Effect of inlet guide vanes and sharp blades on the performance of a turbomolecular pump

Transcription

1 Effect of inlet guide vanes and sharp blades on the performance of a turbomolecular pump G. A. Bird a GAB Consulting Pty Ltd, 144/110 Sussex Street, Sydney, New South Wales 2000, Australia Received 30 August 2010; revised manuscript received 14 October 2010; accepted 15 October 2010; published 4 January 2011 The direct simulation Monte Carlo method is applied to a two-dimensional representation of the flow through a turbomolecular pump. It is shown that a conventional pump with a rotor as the first stage produces a significant disturbance in the vacuum chamber upstream of the pump. This disturbance can be prevented, without any adverse effect on the performance of the pump, by the addition of inlet guide vanes, similar to a stator, upstream of the rotor. In addition, it is shown that the compression ratio across the early stages that are in near free-molecule flow is significantly improved if the blades have sharp leading edges. The calculations required runs of just several hours on an ordinary PC American Vacuum Society. DOI: / I. INTRODUCTION The first or leading stage of existing turbomolecular pumps is almost invariably a rotating stage. In the ultimate operating state with zero mass flux, the initial stages of a turbomolecular pump operate in the near free-molecule flow regime in which the mean free path of the gas molecules is very large in comparison with the turbine diameter. All the molecules that collide with a rotating first stage acquire the circumferential velocity associated with the point of impact. A substantial fraction of these reflected molecules has an axial velocity component that is directed upstream of the rotor. These molecules do not pass through the pump and affect the upstream gas over distances that are very large in comparison with the dimensions of the pump. Should the pump be mounted directly on an ultrahigh vacuum chamber, the flow within the whole of the chamber will be disturbed by the molecules that have collided with the rotating blades. This disturbance would be reduced if there was a pipe between the chamber and pump, but a strong rotation would be imparted to the gas within the tube. Turbomolecular pump performance is adversely affected by the gradual concentration of the gas toward the periphery where blade tip clearance effects become important and any prerotation of the upstream gas exacerbates this problem. The main purpose of this article is to study the magnitude of the disturbance that occurs upstream of a turbomolecular pump and to determine the extent to which the disturbance can be reduced by adding a stator or inlet guide vanes upstream of the first rotor. The study employs the direct simulation Monte Carlo DSMC method, 1 which simulates the flow at the molecular level and is valid in all flow regimes from collisionless or free-molecule flow, through the transition regime, to continuum flow. Any computational fluid mechanics study of an axial compressor has to cope with the relative motion between the rotors and stators. This causes difficulties for methods that solve the continuum Navier Stokes equations because these are Eulerian equations that describe the flow in terms of fixed locations and generally assume that it is time invariant. The introduction of time as an independent variable increases the computation effort by at least an order of magnitude, and the additional requirement to continually change the computational mesh to conform to the moving surfaces makes this an extremely difficult problem. By contrast, a DSMC computation is always a time-accurate unsteady calculation and, because the flow information is entirely carried by the moving molecules, position and velocity transformations can be applied to each molecule whenever it moves between a static and a moving flow region. These transformations are exact and are time dependent in that the positional transformation varies with the relative position of the rotor and stator at the instant that the molecule crosses the boundary between them. In other words, the calculations in the rotor stages are made in a frame of reference that moves with the rotor blades. This means that a fixed mesh can be employed for the flow irrespective of the number of moving stages. Therefore, unlike continuum CFD calculations, the introduction of moving or rotating flow regions into a DSMC calculation leads to a negligible increase in the computational effort. There have already been a number of applications of the DSMC method to turbomolecular pumps. 2 5 Reference 3 is the only application in which the flowfield extends beyond the pump and the compression ratio was found to decrease as a result. Both this reference and Ref. 4 deal only with a single moving three-dimensional blade and, while the other applications deal with multiple stages, it is not clear whether or not they employed the molecular transformations that are described in the preceding paragraph. References 2 and 5 employ a plane or two-dimensional representation of the flow at a given blade radius, and the results compare well with those from corresponding three-dimensional calculations. All the preceding applications evidently employ the simple DSMC procedures that are described in Ref. 1. Almost all of these procedures have now been either signifia Electronic mail: gab@gab.com.au J. Vac. Sci. Technol. A 29 1, Jan/Feb /2011/29 1 /011016/6/$ American Vacuum Society

2 G. A. Bird: Effect of inlet guide vanes and sharp blades cantly enhanced or superseded by new procedures, thus leading to sophisticated DSMC. 6 The most significant developments have been the following. 1 Separate cell or grid systems are used for the sampling of the flow properties and for the computation of representative collisions. The sampling cells affect only the resolution of the flow property displays, while the very much smaller collision cells reduce the separation distances of the molecule pairs that are randomly selected as possible collision partners. The cell structures are readily adaptable to the local gas density in order to produce a specified uniform number of simulated molecules in each cell. 2 The collision pair separation distances are further reduced through the selection of nearest neighbors as possible collision partners. However, these procedures must actively prevent successive collisions of a molecule with the same collision partner. 3 The time step varies over the flowfield and automatically adapts to a specified fraction of the local mean collision time. Time parameters are associated with every simulated molecule and with every collision cell. A molecule is moved when its time falls half a local time step behind the overall flow time and representative collisions are calculated in a collision cell when its time falls half a local time step behind the overall flow time. 4 The computational parameters such as the initial number of simulated molecules, the number of cells, the values of the time step, and the computational grids that define the cells are generated automatically by the program. The user needs only specify the approximate number of megabytes to be used by the program. The program then checks whether the criteria for a good DSMC are being met during the calculation and warns the user if the computational resources are inadequate for the problem that has been specified in the data file. It has been shown 7 that the rate at which a result converges to the correct value with increasing molecule number and sample size is much greater for programs that employ the sophisticated procedures than it is for programs that employ the simple procedures. This means that a result of given accuracy can now be achieved with far fewer simulated molecules. Almost all the statements on the computational requirements of the DSMC method that have appeared in literature relate to the simple procedures and they have often been based on inefficient implementations of these procedures. This means that these statements habitually overstate the real computational requirements of the DSMC method by at least one order of magnitude. II. TURBO2V PROGRAM The results in this article were obtained with the TURBO2V program, which is a dedicated DSMC program for the twodimensional or stream surface representation of the flow through a turbomolecular pump with an arbitrary number of stages. It is based on the DS2V program, 8 which is a well established and freely downloadable general program for FIG. 1. Geometry of the simulated flow. two-dimensional and axially symmetric flows that has a user base in the hundreds. The features of the DS2V program that become redundant in the dedicated program for turbomolecular pumps were removed, and the new capabilities that are necessary for the pump simulation were added. The most important of the latter was to implement the molecular transformations that are required for the exact simulation of moving stages in a simulation with a static grid. The V in the program names indicates that they are visual interactive programs with a graphical user interface that allow the user to monitor the computational variables, the flowfield properties, the surface distributions, and the pumping performance at all stages of the calculation. Figure 1 shows the geometry of a calculation that involves a single rotor followed by a stator. The simulated flow region is delineated by the sampling cells in a simulation that employed simulated molecules with 30 per sampling cell after adaption to the steady flow density. There is no requirement for DSMC cells to be regular and each of these cells is grown from the very small elements of a fine rectangular background grid that are closer to its cell node than to any other node. Note that each cell must be entirely within either a stationary or a moving flow region. This is a grid scheme that requires negligible computer time to automatically adapt to the local density as the flow develops and has been readily scaled to three spatial dimensions in the DS3V Ref. 8 program. The notional axis of the pump is in the horizontal or x direction and the rotating stage moves in the vertical or y direction. The geometry differs from the previous DSMC applications in that the blades are at the center of the simulated regions rather than at the edges. The blade shapes and the axial or x locations of each stage are specified in the blade geometry screen of the data menus, and the program arranges them such that the y coordinate of the leading edge of a blade is at the y coordinate of the trailing edge of the preceding blade. The blade spacing in the radial or y direction is specified in the main data screen, and the program sets the upper and lower boundaries of the simulated region to the centerline of the space between blades. These boundaries are periodic boundaries. The extent of the simulated region upstream and downstream of the blade stages is also set as data, as is the composition of the gas. It is also possible to specify a variation in the cross-sectional area of the flow as a linear function of x. The ultimate operating state of a turbomolecular pump involves an upstream gas density in the near free-molecule J. Vac. Sci. Technol. A, Vol. 29, No. 1, Jan/Feb 2011

3 G. A. Bird: Effect of inlet guide vanes and sharp blades regime. The overall pressure ratio will then be independent of the actual value of the upstream gas density as long as it is within that regime, and the upstream gas can be referred to simply as the stream. The upstream boundary is then selected from the boundary menu as a stream boundary and, because the mass flux is zero, the downstream boundary is set as a plane of symmetry. The temperature of the stream is set by the desired ratio of the blade speed to either the speed of sound or the most probable molecular speed in the gas, and the velocity of the stream is zero. The initial state of the gas in the simulated pump is set to these stream properties, and the rotor blades are impulsively set in motion at zero time. The number of simulated molecules builds up as the gas is compressed, and a steady flow is established when this number levels off to its limiting large-time value. A timeaveraged sample is then made of the flowfield and surface properties. If necessary, separate samples could be made of the properties at various relative blade positions so that the periodic variation of the pressure distributions on the blades could be determined. These variations could be a structural design consideration for the blades in the later stages that are in near continuum flow. There is a finite mass flux through the pump when the downstream pressure is less than the limiting pressure, and the pump characteristics are usually described by the volumetric input flux, or throughput, as a function of the compression ratio. A point on the characteristic curve is most conveniently determined by specifying the downstream pressure and determining the resulting throughput in the DSMC simulation. This can be achieved through the constant pressure boundary conditions that had already been developed in the DS2V program for applications that involve the flow through microchannels at a specified overall pressure ratio. The upstream and downstream pressures and temperatures are specified, but the velocities are initially set to zero. There are therefore two streams and the initial state of the gas employs a boundary at a specified x location within the pump that separates the upstream gas from the downstream gas. The blades are set in motion at zero time and the mass flux across the internal boundary is sampled. This sampled flux is used to set consistent axial velocities at the upstream and downstream boundaries. The desired steady state solution is again the large-time state of the simulation. The upstream and downstream pressures could be controlled interactively so that the pump characteristics could be determined in the course of a single run. The two-dimensional representation in the TURBO2V program effectively assumes that the blade cross-sectional element is at an infinite radius. This should be adequate for the optimization of the shapes, angles, and spacing of the blades and to study the effects of the stage clearances. If there is a finite radius, the calculations in the frames of reference moving with the rotor blades become calculations in rotating frames of reference, and the centrifugal and Coriolis accelerations must be taken into account when the molecules move. The effects due to a finite radius cannot be incorporated in the two-dimensional treatment because they lead to FIG. 2. Time-averaged molecular speeds m/s in a rotor-stator stage. gradients normal to the plane of the calculation. The program can indicate the magnitude of these accelerations if a radius is specified, but an analysis of the extent to which the gas becomes concentrated at the periphery of the pump and the consequential detrimental effects of blade tip clearance requires a full three-dimensional program. A TURBO3V program, based on the DS3V program, is being developed. III. CONVENTIONAL FIRST ROTOR-STATOR STAGE The TURBOV2 program has been used to calculate the limiting or zero mass flux state of a single rotor-stator stage with the geometry of Fig. 1. The rotor and stator blades have the same triangular shape, and the rotor blade moves in the downward y or circumferential direction with a speed of 316 m/s. The width of each blade in the x or axial direction is m, and the flat side of the blades is at an angle of 45. The maximum thickness of the blade is at x =0.0085, and the thickness measured in the y direction is m. The space between the blade rows in the x direction is m, and the blade spacing in the y direction is m. The gas is argon at an initial number density of at a temperature of 298 K, so that the blade speed is 0.9 times the most probable molecular speed in the undisturbed stream. The temperature of the blade surface is also 298 K and is assumed to be diffusely reflecting with complete accommodation. A small fraction of nondiffuse reflection may occur under ultrahigh vacuum conditions, and the TURBO2V program includes options for a specified fraction of specular reflection and for the more realistic Cercignani Lampis Lord gas-surface interaction model. However, because the degree of nondiffuse behavior is unknown and case dependent, it could only be dealt with by a sensitivity study that is beyond the scope of this article. It is assumed that there is no variation in the blade height so that the area of the flow is constant, but a variable area can be specified in the data. The time-averaged speed of the molecules in the fully developed flow is shown in Fig. 2. The moving frame of reference extends from the leading edge of the rotor blade to the midpoint of the gap between the rotor and stator and is indicated by the external bar and arrow. The speeds in this region are relative to the blade, but are otherwise in the laboratory frame of reference. The most striking feature is that JVST A - Vacuum, Surfaces, and Films

4 G. A. Bird: Effect of inlet guide vanes and sharp blades FIG. 3. Scalar pressure N/sq m in a rotor-stator stage. the gas upstream of the rotor has a uniform speed of approximately 200 m/s in the direction of the blade movement, and this is almost two-thirds the speed of the moving blade. Similarly high average speeds occur behind the trailing edge of the moving blade and below the leading edge of the stationary blade, but the stator blade is very effective in preventing any significant speeds downstream of the stator. The figures also show lines of constant average molecular speed and, unlike flows past similar isolated surfaces, there is a distinctive structure to the flow. This is wavelike in this case, but closed vortices appear in less efficient configurations. It is possible that this structure could provide a guide to the most efficient blade shapes and spacing. The density of the moving gas upstream of the rotor is about 3% greater than the density of the initial gas, and the temperature is approximately 350 K. This means that the upstream pressure is about 20% higher than the pressure of N/sq m in the undisturbed upstream gas. The pressure tensor in the upstream gas is anisotropic with the normal component in the y direction being 40% above, and the normal components in the x and z directions 20% below, the average of the three components. This average of the normal components of the pressure tensor is called the scalar pressure, and its distribution is shown in Fig. 3. The highest pressures are in the gas above the forward upper surface of the stator, and the maximum is just under N/sq m. The gas downstream of the stator is at a near uniform pressure of N/sq m so that the ratio to the undisturbed pressure is 6.6. The disturbance due to the molecules that are reflected upstream of the rotor not only reduces the effective compression ratio to 5.3, but can also have an adverse effect on the processes within the vacuum chamber. The calculation was made on a PC with a 2.4 GHz CPU, and the computation time was 2 h. The sample size in each cell was about half a million, and results with sufficient accuracy for design optimization could be obtained from much shorter runs. FIG. 4. Scalar pressure N/sq m with blunt edged blades. obvious that, for a given blade angle, the fraction of upstream moving reflected molecules is minimized by a sharp leading edge. Despite this, most turbomolecular pumps appear to employ blades with blunt edges. An additional calculation was therefore made with flat blades of uniform thickness and cylindrical leading and trailing edges. The calculation was otherwise identical to the calculation with sharp blades, and the resulting pressure distribution is shown in Fig. 4. The pressure in the region downstream of the stator falls from to N/sq m. This is for a blade with rounded edges and a thickness to chord ratio of only 0.06 and, for the thicker blades with flatter leading edges that are sometimes used in real pumps, the pressure reduction could be much greater than the 25% that occurs in this example. The velocity that is induced upstream of the blades is reduced by 5%, while the upstream pressure increase remains at 20%, and there is an unchanged degree of anisotropy in the pressure tensor. V. EFFECT OF INLET GUIDE VANES The calculation that led to Figs. 2 and 3 was repeated with the addition of inlet guide vanes that are identical in shape to the downstream stator in the previous calculation. The timeaveraged molecular speeds in this case are shown in Fig. 5. High speeds are confined to areas adjacent to the boundaries between moving and stationary flow regions. The uniform downward velocity in the region upstream of the three component compressor stage is reduced from 200 to about 2 m/s. Moreover, the number density, the temperature, and the pres- IV. EFFECT OF BLADE BLUNTNESS The preceding calculation employed blades with sharp leading and trailing edges. The most efficient aerofoil shape in free-molecule flow is an infinitely thin flat plate, and it is FIG. 5. Time-averaged molecular speeds m/s with inlet guide vanes. J. Vac. Sci. Technol. A, Vol. 29, No. 1, Jan/Feb 2011

5 G. A. Bird: Effect of inlet guide vanes and sharp blades FIG. 6. Scalar pressure N/sq m with inlet guide vanes. sure of the upstream gas are unchanged from those in the undisturbed upstream gas to within a small fraction of 1%. The scalar pressure distribution is shown in Fig. 6 and, given the magnitude of the upstream velocity that has been removed by the guide vanes, the pressure rise across these vanes is surprisingly small. However, the overall pressure ratio is slightly higher at 6.7 and, because there is no longer an induced pressure upstream of the rotor, this is also the effective compression ratio. VI. EFFECT OF KNUDSEN NUMBER The calculation that led to Figs. 5 and 6 was repeated for a range of upstream number densities. Figure 7 shows the compression ratio as a function of the Knudsen number based on the blade chord. As expected, the compression ratio is uniform at the very high Knudsen numbers that are in the free-molecule regime. The compression ratio falls at an increasing rate through the transition regime, and it is clear that different blade shapes and geometries are needed in the near continuum regime. At the same time, it should be noted that the geometry in Sec. V was chosen arbitrarily, and the optimum configuration has not been determined even for the free-molecule regime. Although these calculations involved a variation of the upstream density through four orders of magnitude, those at the higher number densities required less computation time FIG. 7. Effect of Knudsen number on the compression ratio. than those at the lower number densities. This was because all the calculations commenced with the same number of simulated molecules, and the molecule number in the eventual steady state declined with the compression ratio. The consequent reductions in computational load more than balanced the increases in the load due to the increasing number of collisions. However, at the lowest Knudsen number of 0.055, the number of collisions reached one-tenth the number of molecular moves, and the criteria for a good DSMC simulation require that this fraction should not be allowed to become significantly larger. This means that further reductions in the Knudsen require a proportional increase in the number of simulated molecules. The computational time is directly proportional to this number. The later stages of a turbomolecular pump are in near continuum flow, and a run of some hours on a PC would then be required for a singlestage calculation. Multistage calculations up to the stages with a Knudsen number of about 0.05 are in easy reach of a PC, and simulations that involve all stages could be made with computation times of several days. VII. SUMMARY AND CONCLUSIONS A DSMC program has been produced for the study of a two-dimensional representation of the flow through turbomolecular pumps with an arbitrary number of stages. The flow through a real turbomolecular pump is three dimensional, and there must be some reservations about making conclusions on the basis of two-dimensional calculations. At the same time, variations in the geometry and ordering of the blade stages and the shapes of the blades can be expected to produce similar effects in two- and three-dimensional models. This means that conclusions based on large effects in the two-dimensional simulations can confidently be expected to apply to real pumps. A conventional first stage rotor was shown to produce a velocity in the upstream gas in the direction of the blade movement that was as high as two-thirds the blade speed, together with a 20% increase in the upstream pressure. It was found that the addition of inlet guide vanes with geometry similar to the first stage stator effectively removes these undesirable effects. For the initial stages that are near free-molecule flow, sharp-edged blades have overwhelming advantages over blunt blades. For example, the replacement of a blade with a triangular cross section by a blunt blade with a slightly smaller average thickness led to a 25% reduction in the single-stage compression ratio. There are many other variables and a large number of two-dimensional DSMC calculations would be required to optimize the various design features of turbomolecular pumps. The average computation time of each run would be of the order of several hours on an ordinary PC, so that the cost of the computations would be negligible. 1 G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows Oxford University Press, Oxford, 1994, 450 p. 2 J.-S. Heo and Y.-K. Huang, Vacuum 56, S. Wang and H. Ninokata, Prog. Nucl. Energy 47, JVST A - Vacuum, Surfaces, and Films

6 G. A. Bird: Effect of inlet guide vanes and sharp blades R. Versluis, R. Dorsman, L. Thielen, and M. E. Roos, J. Vac. Sci. Technol. A 27, F. Shapirov, 27th Symposium on Rarefied Gas Dynamics, Pacific Grove, CA unpublished. 6 G. A. Bird, in Rarefied Gas Dynamics, edited by M. S. Ivanov and A. K. Rebrov Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia, 2007, p M. A. Gallis, J. R. Torczynski, D. J. Rader, and G. A. Bird, J. Comput. Phys. 228, G. A. Bird, AIP Conf. Proc. 762, J. Vac. Sci. Technol. A, Vol. 29, No. 1, Jan/Feb 2011