可變壓縮比渦卷式壓縮機之研究

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(1)國立交通大學機械工程學系. 博士論文. 可變壓縮比渦卷式壓縮機之研究 Research on Scroll Compressors with Variable Compression Ratio. 研究生：劉陽光指導教授：洪景華教授. 中華民國九十九年六月.

(2) 可變壓縮比渦卷式壓縮機之研究 Research on Scroll Compressors with Variable Compression Ratio. 研究生: 劉陽光. Student: Yangguang Liu. 指導教授: 洪景華. Advisor: Chinghua Hung. 國立交通大學機械工程學系博士論文. A Dissertation Submitted to Department of Mechanical Engineering College of Engineering National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mechanical Engineering June 2010 Hsinchu, Taiwan, Republic of China. 中華民國九十九年六月.

(3) 可變壓縮比渦卷式壓縮機之研究研究生: 劉陽光. 指導教授: 洪景華. 國立交通大學機械工程學系博士班. 摘要對高效能與可變負載之冷凍空調系統而言，為因應其變動負載之需求，搭配變頻馬達之可變壓縮比渦卷式壓縮機已成為目前發展之主流。本研究第一步先探討渦卷之旁通機構，此為裝於固定渦卷背端之閥機構，用於防止當負載變動時因為渦卷幾何線形之固定容積比限制而在壓縮腔內產生之過壓縮；且其亦有避免冷媒剛進入壓縮腔時因過熱度不足而造成的液阻卻。將此步驟發展之旁通機構數學模型整合於一渦卷式壓縮機模擬軟體中，並針對高壓與低壓外殻之壓縮機架構分別進行模擬分析。結果發現不論在何種負載下，設置之旁通孔皆能有效地排除過壓縮與液阻卻。而配合開發中的渦卷式壓縮機也藉由實驗結果驗證此機構在不同負載條件下，能有效地提升壓縮機之效率。另一方面，針對變動負載之設計需求，也整合最佳化流程於上述渦卷壓縮機模擬軟體中，以輔助產品開發時之參數最佳化設計。並對降低摩擦功之軸承尺寸最佳化問題加以研究，藉最佳化之流程而獲得在考量強度與液動壓潤滑限制條件下，軸承尺寸的最佳參數組合，以作為產品開發時之設計依據。此外，針對提高渦卷對之壓縮比(亦即提高線形之容積比)以匹配更寬廣的負載需求方面，本研究也發展可變基圓半徑圓漸開線之渦卷線形的數學模型，並以此發展一種中心較厚且由內向外厚度漸薄之圓漸開線渦卷渦片，其強度與剛性於高溫與高壓負載條件下極具優勢。同時也驗證了圓弧與直線此兩種用於傳統渦卷對以修整中心干涉並達到完美嚙合的數學模型，也可應用於此可變基圓半徑圓漸開線之渦卷線形上，進而達成嚙合修正，同時亦可提高渦卷對之壓縮比。. i.

(4) Research on Scroll Compressors with Variable Compression Ratio. Student: Yangguang Liu. Advisor: Dr. Chinghua Hung. Department of Mechanical Engineering National Chiao Tung University. ABSTRACT A scroll-type compressor (STC) with variable compression ratio using a variable-speed motor has become a major trend in recent years because of the requirements for variable capacities in high efficient refrigeration and air-conditioning systems. This study first develops a bypass valves mechanism which is disposed to the backside of a fixed scroll of STC. The over-compression resulted from the mismatch between the intrinsic volume ratio (the designed compression ratio) in STC and the variable operating pressure ratio (variable capacity) can be prevented by using this mechanism. In addition, this mechanism can also avoid the liquid slug while the refrigerant flows into the chamber of STC without enough superheat for fully vaporized. The mathematical model for the bypass valves has been constructed and integrated into a STC simulation computer package, and the high and low side structures of STC are also simulated respectively. Simulation results find that the subtle disposition of bypass valves can prevent the over-compression and liquid slug problems regardless various operating conditions. One practical STC product, with bypass valves built in it, has also validated the effect through the actual STC efficiency experiments. In addition, an optimization module had been combined into the simulation package for aiding the design of STC with variable capacities and speeds. One application regarding reducing frictional losses in bearing components of STC is demonstrated and discussed in this dissertation. By using the numerical optimization procedure, the optimum parametric combinations in bearings can be found with considerations of hydrodynamic lubrication and strength constraints. It is expected that this procedure can become the base in developing STC ii.

(5) product design. In order to raise the compression ratio for the scroll pair (i.e. to raise the intrinsic volume ratio) to match even wider loading conditions, a new geometric model of the scroll profiles constructed from an involute of circle with variable radii has been built. One kind of these profiles with greater wrap thickness at the center and decreasing thickness from the inside outwards can provide better strength and rigidity. Finally, this study also demonstrates that the arc and line modifications, used in the conventional scroll profile, can be also applied to this new scroll profile. The two most important purposes, avoiding interference at the center of the scroll pair and boosting the intrinsic volume ratio to match the specified operating conditions, are both achieved by properly choosing one of these two modifications according to design requirements.. iii.

(6) 誌謝在交大已經待了七年，經歷兩千五百多個日子終於能畢業了。當然首先感謝的就是引領我踏入交大的碩士班恩師曾錦煥老師。督促學業之外，老師改變了自信心不足和單線思考的我，激發我對研究的興趣和對知識的渴求，使我在面對問題時能更多面向地思考，進而提出豐富的創意與手段來解決它。在曾老師辭世後的那段日子，博一剛入學的我，更想過是否要放棄學業而進入職場。在這裡深深感謝博士班恩師洪景華老師，辛苦地延續指導我的工作，不斷地給我鼓勵和鞭策。除了幫助我提升在專業領域的學養和知識外，更針對進行博士研究所應具備的精神和態度傾囊相授。在這我想向老師致上我最深的感謝，謝謝老師。此外，我也特別感謝張鈺炯學長在這段期間教導我這個晚輩，傳授在產品設計、系統測試等專業知識，以及科技結合人文與時間管理等觀念。謝謝學長。感謝我的口試委員，蔡忠杓老師、宋震國老師、陳俊勳老師、陳申岳老師以及林聰穎老師。謝謝老師們對學生論文的建議和教導，讓學生能進一步改進論文，提升到更好的層次。謝謝實驗室的各位師兄弟姐妹對我的照顧和幫助，也感謝齒輪實驗室的成員們這幾年與我同在一間研究室中互相砥礪和提攜。同時也特別感謝工研院能環所的先進們，不管是待人接物和研究工作上，對我的指導和照顧，使我獲益良多。謝謝我的父母和家人對我的支持讓我能完成這個目標，在我面對挫折和徬徨失據時成為我的依靠，謝謝你們。最後也謝謝我的愛人，在博士班這段期間始終沒有出現，讓我能專心一致地完成研究。感謝我所愛以及愛我的人，謝謝大家。. iv.

(7) LIST OF CONTENTS 摘要 ...........................................................................................................................................i ABSTRACT ..............................................................................................................................ii 誌謝 .........................................................................................................................................iv LIST OF CONTENTS..............................................................................................................v LIST OF TABLES .................................................................................................................viii LIST OF FIGURES .................................................................................................................ix NOMENCLATURE.................................................................................................................xi CHAPTER 1 INTRODUCTION.............................................................................................1 1.1 Scroll-type compressor (STC)......................................................................................1 1.2 Simulation program for STC.......................................................................................4 1.3 Variable speed and compression ratio.........................................................................5 1.4 Design optimization for variable compression ratio...................................................6 1.5 Scroll profiles based on an involute of circle with variable radii ..............................6 1.6 Dissertation Scope .......................................................................................................7 CHAPTER 2 STC SIMULATION MODEL AND PROCESS .............................................8 2.1 Developments in mathematical models of STC ..........................................................8 2.1.1 Geometry of scroll profiles................................................................................8 2.1.2 Thermodynamics in the scroll pair ....................................................................9 2.1.3 Dynamics of components and bearing models ..................................................9 2.2 Geometrical model of STC ........................................................................................10 2.2.1 Scroll profile design by involute of circle with a fixed radius ........................10 2.2.2 Derivation of volume and its ratio of the scroll chambers ..............................12 2.3 Thermodynamic model ..............................................................................................13 2.3.1 Refrigerant property ........................................................................................13 2.3.2 Suction and discharge superheating calculations ............................................14 2.3.3 Compression and discharge process ................................................................16 2.3.4 Leakage flows..................................................................................................16 2.4 Dynamics in mechanical components and mechanisms in STC .............................18 2.4.1 Dynamic forces and moments .........................................................................18 2.4.2 Counterweight analysis ...................................................................................20 2.5 Frictional losses in mechanical components............................................................21 2.5.1 Thrust bearing..................................................................................................21 v.

(8) 2.5.2 Journal bearings...............................................................................................24 2.6 Simulation process in the STC computer model.......................................................24 2.6.1 Simulation process...........................................................................................25 2.6.2 Output results...................................................................................................26 2.7 Summarization of the STC simulation model ..........................................................28 CHAPTER 3 MATHEMATICAL MODEL OF BYPASS VALVES ..................................29 3.1 Developing history regarding bypass valves .............................................................29 3.2 Geometry of bypass holes ..........................................................................................31 3.2.1 Geometry of bypass holes ...............................................................................31 3.2.2 Uncovered area of the bypass holes ................................................................34 3.2.3 Corresponding chambers to bypass holes........................................................34 3.3 Bypass valve model ....................................................................................................34 3.4 Simulation model .......................................................................................................35 3.5 Remarks .....................................................................................................................37 CHAPTER 4 SIMULATION FOR BYPASS VALVES IN STC.........................................38 4.1 Simulations with low-side structure..........................................................................38 4.1.1 Geometric observation.....................................................................................38 4.1.2 Simulation results during working cycle .........................................................38 4.1.3 Simulation results of two different arrangements............................................41 4.2 Simulations with high-side structure ........................................................................45 4.2.1 Effect of bypass valves behavior .....................................................................47 4.2.2 Effect of bypass valves in thermodynamics ....................................................50 4.3 Verification of bypass mechanism.............................................................................51 4.3.1 Experimental apparatus ...................................................................................51 4.3.2 Verification of bypass action ...........................................................................55 4.4 Remarks .....................................................................................................................55 CHAPTER 5 OPTIMIZATION PROCEDURE APPLIED IN STC MODEL .................56 5.1 Researches about optimum design of STC ...............................................................56 5.2 Verification of simulation results and experiments ..................................................56 5.3 Optimization solver ....................................................................................................60 5.3.1 Formulation of optimization problem .............................................................61 5.3.2 Optimization procedure ...................................................................................61 5.4 Case study: optimization in reducing frictional losses.............................................62 5.4.1 Selection of objective function and design variables ......................................62 vi.

(9) 5.4.2 Selection of constraint conditions....................................................................63 5.5 Case study: optimum results and discussions...........................................................64 5.5.1 Selection of testing conditions.........................................................................64 5.5.2 Optimum results ..............................................................................................64 5.5.3 Discussions ......................................................................................................67 5.6 Remarks .....................................................................................................................68 CHAPTER 6 INVESTIGATION ON INVOLUTE OF CIRCLE WITH VARIABLE RADII IN A STC.....................................................................................................................69 6.1 Various studies for scroll profiles ..............................................................................69 6.2 Geometric model of the scroll profile constructed from an involute of circle with variable radii ....................................................................................................................70 6.2.1 Theorem of planar orbiting mechanisms .........................................................70 6.2.2 Conceptual illustration and formulations ........................................................71 6.3 Application for different k values and volume calculations.....................................75 6.3.1 Different k values ............................................................................................75 6.3.2 Volume calculations.........................................................................................79 6.4 Parametric study and discussion...............................................................................80 6.4.1 Fixed suction volume, volume ratio and housing size.....................................80 6.4.2 Fixed suction volume, volume ratio and wrap height .....................................83 6.5 Remarks .....................................................................................................................84 CHAPTER 7 PERFECTLY MESHING MODIFICATION FOR INVOLUTE OF CIRCLE WITH VARIABLE RADII ....................................................................................85 7.1 Introduction of modifications at the center of the scroll pair ..................................85 7.2 Arc modification of involutes of circle with variable radii.......................................85 7.2.1 Formulation .....................................................................................................86 7.2.2 Discussion........................................................................................................87 7.3 Line modification of involutes of circle with variable radii .....................................89 7.3.1 Formulation .....................................................................................................89 7.3.2 Discussion........................................................................................................90 7.4 Remarks .....................................................................................................................91 CHAPTER 8 CONCLUSIONS AND FUTURE WORKS ..................................................93 8.1 Conclusions................................................................................................................93 8.2 Future works..............................................................................................................95 REFERENCES .......................................................................................................................97 vii.

(10) LIST OF TABLES Table. 4-1 Parameters of the STC.............................................................................39 Table. 4-2 Parameters of bypass hole .......................................................................39 Table. 4-3 Parameters of bypass hole: Case (A).......................................................42 Table. 4-4 Parameters of bypass hole: Case (B).......................................................43 Table. 4-5 Parameters of bypass hole: Case (C).......................................................44 Table. 4-6 Parameters of the STC.............................................................................46 Table. 4-7 STC operating conditions........................................................................46 Table. 4-8 Comparisons of sim. & exp. results of m& , η V & ηC ..........................55 Table. 5-1 Parameters and operating conditions of the STC ....................................58 Table. 5-2 Specifications of calorimeter for measuring STC performance ..............59 Table. 5-3 Comparison of sim. & exp. results in different conditions......................59 Table. 5-4 Bounds of design variables and constraints ............................................62 Table. 5-5 Optimized results with design variables..................................................65 Table. 6-1 Combinations of design parameters and related results ..........................80 Table. 6-2 ODP/GWP values to several refrigerants ................................................82 Table. 6-3 Combinations of design parameters and related results ..........................83. viii.

(11) LIST OF FIGURES Fig. 1-1 A complete cycle of STC: suctionÆcompressionÆdischarge .....................2 Fig. 1-2 A schematic of a STC....................................................................................3 Fig. 2-1 Parametric definition for the involute of circle...........................................10 Fig. 2-2 Inner and outer Profiles of fixed scroll .......................................................12 Fig. 2-3 Illustration of “Refprop7” linking to STC simulation package ..................14 Fig. 2-4 Low-side and High side structure of STC [1].............................................15 Fig. 2-5 Illustration of end-side leakage...................................................................17 Fig. 2-6 Illustration of flank leakage ........................................................................17 Fig. 2-7 Dynamic forces of the orbiting scroll .........................................................19 Fig. 2-8 Dynamic balance of the driving shaft .........................................................20 Fig. 2-9 Schematic of the three journal bearings and the thrust bearing ..................21 Fig. 2-10 Parameters of the thrust bearing ...............................................................22 Fig. 2-11 Nominal height of oil-film ........................................................................23 Fig. 2-12 Motor torque-efficiency-power for specified speed .................................24 Fig. 2-13 Flowchart of the simulation process in STC package...............................25 Fig. 2-14 Flowchart of thermodynamics with leakage model in STC package .......26 Fig. 3-1 Bypass holes during one orbiting scroll [41] ..............................................30 Fig. 3-2 Speed, volume vs. pressure ratio & bypass design [42] .............................31 Fig. 3-3 Scheme of bypass holes (a) range of positions (b) relations from fixed scroll (c) relations from orbiting scroll.............................................................33 Fig. 3-4 Uncovered areas of the orbiting scroll and the bypass hole........................35 Fig. 3-5 The flowchart of the STC simulation process.............................................36 Fig. 3-6 The flowchart of leakage and bypass model in STC ..................................37 Fig. 4-1 Position of bypass holes..............................................................................39 Fig. 4-2 Uncovered interval of bypass holes ............................................................40 Fig. 4-3 Working cycle .............................................................................................40 Fig. 4-4 Working cycle at different condition...........................................................40 Fig. 4-5 Position of bypass holes: Case (A) .............................................................42 Fig. 4-6 Position of bypass holes: Case (B) .............................................................43 Fig. 4-7 Position of bypass holes: Case (C) .............................................................44 Fig. 4-8 Working cycle at different cases .................................................................45 ix.

(12) Fig. 4-9 The scheme of the STC and bypass holes on fixed scroll...........................46 Fig. 4-10 Change of the uncovered area of bypass holes .........................................48 Fig. 4-11 Uncovered area of bypass holes in (a) chamber 2 and chamber 3 (b) chamber 2’ and chamber 3’ ..............................................................................49 Fig. 4-12 Prediction of performance of m& ,η V ,ηC ..................................................50 Fig. 4-13 Prediction of pressure variation with and without bypass valves (a) chamber 2, 3 (b) chamber 2’, 3’ .......................................................................52 Fig. 4-14 CO2 test rig................................................................................................53 Fig. 4-15 Working procedure of CO2 test rig ...........................................................54 Fig. 5-1 Frictional losses in (a) Condition A (b) Condition B (c) maximum loading in Condition A ..................................................................................................60 Fig. 5-2 Flow chart of the optimization procedure...................................................63 Fig. 5-3 Iteration history (a) Objective function (b) Constraints 1–4 (c) Constraints 5–8 ....................................................................................................................66 Fig. 5-4 Frictional losses in all testing conditions ....................................................68 Fig. 6-1 Illustration: theorem of planar orbiting mechanisms ..................................71 Fig. 6-2 Sketch map and parametric definitions of an involute of circle with variable radii...................................................................................................................72 Fig. 6-3 Parametric relations of the scroll profiles constructed from an involute of circle with variable radii ...................................................................................74 Fig. 6-4 Sketch map of the scroll profiles for different k values ..........................78 Fig. 6-5 Structure of orbiting scroll (k=1, α=50° and δ0 = -0.05mm).......................78 Fig. 6-6 Scroll profiles from A0 to A4 .....................................................................82 Fig. 6-7 Reduction ratio in A1 to A4 and A1’ to A4’, A0 as standard ......................84 Fig. 7-1 Parametric definitions of the arc modification ...........................................87 Fig. 7-2 Comparison of differentβvalues to the scroll profile with arc modification ..........................................................................................................................88 Fig. 7-3 Parametric definitions of the line modification ..........................................90 Fig. 7-4 (a) Comparison of different Δr values ( β =105° (21π /36 rad) ) (b) Comparison of different β values ( Δr =0.5 mm )..........................................92. x.

(13) NOMENCLATURE a radius of base circle (mm) a0 initial radius of base circle (mm) A area ABy uncovered cross section area Adis equal flow area Ai, Ao enclosed area by involutes Ain, Aou leakage area from clearances C, Cvalve coefficient C0, C1 contact points Cin, Cou joint points Cb clearance of journal bearing (mm) CC1 convergence criterion d distance d,d’ distance used in arc and line modification (mm) dtu,s, dtu,,d tube diameter D diameter (mm) Db diameter of journal bearing(mm) Dm minimum endplate diameter of the orbiting scroll (mm) Di,th, Do,th inner and outer diameter of thrust bearing (mm) F force (N) Fb force from journal bearing (N) f flow friction factor g gravity h height (m) hin,hou inlet and outlet enthalpies (kJ/kg) htu,s, htu,d coefficient of heat convection (W m-2 K-1) H, H0 oil-film clearance (mm); nominal oil-film clearance (mm) Lb length of journal bearing (mm) length of the tube Ltu,d Li length of the inner involute Lo length of the outer involute li distance between inner involute and bypass hole center lo distance between outer involute and bypass hole center & mass flow rate (kg/s) m m mass (kg) mslo, slope mob mass of orbiting scroll (kg) M moment (N．m) N turn number of scrolls n polytropic index p, P pressure (MPa); power (W) Pb frictional loss from journal bearing (W) Pth frictional loss from thrust bearing (W) Pr Prandtl number pt pitch of scroll Q heat transefer quantity(kW) Qc cooling capacity (kW) xi.

(14) R, R’ RaL Re r req rob T t, th U V Vdis Volr Vsuc x,y R, Θ. radius of the modified arcs (mm) Rayleigh number, Ra L =(g βΔTL3 / αν ) ! Renolds number radius of circular bypass hole (m) equivalent radius (m) orbiting radius (m) temperature (K) time (s); thickness (mm) surface velocity (m．s-1) volume (mm3) discharge volume volume ratio suction volume coordinate cylindrical coordinate. Greek letters. α α lp β β el γ σ Δr. δ. δ0. φ φe , φdis φr θ θe θ dis ψ Φ. λ ρ ρ* ω ε μ ηC, ηV. initial angle of involute tilting angle (rad) modified angle between the outer and inner involute curve (º, rad) directional turning angle (rad) derivative angle (rad) derivative angle (rad) modified distance (mm) clearance (m) corrected increment (mm) involute angle of scroll ended involute angle (rad); corresponding involute angle at discharge (rad) roll angle (or called ended angel) of scroll pair orbiting angle of scroll(rad) modified angle to specified end involute angle (rad) orbiting angle at discharge (rad) independent angle parameter (rad) phase difference (rad) coefficient of heat conductivity (W m-1 K-1) radius of curvature (mm) density (kg/m3) angular velocity (rad．s-1) eccentricity ratio viscosity (Pa．s) compressor efficiency, volumetric efficiency. ∏ f , ∏ m coordinate planes Subscripts ax A. axial direction adiabatic xii.

(15) b back b,cr b,low b,upp C c cur c,csh c,lcw c,ob c,ucw dis dw e f f,in f,ou f,b , f,th g i, j, k in l l,b l,cur l,e l,f l,pre Me motor m,in m,ou ou o_a o_By o_l o_u r R suc scr suc,h t tu up V vg θ ,b. base circle back-side crank journal bearing lower journal bearing upper journal bearing compressor center position current centrifugal direction of crank shaft centrifugal direction of lower counterweight centrifugal direction of orbiting scroll centrifugal direction of upper counterweight discharge downstream end-side flank inner involute of fixed scroll outer involute of fixed scroll friction of the journal and thrust bearing gas index number inner leakage bypass leakage current leakage end side leakage flank leakage previous leakage mechanical motor inner involute of orbiting scroll outer involute of orbiting scroll outer corresponding involute angle for bypass hole bsypass hole coordinate to outer involute lower limit of the outer involute angle for bypass hole upper limit of the outer involute angle for bypass hole radial direction refrigeration suction locus of the turning moment superheat tangent coordinate tube upstream volumetric average tangential direction bearing. xiii.

(16) CHAPTER 1 INTRODUCTION 1.1 Scroll-type compressor (STC) Scroll-type compressor (STC), as one kind of positive displacement pumps, has been developing vigorously since 1980s. With commercial, domestic and automotive applications such as refrigeration, air-conditioning and heat-pump, STC has been regarded as a suitable substitute to other type of rotary compressors because of its higher efficiency, lower noise and fewer numbers of components. In virtue of these characteristics, the STC becomes the research objective in this dissertation. One orbiting motion, as the most important motion in STC, is composed of the two key components, fixed scroll and orbiting scroll. Both these two scrolls have a circular end-plate and a protrusion extended from one surface of it with spiral profile or so called “scroll”. Before proceeding with this motion, the orbiting scroll must be rotated first by 180 degrees ( π rad) relative to the stationary fixed scroll. After that, the orbiting scroll, with a crank mechanism driving it, orbits with a constant radius around a designate point on the fixed scroll and at the same time, an anti-rotation coupling (ex. Oldham ring) is assembled to it for preventing the self-rotation arising from the orbiting scroll. The spiral protrusions of both scrolls can be meshed with each other by this orbiting motion and several crescent and enclosed chambers are generated thereon. These chambers, accompanying with moving inward from the periphery to the center, make their volumes decrease gradually, and vice versa. Hence the scroll pair can be used as compressors, expanders and pumps by virtue of its direction of the orbiting motion. Figure 1.1 shows a complete cycle of the STC. The scroll pair inhales the gas at the periphery into the chambers and compresses it progressively toward the center with gradually smaller chambers and after reaching the discharger step, it is exhausted through the discharge port of the fixed scroll out of the scroll pair. Because of its simultaneous compression and discharge movement in several continuous chambers and without suction or discharge valves, the gas pulsation and flow losses can be reduced for smoother flow pattern. Therefore the STC is operated at lower torque variation, lower noise and vibration levels and easier to start and restart. Besides, the lower flow losses indicate the higher volume efficiency and the smoother operation means the better reliability of STC.. 1.

(17) Fig. 1-1 A complete cycle of STC: suctionÆcompressionÆdischarge The key components for a STC includes a orbiting scroll, a fixed scroll, an Oldham ring, a frame structure, a shaft with eccentric crank, a motor and several counterweight parts, bearing components and compliment mechanisms as shown in Fig1.2. Generally speaking, a STC is more durable than other rotary type compressors because of having fewer parts.. 2.

(18) Fig. 1-2 A schematic of a STC Though the idea of scroll machine was originated since 1905 and has been evaluated for many benefits, it did not be commercialized before 1970’s due to the need of precise manufacturing and assembly technique and the accompanied requirements of keeping the low cost for their key components, such as orbiting and fixed scrolls, Oldham ring and crankshaft. Besides, the sealing mechanism for reducing leakage between the contact regions to promote higher efficiency is also a critical topic. These two issues have been solved since 1975 from lots of disclosed patents and literatures. The mass production of STC products, including resident, commercial and automobile air-conditioning and refrigeration fields, were launched 3.

(19) since 1980. Several well-known companies commercialized STC products, like Sanden Corp., Hitachi Ltd., Trans Corp., and so on (Chang [1]). Among them, Copeland Corp., by means of its innovative technology, has made their STCs with high efficiency, easy fabrication and low cost and this company dominated the STC development for occupying over 50% share of total STC markets (7 millions units per year). With regard to developing STC, many particular designs have been disclosed from patents and research papers. However, except a few experiments and simulation results presented in them to verify their claimed merits, most literatures possessed just a few descriptions to their functions that they claimed without any proof of feasibility. Therefore, a simulation model based on the mathematical foundation must be constructed to analyze the effects regarding those design improvements. This procedure not merely provides useful estimations analyzing from the simulation results, but also reduces the times and cost expending on the experiments and tests. Owning to various design concepts of mechanisms applied to the STC, the general mathematical models about those new and innovative designs must be constructed and the simulation process be programmed, and the analysis to the simulation results be executed to fully evaluate the benefits and defects through those designs.. 1.2 Simulation program for STC One STC simulation program, as a developing tool, is essential to be constructed for estimating various design concepts and assisting engineers in advancing the development of STC. By extending the previous developed one constructed by ITRI, a STC simulation program has be developed for more powerful functions. Up to now, the program has contained 5 main modules as follows; the user interface with I/O handling, the geometrical model with scroll profiles and related mechanisms, the thermodynamics model for compression and discharge in chambers, the dynamics and bearing models dealing with loadings coming from thermodynamic results, and the optimization model for deriving optimum parametric combinations. In addition, several individual mathematical models with regard to mechanism design used in STC have also been constructed to assess their values towards STC. Details of this STC simulation program, including the formulations, procedures, functions and limitations, will be revealed throughout this dissertation.. 4.

(20) 1.3 Variable speed and compression ratio Traditionally, the STC, in consideration of simplifying complicated layout and saving cost, is designed to operate at constant speed by using single or multiple phases AC (alternating current) inductive motor. It means that the STC can suit to a specified operating condition or thermal loading (cooling or heating capacity) with constant-speed operation. That is to say, the STC running at one design point has advantageous performance. But deviated from this specific operating condition, other ones with which a STC is faced, will result in various thermal loadings owing to varied environmental temperatures. Hence the STC must be turned off and on to match different thermal loadings, but these switching on/off actions do result in unnecessary power consumption and bring about loud noise and vibration to the pipes and the frame of STC. Therefore, it is incapable of guaranteeing that the STC has high efficiency when running away from the specific design point. With global warming effect and increasing environmental consciousness, development in STC with higher efficiency and more power saving has been considered. Due to this, a STC that can operate at variable pressure ratio (compression ratio) and speed and can produce optimum efficiency at different operating conditions has been studied in recent years. For variable speeds purpose, several innovative designs of motors have been put in use with STC for providing wider range in operating speed with superior motor efficiency, for example, the 3 phases AC induction motor (Sarma [2] and Engelmann et al. [3]) and PMSM synchronous motor (Engelmann et al. [3] and Igata et al. [4]). The pressure ratio, on the other hand, is defined as the ratio of the saturated condenser pressure to the saturated evaporator pressure ( pdis /psuc ) and is decided for ambient operating conditions. In general, volume ratio is fixed after the geometrical parameters of the scroll pair in STC have been decided. Due to the intrinsic limitation of fixed volume ratio and separated compression chambers in the STC, when the pressure ratio does not match with the volume ratio of the STC, two cases, over-compression and under-compression, will happen (Schein [5]) . For under-compression, the repetitive compression in the final chamber or back flow from discharge chamber unavoidably will occur in different designs of STC. For over-compression, the STC will compress the gas to its design point regardless of the high pressure in chambers and extra power is consumed. Under-compression could not be prevented except by designing a STC with low volume ratio while narrowing the range of operating conditions and eventually reducing the thermal. 5.

(21) capacity. Nevertheless, over-compression could be reduced by using bypass valves added to the fixed scroll. Application of bypass valves can change fixed volume ratio to match up with varied operating conditions. Therefore, STC with variable compression ratio can be designed and produced with superior efficiency. In view of this, the mathematical model of bypass valves has been constructed in this study and comparison with experiment, important discussions to the simulation results of STC program were also investigated.. 1.4 Design optimization for variable compression ratio After constructing of a parametric computer model to the STC, one interesting issue is how engineers or designers use it to aid the design process about STC. Due to a great number of parameters defined in the STC model can be selected as design variables and many important simulations results be provided as constraint conditions and cost function, many engineering experiences in STC fields must be introduced into the optimization procedure for finding not only the optimum, but also the possible and reasonable combination of parameters used in designing STC. This study has developed an optimization module, including the user interface, the procedure flow and input/output formatted files, for this STC design package and the solver “MOST” (Tseng [6]) has been integrated into it as the calculation kernel. After that, one case riveted on reduction of the frictional losses for bearing components in STC and investigation of relevant effects, was analyzed for demonstrating the optimization procedure.. 1.5 Scroll profiles based on an involute of circle with variable radii Based on using an involute of circle with a fixed radius as scroll profiles, numerous technical researches were proposed for advancing the performance of STC. However, the intrinsic volume ratio to this type of scroll profile is constrained because of the considerations on the strength and stiffness to the scroll wrap, the rigidity and difficulty to the cutting tool, and the housing size to the space and disposition. This means that the volume ratio in STC can just be lifted to a limited extent by using the scroll profile based on involute of circle with a fixed radius. Though the volume ratio in STC can be improved by modifying the center portion of the scroll pair, the severer loading could be confronted because of the higher pressure difference generated on the center portion scroll wrap between the adjacent chambers. On the other hand,. 6.

(22) the higher volume ratio means the higher pressure conditions for the needed operating temperature with which the STC must confront. Due to this, developing a scroll pair in a STC with better rigidity and strength to endure these high pressure conditions has become an issue. In this study, a complete geometrical model of the scroll curves constructed from an involute of circle with variable radii will be formulated and proved by utilizing the theorem of planar orbiting mechanisms to overcome the mentioned difficulties. After that, several case studies will be implemented and discussed to distinguish the values of this type of scroll profiles. Furthermore, two types of perfectly meshing modifications, arc and line shapes to the center portion of this new scroll profile will also be developed.. 1.6 Dissertation Scope The content of this dissertation are outlined as below. Chapter 1 briefly introduces the STC simulation computer package, bypass mechanism for variable compression ratio and variable speed consideration, the numerical optimization module and the geometrical model of the scroll profiles based on an involute of circle with variable radii. In Chapter 2, the complete structure about the STC simulation package has been depicted. Several important literatures will be reviewed firstly, and mathematical models such as geometry, thermodynamics and mechanism dynamics, will be built up, based on those ones. Chapter 3 investigates the bypass mechanism used in STC products. Then the mathematical model will be constructed and integrated into the developed STC package. The simulation results of the bypass mechanism regarding different STC structures, described in Chapter 4, are connected to energy saving and liquid-slug protection. The verification for these results is executed by one test platform constructed for an actual STC product using CO2 as refrigerant. Development of an optimization module for design of STC products is introduced in Chapter 5. It includes the user interface and the optimum solver. One case on reduction of the frictional losses for bearing components will be studied to estimate the benefits by means of the optimization procedure. Chapter 6 constructs a mathematical model about the scroll profiles based on an involute of circle with variable radii. Its values will be illustrated. In Chapter 7, two kinds of modification regarding the center portion of the scroll profiles created by an involute of circle with variable radii will be provided. Several case studies will exhibit their uses, defects, and constraints. Finally, conclusions and future works are summarized in Chapter 8. 7.

(23) CHAPTER 2 STC SIMULATION MODEL AND PROCESS Before proceeding with designing the STC products, it goes without saying that a STC simulation process with several important mathematical models must be constructed firstly and integrated into a computer program. The whole models include geometry of the scroll, thermodynamics with refrigerant in compression and discharge processes, leakage through clearances, back pressure mechanism, superheat of suction pipe, and dynamic balance of the mechanical components.. 2.1 Developments in mathematical models of STC Numerous literatures about the mathematical model of STC were published since 1980’s. Among them, Morishta et al. [7, 8] constructed one analytical model for a STC. This included the geometry of scroll profile, the thermodynamics during compression and discharge process and the dynamics of related mechanical components used in STC. Based on this primitive model, a great many technical researches have been issued for the respective mathematical models or concerning the integral process during the two decades.. 2.1.1 Geometry of scroll profiles With regard to the scroll pair (which mean the fixed and orbiting scroll), many papers and patents have been proposed to refine them in order to improve the STC’s performance. In the theoretic study, Lee and Wu [9] proved that several theorems related to planar orbiting mechanisms can be used to design a scroll pair. In addition, the planar curves expressed by the intrinsic equation (Gravesen and Henrisken [10]) have also been developed, to derive the closed analytical expression about several types of scroll profiles (Bukac [11] and Qiang [12]). By way of using differential geometry, many curve profiles, in addition to the conventional involute of circle with a fixed radius, have been used to create the scroll pair and investigate its application continuously. These includes the archimedes spirals (Gagne and Nieter [13]) and segmental arcs (Mahfouz et al. [14], Liu and Liu [15]). Moreover, Wang et al. [16] provided a modified mathematical model for the discrepant angle to the profiles of. 8.

(24) involute. On the other hand, several methods and modifications for reaching the perfectly meshing engagement at the center of the scroll pair and avoiding the mutual interference have been exhibited (Terauchi and Hiraga [17], Hirano and Hagimoto [18] and Lee and Wu [19]). It is expected that these modifications can improve the STC efficiency and durability. Furthermore, Li et al. [20] compared different curves which were used to generate the scroll wrap profiles in a STC, and discussed their advantages and defects.. 2.1.2 Thermodynamics in the scroll pair By referring the rotatory compressor, Yanagisawa et al. [21, 22] estimated the suction heating and leakage losses. Extending their methods and on the basis of polytropic process, Morishita et al. [7, 8], Nieter and Gagne [23] and Morimoto et al. [24] derived their respectively analytical model for STC. By referring to the above references, Chen et al. [25, 26], using the mass and energy conservation equations with the 1st law of thermodynamics, developed a detailed mathematical model of STC with consideration of scroll geometry, compression and discharge process and two kinds of leakage between the adjacent chambers. Several important indices about thermal efficiency were also derived under different operating conditions. Moreover, Schein and Radermacher [5] not only disclosed the simulation results of their STC computer model, but explained the causes of under-compression and over-compression which may happened during the compression stage of STC.. 2.1.3 Dynamics of components and bearing models In addition to the scroll pair, the most important one in STC is the coupling mechanism which is used to prevent the orbiting scroll from revolving on its own axis. The dynamic model of the orbiting scroll coupled with the Oldham-ring [7] can handle this mechanism. In additions, the dynamic forces and moments which generated from the gas pressure in chambers and the inertial effect of components (such as counterweight parts and some eccentric components) were also derived and solved to obtain the needed supporting forces of the bearings through the STC mathematical model disclosed by Morishita et al. [7, 8]. In machine design for STC, the journal and thrust bearings must be used for supporting the lateral and axial loadings since their inexpensive cost as compared with the ball or roller bearings. In light of this, Kulkari [27, 28] and Sato et al. [29] have developed the thrust bearing models with hydrodynamic lubrication theory. For journal bearing, the numerical model utilizing mobility method for journal bearings (Booker [30–32] and Geonka [33]) has 9.

(25) been provided and this study will refer these well-developed models for calculating the frictional losses and predicting some important indices.. 2.2 Geometrical model of STC Most STC products take the involute of circle with a fixed radius for the profiles of scroll wrap because of its inexpensively manufacturing cost. Hence the mathematical formulations of it must be introduced at first.. 2.2.1 Scroll profile design by involute of circle with a fixed radius An involute of circle with a fixed radius and the area enveloped by it is shown in Fig. 2-1. Let the involute angle φ be one variable and the involute can be expressed in Cartesian coordinate system as below:. ⎪⎧ x = a ⎣⎡ cos φ + (φ ) sin φ ⎦⎤ ⎨ ⎩⎪ y = a ⎡⎣sin φ − (φ ) cos φ ⎤⎦. (2.1). where a represents the radius of base circle.. Fig. 2-1 Parametric definition for the involute of circle If adding or subtracting an increment α (also called initial angle) to equation (2.1), the. 10.

(26) outer and inner involute profiles of a fixed scroll can be written as. ⎧⎪ xf,ou = a ⎡⎣ cos φ + (φ − α ) sin φ ⎦⎤ ⎨ ⎪⎩ yf,ou = a ⎡⎣sin φ − (φ − α ) cos φ ⎤⎦. (2.2). ⎧⎪ xf,in = a ⎣⎡cos φ + (φ + α ) sin φ ⎦⎤ ⎨ ⎪⎩ yf,in = a ⎡⎣sin φ − (φ + α ) cos φ ⎤⎦. (2.3). By using the equations (2.2) and (2.3) with consideration of coordinate transformation, the orbiting scroll also can be generated as below:. ⎧⎪ xm,ou = − xf,ou − rob cos (θ + θ e ) ⎨ ⎪⎩ ym,ou = − yf,ou + rob sin (θ + θ e ). (2.4). ⎧⎪ xm,in = − xf,in − rob cos (θ + θ e ) ⎨ ⎪⎩ ym,in = − yf,in + rob sin (θ + θ e ). (2.5). Some important parameters used above and in this geometrical model are defined firstly as follows: ⎧a: radius of base circle ⎪ ; pt ⎨ ⎪⎩a = 2π. ⎧α : intial angle of involute ⎧rob : orbiting radius ⎪ ⎪ ; ⎨ th ⎨ pt ⎪α = 2r ⎪⎩rob = 2 - th b ⎩. ⎧ N : number of chambers ⎪ (φe − φdis ) ⎪ ) +1 ⎨ N = Int( 2π ⎪ ⎪⎩ where φdis the discharged involute angle decided by cutting condition ⎧θ e : modified angle to specified end involute angle ⎨ ⎩θ e = 2π ⋅ ( N + 1/4) − φe ⎧θ dis : discharged orbiting angle at one revolutation ⎨ ⎩θ dis = (φe − φdis ) − 2π ⋅ ( N -1). (2.6). Figure 2-2 shows the fixed and orbiting scrolls when they are just ready to proceed with compression process.. 11.

(27) Fig. 2-2 Inner and outer Profiles of fixed scroll. 2.2.2 Derivation of volume and its ratio of the scroll chambers Referring to Fig. 2-1 again, the enclosed chamber’s area can be formed from the inner involute on one scroll wrap and the outer involute on another one, those areas can be derived by integrating the minute area dS along with the involute angle φ , hence the enclosed areas created by the inner involute is Ai = ∫. φ −α. =∫. φ −α. φ −α −π. φ −α −π. 1 1 [ (aφ ) 2 dφ − (a (φ − 2π )) 2 dφ ] 2 2 [2π a 2 (φ − π )]dφ. (2.7). = π 2 a 2 [2(φ − α ) − 3π ] Similarly, the areas by the outer involute is Ao = ∫. φ −π. =∫. φ −π. φ − 3π. φ − 3π. 1 1 [ (a (φ + α )) 2 dφ − (a (φ − α )) 2 dφ ] 2 2 2a 2αφ dφ. (2.8). = 4π 2 a 2α (φ − 2π ) Assume the scroll warp have symmetric chambers, therefore, the enclosed areas can be expressed as follows. 12.

(28) A = 2(Ai − Ao ) = 2π a (2φ − 3π )(π a − 2α a ) = pt (. pt − th )(2φ − 3π ) 2. (2.9). Nevertheless, another method using differential geometry to derived the enclosed chamber areas can also be adopted and expressed as below A=. d(y ) d(y ) 1 φ xm,in ⋅ m,in − ym,in ⋅ m,in dφ ∫ 2 φ − 2π dφ dφ d(y ) d(y ) 1 φ − ∫ xf,ou ⋅ f,ou − yf,ou ⋅ f,ou dφ 2 φ − 2π dφ dφ. (2.10). Multiplying equation (2.9) by the wrap height h , the chamber volume can be derived as. V (φ ) = h ⋅ A = hpt (. pt − th )(2φ − 3π ) 2. (2.11). If we decide the ended involute angle φe and φdis as the involute angle while the meshing is at the discharge step, the volume ratio can be expressed as. Volr =. 2φe − 3π 2φdis − 3π. (2.12). The details to the fundamental formulas applied in modifying the discharge angle φdis to the involute angle with constant radius for raising the volume ratio and preventing interference can refer to the papers derived by Morishita et al. [7, 8], and Nieter and Gagne [23].. 2.3 Thermodynamic model The thermodynamics including suction/discharge superheat, the compression and discharge with variations in pressure and temperature are considered. In addition, the leakage clearances of the scroll pair, including the end-side and flank leakage, are dynamically and linearly linked to pressure ratio and back-pressure (Chen et al. [25]), and those linear functions are expressed in this model. Also, several important indices related to thermodynamics will also be formulated in the following paragraph.. 2.3.1 Refrigerant property The “Refprop7” published by NIST [34], as a powerful tool, is used in this model to obtain lots of data for most known refrigerants. The STC simulation package integrates it by. 13.

(29) linking the dynamic link libraries (.dll) to acquire the related properties which will be used for calculation during the compression and discharge processes. The linking procedure can refer to Fig. 2-3.. f ( P, T ) = f ( P, T , ρ ,υ ,...) f ( P, ρ ) = f ( P, ρ , T , υ ,...) f ( P, T ) = f ( P, T , ρ , υ ,...). f ( ρ , υ ) = f ( ρ , υ , P, T ...). ρυ. Fig. 2-3 Illustration of “Refprop7” linking to STC simulation package. 2.3.2 Suction and discharge superheating calculations In general, the STC can be divided into two structures—high and low-side structures. The low-side structure means the chamber above the fixed scroll is at discharge pressure, but the other regions in the housing are at suction pressure. The high-side structure is at discharge pressure in the whole housing except the suction and compression chambers. The main benefits and drawbacks of those two structures are summarized as follows:. Low-side: Benefits: (1) Simple motor/load application and radial compliance. (2) Large housing chamber used as suction muffler and good overturning moment control. Drawbacks: (1) Complex thrust bearing design, axial compliance mechanism and small discharge muffler volume. (2) Difficulties in designing and machining intermediate pressure holes and more suction gas. 14.

(30) heating. (3) Exact scroll set machining and fine finishes are need for minimal leak paths.. Fig. 2-4 Low-side and High side structure of STC [1]. High-side: Benefits: (1) Simple pressure control and no intermediate pressure holes can avoid leakage. (2) Simple machining, minimal suction superheat and large housing chamber used as discharge muffler. (3) Simple radial and axial compliance mechanism, and no problem involved passing large quantities of oil through the scroll set. (4) Oil can help control overturning moment and provide to seal leak paths. Drawbacks: (1) Small suction muffling and hardness in motor/load application. (2) Severer housing design to conform the ultimate rest requirements. Owing to the different structure of the two STC, their heat transfer coefficients for the pipe at suction (for low side structure and high side structure)) and discharge (for high side structure) can be expressed by two experienced formulas—the Gnielinski relationship (Winandy et al. [35]) and Squire-Eckert prediction (Fukuta et al. [36])—which expressed as. htu,s =. λ d tu,s. ⋅. ( f / 8)( Re − 1000) Pr 1 + 12.7 f / 8( Pr 2/3 − 1) 15. (2.13).

(31) λ Pr htu,d = 0.678 ⋅ Ra L 0.25 ( )0.25 0.952 + Pr L. (2.14). Then the suction and discharge superheat can be evaluated.. 2.3.3 Compression and discharge process This simulation model provides two ways for users to calculate the compression and discharge processes. Several assumptions must be employed in the model for simplicity, such as neglecting the turbulent flow of refrigerant and oil effects inside the chambers. Those are stated as below: 1. Refrigerant in working chambers is homogeneous. 2. Gravitational, kinetic energy variations are neglected. 3. Oil effects are neglected. 4. Chambers of the scroll pairs are symmetric. (1): The mass conservation of refrigerant in chambers (control volume) with polytropic process consideration. These can be described as ⎧m& = m& in − m& ou ⎪ n ⎛ ρ *cur ⎞ ⎨ ⎟ ⎪ pcur = psuc ⎜ ⎝ ρ *suc ⎠ ⎩. (2.15). where n can be measure by laboratory experiment (DeBlois and Stoeffler [37]). (2): The mass conservation of refrigerant in chambers (control volume) with the simplified first law of thermodynamics, which can be expressed as ⎧ ⎪ ⎪m& = m& in − m& ou ⎪ 1 ⎡ ⎛ ∂p g ⎞ ⎛ dVg 1 ⎛ dm g,in dm g,ou ⎪ dTg − − = ⎢Tg ⎜ ⎟ ⎜ ⎨ ⎜ ⎜ ⎟ ⎜ dθ ⎪ dθ m g c vg ⎣⎢ ⎝ ∂Tg ⎠ v ⎝ dθ ρ *g ⎝ dθ ⎪ dm g,in dQ ⎤ ⎪ +∑ (h g,in − h g )+ ⎥ ⎪ dθ dθ ⎦ ⎩. ⎞⎞ ⎟ ⎟⎟ ⎠⎠. (2.16). 2.3.4 Leakage flows Generally speaking, there are two kinds of leakage paths—end-side and flank leakage—existed in a STC. Fig. 2-5 and 2-6 show the illustration about these two paths and the leakage mass flow can be shown as follow:. 16.

(32) m& l = m& l,e + m& l,f. (2.17). Where m& l,e and m& l,f are end-side and flank leakage mass flow rate.. Fig. 2-5 Illustration of end-side leakage. Fig. 2-6 Illustration of flank leakage Gap sizes of the two kinds of leakage are both dynamically related to the operating pressure, pressure ratio and back-pressure mechanisms acted on scroll pairs. These three ones are dependent on various design structures of STC (such as low side and high side structures). The end-side gap δ e , in meters, can be derived as below:. ⎡. ⎤ −6 ⎛ pdis − pback ⎞ ⎟ − 0.45⎥ ⋅10 psuc ⎝ ⎠ ⎦. δ e = ⎢1.02 ⋅ ⎜ ⎣. (2.18). The end-side flow areas can be calculated [25] as:. Ain = δ e ∫. φk+1. Aout = δ e ∫. φk+1. φk. φk. Lo dφ Li dφ. 17. (2.19).

(33) where φk+1 and φk are the involute angles of the conjugate end points for the scroll pair. For end-side leakage, one-dimensional isentropic compressible flow in a nozzle is used as. ⎧ ⎡ 2n ⎢⎛ pdw ⎪ m& l,e = C ⋅ A ⎨ pup ⋅ ρ *up ⋅ ⎜ n − 1 ⎢⎜⎝ pup ⎪ ⎢⎣ ⎩. 2 n. ⎞ ⎛ pdw ⎞ ⎟⎟ − ⎜⎜ ⎟⎟ ⎠ ⎝ pup ⎠. n +1 n. 0.5. n +1 ⎡ ⎤ ⎛p n −1 2 ⎛ ⎞ ⎢ ⎥ & ml,e = C ⋅ A pup ⋅ ρ *up ⋅ n ⋅ ⎜ ; when ⎜ dw ⎟ ⎜ pup ⎢ ⎝ n +1⎠ ⎥ ⎝ ⎣ ⎦. where. 0.5. ⎤⎫ ⎥ ⎪ ; when ⎛ pdw ⎜⎜ ⎥⎬ ⎪ ⎝ pup ⎥⎦ ⎭. n. ⎞ ⎛ 2 ⎞ n −1 ⎟⎟ ≥ ⎜ ⎟ , + n 1 ⎝ ⎠ ⎠. n. ⎞ ⎛ 2 ⎞ n −1 ⎟⎟ < ⎜ ⎟ ⎠ ⎝ n +1⎠. (2.20). A is Ain or Aout for inflowing or outflowing conditions. Similarly, the flank gap. is expressed in meters as follow:. ⎡. ⎤ −6 ⎛ pdis − pback ⎞ ⎟ + 20 ⎥ ⋅10 psuc ⎝ ⎠ ⎦. δ f = ⎢ −6 ⋅ ⎜ ⎣. (2.21). The flank flow area is Af = hδ f. (2.22). For m& l,f , equation (2.20) also be used with A = Af . Thus the total leakage mass flow rate can be calculated.. 2.4 Dynamics in mechanical components and mechanisms in STC 2.4.1 Dynamic forces and moments The dynamic model of the orbiting scroll coupled with the Oldham ring (Morishita et al. [7]) is adopted in this study. Several dynamic forces caused by gas pressure in chambers and acted on the orbiting scroll are considered. Those forces are classified as tangential, radial and axial ones. As shown in Fig. 2-7, the tangential force ( Fθ ) and radial force ( Fr ) can be expressed as follows: N. N. i =1. i =1. Fθ = ∑ Fθi = ∑ rb ⋅ h ⋅ (2φ − π ) ⋅ ( pi − pi +1 ) N. N. i =1. i =1. Fr = ∑ Fri = ∑ 2 ⋅ rb ⋅ h ⋅ ( pi − pi +1 ). 18. (2.23) (2.24).

(34) Fig. 2-7 Dynamic forces of the orbiting scroll In axial direction, the back pressure mechanism is used to reduce the axial load and the resultant axial force is N. Fax = ∑ pi ⋅ Ai − Fback + mob ⋅ g. (2.25). i =1. where Ai represented the chamber area derived by Morishita et al. [8]. Moreover, by neglecting the inertial force with constant speed consideration, the centrifugal force of the orbiting scroll is. Fc,ob. ⎛ dθ ⎞ = mob ⋅ rob ⋅ ⎜ ⎟ ⎝ dt ⎠. 2. (2.26). The force and moment equations of the orbiting scroll coupled with Oldham ring are used to derive the bearing force of the crank journal bearing ( Fb,cr ) set up on the crank shaft [7]. Furthermore, by assuming the resultant axial force ( Fax ) and turning moment ( M ) must be balanced by reaction components resulted from the thrust surface [7], the turning moment M and the directional turning angle β of the orbiting scroll can be shown as follows: ⎧⎪ M x = Fax ⋅ ( yscr − rob sin θ ) ⎨ ⎪⎩ M y = Fax ⋅ ( xscr + rob cos θ ) M = M x2 + M y2. β el = tan −1 (. My Mx. (2.27). ). where (x scr , yscr ) represents the locus of the turning moment on the thrust surface.. 19.

(35) 2.4.2 Counterweight analysis Another important part is the counterweight analysis. Fig. 2-8 is the force diagram in x-direction, the crank bearing force ( Fb,cr ) and the centrifugal forces from the crank shaft ( Fc,csh ), the upper and lower counterweights ( Fc,ucw & Fc,lcw ) are considered. The upper and lower journal bearings are used to support the driving shaft. By assuming the center of the upper journal bearing as the pivot, the two bearing forces can be derived as follows. ⎧⎪ Fbx,low ⋅ zbx,low = Fbx,cr ⋅ zbx,cr + Fcx,csh ⋅ zcx,csh + Fcx,ucw ⋅ zcx,ucw − Fcx,lcw ⋅ zcx,lcw ⎨ ⎪⎩ Fby,low ⋅ zby,low = Fby,cr ⋅ zby,cr + Fcy,csh ⋅ zcy,csh + Fcy,ucw ⋅ zcy,ucw − Fcy,lcw ⋅ zcy,lcw ⎧⎪ Fbx,upp + Fcx,ucw = Fbx,cr + Fcx,csh + Fbx,low + Fcx,lcw ⎨ ⎪⎩ Fby,low + Fcy,ucw = Fby,cr + Fcy,csh + Fby,low + Fcy,lcw Fb,low = Fbx,low 2 + Fby,low 2 ; Fb,upp = Fbx,upp 2 + Fby,upp 2. Fig. 2-8 Dynamic balance of the driving shaft. 20. (2.28).

(36) 2.5 Frictional losses in mechanical components In addition to the compression power consumed during STC operation, other main expenses are the frictional losses from the mechanical components. Those losses are mainly caused from the thrust surface (as thrust bearing) and the three journal bearings. In addition, by assuming the frictional losses around the other mechanical components (such as Oldham ring) is negligibly small compared with the bearing losses (Ishii et al. [38]), the frictional coefficient of the other mechanical components is set as 0.013 [38] to calculate the friction force and losses in this study.. 2.5.1 Thrust bearing As shown in Fig. 2-9, the thrust bearing is used to support against the resultant axial force Fax and turning moment M . Several models have been investigated in literatures (Kulkarni [27, 28], Sato et al. [29], Akei et al. [39] and Oku et al. [40]). Among those, one rigid-body wobbling model [27, 28, 39] with hydrodynamic lubrication theory is adopted in this developed STC model. The definitions of relevant parameters are showed in Fig. 2-10.. Fig. 2-9 Schematic of the three journal bearings and the thrust bearing. 21.

(37) Fig. 2-10 Parameters of the thrust bearing By assuming sufficient circulation of lubricant through the thrust bearing and the steady temperature, the Reynolds equation, including the wedge and squeeze terms for generating the pressures on the circular thrust surface, can be expressed in cylindrical coordinates as:. 1 ∂ ⎛ 1 ∂ ⎛ 3 ∂p ⎞ 3 ∂p ⎞ ⎜ RH ⎟+ 2 ⎜H ⎟ R ∂R ⎝ ∂R ⎠ R ∂Θ ⎝ ∂Θ ⎠ ∂H ⎞ 1 ∂ ⎛1 ∂ = 6μ ⎜ ( RHU R ) + ( HU Θ ) + 2 ⎟ ∂t ⎠ R ∂Θ ⎝ R ∂R. (2.29). The oil-film clearance H between the tilting orbiting scroll and thrust surface is H = H 0 − tan α lp ( R sin(Θ + β ) − rob cos(θ − β ) ). (2.30). where H 0 is the nominal oil-film clearance (shown in Fig. 2-11), α lp is the tilting angle and β el is the directional tilting angle. The surface velocities and velocity derivatives of the orbiting scroll can be simplified as: U = rob ⋅ ω , ∂U R ⎧ ⎪⎪U R = U cos(Θ + θ ) , ∂R = 0 ⎨ ⎪U = −U sin(Θ + θ ) , ∂U Θ = −U cos(Θ + θ ) ⎪⎩ Θ ∂Θ. (2.31). The following assumptions simplify the squeezing effect:. ∂H ∂H = H& 0 = 0 , α& = 0 , β& = θ& = ω and ω ∂t ∂Θ. 22. (2.32).

(38) Fig. 2-11 Nominal height of oil-film The average Reynolds equation (2.29) is approximated by the finite difference operators and the successive over-relaxation (SOR) method is used as solution approach and the suction pressure is set as the boundary condition of the thrust surface. Once the pressure distribution is known, the force and moment from the lubrication oil can be integrated as 2π Ro. Fax' =. ∫ ∫ p ⋅ RdRd Θ. (2.33). 0 Ri. M = ' x. 2π Ro. ∫ ∫ p ⋅ R sin Θ ⋅ RdRd Θ ,. M = ' y. 0 Ri. 2π Ro. ∫ ∫ p ⋅ R cos Θ ⋅ RdRd Θ 0 Ri. M ' = M '2x + M '2y and β el ' = tan −1 (. M M. ' y ' x. (2.34). ). By equating the force and moment in equations (2.25), (2.33) and (2.27), (2.34), there are two simultaneous equations with two unknowns H 0 and α lp , and the Newton-Raphson method is used to solve the equations. Then the friction force Ff,th and the frictional losses Pth are derived as below: Ff,th =. 2π Ro. ∫. ∫μ. 0 Ri. R 2ω dRd Θ , H. Pth = Ff,th ⋅ U Finally the average frictional losses of thrust bearing can be calculated.. 23. (2.35).

(39) 2.5.2 Journal bearings For journal bearing, selection of the inside diameter Db , the clearance Cb and the length Lb can decide its frictional characteristics. In order to estimate the frictional losses, the numerical model utilizing mobility method for journal bearings [32, 33] is used and the frictional force becomes. Ff,b =. FbCbε 2πμω Lb ⎛ Db ⎞ sin Φ + ⎜ ⎟ Db Cb (1- ε 2 )0.5 ⎝ 2 ⎠. 2. (2.36). where ε is the eccentricity ratio and Φ is the phase difference from eccentric and load direction. The frictional loss Pb is. Pb = Ff,b ⋅. Db ⋅ω 2. (2.37). Then the average frictional loss of the journal bearings can be calculated.. 2.6 Simulation process in the STC computer model The complete simulation model for STC is programmed by C++ Builder combined with “Refprop7”. The inputs include scroll geometry, related mechanisms, operating conditions and motor Torque-Efficiency-Power curves (as shown in Fig. 2-12) from dynamometer test for specified speed used in this dissertation.. Fig. 2-12 Motor torque-efficiency-power for specified speed 24.

(40) 2.6.1 Simulation process The flowchart of the package is shown in Figure 2-13. The geometry and thermodynamic models are solved with numerical iterations. After that, the dynamics and frictional losses of bearings are also calculated with the previous models iteratively and finally, the output results can be obtained.. Start Scroll geometry Related mechanisms Refrigerant properties Operating conditions Motor performance. Input parameters. Initial guess. Various chamber areas & volumes Thermal properties without leakage. Compression & discharge process. Suction model Superheating &heat transfer Polytropic compression & discharge Leakage model. REFPROP 7.0. Dynamic balance. Force & moment balance Bearing loads Friction loss. Efficiency & outputs. Volumetric efficiency Isentropic efficiency Mechanical efficiency Refrigerant mass flow rate. Simulation results Fig. 2-13 Flowchart of the simulation process in STC package Among the process, the leakage model is included in the polytropic compression and discharge process (or the energy conservation with compression and discharge process) and then is solved with the 4th Runge-Kutta method (as shown in Fig. 2-14). The 4th R-K method is the solution with convergent criterion as follow:. m& l,pre − m& l,cur m& l,cur. < CC1. 25. (2.38).

(41) Initial parameters. θ =θstart. Modify inputs. θ =θ +Δθ. Compression & discharge with 4th R-K method (1) End-side leakage (2) Flank leakage. θ =θ end ?. No. Yes. Leakage converge ?. No. Yes Output result Fig. 2-14 Flowchart of thermodynamics with leakage model in STC package. 2.6.2 Output results After carrying on the above-mentioned simulation process, various performance indices as output data can be obtained. These basic but important outputs used in this dissertation are expressed as follows:. ¾ Volumetric efficiency:. ηV =. m& suc,h − m& l m& suc. (2.39). ¾ Mass flow rate: m& = η V ⋅ ωmotor ⋅ ρsuc ⋅ Vsuc. (2.40). where ω is the specified operating speed, ρsuc is the density of the suction refrigerant and Vsuc represents the designated suction volume of the scroll pair in STC.. 26.

(42) ¾ Cooling capacity:. Q& C = m& ⋅ (hin − hout ). (2.41). where hin and hout represent the enthalpies of refrigerant at evaporator inlet and outlet respectively.. ¾ Adiabatic power:. Padiabatic. ⎡⎛ p ⎞(n −1)/n ⎤ ⎪⎧⎛ n ⎞ ⎪⎫ dis ⎢ ⎥ 1 ω p V = ηV ⎨⎜ ⋅ ⋅ − ⋅ ⎟ motor ⎬ ⎟ suc suc ⎜ p ⎪⎩⎝ n − 1 ⎠ ⎪⎭ ⎣⎢⎝ suc ⎠ ⎦⎥. (2.42). ¾ Compression power:. Pcompression = Fθ ⋅ rob ⋅ ωmotor = Tθ ⋅ ωmotor. (2.43). ¾ Adiabatic compression efficiency:. ηA =. Padiabatic Pcompression. (2.44). ¾ Frictional losses:. Pfrictional loss = Pthrust bearing + Pjournal bearings + Pother mechanisms. (2.45). where Pother mechanisms are the power resulted from other mechanisms in STC, such as oldham coupling.. ¾ Mechanical efficiency:. η Me =. Pcompression Pcompression + Pfrictional loss. (2.46). ¾ Motor power: Pmotor =. (Pcompression + Pfrictional loss ). η motor. (2.47). where η motor is the motor efficiency at the specified speed and torque requirement(as shown in Fig. 2-12).. ¾ Compressor efficiency:. ⎛ Padiabatic ⎜ Pcompression ⎝ P = adiabatic Pmotor. ηC = η A ⋅η Me ⋅η motor = ⎜. ⎞ ⎛ ⎞ ⎛ Pcompression + Pfrictional loss ⎞ Pcompression ⎟⎟ ⋅ ⎜⎜ ⎟⎟ ⋅ ⎜ ⎟ Pmotor ⎠ (2.48) ⎠ ⎝ Pcompression + Pfrictional loss ⎠ ⎝. 27.

(43) ¾ COPR (coefficient of performance to the refrigeration on electrical power input): COPR =. Q& C Pmotor. (2.49). 2.7 Summarization of the STC simulation model A complete mathematical model regarding the STC has been illustrated in this chapter, and a parametric computer package has been constructed by virtue of the model. The simulation process for the package includes the geometry of scroll pair, the thermodynamics in chambers of STC, the dynamics about the related mechanisms and the frictional calculation of bearing models. After executing the process, many important indices as output data can be derived. These outputs turn into a design foundation for STC and assist in various analyses further.. 28.

(44) CHAPTER 3 MATHEMATICAL MODEL OF BYPASS VALVES A general bypass valves model, based on computation geometry, will be reviewed at first and completely constructed in this chapter. This model will be integrated into the above-mentioned STC package (exhibited in Chapter 2).. 3.1 Developing history regarding bypass valves The STC of variable pressure ratio was developed in recent years due to higher efficiency and power-saving considerations. The pressure ratio is defined as the ratio of the saturated condenser pressure to the saturated evaporator pressure ( pdis /psuc ), and is decided by operating conditions of temperature. In general, volume ratio of a STC is fixed after the geometric parameters regarding the scroll pair in it have been decided. When the pressure ratio does not match with the volume ratio of the STC, two cases, over-compression and under-compression, will happen [5]. For under-compression, the repetitive compression in the final (or called the central) chamber or back flow from discharge chamber will occur in different designs of STC. For over-compression, the STC will compress the gas to its design point regardless of the high pressure in chambers and extra work is consumed. Under-compression could not be avoided except by designing a STC with a lower volume ratio while narrowing the range of operating conditions it performs. Nevertheless, over-compression could be reduced by using bypass valves added to the fixed scroll. Discussions regarding bypass valves of STC are seldom seen in papers but have been presented in several patents. Murayama et al. [41] designed two groups of bypass holes (Fig. 3-1) for each compression chamber with valves operated by pressure differences to prevent over-compression. Fuji et al. [42] use a plurality of symmetrical bypass holes (Fig. 3-2) to avoid over-compression caused by the open delay of the bypass valves. A STC with a back pressure mechanism for axial seal and bypass valves for over-compression is also exposed (Tsubono et al. [43]). In addition, a study using bypass mechanism and optimization of the volume ratio in STC was presented to improve efficiency by 10 to 20 % under the conditions of both low speed and low pressure ratio (Morimoto et al. [24]). Even so, the examples above 29.