MODELLING AND SIMULATION OF A DISH STIRLING SOLAR ENGINE. Sergio Bittanti Antonio De Marco Marcello Farina Silvano Spelta

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1 MODELLING AND SIMULATION OF A DISH STIRLING SOLAR ENGINE Sergio Bittanti Antonio De Marco Marcello Farina Silvano Selta Diartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34, Milan, Italy CESI, Via Rubattino 54, Milan, Italy Abstract: Motivated by the necessity to study a Dish Stirling solar real lant, in this aer a model suitable for simulation and control is orked out. The model obtained from basic thermodynamical equations has the main disadvantage of being a stiff system. A handy and significant model is therefore derived by an analytical aroximation aroach. The comarison ith exerimental data is very satisfactory. Coyright 2005 IFAC Keyords: Control system design, dynamic behavior, engine modelling, model aroximation, oer generation, simulators. 1. INTRODUCTION In the field of solar based energy suliers, a device of major interest in our days is the so-called Dish Stirling engine. In such engine, a arabolic mirror focuses the incoming sun rays toards a receiver acting as a thermal source for a thermodynamical machine based on the Stirling cycle (to isovolumic and to isothermal transforms). In the receiver and in the hole thermodynamical machine, helium gas circulates, ushed by the alternate movement of istons oscillating inside to cylinders in V configuration. In this aer, it s first sulied a model of such an engine by describing the behavior of the various comonents ith ordinary differential equations. A feature of this system is that the involved state variables can be groued in to categories: those related to the alls dynamics and those related to the gas behavior. The former exhibit the tyical (slo) dynamics of thermal exchanges beteen metal and gas, metal and thermal source, metal and environment. The latter are also subject to mechanical effects, so that they can be seen as fast variables. In fact, they are eriodically varying at 30Hz as a consequence of the iston oscillations taking lace at that frequency. The different behavior of these to grous of state variables is a remarkable obstacle for the use of the model in ractice. Indeed, for the fast variables one can conceive only the control of their average behavior. Furthermore, the model does not lend itself to exlicate the correlations beteen the exogenous variables (such as emitted oer) and internal state variables. Last, the model erforms very sloly in simulation due to the fast samling required for the gas variables. To overcome these roblems, e roose to aroximate the fast oscillating variables ith sinusoidal signals ith average values, amlitude and hase varying in time (cisoids). By solving the differential equations of the original model ith such an aroximation, it is ossible to cature the slo average behavior of the fast variables. In this ay, one can ork out a ne model comrising the slo state variables of the original model lus the average variables describing the thermal dynamics of the gas. The so obtained model is suitable for simulation, since the dynamics of all variables have the same rate, and for control, since it describes the essential features of the gas variables. In this ay, as it ill be described in the sequel, it is ossible to ork out correlations

2 among the variables hich clarify the theoretical behavior of the Dish Stirling solar engine. Desite the roblem of modelling and simulate the behavior of the Stirling Engine has been dealt for a long time (see, for examle ((Rodgakis et al., 2002)), or ((Hirata et al., 1997))), to the author s knoledge this is the first aer here the cisoid aroach is adoted to such a device leading to these general results. The results obtained by our model have been comared to exerimental trials erformed ith the Dish Stirling lant of CESI-Milan. Indeed, the research activity resented herein is the outcome of a collaboration beteen CESI and the Milan Institute of Technology (Politecnico di Milano). The CESI engine is a 10kW lant ith a dish of 8.5m. This aer is organized as follos: in Section 2, the main comonents of the lant are described; the key equations required to describe the "thermomechanical" subsystems of the engine are resented in Section 3; in Section 4 e aly the cisoid aroach to ass from the original model to the simlified one; thanks to such a model one can ork out an exression for the generated mechanical oer; moreover, it is ossible to design a control system satisfying the secifications of the lant (Section 5); finally, some simulation trials are erformed in order to comare the model behavior to the CESI lant data (Section 6). The comarison indicates that the model orked out is very satisfactory. 2. THE DISH STIRLING ENGINE PLANT The thermo-mechanical art of the Dish Stirling Engine is an external combustion thermodynamical machine, hose thermal source is reresented by insulation. The sun radiation is concentrated on the receiver by a concentrator, ith a arabolic mirror (Dish) that rotates according to the sun osition (Figure 1). The engine (see (Walker, 1980)) is based on the Stir- temerature is about 20 C (lo temerature). The first (isovolumic) hase of the thermodynami- HOT CYLINDER RECEIVER COLD CYLINDER REGENERATOR HELIUM BOTTLE Figure 2. Scheme of the thermo-mechanical art of the engine cal cycle takes lace hen the gas, in the comression cylinder (cold cylinder) is ushed by the corresonding iston to the chamber of the second exansion cylinder (hot cylinder), assing through a device called regenerator (hich gains heat hile gas asses from the heat source to the cold cylinder, and looses heat hile gas asses from the cold cylinder to the heat source) and then the receiver (heat source). Along this ath, the temerature of the gas increases from about 20 C (in the cold cylinder) to about 600 C-800 C (in the hot cylinder). The second (isothermal) hase is the exansion of the hot gas in the exansion cylinder. Such exansion is due to the heat accumulated in the receiver and roduces mechanical ork. Then the hot gas goes back on the same ath loosing heat, so erforming another isovolumic transformation. The last hase is the isothermal comression taking lace in the cold cylinder. The cold cylinder s alls are ket at a lo temerature thanks to a suitable cooling system. The roduced mechanical energy is transformed into electrical oer by an induction motor, by hich the lant sulies energy to the oer netork. The hole engine is equied ith a control system the main scoe of hich is to kee the receiver s temerature constant, at a given set oint value. Figure 1. Plant scheme ENGINE CONCENTRATOR STRUCTURE PARABOLIC MIRROR (DISH) ling thermodynamical cycle (consisting of to isovolumic and to isothermal transformations) here the orking gas is helium. The engine is a to-cylinders engine (in "V configuration", see Figure 2). The helium gas is delivered by to valves from the so-called helium bottle to the comression cylinder here the 3. THE BASIC MATHEMATICAL MODEL OF THE PLANT 3.1 Main Variables To ork out the model of the lant, the folloing assumtions are introduced. First of all, the temerature of the gas ithin each chamber of the to cylinders is considered uniform, as ell as the temerature of the alls of the cylinders. Furthermore, the uniformity assumtion is made for the receiver too. What about the regenerator, hose thermodynamical behavior is more comlex than that of the other comonents (see (Mayzus et al., 2002) and (Organ, 2000)), it is subdivided it into 10 "slices", in each of hich the temerature is assumed to be uniform. As far as the ressure is concerned, the satial variation along the engine is

3 relatively small (due to gas friction) and it is neglected for the equations. This corresonds to considering a uniform mean ressure in the hole engine. Of course, the gas friction is taken into account in the exression of the generated oer. The adoted symbols are:. T c,g (t): cold cylinder gas temerature,. T reg,g,i (t): regenerator i-th slice s gas temerature (i = 1,2,...,10),. T rec,g (t): receiver homogeneous gas temerature,. T h,g (t): hot cylinder gas temerature,. (t): engine s mean gas ressure,. T c, (t): cold cylinder alls temerature,. T reg,,i (t): regenerator i-th slice s alls temerature (i = 1,2,...,10),. T rec, (t): receiver homogeneous alls temerature,. T h, (t): hot cylinder alls temerature. Moreover, there are further variables reresenting the helium flo in the chambers. For instance:. z c (t) = z c cos(ωt), z h (t) = z h cos(ωt + ϕ h ): istons ositions in the cold and hot cylinder (ω = 30Hz),. u c (t), u h (t): istons seeds (u c (t) = ż c (t), u h (t) = ż h (t)). It s assumed that u c > 0 imlies: cold iston ingoing, u h > 0 imlies: hot iston outgoing),. c (t), h (t): the gas mass flos [Kg/s] beteen the comression cylinder and the receiver, and beteen the receiver and the hot cylinder, resectively. Obviously c (t) = A c ρ c u c (t), and h (t) = A h ρ h u h (t), here A c e A h are the sections of the cold and hot cylinders, ρ c and ρ h are the gas den- RT h,g ), sities in the cylinders (ρ c = and ρ RT c,g h =. in, out : gas mass flos from the gas bottle to the cold cylinder ( in 0) and from the cold cylinder to the helium bottle ( out 0), resectively. 3.2 Basic equations The tyical aroach used in modelling such a device is formally describing his thermodynamical behavior using conservation las (see, for examle (Organ, 2002)). Account taken of the above simle schematization of the thermodynamical machine, e have led to consider 1 state equation related to the gas ressure, 13 state equations for the alls temeratures, 13 state equations for the gas temeratures. The gas ressure equation is obtained by the mass conservation rincile for a gas in a ie. Precisely, let A the section area of the ie, T g the gas temerature (uniform in the ie). Furthermore, denote by 1 and 2 the gas mass flos assing through the extremal oints of the tube ith coordinates z 1 and z 2. The conservation la is exressed as follos: A(z 2 (t)) ṗ RT g RT g 2 A(z 2 (t)) T g = (1) = 1 2 here R = R/ m, R being the Boltzmann constant and m being the molar eight of helium. Note that in this equation, hen alied to the cylinder chambers, ill resent time varying coordinates z 1 and z 2. Passing to the 13 all temeratures, the basic equation is the energy conservation la for a metal (at temerature T, uniform in the ie) exchanging heat ith a gas and a cooling system of ith the external environment, and subject to a generic source sulying heat at the rate Q in, and ith a generic heat release Q out : c all M all Ṫ = = γ mg Ω(T T g )(z 2 (t))+ (2) γ c S c (T T ater )+ +Q in Q out Obviously, in this equation c all is the secific heat of the metal, M all the corresonding mass, T ater the temerature of the cooling ater, γ mg and γ c are the heat exchange coefficients beteen the gas and the metal, and the cooling ater and the metal, S c being the exchange surface beteen ater and metal. Finally, for the third grou of variables the basic equation is the energy conservation la for a gas in a ie. Precisely, let c v the secific heat of the gas, and denote by T i and ρ i the temerature and density of gas at the extremal oint of the tube, denoted by the coordinate z i (i = 1,2). c v AρT g (z 2 (t)) = = γ mg Ω(T T g )(z 2 (t))+ c v 2 (T 2 T g ) + c v 1 (T 1 T g )+ + [( ρ ) The overall basic model ( ρ ) ] (3) By exloiting the above equations, the overall model is a state sace lumed arameter model ith 27 state equations. Focus no on the last 13 equations. Among them, to equations refer to the gas temeratures in the cylinders. These equations are affected by the main exogenous variables, the oscillating (30Hz) istons ositions. This imlies that the temeratures of the cylinders resent an oscillatory behavior too. Moreover, the gas mass flos c and h obviously deend on the iston velocities, and therefore they are both oscillating variables. Finally, c and h act as exogenous variables in the remaining 11 gas temerature equations. This means that all temeratures in the regenerator lus the temerature in the receiver exhibit oscillatory behavior. In conclusion all 13 gas temerature variables have oscillations at 30Hz. In turn, this imlies that the ressure is oscillating too. As for the temerature of the cylinders alls and the temerature of the receiver, the thermal inertia is so high that the effects of oscillations of the internal gas temeratures are filtered out, so that the 3 temeratures have slo dynamics. As for the last 10 2

4 temeratures (regenerator), the oscillatory behavior is maintained since the thermal inertia is not extremely high. Summing u, of the 27 variables in the model, 24 resent oscillations at 30Hz, and 3 have slo dynamics. 3.4 Poer One main challenge is to find an exression describing thermal conversion into kinetic energy, so to derive the exression for the generated oer (see, for examle (Wei et al., 2002)). To do this, denote by c (t) ( h (t)) the ressure of gas in the cold (hot) cylinder. Then, the instantaneous oer absorbed by the comression iston is W c (t) = A c c (t)u c (t) (4) hereas the instantaneous oer generated in the exansion iston is: W e (t) = A h h (t)u h (t) (5) So that the gross generated oer is: W mech (t) = W e (t) W c (t) (6) Notice that, above, it is assumed that the ressure is everyhere uniform ((t)). Hoever, hen dealing ith the comutation of the oer, one cannot neglect friction and therefore one should distinguish beteen the to ressures c (t) and h (t). 4. A SIMPLE MODEL OF THE PLANT 4.1 Working out a mean value model by a cisoid aroach The basic model has been imlemented in a SIMULINK frameork and several simulations trials have been erformed. Such simulator requires a huge comutational effort, mainly due to the stiffness of the system. Therefore the basic model is not only difficult to deal ith in samling and simulation, but it is also of no use for control design uroses. Hoever, the erformed simulations sho that the oscillating variables have a sinusoidal-like behavior, ith higher harmonics ractically negligible, as it is also suorted by exerimental results (see (Bonnet et al., 2002)). So e have exlored the ossibility of modelling the fast variables as if they ere in a cisoid regime. Indicating by x i (t) a generic fast variable, the folloing has been imosed: x i (t) = x i (t) + x i (t)cos(ωt + ϕ xi (t)) (7) As it can be seen, this corresonds to consider the fast variable as comosed by a sloly varying average value x i (t) lus a sinusoid ith given frequency (30Hz) and sloly varying amlitude and hase x i (t) and ϕ xi (t). Then, by analytic comutations one can ork out algebraic exlicit exressions sulying the amlitude and hase x i (t) and ϕ xi (t) in terms of the average values of the various fast variables { x j (t)}, as it ill be illustrated in the subsequent oint ith reference to the ressure variable. Second, it is ossible to derive a ne set of differential equations here the derivatives of x i (t) are given in terms of the variables { x j (t)}. In this ay, the original basic model, consisting of 3 slo equations lus 24 fast equations (in the state variables x i (t)), is relaced by a ne system here the 3 slo equations are comlemented by 24 ne equations in the mean variables { x j (t)}. To such a system, a number of algebraic equations have to be added for the derivation of x i (t) and ϕ xi (t) in terms of { x j (t)}. The remarkable advantage of such a model is that the dynamics of all variables are comarable, so that this model is effective for simulation and control design. Furthermore, the mean value model enables to evidence the effects of the geometry of the lant comonents and of the thermodynamical characteristics of the gas on the generated oer. This is most useful in the early hase of the Dish Stirling solar lant design. 4.2 Pressure and oer By alying the cisoid aroach to the ressure equations, taken as a significant case, the folloing exressions for and ϕ in terms of can be derived. here ϕ = arctan = A h z h sin(ϕ h ) R T h,g A h z h sin(ϕ h ) R T h,g A c z c A h z h sin(ϕ h ) R T c,g R T h,g M He sin(ϕ ) = K( T gas,ϕ h ) M He = A c z cm ρ c +V reg,i 10 i=1 ρ h,i + (8) +V ric ρ ric + A h z hm ρ h Such results are useful to obtain a simlified formula for the mean oer generated by the engine W mech. Precisely, in equations (4) and (5) one can adot a cisoid aroach for the fast variables h and c, to find out the folloing exression: W mech = ω 2 A h h z h sin(ϕ h ϕ h )+ ω 2 A c c z c sin(ϕ c ) Note that c and h can be exressed in terms of, hich in turn is a function of, as shon in equation (8). In this ay, one comes to the folloing exression: W mech = W mech,id W mech,lam W mech,turb

5 here W mech,id is the ideal oer, generated as if the friction ere absent, W mech,lam ( W mech,turb ) is the loss in the generated oer caused by the friction due to laminar flo (turbolent flo). As for the ideal oer, one obtain: W mech,id = 1 2 A c z c A h z h R ( ω M He ) 1 1 T c,g T h,g (9) here A c z c and A h z h are the volumes set by the istons inside the cylinders. This formula oints out the deendence of the oer uon the main variables of the lant. In articular, it shos ho the oer is determined by the mean gas ressure, the hot and the cold cylinder gas temeratures T h,g and T c,g and the geometric characteristics of the lant. A further analytical elaboration leads to the folloing equivalent exression: W mech,id = = 1 M He,set,c M ( ) He,set,h ω R T 2 M h,g T c,g He (10) here M He,set,c and M He,set,h are the masses of Helium gas set by the istons in the cold cylinder and in the hot cylinder. The to terms describing the losses in oer due to friction effect are: W ( ) mech,lam = ω 2 k f l1 A 2 c z 2 c + k f l4 A 2 h z2 h (11) ( W mech,turb = ω3 k 2 R ft1 A 3 c z 3 c + k ) ft2 A3 h z3 h (12) T c,g T h,g 5. CONTROL SYSTEM The engine can be seen as a controlled rocess as outlined in Figure 3. The inut variables are the ingoing helium flo in and the outgoing helium flo out. The outut variables are the gas mean ressure the temerature T rec, of the receiver, and of course the roduced mechanical oer W mech. Observe that, among the three outut variables, only to are to be regulated, namely and T rec,. As e intend the lant to suly energy to a oer netork, it is a common assumtion to consider infinite load. Load variations are thus not consider. Therefore, the only disturbance that affect the rocess is insulation, hich is measurable. Normally, the goal of the control system is to in out Q insulation Stirling Engine W T mech rec, Figure 3. The Stirling Engine seen as a controlled rocess kee the receiver s alls temerature (T rec, ) constant at a given set oint value. The control scheme that has been alied is the cascade control system of Figure 4. (T re f ). The adoted Tref + - Temerature Controller Disturbance Comensator ref, closed loo + + ref,oen loo - Pressure Controller in out Qinsulation Figure 4. The Stirling Engine control system Stirling Engine rationale is that the inner loo is designed to control the helium mean ressure, ith the ressure set oint imosed by the outer loo. Moreover, being the disturbance measurable, the control system is also constituted by an insulation comensator. In conclusion, the ressure set oint is given by the suerosition of a reference ressure value decided by means of a look u table lus an additional term to deal ith non standard situations, esecially transient behaviors. The inner and the outer loo regulators are designed ith a standard frequency domain aroach. 6. SIMULATED MODEL VS REAL DATA The overall model of the lant has been simulated in C language, in the Matlab SIMULINK environment. It consists of various submodels for the thermodynamic art (described by the mean value model above outlined), the helium bottle, the cooling system, the induction motor, valves and transducers, controller. Some heat exchange or geometric arameters have been numerically calculated analyzing the hysics of the model and of the henomena that take lace in the engine (see, for examle, (Thomas and Bolleber, 2000)). Some uncertain arameters have been tuned by comaring the simulation outcoming ith the real data measured on the CESI Dish Stirling lant. Note that the CESI lant is equied ith a controller of unknon characteristics. The comarison beteen the so obtained simulator and the real lant has been made ith reference to an interval of time ith enough insulation as deicted in Figure 5. The comarison Insulation (W/m 2 ) Figure 5. Insulation real data W T rec, mech

6 beteen the simulated signals and the true ones is erformed in terms of helium mean ressure (Figure 6) and roduced oer (figure 7), ith very satisfactory results. In the Figure 8 the simulated nominal control gas ressure simulated data real data Figure 6. Simulated gas ressure versus real ressure of the Helium gas in the engine Produced Poer simulated data real data Figure 7. Simulated roduced oer versus really roduced oer by the lant erformances of the designed regulator is shon by the lot of the controlled receiver temerature T ric. T ric 10 o C simulated data Figure 8. Simulated and real temerature of the receiver s alls 7. CONCLUSIONS This aer discuss the main results obtained as far as the modelling of the Dish Stirling Solar Engine is concerned. While doing this, the model is simlified using steady state eriodic behavior to remove stiffness introduced by fast dynamics that are not relevant in the time scale of interest. The so obtained model is suitable for numerical solution, to desume fundamental correlations that clarify the behavior of the considered engine, and to design a control system hich could be successfully alied to the rocess. The overall control system has been extensively tested in simulation and comarisons of simulated results to real measurements are satisfactory. Exerimental results are shon in the aer. 8. ACKNOWLEDGEMENTS This ork has been develoed in the frame of the research on the Italian Electrical System "Ricerca di Sistema", Ministerial Decrees of January and Aril Research has been suorted also by The Italian National Research Project "Ne Techniques of Identification and Adative Control for Industrial Systems, by CNR-IEIIT and by CESI- Milano. REFERENCES Bonnet, S., M. Alahilie and P. Stouffs (2002). Exerimental study of the thermodynamic rocesses and the instantaneous temerature field in a small stirling engine. Proceedings of Euroaisches Stirling Forum Hirata, K., S. Iamoto, F. Toda, K. Hirata and K. Hamaguchi (1997). Performance evaluation for a 100 stirling engine. Proceeding of 8 th International Stirling Conference Mayzus, P., L. Fang, X. Deng, O.R. Fauvel and L. Bauens (2002). Pressure gradients in the regenerator and overall ulse-tube refrigerator erformance. AIAA Journal 40, Organ, A.J. (2000). Regenerator analysis simlified. Proceedings of Euroaisches Stirling Forum , Organ, A.J. (2002). The equations of temeraturedetermined gas-dynamics. Proceedings of Euroaisches Stirling Forum Rodgakis, E.D., N.A. Borbilas, E.A. Paradissis and A.P. Nikolaidis (2002). A mathcad rogram (amoco) as simle tool for the study of the stirling engines. Proceedings of Euroaisches Stirling Forum Thomas, B. and F. Bolleber (2000). Evaluation of 5 different correlations for the heat transfer in stirling engines regenerators. Proceedings of Euroaisches Stirling Forum , Walker, G. (1980). Stirling Engines. Clarendon Press. Oxford. Wei, D., M. Lucentini and V. Naso (2002). Processes of thermal energy conversion into kinetic energy in stirling engines. Proceedings of Euroaisches Stirling Forum 2002.

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