1. PROCESS MODELLING AND MODEL ANALYSIS © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Conservation Principles

2. PROCESS MODELLING AND MODEL ANALYSIS 2 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Overview of Chapter   Thermodynamic system principles   Balance volumes in process systems   Extensive and intensive variables   Principle of conservation   Conservation equations   Induced algebraic equations

3. PROCESS MODELLING AND MODEL ANALYSIS 3 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Thermodynamic system principles Spaces and their characteristics open systems closed systems isolated systems Mass Energy

4. PROCESS MODELLING AND MODEL ANALYSIS 4 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Scalar field P(x,y,z,t) examples? Vector field v(x,y,z,t) examples? Scalar and Vector Fields

5. PROCESS MODELLING AND MODEL ANALYSIS 5 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Scalar field P(x,y,z)  P = grad P = (  P/  x,  P/  y,  P/  z) (vector field)  2 P = div grad P =  2 P/  x 2 +  2 P/  y 2 +  2 P/  z 2 (scalar field) Vector field v(x,y,z)  v = div v =  v x /  x +  v y /  y +  v z /  z (scalar field) Operations on Scalar and Vector Fields

6. PROCESS MODELLING AND MODEL ANALYSIS 6 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Balance or “Control” Volumes V F  F n  J General principle Particular application ? Copper converter Gas compressor

7. PROCESS MODELLING AND MODEL ANALYSIS 7 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Balance Volumes  Defined by physical equipment  Defined by distinct phases  Dictated by the modelling goal  Need for a co-ordinate system  rectangular  cylindrical  spherical

8. PROCESS MODELLING AND MODEL ANALYSIS 8 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Principal Co-ordinate Systems

9. PROCESS MODELLING AND MODEL ANALYSIS 9 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Relation Between Balance Volumes  Mass balance volume set: primary set  Energy balance volume set can span or encapsulate mass balance sets  Balance volume manipulations  coalescence  division

10. PROCESS MODELLING AND MODEL ANALYSIS 10 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Balance Volumes for a Heated CSTR

11. PROCESS MODELLING AND MODEL ANALYSIS 11 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Relationship Between Balance Volumes

12. PROCESS MODELLING AND MODEL ANALYSIS 12 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Balance Volumes for a Vaporizer

13. PROCESS MODELLING AND MODEL ANALYSIS 13 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Balance Volumes for Copper Converter

14. PROCESS MODELLING AND MODEL ANALYSIS 14 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Extensive and Intensive Properties  Extensive Properties  depend on the extent of the system  Intensive Properties  do not depend on extent

15. PROCESS MODELLING AND MODEL ANALYSIS 15 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Extensive Properties Canonical set – overall mass – component masses (n-1 independent) – energy or enthalpy E = M f(I j, I i ) Extensive properties are conserved

16. PROCESS MODELLING AND MODEL ANALYSIS 16 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Intensive Properties Intensive – P, T, compositions – mass specific properties – key variables for constitutive relations Potentials – driving forces for diffusive flow of their extensive counterpart

17. PROCESS MODELLING AND MODEL ANALYSIS 17 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Principal Variables for Process Models Canonical set: intensive properties and total mass for every balance volume  pressure (P)  temperature (T)  compositions (x 1, x 2, …, x n-1 )  total mass (M)

18. PROCESS MODELLING AND MODEL ANALYSIS 18 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Balance or “Control” Volume V F  F n  J Conservation for the general balance volume

19. PROCESS MODELLING AND MODEL ANALYSIS 19 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Principle of Conservation  Word form  Integral equation form

20. PROCESS MODELLING AND MODEL ANALYSIS 20 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Differential Form of Conservation (V to 0) J C = convective flows J D = diffusive flows Source/Sink terms – chemical reaction – rate processes – external sources

21. PROCESS MODELLING AND MODEL ANALYSIS 21 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Differential Form of Conservation (expanded) Flow terms expanded  convective flows: J C =  v  diffusive flows: J D = -D grad  (assuming D=constant and no interaction effects) (co-ordinate system independent form)

22. PROCESS MODELLING AND MODEL ANALYSIS 22 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Conservation in Rectangular Co-ordinates Expand the differential operators in rectangular co-ordinate system Co-ordinate independent form

23. PROCESS MODELLING AND MODEL ANALYSIS 23 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Conservation Equation in Rectangular Co-ordinates Parabolic type partial differential equation+ Induced algebraic equations - extensive-intensive relations - rate equations (transfer and reaction)

24. PROCESS MODELLING AND MODEL ANALYSIS 24 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences Example: Spherical Catalyst Pellet  Spherical geometry assumed  Coupled equation system via reaction term  Parabolic partial differential equations

25. PROCESS MODELLING AND MODEL ANALYSIS 25 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences For Discussion (1)  Double pipe heat exchanger

26. PROCESS MODELLING AND MODEL ANALYSIS 26 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences For Discussion (2) Closed tank system

27. PROCESS MODELLING AND MODEL ANALYSIS 27 © CAPE Centre, The University of Queensland Hungarian Academy of Sciences For Discussion (3)