In this example, we apply GOAL  to find the parameters of a Activity Coefficients Model which best fits the Vapor - Liquid Equilibrium (VLE) data of a binary mixture. 

The start point for this example are the experimental VLE data for the binary mixture carbon sulfide + acetone at P = 1 atm:

If we suppose that the gas phase is ideal, while the liquid phase can be described by an activity coefficients model, the composition of a gas in phase equilibrium with a liquid  at a temperature T is:

Were x_i and y_i are the compositions of liquid and vapor, P^TOT is the ambient pressure, P^SAT is the saturation pressure of the pure component, that can be calculated with the Antoine equation: 

And g is the activity coefficient. If this coefficient is calculated with Van Laar model, the following equation must be applied:

With these equations we can calculate y if x and T are known. To find the best parameters of the Van Laar model, we can calculate y for each experimental value of x and T, and try to minimize the sum of squares of the differences between experimental and predicted results.

The implementation of this minimization problem in GOAL is:

PARAMETERS {{ REAL: A12[0;10] REAL: A21[0;10] }} MODEL {{ 'VLE data fit 'Binary mixtures carbol sulfide (1) + acetone (2) 'Van Laar Activity Coefficients Model '*************************************************************** '***** DECLARATIONS ******************************************** '*************************************************************** Dim x(11), y(11), T(11) '*************************************************************** '***** PREPROCESSING ******************************************* '*************************************************************** Sub PreProcessing() x(0) = 0.0476: x(1) = 0.1340: x(2) = 0.1858: x(3) = 0.2912: x(4) = 0.3798: x(5) = 0.4477: x(6) = 0.5360: x(7) = 0.6580: x(8) = 0.7894: x(9) = 0.8023: x(10) = 0.8799 'Liquid molar fractions y(0) = 0.1850: y(1) = 0.3510: y(2) = 0.4430: y(3) = 0.5275: y(4) = 0.5740: y(5) = 0.5980: y(6) = 0.6270: y(7) = 0.6610: y(8) = 0.7050: y(9) = 0.7230: y(10) = 0.7600 'Vapor molar fractions T(0) = 51.4: T(1) = 46.6: T(2) = 44.0: T(3) = 41.4: T(4) = 40.3: T(5) = 39.8: T(6) = 39.3: T(7) = 39.1: T(8) = 39.3: T(9) = 39.6: T(10) = 40.5 'Temperatures End Sub '*************************************************************** '***** OBJECTIVE FUNCTION ************************************** '*************************************************************** Function ObjectiveFunction() dif = 0 for i = 0 to 10 'Liquid phase compositions x1 = x(i) x2 = 1 - x1 'Pure component vapor pressures psat1 = Antoine(15.9844, 2690.85, -31.62, T(i)) 'Activity coefficients if A12*x1 + A21*x2 <> 0 then 'Avoid divisions by zero g1ln = A12*(A21*x2/(A12*x1+A21*x2))^2 if g1ln < 60 then 'Avoid flooding g1 = exp(g1ln) else g1 = 100 end if else g1 = 100 end if 'Capculated vapor compositions ycalc = x1*g1*psat1/760 'Error dif = dif + (ycalc - y(i))^2 next ObjectiveFunction = dif End Function '*************************************************************** '***** POSTPROCESSING ****************************************** '*************************************************************** Sub PostProcessing() 'TODO: Add the code of the postprocessing function here 'This code will be executed only once at the end of the execution End Sub 'Antoine function - calculates pure component vapor pressures Function Antoine(A, B, C, T) Antoine = exp(A - B/(T + 273.15 + C)) End Function }} CONSTRAINTS {{ }} ALGORITHM {{ POPULATION: 30 GENERATIONS: 100 REPRODUCTION_TYPE: 1 REPRODUCTION_PROBABILITY: 0,85 SELECTION_TYPE: 2 SELECTION_PROBABILITY: 0,85 MUTATION: 0,005 ELITISM: 1 REFRESH_PERIOD: 10 REFRESH_PERCENT: 10 }}

Send an e-mail to geneticoptimization@yahoo.es with questions or comments about this web page
Last modification: 01/27/03