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, H 1 = 200 bar. Estimate the equilibrium mole fraction of methane in the liquid phase. The second virial coefficient of methane at 200K is -105 cm 3 /mol

solution

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

Solution (cont) How do we solve for the mole fraction of the solute in the liquid phase? x 1 =0.118 How would you solve this problem if the liquid solution were not ideal, For example if G E /RT = A x 1 x 2

Another example For chloroform(1)/ethanol(2) at 55 o C, the excess Gibbs energy is The vapor pressures of chloroform and ethanol at 55 o C are P 1 sat = kPa, and P 2 sat = kPa Make BUBLP calculations, knowing that B 11 =-963 cm 3 /mol, B 22 = cm 3 /mol, B 12 =52 cm 3 /mol

Need equations for the activity coefficients

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

For example, at x 1 =0.25, solve for y 1, y 2, P y 1 = y 2 = P = kPa

VLE from cubic EOS

Vapor pressure pure species Liquid branch Vapor branch L  V transition Cubic EOS is P i sat = f(T)

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

How to calculate  i from a cubic EOS We solve for the saturation pressure such that the fugacity coefficients are equal in the two phases: 2 compressibility eqns, two fugacity coeff. eqns., 4 unknowns

Mixture VLE from a cubic EOS Equations for Z l and Z v have the same form However the parameters a and b are functions of composition The two phases have different compositions, therefore we could think about two PV 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  i is related to the partial molar property of G R (residual G)

example Vapor mixture of N 2 (1) and CH 4 (2) at 200K and 30 bar contains 40 mol% N 2. Calculate the fugacity coefficients of nitrogen and methane using the RK equation of state. For RK,  =0 and  =1

Calculate P-x-y diagram at 100 o F 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 y i can be obtained from experimental data First read critical constants, , from Table B.1 and  from Table 3.1 Calculate b1, b2, a1, a2 In this case T > Tc1

K value given by Step 1

The equations for  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 x i mole fractions

Follow diagram Fig Assume P and y i Calculate Z l and Z v, and the mixture fugacity coefficients Calculate K 1 and K 2 and the  K i x i Calculate normalized y i =K i x i /  K i x i Reevaluate fugacity coefficients vapor phase, etc If  K i x i > 1,  P is too low; if  K i x i < 1, then reduce P

Results: Rms % difference between calculated and exp. P is 3.9% Rms deviation between calculated and exp. y 1 is Note that the system consists of two similar molecules Where are the largest discrepancies with the experimental data?

Back to mixing rules Quadratic mixing rule for a sometimes fails: A different mixing rule is based in the relationship between activity coefficient and fugacity coefficient

Relate activity and fugacity coefficients This equation can be solved for And then use the summability condition

HW 9. Due Wednesday, October 31st Problem 1. Use the Peng-Robinson equation to determine the bubble point pressure of an equimolar solution of N nitrogen (1) and methane (2) at 100 K. Use NRTL for the activity coefficients. Problem 2. A distillation column is to produce overhead products having the following compositions: z 1 = 0.23 (propane), z 2 = 0.67 (isobutane), z 3 = 0.1 (n-butane). Suppose a partial condenser is operating at 320 K and 8 bars. What fraction of the liquid would be condensed according to the Peng-Robinson equation?

HW 9. Due Wednesday, October 31st Problem 3. Use the SRK equation to determine the phase envelope of ethane and n-heptane at compositions of x C7 = 0., 0.1, 0.2, 0.3, 0.5, 0.7, 0.9, 1.0. Plot P vs T at each composition by performing BP calculations to their terminal point and dew T calculations until the temperature begins to decrease significantly and the pressure approaches its maximum. If necessary, close the phase envelope by starting at the last dew-temperature state and performing dew-P calculations until the temperature and pressure approach the terminus of the BP curve. For each composition, mark the points where the bubble and dew curve meet with Xs. These Xs designate the mixture critical points. Connect the Xs with a dashed curve. This curve is known as the “critical locus” of the mixture. Note that to generate these phase envelopes, you change T instead of composition along each curve. Both dew and bubble calculations must be performed to generate each curve. First, generate the diagram assuming that the activity coefficients are equal to 1 in all cases. Second, repeat the problem using activity coefficients as given by the UNIQUAC equation.