Systems Physics Library

Transcription

1 Werner aurer Elsabeth Dumont Insttut für Angewandte athematk und Physk Zürcher Hochschule für Angewandte Wssenschaften Technkumstrasse 9, Wnterthur, 8401 Swtzerland Abstract In ths poster the odelca Systems Physcs lbrary s presented. The Systems Physcs lbrary s a free open-source lbrary wth models for dfferent areas of physcs [1]. The prmary use of the lbrary s for educatonal purpose n Physcs courses at medum level. The lbrary contans models from fve dfferent domans (s, translatonal and rotatonal mechancs, electrodynamcs and thermodynamcs). In the future, we plan to add chemstry as a sxth doman. Each doman contans connectors contanng a substance-lke quantty and the correspondng potental; basc models (capactance and resstance); sensors and actuators as well as some doman specfc elements, such as nonlnear accumulator, nonlnear resstors, valves, sprngs or nductances. In addton to the consttutve equatons each model also comprses the energy balance. For example, dsspatve elements calculate the loss energy and even the entropy producton wth the help of an addtonal thermodynamc connector. Keywords: physcs; educaton; system dynamcs 1 Introducton Systems Physcs s a novel approach to physcs wth whch begnners are able to grasp the fundamental concepts underlyng processes n nature and technology [2]. It s based on everyday concepts known from fluds whch are famlar to everybody. The analogy between physcal quanttes and fluds offers a very ntutve approach to physcs [3]. The powerful pctoral modelng offered by odelca helps students to understand basc physcal processes. oreover, here s an mmense number of problems that can be addressed by ths approach whch are usually not ncluded n the standard physcs textbooks at undergraduate level. Wth the help of ths odelca lbrary, students are able to model rather complcated physcal systems (e.g. frcton) wth lttle mathematcal knowledge. In the frst two semesters at the ZHAW School of Engneerng students learn the modelng concept of system dynamcs wth Stella or Berkeley adonna. Thereby they learn the basc structures and formulas (balance equaton, consttutve laws and the energy carrer concept) of physcs. The Systems Physcs lbrary could then be used n subsequent semesters n order to deepen ths knowledge. Systems Physcs combnes the modelng concept of System Dynamcs wth a unfed descrpton for all branches of classcal physcs known from Bond Graph theory [4]. Our concept of energy carrer s smlar to that of the Bond Graph theory. But there s a crucal dfference. In Bond Graph theory, force and torque are potental quanttes (effort quanttes) and the knematc varables velocty and angular velocty are seen as flow quanttes. In Systems Physcs however, ths approach s not possble because force and torque are part of the balance equaton and velocty and angular velocty are the drvng force for the approprate currents. Therefore force and torque are flow varables and the two veloctes are effort or potental quanttes. In our model based approach students start by formulatng the balance of a fundamental quantty (bathtub-thnkng for volume, mass, electrcal charge, momentum, angular momentum, entropy or amount of substance). Then they have to specfy the currents and the rates of change (feedback-thnkng). On a second layer they can add the balance of energy (bookkeeper-thnkng). Our vson of a new physcs course for engneers or natural and medcal scentsts covers system dynamcs modelng n the frst year and object-orented modelng n the followng years of studes [5]. 1137

2 2 Structure The systems physcs lbrary ncludes the domans s, electrodynamcs, translatonal mechancs, rotatonal mechancs, thermodynamcs and chemstry. Each doman ncludes a substance lke quantty and a correspondng potental. All connectors are constructed accordng to ths basc structure (Table 1) In all branches of physcs there are many dfferent storage systems such as tanks, plastc bottles, capactors, movng or rotatng bodes, heat accumulators. For each storage system we can wrte down two balance equatons: one for the basc quantty (volume, electrc charge, momentum, angular momentum, entropy); and one for the Energy W. The balance equatons for and W are gven by d I (1) dt dw IW (2) dt where I are substance currents and I W are energy currents. For the entropy balance equaton a producton rate Π S s ntroduced n addton to entropy currents. The doman specfc potental ϕ (pressure, voltage, velocty, angular velocty and absolute temperature) connects the current for the basc quantty wth the energy flux IW ϕi (3) A odel wth two ports and a conserved quantty (deal volume, electrc charge, momentum or angular momentum) contans at least one equaton for conservaton and one for the power I1 + I2 0 (4) P ϕ I (5) These equatons are formulated n separate partal models. In thermodynamcs we need two dfferent partal models, one for heat conductance whch conserves energy TI 1 S1 + TI 2 S2 0 (6) and one for deal heat engnes whch conserve entropy IS1 + I S2 0 (7) An addtonal equaton calculate the producton of entropy n the partal model for heat conductance IS1 + I S2 ΠS (8) or the power n a partal model for deal heat engnes P TI S1 (9) Ideal heat s are modeled smlar to s: n the same way as a s water or ol heat s entropy. Ths s one of the man messages of Systems Physcs. Table 1: flow and potental varables n the dfferent domans of Systems Physcs doman quantty potental s volume pressure electrodynamcs charge voltage translatonal momentum velocty mechancs rotatonal mechancs angular momentum angular velocty thermodynamcs entropy temperature 2.1 Hydraulcs The volume of an ncompressble flud s the basc quantty n s and pressure s the assocated potental. Because pressure multpled by volume flow equals the flux of energy, the conservaton of energy s guaranteed. The s lbrary ncludes dfferent storage elements lke open vessels, sprng-loaded tanks and other accumulators, ppes wth lamnar and turbulent flow characterstcs as well as varous valves. The nerta of the flud flowng through the ppe s responsble for the nductve effect. In all three system categores (capactance, resstance and nductance) the stored or dsspated energy s calculated. In an addtonal resstor element, the entropy s recalculated and s connected by a thermal connector. The temperature of ths connector determnes the vscosty of the flud. Two deal s are modeled, one wth a sgnal nput for pressure dfference and one wth a sgnal nput for the volume flow rate. In addton, the lbrary contans sensors for pressure and volume flow. 2.2 Electrodynamcs The electrodynamcs lbrary based on the odelca standard lbrares wth capactor, resstor, dode and nductor. In addton, an solated metal sphere s modeled for charge storage n experments wth hgh voltage. The energy balance s calculated n all elements. Ths calculaton s made for ddactc purpose and for energy check n complex systems. A resstor wth a thermal connector wheren the entropy s calculated enables the modelng of electro-thermal elements such as resstance heatng or lght bulb. 1138

3 Poster Sesson 2.3 echancs The mechancs lbrary ncludes two parts: one for moton along a straght lne and one for rotaton around a fxed axs. Therefore, the assocated quanttes momentum and angular momentum can be treated as scalars. The smlar statement s true for the assocated potentals (velocty and the angular velocty). The tght separaton of the balance equaton for momentum or angular momentum from knematcs has some mplcaton: mass and moment of nerta have only one connector and dstance or angle can only be calculated n elements that descrbe the flow of momentum or angular momentum. In addton to the lnear systems some more elements, such as frcton, ar resstance or elastomer sprng are modeled. An ordnary rope or strng s a further element often used n physcs nstructon. It s modeled as a sprng-damper-system wth predetermned breakng pont. As n s and electrodynamcs some elements for momentum or angular momentum flow are provded wth a heat connector. The produced entropy s calculated and the temperature has an nfluence on the consttutve laws of these elements. energy and momentum are connected wth the help of the famous Ensten equaton. The model of a smple rocket engne completes the model zoo, although ths element belongs to the open systems, whch are not ncluded n ths lbrary. Fgure 2 shows the velocty-tme behavor of a rocket another system that can be easy modeled wth our approach. Fgure 2: Velocty-tme-dagram of a rocket ascendng n an sentropc atmosphere wth constant gravtatonal feld Translatonal and rotatonal mechancs are connected to each other by means of pulleys. The correspondng model has four connectors, three for translaton and one for rotaton. Wth two bodes, two strngs, a pulley and a bearng frcton, we can model Atwood's machne (Fgure 1). 2.4 Thermodynamcs Fgure 1: A model of an Atwood machne wth two weghts and bearng frcton. The symbolc earth wth fve connectors for the flow of volume, momentum, angular momentum, electrc charge and entropy stands for the surroundng. The mass element has a momentum source whch strength corresponds to the tangental component of the weght force. A further element contans the equaton for the relatvstc mass. In ths element, If we take entropy as the basc quantty of thermodynamcs t s easer to wrte down the correct equatons than f we take the energy as a conserved quantty [6]. However, for the equatons themselves, t does not matter whether we start from the energy or from entropy as we have shown n the ntroducton. There are two dfferent models whch specfy the transport of heat, the heat conducton and the deal heat engne. We descrbe homogeneous systems, whch are heated at a constant pressure, wth the state varable enthalpy H. Enthalpy s a specal form of energy and a thermodynamc potental. Although entropy flow and temperature are calculated n the connector, ths s no problem wth respect to the balance equaton for energy dh TIS (10) dt ore generally, a homogeneous thermodynamc system can at least change entropy and volume. Therefore the system has temperature and pressure as two assocated potentals. To dscuss and model 1139

4 such a system we have developed the Carnotor, a smple machne wth a thermal and a connector (Fgure 3). Carnotor s a portmanteau composed of Carnot and otor (German word for engne). The Carnotor conssts of a double-actng cylnder flled wth the substance to be examned on one sde of the pston and an deal flud on the other sde. To each port we can add a, a closng-off or a bg storage tank. Wth ths equpment students can analyse all four basc processes of thermodynamcs (Table 2). The correspondng model calculates pressure and temperature from the change of volume and entropy. For the deal Gas, the consttutve laws are as follows V f T S S0 + nrln + nrln ; (11) V 2 T pv nrt 0 0 (12) f stands for the degrees of freedom of gas molecules, n for the amount of substance and R for the gas constant. Equaton (12) s known as deal gas low. Table 2: The four basc processes n thermodynamcs process heat port sochorc heat sobar heat sentropc closngoff sotherm storage tank port closng-off storage tank unchanged volume pressure entropy temperature Fgure 3: The Carnotor has two ports, one for heat and one for an deal flud. Both ports can be combned wth a closng-off, a storage tank or a Fgure 4: Ths system contans four models for deal gas, two deal heat s, two deal s, a heat flow and a volume flow element. The Carnotor can be taken as the core element for a lot of thermodynamcs engnes. Fgure 4 shows a model wth whch one can smulate all four basc processes smultaneously 3 Conclusons Systems Physcs provdes a consstent, coherent and relevant structure of physcs. A huge number of dynamcal systems can be modeled wth the same heurstc approach. The equaton of balance for substance-lke quanttes lke volume, mass, electrc charge, momentum, angular momentum and entropy yelds the backbone for such models. By addng the consttutve laws for accumulators and conductors we get the basc equatons. In a thrd step we can add energy as a second substance-lke quantty. The energy balance analyss s often useful but not necessary for smple systems. But energy conservaton becomes an nevtable requrement n more complex systems lke thermodynamc accumulators. Systems Physcs has been developed on the bass of the Karlsruher Physkkurs [7] and taught n dfferent physcs courses at Zurch Unversty of Appled Scences. Wth the help of the Systems Physcs lbrary we hope that we can convnce more and more teachers of the usefulness of ths method. A countless number of dynamc models are watng to be modeled wth a System Dynamcs or a odelca tool. On Youtube you can fnd some tutorals on specfc topcs of the Systems Physcs lbrary [8] 1140

5 Poster Sesson References [1] [2] Borer, T et al.(2010). Physk En systemdynamscher Zugang für de Sekundarstufe II. hep, Bern. [3] aurer, W (2002). Systemdynamk En möglcher Pfad durch den Irrgarten von Fehlvorstellungen. PdN-PhS. 7/51, Auls, ünchen. [4] Karnopp, D., argols, D., Rosenberg, R. (1990). System dynamcs: a unfed approach. Wley, New York. [5] aurer W. Physk und Systemwssenschaft n Avatk. Proceedngs of the 20th Symposum on Smulatontechnque ASI 2009, Cottbus,Germany, ASI September 23-25, [6] Fuchs H (2010). The dynamcs of heat. Sprnger, New York. [7] 2005.Fredrch Herrmann et al. (2010): Der Karlsruher Physkkurs. Auls, ünchen. [8] ker (playlsts odelca-kurs and odelca). 1141