1. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Modeling Multi-Element Systems Using Bond Graphs Modeling Multi-Element Systems Using Bond Graphs Jürgen Greifeneder François E. Cellier 18.10.2001

2. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier 1.Introduction Review Basics of Multi-Element Systems 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions Contents Jürgen Greifeneder: Review on paper the main aspects of paper 1 and 2 Pressure Cooker already diskussed in 2, however, only in a really short way, as the model is based on the multi-element system theory also. Jürgen Greifeneder: Review on paper the main aspects of paper 1 and 2 Pressure Cooker already diskussed in 2, however, only in a really short way, as the model is based on the multi-element system theory also.

3. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Introduction Describing a thermodynamical problem necessitates 3 variables. Separation in storage and dissipative elements. Storage elements calculate the potentials and therefore need to know about the matter, they are representing. Dissipative elements calculate flows and do not care, which matter they are dealing with (network theory). Elements do not know about each other. No quasi-stationary or flow-equilibrium assumptions were made. Contrary to earlier efforts in this field, this work delt with real, rather than pseudo bond graphs. }

4. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier p q T S g M.. 0 0 0 CF CC C Ø 3 CF Icon: The C-field (storage element) } Jürgen Greifeneder: 3 storage elements, but none of them can calculate its potential on ist own Bus- vs. Vektorbond Jürgen Greifeneder: 3 storage elements, but none of them can calculate its potential on ist own Bus- vs. Vektorbond

5. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier CF 1 CF 2 33 CD CF 1 CF 2 33 DVA Volume work: Conduction: ØØ 3 RF 3 33 CF 1 CF 2 , s 11  22 Convection: Basic dissipative Elements } Jürgen Greifeneder: Unterscheidung zwischen RF-Element und RF-Konzept !!! Hinweis, daß es sich um Dichte und spezifische Entropie handelt Jürgen Greifeneder: Unterscheidung zwischen RF-Element und RF-Konzept !!! Hinweis, daß es sich um Dichte und spezifische Entropie handelt

6. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Traditional Thermodynamics One Temperature, one pressure and n partial mass’ => n+2 equations. } Jürgen Greifeneder: Advantages of n+2-CF-Element: no constraint equations topological model of a complex system would be simpler and more easy to understand Disadvantages The previously introduced structures would have to be extended ?Internal equations of the C-field would change in accordance with the composition of the mixture unnecessary complexity, especially in the case of simple systems Processes would be hidden, that the authors would like to make visible However, as this is the classical thermodynamical approach, one would certainly have done so also Jürgen Greifeneder: Advantages of n+2-CF-Element: no constraint equations topological model of a complex system would be simpler and more easy to understand Disadvantages The previously introduced structures would have to be extended ?Internal equations of the C-field would change in accordance with the composition of the mixture unnecessary complexity, especially in the case of simple systems Processes would be hidden, that the authors would like to make visible However, as this is the classical thermodynamical approach, one would certainly have done so also p q 0 0 0 CF CC C T S. 000.... g1g1 M1M1. g2g2 M2M2. g3g3 M3M3. gngn MnMn. 123 n n Ø ? CF

7. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Multi-Element Mono-Phase Systems CF 1 Ø 3 3 2 3 Ø 2 2 DVA CD CF 3 Ø DVA CD 2 2 DVA CD 2 2 } Jürgen Greifeneder: Each matter has its own CF-Element. Each CF-Element is assumed to be a direct neighbor of each other element The contact surfaces between the different matters are assumed to be infinitely large => temperature and pressure may equilibrate infinitely fast. However, the corresponding transfer rates cannot be chosen infinitely large and therefore, the temperature and the pressure of the different components can assume somewhat different values (in the simulation). Although, if let alone, the will equilibrate, eventually. Jürgen Greifeneder: Each matter has its own CF-Element. Each CF-Element is assumed to be a direct neighbor of each other element The contact surfaces between the different matters are assumed to be infinitely large => temperature and pressure may equilibrate infinitely fast. However, the corresponding transfer rates cannot be chosen infinitely large and therefore, the temperature and the pressure of the different components can assume somewhat different values (in the simulation). Although, if let alone, the will equilibrate, eventually.

8. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Ideal and Non-Ideal Mixtures In the process of mixing, additionally entropy will be created, which must be distributed among the participating components Distribution is a function of the molar fractions CF-Elements are not supposed to know about each other Þ only necessary information will be provided CF 1 2 MI {M 1 } {x 1  {M 2 } {x 2  } Jürgen Greifeneder: Ideally mixed = molecules are distributed at random (prediction, which molecul becomes a neighbor of which other molecules is not possible Jürgen Greifeneder: Ideally mixed = molecules are distributed at random (prediction, which molecul becomes a neighbor of which other molecules is not possible

9. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Ideal and Non-Ideal Mixtures In the process of mixing, additionally entropy will be created, which must be distributed among the participating components Distribution is a function of the molar fractions CF-Elements are not supposed to know about each other Þ only necessary information will be provided CF 1 2 MI {M 1, V 1, S 1 } {x 1  s 1 Ex  v 1 Ex  {M 2, V 2, S 2 } {x 2  s 2 Ex  v 2 Ex  } Jürgen Greifeneder: Non-Ideal Mixtures: Volume will change also; specific excess volume and entropy of a non-ideal mixture are tabulated in the literature Jürgen Greifeneder: Non-Ideal Mixtures: Volume will change also; specific excess volume and entropy of a non-ideal mixture are tabulated in the literature

10. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Entropy of Mixing T S. p q 1 1 1 1 g 1 (T,p) 1 M. 1 T S. p q 1 1 M. 1 mix T RS M. 1 g1g1  S id mix 1 CF 11 CF 12 T S. p q 1 2 2 1 g 2 (T,p) 1 M. 2 T S. p q 2 2 M. 2 mix T RS M. 2 g2g2  S id mix 2 CF 21 CF 22 MI x 21 x 11 M 21 M 11 CD DVA.. } Jürgen Greifeneder: Ideal Mixture: Temperature and pressure do not change. Free enthalpy does change => difference creates an entropy flow Jürgen Greifeneder: Ideal Mixture: Temperature and pressure do not change. Free enthalpy does change => difference creates an entropy flow

11. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier } Jürgen Greifeneder: Non-ideal Mixtures or general case (cold milk poured into hot coffee): Differences also in the values of temperature and pressure Jürgen Greifeneder: Non-ideal Mixtures or general case (cold milk poured into hot coffee): Differences also in the values of temperature and pressure An François: eigentlich müßte ich hier erwähnen, daß die drei (m)RS-Elemente einem RF- Element entsprechen (vor allem, weil dies auf der nächsten Folie verwendet wird). Allerdings habe ich die interne RF-Struktur nie verwendet und sehe dies auch nicht als erforderlich an. Daher: Was tun? Die nächste Folie rauslassen? An François: eigentlich müßte ich hier erwähnen, daß die drei (m)RS-Elemente einem RF- Element entsprechen (vor allem, weil dies auf der nächsten Folie verwendet wird). Allerdings habe ich die interne RF-Struktur nie verwendet und sehe dies auch nicht als erforderlich an. Daher: Was tun? Die nächste Folie rauslassen?

12. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier CF 12 CF 22 MI CD DVA Ø Ø 3 RF 3 CF 11 Ø Ø 3 RF 3 CF 21 3 3 3 3 3 CD DVA 3 CD DVA Ø outside 3 3 } Jürgen Greifeneder: Pressure and Temperature may adjust to their corresponding outside values => volume „increases“ => additionally Entropy will be created. More volume and a higher Entropy leads to a higher temperature („Mischungswärme“) Jürgen Greifeneder: Pressure and Temperature may adjust to their corresponding outside values => volume „increases“ => additionally Entropy will be created. More volume and a higher Entropy leads to a higher temperature („Mischungswärme“)

13. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Convection in Multi-Element Systems CF 12 CF 13 CF 11 Ø 3 3 3 3 3 Ø Ø 3 DVA CD 3 3 DVA CF 22 CF 23 CF 21 3 3 3 3 3 Ø Ø Ø 3 DVA CD 3 3 DVA CD 3 DVA 3 CD 3 RF DVA CD 3 3 3 RF DVA CD 3 3 RF DVA CD horizontal Exchange (transport) vertical Exchange (mixture) } Jürgen Greifeneder: Vertical exchange as discussed before Horizontal exchange: coupled RF-Elements. Only one of them is independent. The flows of the others are fixed by the composition of the emitting mixture. Jürgen Greifeneder: Vertical exchange as discussed before Horizontal exchange: coupled RF-Elements. Only one of them is independent. The flows of the others are fixed by the composition of the emitting mixture.

14. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Two-Element, Two-Phase, Two-Compartment Convective System Gas CF 11 Fl. CF 11 Fl. CF 21 Gas CF 21 Ø 3 3 3 3 3 Ø Ø Ø 3 DVA CD 3 Condensation/ Evaporation DVA 3 3 3 3 CD Condensation/ Evaporation DVA Gas CF 12 Fl. CF 12 Fl. CF 22 Gas CF 22 Ø 3 3 3 3 3 3 Ø Ø Ø 3 DVA CD 3 Condensation/ Evaporation DVA 3 3 3 CD 3 Condensation/ Evaporation DVA 3 CD 3 RF DVA CD 3 3 3 3 3 DVA RF CD DVA RF 3 3 DVA CD phase- boundary 3 DVA CD 3 3 DVA CD 3 3 DVA CD 3 3 DVA CD 3 MI {x 21,  S E 21,  V E 21 } {M 21, T 21, p 21 } MI 1 2 + V ges + {M 11, T 11, p 11 } {x 21,  S E 21,  V E 21 } {M 12, T 12, p 12 } {x 12,  S E 12,  V E 12 } {M 22, T 22, p 22 } {x 22,  S E 22,  V E 22 } } Jürgen Greifeneder: Top of figure: Gas phase; bottom of figure: fluid phase On the left hand side one compartment on the right hand side the other one Gas phase = ideal gases => no MI-Elements necessaire. However, total volume is needed to calculate the partial pressures, which is needed for the condensation element Fluid phase needs MI-Elements Evaporation and Condensation are two independent processes!! Jürgen Greifeneder: Top of figure: Gas phase; bottom of figure: fluid phase On the left hand side one compartment on the right hand side the other one Gas phase = ideal gases => no MI-Elements necessaire. However, total volume is needed to calculate the partial pressures, which is needed for the condensation element Fluid phase needs MI-Elements Evaporation and Condensation are two independent processes!! An François: Wie genau muß ich auf diese Abbildung eingehen? An François: Wie genau muß ich auf diese Abbildung eingehen?

15. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Equilibration of Concentrations CF i 3 Ø 33 CD DVA KA 3 Ø CF i+1 33... } Jürgen Greifeneder: The concentrations in two neighboring compartments may become different, as each compartment can be connected to any other compartment or an outside source. Internal structure is RF-Element However, RF-Elements were only provided with the state information of the emitting CF-Element Jürgen Greifeneder: The concentrations in two neighboring compartments may become different, as each compartment can be connected to any other compartment or an outside source. Internal structure is RF-Element However, RF-Elements were only provided with the state information of the emitting CF-Element

16. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Model of a Pressure Cooker water air steam SE : 393 K CD (t) KV DVA CD DVA } Jürgen Greifeneder: Air is needed, to provide the pressure cooker at room temperature with the pressure of the environment. Having the same volume, without the air, some water would have to evaporate even at room temperature in order to produce equilibrium pressure, which would be considerably lower than 1 bar. Explain the used components (animation) Jürgen Greifeneder: Air is needed, to provide the pressure cooker at room temperature with the pressure of the environment. Having the same volume, without the air, some water would have to evaporate even at room temperature in order to produce equilibrium pressure, which would be considerably lower than 1 bar. Explain the used components (animation)

17. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier water air steam KV DVA CD DVA KV Air in boundary layer Steam in boundary layer CD RF:  p RF:  p CD DVA CD SE : 293 K CD (t) SE : 393 K CD (t) Model of a Pressure Cooker }

18. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Temperature Graphs } Jürgen Greifeneder: Show heating phase (almost identical temperatures) Cooking phase Cooling phase: boundary layer cools down most rapidly, the bulk follows somewhat slowly, and the water cools down last. Jürgen Greifeneder: Show heating phase (almost identical temperatures) Cooking phase Cooling phase: boundary layer cools down most rapidly, the bulk follows somewhat slowly, and the water cools down last.

19. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Pressure Graphs } Jürgen Greifeneder: Pressure in bulk is indistinguishable from that of the fluid Heating phase: no differences. Knee in the curve (roughly at time 150s) represents the point where the water begins to boil (~380K; 130 kPa). Cooling phase: pressure in boundary layer drops temporarily below that of the bulk, because water condensates more rapidly in the boundary layer and because the two RF-elements cannot resupply the boundary layer with air/steam frum the bulk arbitrarily fast. Jürgen Greifeneder: Pressure in bulk is indistinguishable from that of the fluid Heating phase: no differences. Knee in the curve (roughly at time 150s) represents the point where the water begins to boil (~380K; 130 kPa). Cooling phase: pressure in boundary layer drops temporarily below that of the bulk, because water condensates more rapidly in the boundary layer and because the two RF-elements cannot resupply the boundary layer with air/steam frum the bulk arbitrarily fast.

20. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Humidity Graphs } Jürgen Greifeneder: Humidity = partial pressure of the steam / saturation pressure of water Decrease of humidity during first heating phase, as the saturation pressure – located in the denominator of the humidity – has the same gradient as the rising temperature. Small differences can be seen, as the boundary layer heats up a little faster than the bulk At time 150s the humidity starts climbing again, because - just like the knee in the pressure trajectories – the water starts to boil and therefore steam is being created by evaporation. Equilibrium state reached at 32% Begin of cooling phase, the temperature in the boundary layer drops down rapidly, the corresponding humidity quickly reaches 100% and dew starts to form on the cold surface of the pressure cooker. Now, the two gaseous phases are no longer identical in their composition and therefore, diffusion occurs. The (slower) cooling down of the bulk together with the diffusion between the bulk and the boundary layer pull the humidity of the bulk up, until it reaches 100% and steam starts to condensate directly via the phase boundary. The humidity inside the pressure cooker will remain at 100% until the end of the simulation, as the only way to lower the humidity would be, to raise the temperature again – or open the pressure cooker. Jürgen Greifeneder: Humidity = partial pressure of the steam / saturation pressure of water Decrease of humidity during first heating phase, as the saturation pressure – located in the denominator of the humidity – has the same gradient as the rising temperature. Small differences can be seen, as the boundary layer heats up a little faster than the bulk At time 150s the humidity starts climbing again, because - just like the knee in the pressure trajectories – the water starts to boil and therefore steam is being created by evaporation. Equilibrium state reached at 32% Begin of cooling phase, the temperature in the boundary layer drops down rapidly, the corresponding humidity quickly reaches 100% and dew starts to form on the cold surface of the pressure cooker. Now, the two gaseous phases are no longer identical in their composition and therefore, diffusion occurs. The (slower) cooling down of the bulk together with the diffusion between the bulk and the boundary layer pull the humidity of the bulk up, until it reaches 100% and steam starts to condensate directly via the phase boundary. The humidity inside the pressure cooker will remain at 100% until the end of the simulation, as the only way to lower the humidity would be, to raise the temperature again – or open the pressure cooker.

21. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Mass Fraction Graphs } Jürgen Greifeneder: Mass fraction = mass of steam / (mass of steam and air) Beginning: No change. After that, evaporation increases the mass fraction until the equilibrium point of approximately 23.6% When cooling starts, the boundary layer cools down more rapidly than the bulk. Also the pressure of the boundary layer drops down more rapidly than that of the bulk. However, the pressure euqilibrates much more rapidly than the temperature. Thus, the pressure in the bulk (and in the water!) decreases more rapidly than the corresponding temperature, the boiling point of the water decreases, and consequently, additional water boils off. As a consequence, the mass fraction of steamin the bulk rises temporarily. However, the mass fractions starts dropping again due to pressure equilibration and diffusion. At time 1315 sec, steam starts to condensate from the bulk, and consequently, the mass fraction drops sharply. The final equilibration of the two mass fractions occurs primarily by means of diffusion. Jürgen Greifeneder: Mass fraction = mass of steam / (mass of steam and air) Beginning: No change. After that, evaporation increases the mass fraction until the equilibrium point of approximately 23.6% When cooling starts, the boundary layer cools down more rapidly than the bulk. Also the pressure of the boundary layer drops down more rapidly than that of the bulk. However, the pressure euqilibrates much more rapidly than the temperature. Thus, the pressure in the bulk (and in the water!) decreases more rapidly than the corresponding temperature, the boiling point of the water decreases, and consequently, additional water boils off. As a consequence, the mass fraction of steamin the bulk rises temporarily. However, the mass fractions starts dropping again due to pressure equilibration and diffusion. At time 1315 sec, steam starts to condensate from the bulk, and consequently, the mass fraction drops sharply. The final equilibration of the two mass fractions occurs primarily by means of diffusion.

22. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Conclusion The elements introduced suffice to model most thermodynamical problems Modeling each matter separately as a storage element and connecting them by means of dissipative elements (RF-concept) simplyfies the modeling task, offers insight into physical functioning of multi-element systems and leads to mathematical models that can be simulated in a numerically stable and highly accurate fashion. Models are still limited to systems without chemical reactions }

23. Universität Stuttgart Modeling Multi-Element Systems Using Bond Graphs 1.Introduction 2.Mixture Properties 3.Transport Phenomena 4.Model of a Pressure Cooker 5.Conclusions University of Arizona 18. Oktober 2001 Jürgen Greifeneder, François Cellier Danke Ende Thank you! Remerciement! Jürgen Greifeneder François E. Cellier