Chamber model simulation is a common approach to simulate rotary positive displacement vacuum pumps. Therefore the pump is abstracted into working chambers and connecting clearances, whereby the clearance leakages can be identified as the major loss mechanism in such machines. The clearance mass flow rates are calculated with respect to the thermodynamic states in the adjacent chambers, which are normally considered to be homogeneous. In this work it is shown, that the chamber state is inhomogeneous for rarefied gases due to the movement of the pistons which causes a pressure gradient within the chamber. This effect increases with higher Knudsen numbers, because of the increasingly dominant friction. An efficient way to model these inhomogeneous states with a one-dimensional approach for geometrically abstracted chambers is shown. The approach is applied to chamber model simulations of a screw spindle vacuum pump (SSVP) and the results are compared to simulations with homogeneous chamber states and to measurements.
Clearance mass flows are a major loss mechanism in dry running rotatory positive displacement vacuum pumps. Therefore, a detailed knowledge of the clearance mass flow is crucial to calculate the operation of those pumps. The small clearance heights and the large pressure range of such pumps require a wide range of gas rarefaction parameters to be taken into account. The flow in the clearance can be described as a combined Couette–Poiseuille flow with variable cross section. This is typically done by solving the Stokes equation, but especially at high gas rarefaction parameters, the inertia cannot be neglected any more, which can lead to choking of the flow. A one-dimensional approach for the compressible fluid flow was provided by Shapiro. It is shown that this approach can be carried over for the rarefied gas flow. The problem is solved in bounds for the constant total temperature and compared to experimental investigations by varying the pressure ratio and the circumferential speed of the clearance boundary in a wide range of gas rarefaction parameters.
Multi-dimensional numeric flow simulation is a common but non-trivial approach to simulate the operation behavior of screw machines. Current challenges of computational fluid dynamics (CFD) simulations of screw machines are especially related to turbulence modelling and meshing. An adaption of CFD simulations to screw vacuum pumps is difficult since the Navier-Stokes equations have limited validity for low pressure regimes. However, the scope of application can be extended to higher Knudsen numbers (e.g. lower pressure regimes) by the use of velocity slip and temperature jump boundary conditions at solid surfaces. Even if the complete simulation of screw vacuum pumps is still challenging, these boundary conditions can be used to examine isolated effects like clearance flows or inhomogeneous pressure distributions in working chambers. A common commercial CFD-Solver is ANSYS CFX, which also has been successfully applied to screw machines, but it does not provide a Maxwell velocity slip and a Smoluchowski temperature jump boundary condition. In the presented work these boundary conditions are implemented in Ansys CFX using user-defined expressions in a similar way as it has been implemented in the rhoCentralFOAM solver of the open source toolbox OpenFOAM. The boundary conditions are validated by comparison with stationary OpenFOAM results, DSMC results and measurements for a hypersonic plate flow. In addition, the boundary conditions are verified by experimental and DSMC results of a gas flow in a clearance between a rotational shaft and a plane counter plate.
Clearance flows do strongly influence the efficiency of dry-running positive displacement machines. To model the operation behaviour of the machines an accurate prediction of the clearance mass flows is crucial. If a chamber model simulation is used, the clearance mass flows are commonly estimate with simple equation to keep the computational effort low. Therefore, regression functions or coefficient databases are built into simulation. However, finding such functions or coefficients is a challenging and expensive task since many boundary conditions and geometry parameters must be varied in experiment or simulation. To reduce the effort of clearance analysis this paper presents a one-dimensional approach to estimate the clearance mass flow in e.g. dry-running screw-type compressors, based on the well-known differential equation of A.H. Shapiro for compressible flow of ideal gases. The friction is modelled with respect to laminar and turbulent flow. The flow separation and the friction of laminar flow are modelled based on the solutions of the Jeffery-Hamel flow. The friction in the turbulent regime is modelled by the Blasius expression, while the flow separation criteria is assumed to be the same as for laminar flow. The moving clearance wall is modelled assuming a superposition of Couette and Poiseuille flow. In addition, the heat transfer between gas and clearance boundary as well as gas rarefaction can also be optionally modelled. Results of the one-dimensional approach are compared to measurements and 3D-CFD simulations from the literature, varying Reynolds number and wall velocity.
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