«aunt-“.2! THE". a. ‘- 1' .. (1‘ . 4' l '__V,..'_ L175"? A..- m 97 ‘-~‘+" £14.23 (A _ 3‘3...) '3‘. ."c v, . . I C‘ I‘ V“‘}".(,* fixo'.t-.,f. ‘..~’a 3h». ‘1. y L n ‘ L 1 v . q ‘ u 'n . '.'-a ' ' kg ’ E“: to; -» 2/ '.. - :v y - \ . g ‘ ' ”-Mq'rbfi'HV'-< “1" "1Q'fi‘, ';’ . ‘ 3' lllllllllllllllllllllllllllllllllllllillllllllllllllllllllllll 3 1293 10388 2365 'rvlfSIuj BEIURNING MATERIALS: PIace in book drop to man/sales remove this checkout from .a-luzs-IIL. your record. FINES wi11 be charged if book is returned after the date stamped be10w. SIMULATION OF THE DYNAMICS OF A SOLAR COLLECTOR SYSTEM USING BOND GRAPHS by Syed Asif Nasar A Thesis Submitted to Michigan State University in partia] fquillment of the requirements for the degree of Master of Science Department of Mechanical Engineering 1979 ABSTRACT Simulation of the Dynamics of a Solar Collector System Using Bond Graphs by Syed Asif Nasar Using bond graphs, an effort has been made to study the dynamics of a solar collector system. The method is applied to a particular flat- plate solar collector typically used in residential heating and cooling. The dynamic performance of the flat-plate solar collector predicted by the bond graph model was similar to that derived from the conventional nodal. The conductive processes represented by the bond graph model are valid for all conditions but the convective model is only valid at low mass "flow" rate. To Kamran, Rizwan and Sabuhi "Dignity at all cost" ii ACKNOWLEDGEMENTS I would like to express my appreciation and thanks to my advisor, Professor Ronald C. Rosenberg for his guidance and support not only on this thesis but throughout my graduate and undergraduate programs. Thanks also to many friendly people in the Department of Mechanical Engineering for making my stay at Michigan State University a worth- while and enjoyable experience. My humble thanks to my father, for his untiring effort in providing me with the best of everything; to my mother, for teaching me love and patience; and to both, for their continued love and prayers for me, without which this would not be possible. My sincere appreciation to my sister and brother-in-law fbr putting up with me for six long years. And last but definitely not the least, my many thanks and love to my little sister, my "gupya" Sabuhi, for her encouragement, love and prayers, without which I could not have achieved my goals. iii TABLE OF CONTENTS List of Tables List of Figures Chapter 1. Introduction 1.1 Objectives 1.2 Organization 1.3 Bond Graphs Chapter 2. Solar Collector System Model 2.1 Solar Energy System 2.2 Details of the Solar Collector System 2.3 Bond Graph Model 2.3.1 Improvement of the Convective Model 2.4 Conventional Approach Chapter 3. Dynamic Performance of the Solar Collector System 3.1 Simulation Procedure 3.2 Simulation Studies 3.3 Results and Performance Evaluation Chapter 4. Conclusions and Recommendations 4.1 Utility of Bond Graphs for Modeling and Simulation 4.2 Solar Collector Models 4.3 Next Steps References Appendices Appendix A. Glossary of Terms Appendix B. A Definition of the Bond Graph Language 1V 43 44 LIST OF TABLES Page Table l 32 Table 2 34 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES 1. Solar Energy System and Residence Solar Collector Thermal Network for Flat-Plate Solar Collector Word Bond Graph of Solar Collector Bond Graph with Causality Solar Collector Subdivided into Four Segments Bond Graph with Causality Convective Path Within the Solar Collector Tubes (Heat removal path) 9. Bond Graph with Causlity for both Conductive and Convective Process 10. A Simple Flow Chart 11. Comparison of Results 12. Collector Efficiency versus Time vi Page 12 13 l4 17 20 23 3O 35 37 CHAPTER I INTRODUCTION Renewed interest in solar energy has developed since 1973 as a result of increasing costs of energy from conventional resources and the problems of importing and extracting fuels that are acceptable from environmental standpoints. The engineering design of solar extraction processes presents unique problems, due to the intermittent and diffuse nature of the resource and the capital-intensive (high initial cost) nature of the processes. Although solar energy may be considered a new or unconventional resource, it has been in use historically for several applications. Solar evaporation of salt brines to recover the salts has been practiced for many centuries. as has solar drying of agricultural products. Solar water heating is a standard method of providing domestic hot water in parts of Australia and Japan (1)} and small viable industries are based on the manufacture, sale and installation of solar water heating equip- ment in these countries. 1.1 Objectives The purpose of this thesis is to investigate the feasibility of using a bond graph approach to study the dynamics of a solar collector system. The objective is threefbld: (1) to derive a mathematical model of the solar collector using a bond graph, (2) to study the dynamics of a solar collector system once the mathematical model is derived, and (3) to consider the utility of the bond graph approach in the study of solar collector systems. 1 Numbers in parentheses designate references. It should be noted that this thesis does not attempt to be com- prehensive, in that it concentrates on the solar collector rather than the use of solar energy for residential heating and cooling. 1.2 Organization A description of the solar collector system is presented in Chapter 2. Using the bond graph techniques as the principal tools, the solar collector system was modeled. The mathematical model was also derived by the conventional approach. In Chapter 3 the procedure used in simulating the operation of this system is outlined. This simulation involved the use of hourly weather bureau data from Madison, Wisconsin. The performance of solar collector systems is also discussed. Chapter 4 presents conclusions and recomnendations derived from this study. 1.3 Bond Graphs A bond graph is a topological diagram, clearly representing any physical network. It was invented in the 1960's by H.M. Paynter (2) after intensive study and use of block diagrams. However, many users of bond graphs have brought them to their present state of utility. As an example of the present state of the art using the Enport program, the elements of a bond graph and their numerical values can be directly typed in a computer and the program will derive the system state equations and analyse it. For those interested in bond graphs and bond graphs computer analysis, informative references are "System Dynamics: A Unified Approach“ by Karnopp and Rosenberg (3), the ASME "Journal of Dynamic Systems, Mea- surement, and Control" (4), "Introduction to Bond Graphs and Their 2 Applications" by Thoma (5), and "A User's Guide to Enport-4" by Rosenberg (6). In addition, Karnopp has investigated the convective aspect of heat transfer in his papers titled "A Bond Graph Modeling Philosophy for Thermofluid Systems" (7) and "Pseudo Bond Graphs for Thermal Energy Transport“ (8). In a compact but general form a definition of the Bond Graph Language is also given in Appendix B. 1.4 Literature Survey The study of solar energy for use in residential heating and cooling has been done on only a very limited basis. Previous work is more substantial in the area of solar energy use in providing service hot water. In the last twenty years, quite a few solar energy houses have been constructed. For example, M.I.T. has been studying solar houses for a number of years and Engebretson (9) summarizes the operation of the fourth and last solar house for the years of 1959 and 1960. The perfbrmance of this house was felt to be satisfactory with no major problems encountered. An experimental system which provides heating and cooling fOr a house in Tucson, Arizona, was studied by Bliss (10). This house and solar energy system is not the type that would prove economically feasible for widescale use, since its cost is quite large. However, as pointed out by the author, the house is constructed for the main purpose of studying the conversion and use of solar energy for residential heating and cooling. Another variation of the solar energy home is presented by Thomason and Thomason (ll, 12), one which uses a simple type of solar collector and energy storage by means of a large water tank, which is surrounded by a layer of stones. Most of the early work in the field of solar energy fbr residential use was summarized in a report of the proceedings of a symposium held in 1950 at M.I.T. (13). Collector models and system models presented at this symposium are still being used today. One of the earlier computer studies of the dynamics of a solar collector system was made by Buchberg and Roulet (14). Equations modelling a solar energy system were used with hourly weather bureau data to accomplish the simulation. One of the earlier references to solar heating and cooling was made by Lof (15) in 1955 in which he discussed the use of triethyl glycol for dehumidification of air. In a more recent paper Tybout and Lof (16) discuss the results of a computer simulation of house heating for various sites in the United States. Equations with eight design para- meters were used to describe the system, and hourly weather bureau data were employed for use on a digital computer. Of the eight parameters studied, the effects of collector size, geographic location, building construction and number of glass covers on the collector were most significant. Klein, Beckman and Duffie (17) made a computer study of a model in Albuquerque, New Mexico, for a solar heating and cooling system. The result was positive in the sense that an economical solar heating system could be designed by a simple graphical method requiring monthly average meteorological data. In a report of the proceeding of a symposium held in 1977 at the Winter Annual Meeting of the American Society of Mechanical Engineers at Atlanta, Georgia (18), the papers deal wflth three broad topics: analysis and experimental behaviour of heat transfer in solar collectors and in storage systems; experimental evaluation of complete solar systems; and computer analysis and simulation of complete solar systems. The volume gives a good overview of on-going work in these areas, and serves as both a reference and a survey. CHAPTER 2 SOLAR COLLECTOR SYSTEM MODEL 2.1 Solar Energy System The use of solar energy for providing heating, cooling and service hot water to a typical residence is illustrated in Figure l (19). The study of such a system is made possible by describing all of its come ponents by mathematical models. In addition, a building mathematical model is also required to determine the heating and cooling loads en- countered by a typical residence. This mathematical model can be derived either by using a bond graph approach or by conventional approach. In this study the bond graph approach is used to derive the mathematical model for the solar collector system. The system, illustrated in Figure 1, includes a solar collector, an energy storage tank, a space heating system, a central air conditioner, a service hot water system andamain auxiliary heater. The energy transfer medium for the system is water. This was chosen for simplicity since the variations in boiling points, specific heats and other properties of, mixtures containing various amounts of anti-freeze and anti-rust additives was beyond the scope of this research project. The system is a closed, pressurized system at 21 psia, which allows the water to reach 230°F without boiling. As mentioned earlier, this study is restricted to the considera- tion of thermal processes in which solar radiation is absorbed by a surface,the solar collector, and converted to the internal energy of the working fluid. (A) (H) (c) if {3 4 + (e) __1—$&)f’r——i (E) (F) (a) .2. _.~—— r1. 4 Figure 1. Solar Energy System and Residence* (A) - Solar Collector, (B) - Energy Storage Tank, (C) - Service Hot Water System, (D) - Air Heater, .,(E) - Air Conditioner, (F) - Cooling Tower, (G) - Main Auxiliary Heater, (H) - Residence * Taken from (19), Figure 1 2.2 Details of the Solar Collector System The solar collector is one essential item of equipment whose function is to transform the solar radiation incident on its surface into thermal energy of a working fluid which is transferred to the energy storage tank. The collector is modelled with a "multi-node” capacitance (20, 21). Such a model is developed by considering the collector to consist of multiple nodes, each with a single temperature and capacitance. The nodes are positioned at a single glass cover, at the collector plate and at each of the four “lumps" of the fluid. The multi-node model can be extended to simulate the performance of flat plate collectors with more than one glass cover by the addition of a node at each cover. The important parts of a typical flat-plate solar collector as shown in Figure 2 are: the "black" solar energy-absorbing surface, with means for transferring the absorbed energy to water; one transparent glass cover to shield solar radiation over the solar absorber surface in order to reduce convection and radiation losses to the atmosphere; and back insulation which causes the conduction and convection losses to be negligible. Water is circulated through the collector by means of a pump whenever conditions are such that the value of the energy transferred to the working fluid is positive. Keeping the cost in mind, realistically speaking, pump activation should occur when the value of the energy obtained is greater than the cost of operating the pump. However, I chose pump activation criterion to be a positive Qu because the main purpose of this research is to study the conversion and use of solar energy. Whatever reasonable GLASS COVERS ABSORBI NG SURFACE TOUT INSULATION F LUIDTUBE Tuvrh Figure 2. Solar Collector criterion is used to activate the pump, this point is minor in compari- son to the overall performance of the solar collector. To prevent overpressurization of the closed system with water in the collector, the temperature of the exiting water from the collector is monitored such that whenever its value reaches 230°F an energy dissipation mechanism is activated. This simulated system allows the collector to get rid of the energy when it is operating at conditions which would cause an exiting temperature greater than 230°F. When operating at these overflow conditions the exiting temperature of the water is constant at its maximum value of 230°F. Throughout this study the following assumptions are made: The flow is steady state and incompressible with the parameter AM/M always less than one. The glass cover and the collector absorbing plate are uniform thermally speaking. Each "lump" of fluid is also uniform in its temperature. The temperature of sky is considered to be the same as temperature ambient. Only the beam radiation is con- sidered and the black loss is assumed to be negligible. 2.3 Bond Graph Model Bond graphs have been shown to be useful in the modelling of a wide variety of physical dynamic systems, but open systems in which several types of energy are transported across boundaries with mass flow have not been modelled as elegantly as fixed mass systems and their analogs. For the solar collector system a bond graph approach is outlined here which allows most of the conceptual and practical advantages of normal bond graph techniques to be retained for systems like this in which thermal energy transported by a flowing fluid is important. 10 The aim of the modeling procedure outlined below is to create reasonable lumped parameter models for performance prediction. Thus it is not possible to give firm rules for selecting the number and size of the lumps nor is it possible to say in advance whether the perfect and instantaneous mixing assumption is satisfactory. However, the model is at least rational in the sense of energy conservation. The thermal network for the solar collector is shown in Figure 3. The loss from the bottom of the collector is assumed to be negligible. If the solar collector is considered without being subdivided into smaller segments, it could be represented as shown in Figure 4 by the word bond graph. The effort and flow variables are temperature (T) and heat flow rate (O) respectively for the heat transfer model. When the variables T and O are used fOr effort and flow, the product T - O has no special significance, since O is already a power term. For this reason, the thermal bond graph is called a pseudo bond graph. In many respects, the thermal graphs resemble normal bond graphs, however. For example, the thermal state variable is Q, a displacement representing stored energy. The most fundamental element in any bond graph is the bond itself or the part of ports which allow subsystems to be conjoined. Figure 5 shows the thermal bond graph model of the solar collector. In Figure 5, the hydraulic part of the bond graph is not shown. This bond graph model would work very well for a purely conductive system. To allow for the convective nature of the heat removal from the collector plate, the hydraulic part of the model must be taken into account. Furthermore, in the model shown, the temperature of the water 11 TAMBIENT Fi9 or; LASS COVER REFLECTION LOSS / TFLUID ’QUSEFUL Figure 3. Thermal Network for Flat-Plate Solar Collector 12 AMBIENT TEMF.> RESISTANCE SOLAR RADIATION TEMP. GLASS COVERS RESISTANCE TEMP. PLATE RESISTANCE TEMP. FLUID Figure 4. Word Bond Graph of Solar Collector 13 U) (D (D :o C3:04 Sf 7i. IRIO (:21 T j :19” 4:: J C3 10 Figure 5. Bond Graph with Causality throughout the entire length of the solar collector is taken to be the same. To improve on the idea of instantaneous mixing of water through- out the collector length and improve the overall prediction accuracy of the model a major reticulation is made. As shown in Figure 6, the solar collector is subdivided into four segments. The bond graph model for the conductive path is not altered; it is the convective path of the system that is greatly altered. At this time there is no novel way to represent the heat transfer be- tween a solid and a fluid in a bond graph (fbr those interested in further investigating this problem see references 7 and 8). To incorporate the convective path of the solar collector, the system is divided into two major models. The first model in the bond graph language is represented by Figure 7. Figure 7 represents the purely conductive path of the system. To analyze the conduction dynamics of the system, the bond graph of Figure 7 presents a straightforward model fer deriving the system state equations by means of existing rules. Since there are six inde- pendent thermal energy variables associated with the bond graph of Figure 7, we will get six system state equations. The six system state equations are: (ncp)gT1 = AC[R9(T7-T])-R10(T1-T2)] (2.1) ("cp)pTz‘Ac[°soi'Rii(Tz'T3)'R12(T2'T4)’Ri3(T2’Ts)' “14(72'T5)*R10(T1'T2)3 (2'2) 15 AM DISPLACED FROM ONE SEGMENT IINTO ANOTHER COLLECTOR MTE ] I l Tm \ j I T 7 i FLUID IN EACH OF THE 4 SEGMENTS Figure 6. Solar Collector Subdivided into 4 Segment 16 Rl‘l-‘I R 1——>|'4 R RIZZj' . 23 29-2—42]: 25]” 5.10. Figure 7. Bond Graph with Causality 17 where (MCp)wT3=(Ac/4)R11(Tz-T3) _ (2.3) (MCp)wT4=(Ac/4)R12(T2-T4) (2.4) (MCPIWT5=(Ac/4)R13(T2-T5) (2.5) (Mcp)wT6=(Ac/4)R14(T2'T6) (2.6) (MCp)g is the capacitance of the glass cover (BTU/F). R9 is the heat transfer coefficient (loss) between the ambient and the glass cover (BTU/hr-ftZ-F). . dT T1=afl , is temperature "flow" rate of the glass cover (F/hr). T7 is the ambient temperature (F). T1 is the temperature of the glass cover (F). R10 is the heat transfer coefficient (loss) between the flat solar plate and the glass cover (BTU/hr-ftz-F). T2 is the temperature of the flat solar collector plate (F). (MCp)p is the capacitance of the solar collector plate (BTU/F). dT2 Tz‘af' 0501 is the rate of the solar radiation intercepted by the solar collector plate (BTU/hr-ftz). R1, i=11,12,l3,l4: heat transfer coefficient (loss) of cell i (BTU/hr-ftz-F). 18 T1, i=3,4.5.6: temperature of cell i (F). (MCp)w is the capacitance of water in each of the four cells (BTU/F). T6 is also the temperature of water exiting from the collector. dTi Tfar . 1=3,4,5,a (F/hr). Ac=collector are (ftz). In order to get the solution of the six equations above we have to know the value of T1, i=l,2,3,4,5,6. Since there are four unknowns due to the convective nature of heat transfer from the plate surface to the fluid, we must get four algebraic equations to find these unknowns. Here we take a little deviation from the normal bond graph approach. As shown in Figure 8, each segment of the solar collector is numbered. In order to get the equation for the convective path, the following rules are followed. Let the location of the fluid be represented by K and the time stage by i. “Freeze V';that is, there is no time advance. Then allow energy to redistribute itself by conduction thereby redistributing the tem- perature TK,i. This would account for the mass flow rate (M) effect instantaneously. Mass conservation is automatically satisfied for steady state fluid flow, since mass leaving K-l cell enters K cell and mass leaving K cell enters K+l cell. Also energy conservation is also satisfied if energy leaving K-l cell enters K cell and energy leaving K cell enters K+1 cell. Since energy redistributes itself during each stage i, the "temperature" is also redistributed. 19 :1, 11-; um IFM Tan CK+I cK+2 Figure 8. Convective Path Within the Solar Collector Tubes (Heat Removal Path) 20 An energy balance is made on each cell. In words the energy balance is written as: The new energy in cell K = (the energy already present in call K) + (the energy that enters cell K from cell K-l) - (the energy that leaves cell K and enters cell K+l). In equation fOrm the above could be expressed as: New UK,i = Present UK,i + AU _],i - AUK,i K where AU = AMCpT. Once an energy balance has been made on a particular cell, the new temperature of that cell is determined immediately as: CpMTnew = New U. Using the above technique the four algebraic equations for the convective path are found to be: where - A". AM Tnew3 ' T3(1 ' M ) + FF'THn (2.7) - AM AN Them - T4(1 - 74—) + M— T3 (2.8) .. an AM TneWS - T5(1 - M ) + M— T4 (2.9) AM - AM Tnewfi - T6(1 - FF' + MT T5 (2.10) T i=3,4,5,6: new temperature of cell i(F). newi’ 21 Ti’ i=3,4,5,6: Temperature of cell i before an energy balance was made (F). M is the amount of water in cell (lbm). AM is the amount of water "displaced" from each cell (lbm). Tin is the inlet temperature of the fluid (F)- The set of ten equations derived above is simulated on a CDC 6500 digital computer to analyze the dynamic response of the flat-plate solar collector. 2.3.1 gImprovement of the Convective Model The methodology described to model the convective heat transfer, from the solar collector plate into the working fluid, could be improved by inserting a conductance effect on a common flow junction between pairs of common effort junctions representing the fluid "lumps". This is shown in the bond graph model of Figure 9. The conductance parameter is MCp, and increases with the mass flow rate. Activation of the upstream bond indicates that there is no direct flow effect. (For a discussion of activation see reference 3, page 30). The concise model shown in Figure 9 is possible because MCp is constant for each "lump". Otherwise a more general two-port R must be used (see reference 8). As illustrated in Figure 9, the bond graph model of Figure 7, after the necessary modifications,is now able to account for the mass 22 /\ RI‘JJ—‘l .' R|<'—2-31 22 .. 1—3-73Ho—3H1F3—5—73 —§>;|137+——2;o-3-7111-3—::o—-57121|4—WJ$3 223131 29R 41- 39L 3161-32 R R mm» C (then) (Inca) C (file) C :59@ Figure 9. Bond Graph with Causality for Both Conductive and Convective Processes flow effect of the fluid within the collector tube. By applying the existing rules to Figure 9, equation 2.3 to equation 2.6 could be modified to include the convective heat transfer within the collector tube. where The modified equations are: (MCp)3T3=R28(Tin-T3)+R]1Ac(T2-T3) (2.3.1) (ncp)4i4=nzg(13-T4)+R12Ac(12-14) (2.4.1) (MCp)515=R30(T4-T5)+R13AC(T2-T5) (2.5.1) (MCp)616=R31(Ts-T6)+R14AC(T2-T6) (2.6.1) (MCp)1, i=3,4,5,6: Capacitance of water in each of the four "lump" (BTU/F). dT - i - 1,, "3E‘ , i-3,4,5,6 (F/hr). T. 1n inlet temperature of the fluid (F). Tout R1=ncp, i=28,29,30,31,32 (BTU/hroF). M = mass flow rate (lbm/hr). Cp = specific heat of fluid (BTU/lbm-F). Ri=Ui, i=11,12,l3,l4, is the heat transfer coefficient (loss) = outlet temperature of the fluid (F). of cell 1 (BTU/hr-ftz-F). 24 Ac is the collector area (ftz). It should be noted that the conduction loss between the two adjacent "cells” is so small, that for all practical purposes it should be neglected without having any serious effect on the pre- diction of the dynamic performance of the collector. Equations 2.1 and 2.2 together with equation 2.3.1 to equation 2.6.1 are the six state equations derived from the bond graph model of Figure 9. These state equations represent a complete mathematical model for the dynamic performance of the solar collector and could be programmed in a digital conputer to predict the system perfbrmance. 25 2.4 Conventional Approach Conventionally the solar collector is modelled with zero thermal capacitance. A zero capacitance model is one in which the effects of thermal capacitance on collector performance are neglected. The collector is considered to be in equilibrium with its environment at any instant of time. The energy absorbed by the solar collector is determined by the following relation: HR = HSRTTgap (2.11) where H is the energy absorbed by the solar collector (BTU/hr-ftz). R H is the radiation on a horizontal surface (BTU/hr-ftz). S RT is the correction factor which converts the radiation on a horizontal surface to that on the inclined surface of the collector. This factor is a function of the latitude, declination, time of day, tilt angle of and the azimuth angle. T9 is the transmittance of the glass cover over the collector plate as a function of the angle of incidence of the solar radiation. up is the absorbtivity of the blackened collector plate as a function of the angle of incidence. Not all the absorbed energy, however, is transferred to the working fluid. A small amount of the absorbed energy is lost to the 26 surroundings with the remaining energy transferred to the water circu- lating through the tubes which are attached to the collector plate. The relationship between the absorbed energy. losses to the surroundings and the useful energy is given by: Qu = FACEHR-UL(Tin-Tamb)] (2.12) where Qu is the rate of energy transfer to the water circulating through the collector (BTU/hr). F is an efficiency factor that accounts for heat losses occurring at a mean temperature greater than the entering water temperature. Ac is the collector area (ftz). U is the heat loss coefficient for heat losses to the surroundings L (BTU/hr-ftzF). Tin is the temperature of water entering the collector (F). Tamb is the ambient air temperature (F). From a purely convective point of view the rate of energy addition to the water is also described by an energy balance. -T. ) (2.13) where M is the mass flow rate of the water through the collector (lbm/hr). 27 Cp is the specific heat of water (BTU/lbm-F). Tout is the temperature of the water coming out of the collector (F)- Once HR is calculated from equation (2.11), its value could be used in equation (2.12) to get Qu' From equation (2.13) with the value of Ou already known Tout could be calculated. 28 CHAPTER 3 Dynamic Perfbrmance of the Solar Collector System 3.1 Simulation Procedure The mathematical equations described in Chapter 2, section 2.3, were programmed on a digital computer to study the dynamics of a flat- plate solar collector. The Euler method of integration was used to perform the inte- gration of all differential equations. The integration interval (At) was picked to be 0.001 of an hour. The value of AM/M must always be less than 1 for the system to be stable. The program goes through three 100ps (see Figure 10). The first loop allows the distribution of temperature due to conduction. On the completion of the first loop which accounts for 0.005 of an hour, the temperature of the fluid within the cell is adjusted for the con- vective process, that is, for the mass flow rate effect. On completion of the second loop which accounts for 0.05 of an hour the temperature, t1, i=l,2,3,4,5,6 for the conductive process and the subsequent adjusted temperatures for the convective process is printed. On completion of the third loop the program has accounted for all the data for that hour and is now ready fOr a new set of data for the next hour. The data available to this program is the hourly weather bureau data. This program is constructed such that the system can be operated in any location by providing the appropriate weather bureau data for that site. 29 START ‘ SET INITIAL commons a - HOURLY DAT , DO I 81.20 DO J=I,IO EQUATIONS TIME ITIME +0001 ADJUST FOR gowvecnow Figure 10. A Simple Flow Chart 30 Incidently, the bond graph of Figure 7 could be directly fed into the computer by using the Enport program. The output would be the equations and the dynamics of the conductive process. 3.2 Simulation Studies The performance of the solar collector system was studied for the site of Madison, Wisconsin, using hourly weather bureau data. This site was chosen because the performance evaluation of the solar collector was available along with various data by means of the conven- tional approach. Since the purpose of this study was to see whether it is feasible to use bond graphs as a modelling tool, the output of the bond graph model was compared to the output obtained from conventional models for a one-day cycle. No optimization was attempted. Table 1 provides values and definitions for the main parameters used in the study. These values and values for many other parameters were either available with the problem or calculated by the authors of reference 1. For each hour of the study the correction factor R was calculated for beam (specular) radiation only. The error resulting from neglecting the diffuse radiation is not of major concern, since we are seeking validation of the basic modeling approach. However, the diffused radiation could easily be included in the model by adding a resistance between Tsky(=T ) and T amb plate parallel to R10 of Figure 3. Also the correction factor R is calculated for the midpoints of the hours studied. 31 Parameter Collector Tilt R R9 R10 R11=R12=R13=R14 (CIDM)g (€lep (cpM)w Delta M Full M NO Ac Tin Ta TABLE 1 Main Parameters Value 55 degrees 158-59 1bm 5.20 BTU/hr-ftz-F 1.64 BTU/hr-ftz-F 6.87 BTU/hr-ft -F 2.19 BTU/F 3.23 BTU/F 1.13 BTU/F 0.79 1bm 1.1321bm 1 21.52 ft2 l40°F 0.84 /hr 2 32 Conments angle between plane of collector and horizontal mass flow rate of water through collector heat loss coefficient from glass cover to ambient air sum of the convection and radiation coefficients from the collector plate to the glass cover heat transfer coefficient from the water to the collector plate for each cell capacitance of the glass cover capacitance of the collector plate capacitance of water in each cell amount of water allowed to enter the solar collector at every .005 of an hour amount of water present in each cell of the solar collector at every instance number of glass covers on collector collector area temperature of water entering the solar collector is constant the transmittance absorptance product of this cover and absorber plate system 3.3 Results and Performance Evaluation The performance of the flat-plate solar collector for one day selected from the month of January is presented in Table II and illus- trated in Figure 11. Figure 11 compares the result of this study with the one available in reference 1. As illustrated the performance of the flat plat solar collector using the bond graph appraoch tends to be very close to that of the conventional approach. The small discrepancy between the two results could be attri- buted to various factors. One such factor is that the data available for various heat transfer coefficients is highly questionable. The authors of reference 1 admit that these values are simply crude estimations. The difference in the two results may be attributed to these two major factors: First not all the data was available with the original problem. The values of R9, R10, R11 were not available with the problem. It was estimated by the authors and they admit that these were crude estimations. The higher the value of the heat transfer coefficients (u), 2 the less the loss. The value of u given in the problem is 1.409 BTU/hr-ft -F 2-F. where as the value used in the bond graph model is 1.247 BTU/hr-ft It is this difference in the value of overall heat transfer coefficients that has caused the slight difference in the two calculations. Furthermore the conventional model is a zero-capacitance model, in which the effects of thermal capacitance on the collector performance are neglected. The collector is considered to be in equilibrium with 33 TABLE II Time TAMB S Tout Qusefull Efficiency (HR) (F) BUT/hr (calculated) (calculated) (F) (BTU/hr) 3 - - - - - 9 32.0 1733.22 136.14 0.00 0.00 10 35.6 2790.93 139.98 0.00 0.00 11 39.2 4278.61 145.25 840.14 0.196 12 50.0 5998.27 151.94 1910.07 0.318 13 50.0 6155.15 152.46 1993.62 0.324 14 46.4 5438.75 149.76 1560.76 0.287 15 46.6 4415.04 146.35 1015.55 0.230 16 42.8 2893.36 140.96 153.80 .053 34 T (CONVENTIONAL I60 T— °”T MODEL) TOUTIBOND GRAPH To 140‘ U ' ’ I éQUSEFUL (cONVENNON A I20 - MODEL) '1" E: 9 LU — — g 100 2 a: EI- QUSEFUL 8 O: 80—— (BONDGRAPH IE LIJ MODEL) \ “é a [.— l'iJ so —- m I 40 A 20 I l l J I 9 IO II 12 I3 I TIME (HOURS) Figure 11. Comparison Of Results * The result of the conventional model is taken from reference (1) 35 its environment at any instant of time, whereas the bond graph model accounts for capacitance effects by dividing the collector into number I of isothermal segments or nodes. Thus there is a slight difference in temperature at each node of the "multi-node" model. Also the real advantage of this model would be more prominent in a larger system. The idea of modeling the whole system into lumps deals with the philosophy that for a larger system the instantaneous mixing effect is not reasonable and as such it should be modelled by smaller lumps. The system studied here was much too small for such a study to have any significant impact. The efficiency Of utilizing available solar energy for every hour studied is plotted in Figure 12. The overall efficiency of the solar collector for this study is 23%. 36 '46 I 1" .J o (0 C3 \ 3 3 33 z .3— L_I_.I - 82 LL LL LIJ O: C) B .2- LIJ .1 ._l C) U '— I . I I I I 9 IO 11 12 13 I4 15 TIME (HOURS) Figure 12. Collector Efficiency vs Time 37 CHAPTER 4 Conclusions and Reconmendations 4.1 Utility of Bond Graphs for Modeling_and Simulation The three objectives of this research were (1) to derive a mathematical model of the flat-plate solar collector using bond graphs, (2) to study the dynamics of a flat-plate solar collector system once the mathematical model was derived, and (3) to consider the utility of the bond graph approach in the study Of the flat-plate solar collector system. The first Objective was accomplished by lumping the thermal capacitance effects of each of the components Of the flat-plate solar collector into a separate node, and then representing each node by bond graphs. The bond graph approach as a modeling tool has a reasonable amount of versatility with respect to the study of the dynamics Of a flat-plate solar collector irrespective of the model chosen. That is, by removing the l'C"-e1ement from the bond graph of Figure 7, the thermal capacitance effect would be eliminated. Consequently, the present approach could easily incorporate studies of the flat-plate solar collector without any major change to the conductive model. However, one of the major advantages Of the present approach is its ability to study a wide range Of possible "hook-ups" Of the components to Obtain the models not only of the flat-plate solar collector but many other, more complete systems, which have been used and proposed fOr solar energy use, including the entire building. 38 4.2 Solar Collector Models The flat plate solar collector studied in this project could be improved for better dynamic performance by making certain changes in its design. One such change would be the addition Of one more glass cover. This would minimize the losses caused by both convective and radiative processes. The back losses would be minimized if the temperature and the capacitance effect of the bottom insulation is included in the model. For better performance of the collector, the tube spacing within the collector should be reduced from its present 6 inch center to center spacing. Consideration should be given to low temperature Operation. If the system operates in low temperate climates a freezing problem exists and the system must either be drained or anti-freeze must be added. Other types of weather problems exist. The system must be able to withstand wind loads, rain and hail. The collector must be designed so that snow does not interfere with winter time Operation. All these could be readily included in the present model. This shows the strength and the versatility of the bond graph model. 4.3 Next Steps Since this is the first such study of its kind, the bond graph model representing the solar collector leaves much to be desired. One of the first things to do to increase the usefulness of bond graphs as a modeling tOOl is to improve the model for the convec- tive processes. One way this might be done is by treating MCp (mass flow rate times the specific heat of the fluid flowing through the 39 collector) as resistance, R, representing loss effects from the collector plate into the fluid passing through the collector tube. Once this is done, the next step would be to model each of the components involved in the solar energy residential heating and cooling. Once the components are represented successfully, they should be "hooked-up" in the right order to describe the entire system. The bond graph should be used directly by the computer in order to study the collector performance. To achieve this, the Enport program must be able to handle a larger and more complex set Of system models than its current capabilities allow. 40 REFERENCES 10. 11. 12. 13. 14. 15. 16. Duffie, J.A. and Beckman, W.A. Solar Energy Thermal Processes. New York: Wiley and Sons, 1974. Dixhoorn, J.J. Van, and Evans, F.J. (eds) Physical Structure in Systems Theory. London: Academic Press, 1974. Karnopp, 0.0. and Rosenberg, R.C. System Dynamics: A Unified Approach. New York, Wiley and Sons, 1975 ASME, Journal Of Dynamic Systems, Measurements, and Control. Sept. 1972. Thoma, J.U. Introduction to Bond Graphs and Their Applications. Oxford: Pergamon Press, 1975. Rosenberg, R.C. A User's Guide to Enport-4. New York: Wiley and Sons, 1974. ASME, Journal Of Dynamic Systems, Measurements, and Control. March 1978. Karnopp. 0.0. Pseudo Bond Graphs for Thermal Energy Transport. Contact the author. Space Heating_with Solar Energy. Proceedings of a Course Sym- posium, M.I.T., 1954. Bliss, R.W. The PerfOrmance of an Experimental System Using Solar Energy fOr Heatihg_and NightTRadiation for Cooling a Building. Paper presented at U}NZ Conference on New Sources of Energy, 1964. Thomason, H.E. Solar Space Heating, Water Heating, Cooling in the Thomason House. Proceedings Of the’COnference on New Sources Off Energy, August 1961, Vol. 5. Thomason, H.E. and Thomason, H.J.L., Jr. Solar Houses/Heating and Cooling Progress Report. Solar Energy, Vol. 15, No. 1, May 1973. Space Heating with Solar Energy, Proceedings of a Course Symposium, NLI.T.,’1954. Buchberg, H. and Roulet, J.R. Simulation and Optimization Of Solar Collection and Storage for House Heating. Solar Energy, Vol. 12, NO. 1, SEpt. 1968. Lof, G.O.G. House Heating and Cooling with Solar Energy. Paper in Solar Energy Research, University OTCWISCOHSIH Proceedings, 1955. Tybout, R.A. and Lof, G.0.G. Cost of House Heating with Solar Energy. Solar Energy, Vol. 14, No. 3, Feb. 1973. 41 17. 18. 19. 20. 21. Klein, S.A., Beckman, W.A. and Duffie, J.A. A Design Procedure for Solar Heating Systems. Solar Energy, Vol.’l8, NO. 4, Nov. 1975. ASME, Heat Transfer in Solar Energy Systems. Proceedings of the WinterfiAnnuil Meeting of thErAmerican Society of Mechanical Engineers, Atlanta, Georgia, Nov. - Dec. 1977. Butz, L.W. Use of Solar Energy for Residential Heating and Cooling. M.S. Thesis inMechanical‘Engineerihg, Madison, University of Wisconsin, 1973. Close, 0.0. A Design Approach for Solar Processes. Solar Energy, Vol. 11, 1967. Klein, S.A., Duffie, J.A. and Beckman, W.A. Transient Considera- tions of Flat-Plate Solar Collectors. ASME, Journal of Engineering fOr Power, Apri171974. 42 APPENDICES APPENDIX A Glossary of Terms Glossary Ac is the collector area (ftz) Beam Radiation is that solar radiation from the sun without change of direction. 01=(Mcp)g, is the capacitance Of the glass (BTU/F). Cz=(MCp)p, is the capacitance of the solar plate (BTU/F). C3=C4=CS=06=(MCP)W, is the capacitance of water in each cell (BTU/hr). Diffuse Radiation is that solar radiation received from the sun after its direction has been changed by reflection and scattering by the atmosphere. HR is the energy absorbed by the solar collector (BTU/hr-ftz). H is the radiation on a horizontal surface (BTU/hr-ftz). S M is the mass flow rate (1bm/hr). Q is the solar radiation from the sun (BTU/hr-ftz). sol 0u is the rate of energy transfer to the water circulating through the collector (BTU/hr). Ri=Ui is the heat transfer coefficient loss (BTU/hr-ftz-F). Ti is the temperatures (F). RT is the tile factor for beam radiation (see reference 1, page 49, equations 3.6.1 and 3.6.2 for detail). 43 APPENDIX B A Definition Of the Bond Graph Language R.C.ROSENBERG Aasocmte Protesaor. Department at Mechanical Engineering, Michigan State University. East Lanamg. Mich. D.C.KARNOPP Professor. Department of Mechenlcal Engineering. UniversIty of California. Davis. Calif. language Introduction Tm: purpose of this paper is to present the basic definitions of the bond graph language in a compact but general form. The language presented herein is a formal mathematical system of definitions and symbolism. The descriptive names are stated in terms related to energy and power, because that is the historical basis of the multiport concept. It is important that the fundamental definitions of the lan- gunge be standardized because an increasing number of people around the world are using and developing the bond graph language as a modeling tool in relation to multiport systems. A common set of reference definitions will be an aid to all in promoting ease of communication. Some care has been taken from the start to construct defini- tions and notation which are helpful in communicating with digital computers through special programs, such as EXPORT (5).! It is hoped that any subsequent modifications and exten- sions to the language will give due consideration to this goal. Principal sources of extended descriptions Of the language and physical applications and interpretations will be found in Paynter [l]. Karnopp and Rosenberg (2, 3], and Takahashi, et al. (4]. This paper is the most highly codified version Of language definition, drawing as it does upon all previous efforts. Basic Definitions Multiport Elementa, Ports, and Bonds. Alultiporf (Irmrnla are the nodes of the graph, and are designated by alpha-numeric characters. They are referred to as elements, for convenience. For example, in Fig. 1(a) two multiport elements, 1 and R, are shown. Port: of a multiport element are designated by line 'Nurnbera in breeketa designate Reference- at end of paper. Contributed by the Automatic Control Division for publIcatIon (without pmenution) in the Jocasat. or DYNAMIC Svercue. .‘Ilzaet'nrwrsr. awn Common. Manuscript received at ASME lleadquartera. May 9. 1972. Paper NO. 72-Aut-T. 44 A Definition of the Bond Graph segments incident on the element at one end. Porto are pleoaa where the element can interact with its environment. For example, in Fig. 1(b) the 1 element has three porta and the R element has one port. We say that the I element in a 3- port, and the R element is a l-port. Bonds are formed when pairs of ports are joined. Thus bonds are connections between pairs of multiport elements. For example. in Fig. 1(c) two ports have been joined, forming a bond between the l and the R. lead Graphs. A bond graph is a collection of multiport elements bonded together. In the general sense it is a linear graph whose nodes are multiport elements and whose brenchca are bonds. A bond graph may have one part or several parta, may have no loops or several loops, and in general has the characteristia of any linear graph. An example of a bond graph is given in Fig. 2. In part (a) a bond graph with seven elements and six bonds is shown. In part (b) the same graph has had its powers directed and bonda labeled. . A bond graph fragment is a bond graph not all of whose porte have been paired as bonds. An example of a bond graph fragment is given in Fig. 1(c), which has one bond and two Open, or unconnected, porta. Pen Variablea. Associated with a given port are three direct and three integral quantities. Eflort, ((0, and flow, f0), are directly associated with a given port, and are called the port power variables. They are assumed to be scalar functions of an independent variable (t). Power, P(t), is found directly from the scalar product of effort and flow, on P0) - «01(1). The direction of positive power is indicated by a half-arrow on the bond. Momcntum, p(t), and displacement, q(t), are related to the eflort and flow at a port by integral relations. That is, smeci otiiiiitioi. NANf 0 SE——; e I eltl scarce of effort ") (b) (H SF-—f—; l - fttl scarce of tin. rig. t Initipofl elements. ports. and bends: (a) two mumport elements; an the eiernenta and their ports: (c) tormation ot a bond I: I i e ' 9(0) "on a ”n“ eiii - giro). {not " I I 1 5 i‘—:—- f I 0(9) inertanu- I t. pti)-o(t°io {e-di "---—C--—T$—--I se—eo—vtr—yi l 2l J 6 .1_____: Ole.” ' 0 resistance . i C l ' 7 ——,TF-———y e - the transformer l“ lb) l in I 2 fig. 2 An eaampio oi a bond graph: (a) a bond graph; (h) the bond "'i ’ '2 graph with powers directed and bonds labeled l 2 _' cv__.’ .‘ n “(2 gyrator ' .2 C 'ef‘ p(ii - pltu + f. #de , _"o .i - e2 - c3 cmn effort and qtii - qtto.‘ + j... {(Mdh, respectively 2 junuion fl 9 f2 - f3 - 0 Momentum and displacement are sometimes referred to a~ energy variables. . 3 ‘ Ermgu. Eu... is related to the power at a port by —-—r '——r '. ' '2 ' '3 (0.2:! “if" t 2 eI 0 e2 - e . O Elli - EU.) + f. PUMA. 3 The qugntuy E“) ._ Elloi represen‘g the net energy it"TSICrTCd F... 3 DOHI|NOHI at "h. .llf‘ MU'tIpO" .IOMOI‘I through the port in the direction of the hali~arrow (i.e., positive power! over the interval (ta. 1). in common bond graph usage the eflort and the flow are often shown explicitly next to the port (or bond). The power. dis- placement. momentum. and energy quantities are all implied. That is, a static relation exists between the effort and flow at the port. Junctions: z-M . . . f c lute atottiport tion-onto. There “are nine basic multiport Transformer, written .1 TF 3. i“ ‘ linear 21,0" element deb elements. grouped into four categories according to their energy . 1, characteristics. These elements and their definitions are sum- fined by iiiarircd iii Fig. 3 'l -' M’C’ Sources. - and tit-f. - fg, Noun-r of rflnri, written SE c. is defined by r - rm. h '. ih d i . Samu- affair. written SF! is defined by I = fill. ' "e m n 0 mo u us Gyrator, written 9 G)’ 9, is a linear 2-port element defined Storagea. It It by t o - rarurflanro. written C, is defined by I h "' "fr r = We) and q") - w.) + f“ MAMA. and r, - r-]., where r is the modulus. Both the transformer and gyrntor preserve ism-er (i.e., I’. s P, in each case shown), and they must each have two ports. so they are called es~ciitial Q-port junctions. Thai is. the effort is a static function of the displm‘ctitelil and the ilispl'ii'eiiietii is llif.‘ time ilil"2rli (if the flow Inorlrinrr. written{ I, is defined by I Junctions: P~Port. :d‘ndme i) fl in. I (P " P Pl. + '. (( ) Common rflort junction, written ’ *7] O 3‘7 That is, the flow is a static function of the. momentum and the 2 1 momentum is the time integral of the efiort. is a linear 3-port element defined by Dissipation. r en - c: = r. (common effort) Resistant-r. written- R. is defined by . f and f. +1. — f. - 0. (flow summation) ‘chl s? 0. Other names for this element are the flow junction and the 45 I 3 —-7 l -—-7. 2P zuo )llflt‘llltll Cowman flow junrhnn, written is e lineer :t-purt element defined by It ‘ f: " I) '|+"—r.-". (common flow) and (effort summation) Uther name- for this element are the effort junction and the an: junction. Both the common eflnrt junction end the common flow junc. tion preserve power (i.e.. the art power in is zero at ell times). so they are called junction-t. If the reference power directions are changed the signs on the summation relation must change accordingly. Extended Definitions lumped Fields . Storage Fields. Multr’port oaparnancn. or C—firids, are written 1 -7C Tn- . and characterized by 213 'I 3 ¢s(9h9u tight - 1‘0". t end q.(t\ - q.(t.) + 111;“)th - l to n. . . l .tlullrport tmrfnntcl. or l-firlds. nre written — 7 l e~n—. l1 0 ., . :tml charm-t wired lift' I‘ ‘ ¢|°~IP3r --- ”I'l‘. - 1‘0”, and C 12.0) - p.00) + L «max. t - l to a. If e C-field or l-field is to have so nxsocieted “energy" state function then certain integrability conditions must he met by the 4’. functions. In multiport terms the relations given in the fun-going are .‘lllrtt‘ll'lll to define a F-field and l-field. respectively. .tltrrd multiport stamgr fill!“ can arise when both C and [- typc i-‘tutnge effects are pre~ent simultaneously. The symbol for uurh an element consists of a .~et of (“9 and l'.~ with appropriate ports indicated. 1 For example, - , l(‘l (53r- indicetes the existence of e set ’1 2 of relation- Ii I ¢tiph Qt. Pal. r: = 02(1):.“ pa). It ' ¢i(Pi. 0:. Pi). and I PI“) ' Pia.) + J... “(Aldk I Qt“) " 'hfle) + j... fiixldh. Pi“) ‘ Prue) + I. f;(h)dh, .lluftiport disupalars. or R-field.~. ere written ~17 R "1" 2 l‘." end are characterized by 4M!» It. 0:. In. . - If the R-field is to represent pure dieipstion. then the power function e-societed with the R—field must be positive definite. .llultiport junctions include 0 junctions and I ~ junctions with n ports. a 2 2. The generel case for each junction is given in the following. .e..l.)-O.i‘-ttoe. l -._7( ) 1."- e.-e.-....-'. It'll- '1- iId-U i“'0 0.! 0‘l neeutetea z-Pert Junctions. The modulated transformer, or i M!) J! TF written I M TF 2 implies the reeltions ._7 -7 e. - m(s)-eg end mill'fi - Is. where rats) is e function of e set of variables. I. The modulated transformer preserves power; i.e.. P.(t) - P.(t). '(I) t The modulated water. or MGY. written 1 MGY 2 "7 '7 implies the relation-s o. - NIP], Os - '(fl'fi. where r(s) is e function of set of variables, I. The modulated gyrstor preserves power; i.e.. P.(t) - P.(t). end Junction Structure. The juadi‘on structure of a bond graph is the set of all 0, l, GY. end 7'" elements end their bonds end ports. The junction structure is en n-port that preserves power (i.e., the ac! power in is zero). The junction structure mey be modulated (if it contains any .l! G Y‘s or M T F 's) or unmoduleted. For example. the junction structure of the graph in Fig. 20)) is e 4-port element with ports 1. 2. 5, end 6 end bonds 3 end 4. It contains the elements 0. TF. end 1. Physical Interpretations The physical interpretations given in this section ere very succinctly stated. References ll]. [2], and [3] contain extensive descriptions of physical applications end the interacted reader is encouraged to oomult them. Iechenleel Translation. To represent mechanical translational phenomena we may make the following variable associations: 1 eflort, e, is interpreted es force; 2 flow. I, is interpreted as urlocity; 3 momentum, p, is interpreted as impulsemomentutn; 4 di~plscetnent, q, is interpreted as mechanical displacement. Then the bait. bond graph elements have the following in- terpretetions: 1 source of efiort, SE. is e force source; 2 wuroe of flow, SP, is e Velocity source (or may be thought of as a geometric miistrsint); 46 3 mistance. R, represents friction and other mechanical In“ mechanisms; 4 capacitance, C, represents potential or elastic enernr storage effects (or spring-like behavior); 5 inertance. I. represents kinetic energy storage (or mass effects); 6 transformer. T F . represents linear lever or linkage action (motion restriCted to small angles); 7 gyrator, G Y, represents gryationai coupling or interaction between two ports; 8 O-junction represents a common force coupling among the several incident. ports tor among the ports of the system bonded to the O-junction); and 9 l-junction represents a common velocity constraint among the several incident ports (or among the ports of the system bonded to the l-junction). The extension of the interpretation to rotational mechamn is a natural one. it is based on the following [\‘KOCinlltil‘t‘ l effort. c, is associated with torque; and 2 flow, I. la !L\~iit't3lttd with angular velocity. Because the development is so similar to the one for translational mechanics it will not be repeated here. Electrical Networks. In electrical networks the key step is to interpret a port as a terminal-pair. Then variable axsociations may be made as follows; I rflort. c, is interpreted as voltage: 2 flow. I. is interpreted as current. 3 momentum. p, is interpreted as flux linkage; 4 displacement. q. is interpreted a~ charge. The basic bond graph elements have the following interpreta- tions: l source of effort. SE. is a Voltage source: 2 source of flow. SF. is a current source. 3 resistance, R, represents electrical resistance; 4 capacitance. C. represents capacitance effect electric energy}; 5 inertanoe. I. energy): ti transformer. TF. represents ideal transformer coupling; 7 gyrator. Gl'. represents gyrational coupling; *3 (i-junction represents a parallel connection of ports (com- mon voltage across the terminal pairs); and 9 l-junciion represent: a series connection of ports (common current through the terminal pairsi. (stored represents inductance (stored magnetic Hydraulic Circuits. For fluid S)‘.\‘lt’fli~ in which the significant fluid power is given as the product of prcuure times Volume flow. the following variable associations are u~eful: 1 (fort. c. is interpreted as pnssnrc; 2 flair. f. is interpreted as rolmnr flow. 3 momentum. p. is interpreted as pressure-momentum; 4 displacement. q. is interpreted as volume. The basic bond graph elements have the following interpreta- liiilis’i l source of effort, SE. is a pressure source; '2 source of flow. SP. is a VOIUmc flow «iiirce; 3 resistance, R, represents loss effects (e.g . due to leakage. valves. orifices. etc); 4 capacitance. C, represents accumulation or tank-like effect. (head storage); 5 inertanoe, I . represents slug-flow inertia effects; 6 o-junction represents a act of ports having a common pressure (e.g, a pipe tee); ~ 7 l-junction represents a net of ports having a common volume flow (i.e.. series). Other interpretations. This brief listing of physical interprets. tions of bond graph elements is restricted to the simplest. most direct, applications. Such applications came first by virtue of historical development. and they are a natural point of de- parture for most Clfi‘ilCfllly trained scientists and engineers. As references- [14) and the special issue collection in the Jot'axu. or l)i‘.\‘\\tl(‘ Srs‘rnus. .\li.\st;m:sii:.\"r. AND Con- rnrii.. Trims. ASMI-I. Sept. 1972. indicate. bond graph elements can be ii~ed to describe an amazingly rich variety of complex dynamic systems. The limits of applicability are not bound by energy and power in the sense of physics; they include any areas in which there exist useful analogous quantities to energy. Concluding Remarks in this brief definition of the bond graph language two im- portant concepts have been omitted. The first is the concept of bond activation. in which one of the two power variables is tltlp- pressed. producing a pure signal coupling in place of the bond. This is very useful modeling device in active system» Further discussion of activation will be found in reference [3). Motion 2.4, as well as in references [ll and [2]. Another concept omitted from discussion in this definitional paper is that of oprralional causality. It is by means of causality operations- applied to bond graphs that the algebraic and dif- ferential relations implied by the graph and its- elements may be organized and reduced to state-space form in a systematic manner. Exteiwive discus-tion of causality will be found in reference [3}. section 3.4 and chapter 5. Systematic formulation of relations- is presented in reference [6]. References l' Paynter. H. 31.. Analysis and Drsign of Engrnccri'ng Syslrms. MlT. Press. 1961. Karnopp. l). C.. and Rosenberg. R. C.. Analysis and Simulation of . Iufli’port Systems. .\l.l.T. Press. 1968. Karno p. l). C.. and ltmenberg. It. C.. "System Dy- namics: A nified Approach.” Division of Engineering lie- search. College of lingincering. Michigan State University. East Lansing. .‘lii‘h.. l97l. 4 Takuhaski. Y.. Rabins. .\1.. and Aiislander. 1).. Control. Add|i~on~We~ley. lit-ruling. his. lii'iti (see chapter 6 in par- lii‘ii ar). 5 Rosenberg. R. C.. "ENPtlltT U~er's Guide." Division of Engineering lieu-arch. College of Engineering. Michigan State University. East Lansing. .\lich.. W72. lto~cnberg. ll 0. “State-Space Formulation for Bond Graph Models of Multiport Sy~tcin~." Jot'nsat. or DYNAMIC Si~Tl.\l\‘. .\li'\