Part 4 Nonlinear Programming 4 8 Examples Example

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=298 K and P=750 Hg. • 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 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.

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, 6 unmeasured 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 64. 25 64. 45 - 5 36. 24 37. 46 - 6 100. 49 101. 91 100. 39