Stirling Engine Marten Byl 12/12/02 1xTeThTc=0RFigure 1: Schematic of Stirling Engine with key variables noted. Introduction In the undergraduate class 2.670 at M.I.T., the students explore basic manufacturing tech-niques by building a stirling engine. The class is concluded by all of the students running their engines at the same time. As the students discover, the stirling engine is very sensi-tive to manufacturing tolerance, specifically the fit of the components determines both the friction in the engine and air leakage out of the engine. The purpose of this project was to develop a model of the stirling engine that accurately predicts the effects of leakage and friction on engine performance. 1 Stirling Engines Figure 1 shows a simple schematic of a stirling engine with key parameters noted. The concept of a stirling engine is fairly simple. The engine consist of heat source, in our case an alcohol flame, and a heat sink, ambient air, an enclosed cylinder, a ”heat” piston, a ”power” piston, and a flywheel connected to the two pistons by a set of linkages. The concept is that the heat flowing through the air in the enclosed cylinder is modulate by the position of the ”heat” piston. When the ”heat” piston is located directly over the flame the heat flow into the engine is minimized while the heat flow out of the cylinder to the heat sink is maximized. Similarly, when heat flow in is maximize, heat flow out is minimized. While the ”heat” piston is moving, the ”power” piston is also moving thus converting the thermal energy being captured by the air into mechanical motion. The flywheel then stores this mechanical energy, thus allowing the mechanical power to flow both in and out of the engine. The geometry of the linkages determines the relationship between the motion of the ”power” piston and the ”heat” piston. Figure 2 shows an animation of the stirling engine in operation. Frame A shows the engine in the starting angular position. In the starting position, we see that the ”heat” cylinderS SS SSSABCDEFFigure 2: Animation of stirling engine in operation. is positioned to maximize the heat in-flow while at the same time the ”power” piston is positioned to maximize output power. In frame B, we see the engine has rotated such that output power is minimized while the heat input area is reducing. In frame C, we see that heat outflow is nearing maximum while mechanical power may actually be flowing back into the engine. Frames D and E, show the transition back to heat in flow and mechanical power outflow. Frame F shows the engine moving back into the maximum thermal power in and mechanical power out position. I would like to thank Katherine Lilienkamp for allowing me to use her matlab code to generate these animations. From the 2.670 class notes by Prof. David Hart [1], the stirling engine built in the class operates with a hot temperature, Th, of 600 K and a cold temperature, Tc, of 300 K. The typical engine will produce 1 W at 400 Rpm. The typical engine will operate at between 400-600 Rpm, with exceptional engines running at speeds up to 1200 Rpm.C 1 TF1MTF1RI00RSe:PaRSe:ThRSe:TaR*cosxApVSaNaTeShScAh( )Ac()Figure 3: Bond graph model of stirling engine. 2 Stirling Engine Model Figure 3 shows the bond graph model developed for the stirling engine. The heat source is modelled a constant temperature effort source, Th, which transfers entropy to the air in the cylinder, modelled as a multi-port capacitor, through a variable resistor. Similarly, the heat sink is modelled as a constant temperature effort source, Tc, which also transfers entropy to the air in the cylinder through a different variable resistor. As mentioned earlier, the air in the cylinder is modelled as a multi-port capacitor. Since leakage from the cylinder is impor-tant, one port on the multi-port capacitor tracks the mass loss through a resitor to ambient conditions, modelled as a constant pressure effort source. A second port on the capacitor tracks the entropy flows too and from the heat sources and the entropy loss due to mass flow. The final port on the capacitor is associated with the volume change. The pressure in the cylinder acts upon the power piston which is modelled as a constant transformer. The piston then acts upon the linkage to the flywheel, modelled as a modulated transformer. Finally, the flywheel is modelled as an inertia, while all of the friction losses in the system are modelled as a resistor with damping b. The major modelling assumption used in this bond graph are: • No power transfer through the ”heat” piston. • Mass-less pistons. • Uniform temperature for air in engine. • Lumped friction element to govern engine speed. • Uniform constant temperature sources. • All leakage from engine through power cylinder. • Motion of ”heat” piston is sinusoid 90 degrees ahead of power piston. There are four state variable in this system.θ the angular position of the flywheel θ˙ the angular velocity of the flywheel Se the total entropy of the air in the cylinder Ne the number of mols of air contained in the cylinder The model results in the following formulation equations: x = Re(1 + sin θ) Ah = Asc(1 + cos θ) Ac = Asc(1 − cos θ) + Ppsx S˙h = Ahµ(Th − Te) Te S˙c = Acµ(Te − Tc) Te ˙Ne = −Al �2ρe(Pe − Pa) or Al �2ρa(Pa − Pe) ˙Se ˙Sa = NaNe S˙e = S˙h − S˙c + S˙a Ve = Vc + Apx Ve v¯e = mNe �−R� v¯e Cv s¯e − s¯oTe = To exp v¯o Cv � v¯e �−(−R +1) s¯e − s¯oCv Pe = Po exp v¯o Cv Fe = (Pe − Pa)Ap τe = FeRe cos θ τI = τe − bθ˙ ¨ τe − bθ˙ θ = I Where:Re = 1.25 cm = Radius of linkage pivot on flywheel Asc = 40 cm2 = Heat transfer surface area of cylinder Ppc = 4.9 cm = Perimeter of power piston Ah = variable = Hot heat transfer area Ac = variable = Cold heat transfer area Te = variable = temp of air in engine µ = 100000 W/m2 = heat transfer constant of cylinder, this was calculated as the thermal conductance of steel with the cylinder wall thickness Al = nominally 0.06 mm2 = area of leak Ap = 1.9 cm2 = area of power piston Vc = 40 cm3 = volume of air cylinder m = 29 kg/kmol = molar mass of air R = 287 J/kg = mass gas constant air s¯o = 2800 J/K*kg = specific entropy of air at T=300 K To =