Chapter 14: Phase Equilibria Applications Part II

If the component is supercritical, then the vapor pressure is not defined

example A binary methane (1) and a light oil (2) at 200K and 30 bar consists of a vapor phase containing 95% methane and a liquid phase containing oil and dissolved methane. The fugacity of methane is given by Henry’s law, and at 200 K, H1 = 200 bar. Estimate the equilibrium mole fraction of methane in the liquid phase. The second virial coefficient of methane at 200K is -105 cm3/mol

solution

Solution (cont) Need equation for the fugacity coefficient vapor phase

Solution (cont) How do we solve for the mole fraction of the solute in the liquid phase? x1 =0.118

Another example For chloroform(1)/ethanol(2) at 55oC, the excess Gibbs energy is The vapor pressures of chloroform and ethanol at 55oC are P1sat = 82.37 kPa, and P2sat = 37.31 kPa Make BUBLP calculations, knowing that B11=-963 cm3/mol, B22 =-1,523 cm3/mol, B12 =52 cm3/mol

Need equations for the fugacity coefficients

But we don’t know P  Guess P (avg sat. pressures)

For example, at x1 =0.25, solve for y1, y2, P y1 = 0.558 y2 = 0.442 P = 63.757 kPa In our web site, there are model spreadsheets that you can download

VLE from cubic EOS

Vapor pressure pure species (1) Liquid branch Vapor branch LV transition Using equation (1), the cubic EOS yields Pisat= f(T)

Compressibility factors For the vapor phase there is another expression, (14.36)

How to calculate vapor pressure from a cubic EOS We solve for the saturation pressure at a given T such that the fugacity coefficients are equal in the two phases: 2 compressibility eqns., two fugacity coefficient eqns., equality of fugacity coefficients, 5 unknowns: Psat, Zl, Zv, fl, fv

Mixture VLE from a cubic EOS Equations for Zl and Zv have the same form as for pure components However the parameters a & b are functions of composition The two phases have different compositions, therefore we could think in terms of two P-V isotherms, one for each composition

Mixing rules for parameters

Also we need “partial” parameters

The partial parameters are used for the calculation of fugacity coefficients Because fi is related to the partial molar property of GR (residual G)

example Vapor mixture of N2(1) and CH4 (2) at 200K and 30 bar contains 40 mol% N2. Calculate the fugacity coefficients of nitrogen and methane using the RK equation of state. For RK, e =0 and s =1

Calculate P-x-y diagram at 100 oF for methane(1)-n butane (2) using SRK and mixing rules (14.42) to (14.44) Compare with published experimental data (P, x, y) Initial values for P and yi can be taken from given experimental data First read critical constants, w, from Table B.1 and a from Table 3.1 Calculate b1, b2, a1, a2 In this case T > Tc1

Step 1 K value given by

The equations for a are valid only up to the critical temperature; however is OK to extend the correlation slightly above the critical temperature Lets calculate the mixture parameters (for step 1). When applied to the liquid phase we use the xi mole fractions

Follow diagram Fig. 14.9 Assume P and yi Calculate Zl and Zv, and the mixture fugacity coefficients Calculate K1 and K2 and the SKixi Calculate normalized yi=Kixi/ S Kixi Reevaluate fugacity coefficients vapor phase, etc If S Kixi > 1, P is too low so increase P; if S Kixi < 1, then reduce P

Results: Rms % difference between calculated and exp. P is 3.9% Rms deviation between calculated and exp. y1 is 0.013 Note that the system consists of two similar molecules Where are the largest discrepancies with the experimental data?