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Part 4 Nonlinear Programming
4.8 Examples
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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.
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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:
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Gibbs free energy per mole of species s
1 2 3 4 5 -9.846 s 6 7 8 9 10
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Total Gibbs Free Energy
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Total Number of Moles of Each Element
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NLP Model
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Linearly Constrained NLP Problem
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Example 2 Data Reconciliation
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Data Reconciliation
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Linearly Constrained NLP
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Always unknown!
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Case 1: Flows of streams 1, 2, 5 and 6 are measured
Case 1 is a redundant and observable system.
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Case 2: Flows of streams 1 and 2 are measured
Case 2 gives rise to a non-redundant observable system!
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Case 3: Flows of streams 1 and 6 are measured
Case 3 is a redundant unobservable system.
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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
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