Equilibrium relations phase

When only the total system composition, pressure, and temperature (or enthalpy) are specified, the problem becomes a flash calculation. This type of problem requires simultaneous solution of the material balance as well as the phase-equilibrium relations. 

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. 

Cost estimation and screening external MSAs To determine which external MSA should be used to remove this load, it is necessary to determine the supply and target compositions as well as unit cost data for each MSA. Towards this end, one ought to consider the various processes undergone by each MSA. For instance, activated carbon, S3, has an equilibrium relation (adsorption isotherm) for adsorbing phenol that is linear up to a lean-phase mass fraction of 0.11, after which activated carbon is quickly saturated and the adsorption isotherm levels off. Hence, JC3 is taken as 0.11. It is also necessary to check the thermodynamic feasibility of this composition. Equation (3.5a) can be used to calculate the corresponding... 

An application of Eq. (19) is shown in Fig. 4, which gives the solubility of solid naphthalene in compressed ethylene at three temperatures slightly above the critical temperature of ethylene. The curves were calculated from the equilibrium relation given in Eq. (12). Also shown are the experimental solubility data of Diepen and Scheffer (D4, D5) and calculated results based on the ideal-gas assumption (ordinate scale is logarithmic and it is evident that very large errors are incurred when corrections for gas-phase nonideality are neglected. 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

Simultaneous solution of these equilibrium relations (coupled with the conservation equations x+ x-f = 1 and x/ + x/ = 1) gives the coexistence curve for the two-phase system as a function of pressure. 

Upon substitution into either one of the equations of stability [Eq. (98) or (99)], we can then determine whether the gas mixture exists in one or two stable phases. If two phases exist at some temperature and pressure, we can calculate the two phase compositions by utilizing the two equilibrium relations... 

Then, in this two-term unfolding model remains to define this exponent 2q, since all other quantities and especially the r-radius are either given, or evaluated from the thermodynamic equilibrium relations. Then, in this model the 2q-exponent is the characteristic parameter defining the quality of adhesion and therefore it may be called the adhesion coefficient. This exponent depends solely on the ratios of the main-phase moduli (Ef/Em), as well as on the ratio of the radii of the fiber and the mesophase. 

In this approach, it is assumed that turbulence dies out at the interface and that a laminar layer exists in each of the two fluids. Outside the laminar layer, turbulent eddies supplement the action caused by the random movement of the molecules, and the resistance to transfer becomes progressively smaller. For equimolecular counterdiffusion the concentration gradient is therefore linear close to the interface, and gradually becomes less at greater distances as shown in Figure 10.5 by the full lines ABC and DEF. The basis of the theory is the assumption that the zones in which the resistance to transfer lies can be replaced by two hypothetical layers, one on each side of the interface, in which the transfer is entirely by molecular diffusion. The concentration gradient is therefore linear in each of these layers and zero outside. The broken lines AGC and DHF indicate the hypothetical concentration distributions, and the thicknesses of the two films arc L and L2. Equilibrium is assumed to exist at the interface and therefore the relative positions of the points C and D are determined by the equilibrium relation between the phases. In Figure 10.5, the scales are not necessarily the same on the two sides of the interface. 

The number of degrees of freedom is represented by/. These are chosen from the list of all quantitatively related aspects of a system that can change. This includes T, P, and the concentrations of c components in each phase, c is the minimum number of components necessary to reproduce the system (ingredients), and p is the number of phases present at equilibrium. A phase is a domain with uniform composition and properties. Examples are a gas, a liquid solution, a solid solution, and solid phases. 

More complex situations where ideal behaviour can no longer be assumed require the incorporation of activity coefficient terms in the calculation of the equilibrium vapour compositions. Assuming ideal behaviour in the gas phase, the equilibrium relation for component i is... 

The equilibrium relation for the system may be taken as Ye = 3.OX, where Ye and X are expressed in mole ratios of pentane in the gas and liquid phases respectively. 

It can be assumed for almost all practical cases that equilibrium exists at the interface between the two phases. The concentrations of solute A at the interface, C A G and CAL, are related by the equilibrium relation of Eq. (5) and which, for gases in general and hydrogen in particular, is often described using simplified Henry s law applied at the interface (Eq. (6)). 

The fundamentals of the first aspect are dealt with in Sections 4.1 and 4.2, concerning the equilibrium relations and the transport processes, respectively. Furthermore, equilibrium aspects of the emission of hydrogen sulfide from the water phase into the sewer atmosphere are included in Section 4.1 as relevant and illustrative. 

Equilibrium relations are required to calculate the values of Cs, the solid phase equilibrium concentrations, for each component. For very dilute systems these relations may be of linear form... 

The phenomena of surface precipitation and isomorphic substitutions described above and in Chapters 3.5, 6.5 and 6.6 are hampered because equilibrium is seldom established. The initial surface reaction, e.g., the surface complex formation on the surface of an oxide or carbonate fulfills many criteria of a reversible equilibrium. If we form on the outer layer of the solid phase a coprecipitate (isomorphic substitutions) we may still ideally have a metastable equilibrium. The extent of incipient adsorption, e.g., of HPOjj on FeOOH(s) or of Cd2+ on caicite is certainly dependent on the surface charge of the sorbing solid, and thus on pH of the solution etc. even the kinetics of the reaction will be influenced by the surface charge but the final solid solution, if it were in equilibrium, would not depend on the surface charge and the solution variables which influence the adsorption process i.e., the extent of isomorphic substitution for the ideal solid solution is given by the equilibrium that describes the formation of the solid solution (and not by the rates by which these compositions are formed). Many surface phenomena that are encountered in laboratory studies and in field observations are characterized by partial, or metastable equilibrium or by non-equilibrium relations. Reversibility of the apparent equilibrium or congruence in dissolution or precipitation can often not be assumed. 

Equation 27 is similar to the solid-liquid equilibrium relation used for non-electrolytes. As in the case of the vapor-liquid equilibrium relation for HC1, the solid-liquid equilibrium expression for NaCl is simple since the electrolyte is treated thermodynamically the same in both phases. 

The phase-equilibrium relation for volatile electrolytes, such as HC1, has the advantage that the electrolyte in aqueous solution... 

Equations 2.15 and 2.16 are the general equilibrium relations for two coexisting phases. 

Sorption has been commonly described as an equilibrium process, in which the pesticide molecules are rapidly and readily exchanged between the sediment and aqueous phases. In this approach ( ), the equilibrium water phase concentration, (expressed relative to suspension volume) is related to the sediment phase concentration, (expressed relative to dry weight sediment), through... 

Let us now consider in detail some of the theoretical possibilities of solid solution. We will look at the microscopic and sub-microscopic effect of each type and how this determines the observed X-ray properties and treatment of phase equilibrium relations of the minerals. 

It would lie far beyond the aim of this chapter to introduce the state-of-the art concepts that have been developed to quantify the influence of colloids on transport and reaction of chemicals in an aquifer. Instead, a few effects will be discussed on a purely qualitative level. In general, the presence of colloidal particles, like dissolved organic matter (DOM), enhances the transport of chemicals in groundwater. Figure 25.8 gives a conceptual view of the relevant interaction mechanisms of colloids in saturated porous media. A simple model consists of just three phases, the dissolved (aqueous) phase, the colloid (carrier) phase, and the solid matrix (stationary) phase. The distribution of a chemical between the phases can be, as first step, described by an equilibrium relation as introduced in Section 23.2 to discuss the effect of colloids on the fate of polychlorinated biphenyls (PCBs) in Lake Superior (see Table 23.5). 

In any event, there is always a strict, equilibrium relation between the charge density, qM, on the electrode sur ce and the total potential difference, E, between the bulk phases of electrode and solution. This relation is often characterized by the differential double-layer capacity, Cd, defined as... 

Kay, W.B. Liquid-Vapor Phase Equilibrium Relations in the Ethane-n-Heptane System, hid. Eng. Chem. (1938) 30, 459-465. 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

These distinctions between the two operations are partly traditional. The equipment is similar, and the mathematical treatment, which consists of material and energy balances and phase equilibrium relations, also is the same for both. The fact, however, that the bulk of the liquid phase in absorption-stripping plants is nonvolatile permits some simplifications in design and operation. 

E equations—phase Equilibrium relation for each component (C equations for each stage) ... 

Calculations of the relations between the input and output amounts and compositions and the number of extraction stages are based on material balances and equilibrium relations. Knowledge of efficiencies and capacities of the equipment then is applied to find its actual size and configuration. Since extraction processes usually are performed under adiabatic and isothermal conditions, in this respect the design problem is simpler than for thermal separations where enthalpy balances also are involved. On the other hand, the design is complicated by the fact that extraction is feasible only of nonideal liquid mixtures. Consequently, the activity coefficient behaviors of two liquid phases must be taken into account or direct equilibrium data must be available. 

Equilibrium relations in leaching usually are simpler than in liquid-liquid equilibria, or perhaps only appear so because few measurements have been published. The solution phase normally contains no entrained solids so its composition appears on the hypotenuse of a triangular diagram like that of Example 14.8. Data for the raffinate phase may be measured as the holdup of solution by the solid, K lb solution/lb dry (oil-free) solid, as a function of the concentration of the solution, y lb oil/lb solution. The correspond-... 

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