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http:/ Engineering Science Engineers, Part C: Journal of Mechanical Proceedings of the Institution of Mechanical http:/ 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: http:/ 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 http:/ Alerts: http:/ http:/ http:/ http:/ What is This? - May 25, 2011Version of Record at ZHEJIANG UNIVERSITY on March 28, Downloaded from Geometricalabstractionofscrewcompressors forthermodynamicoptimization JHauser and ABrmmer 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 profiles is an important area in connection with the development of screw compressors for specific applications. Geometrical performance figures (using criteria to describe interdependencies of geometrical parameters for screw compressors) for profile optimization are used in order to achieve specific improvements in 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 take spatial parameters into account provide a better evaluation of gap conditions and operating efficiency of the compressors under examination. Keywords: screw compressor performance, profile optimization, new design concepts 1 INTRODUCTION 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 finding a compromise between pre- cision and calculation time, evaluation and analysis by means of abstracted geometrical performance fig- ures is a reasonable approach. The main feature of this approach is intended to be a reduction in develop- ment time by using geometrical codes to characterize the thermodynamic performance behaviour of a screw compressor. Within the field for developing geometrical per- formance figures, 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 profile modification 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 profile abstraction inevitably ignores spatial geometrical parameters. These exercise con- siderable influence 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 efficiency 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 whether simple geometrical data are capable of replac- ing extensive measurements and simulations in a first assessment of the energy conversion efficiency 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, Downloaded from JHauserandABrmmer Fig.1 Gap situation of a screw compressor. Above: casing and front gap, below: profile intermesh gap (left), blow hole (right) 2 INTERRELATIONSHIPSBETWEEN GEOMETRICALPARAMETERS The selection of the geometrical parameters of a com- pressor, such as rotor diameter, length, rotor wrap angle, and compression ratio, influences gap interac- tion and the resulting degree of energy conversion. The settings for the casing, front, and profile intermesh are responsible for the internal leakage characteristics of the compressor, and are mainly responsible for gap flow losses (Fig. 1). If gap flow losses are increased, efficiency suffers. The types of gaps have different influences on the energy conversion efficiency of a screw compressor. A general view of gap priorities with regard to the relative rates of mass flow at the gaps with respect to the current pressure situation and the respective gap lengths is provided by 1,4 the following. 1. Profile 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 confirmed 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 efficiency as the principal performance figure. A fur- ther important item is the volumetric efficiency for rotational displacement machines. The framework of the geometrical assessment does not include factors which have no direct influence on the rotor geometry. The level of volumetric effi- ciency thus represents the influence of the gap mass flows. Volumetric efficiency is influenced 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, profile design plays a major role, because it is the form of the profile which influences the main types of gap 3,4. Based on a reference compressor, the influence of the profile form on the volumetric efficiency of the compressor is variable. Every change in the profile form directly influences the gap configuration and the order of priority among the gaps, which in turn has a direct influence on leak- ages in the compressor. The fact that changing the profile 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, Downloaded from Geometricalabstractionofscrewcompressors 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. 3 GAPS:AGEOMETRICALEXAMINATION Within the framework of profile development for dry- running rotational displacement machines, geometri- cal performance figures are helpful in carrying out a comparative assessment of different profiles. This can be done using either thermodynamic or mechanical flow values. Ignoring rotor length and wrap angles of the rotor under examination, rotor profiles with 2D performance figures (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 profile contour itself. Conse- quently, various approaches to a description of the gap conditions should be set up and validated within the framework of a computer-aided profile optimization procedure. 3.1 Geometricalgapsituation 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 influence of the front gap on the low-pressure side is small, this is not taken into account in the performance figures. A comparison of different wrap angles shows that as the angles increase, the num- bers and also the total lengths of the gaps increase. Where there is an identical pressure ratio, greater wrap angles will result in a more constant pressure gradient, which should result in higher volumetric efficiency. This assessment only applies if there is a constant theoretical 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 the length or the height), changes its priority and its influ- ence on volumetric efficiency. 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 Pi1 1 , because it is assumed that the volumet- ric efficiency will fall in proportion to the gap area (equation (1). The gap area A Gap is arrived at by adding the gap lengths, multiplied in each case by the gap height. Comparability of different compressors can be Male Rotor (MR) Female Rotor (FR) CG 1 HP-Port CG 2 CG 3 CG 1 CG 2 CG 3 CG 4 IMC 2 BH 1 IMC 1 CG Male Rotor (MR) Female Rotor (FR) CG 1 HP-Port CG 2 CG 3 CG 1 CG 2 CG 3 CG 4 IMC 3 BH 2 CG 5 BH 1 IMC 2 CG 4 IMC 1 CG Fig.2 Rotor 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 pressure Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science 1401 at ZHEJIANG UNIVERSITY on March 28, Downloaded from JHauserandABrmmer achieved with reference to the delivery volume (i.e. depending on the number of teeth) Pi1 1 (Pi1 1 ) 11 = A Gap (A Gap ) 11 (V max z MR ) 11 V max z MR (1) with A Gap = summationdisplay (h IMC l IMC ) + Sigma1A BH + parenleftBig summationdisplay (h CG l CG ) + summationdisplay (h FG l FG ) parenrightBig MR + parenleftBig summationdisplay (h CG l CG ) + summationdisplay (h FG l FG ) parenrightBig FR This approach only provides a rough estimate of the volumetric efficiency in an assessment of profile char- acteristics with different wrap angles. This is because gap area does not necessarily change along with the wrap angle, whereas volumetric efficiency develops in an approximately anti-proportional way. Gap influences 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 Pi1 2 (Pi1 2 ) 11 = Pi1 1 (Pi1 1 ) 11 (i Gap ) 11 i Gap (2) with i Gap = summationdisplay (i IMC + i BH + (i CG + i FG ) MR + (i CG + i FG ) FR ) The performance code Pi1 2 thus represents the rela- tionship between a moderate gap area for all gaps of the compressor and the theoretical delivery volume. The number of gapsi Gap is arrived at by adding together 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. 3.2 Assessmentofgaptypes The previous calculations do not directly cater for gap areas dependent on wrap angles and rotor length. This means that the actual significance of the different gaps is not taken into account. The performance figures will be augmented by internal and external weighting factors for the respective gap areas (3) Pi1 3 (Pi1 3 ) 11 = summationtext ( Gap,e summationtext ( Gap,i A Gap,Type,i ) parenleftbigsummationtext ( Gap,e summationtext ( Gap,i A Gap,Type,i ) parenrightbig 11 (V max z MR ) 11 V max z MR (3) with Gap,e = Sigma1A Gap,Type A Gap i Gap i Gap,Type and Gap,i = A Gap,Type,i Sigma1A Gap,Type Internal weighting factors are revealed by examining 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 specific 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 factor Gap,e therefore rep- resents the relationship between the mean area of each 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 profile form and intermesh charac- teristics. As by the creation of these codes the area of a gap type is entered in quadratic form, the surface com- 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. 3.3 Evaluationofasingle-chamberexamination The working chamber, which is mainly responsible for the compression process, exercises a decisive influ- ence on volumetric efficiency.With an increase in wrap 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 defined rotor position can help to provide further gap performance values. The generation of these values, referred to as Pi1 1,OCM and Pi1 3,OCM , is carried out in the same way as Pi1 1 and Pi1 3 . Code Pi1 1,OCM evaluates only the gap surfaces of the HP chamber, and does not take into account variations in rotor length with suitably modified wrap ratios (4) Pi1 1,OCM (Pi1 1,OCM ) 11 = A Gap,1 (A Gap,1 ) 11 (V max z MR ) 11 V max z MR (4) with A Gap,1 = A IMC,1 + A BH,1 + (A CG,1 + A FG,1 ) MR + (A CG,1 + A FG,1 ) FR The gap area A Gap,1 results from combining the indi- vidual gaps of the HP chamber, but this does not allow gap priorities to be ascertained. This influence is included via code Pi1 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). With constant wrap angles, increasing the rotor length Proc. IMechE Vol. 225 Part C: J. Mechanical Engineering Science 1402 at ZHEJIANG UNIVERSITY on March 28, Downloaded from Geometricalabstractionofscrewcompressors inevitably leads to a more uniform pressure distri- bution throughout the machine, as there are more chambers between the high- and low-pressure sides Pi1 3,OCM (Pi1 3,OCM ) 11 = summationtext ( Gap,i A Gap,Type,1 ) parenleftbigsummationtext ( Gap,i A Gap,Type,1 ) parenrightbig 11 (V max z MR ) 11 V max z MR (5) with Gap,i = A Gap,Type,1 summationtext A Gap,Type The weighting factor Gap,i thus 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 number of teeth increases, the weighting factor Gap,i is reduced, which corresponds to a reduction in the gap priority of the individual gap types, and a consequent improvement in volumetric efficiency. All these performance figures 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 efficiency or the overall efficiency of the compressor. They serve as a first step in the relative geometrical assessment of dif- ferent compressor designs (e.g. in a computer-aided optimization process). 4 APPLICATIONINPROFILEOPTIMIZATION After implementing the performance codes defined above, it is necessary to examine their validity in computer-aided profile generation. For this purpose, an optimization strategy for screw rotor profiles using evolutionary approaches is employed 5. The repre- sentation of the rotor flanks is carried out by means of a non-uniform rational basis spline (NURBS) curve 6. The geometrical parameters of the reference machine are listed in Table 1. Further the general requirements for the optimiza- tion process are a sample size of 30 rotors and a maximum of 100 000 optimization steps. The curve of Table 1 Parameters of the reference machine (sub- script 11) Rotor length 100 mm Tooth relation (malefemale rotor) 3/5 Wrap angle (malefemale rotor) 200 /120 Gap height setting 0.1 mm Parameters for optimization Male rotor profile, curve with 12 control points, polynomial degree: 3, rolling circle fixed, crown and root circles not fixed the rotor profile of a tooth is represented by a polyno- mial degree of 3, and 12 control points. The intermesh conditions for profile generation follow the general gearing law. Because of the varying influences of the gap types on the general course of the process dur- ing the compression phase, the front rotor gaps on the low-pressure side are ignored during the generation of the performance figures. These only have a marginal influence on the volumetric efficiency of the compres- sor compared to the other gap types which are taken into account. In order to check the validity of the figures, the opti- mization process deliberately began with a reference profile which diverged very considerably from a mod- ern standard profile, see Fig. 3. This is reflected in the very large relative area of the blow hole. The task was to check whether the minimized performance figures in use would modify the profile generation towards mod- ern rotor profiles, changing the relation between the gap areas to bring them in line with normal relations in a modern screw compressor. The optimized profiles can be seen in Fig. 3. The assessment of the relevant data Pi1 1 to Pi1 3,OCM was carried out by evaluating the percentage change in the gap area in relation to the volumetric efficiency of the compressor, see Fig. 4. Compared with the refer- ence machine, which has its gap characteristics set at 100 per cent, basic differences between the individual figures can be seen. The optimization results show that the profile lengths are increased by up to 10 per cent in all cases, while the area of the blow hole is considerably reduced. Minimizing Pi1 1 reduces the blow-hole area by c. 65 per cent, the second code by c. 70 per cent while the last code reduces it by up to 90 per cent compared to the reference machine. It is clear that the profile intermesh gap has opposing characteristics to the blow-hole area, because reducing the blow-hole area basically results in enlarging the intermesh clearance. This does not necessarily mean that there is a linear interconnec- tion between these gaps, as code Pi1 2 allows a greater change in the intermesh profile in relation to Pi1 3,OCM , resulting only in a smaller per