Part 4 Nonlinear Programming 4.8 Examples

Example 1: Chemical Equilibrium From thermodynamics of chemical reaction equilibrium, the equilibrium state of a closed system at constant temperature and pressure is the state at which its total Gibbs free energy is at a minimum. This criterion can be used to obtain the equilibrium composition of a given mixture by minimizing its free energy w.r.t. its composition.

Given Data An ideal gas mixture of 10 chemical species is maintained at T=298K and P=750Hg. The 10 species are made of 3 atomic elements (e.g., H, O and C), and they are denoted as A, B and C. The species formulas are:

Gibbs free energy per mole of species s 1 2 3 4 5 -10.021 -21.096 -37.986 -9.846 -28.653 s 6 7 8 9 10 -18.918 -28.032 -14.640 -30.594 -26.111

Total Gibbs Free Energy

Total Number of Moles of Each Element

NLP Model

Linearly Constrained NLP Problem

Example 2 Data Reconciliation

Data Reconciliation

Linearly Constrained NLP

    Always unknown!

Case 1: Flows of streams 1, 2, 5 and 6 are measured Case 1 is a redundant and observable system.

Case 2: Flows of streams 1 and 2 are measured Case 2 gives rise to a non-redundant observable system!

Case 3: Flows of streams 1 and 6 are measured Case 3 is a redundant unobservable system.

Reconciled Flow Values Table 2.2: Flow reconciliation of partially measured process Stream Reconciled Flow Values   Case 1 - Streams 3 and 4 unmeasured Case 2 – Streams 3,4,5,6unmeasured Case 2 - Streams 2,3,4,5 unmeasured 1 100.49 101.91 100.39 2 64.25 64.45 - 3 36.24 37.46 4 5 6