Dynamic simulation and performance prediction of free displacer Stirling engines

ABSTRACT

In this study, a Stirling engine with a free-displacer and a kinematically controlled power piston was proposed and analyzed from thermodynamic and dynamic points of view. The analysis intended to reveal the dynamic behaviors of moving components of the engine as well as predicting global thermal performance of it. A dynamic-thermodynamic mathematical model of the engine involving the isothermal gas pressure equation and motion equations of the displacer, power piston and crankshaft was developed. For the solution of the dynamic-thermodynamic model equations, and simulation of the engine’s running, a computer program was prepared in FORTRAN language. By considering a hot-end temperature of 1,000 K and a cold-end temperature of 350 K, dimensions of mechanic, volumetric and thermal components of the engine were quantified interactively. Variations of engine speed, engine power, displacer stroke, and engine torque were examined with respect to the spring constant, displacer mass, displacer damping constant and external load and, results were graphically presented. In comparison with engines having free-piston and kinematically driven displacer, the thermodynamic performance of the free-displacer engine was found to be lower. The engine was found to be able to work at constant speed and power. The values of the displacer mass and spring constant were optimized as 1,500 g and 1,30,000 N/s, respectively and the global speed of the engine was determined to be 47.75 Hz for these values. The effective and the indicated work of the engine were determined to be 113 and 126 J, respectively.

KEYWORDS:

• Free displacer Stirling engine
• dynamic model
• isothermal-nodal analysis
• numerical solution of dynamic model equations
• estimation of dynamic behaviors

Nomenclature

$A_R$ area of the displacer piston (m²)

$C_{km}$ hydrodynamic damping constant at crankshaft main journals (Nms/rad)

$C_{mj}$ hydrodynamic damping constant at crankshaft pin journals (Nms/rad)

$C_p$ hydrodynamic damping constant at power piston (Ns/m)

$F_b$ connecting rod force (N)

$F_{bx}$ dry friction on the piston surface (N)

$F_{ch}$ the force caused by the casing pressure exerting to the back surface

of the piston (N)

$F_w$ gas force exerting on the piston top (N)

$F_{\mathrm{\infty }}$ dry friction on the piston surface (N)

$H$ length of connecting rod (m)

$h_p$ distance between piston top and piston pin (m)

$I$ inertia moment of crankshaft and flywheel (m²kg)

$k$ spring constant (N/m)

$m$ gas mass in the working space (kg)

$m_p$ mass of piston (kg)

$M_c$ moment of the piston force (Nm)

$M_s$ starter moment (Nm)

$M_q$ external load (Nm)

p gas pressure in the expansion volume (Pa)

$p_{{ }_{ch}}$ charge pressure (Pa)

$R$ radius of crankshaft (m)

$\mathrm{\Re }$ gas constant (J/kgK)

$T_i$ gas temperature in nodal volumes (K)

$V_i$ volume of cells (m³)

x position of the displacer (m)

$y$ displacement of the power piston (m)

$\dot{y}$ velocity of the power piston (m/s)

$\ddot{y}$ acceleration of the power piston (m/s²)

$z$ position of displacer top (m)

$\dot{z}$ velocity of the displacer piston (m/s)

Ź acceleration of the displacer piston (m/s²)

$\theta$ crankshaft angle (rad)

$\psi$ the angle between connecting rod and piston axis (rad)

Additional information

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.