Example 6.8 Chemical Equilibrium

The following example is used in many test libraries for nonlinear programming and was taken originally from Bracken and McCormick (1968).

The problem is to determine the composition of a mixture of various chemicals satisfying its chemical equilibrium state. The second law of thermodynamics implies that a mixture of chemicals satisfies its chemical equilibrium state (at a constant temperature and pressure) when the free energy of the mixture is reduced to a minimum. Therefore the composition of the chemicals satisfying its chemical equilibrium state can be found by minimizing the function of the free energy of the mixture.

Notation:

Constraints for the Mixture:

• The number of moles must be positive:

• There are mass balance relationships,

Objective Function: Total Free Energy of Mixture

with

where is the model standard free energy function for the th compound (found in tables) and is the total pressure in atmospheres.

Minimization Problem:

Determine the parameters that minimize the objective function subject to the nonnegativity and linear balance constraints.

Numeric Example:

Determine the equilibrium composition of compound at temperature and pressure .

Example Specification:

proc nlp tech=tr pall;
      array c[10] -6.089 -17.164 -34.054  -5.914 -24.721
              -14.986 -24.100 -10.708 -26.662 -22.179;
   array x[10] x1-x10;
   min y;
   parms x1-x10 = .1;
   bounds 1.e-6 <= x1-x10;
   lincon 2. = x1 + 2. * x2 + 2. * x3 + x6 + x10,
          1. = x4 + 2. * x5 + x6 + x7,
          1. = x3 + x7  + x8 + 2. * x9 + x10;
   s = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10;
   y = 0.;
   do j = 1 to 10;
      y = y + x[j] * (c[j] + log(x[j] / s));
   end;
run;

Displayed Output:

The iteration history given in Output 6.8.1 does not show any problems.