Usually we require higher precision than obtained by graphing the Gibbs energy. Furthermore, we may encounter multicomponent mixtures, for which the extension of the above method is not straightforward. We can develop an entirely general method for computing the phase partitioning given relative activities in Eqn 14.1. In Fig. 14.4 are plotted the activities for the water + MEK system of Example 14.3. The extrema in the activity plot are characteristic of LLE. The vertical lines indicate the compositions where the activities are equal in each phase. The horizontal lines indicate the activity values. This analysis is a graphical solution to Eqn. 14.1. We need a method to search for this condition numerically. Rearranging Eqn. 14.1 we have,
Note that the K-ratios calculated using the ratio of mole fractions should be identical to the value calculated using the ratio of activity coefficients at the stable LLE condition. This form is entirely analogous to the K-ratios in VLE. To find this condition, we can use an LLE flash calculation. We can iterate on the system by assuming values for mole fractions, generating activity coefficients at those x values to get K-ratios and then generating new values for mole fractions from the K-ratios. If the loop is constructed properly, it will create a successive substitution algorithm that will converge.
We can develop such a procedure by noting for a binary mixture that 
must sum to unity. We can calculate the concentrations using the compositions 
in the other phase, and use the K-ratio to generate a new guess of composition. The principle balance equation is 
which leads to
Iterative flash procedure for binary LLE.
The method is initialized by assuming the two phases are virtually immiscible with an infinitely dilute trace of the other component. The method is as follows.
1. Assume that phase β is nearly pure 1, 
, and α is nearly pure 2, 
. These represent initialization of the iteration procedure. The procedure is most stable with an initial guess of mutual solubility outside the two-phase region.
2. Calculate 
where the γi′s are evaluated at the initial compositions.
3. Calculate 
.
4. Calculate 
.
5. Determine γi,new values for each liquid phase from the xi,new values.
6. Calculate 
.
7. Replace all xi,old and Ki,old values with the corresponding new values.
8. Loop to step 3 until calculations converge. The calculations converge slowly.
A similar method for ternary systems is explored in a homework problem. Note that the Rachford-Rice flash method given for VLE in Section 10.3 can be adapted and provides an even more robust solution, but it is not as easy to implement in Excel without a macro. The method is provided in Matlab/Chap14/LLEflash.m.
LLEflash.m implements the Rachford-Rice flash method.
Let us apply the binary algorithm above to the water and MEK system studied in the previous example.
Example 14.4. The binary LLE algorithm using MAB and SSCED models
Compute the mutual solubilities of water and MEK at 298 K and compare to the experimental data of Example 14.3 assuming the following models: (a) MAB (b) SSCED.
Solution
a. From Example 14.3, A12 = 2.931. The symmetry of the MAB model gives x1α = x2β = 1/exp(2.931) = 0.05335. Computing γi′s at these compositions, KW = 1.0084/13.83 = 0.0729; KMEK = 13.83/1.0084 = 13.72. Then Eqn. 14.5 gives 
; 
for the first iteration. Unfortunately, the LLE calculations converge more slowly than VLE flash calculations. The calculations may drift a couple mole percent in compositions after they are changing at step sizes in the tenths of mole percents, so patience is required in converging the calculations. Section 14.9 provides details on setting up a macro or circular calculation. The table below summarizes the initial iterations. This same model is used above and the results are the same, but numerically known to better precision than the graphical method.
b. The SSCED model gives:
k12 = (50.13 – 0)(15.06 – 9.70)/(4·27.94·18.88) = 0.1274.
From lnγ1∞ = 18[(27.94-18.88)2+2(0.1274)27.94(18.88)]/(8.314·298) = 1.573, x1α = 1/exp(1.573)= 0.2072.
Applying the same formulas to MEK: 
.
The table below shows the improved predictions from SSCED relative to MAB. Note how the molecular size difference is reflected by the much greater activity of trying to squeeze the large molecule among the small ones. This reflects a significantly improved insight for SSCED relative to the MAB model.
Iterating further on x1α through Eqn. 14.5 gives x1α = 0.2509.
A similar approach to this example could be applied to solve for the ternary problem of partitioning of the propanoic acid between water and MEK, starting with the above result and infinite dilution of the propanoic acid. By steadily increasing the propanoic acid and performing flash calculations each time, the impact of the propanoic acid on the water + MEK partitioning could be studied. Can you guess whether the propanoic acid causes relatively more MEK to dissolve into the water phase or vice versa? The answer is explored later in a homework problem.
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