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Geometricalperformance figures (using criteria to describe interdependencies of geometrical parameters forscrew compressors) for profile optimization are used in order to achieve specific improvementsin performance. During this process, rotor profiles and spatial parameters are the main factors.Compared to data derived from the front section of rotor profiles, these figures which also takespatial parameters into account provide a better evaluation of gap conditions and operatingefficiency of the compressors under examination.Keywords: screw compressor performance, profile optimization, new design concepts1INTRODUCTIONThe operational behaviour of a screw compressorcan be examined experimentally but also by meansof comprehensive simulation procedures 1,2. Thecomplexity 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 istherefore undesirable, despite leading to valid results.With a view to finding a compromise between pre-cision and calculation time, evaluation and analysisby means of abstracted geometrical performance fig-uresisareasonableapproach.Themainfeatureofthisapproach is intended to be a reduction in develop-ment time by using geometrical codes to characterizethethermodynamicperformancebehaviourofascrewcompressor.Within the field for developing geometrical per-formance figures, there have been attempts to eval-uate the gap conditions for various front-sectionCorresponding author: Department of Fluidics, Technical Uni-versity Dortmund, Leonhard-Euler-Strasse 5, D-44227 Dortmund,Germany.email:jan.hausertu-dortmund.degeometries 3. This approach has proved useful inspite of the two-dimensional (2D) viewpoint, anddemonstrates that direct profile modification tak-ing into account application-orientated requirementsseems to be effective. However, this approach doesnot provide direct comparability of 3D rotor geome-tries, as a pure profile abstraction inevitably ignoresspatial geometrical parameters. These exercise con-siderable influence on gap conditions, and thus onthermodynamic processes in the compressor.The aim of this study is to link geometrical param-eters with their effects in terms of performance. Theinterrelationships will be integrated in geometricalperformance codes, which can then be used to reduceinternal leakages and, to some extent, improve theenergy conversion efficiency of screw compressors.Ascertaining general trends for geometrical optimiza-tion will be carried out via a purely geometricalevaluation of the compressors, independent of anyactual operating situation. In order to compare thegeometries, it is only necessary to decide whichmechanical and operational parameters it is advis-able to keep constant. It will be necessary to ascertainwhethersimplegeometricaldataarecapableofreplac-ing extensive measurements and simulations in afirst assessment of the energy conversion efficiencyof differing compressor designs, for example, withinthe framework of a computer-aided optimizationprocess.Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science1399 at ZHEJIANG UNIVERSITY on March 28, Downloaded from J Hauser and A BrmmerFig.1Gap situation of a screw compressor. Above: casing and front gap, below: profile intermeshgap (left), blow hole (right)2INTERRELATIONSHIPS BETWEENGEOMETRICAL PARAMETERSThe selection of the geometrical parameters of a com-pressor, such as rotor diameter, length, rotor wrapangle, and compression ratio, influences gap interac-tionandtheresultingdegreeofenergyconversion.Thesettingsforthecasing,front,andprofileintermeshareresponsible for the internal leakage characteristics ofthe compressor, and are mainly responsible for gapflow losses (Fig. 1). If gap flow losses are increased,efficiency suffers. The types of gaps have differentinfluences on the energy conversion efficiency of ascrewcompressor.Ageneralviewofgapprioritieswithregard to the relative rates of mass flow at the gaps with respect to the current pressure situation andthe respective gap lengths is provided by 1,4 thefollowing.1. Profile intermesh gap (between male and femalerotor).2. Casing gap at (depending on number of teeth):(a) male rotor;(b) female rotor (larger number of teeth than themale 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 ofthis type used in mechanical compression applica-tions, this order of priorities is confirmed by Kauderand Janicki 4. Results for gap priorities in expansionapplications are not available at present. The energyconversion assessment is carried out via mechanicalefficiency as the principal performance figure. A fur-ther important item is the volumetric efficiency forrotational displacement machines.The framework of the geometrical assessment doesnot include factors which have no direct influenceon the rotor geometry. The level of volumetric effi-ciency thus represents the influence of the gap massflows. Volumetric efficiency is influenced by geomet-rical parameters such as the way rotor tooth countsare paired, length of the rotor, wrap angle, lengthdiameter ratio, and the settings of the gap heights.In addition, profile design plays a major role,because it is the form of the profile which influencesthe main types of gap 3,4. Based on a referencecompressor, the influence of the profile form on thevolumetric efficiency of the compressor is variable.Every change in the profile form directly influencesthe gap configuration and the order of priority amongthe gaps, which in turn has a direct influence on leak-ages in the compressor. The fact that changing theprofile directly affects the size of the machine (i.e. themaximum delivery volume) should also be taken intoProc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science1400 at ZHEJIANG UNIVERSITY on March 28, Downloaded from Geometrical abstraction of screw compressorsaccount. The smaller the delivery volume, the largerthe effect of the gaps, as the increase in length is lin-ear, while the delivery volume increases at the powerof three related to the size of the compressor.3GAPS:A GEOMETRICAL EXAMINATIONWithin the framework of profile development for dry-runningrotationaldisplacementmachines,geometri-cal performance figures are helpful in carrying out acomparative assessment of different profiles.This canbe done using either thermodynamic or mechanicalflow values. Ignoring rotor length and wrap angles ofthe rotor under examination, rotor profiles with 2Dperformance figures (e.g. with 2Dgap lengths in rela-tion to the scoop surface), have so far been physicallycharacterized. However, the part played by the spatialgap 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 gapsituation. Gap priority now depends on the settingsof the gap height and the profile contour itself. Conse-quently,variousapproachestoadescriptionofthegapconditions should be set up and validated within theframework of a computer-aided profile optimizationprocedure.3.1Geometrical gap situationAn assessment of 3D gap interrelationships can beeffectively 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 influence of the front gap on the low-pressureside is small, this is not taken into account in theperformance figures. A comparison of different wrapangles shows that as the angles increase, the num-bers and also the total lengths of the gaps increase.Wherethereisanidenticalpressureratio,greaterwrapangleswillresultinamoreconstantpressuregradient,which should result in higher volumetric efficiency.This assessment only applies if there is a constanttheoretical mass flow in comparable compressors.Gap conditions can be derived from the representa-tion of the rotors, as a single gap alteration (e.g. in thelength or the height), changes its priority and its influ-ence on volumetric efficiency. Based on the referencecompressor (compressor with subscript 11), simplycombining values for the surfaces of the respectivegaps produces an approximation of the performancerelation ?1, because it is assumed that the volumet-ric efficiency will fall in proportion to the gap area(equation(1).ThegapareaAGapisarrivedatbyaddingthe gap lengths, multiplied in each case by the gapheight. Comparability of different compressors can beMale Rotor (MR)Female Rotor (FR)CG1HP-PortCG2CG3CG1CG2CG3CG4IMC2BH1IMC1 CGMale Rotor (MR)Female Rotor (FR)CG1HP-PortCG2CG3CG1CG2CG3CG4IMC3BH2CG5BH1IMC2CG4IMC1 CGFig.2Rotor intermesh diagram demonstrating the gap analysis for a specified rotor position.Above: small wrap angle, below: large wrap angle. CG, casing gap; BH, blow hole; IMC,intermesh clearance; HP, high pressureProc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science1401 at ZHEJIANG UNIVERSITY on March 28, Downloaded from J Hauser and A Brmmerachieved with reference to the delivery volume (i.e.depending on the number of teeth)?1(?1)11=AGap(AGap)11(VmaxzMR)11VmaxzMR(1)withAGap=?(hIMClIMC) + ?ABH+?(hCGlCG) +?(hFGlFG)?MR+?(hCGlCG) +?(hFGlFG)?FRThis approach only provides a rough estimate of thevolumetric efficiency in an assessment of profile char-acteristics with different wrap angles. This is becausegap area does not necessarily change along with thewrap angle, whereas volumetric efficiency develops inan approximately anti-proportional way.Gap influences tend to reduce as wrap anglesincrease, with the total number of gaps increasing.We can conclude from this that the gap count tendsto develop in an approximately anti-proportional waycompared with the gap area, so the equation can beextended as follows?2(?2)11=?1(?1)11(iGap)11iGap(2)withiGap=?(iIMC+ iBH+ (iCG+ iFG)MR+ (iCG+ iFG)FR)The performance code ?2thus represents the rela-tionship between a moderate gap area for all gaps ofthe compressor and the theoretical delivery volume.ThenumberofgapsiGapisarrivedatbyaddingtogetherthe total gaps for the machine. The total gap countbroadly corresponds with the weighting of the gaps ina screw compressor. As the number of individual gaptypes does not change at the same rate (e.g. the wrapangle), it is desirable to carry out an assessment of theindividual gap types.3.2Assessment of gap typesThe previous calculations do not directly cater for gapareasdependentonwrapanglesandrotorlength.Thismeansthattheactualsignificanceofthedifferentgapsis not taken into account. The performance figureswill be augmented by internal and external weightingfactors for the respective gap areas (3)?3(?3)11=?(Gap,e?(Gap,iAGap,Type,i)?(Gap,e?(Gap,iAGap,Type,i)?11(VmaxzMR)11VmaxzMR(3)withGap,e=?AGap,TypeAGapiGapiGap,TypeandGap,i=AGap,Type,i?AGap,TypeInternalweightingfactorsarerevealedbyexamininga single gap, with changes in the machine parametersalso leading to changes in the number of gaps, andalso in the total area of the gap under examination.The factor Gap,irelates the specific area of a gap typeto 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 particulargap type result from the machine gaps, evaluated viathe gap area and count.The factorGap,etherefore rep-resentstherelationshipbetweenthemeanareaofeachgap 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 varyaccording to the profile form and intermesh charac-teristics. As by the creation of these codes the area of agaptypeisenteredinquadraticform,thesurfacecom-ponent of the gap as a whole only in linear form, thereis an extreme lack of proportion between the areasof the gap types, which results in an unsatisfactoryrepresentation of gap priorities.3.3Evaluation of a single-chamber examinationTheworkingchamber,whichismainlyresponsibleforthe compression process, exercises a decisive influ-enceonvolumetricefficiency.Withanincreaseinwrapangle, the total area of the gap increases, but the gaparea of the process chamber is further reduced. Con-sequently, an examination of the high-pressure (HP)chamber in the previously defined rotor position canhelp to provide further gap performance values.The generation of these values, referred to as ?1,OCMand ?3,OCM, is carried out in the same way as ?1and?3. Code ?1,OCMevaluates only the gap surfaces of theHPchamber,anddoesnottakeintoaccountvariationsin rotor length with suitably modified wrap ratios (4)?1,OCM(?1,OCM)11=AGap,1(AGap,1)11(VmaxzMR)11VmaxzMR(4)withAGap,1= AIMC,1+ ABH,1+ (ACG,1+ AFG,1)MR+ (ACG,1+ AFG,1)FRThe gap area AGap,1results from combining the indi-vidual gaps of the HP chamber, but this does notallow gap priorities to be ascertained. This influenceis included via code ?3,OCM, with a weighting factorproduced by the relationship between the HP side ofthe gap and the total area of the gap (equation (5).Withconstantwrapangles,increasingtherotorlengthProc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science1402 at ZHEJIANG UNIVERSITY on March 28, Downloaded from Geometrical abstraction of screw compressorsinevitably leads to a more uniform pressure distri-bution throughout the machine, as there are morechambers between the high- and low-pressure sides?3,OCM(?3,OCM)11=?(Gap,iAGap,Type,1)?(Gap,iAGap,Type,1)?11(VmaxzMR)11VmaxzMR(5)withGap,i=AGap,Type,1?AGap,TypeThe weighting factor Gap,ithus expresses the rela-tionship of the HP gap-type area to the total gap-typearea of the machine. It seems desirable to work thisout, as with constant wrap angles, tooth pairings withdifferent numbers of teeth can be compared. As thenumberofteethincreases,theweightingfactorGap,iisreduced, which corresponds to a reduction in the gappriority of the individual gap types, and a consequentimprovement in volumetric efficiency.All these performance figures are basically suitablefor a qualitative assessment of machines based ontheir gap areas and values, but not for a quantita-tive representation of the volumetric efficiency or theoverall efficiency of the compressor. They serve as afirst step in the relative geometrical assessment of dif-ferent compressor designs (e.g. in a computer-aidedoptimization process).4APPLICATION IN PROFILE OPTIMIZATIONAfter implementing the performance codes definedabove, it is necessary to examine their validity incomputer-aided profile generation. For this purpose,an optimization strategy for screw rotor profiles usingevolutionary approaches is employed 5. The repre-sentation of the rotor flanks is carried out by means ofanon-uniformrationalbasisspline(NURBS)curve6.The geometrical parameters of the reference machineare listed inTable 1.Further the general requirements for the optimiza-tion process are a sample size of 30 rotors and amaximum of 100000 optimization steps. The curve ofTable 1Parametersofthereferencemachine(sub-script 11)Rotor length100mmTooth relation (malefemale rotor)3/5Wrap angle (malefemale rotor)200/120Gap height setting0.1mmParameters for optimizationMale rotor profile, curvewith 12 control points,polynomial degree:3, rolling circle fixed,crown and root circlesnot fixedthe rotor profile of a tooth is represented by a polyno-mial degree of 3, and 12 control points.The intermeshconditions for profile generation follow the generalgearing law. Because of the varying influences of thegap types on the general course of the process dur-ingthecompressionphase,thefrontrotorgapsonthelow-pressuresideareignoredduringthegenerationofthe performance figures. These only have a marginalinfluenceonthevolumetricefficiencyofthecompres-sor compared to the other gap types which are takeninto account.In order to check the validity of the figures, the opti-mization process deliberately began with a referenceprofile which diverged very considerably from a mod-ern standard profile, see Fig. 3. This is reflected in theverylargerelativeareaoftheblowhole.Thetaskwastocheck whether the minimized performance figures inusewouldmodifytheprofilegenerationtowardsmod-ern rotor profiles, changing the relation between thegap areas to bring them in line with normal relationsinamodernscrewcompressor.Theoptimizedprofilescan be seen in Fig. 3.The assessment of the relevant data ?1to ?3,OCMwas carried out by evaluating the percentage changein the gap area in relation to the volumetric efficiencyofthecompressor,seeFig.4.Comparedwiththerefer-ence machine, which has its gap characteristics set at100 per cent, basic differences between the individualfigures can be seen.The optimization results show that the profilelengths are increased by up to 10 per cent in all cases,whiletheareaoftheblowholeisconsiderablyreduced.Minimizing ?1reduces the blow-hole area by c. 65 percent, the second code by c. 70 per cent while the lastcode reduces it by up to 90 per cent compared to thereferencemachine.Itisclearthattheprofileintermeshgaphasopposingcharacteristicstotheblow-holearea,because reducing the blow-hole area basically resultsin enlarging the intermesh clearance. This does notnecessarily mean that there is a linear interconnec-tion between these gaps, as code ?2allows a greaterchange in the intermesh profile in relation to ?3,OCM,resulting only in a smaller percentage gap change inthe blow hole.The operating code, which did not significantlyreduce the area of the blow hole, shows largerchangesin the casing gap instead. This can be explained bythe increased crown circle diameter of the rotors.Thisresults in a reduction in the area of the rotor front gap.Code?3,OCMreducestheareacomponent,particularlyon the female rotor side, as a result of a reduction inthefemalecrowncircleandalsoofanarrowerprofileofthe female rotor itself. Using code ?3,OCMa maximumreduction in the blow-hole area can only be achievedby configuring a narrow, pointed female rotor form.The percentage area distribution for each of thegap types in the compressor is shown in Fig. 5. Theprofile form of the reference machine has a gap-typeProc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science1403 at ZHEJIANG UNIVERSITY on March 28, Downloaded from J Hauser and A BrmmerFig.3Representation of the reference profile and the solutions provided by the profile optimiza-tion using the operational data. Left: ?1, middle: ?2, right: ?3,OCM(black), with referenceprofile (grey)0%20%40%60%80%100%120%gap area /delivery volume %123,OCMreference machineIntermeshclearance Blow Hole Casing Gap(male rotor)Front Gap (male rotor)Casing Gap(female rotor)Front Gap(female rotor)F