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1 Modeling Multi-Element Systems Using Bond Grahs Jürgen Greifeneder Franοcois E Cellier Fakultät Verf-technik und echn Kybernetik Det of Electr & Com Engr Universität Stuttgart he University of Arizona POBox Stuttgart ucson, Arizona Deutschland USA J@Greifenederde Cellier@ECEArizonaEdu htt://wwwgreifenederde/jg/ htt://wwwecearizonaedu/~cellier/ Abstract his is the last in a series of three aers he first aer [] discussed the modeling of conductive as well as convective flows of a single homogeneous substance through a homogeneous medium he second aer [4] discussed the henomenona associated with hase change, ie, it discussed, from a bond-grahic ersective, henomenona such as evaoration and condensation, solidification, melting, and sublimation his third aer deals in articular with the difficult roblem of modeling multi-element systems ogether they offer a general methodology for modeling thermodynamical henomenona using true, rather than seudo-, bond grahs No uasi-stationary or flow-euilibrium assumtions were made, such that the models generated using the roosed methodology would be ket as general as ossible Keywords: hermodynamics; multi-element systems; chemical reactions; ressure cooker; bond grahs INRODUCION Bond grahs enable the modeler to describe the dynamics of a hysical system in a modular fashion using energy storage, dissiative ower flow, energy source, and transformation elements he basic bond-grahic icon library enables the user to model, in a systematic fashion, hysical systems, the dynamic behavior of which is governed by ower flows alone Systems with macroscoic mass flows add additional comlexity to the modeling task, since the mass that flows through the system carries with it its stored internal free energy, which is thus transorted from one location to another in a non-dissiative fashion In the most general sense, thermodynamics ought to be described by distributed arameter models Since bond grahs are geared to be used for the descrition of lumed arameter models only, a simlifying assumtion will be made, in that the system to be modeled is comartmentalized, whereby each comartment is considered to be homogeneous New bond-grahic macro-elements were introduced in the first two aers to describe the energy storage within a comartment as well as the mass (and energy) flows between neighboring comartments his aer discusses rimarily issues surrounding the thermodynamic modeling of multi-element systems BACKGROUND In the revious aers [], [4], systems were discussed that only included a single comonent For dealing with multi-element systems, this aer needs the assumtion of homogeneous, ideally ed comartments `Ideally ed' here means that the molecules are distributed at random, ie, a rediction of what molecule becomes a neighbor of which other molecules is not ossible Aossibleway of modeling an ideally ed comartment would be, to define a single C-field (CF-element) reresenting the ture, while changing its internal euations as a function of the comosition of the ture However, such a decision would necessitate the definition of C-fields with n + instead of external

2 cuts (n being the number of comonents in the ture) 0 g M n g M C M g C CF C 0 0 Figure : Modified CF-element, using n + external cuts Such an aroach would carry the advantage, that no constraint euations wouldneedtobeintroduced and that the toological structure of the model of a comlex system would be simler and more easy to comrehend However, the aroach would also be characterized by serious disadvantages, namely that the reviously introduced structure of the C-field would have to be extended, that the internal euations of the C-field would change in accordance with the comosition of the ture an unnecessary comlexity esecially in the case of simle systems, and that rocesses would be hidden that the authors would like to make visible However, as this would have beenthe classical thermodynamic aroach tomodelingmulti- element systems, one could certainly have doneso MONO PHASE SYSEMS he model discussed in this aer is based on the following idealizations Let us assume the system to be considered contains n comonents that are mutually searated by moveable membranes Let us further assume that the eigendynamics and the friction are neglected We now make each comartment infinitely thin while reserving its volume hereby the surface of the boundary layers to the neighbors become infinitely large We want to assume that each of these infinitely thin comartments is in contact with every other comartment, such that all comartments are n M n g n direct neighbors of each other hence, if the initial temerature and ressure in the different comartments were distinct, the exchange between the comartments would guarantee that they would euilibrate infinitely fast A system modeled in this fashion can be considered ideally ed his concetual system has been modeled by introducing one CF-element for each comonent of the ture and by connecting them, using - and elements Stability analysis shows that the corresonding transfer rates cannot be chosen infinitely large In fact, they must be limited such that the exchange between neighboring comonents is relatively slow for the simulation to work Conseuently, the temerature and ressure of different comonents of the ideal ture can assume, in the simulation, somewhat different values, although if left alone, they will euilibrate eventually he rimary advantage of this kind of modeling lies in the fact that the reviously introduced structure of the CF-elements can be retained and that the disclosure of imagined" flows among the comonents may imrove our understanding of what is going on in tures 4 IDEAL GASES Let us start with the simlest case: a ture of ideal gases According to Dalton's law[], ideal gases do not influence one another When ing such gases, their masses, volumes, and entroies simly add u herefore, the above model can be used for the most general case of such a ture However, the modeler must rovide a consistent set of initial conditions, as the sum of all artial volumes must be identical to the volume itself Furthermore, some elements (eg hase transitions) are more conveniently exressed in terms of artial ressures than artial volumes In multi-element systems, these elements have to be extended by an interface euation that determines the artial ressures as a function of the artial volumes and the overall ressure he - and the -element [] reresent the exchange of volume and entroy, resectively, driven by the difference of the corresonding otentials (ressure and temerature, resectively)

3 6 IDEAL MIXURES artial V artial i i = P n j= V artial i (4) j 5 -ELEMENS IN MULI- ELEMEN SYSEMS he R-field (-element), describing dissiative henomena of mass and energy transfer between C-fields, can easily be generalized to multi-element systems If a ture is being transorted from one set of C-fields to another, each comonent of that ture is transorted as well Conseuently, an-element can be laced between each air of CF-elements reresenting the same comonent of the two tures, and the flow rates across these -elements are comuted in accordance with the volume fractions ( P V i Λ V Λ ) of the ture i of the emitting C-field herefore, the multi-element transortation system can be identified as a arallel connection of singleelement transortation systems, the throughut of which is determined bythecomosition of the emitting system (fig ) Let us now roceed to the next more difficult case of a ture It is called ideal ture" An ideal ture is defined in such away that masses and volumes still add u, but additional entroy is roduced in the rocess of ing the comonents ogether with the entroy, also the free enthaly changes At this oint, the reviously roosed way of modeling leads to a roblem: CF-elements are not suosed to know about each other Yet, the additionally generated entroy and free enthaly need to be distributed among the articiating comonents o this end, we either need an additional bond grah element that distributes the newly generated entroy and free enthaly among the articiating CF-elements, or alternatively, each CF element needs to know its own molar fraction to comute the aroriate correction terms In our imlementation, we chose the latter route he reuired information is rovided to the articiating CF-elements by the Mixture Information (MI) element Contrary to all reviously introduced elements, the MI-element does not carry any energy (flows) It only rovides information flows (Fig ) Pure information flows are dubious from a hysical ersective, but they are justified when they are used locally among bond grah elements that, in the hysical reality, model different asects of one and the same henomenon, as is the case here CF CF {M } {M } CF MI CF {x } {x } CF CF CF vertical Exchange (ture) horizontal Exchange (transort) Figure : Multi-element transortation system as a sum of couled single-element systems he decomosition illustrates the difference between horizontal (satial transort) and vertical (ture) exchange CF Figure : Mixture information element for ideal tures shown here for the case of only two comonents When ing two or more different comonents (chemical reactions are not being considered in this aer), their energy must be reserved However in the case of an ideal ture, irreversible rocesses are being considered, leading to an increase in entroy Since the volume, ressure, temerature, and mass are conserved in an ideal ture, the energy balance can be written as: S id ed = gid ed M (6) (=) H id ed = g id ed M + sid ed =0) (6) As reuested, the enthaly H is conserved he change in secific entroy due to ing can be writ-

4 ten as [5]: s id ed = R Xk x k ln(x k ) (6) where x k is the molar fraction, and conseuently, the change of free (Gibbs) enthaly can be evaluated as: g id ed = R Xk x k ln(x k ) (64) Howarethesechanges modeled using bond grahs? Evidently, as additional entroy is being generated irreversively,thus using an -element, the corresonding heat flow must come from somewhere he above euation shows, where the energy comes from It is deducted from the mass flow, as the free enthaly of the mass undergoing the ture is reduced his is shown in Figure 4 CF CF g (,) Ṁ g (,) Ṁ g Ṁ g Ṁ g (,) Ṁ g (,) Ṁ S id S id CF x M MI x CF M Figure 4: Entroy of ing (the 0-junctions associated with each of the C-fields were suressed for simlicity) he left hand side of Figure 4 shows two comonents before ing, reresented by their corresonding CF-elements he right hand side shows the same two comonents after ing he two -elements reresent the generation of ing entroy at the exense of free enthaly What haens to temerature and ressure? It would be ossible to comute the temerature and the ressure as functions of the secific enthaly h and the secific volume v Since neither h nor v are changed in the rocess, and cannot change either In our code, we comute and as functions of the secific entroy s and the secific volume v: = (s; v) (65) = (s; v) (66) Since s changes its value, the formulae (s; v) and (s; v) must get modified in the rocess of ing, suchthat and remain constant Alternatively, itis ossible to kee the formulae the same, but calculate a modified entroy value (a coordinate transformation) that corrects for the different formulae being used: where: = (v; s intern ) (67) = (v; s intern ): (68) s intern := s s id ed : (69) It may beof interest to briefly discuss causes and effects in Figure 4 Let us assume we start out with a ure substance M, andwenow start to to it a substance M Initially, the C-field CF inherits all of the roerties of CF However, as material flows in, CF comutes the change in free enthaly ged id in accordance with Euation 64, and thereby, the free enthaly g of CF changes his change roduces the reuired g, which in turn leads to an entroy flow _ S id into CF, which changes the secific entroy of the ture What haens to CF? Since there is no mass flow of comonent M, no ing entroy is roduced Yet, the entroy of the ture changes, and conseuently, and of CF will temorarily change Now the euilibration elements between CF and CF become active, which euilibrate and, and in the rocess generate the additional ing entroy reuired In steady state, the temerature and the ressure of both CF and CF resume aroximately their original values, and the total ing entroy added to the ture corresonds to that needed in accordance with Euation 6 he euilibration acts as a P-controller here may remain a final bias, because, as soon as the total mass transfer into the ture has taken lace, there is no mechanism left to euilibrate the free enthaly values, and a g 6= 0may remain that does not generate any entroyflow, since by now, the mass flow M _ is eual to zero In the worst case, the entire mass M is added to the ture in a single Dirac event Conseuently, the mass flow _M is zero excet during the event, when it is infinitely large During the event, the entire ing entroy should be generated, but numerically, our code has no way of accomlishing this In the simulation, the ing entroy will remain zero, and the temerature of the ture will assume a lower value accordingly

5 7 NON-IDEAL MIXURES Non-ideal tures are defined in such a way that now, changes in volume due to the ture are also considered For examle, if wine is ed with water, the volume of the ture is smaller than the sum of the volumes of the comonents However, if the ing takes lace in a closed system, the volume of the ture must be the sum of the individual volumes, and conseuently, the ressure of the ture will now be smaller than the ressure of its comonents Figure 5 shows the generation of ing entroy in this case Since and are no longer constant, additional energy flows and _S are generated that result in two additional entroy flows into the CF-element of the ture CF g (, ) Ṁ Ṁ g S m 0 g (, ) M S CF MI Notice that the entroy consumtion" in one of the m-elements of the thermal domain is erfectly consistent with thermodynamics, since it is always overcomensated for by the generation of entroy in the other m-element he reader is reminded that the searation of a ture into comonent C-fields is only a mathematical construct, not hysical reality In ractice, the non-ideal ture usually takes lace in an oen environment, ie, the ressure remains constant, whereas the volume gets reduced his rocess can be modeled as deicted in Figure 6 CF CF CF MI CF outside CF g (, ) Ṁ Ṁ g S m 0 g (, ) M S CF Figure 5: Entroy of ing in non-ideal closed tures It can be seen that the rocess of ing makes use of the general -element as introduced in [] Evidently, entroy gets generated by means of the melements until the temerature of the ture euals that of the comonents If the comonents are at different temeratures (cold milk getting oured into hot coffee), the temerature of the ture assumes a temerature in between those of the two comonents Since the entroyflow is dictated by the mass flow, one of the m-elements consumes" entroy (assumes a negative R-value), whereas the other generates it Figure 6: Entroy of ing in non-ideal oen tures he -elements to the environment ensure that the temerature of the ture aroaches that of the environment, whereas the -elements ensure that the ressure of the ture, in steady state, euals that of the environment In the simulation code, these bond grahs are imlemented as follows he secific excess volume v Ex and the secific excess entroy s Ex of a non-ideal ture is tabulated in the literature as a function of temerature and ressure Given the current temerature and the ressure of the ture, these uantities can be comuted Using these values, and can be corrected as follows: = (v + v Ex ;s s Ex ) (7) = (v + v Ex ;s s Ex ) (7) A non-linear iteration takes lace until and no longer change hese are the values that the ture CF-elements ut out initially In general, both the

6 temerature and the ressure will reduce their values during the initial iteration As a conseuence, the -elements start to generate entroy, which flows into the ture, and raises both the temerature and the ressure again As long as the mass flow into the ture continues, the temerature and ressure will rise, as additional entroy gets generated In the steady state, the sum of the three entroy flows into the CF-element, S, _ adds u to zero At that time, the temerature is above thatof the ideal ture, whereas the ressure is still below that of the ideal ture he lower ressure is what is exected to haen in a closed system he higher temerature is needed to comensate for the entroy generated by the difference in ressures of the uned and the ed comonents his henomenon is often referred to as the heat of ing emerature and ressure only get adjusted to the values of the environment if the CF-elements of the ture exchange energy with the environment, as shown in Figure 6 8 MULI-ELEMEN MULI- PHASE SYSEMS he authors roose, to model each hase indeendent of the others (cf Section 5), and neglect influences of secondary imortance as needed his aroach is meaningful, as the euilibration elements (the -elements and -elements) have a tendency to stabilize the system and to reduce the imortance of secondary effects By decouling the hases, the fluid and gaseous hases may assume different flow velocities, which is commonly the case Figure 7 shows a system consisting of two comonents, each with a fluid and a gaseous hase traveling along a ie that is comartmentalized Only two comartments are shown he bottom half of the grah shows the fluid hase, whereas the to half deicts the gaseous hase he left half shows one comartment, whereas the right half shows the other he vertical bonds show the exchange of energy between the different hases of the same comonents Beside the usual euilibration elements, thermal conduction and volume euilibration, also condensation and evaoration are being modeled [4] Activated bonds are used to indicate that the condensation and evaoration elements reuire knowledge of the overall volume to determine their own artial ressures CF Fl CF V ges + {x, S E, V E } CF Fl CF {M,,} haseboundary {M,,} CF Fl CF {x, S E, V E } MI {M,, } MI {x, S E, V E } + V ges {x, S E, V E } {M,, } CF Fl CF Figure 7: Model of a two-element two-hase twocomartment convective system he horizontal bonds show the convective flows from one comarment to the next Beside from the euilibration elements, each hase uses couled elements to describe the convective flows he diagonal bonds show secondary effects of temerature and volume interchange between two comonents, one in its fluid hase and the other in its gaseous hase Mixture information (MI) elements are shown for the fluid hase only, as the gases are assumed ideal 9 MODEL OF A PRESSURE COOKER As an examle of the theory discussed in this aer, the model of a ressure cooker is being resented he ressure cooker reresents a multi-comartment, multi-element, multi-hase system In a first aroximation, the ressure cooker can be reresented as shown in Figure 8 When the ressure cooker is heated, more and more of the water turns into steam by means of evaoration he air comonent of the gas hase is needed to rovide the ressure cooker at room temerature with the ressure of the environment Without the air, some water would have to evaorate even at room temerature in order to roduce euilibrium ressure, which would be considerably lower than bar A more comlete, yet more stylized, model is shown in Figure 9 Here, each bubble reresents one com-

7 CF steam haseboundary air CF layer Only -elements were used between the two comartments (bulk and boundary layer), because the convective transort between these two layers (elements) is driven rimarily by the ressure difference between these two layers Hence the -element and the -element would be cometing for the euilibration of ressure, and a -element is not needed since the -element will accomlish the desired euilibration of ressure Additional -elements were Air in Boundary layer (t) SE: 9 K (t) Steam in Boundary layer Fl CF water Figure 8: Simlified ressure cooker model with two gaseous and one fluid hases : : onent, ie, one C-field together with its vector 0- junction Beside from water, steam, and air, twomore gaseous comonents were added, reresenting both air and steam in the boundary layer close to the wall of the ressure cooker hus, the gaseous hase has now two comartments, one reresenting the inside of the ressure cooker (the bulk), the other reresenting the boundary layer close to the wall his comartmentalization was necessary, since the assumtion of the entire gaseous hase being homogeneous is untenable, as it would lead to oor simulation results At some oint in time, the authors also considered the inclusion of air bubbles in the water and of water drolets in the air, but it was decided that the influences of these additional comonents were of second order small Air and steam were modeled as ideal gases he inaccuracies associated with this assumtion are of second order small he assumtion has the advantage, that the artial ressure of the steam can be comuted easily as steam = V V steam : he disadvantage of this solution is that the temerature and ressure tables for liuid water had to be slightly corrected in the vicinity of the boiling oint in order to rovide a clean transition to the gaseous hase he usual euilibration elements (the -elements and -elements) were included between water and steam, between water and air, as well as between steam and air, both in the bulk and in the boundary (t) SE: 9 K Air (t) KV water steam KV (t) Figure 9: Stylized bond grah model of the ressure cooker included between steam in the bulk and air in the boundary layer and vice-versa and condensation (KV-element) take lace between the water and the steam in the bulk Condensation takes also lace along the wall, ie, between the steam in the boundary layer and the water he SE-element at the bottom of Figure 9 reresents the heating element, whereas the SE-element at the to reresents

8 the cooling of the ressure cooker by letting cold water flow over it he following simulation exeriment was conducted At time = 0, the ressure cooker, which at that time is at room temerature, ( ß 9K), and has just been closed, ( ß 0kPa, humidity ß 0:5), is laced on a hot surface ( =9K) As exected, both ressure and temerature rise he ressure release valve, that would revent arealressure cooker from exloding, was omitted from the model It was assumed that the walls are strong enough to withstand arbitrarily high ressure values After half of the simulation eriod, the ressure cooker is removed from the stove and is laced under cold water (=9K) Dew forms immediately in the boundary layer, whereas condensation in the bulk starts somewhat later Figure 0 shows the temerature trajectories of the five comonents as functions of time During the heating hase, the five temeratures are almost identical During the cooling hase, the boundary layer cools down most raidly, the bulk follows somewhat more slowly, and the water cools down last Pressure/kPa oint where the water begins to boil At that oint in time, the temerature has reached a value of roughly 8 K, which, at a ressure of 0 kpa, indeed corresonds to the boiling oint of water During the cooling hase, the ressure in the boundary layer dros temorarily below that of the bulk, because water condensates more raidly in the boundary layer, and the two -elements cannot resuly the boundary layer with air/steam ture from the bulk arbitrarily fast Pressures Boundary layer Air, Steam, Water emeratures Air Steam Water Boundary Air Boundary Steam ime/s Figure : Pressure grahs of the simulated ressure cooker emerature/k 40 0 CONCLUSIONS ime/s Figure 0: emerature grahs of the simulated ressure cooker Figure shows the ressure trajectories he ressure of the bulk is indistinguishable from that of the fluid hase Only the boundary layer exhibits temorarily a ressure different from that of the other comonents During the heating hase, all ressures are indistinguishable from each other he knee in the curve (roughly at time 50 sec) reresents the he aer discusses different asects of the rather comlicated rocesses of modeling multi-element systems It was found that modeling each element searately as a storage element (a C-field), and connecting them by means of dissiative elements (R-fields) simlifies the modeling task, offers insight into the hysical functioning of multi-element systems, and leads to mathematical models that can be simulated in a numerically stable and highly accurate fashion he modeling methodology, as discussed in this aer, is still limited to multi-element systems without chemical reactions he discussion of the latter may, in the future, form a aer on its own he thermodynamics of chemical reaction systems were discussed already in [] he examle of a ressure cookerwas used to illustrate the functioning of the methodology, and to demon-

9 strate the racticality of the advocated aroach to modeling References [] Cellier, FE (99), Continuous System Modeling, Sringer-Verlag, New York [] Greifeneder, J (00), Modellierung thermodynamischer Vorgänge unter Verwendung von Bondgrahen, Dilomarbeit, Universität Stuttgart, Germany [] Greifeneder, J and FE Cellier (00), Modeling ConvectiveFlows Using Bond Grahs," Proceedings ICBGM'00, International Conference on Bond Grah Modeling and Simulation, Phoenix, AZ, [4] Greifeneder, J and FE Cellier (00), Modeling Multi-Phase Systems Using Bond Grahs," Proceedings ICBGM'00, International Conference on Bond Grah Modeling and Simulation, Phoenix, AZ, 85-9 [5] Stehan, K and F Mayinger, hermodynamik, Vol, Mehrstoffsysteme und chemische Reaktionen; Grundlagen und technische Anwendungen, Sringer-Verlag, Berlin, Heidelberg