Simulation of a Reciprocating Compressor Accounting for Interaction between Valve Movement and Plenum Chamber Pressure

A mathematical model is described which will simulate a reciprocating compressor on a digital computer. Initially it is assumed in the model that the pressures in the valve plenum chambers remain constant. A sufficiently converged solution is obtained in about three compressor cycles. The inherently intermittent and unsteady flow is then calculated by integrating the solution to the simultaneous differential equations which describe the flow through the suction and discharge valves and the valve movement. A Fourier analysis is performed on this predicted flow through each valve and the resulting Fourier coefficients are used in an acoustic analysis of the compressor suction and discharge systems in order to predict the pressure variation in the appropriate plenum chamber. These pressure variations are then used to revise the initial flow rates and the mutually interdependent movement of the valves. An iterative process requiring about four cycles of computation was generally successful in obtaining a cyclically repeatable solution. In some instances resonance oocured between natural frequences of the gas vibration in the suction or discharge systems and a harmonic of the compressor speed. Under these conditions it became necessary to introduce mathematical damping to simulate the viscous damping in the real system. It was difficult to select a value of damping coefficient appropriate to achieving adequate agreement between analytical and experimental plenum pressure records. Application of the model resulted in little change in compressor performance compared with values predicted by a simpler model in which the plenum chamber pressures are assumed constants. However the interaction between valve movement and non constant plenum chamber pressures modified the valve action: hence the new model is of use when making a detailed analysis of valve behaviour. INTRODUcriON In a compressor fitted with automatic valves, the valve movement during the suction and discharge process is afuncticinof the pressure difference across the vGlve. Both the valve movement and the pressure difference are dependent on compressor speed, compressor pressure ratio, valve inertia, 354 valve spring stiffness, fluid properties, etc. In early models, which assume that the valve plenum chamber pressures remain constant, (1, 2, 3,) two simultaneous differential equations, which relate the more important relevant variables, provide a mathematical model to describe the suction and discharge events in the cycle. One is a nonlinear equation which expresses the flow through the valve as a function of the pressure difference across the valve and the displacement of its moving element. The other is an equation which describes the valve displacement in terms of the relevant gas and spring forces acting upon it. The valve is considered to be a single-degree-of-freedom spring mass system. Both equations are expressed as functiornof time and hence of crankangle. These equations, solved either by graphical, analog, or on digital computer by numerical methods, yield the displacement of each valve and the pressure difference across it. The pressure in the cylinder is obtained by subtracting (during suction) or adding (during discharge) the pressure difference across the valve to the appropriate plenum ~hamber (constant) pressure. Integration of the flow and pressure variations over a cycle permits the evaluation of volumetric efficiency and power consumption. The assumption that the suction and discharge plenum chamber pressures remain constant is a simplification which provides a relatively simple model which is economical to use. Such a model is still of use as an aid to design despite the availablility of more complete, but more complex, models. Since the mass flow of gas through positive displacement compressors is inherently intermittent, a fluctuation of pressure must occur in the finite volume plenum chambers and associated piping. This fluctuation is coupled with the valve action and each can have a significant effect on the other and on compressor performance. Simulation models which account for these complex pulsation phenomena have been developed for use on analogue computers (5) and hybrid (analogue/digital) computers (6). Most models, however, have been used with digital computers. References (7, e, 9, 10) are examples of papers presented which contain descript~ons of these more complete models at the four previous Compressor Technology Conferences at Purdue University.