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9. LLE Calculations

9. LLE Calculations. For two liquid phases at equilibrium the fugacity of each component in the phases must be equal. For the binary case shown: are the two relationships that govern the partitioning of species 1 and 2 between the two phases. Feed @ T, P a L a , x 1 a , x 2 a

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9. LLE Calculations

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  1. 9. LLE Calculations • For two liquid phases at equilibrium • the fugacity of each component in • the phases must be equal. • For the binary case shown: • are the two relationships that • govern the partitioning of species • 1 and 2 between the two phases. J.S. Parent

  2. Feed @ T, P a La, x1a, x2a z1, z2b Lb, x1b, x2b Binary LLE Separations • The equivalent of a VLE flash calculation can be carried out on liquid-liquid systems. • Given: T, P and the overall composition of the system • F, z1, z2 • Find: La, x1a, x2a • Lb, x1b, x2b J.S. Parent

  3. Binary LLE Separations - Governing Eqn • Solving these problems requires a series of material balances: • Using a unit feed as our basis, an overall material balance yields: • (A) • A material balance on component 1 give us: • (B) • Substituting for Lb from A into equation B: • (C) • An analogous material balance on component 2, yields: • (D) • We have two equations (C,D) and three unknowns (La, x1a and x1b). • We need an equilibrium relationship between xia and xib J.S. Parent

  4. Binary LLE Separations - Governing Eqn • Our LLE expression is: • (14.10) • or • and (E) • The governing equation we require to solve the problem is generated from a final material balance on one of the liquid phases: • (F) • Substituting equations C, D, E into the material balance F gives us the final equation: J.S. Parent

  5. Solving Binary LLE Separation Problems • Given: T, P,F, z1, z2 Find: La, x1a, x2a • Lb, x1b, x2b • The solution procedure follows that of binary VLE flash calculations very closely. • You can immediately solve for x1a and x1b using the LLE relationships • Or • You can solve the governing equation by iteration, starting with estimates of x1a and x1b to determine activity coefficients, and refining these estimates and La by successive substitution. J.S. Parent

  6. Vapour-Liquid-Liquid Equilibrium (VLLE) • In some cases we observe • VLLE, in which three • phases exist at • equilibrium. • F = 2 - p + C • = 2 - 3 + 2 = 1 • Therefore, at a given P, • all intensive variables • are fixed, and we have • a single point on a binary • Tx,x,y diagram J.S. Parent

  7. Vapour-Liquid-Liquid Equilibrium (VLLE) • At a given T, we can • create Px,x,y diagrams • if we have a good • activity coefficient • model. • Note the weak • dependence of the • liquid phase • compositions on the • system pressure. J.S. Parent

  8. 10. Chemical Reaction Equilibrium SVNA 15 • If sufficient data exists, we can describe the equilibrium state of a reacting system. • If the system is able to lower its Gibbs energy through a change in its composition, this reaction is favourable. • However, this does not imply that the reaction will occur in a finite period of time. This is a question of reaction kinetics. • There are several industrially important reactions that are both rapid and “equilibrium limited”. • Synthesis gas reaction • production of methyl-t-butyl ether (MBTE) • In these processes, it is necessary to know the thermodynamic limit of the reaction extent under given conditions. J.S. Parent

  9. Reaction Extent • Given a feed composition for a reactive system, we are most interested in the degree of conversion of reactants into products. • A concise measure is the reaction extent, e. • Consider the following reaction: • In terms of stoichiometric coefficients: • where, nCH4 = -1, nH20 = -1, nCO = 1, nH2 = 3 • For any change in composition due to this reaction, • 15.2 • where de is called the differential extent of reaction. J.S. Parent

  10. Reaction Extent • Another form of the reaction extent is: • (i=1,2,…,N) 15.3 • The second part of our definition of reaction extent is that it equals zero prior to the reaction. • Given that we are interested in the reaction extent, and not its differential, we integrate 15.3 from the initial, unreacted state to any reacted state of interest: • or • 15.4 J.S. Parent

  11. Reaction Extent and Mole Fractions • Translating the reaction extent into mole fractions is accomplished by calculating the total number of moles in the system at the given state. • Where, • Mole fractions for all species are derived from: • 15.5 J.S. Parent

  12. Multiple Reactions and the Reaction Extent • The reaction extent approach can be generalized to accommodate two or more independent, simultaneous reactions. • For j reactions of N components: • (i=1,2,…,N) • and the number of moles of each component for given reaction extents is: • 15.6 • and the total number of moles in the system becomes: • where we can write: J.S. Parent

  13. Chemical Reaction Equilibrium Criteria • To determine the state of a • reactive system at equilibrium, • we need to relate the reaction • extent to the total Gibbs • energy, GT. • We have seen that GT of a • closed system at T,P • reaches a minimum at • an equilibrium state: • Eq. 14.4 J.S. Parent

  14. Reaction Extent and Gibbs Energy • For the time being, consider a single phase system in which chemical reactions are possible. • The changes in Gibbs energy resulting from shifts in temperature, pressure and composition are described by the fundamental equation: • At constant temperature and pressure, this reduces to: • and the only means the system has to lower the Gibbs • energy is to alter the number of moles of individual • components. • What remains is to translate changes in moles to the reaction extent. J.S. Parent

  15. Criterion for Chemical Equilibrium • For a single chemical reaction, we can apply equation 15.3 which relates the reaction extent to the changes in the number of moles: • 15.3 • Substituting for dni in the fundamental equation yields: • At equilibrium, we know that d(nG)T,P, = 0. Therefore, for the above equation to hold at any reaction extent, we require that • 15.8 J.S. Parent

  16. Reaction Equilibrium and Chemical Potential • We have developed a criterion for chemical equilibrium in terms the chemical potentials of components. • 15.8 • While this criterion is complete, it is not in a useable form. • Recall our definition of fugacity which applies to any species in any phase (vapour, liquid, solid) • In dealing with reaction equilibria, we need to pay particular attention to the reference state, Gi(T). We can assign a standard state, Gio, as: J.S. Parent

  17. Standard States 4.4 SVNA • For our purposes, the Gibbs energy at standard conditions is of greatest interest. • This is the molar Gibbs energy of: • pure component i • at the reaction temperature • in a user-defined phase • at a user-defined pressure (often 1 bar) • A great deal of thermodynamic data are published as the standard properties of formation at STP (Table C.4 of the text) • DGfo is standard Gibbs energy of formation per mole of the compound when formed from its elements in its standard state at 25oC. • Gases: pure, ideal gas at 1 bar • Liquids: pure substance at 1 bar J.S. Parent

  18. Chemical Potential and Activity • Substituting our standard Gibbs energy (Gio) in the place of Gi(T), the chemical potential of component i in our system becomes: • 15.9 • We define a new parameter, activity, to simplify this expression: • 15.11 • where, • The activity of a component is the ratio of its mixture fugacity to its pure component fugacity at the standard state. J.S. Parent

  19. Reaction Equilibrium and Activity • When a reactive system reaches an equilibrium state, we know that the equilibrium criterion is satisfied. Recall that chemical reaction equilibrium requires: • where ni is the stoichiometric coefficient of component i and mi is the chemical potential of component i at the given P,T, and composition. • Substituting our expression for chemical equilibrium into the above equation gives us : • Or, J.S. Parent

  20. The Equilibrium Constant • Our equilibrium expression for reactive systems can be expressed concisely in the form: • 15.12 • where P signifies the product over all species. • The right hand side of equation 15.12 is a function of pure component properties alone, and is therefore constant at a given temperature. • The equilibrium constant, K, for the reaction is defined as: • 15.13 • K is calculated from the standard Gibbs energies of the pure components and the stoichiometric coefficients of the reaction. J.S. Parent

  21. Standard Gibbs Energy Change of Reaction • The conventional means of representing the equilibrium constant uses DGo, the standard Gibbs energy change the reaction. • Using this notation, our equilibrium constant assumes the familiar form: • 15.14 • When calculating an equilibrium constant (or interpreting a literature value), pay attention to standard state conditions. • Each Gio must represent the pure component at the temperature of interest • The state of the component and the pressure are arbitrary, but they must correspond with fio used to calculate the activity of the component in the mixture. J.S. Parent

  22. Temp. Dependence of Reaction Equilibrium • Defined by the following relationship, • the equilibrium constant is a function of temperature. • Recall that DGo represents the standard Gibbs energy of reaction at the specified temperature. • We know that: 15.15 • From which we can derive the temperature dependence of K: • 15.16 • If we assume that DHo is independent of temperature, we can integrate 15.16 directly to yield: • 15.17 J.S. Parent

  23. K vs Temperature • Equation 15.17 predicts that ln K • versus 1/T is linear. This is based on • the assumption that DHo is a weak • function of temperature over the • range of interest. • This is true for a number of • reactions, including those • depicted by Figure15.2. • A rigorous development of • temperature dependence • of K may be found in the text • (Equation 15.20) J.S. Parent

  24. Equilibrium State of a Reactive System • Given that an equilibrium constant for a reaction can be derived from the standard state Gibbs energies of the pure components, we can define the composition of the system at equilibrium. • 15.13 • Consider the gas phase reaction: • The equilibrium constant gives us: • Or J.S. Parent

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