Mathematical Model and Energy Efficiency Analysis of a Scroll-type Air Motor

Transcription

1 IENG Intenational Jounal of pplied Mathematic, 38:, IJM_38 3 Mathematical Model and Enegy Efficiency nalyi of a Scoll-type i Moto Li Yang, Jihong Wang, Stephen Mangan 3, Jame W Deby 3, Nan Lu btact The pape peent a implified mathematical model fo a coll-type ai moto. Though ai powe/enegy analyi, it ha hown that a coll-type ai moto with a popely pecified tuctue and uitable peue of compeed ai upply i able to achieve high enegy efficiency and atifie ai powe equiement. The featue of a coll-type ai moto in geomety, mechanical tuctue, and pneumatic powe tanmiion ae tudied in the pape. Diffeential geomety i ued a the mathematical tool to paameteie the coll. In thi tudy, Matlab Symbolic Math Toolbox i found vey efficient in finding analytic expeion of coll chambe volume and genealied toque. State hifting i employed to olve the dynamic ytem equation numeically. Enegy efficiency of the coll-type ai moto i analyed baed on the mathematical model. Index Tem Enegy efficiency, coll-type ai moto, pneumatic actuato, mathematical modeling. I. INTRODUCTION Since 96, compeed ai, a a kind of clean and afe enegy, ha been ued in moden indutial manufactue. Compaed with hydaulic and electical countepat, pneumatic ytem ae envionmental fiendly, cleane, imple, eaie in maintenance. They can wok in hah envionment, wok without pak, tall without damage. Nowaday, compeed ai ytem take up a ignificant pat of the total electicity conumption in manufactuing induty. In 998, in Japan, pneumatic ytem conumed % to % of the total electicity upplied to factoie, eaching 4 billion kilowatt hou [. That wa appoximately 5% of the national total electicity conumed in Japan [. Cutome ae becoming awae of that pneumatic actuato ae compaatively expenive to opeate due to it lowe enegy efficiency. In fact, enegy efficiency of pneumatic actuato i found to be 3~3% and often lowe than %, which i much lowe than it electical countepat, 6% [. Ue have ealied that ome ieveible pocee, uch a ai leak, fiction, peue loe though dye, filte, etc a well a Manucipt eceived 9 th Octobe, 7. Li Yang and Nan Lu ae with Depatment of Electic Engineeing and Electonic, Univeity of Livepool, Livepool, L69 3GJ UK (D Jihong Wang i the autho fo coepondence. She i with the Depatment of Electonic, Electical and Compute Engineeing, Univeity of imingham, Edgbaton, imingham 5 TT, UK, Telephone: +44 ( 44358; j.h.wang@bham.ac.uk 3 Stephen Mangan and Jame W. Deby ae both with Enegetix Goup PLC, Capenhut, CH 6EH, UK. impope etting and opeation account fo enegy wate of pneumatic ytem. coll type ai moto, alo known a a coll expande i a elatively new to pneumatic actuato. It unique tuctue featue it many advantage a well a highe ability of enegy conveion than conventional pneumatic actuato, uch a cylinde, vane-type ai moto, and etc. The coll technique i now mainly and widely implemented in ai conditione and efigeato compeo, becaue it compact deign matche the equiement of mall, quiet, and highly efficient efigeation compeo. Recently, the concept wa e-invented to ai moto. In tead of compeing ai to aie peue, the ai moto o expande eleae the high peue ai enegy to geneate a diving foce, which lead to a uitable actuato fo diffeent application. Fo example, coll ai moto can be ued to ecove wok in fuel cell [4. Thi pape aim to explain why coll ai moto have highe enegy efficiency than othe type of ai moto. The fundamental poge to the explanation i a newly developed implified mathematical model of a coll-type ai moto. The model tat fom the baic geomety deciption to the deciption of dynamic elationhip of compeed ai, genealied toque, enegy conveion, and dynamic epone. The pape epot a implified mathematical model and it enegy efficiency analyi baed on the popoal of ai powe calculation given in [ and [. II. WORKING PROCESSES OF SCROLL IR MOTOR coll-type ai moto i eentially a efigeation coll compeo woking backwad. It conit of two intemehed identical coll, one of which i otated though 8 degee with epect to the othe, that i, one i mioed with epect to the othe [5. The moving coll can evolve eccentically with epect to the fixed one to fom eveal ealed cecent chambe. The moving coll wobble inide the fixed coll, which doe not otate but jut wobble on a cam, whee a haft i located. Compeed ai i intoduced into the cente of a coll ai moto though the inlet, then the potential enegy foce the moving coll wobble in the diection that the chambe ae getting bigge and towad the oute peiphey to dive the cankhaft. Thi tuctue featue coll ai moto deiable advantage elative to othe type of ai moto, becaue the motion i otay and can be completely balanced, which will educe vibation and noie. ll thee chaacteitic (dvance online publication: 9 Febuay 8

2 IENG Intenational Jounal of pplied Mathematic, 38:, IJM_38 3 Fig. aic geomety of a pial [6 III. GEOMETRY OF SCROLL IR MOTOR Fig. Schematic diagam of a coll-type ai moto make a coll ai moto a imple, quiet and eliable machine at a competitive manufactuing cot. typical coll ai moto in motion i hown in Fig.. It i hown that both coll ae cicula involute of thee wap. The gey coll i the fixed coll and the black one epeent the moving coll that tavel along the obit anticlockwie when compeed ai come into Chambe in the cente though the inlet. Each coll i fitted to a back plate. the moving coll tavel along the cicula obit, thee two coll keep contacting at ome point, o that thee ae even numbe of cecent chambe. When a coll ai moto i unning, it goe ucceively and peiodically though the fou tate in the equence indicated by the aow a hown in Fig.. n ai moto woking poce include thee phae, chaging, expanion, and dichaging phae. chaging phae tat at the uppe left diagam in Fig., when compeed ai tat enteing the cental chambe maked with. Thee ae now fou ealed chambe,,, 3, and 3. fte one quate of a cycle, the coll ai moto come to the uppe ight diagam in Fig. diagam of a implified coll-type ai moto, the chambe i getting bigge; chambe,, 3, and 3 have moved anticlockwie and inceaed in ize. It continue though anothe quate of cycle. Then afte one complete cycle, the coll poition come back to the diagam in uppe ight. ut the ai tated life in chambe i now patially in chambe and and get into the ealed chambe. fte the ai i ealed, it tat to expand with expanion of chambe. the ai moto continue though the econd cycle, the ai in chambe and expand and ente into chambe 3 and 3. Though the thid cycle, the ai in chambe 3 and 3 expand futhe and finally dichage immediately to the ophee when 3 and 3 ae not ealed any moe afte the tate hown in the lowe ight diagam in Fig.. Compeed ai alway puhe the moving coll to go along the obit though the thee phae o that thee i output wok deliveed though the eccentic haft.. The mathematical deciption fo a pial pial i the fundamental geomety cuve of a coll. cicula pial i hown in Fig., in whichϕ i the tangential angle of a point on Spial. Tangential angle i the angle fomed fom the hoizon to the tangent line at a point on the e ϕ i the unit cuve. epeent the length of the ac MN, ( tangent vecto at point M, f ( ϕ i the unit nomal vecto at point M, f ( ϕ = ( inϕ,coϕ and e ( ϕ = ( coϕ, inϕ ae a pai of othonomal fame, ρ i the adiu of the cuvatue, and C i the cente of cuvatue. point ( x, y on a pial i decibed by the following pai of equation: x ϕ cou = ρ ( u du y in u If the initial point of the pial i = ( x ρ = ρ + kϕ, then ( become, y ( and x ϕ ( ϕ cou = + ( ρ + ku du ( y in u whee k = d / dϕ detemine the hape of a pial. Fo a tandad involute coll with a contant wall thickne, k = ( + d /π, whee i the adiu of the obit and d i the thickne of the wall [7.. Relationhip between the moving and the fixed coll To implify analyi, in thi pape, the thickne of coll wall ae not conideed, that i, they have a zeo thickne. Figue 3 how a implified coll ai moto tuctue with thee wap of coll. In thi claical deign, both the moving coll and the fixed coll ae cicula involute. The gey coll i the moving coll; the black one i the fixed coll; the dahed cicle in the cente i the obit along which the moving coll (dvance online publication: 9 Febuay 8

3 IENG Intenational Jounal of pplied Mathematic, 38:, IJM_38 3 wobble; i the obit angle that indicate the location of the moving coll. The obit angle vaie at the peiod of π. The chambe maked with i the cental chambe whee the inlet i located. The chambe maked with,, 3 and 3 ae two pai of cecent ide chambe that ae ealed. Side chambe ae located at the ide of the cental chambe. coll ai moto, nomally, ha only one cental chambe, but a cetain pai of ide chambe. Fig. 3. diagam of a implified coll-type ai moto When a coll ai moto i unning, the moving coll move anticlockwie along the obit. If we poject ome moment duing one cycle onto one ingle figue, it i eay to ay that the moving coll fom a family of cuve, and the fixed coll i the envelope of the family a hown in Fig.. The obit i given a ( in, co D = f = (3 and the equation of the fixed coll i epeented by ( ϕ the family of the moving coll i, then ide of the moving coll, +, contact the inne ide of the fixed coll, +, at ϕ = nπ ( n =,,,3L. The inne ide of the moving coll,, contact the oute ide of the fixed coll,, at ϕ = nπ ( n =,,,3L. The ubciption + and - indicate the two ide of a coll. ecaue thi pape peent a implified model of a coll ai moto and the aumed wall thickne value i zeo, ( ϕ = + ( ϕ + π = ( ϕ + π and ( ϕ ( ϕ π = ( ϕ π. = + Fig. 4. family of pial Thee i one cental chambe and two ide chambe (one pai a hown in Fig.5. The boundaie of the chambe have been paameteized a (4 and (5. ( ϕ = ( ϕ D(, + (4 Fom (3 and (4, the fixed coll can be decibed by: ( ϕ = ( ϕ + D( ϕ + nπ, n Z (5 ecaue D ( ϕ i peiodic with a peiod of π, we have in fact two diffeent envelop, one fo each ide of ( ϕ,, i.e. ( ϕ = ( ϕ + ( ϕ ( ϕ = ( ϕ ( ϕ + + D D. (6 C. Calculation of chambe volume Geen' Theoem give the elationhip between a line integal aound a imple cloe cuve and a double integal ove the plane egion bounded by a cloed cuve. It can be ued to find aea. In Fig.5, at the obit angle, the moving coll and the fixed coll contact each othe at ome point. The oute Figue 5. Illutation of the aea bounded by the fixed and moving coll (dvance online publication: 9 Febuay 8

4 IENG Intenational Jounal of pplied Mathematic, 38:, IJM_38 3 ccoding to Geen Theoem, the volume of the cental chambe i V c ( = + π y y π z ( ϕ d( x ( ϕ + x ( ϕ d( y ( ϕ + z ( ϕ d( x ( ϕ + x ( ϕ d( y ( ϕ Z tand fo the height (o chambe thickne of the coll. (6 angle between the diection of movement and that of the peue at the point. Thu the total genealized foce in the diection of the motion to dive the moto moving i integal of (8. The volume of the ide chambe bounded by [, + π and [, + π i calculated by V ( = z y ( ϕ d( x ( ϕ + x ( ϕ d( y ( ϕ + z + π + π y ( ϕ d( x ( ϕ + x ( ϕ d( y ( ϕ (7 IV. MOTION NLYSIS When a coll ai moto i unning, evey paticle on the moving coll i taveling along a cicle, which i identical to the obit. Thu, at time t, the diection of the motion, i.e. the diection of moving velocity, i the tangent diection of the point D ( ( t.. The diving egment and the balanced egment Fo a coll-type ai moto, ome chambe ae identical, o ome egment of the coll ae balanced by the peue in the identical chambe. In Fig., thee ae two pai of identical chambe, and, 3 and 3. The peue in the identical chambe ae of the ame value. Thu, the egment between and i balanced. Futhemoe, the peue value in 3 i lowe than that in, o the egment between and 3 i a diving egment. Fo the ame eaon, the egment between 3 and the ophee i alo a diving egment.. nalyi of the genealized foce o toque Tanient peue in evey chambe can be obtained. y integating the peue along the wall of the chambe individually, the diving foce can be deived. Taking into conideation of fiction, tiction and the pay load, a mathematical deciption fo the coll diving foce o toque i developed, which can epeent the mechanical tuctue and the dynamical poce of a coll-type ai moto. In Fig.6, the bigge aow indicate the diection of motion of the moving coll, the malle aow indicate the diection of the um peue at the point on the moving coll. The um foce at a point i: df = zpd coθ (8 whee z i the height of the coll wall, p i the um peue at the point, d i the ac length, and θ = ϕ π / i the Fig. 6 Relationhip between the diection of motion and the diection of the um of the peue V. MTHEMTICL DESCRIPTION OF THE IR MOTOR DYNMIC PROCESS. State-pace deciption To deive a mathematical model of a implified coll ai moto, ome aumption ae made a thee i no ai leakage of the moto, the compeed ai i ideal ga, the tatic fiction can be neglected, thee i no heat tanfe, moto uounding tempeatue i contant, and the upply peue i kept at a contant level. Let x be obit angle, x angula velocity, x 3 peue in the cente chambe, x 4 peue in the fit pai of ide chambe, and x 5 peue in the econd pai of ide chambe. aed on the above deciption, by applying Newton econd law of motion and the tandad oifice theoy [9, involving the mathematical model fo contol volume, a tate-pace deciption of a coll-type ai moto with thee-wap coll can be deived. Equation (9 peent the tate equation of a implified coll ai moto with a popotional contol valve. = x = J V& c 3 = Vc V& = 4 V V 5 = V ( τ M x ( x γx + 3x ( x ( Vc x ( x, γx4 x ( x, & ( x, γx5x ( x, p f Rγc c c d [, π ( π,π k T p XX max f ( p / x 3 (9 (dvance online publication: 9 Febuay 8

5 IENG Intenational Jounal of pplied Mathematic, 38:, IJM_38 3 whee f ( p = / γ ( + γ z + + τ = z + / γ ( p p / p [( ρ + k + kπ ( x3 x4 ( ρ + k + 5kπ ( x4 x5 ( ρ + k + 9kπ ( x p [( ρ + k + kπ ( x3 x4 ( ρ + k + 5kπ ( x p 5 4 / p c u p J i the total inetia of the moto; < p < < c [, π ( π,π M f i the fiction coefficient; V c i the volume of the cental chambe; V i the volume of a ide chambe; γ =. 4 i the atio of pecific heat; c =.4 ; c =. 58 ; c k = ; c d =. 8 ; X i the effective valve width; X i the maximal effective valve max width; p i the upply peue; p i the peue of ophee; p = p / p ; τ tand fo toque. d u Fig. 7 State hifting of equation (9. State hifting In ode to olve the dynamic ytem (9, a method called tate hifting mut be intoduced. hown in Fig.7, a coll ai moto un epeatedly in a peiod of π. Compeed ai goe into the coll-type ai moto. The fit chambe it eache i the cental chambe. long with motion of the ai moto, at x = π, a cetain ma of compeed ai i ealed in the fit pai of ide chambe. The initial value of x 4 i the final value of x 3. fte anothe cycle, thi cetain ma of ai i in the econd pai of ide chambe, o the initial value of x 5 i the final value of x 4 of the lat cycle. The pinciple of the tate hifting i that the tate vaiation of a cetain ma of compeed ai i obeved a it tavel fom the cental chambe though the ide chambe to the outlet. If thee ae N pai of ide chambe, at evey x = π, the peue of the th n chambe become the peue of the ( n + th chambe and then the ai moto goe into anothe cycle [8. VI. ENERGY EFFICIENCY NLYSIS. Enegy caied by ai Enegy epeent the potential fo poducing wok that can be extacted fom a ubtance. Fo compeed ai, the potential i the available enegy fom it high peue. It i the maximum wok that can be extacted fom ai a it undegoe a eveible poce fom a given high peue tate to the opheic tate on the uounding of the ophee [. When the ai tempeatue i equal to opheic tempeatue, the available enegy pe unit ma of ai can be expeed by e = pv m ln p p The definition of ai powe i p p P = GRT ln = pq ln ( p p whee G tand fo ma flow ate; R i the ga contant; Q tand fo volumetic flow ate; T i opheic tempeatue. Fom (, we know that if the ai peue p i malle o equal to the opheic peue p, the powe P i not geate than zeo. That mean ai convey available enegy if and only if it peue i geate than p. i powe conit of two pat. One i tanmiion powe that epeent puhing powe fom the upteam to the downteam. It i calculated by uing the ame expeion a hydaulic powe. Pneumatic cylinde nomally utilie thi pat only. Howeve, compeed ai contain not only tanmiion powe, but alo, becaue of compeibility of ai, expanion powe. Even when the upteam i hut off, compeed ai can till pefom wok by expanding. Thu anothe ignificant pat of ai powe i expanion powe that how the wok ability by ai expanion [. Fig.8 how the pecentage of expanion enegy inceae a the ai peue inceae, while the potion of tanmiion enegy deceae. The tempeatue of ai hadly influence ai enegy when we ae dicuing enegy efficiency of a pneumatic actuato becaue the ai powe inceae about 5% when the tempeatue diffeence i K. eide tanmiion enegy and expanion enegy, kinetic enegy can alo be conveted into mechanical wok. Howeve it account fo le than 5% of available enegy when the aveage velocity i below m/ at 5 the peue above 3 Pa [. Theefoe, kinetic enegy can nomally be neglected when we ae not analying the intenal enegy ditibution in pneumatic component. (dvance online publication: 9 Febuay 8

6 IENG Intenational Jounal of pplied Mathematic, 38:, IJM_38 3. Enegy efficiency analyi of a coll-type ai moto Enegy efficiency of a coll-type ai moto i calculated by: Powe geneated by the coll ai moto η = ( i powe Powe geneated by coll ai moto i calculated by: Toque (Nm Speed (pm Powe (kw = ( 955 Fig. how that, fo the ytem (9, while the upply peue inceae fom.35 MPa to 3 MPa, enegy efficiency dop fom 84% down to 46%. Fig. 9 Enegy efficiency v. upply peue Fig. 8 Compoition of ai enegy The enegy efficiency, i.e. ability of enegy conveion, of the coll-type ai moto i much highe than the aveage enegy efficiency of conventional pneumatic actuating ytem, which i often lowe than %. In fact, ieveible pocee uch a fiction, heat tanfe, and ai mixtue will caue ai powe lo, o the actual enegy efficiency of a coll-type ai moto hould below the theoetical value. In fact, enegy efficiency of a coll-type ai moto mainly depend on upply ai peue, uounding ai peue and atio of expanion. Futhemoe, high enegy efficiency doe not mean high powe. Howeve, it i till poible fo a coll-type ai moto to have both high enegy efficiency and high powe. One can feed high peue to a coll-type ai moto with a pope expanion atio. Fom the analyi above, a coll-type ai moto i able to convet moe available enegy of compeed ai. In ode to utilie ai enegy a much a poible, expanion atio of ide chambe i vey impotant. Expanion atio detemine how much expanion enegy i utilied. Howeve the highe expanion atio doe not mean the highe enegy efficiency. It mut be enue that the exhaut peue i above ophee to avoid ove expanion. VII. CONCLUSION Thi pape analyed the ability of a coll-type ai moto to convet ai enegy. The analyi i baed on the implified mathematical model of a coll-type ai moto deived in the pape. vailable enegy of compeed ai wa defined a a elative potential that only exit when the compeed ai tate diffe fom opheic uounding. i powe conit of two pat: tanmiion powe and expanion powe. Nomally, a pneumatic actuato like a cylinde ue only the tanmiion powe. That lead to low enegy efficiency of a pneumatic actuato. The analyi how that a coll-type ai moto utilie both tanmiion powe and expanion powe. That eult in high enegy efficiency. REFERENCES [ Maolin Cai, Kenji Kawahima, Tohihau Kagawa, Powe aement of flowing compeed ai, Jounal of Fluid Engineeing, vol. 8, Ma. 6, pp [ M. Cai, T. Kagawa, Enegy conumption aement of pneumatic actuating ytem including compeo, IMechE,, pp [3 T Cozie, G Jone, The Potential Maket fo MicoCHP in the UK, Enegy fo Sutainable Development Limited,. [4 Gao Xiaojun, Li Lianheng, Zhao Yuanyang, Shu Pengcheng, Shen Jiang, Reeach on a coll expande ued fo ecoveing wok in a fuel cell, Int. J. Themodynamic, vol. 7, 4, pp. -8. [5 aolong Wang, Xianting Li, Wenxing Shi, geneal model of coll compeo baed on dicetional initial angle of involute, Intenational Jounal of Refigeation, vol. 8, 5, pp [6 Jen Gaveen, Chitian Heniken, The geomety of the coll compeo, SIM Review 43,, pp [7 Li Yang, Jihong Wang, Jia Ke, Development of a mathematical model of a coll type ai moto, ICSE 6, 5-7 Sept. 6, pp [8 Pete Howell, Fluid Mechanical Modelling of the Scoll Compeo, Cambidge Univeity Pe, 999. [9 William D. Wolanky, John Nagohoian, Ruell W. Henke, Fundamental of Fluid Powe, Houghtom Mifflin Company, 977. (dvance online publication: 9 Febuay 8