomponent equilibrium and mass-transfer rate performance will allow significant improvements ia the design of new adsorption systems and ia the energy efficiency of existing systems. 

Interfacial Contact Area and Approach to Equilibrium. Experimental extraction cells such as the original Lewis stirred cell (52) are often operated with a flat Hquid—Hquid interface the area of which can easily be measured. In the single-drop apparatus, a regular sequence of drops of known diameter is released through the continuous phase (42). These units are useful for the direct calculation of the mass flux N and hence the mass-transfer coefficient for a given system. 

Consider a closed PVT system, either homogeneous or heterogeneous, of uniform T and P, which is in thermal and mechanical equilibrium with its surroundings, but which is not initially at internal equilibrium with respect to mass transfer or with respect to chemical reaction. Changes occurring in the system are then irreversible, and must necessarily bring the system closer to an equihbrium state. The first and second laws written for the entire system are... 

Mass-Transfer Principles Dilute Systems When material is transferred from one phase to another across an interface that separates the two, the resistance to mass transfer in each phase causes a concentration gradient in each, as shown in Fig. 5-26 for a gas-hquid interface. The concentrations of the diffusing material in the two phases immediately adjacent to the interface generally are unequal, even if expressed in the same units, but usually are assumed to be related to each other by the laws of thermodynamic equihbrium. Thus, it is assumed that the thermodynamic equilibrium is reached at the gas-liquid interface almost immediately when a gas and a hquid are brought into contact. 

Kl Xbm /cl x bm kc y yt When the equilibrium cui ve is a straight line, the terms in parentheses can be replaced by the slope m as before. In this case the overall mass-transfer coefficients for concentrated systems are related to each other by the equation... 

For the liquid-phase mass-transfer coefficient /cl, the effects of total system pressure can be ignored for all practical purposes. Thus, when using Kq and /cl for the design of gas absorbers or strippers, the primary pressure effects to consider will be those which affect the equilibrium curves and the values of m. If the pressure changes affect the hydrodynamics, then Icq, and a can all change significantly. 

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-... 

Wet-bulb temperature is the dynamic equilibrium temperature attained by a water surface when the rate of heat transfer to the surface by convection equals the rate of mass transfer away from the surface. At equilibrium, if neghgible change in the dry-bulb temperature is assumed, a heat balance on the surface is... 

Tti e wet-bulb temperature is established by a dynamic equilibrium between heat and mass transfer when liquid evaporates from a small mass, such as the wet bulb of a thermometer, into a veiy large mass of gas such that the latter undergoes no temperature or humidity change. It is expressed by the relationship... 

The general expression given by Eq. (14-8) is more complex than normally is required, but it must be used when the mass-transfer coefficient varies from point to point, as may be the case when the gas is not dilute or when the gas velocity varies as the gas dissolves. The values of yi to be used in Eq. (14-8) depend on the local hquid composition Xi and on the temperature. This dependency is best represented by using the operating and equilibrium lines as discussed later. 

All these processes are, in common, liquid-gas mass-transfer operations and thus require similar treatment from the aspects of phase equilibrium and kinetics of mass transfer. The fluid-dynamic analysis ofthe eqmpment utihzed for the transfer also is similar for many types of liquid-gas process systems. 

The separation of components by liquid-liquid extraction depends primarily on the thermodynamic equilibrium partition of those components between the two liquid phases. Knowledge of these partition relationships is essential for selecting the ratio or extraction solvent to feed that enters an extraction process and for evaluating the mass-transfer rates or theoretical stage efficiencies achieved in process equipment. Since two liquid phases that are immiscible are used, the thermodynamic equilibrium involves considerable evaluation of nonideal solutions. In the simplest case a feed solvent F contains a solute that is to be transferred into an extraction solvent S. 

The concept of a mass-transfer unit was developed many years ago to represent more rigorously what happens in a differential contactor rather than a stagewise contactor. For a straight operating line and a straight equilibrium line with an intercept of zero, the equation for calculating the number of mass-transfer units based on the overall raffinate phase N r is identical to the Kremser equation except for the denominator when the extraction factor is not equal to 1.0 [Eq. (15-23)]. 

Methods for analysis of fixed-bed transitions are shown in Table 16-2. Local equilibrium theoiy is based solely of stoichiometric concerns and system nonlinearities. A transition becomes a simple wave (a gradual transition), a shock (an abrupt transition), or a combination of the two. In other methods, mass-transfer resistances are incorporated. 

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

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