Geometrical abstraction of screw compressors for thermodynamic optimization1.pdf

/ Engineering Science Engineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical /content/225/6/1399 The online version of this article can be found at: DOI: 10.1177/0954406210395884 1399 2011 225:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science J Hauser and A Brmmer Geometrical abstraction of screw compressors for thermodynamic optimization Published by: On behalf of: Institution of Mechanical Engineers can be found at:Science Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical EngineeringAdditional services and information for /cgi/alertsEmail Alerts: /subscriptionsSubscriptions: /journalsReprints.navReprints: /journalsPermissions.navPermissions: /content/225/6/1399.refs.htmlCitations: What is This? - May 25, 2011Version of Record at ZHEJIANG UNIVERSITY on March 28, 2012Downloaded from Geometrical abstraction of screw compressors for thermodynamic optimization J Hauserand A Brmmer Department of Fluidics,Technical University Dortmund, Dortmund, Germany The manuscript was received on 17 May 2010 and was accepted after revision for publication on 17 November 2010. DOI: 10.1177/0954406210395884 Abstract: The construction and development of different rotor profi les is an important area in connection with the development of screw compressors for specifi c applications. Geometrical performance fi gures (using criteria to describe interdependencies of geometrical parameters for screw compressors) for profi le optimization are used in order to achieve specifi c improvements in performance. During this process, rotor profi les and spatial parameters are the main factors. Compared to data derived from the front section of rotor profi les, these fi gures which also take spatial parameters into account provide a better evaluation of gap conditions and operating effi ciency of the compressors under examination. Keywords: screw compressor performance, profi le optimization, new design concepts 1INTRODUCTION The operational behaviour of a screw compressor can be examined experimentally but also by means of comprehensive simulation procedures 1,2. The complexity of the simulation, the need for trial runs, and the need to achieve the closest possible par- allels to a real compressor do lead to valid results, but the process is very time-consuming. Computer- aided optimization by comprehensive simulation is therefore undesirable, despite leading to valid results. With a view to fi nding a compromise between pre- cision and calculation time, evaluation and analysis by means of abstracted geometrical performance fi g- uresisareasonableapproach.Themainfeatureofthis approach is intended to be a reduction in develop- ment time by using geometrical codes to characterize thethermodynamicperformancebehaviourofascrew compressor. Within the fi eld for developing geometrical per- formance fi gures, there have been attempts to eval- uate the gap conditions for various front-section Corresponding author: Department of Fluidics, Technical Uni- versity Dortmund, Leonhard-Euler-Strasse 5, D-44227 Dortmund, Germany. email:jan.hausertu-dortmund.de geometries 3. This approach has proved useful in spite of the two-dimensional (2D) viewpoint, and demonstrates that direct profi le modifi cation tak- ing into account application-orientated requirements seems to be effective. However, this approach does not provide direct comparability of 3D rotor geome- tries, as a pure profi le abstraction inevitably ignores spatial geometrical parameters. These exercise con- siderable infl uence on gap conditions, and thus on thermodynamic processes in the compressor. The aim of this study is to link geometrical param- eters with their effects in terms of performance. The interrelationships will be integrated in geometrical performance codes, which can then be used to reduce internal leakages and, to some extent, improve the energy conversion effi ciency of screw compressors. Ascertaining general trends for geometrical optimiza- tion will be carried out via a purely geometrical evaluation of the compressors, independent of any actual operating situation. In order to compare the geometries, it is only necessary to decide which mechanical and operational parameters it is advis- able to keep constant. It will be necessary to ascertain whethersimplegeometricaldataarecapableofreplac- ing extensive measurements and simulations in a fi rst assessment of the energy conversion effi ciency of differing compressor designs, for example, within the framework of a computer-aided optimization process. Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science 1399 at ZHEJIANG UNIVERSITY on March 28, 2012Downloaded from J Hauser and A Brmmer Fig.1 Gap situation of a screw compressor. Above: casing and front gap, below: profi le intermesh gap (left), blow hole (right) 2INTERRELATIONSHIPS BETWEEN GEOMETRICAL PARAMETERS The selection of the geometrical parameters of a com- pressor, such as rotor diameter, length, rotor wrap angle, and compression ratio, infl uences gap interac- tionandtheresultingdegreeofenergyconversion.The settingsforthecasing,front,andprofi leintermeshare responsible for the internal leakage characteristics of the compressor, and are mainly responsible for gap fl ow losses (Fig. 1). If gap fl ow losses are increased, effi ciency suffers. The types of gaps have different infl uences on the energy conversion effi ciency of a screwcompressor.Ageneralviewofgapprioritieswith regard to the relative rates of mass fl ow at the gaps with respect to the current pressure situation and the respective gap lengths is provided by 1,4 the following. 1. Profi le intermesh gap (between male and female rotor). 2. Casing gap at (depending on number of teeth): (a) male rotor; (b) female rotor (larger number of teeth than the male rotor). 3. Blow hole. 4. Front gap at: (a) high-pressure side; (b) low-pressure side. The order above refers basically to the overpres- sure area of a screw compressor. For a machine of this type used in mechanical compression applica- tions, this order of priorities is confi rmed by Kauder and Janicki 4. Results for gap priorities in expansion applications are not available at present. The energy conversion assessment is carried out via mechanical effi ciency as the principal performance fi gure. A fur- ther important item is the volumetric effi ciency for rotational displacement machines. The framework of the geometrical assessment does not include factors which have no direct infl uence on the rotor geometry. The level of volumetric effi - ciency thus represents the infl uence of the gap mass fl ows. Volumetric effi ciency is infl uenced by geomet- rical parameters such as the way rotor tooth counts are paired, length of the rotor, wrap angle, length diameter ratio, and the settings of the gap heights. In addition, profi le design plays a major role, because it is the form of the profi le which infl uences the main types of gap 3,4. Based on a reference compressor, the infl uence of the profi le form on the volumetric effi ciency of the compressor is variable. Every change in the profi le form directly infl uences the gap confi guration and the order of priority among the gaps, which in turn has a direct infl uence on leak- ages in the compressor. The fact that changing the profi le directly affects the size of the machine (i.e. the maximum delivery volume) should also be taken into Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science 1400 at ZHEJIANG UNIVERSITY on March 28, 2012Downloaded from Geometrical abstraction of screw compressors account. The smaller the delivery volume, the larger the effect of the gaps, as the increase in length is lin- ear, while the delivery volume increases at the power of three related to the size of the compressor. 3GAPS:A GEOMETRICAL EXAMINATION Within the framework of profi le development for dry- runningrotationaldisplacementmachines,geometri- cal performance fi gures are helpful in carrying out a comparative assessment of different profi les.This can be done using either thermodynamic or mechanical fl ow values. Ignoring rotor length and wrap angles of the rotor under examination, rotor profi les with 2D performance fi gures (e.g. with 2Dgap lengths in rela- tion to the scoop surface), have so far been physically characterized. However, the part played by the spatial gap lengths is not taken into account. It seems desir- able to transfer the geometrical compressor parame- ters (wrap angle and rotor length) to the resulting gap situation. Gap priority now depends on the settings of the gap height and the profi le contour itself. Conse- quently,variousapproachestoadescriptionofthegap conditions should be set up and validated within the framework of a computer-aided profi le optimization procedure. 3.1Geometrical gap situation An assessment of 3D gap interrelationships can be effectively represented by means of a rotor diagram (Fig. 2). The rotor position is determined by the lead- ing chamber, where the volume has just reached zero. As the infl uence of the front gap on the low-pressure side is small, this is not taken into account in the performance fi gures. A comparison of different wrap angles shows that as the angles increase, the num- bers and also the total lengths of the gaps increase. Wherethereisanidenticalpressureratio,greaterwrap angleswillresultinamoreconstantpressuregradient, which should result in higher volumetric effi ciency. This assessment only applies if there is a constant theoretical mass fl ow in comparable compressors. Gap conditions can be derived from the representa- tion of the rotors, as a single gap alteration (e.g. in the length or the height), changes its priority and its infl u- ence on volumetric effi ciency. Based on the reference compressor (compressor with subscript 11), simply combining values for the surfaces of the respective gaps produces an approximation of the performance relation ?1, because it is assumed that the volumet- ric effi ciency will fall in proportion to the gap area (equation(1).ThegapareaAGapisarrivedatbyadding the gap lengths, multiplied in each case by the gap height. Comparability of different compressors can be Male Rotor (MR)Female Rotor (FR) CG1 HP-Port CG2 CG3 CG1 CG2 CG3 CG4 IMC2 BH1 IMC1 CG Male Rotor (MR)Female Rotor (FR) CG1 HP-Port CG2 CG3 CG1 CG2 CG3 CG4 IMC3 BH2 CG5 BH1 IMC2 CG4 IMC1 CG Fig.2 Rotor intermesh diagram demonstrating the gap analysis for a specifi ed rotor position. Above: small wrap angle, below: large wrap angle. CG, casing gap; BH, blow hole; IMC, intermesh clearance; HP, high pressure Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science 1401 at ZHEJIANG UNIVERSITY on March 28, 2012Downloaded from J Hauser and A Brmmer achieved with reference to the delivery volume (i.e. depending on the number of teeth) ?1 (?1)11 = AGap (AGap)11 (VmaxzMR)11 VmaxzMR (1) with AGap= ? (hIMClIMC) + ?ABH + ? (hCGlCG) + ? (hFGlFG) ? MR + ? (hCGlCG) + ? (hFGlFG) ? FR This approach only provides a rough estimate of the volumetric effi ciency in an assessment of profi le char- acteristics with different wrap angles. This is because gap area does not necessarily change along with the wrap angle, whereas volumetric effi ciency develops in an approximately anti-proportional way. Gap infl uences tend to reduce as wrap angles increase, with the total number of gaps increasing. We can conclude from this that the gap count tends to develop in an approximately anti-proportional way compared with the gap area, so the equation can be extended as follows ?2 (?2)11 = ?1 (?1)11 (iGap)11 iGap (2) with iGap= ? (iIMC+ iBH+ (iCG+ iFG)MR+ (iCG+ iFG)FR) The performance code ?2thus represents the rela- tionship between a moderate gap area for all gaps of the compressor and the theoretical delivery volume. ThenumberofgapsiGapisarrivedatbyaddingtogether the total gaps for the machine. The total gap count broadly corresponds with the weighting of the gaps in a screw compressor. As the number of individual gap types does not change at the same rate (e.g. the wrap angle), it is desirable to carry out an assessment of the individual gap types. 3.2Assessment of gap types The previous calculations do not directly cater for gap areasdependentonwrapanglesandrotorlength.This meansthattheactualsignifi canceofthedifferentgaps is not taken into account. The performance fi gures will be augmented by internal and external weighting factors for the respective gap areas (3) ?3 (?3)11 = ? (Gap,e ? (Gap,iAGap,Type,i) ? (Gap,e ? (Gap,iAGap,Type,i)?11 (VmaxzMR)11 VmaxzMR (3) with Gap,e= ?AGap,Type AGap iGap iGap,Type andGap,i= AGap,Type,i ?AGap,Type Internalweightingfactorsarerevealedbyexamining a single gap, with changes in the machine parameters also leading to changes in the number of gaps, and also in the total area of the gap under examination. The factor Gap,i relates the specifi c area of a gap type to the total area of a gap type. At constant pressure, gaps with higher values have a positive effect on per- formance. External weighting factors for a particular gap type result from the machine gaps, evaluated via the gap area and count.The factorGap,etherefore rep- resentstherelationshipbetweenthemeanareaofeach gap of a particular type and the mean area of all gaps. This performance code therefore combines all impor- tant gap-geometrical and variable values, which vary according to the profi le form and intermesh charac- teristics. As by the creation of these codes the area of a gaptypeisenteredinquadraticform,thesurfacecom- ponent of the gap as a whole only in linear form, there is an extreme lack of proportion between the areas of the gap types, which results in an unsatisfactory representation of gap priorities. 3.3Evaluation of a single-chamber examination Theworkingchamber,whichismainlyresponsiblefor the compression process, exercises a decisive infl u- enceonvolumetriceffi ciency.Withanincreaseinwrap angle, the total area of the gap increases, but the gap area of the process chamber is further reduced. Con- sequently, an examination of the high-pressure (HP) chamber in the previously defi ned rotor position can help to provide further gap performance values. The generation of these values, referred to as ?1,OCM and ?3,OCM, is carried out in the same way as ?1and ?3. Code ?1,OCMevaluates only the gap surfaces of the HPchamber,anddoesnottakeintoaccountvariations in rotor length with suitably modifi ed wrap ratios (4) ?1,OCM (?1,OCM)11 = AGap,1 (AGap,1)11 (VmaxzMR)11 VmaxzMR (4) with AGap,1= AIMC,1+ ABH,1+ (ACG,1+ AFG,1)MR + (ACG,1+ AFG,1)FR The gap area AGap,1results from combining the indi- vidual gaps of the HP chamber, but this does not allow gap priorities to be ascertained. This infl uence is included via code ?3,OCM, with a weighting factor produced by the relationship between the HP side of the gap and the total area of the gap (equation (5). Withconstantwrapangles,increasingtherotorlength Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science 1402 at ZHEJIANG UNIVERSITY on March 28, 2012Downloaded from Geometrical abstraction of screw compressors inevitably leads to a more uniform pressure distri- bution throughout the machine, as there are more chambers between the high- and low-pressure sides ?3,OCM (?3,OCM)11 = ? (Gap,iAGap,Type,1) ? (Gap,iAGap,Type,1)?11 (VmaxzMR)11 VmaxzMR (5) with Gap,i= AGap,Type,1 ? AGap,Type The weighting factor Gap,ithus expresses the rela- tionship of the HP gap-type area to the total gap-type area of the machine. It seems desirable to work this out, as with constant wrap angles, tooth pairings with different numbers of teeth can be compared. As the numberofteethincreases,theweightingfactorGap,iis reduced, which corresponds to a reduction in the gap priority of the individual gap types, and a consequent improvement in volumetric effi ciency. All these performance fi gures are basically suitable for a qualitative assessment of machines based on their gap areas and values, but not for a quantita- tive representation of the volumetric effi ciency or the overall effi ciency of the compressor. They serve as a fi rst step in the relative geometrical assessment of dif- ferent compressor designs (e.g. in a computer-aided optimization process). 4APPLICATION IN PROFILE OPTIMIZATION After implementing the performance codes defi ned above, it is necessary to examine their validity in computer-aided profi le generation. For this purpose, an optimization strategy for screw rotor profi les using evolutionary approaches is employed 5. The repre- sentation of the rotor fl anks is carried out by means of anon-uniformrationalbasisspline(NURBS)curve6. The geometrical parameters of the reference machine are listed inTable 1. Further the general requirements for the optimiza- tion process are a sample size of 30 rotors and a maximum of 100000 optimization steps. The curve of Table 1Parametersofthereferencemachine(sub- script 11) Rotor length100mm Tooth relation (malefemale rotor)3/5 Wrap angle (malefemale rotor)200/120 Gap height setting0.1mm Parameters for optimization Male rotor profi le, curve with 12 control points, polynomial degree: 3, rolling circle fi xed, crown and root circles not fi xed the rotor profi le of a tooth is represented by a polyno- mial degree of 3, and 12 control points.The intermesh conditions for profi le generation follow the general gearing law. Because of the varying infl uences of the gap types on the general course of the process dur- ingthecompressionphase,thefrontrotorgapsonthe low-pressuresideareignoredduringthegenerationof t