Nonsmooth Modelling Methods in Chemical Engineering 21 st

Nonsmooth Modelling Methods in Chemical Engineering 21 st Nordic Process Control Workshop Marius Reed, Marlene Lund and Johannes Jäschke 19 th January 2018 Department of Chemical Engineering - Process Systems Engineering

Nonsmoothness occurs many places • Flow regime changes • Vanishing or appearing phase – Flashing – Hydrate formation • Flow direction changes • High pressure release valve • Nonsmooth physical property models – max/min operators • Can be challenging for traditional numerical simulations/optimizations • Solution: Nonsmoothness usually avoided if possible • Take home message: Can be handled using recently developed methods 3

Smooth equation solving • 4

Nonsmooth equation solving • 5

Generalized derivatives • 0 6

LD-derivatives • 7

Functions that we can use for modelling • 8

The oil-well system 9 Manifold/flow Valves Composition Bi-directional flow

Valve equations 10

Composition in Manifold 11

Composition in connection flows 12

Results 13

Composition in Manifolds and connecting flows 14

Model surface plots 15

Solvers • Levenberg-Marquardt Algorithm • Semi-smooth Newton method

Convergence 17

Conclusion • Nonsmooth techniques give additional modelling flexibility – E. g. switching between equation system (replace if-else) – Flow reversal • Recent development in efficient algorithms – Quadratic convergence • Computationally relevant generalized derivative information can be computed using Automatic Differentiation Thank you! 18

LD-derivatives 20

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Clarke Jacobian – Chain rule (Inclusion) 24

Automatic Differentiation 25