Quasi One-Dimensional Steady-State Models for Gas Leakage Part I: Comparison and Validation

Most theoretical work done on gas leakage in compressors is concerned with quasi one-dimensional steady-state models. Four of these, each quite different in mathematical formulation and solution strategy, have been implemented and tested. Two experiments reported in literature were used to validate the models. The results show that chocking is likely to occur. Unfortunately none of the models predicts the mass flow rate with sufficient accuracy and reliability, mainly due to poor modeling of viscous effects. INTRODUCTION The investigation presented here is part of a larger project which seeks to improve the modeling of leakage flows in twin screw compressors, but the same problem is relevant to other types of compressors. Fleming & Tang [3] estimate the efficiency losses due to leakage well over 10%. A thorough knowledge of the process might help to decrease these losses. Models are valuable for gaining insight and as design tools. Many models of various types have been reported in literature and in this paper four of these are compared and validated with experiments. The selection confines to quasi one-dimensional models for steady-state flows of pure gas. The relative movement of the walls is not taken into account. This paper will start off with a discussion of the physics of the leakage flows, followed by a brief presentation of the models. Two experiments were selected from literature for validation. These will be explained, together with the results in the sections before the conclusions. nomenclature specific heat ratio "( dynamic viscosity 'l} friction coefficient e density p wall shear stress T flow area A hydraulic diameter D Mach number M wetted perimeter p pressure p specific gas constant R Reynolds number Re temperature T velocity u flow path coordinate X 1 Pa·s 1 kg/m3 Pa m2 m 1 m Pa Jjkg-K 1 K m/s m For the sake of completeness it should be noted that two other types of models are reported in literature. Thermodynamic simulations often use zero-dimensional models, ie. algebraic equations that relate the mass flow rate to the smallest flow area and the thermodynamic state of the incoming gas (see eg. Fujiwara et al. [4] or Caillat et a!. [2]). Others incorporate one-dimensional models into their thermodynamic models ( eg. Sangfors [12]). And some work has been done on a two-dimensional model by Neumann [8). PHYSICS OF THE FLOW The main leakage paths in twin screw compressors have a converging-diverging geometry. This means that supersonic flow might occur for sufficiently high pressure ratio's. The transition back to subsonic takes place in one or more shock waves. In slender channels the shock is likely to be normal, ie. perpendicular to the direction of the flow. This type of shock always has supersonic flow upstream and subsonic downstream. For subsonic flow the boundary conditions are the thermodynamic state at the inlet and the pressure at the outlet (:fig.1, case A). In the supersonic flow there is no information traveling upstream, thus the outlet pressure cannot be felt at the inlet. In this case an extra boundary condition is needed for the subsonic flow at the inlet: the sonic point must be as far downstream as possible. This is equivalent with the lowest inlet velocity that results in supersonic flow (cases B .. E), a phenomenum known as chocking. The outlet pressure is still relevant since it