Phase Equilibrium Relationships

The additional equations that will be needed to complete the mathematical model of the process (i.e., transport and kinetic rate expressions, reaction and phase equilibria relationships, etc.)... 

Tridymite In his classical effort to determine phase equilibria relationships among the silica polymorphs, Fenner (1913) observed that tridymite could be synthesized only with the aid of a mineralizing agent or flux such as Na2WQj. If pure quartz is heated, it bypasses tridymite and transforms... 

Vapor/liquid equilibrium (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

In processing, it is frequently necessary to separate a mixture into its components and, in a physical process, differences in a particular property are exploited as the basis for the separation process. Thus, fractional distillation depends on differences in volatility. gas absorption on differences in solubility of the gases in a selective absorbent and, similarly, liquid-liquid extraction is based on on the selectivity of an immiscible liquid solvent for one of the constituents. The rate at which the process takes place is dependent both on the driving force (concentration difference) and on the mass transfer resistance. In most of these applications, mass transfer takes place across a phase boundary where the concentrations on either side of the interface are related by the phase equilibrium relationship. Where a chemical reaction takes place during the course of the mass transfer process, the overall transfer rate depends on both the chemical kinetics of the reaction and on the mass transfer resistance, and it is important to understand the relative significance of these two factors in any practical application. 

Each of these processes is characterised by a transference of material across an interface. Because no material accumulates there, the rate of transfer on each side of the interface must be the same, and therefore the concentration gradients automatically adjust themselves so that they are proportional to the resistance to transfer in the particular phase. In addition, if there is no resistance to transfer at the interface, the concentrations on each side will be related to each other by the phase equilibrium relationship. Whilst the existence or otherwise of a resistance to transfer at the phase boundary is the subject of conflicting views"8 , it appears likely that any resistance is not high, except in the case of crystallisation, and in the following discussion equilibrium between the phases will be assumed to exist at the interface. Interfacial resistance may occur, however, if a surfactant is present as it may accumulate at the interface (Section 10.5.5). 

The relation between CAi[ and CAi2 is determined by the phase equilibrium relationship since the molecular layers on each side of the interface are assumed to be in equilibrium with one another. It may be noted that the ratio of the differences in concentrations is inversely proportional to the ratio of the mass transfer coefficients. If the bulk concentrations, CAt> and CA02 are fixed, the interface concentrations will adjust to values which satisfy equation 10.98. This means that, if the relative value of the coefficients changes, the interface concentrations will change too. In general, if the degree of turbulence of the fluid is increased, the effective film thicknesses will be reduced and the mass transfer coefficients will be correspondingly increased. 

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. 

Only two of the four state variables measured in a binary VLE experiment are independent. Hence, one can arbitrarily select two as the independent variables and use the EoS and the phase equilibrium criteria to calculate values for the other two (dependent variables). Let Q, (i=l,2,...,N and j=l,2) be the independent variables. Then the dependent ones, g-, can be obtained from the phase equilibrium relationships (Modell and Reid, 1983) using the EoS. The relationship between the independent and dependent variables is nonlinear and is written as follows... 

The use of phase equilibrium relationships and other constraints in determining stream compositions and flows is discussed in more detail in Chapter 4. 

This example illustrates the use of phase equilibrium relationships (vapour-liquid) in material balance calculations. 

Dewpoint calculations must be made when we know the composition of the vapor yj and P (or T) and want to find the liquid composition Xj and T (or P). Flash calculations must be made when we know neither Xj nor yj and must combine phase equilibrium relationships, component balance equations, and an energy balance to solve for all the unknowns. 

The digital simulation of a distillation column is fairly straightforward. The main complication is the large number of ODEs and algebraic equations that must be solved. We will illustrate the procedure first with the simplified binary distillation column for which we developed the equations in Chap. 3 (Sec. 3.11). Equimolal overflow, constant relative volatility, and theoretical plates have been assumed. There are two ODEs per tray (a total continuity equation and a light component continuity equation) and two algebraic equations per tray (a vapor-liquid phase equilibrium relationship and a liquid-hydraulic relationship). 

Aqueous Solubility. Solubility of a chemical in water can be calculated rigorously from equilibrium thermodynamic equations. Because activity coefficient data are often not available from the literature or direct experiments, models such as UNIFAC can be used for structure—activity estimations (24). Phase-equilibrium relationships can then be applied to predict miscibility. Simplified calculations are possible for low miscibility, however, when there is a high degree of miscibility, the phase-equilibrium relationships must be solved rigorously. 

In addition, the predicted orders in eqs (7.18) and (7.19) do not take into account phase equilibrium relationships involving coexisting minerals. For example, olivine should always be enriched in Fe2+ ions relative to orthopyroxene, according the the CFSE data plotted in fig. 7.6. Moreover, studies of the system MgO-FeO-Si02 at atmospheric pressure (Bowen and Schairer, 1935) showed that magnesium olivine crystallizes first and becomes increasingly enriched in iron before pyroxene commences to crystallize. The olivine main-... 

Relate the gas phase to the liquid phase. Using phase-equilibrium relationships (see Section 3), the following equation can be set out for each of the two components ... 

For a first approximation to the solution, we will assume that essentially all the methanol condenses, with only trace amounts appearing in the recycle line. We will also assume that most of the water condenses and that very small amounts of carbon monoxide, carbon dioxide, hydrogen, methane, and nitrogen dissolve in the condensate. To account for methanol and water vapor in the recycle gases and the solubility of the gases in the crade methanol, we would have to include phase equilibrium relationships in the analysis. As stated earlier, several condensable byproducts, high and low-boiling compounds in the cmde methanol, are present in small amounts, as shown in Table 3.5.1. We will not consider these compounds in the synthesis-loop analysis. 

To perform energy balance calculations on a reactive system, proceed much as you did for nonreactive systems (a) draw and label a flowchart (b) use material balances and phase equilibrium relationships such as Raoult s law to determine as many stream component amounts or flow rates as possible (c) choose reference states for specific enthalpy (or internal energy) calculations and prepare and fill in an inlet-outlet enthalpy (or internal energy) table and (d) calculate AH (or AC/ or A/C), substitute the calculated value in the appropriate form of the energy balance equation, and complete the required calculation. 

Once this interpretation has been established, MODEL.LA. (a) generates all the requisite modeling elements and (b) constructs the modeling relationships, such as material balances, energy balance, heat transfer between jacket and reactive mixture, mass transport between the two liquid phases, equilibrium relationships between the two phases, estimation of chemical reaction rate, estimation of chemical equilibrium conditions, estimation of heat generated (or consumed) by the reaction, and estimation of enthalpies of material convective flows. In order to automate the above tasks, MODEL.LA. must possess the following capabilities ... 

This is the Gibbs-Duhem equation, which relates the variation in temperature, pressure, and chemical potentials of the C components in the solution. Of these C + 2 variables, only C + 1 can vary independently. The Gibbs-Duhem equation has many applications, one of which is providing the basis for developing phase equilibrium relationships. 

The calculation of multi-stage separation processes involves the solution of phase equilibrium relationships, mass balances, and energy balances. Energy balances require the computation of enthalpies of streams entering and leaving an equilibrium stage. The enthalpy is a function of state, defined in Section 1.1.2 as H = U -i- PV. It is a function of the stream composition, its temperature, and its pressure. 

Phase equilibrium relationships are expressed in terms of Vj, Lj, and Lj. Since... 

To solve this problem uniquely, one would have to specify 2C -I- 6 variables. One example would be to specify the composition, flowrate, T and P of the two feed steams. If the exit streams are not in thermodynamic equilibrium, then the phase equilibrium relationships cannot be used and the exit streams may not be at the same T and P. 

Application of Equation (12.6) over a range of relative volatihties gives results as shown in Table 12.1. Clearly, the phase equilibrium relationship is of paramount importance in distillation... 

For the DOF, equations and constraints include the independent material balances for each species or a total flow balance and (N,p - 1) species balances, the energy balance, the phase equilibrium relationships that link the compositions between phases, and the chemical equilibrium relationships. The unit and composition constraints may be explicit (e.g., a given stream fraction is condensing) or implicit (e.g., a species concentration is zero). Flimmelblau (1996) provides the full derivation for the degrees of freedom for the following Examples 16.11-16.14. 

Furthermore, if ihe molecules condeaae, the total enthalpy given up would include the latent heat. Sherwood ei al. descriha how the above equations along with appropriate mass transfer expressions aed phase equilibrium relationships are employed in the analysis of condensers and other processes involving simultaneous heal transfer and mass transfer. 

Although the same enclosures are used for the component-material balances in the formulation of the Almost Band Algorithm as were used in the formulation of the 2N Newton-Raphson method, it is convenient in this case (because of the form of the phase equilibrium relationships) to include Tr+l, Ts, vr+lt,, and usl in the set of independent variables. Thus, in the formulation of the Almost Band Algorithm, the following choice of independent variables is made. 

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