Modeling Chemical Reactions in Modelica By Use of Chemo-bonds

1. Modeling Chemical Reactions in ModelicaBy Use of Chemo-bonds Prof. Dr. François E. Cellier Department of Computer Science ETH Zurich Switzerland Dr. JürgenGreifeneder Corporate Research Center ABB Germany

2. Most researchers model chemical reactions using (molar) mass flow equations only, not taking into account energy flows at all. This is possible for isothermal and isobaric reactions, but in general doesn’t work, because the reaction rate constants depend on temperature and in the case of gaseous reactions also on pressure. A better approach to modeling chemical reaction systems is by means of bond graphs. This shall be demonstrated here. Chemical Reactions and Convective Flows

3. Hydrogen-Bromine Reaction I • Given the following balance reaction: • Its individual step reactions are known and well understood: H2 + Br2⇌ 2HBr

4. k1 = k1· nBr2 k5 = k5· nH·· nBr2 /V Hydrogen-Bromine Reaction II • The mass flow equations can be written as follows: • where: Br2 = –k1 + k2 – k5 Br· = 2k1 – 2k2 – k3 + k4 + k5 H2 = –k3 + k4 H· = k3 –k4 –k5 HBr = k3 –k4 + k5 k2 = k2· (nBr·)2 /V k3 = k3· nH2· nBr· /V k4 = k4· nHBr · nH· /V

5. Chemical Energy Flow • Each mass flow is accompanied by a chemical energy flow: • such that: chemical potential Gibbs potential m n molar flow rate mass flow rate m [J/mol] g[J/kg] TF . n [mol/sec] m[kg/sec]

6. Hydrogen-Bromine Reaction III • The step reactions can be interpreted as a bond graph: m k1 = 2mBr– m Br2 energy flow equations Br2 = –k1 + k2 – k5 reaction rate equations

7. Hydrogen-Bromine Reaction IV • Programmed in BondLib:

8. Hydrogen-Bromine Reaction V • Simulation results:

9. The reaction rate equations can be rewritten in a matrix-vector form: or: Hydrogen-Bromine Reaction VI Br2 –1 +10 0 -1 k1 Br· +2 –2 –1+1+1 k2 H2 = 0 0 –1+10 ·k3 H· 0 0 +1–1–1 k4 HBr 0 0 +1–1+1 k5 mix = N · reac

10. The energy flow equations can also be written down in a matrix-vector form: or: Hydrogen-Bromine Reaction VII mk1 –1 +20 0 0 mBr2 mk2+1–2 000mBr· mk3 = 0 –1–1+1+1·mH2 mk40 +1 +1–1–1 mH· mk5-1 +1 0–1+1 mHBr mreac = NT · mmix

11. Thus, the bond graph describing the chemical reaction network can be reinterpreted as a multiport transformer: where:  MTF mix reac N  mix reac   The Chemical Reaction Network I nmix = N ·nreac mreac = NT·mmix

12. We can now plug everything together:  mix reac MTF CF RF  N mix reac   Capacitive storage of all reactants in the mixture. Transformation of the reactants into each other in the chemical reaction. The Chemical Reaction Bond Graph

13. Hydrogen-Bromine Reaction VIII • Programmed in MultiBondLib:

14. Hydrogen-Bromine Reaction IX

15. The chemical reaction bond graph, as shown until now, still doesn’t reflect the physics of chemical reactions in all their complexity. The problem is that a substance that undergoes a transformation does not only carry its mass along, but also its volume and its heat. Hence we should describe each step reactions by three parallel bonds, one describing mass flow, a second describing volumetric flow, and a third describing heat flow. Thus, we should really use thermo-bonds. Chemical Reactions and Convective Flows

16. Thermo-bonds • A (red) thermo-bond represents a parallel connection of three (black) regular bonds.

17. Thermo-bonds and Chemo-bonds • The (green) chemo-bonds and the (red) thermo-bonds are essentially the same thing. However, the thermo-bonds have been designed for convective flows, and therefore, operate on regular mass flows (measured in kg/sec), whereas the chemo-bonds were designed for chemical reactions, and therefore, operate on molar mass flows (measured in mol/sec).

18. Hydrogen-Bromine Reaction X • This version of the chemical reaction network contains more information than the original one. Yet it is simpler, because the thermal and pneumatic ports don’t need to be carried separately any longer. They are now integrated into the chemical reaction network. Each mass flow carries its own volumetric and heat flows along.

19. Also the thermo-bond graph describing the chemical reaction network together with its volumetric and heat flows can be reinterpreted as a multiport transformer: This transformer is now of cardinality 15, as there are 5 step reactions, each represented by a thermo-bond of cardinality 3. The Chemical Reaction Network II

20. Hydrogen-Bromine Reaction XI

21. Hydrogen-Bromine Reaction XII

22. Efficiency Considerations • The simulations are so fast that there is very little difference between all four of these models.

23. Chemical reactions are quite tricky. To model chemical reactions down to their physical and, in particular, thermodynamic properties quickly leads to models that are rather bulky. The bond graph methodology offers us a unified framework to deal with this complexity in an orderly and organized fashion. To this end, we used all three of our bond graph libraries; BondLib, featuring (black) regular bonds, MultiBondLib, featuring (blue) vector bonds, and ThermoBondLib, featuring (red) special vector bonds for the description of convective flows. Conclusions I

24. To deal with chemical reactions more conveniently, we introduced even a fourth bond graph library: ChemBondLib, featuring (green) special vector bonds for the description of chemical reaction flow rates. The chemo-bonds are identical to the thermo-bonds, except that they describe their internal mass flows with molar flow rates and chemical potentials, rather than with regular mass flow rates and Gibbs potentials (the specific Gibbs energy). Conclusions II