Calculation of Binary Interaction Parameters for Wilson Model
Case of Methanol-Water binary system
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Total Pressure and Gaz Constant

P = 760; R = 1.987;

Vapor Pressure of Methanol Using Antoine Equation

Aa = 8.08097; Ba = 1582.271; Ca = 239.726;

Psat = 10^(Aa - Ba/(T + Ca))

Equilibrium Equation: We would like to determine d1 and d2

A12[T_] := 
        18.07/40.73 Exp[-d1/(R (T + 
                      273.15))];
A21[T_] := 
        40.73/18.07 Exp[-d2/(R (T + 273.15))];

y[x_, T_] := 
      x 10^(Aa - 
              Ba/(T + Ca)) Exp[-Log[
                  x + A12[T] (1 - x)]
 + (1 - 
                    x) (A12[T]/(x + A12[T] (1 - x)) - 
                    A21[T]/(A21[T] x + 1 - x))]/P

Sum of Squares Function:
Interaction parameters d1 and d2 are chosen in order to minimize this function.

sumOfSquares[data_] := 
      Apply[Plus, 
        Apply[Plus, 
          Map[{(y[#[[1]], #[[2]]]  -  #[[3]])^2} &,  
            data]]]

Experimental data {Methanol liquid mole fraction, Temperature, Methanol Vapor Mole fraction} from P. C. Wankat, Equilibruim Staged Separations, page 78, Prentice Hall 1988.

mydata = {{0, 100, 0}, {0.02, 96.4, 0.134}, {0.04, 93.5, 
            0.23}, 
{0.06, 91.2, 0.304}, {0.08, 89.3, 
            0.365}, {0.1, 87.7, 0.418}, 
{0.15, 84.4, 
            0.517}, {0.2, 81.7, 0.579}, {0.3, 78, 0.665}, {0.4, 75.3, 
            0.729}, {0.5, 73.1, 0.779}, 
{0.6, 71.2, 
            0.825}, {0.7, 69.3, 0.87}, {0.8, 67.6, 0.915}, {0.9, 66, 
            0.958}, {0.95, 65, 0.979}, {1, 64.5, 1}};

param1 = 
      FindMinimum[sumOfSquares[mydata], {d1, 0.1, 90}, {d2,  .01, 1000}, 
        MaxIterations -> 300]

Checking result by computing values of vapor composition for given liquid composition and temperature

T = 87.7;

y[0.1, 87.7] /. d1 -> 127.62425304109692 /. 
      d2 -> 484.17839190370023

T = 69.3;

y[0.7, 69.3] /. d1 -> 127.62425304109692 /. 
      d2 -> 484.17839190370023