Equilibrium relation

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. 

Both equilateral and right triangular diagrams have the property that the compositions of mixtures of all proportions of two mixtures appear on the straight line connecting the original 

Experimental data on only 26 quaternary systems were found by Sorensen and Arlt (1979), and none of more complex systems, although a few scattered measurements do appear in the literature. Graphical representation of quaternary systems is possible but awkward, so that their behavior usually is analyzed with equations. To a limited degree of accuracy, the phase behavior of complex mixtures can be predicted from measurements on binary mixtures, and considerably better when some ternary measurements also are available. The data are correlated as activity coefficients by means of the UNIQUAC or NRTL equations. The basic principle of application is that at equilibrium the activity of each component is the same in both phases. In terms of activity coefficients this 

Several tieline correlations in equation form have been proposed, of which three may be presented. They are expressed in weight fractions identified with these subscripts  

CA solute C in diluent phase A CS solute C in solvent phase S SS solvent S in solvent phase S AA diluent A in diluent phase A AS diluent A in solvent phase S SA solvent S in diluent phase A. 

When an adsorbent is in contact with the surrounding fluid of a certain composition, adsorption takes place and after a sufficiently long time, the adsorbent and the surrounding fluid reach equilibrium. In this state the amount of the component adsorbed on the surface mainly of the micropore of the adsorbent is determined as shown in Fig. 3.1. The relation between amount adsorbed, q, and concentration in the fluid phase, C, at 

The relation between concentration and temperature yielding a given amount adsorbed, q, is called the adsorption isostere (Fig. 3 2), 

The distribution coefficient A) is the ratio of activity coefficients and may be estimated from binary infinite dilution coefficient data. 

Binary interaction parameters (Ay) and infinite dilution activity coefficients are available for a wide variety of binary pairs. Therefore the ratio of the solute infinite dilution coefficient in solvent-rich phase to that of the second phase ( ) will provide an estimate of the equilibrium distribution coefficient. The method can provide a reasonable estimate of the distribution coefficient for dilute cases. 

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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. 

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

Substitution of Equations (2) and (3) into the equilibrium relations dictated by Equation (2-l)[Pg.99]

Subroutine MULLER. MULLER iteratively solves the equilibrium relations and computes the equilibrium vapor composition when organic acids are present. These compositions are used by subroutine PHIS2 to calculate fugacity coefficients by the chemical theory. 

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. 

Pi =f Ci) or Pi = HCi, equilibrium relation at the interface a = interfacial area/iinit volume Zg, Z-L = film thicknesses The steady rates of solute transfer are... 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

References A variety of mathematical methods are proposed to cope with hnear (e.g., material balances based on flows) and nonhnear (e.g., energy balances and equilibrium relations) constraints. Methods have been developed to cope with unknown measurement uncertainties and missing measurements. The reference list provides ample insight into these methods. See, in particular, the works by Mah, Crowe, and Madron. However, the methods all require more information than is tvpicaUy known in a plant setting. Therefore, even when automated methods are available, plant-performance analysts are well advised to perform initial adjustments by hand. 

One way of calculating the number of equilibrium stages (or number of theoretical plates, NTP) for a mass exchanger is the graphical McCabe-Thiele method. To illustrate this procedure, let us assume that over the operating range of compositions, the equilibrium relation governing the transfer of the pollutant from the... 

A A multistage extraction column uses gas oil for the preliminary removal of phenol from wastewater. The flowrate of wastewater is 2.0 kg/s and its inlet mass fraction of phenol is 0.0358. The mass fraction of phenol in the wastewater exiting the column is 0.0168. Five kg/s of gas oil are used for extraction. The inlet mass fraction of phenol in gas oil is 0.0074. The equilibrium relation for the transfer of phenol from wastewater to gas oil is given by... 

The equilibrium relation for stripping TCE from water is theoretically predicted using Eq. (2.5) to be... 

The absorption operation is assumed to be isothermal with the following equilibrium relation (King, 1980) ... 

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... 

Several interception techniques should be considered to remove methanol. Air stripping is among these methods. The equilibrium relation for stripping from methanol aqueous streams may be approximated by the following expression... 

Two external MSAs may be considered for removal of chlorine ions activated carbon (Si) and ion exchange resin (Si). The equilibrium relation for removing chlorine by these MSAs is given by y =... 

Consider an aqueous caustic soda solution whose molarity mi = 5.0 kmol/m (20 wt.% NaOH). This solution is to be used in >scH(t>ing H2S from a gaseous waste. The operating range of interest is 0.0 < xi kmoUn ) < 5.0. Derive an equilibrium relation for this chemical absorption over the operating range of interest. 

Now that a procedure for establishing the corresponding composition scales for the rich lean pairs of stream has been outlined, it is possible to develop the CID. The CID is ccHistructed in a manner similar to that described in Chapter Five. However, it should be noted that the conversion among the corresponding composition scales may be more laborious due to the nonlinearity of equilibrium relations. Furthermore, a lean scale, xj, represents all forms (physically dissolved and chemically combined) of the pollutant. First, a composition scale, y, for component A in... 

Heretofore, the presented MEN/REAMEN synthesis techniques were applicable to the cases where mass-exchange temperatures are known ahead of the synthesis task. Mass-exchange equilibrium relations are dependent upon temperature and the selection of optimal mass-exchange temperatures is an important element of design. In selecting these temperatures, there is a tradeoff between cost of MSAs and cost of heating/cooling utilities. 

In the range of operating temperatures and compositions, the equilibrium relations are monotonic functions of temperature of the MSA. This is typically true. For instance, normally in gas absorption Henry s coefficient monotonically decreases as the temperature of the MSA is lowered while for stripping the gas-liquid distribution coefficient monotonically increases as the temperature of the stripping agent is increased. 

Within the considered range of operation, the equilibrium relation for anunonia scrubbing in water is dependent on the temperature as follows ... 

Because the interfacial slippage is assumed to be caused by the thin TLCP-rich interlayers, we can form a stress equilibrium relation as ... 

A typical example is as follows. Benzoic acid, C6H5COOH, is a solid substance with only moderate solubility in water. The aqueous solutions conduct electric current and have the other properties of an acid listed in Section 11-2.1. We can describe this behavior with reaction (42) leading to the equilibrium relation (43) ... 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

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... 

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