C 12-6 MODELING COMPRESSOR START UP

Frequently, a design or R&D engineer is asked a question: "Will that compressor start if we change, if we use, etc." Usually the answer may be given after long testing. This paper offers a method of fast assessment of compressor startability by means of computer simulation. The core of the paper is a simple fractional-polynomial model of an electrical induction motor that is suitable for modeling compressor start up and run. The torque-slip or torque-rpm dynamometer test data are fitted to the model that is based on the equivalent circuit diagram of an induction motor. Only two torque-slip or torque-rpm pairs that are measured at only one voltage and one frequency suffice to find all constants of the model. The identified model generates full range continuous, from zero rpm to synchronous rpm, torque-slip or torque-rpm function for any other voltage and frequency. The model constants have physical meaning of resistance and reactance of the circuit diagram. The model was also successfully used to identify friction loses of the compressor. EQUATIONS OF MOTION OF A COMPRESSOR Most frequently, a hermetic compressor is flexibly supported on suspension springs inside the shell, and the shell is flexibly supported by rubber grommets. This mechanical system of three rigid bodies has fourteen degrees of freedom. In this analysis, we restrict our analysis to the rotational motion of the rotor only as it is depicted in Figure A1 in the appendix. When such a hermetic compressor that is driven by an electric induction motor is switched on, the torque that develops between the stator and the rotor accelerates the rotor in accordance with the equation (1). ( ) ( ) 1 d d J T , ; ; dt dt ω φ φ ⋅ω = φ ω ω = ω = φ = & & & (1) Where is J1(φ) instantaneous moment of inertia of the rotor about the axis of rotation [kg.m] T is torque acting upon the shaft [N.m] φ is angle of rotation as shown in Figure A1 [rad] t is time [s]. Instantaneous moment of inertia of the rotor is Where mC mass of connecting rod [kg] mP mass of piston [kg] JR is moment of inertia of the rotor proper [kg.m] JC moment of inertia of connecting rod with respect to its center of gravity [k.m] Coordinates of the position of the piston xB, position of the CG of connecting rod xc and yc, and the corresponding velocities can be found from the geometry of the slider crank mechanism that is shown in Figure A1. The torque acting between the rotor and the stator consists of several terms ( ) 2 2 2 2 C C B 1 R C C P x y x J J J m m (       + ψ φ = + + +         φ φ φ       & & & & & & & 2) 3) ( ) m C P B T T T T T ( φ = − − − Where Tm is torque of electric motor [N.m] TC is compression torque [N.m] TP is torque due to piston friction [N.m] TB is torque of bearing friction [N.m] For the purpose of the simulation of compressor start up, the compressor is assumed to have ideal valves. This means that the maximum pressure in the cylinder during the discharge period is constant and equal to the condensing pressure. Similarly, the minimum pressure in the cylinder during the suction period is also constant and equal to the evaporating pressure. To take into account function of valves, one can increase condensing pressure and decrease evaporating pressure by several percentage points. The torque TB due to bearing friction is 3 i Bi B i i d L T (4) 2 c π⋅μ ⋅ ⋅ = ⋅φ ⋅ ∑ & Where di is shaft diameter [m] LBi is length of bearing [m] μ is dynamic viscosity of oil [Pa.s] c is radial bearing clearance [m] The torque due to the viscous friction of piston movement is P P P P P B C P d L R cos T F R sin 1 ; F X (5 L cos d d   π ⋅μ ⋅ ⋅ φ = ⋅ ⋅ φ⋅ + ⋅ = ⋅   ψ −   & ) The compression torque is