The Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System 1Du Huiyong, 2Zhao Kaibin, 3Liu J ianxin, 4Li Min and 5Wang Zhancheng 1, Henan University of Science and Technology, Luoyang, Henan, China, dhy@mail.haust.edu.cn *2, Henan University of Science and Technology, Luoyang, Henan, China, zkb45@126.com 3,4,5, Henan University of Science and Technology, Luoyang, Henan, China, liujianxin_01@mail.haust.edu.cn, dhy@mail.haust.edu.cn, limin_haust@126.com, zhzcw@126.com Abstract Comparedwiththeotherdynamicanalysismethods,BondGraphtheoryisapowerfultoolinthe field of dynamics and multiple-energy domains analyses of complex engineering systems which using a simplestructurecontainingagreatdealofinformation.Thispapershowstheprocessofmodelinga Stirling Engine fuel supply system based on Bond Graph theory on the purpose of analyses the dynamic characters of Stirling Engine fuel supply system, deriving state equations which reveal the interaction betweencomponentsandenergyconversionandbuildingaSimulinkmodel,finally,simulatingthe dynamic characteristic of the system and verified the accuracy of the model through experiment. Keywords: Bond Graph, Stirling Engine, Pressure control, Matlab/Simulink, Algebraic loop 1. Introduction Stirling engine is one external heating (or combustion) piston engine that operates on a regenerative thermodynamic cycle. It covers machines capable of operating a prime mover, heat pump, refrigerating engine, etc [1, 2, 3]. A typical Stirling engine consisted of external combustion system, circulatory system, and transmission system. The fuel nozzle in the external combustion system is usually cyclone atomizing nozzle which works in the mode of open and continuous injection as shown in Fig. 1 [4]. To improve stability of engine operation, the fuel flow rate should be controlled accurately [1, 3]. In order to automate the design or revision of multi-domain systems, the Bond Graph theory has been chosen.Bond Graph theory was invented by Professor H. Paynter in 1954, which was proved to be a very efficient tool for modeling, analyzing and designing mechatronic systems from an energy and dynamic point of view, when worked on the development of complex systems which suffered from thermal problems, structural problems, vibration and noise problems, and control and stability issues that do not fit into a single disciplines, Bond Graphs provide the link by which all these different disciplines can be represented in an overall system model, and now there are many computer programs that can go from a bond graph picture to a simulation of a nonlinear system in a straightforward, systematic approach.[5, 6,7,8,10]. Basedonbasicprincipleofenergyconservationandthesimilarityrelationshipofthe power transport, storage, dissipation and transformation, Bond Graph theory can convert the machinery, circuit, hydraulic, thermal and the other physical quantities into four kinds of generalized variables whichareeffort,flow,displacementandmomentum[9,12,13].Tab.1showstheeffort/flow definitions in different engineering domains. The Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang ZhanchengInternational Journal of Digital Content Technology and its Applications(JDCTA) Volume6,Number16,September 2012 doi:10.4156/jdcta.vol6.issue16.55453 Figure 1. Cyclone Atomizing Nozzle Table 1. Effort/Flow definitions in different engineering domains System Domaineffortflow ElectricalVoltage (V)Current (A) TranslationalForce (N)Velocity (m/s) RotationalTorque (N*m)Angular Velocity(rad/s) HydraulicPressure (N/m2)Volumetric Flow(m3/s) ChemicalChemical Potential(J /mol)Molar Flow(mol/s) ThermodynamicTemperature(K)Entropy Flow(W/K) The main work presented is to build a model of the Stirling Engine fuel supply system based on BondGraphtheoryandderivethesystemstateequations.Forresearchingthesystemdynamic characteristics and optimizing the design of electric control system, a Simulink model of the system was built and the accuracy of digital simulation results was verified through an experiment. 2. Stirling engine fuel supply system The Stirling Engine fuel supply system schematic diagram was shown in Fig. 2, the control mode is based on speed regulating. As shown, the fuel was pumped from a fuel tank by a low pressure pump, and the pressure maintained at about 0.2MPa. After filtering, the low pressure fuel was pressurized to working pressure by a high pressure plunger pump which was driven by a servo-motor. The plunger pump has three pistons which pumped fuel in turn in every rotation of a eccentric shaft. There is one check valve at inlet and outlet of each compressive cylinder of the piston. The high pressure fuel will pass through a short pipeline after the check valve, and then goes into an accumulator which could works as a buffer to smooth the pressure pulse that caused by the plunger pump. After crossing a long fuel pipeline, the fuel was sprayed into engine burner through a cyclone atomizing nozzle. According to energy domain, this system can be divided into mechanical systems and hydraulic system. Figure 2. Stirling Engine Fuel Supply SystemThe Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang Zhancheng4543. Bond graph mode In accordance with the form of fuel flow, the model of the whole system could be divided into two small parts: high pressure plunger pumps model and hydraulic systemmodel. The building procedure of their bond graph and the processes of driven state equations were set out below. 3.1. High pressure plunger pumps model When starting to build a model in Bond Graph, first, the practical components should be replaced by standardBondGraphelements.Second,connectingmultiportsofelementsbybondsindicatesthe powerorientationsandcausalstroke.Finally,assigningnumberstoeachbondstoaccomplishthe computable Bond Graph model. According to the Bond Graph theory rules, two generic variables effort (ei) and flow (fi) which were assumed to transfer bi-directional signals were associated with every power through each bond, they follow the Eq. (1) [11]. i ie f power = (1) 0- and 1-junctions (common effort and common flow junction, respectively) were used to connect immediate relevance physical effects which had the same energy forms and equal in numeric values [14, 15]. Eq. (2) and (3) show the characteristics of 0- and 1-junctions. 1 2 31nni iie e e ef o== = = = (2) 1 2 31nni iif f f fe o== = = = (3) In this high pressure plunger pumps model, the internal and external fuel leakage, friction between movingpartsandelasticdeformationwereignored.Dependingonthesymmetryofthesystem structure, high pressure fuel pump can be expressed as combinations of three parallel single piston pumps.

Figure 3.0- and 1-junctions in Bond GraphFigure 4.Single piston pump in Bond Graph The Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang Zhancheng455According to the basic Bond Graph principles above-mentioned, one single piston pump could be built as shown in Fig 3. Each power arrow or bond was assigned a number to mark and distinguish Bond Graph elements. In Fig 4, flow source Sf1 represented constant rotation the speed of servo motor, effort source Se6 represented the pressure fuel after filter, element C3represented the liquid capacity effect of fuel in piston compressive cylinder. The liquid capacity C in piston is nonlinear because the cylinder system changes from closed to open system in one compression stroke which could be calculated by Eq. (4). 03VCB= (4) 0Vinitial volumeBbulk modulusBecause this research is not on the details of the valves dynamic characteristics, the check valve was simplified to a modulated nonlinear resistive element R, the flow characteristic of the check valve would meet Eq. (5). Then we can get the elements R5and R8which were used to simulate inlet and outlet the nonlinear liquid resistance of the cylinder they both follow the Eq. (6). vvvpQRc A = (5) vp Apressure between upstream and downstream cresponse pressure of mechanical vR damping characteristic 1212( )vdp RRC A ppccc cA s = A A >(6) Rc hydraulic leaking resistance dC orifice discharge coefficient Aorifice area fluid density 3.2. Hydraulic system model As the result of simulation accuracy, hydraulic models can be divided into three levels: ideal model, static model and dynamic model [16, 19]. In this paper, it is necessary to build the dynamic model for analysis and optimization because of the hydraulic dynamic characteristics had a significant impact on StirlingEngineperformance.StirlingEnginehydraulicsystemincludedpipeline,accumulatorand cyclone atomizing nozzle, though these three parts had its different geometric dimension, they can be described by the same hydraulic unit with configurable resistive, capacitive and inertial parameters in Bond Graph theory Inaddition,consideringthepipelineslengthandthefuelpumpshighfrequencymovement,a distributing parameter model has to be built. The principle of the distributing parameter model is to The Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang Zhancheng456divide the whole pipe into some length-limited segments, and each segment is a dynamic cell as shown in Fig 5. Figure 5.Dynamic Cell When the moving media in the pipeline is diesel fuel, the segment number could be calculated by an empirical formula, the optimal number of segments was proportional to the length of pipe as Eq. (7). The pressure wave and attenuation of simulation results were similar to the experimental result. 0slnl>(7) 0ltotal pipe length slbasic pipe length, sl =1m nsegment number,2 n >Based on Bond Graph theory, a hydraulic unit is built as shown in Fig. 6, in which each unit has two segments. The element I in the graph represented the liquid inertia effect which associated with the pressuremomentumandflowEq.(8),(9)and(10)showthecalculationprocedureofeachbond variable.

0ilIA=(8)

4128ilRdt=(9) 0iVCB= , (10) lpipe length 0A cross-sectional area fuel densityfuel dynamic viscosity dpipe diameter 3.3. State equations Physicaleffects,componentsconnectionsandpowertransmissionsweredescribedinthemodel aforesaid clearly. According to Bond Graph theory, the model is not only the graphics display of power transmissionandtransformationbutalsocontainsstateequationswhichdescribedthedynamic performance of the system.As shown in Fig. 5 and Fig. 6, the bond graphs are all causal relations, according to the Bond Graph rules,thenumberofstatevariablesshouldbeequaltothenumberofsystem energystorage components[17].Inthiscase,theexpressionofthesystemstateequationscouldfollowthesteps below1) Choosing generalized momentum p of the I component and generalized displacements q of Pi Qi PoQoThe Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang Zhancheng457the c components as state variables, choosing the effort of SE elements and the flow of SF elements as input variables. 2) Computing output variables by the characteristics of Bond Graph energy storage componentsandRcomponents.3)Usingdisplacementqiandmomentumpiasthesystemstate variables and list effort and flow equation to expresspandq . 4) Substituting the Bond graphs energy storage components and output variables of R element in the flow and effort equations, consolidating and finishing system state equation. Figure 6.Hydraulic unit in Bond Graph According to the steps above and the Fig 4 and Fig 6, we could derivate basic systemstate equations. The Eq. (11) and (12) show the single piston pump state equations and output equations: 3336 3 3 93 15 8efqeCS e e eq S AR R= = + (11) 3 998e efR= (12) Eq. (13) and (14) show the hydraulic unit state equations and output equations: 3133 6 13 41 3 63 863 86 88 96 8__pq Q inIp q qp RC I Cp pqI Iq pp R p outC I= = = = (13) 1188__qp inCpQ outI== (14) 4. Simulation model in simulink There are three main simulation algorithms for hydraulic system: signal flow method [18], based on equationmethodandenergyportsmethod[19].ThemodelwasbuiltintheMatlab/Simulink environment in this paper is belongs to signal flow method. In Simulink, each functional module could be built according to state equation and then connecting the function modules in accordance with the signal flow and adopted scope blocks to monitor the simulation process in real-time at key points. The The Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang Zhancheng458simulation parameters which were used in this paper are shown in Tab. 2. And Fig. 7 shows the top level of the Simulink model according to the systemstate equations. Figure 7. Simulation model in Simulink Through the description in the part 3, we can see that the state equations of the module express only the properties themselves, and only coupled through input and output variables. Therefore each module hasastrongindependenceandreusability,thesecharacteristicshelpusshareouttheworkand cooperate with one another when modeling, accurate and fast location of fault source when an error occurs, and more convenient learning and improvement models for the new developer. The feedback loops are widely appeared in the system model established in accordance with the state equations. And because of the simulation time sequence of the computer, if the input signal directly depends on the output signal and output signal directly depends on the input signal at the same time, there would be algebraic loops in some of the feedback loops. Algebraic loops would seriously reduce the speed and precision of simulation. Toremovealgebraicloops,wecanusesomenon-direct-throughmodulestoreplacethedirect-through modules in same function or insert Memory blocks, the Memory blocks would add some extra delay to the system, if it do not have much stability margin would become unstable, so it need more test to validate its rationality. 5. Comparison of simulation and experiment TheSimulinksolutionusedinthismodelwasvariablestep-sizeode45algorithm.Withthe increasing of motor speed, the system disturbance frequency increase too, so the simulation maximum step and error threshold should be decreased to avoid reducing precision. By comparison, 1e-6 will be reasonable.The high pressure fuel pump was driven by a three-phase servo motor, and the servo system was controlled by a particular ECU. When receiving a voltage signal from external controller, it would stable the motor speed through a closed-loop control. The pressure sensor which was used to get the accumulators pressure was Kistler 4262a, the data acquisition facility is consist of WaveBook/512 and DasyLab. The Sampling frequency was 100 KHz. The experimental environment is shown as Fig. 8. The Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang Zhancheng459 Figure 8.Experimental Environment (1. servo-motor; 2. high pressure pump; 3. accumulator; 4. press sensor; 5. flowmeter; 6. fuel nozzle;) The following experiment had designed to compare the pressures fluctuations in the accumulator in case of static and dynamic. To verify the static property, the speed of servo motor was set constant at 800r/min and 380r/min. Fig. 9 shows the waveform at speed of 800 r/min (a), 380r/min (b), the experiment waveform had been commutated by a low-pass filter. As shown the waveform of simulation model based on Bond Graph theory can approximate to the actual waveform, and the stability at high speed is better than low speed. Fig. 10 shows a manoeuvring condition which speed change from 460r/min to 560r/min. we can see that the system response time of the simulation model is short than the actual system response time, andthefluctuationofsimulationmodelinmanoeuvringconditionsofisevenmoreacute.Thatis becausethesimulationmodelwasthesimplifiedoftherealsystemandithadnotconsiderofthe mechanical inertia of servo motor and reaction to high pressure oil pump. (a) (b) Figure 9. Measured and simulated pressure waveform at 800 r/min(a), 380r/m(b) (blue continuous line is measured pressure and the smooth red dash line is simulation result) ExperimentSimulationExperimentSimulation1 23 456The Application of Bond Graph Theory in Modeling and Simulation of Stirling Engine Fuel Supply System Du Huiyong, Zhao Kaibin, Liu Jianxin, Li Min and Wang Zhancheng460 Figure 10. Measured and simulated pressure waveform at manoeuvring condition 6. Conclusion Compared with the other dynamic analysis methods, the Bond Graph theory could solve multiple energy domains systems using a simple structure containing a great deal of information. Based on this theory, Bond Graph model which reveals the interaction between components and energy conversion was built in this paper, state equations of the Stirling Engine fuel supply systemand a Simulink model were built. Finally, we simulated the dynamic characteristic of the system and verified the accuracy of the model through experiment. 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