Analysis of pressure wave dynamics in fuel rail system

Transcription

1 It. Jl. f Multiphysics Vlume Number Aalysis f pressure wave dyamics i fuel rail system Basem Alzahabi ad Keith Schulz Ketterig Uiversity Siemes Autmtive ABSTRACT A mdel f a amplified cmm rail fuel system is simulated i Matlab t aalyze the wave mechaics i the rail. The ijectrs are mdeled as a system f liear ad -liear ODE s csistig f masses, a helical sprig, cmpressibility effects frm fluid vlumes, ad hydraulic flw thrugh rifices. The ijectr simulati the predicts the rate f il csumpti, which is the iput it the rail mdel. The rail is mdeled i three sectis which are cupled tgether. The pits where the cuplig ccurs are the lcatis where the curret firig ijectr ad the pump supply are cected t the rail. This allws the mdel t ctrl the pressure ad velcity (as budary cditis) at these pits. The rail mdel is based the D, udamped wave equati, i a -dimesial frm [] (i the psiti variable,.) The Reducti f Order methd was used t slve the wave equati with the Matlab fucti PDEPE. The mdel was ru with tw differet sets f iitial cditis - mial (cstat pressure ad zer velcity,) ad wrst case usig a simplified represetati f the pressure ad velcity distributi at start f ijecti. This was de t determie the effect f rail waves at the start f ijecti, the utput f the mdel. The variati i fuel delivery, due t the variati i rail pressure, was the evaluated at three peratig cditis - Idle, Peak Trque (PT) ad High Speed ight ad (HS.) The simulati utput is the cmpared t aalytical slutis f tw frms f simplified gemetry, usig the prduct methd t slve the system [.]. INTRODUCTION Mder Diesel fuel systems utilize a cmm rail t distribute pressurized fuel t uit ijectrs. Previus geerati techlgy cmmly used uit pumps, which were drive frm camshafts. This had the disadvatage f limitig ijecti pressure ctrl (because the ijecti pressure is apprimately prprtial t egie speed.) Cmm Rail fuel systems perate idepedet f egie speed. This adds greater ctrl fr egie maagemet ad emissis reducti. A hybrid apprach t the cmm rail ijectr is the Amplified Cmm Rail (ACR.) Desigs utilizig a hydraulic amplificati system t bst the ijecti pressure (secdary) typically use the eergy frm a supply pressure (primary.) The advatage with a ACR desig is that higher ijecti pressure (> Mpa) ca be achieved. The higher ijecti pressures prduce a better atmizati f the fuel which the reduces the frmati f NO ad particulates durig cmbusti. Variati i the rail pressure hwever icreases the variati i fuel delivery [, 3] ad icreases emissis utput frm the egie.

2 4 Aalysis f pressure wave dyamics i fuel rail system ECU Primary pump ad IPR Secdary fuel pump Rail Ijectrs/Cyliders Figure Schematic f a ACR system. Aalyzig the system behavir at Idle, PT ad HS is useful because these represet cmm cditis i vehicle ad egie lad histries.. SYSTEM MODE A schematic f a ACR system is shw i Figure. A primary system pump supplies pressurized il t the rail (up t 4 psi.) A secdary system pump supplies lw pressure (6 psi) fuel t the ijectrs. The ijecti pressure regulatr (IPR,) ad the ijectr ctrl valves are ctrlled by the Egie Ctrl Uit (ECU.) The primary supply pump is a variable displacemet pump. 3. INJECTOR MODE A secti view f a ACR ijectr is shw Figure with the ctrl valve spl is shw i a eutral psiti. Whe the ECU supplies curret t the pe cil, the magetic frce pulls the spl t the pe psiti (left.) Fluid the flws frm the supply prt, thrugh the valve ad it the itesifier. The supply pressure the prduces a frce the tp surface f the pist which drives the pist dwward. The frce trasmits thrugh t the pluger which the cmpresses the diesel fuel i the lwer secti f the itesifier. The zzle sprig eerts a dwward frce the eedle. The pressure ecessary t vercme the zzle sprig frce, ad lift the eedle, is the Valve Opeig Pressure (VOP.) As the fuel pressure i the lwer ed f the itesifier icreases abve VOP, the eedle lifts ff f the seat i the zzle, ad fuel flws thrugh the rifices i the zzle ad it the egie cylider. At the ed f ijecti, the ECU supplies curret t the clse cil. This causes the spl t shift t the clse psiti (right.) The itesifier sprig the pushes the pist ad pluger upward, which causes the primary fluid t vet frm the space abve the pist. This als causes the ilet check ball t lift ff f its seat ad fuel t flw it the lwer itesifier bre. After the pist has reached the tp f the bre (i ctact with the ctrl valve bdy) the ijectr is ready fr the et ijecti. A schematic represetati f the ijectr ad rail is shw i Figure 3. The pressure i the rail ca be mdeled frm the cmpressibility equati, where P r is the rail pressure, B is the Bulk Mdulus, V r is the rail vlume, Q p is the pump flw, Q is the ijectr flw, ad t is time.

3 It. Jl. f Multiphysics Vlume Number Ctrl valve Fill prt Drai prt Ope cil Clse cil Itesifier bdy Pist Spl Itesifier sprig Pluger Check ball Check disk Sprig cage Nzzle sprig Needle pi Needle Nzzle Figure Secti view f a ACR ijectr. dp r B = V Q p Q dt r ( ) () The flw thrugh the ctrl valve ca be mdeled as rifice flw, Q = A c ( P P r ) ρ () where A c is the flw area acrss the spl, P is the pressure abve the pist ad ρ is the desity f the fluid. The pressure abve the pist ca be mdeled as Qdt Ad dp = B V + A (3) where A is the area f the pist, is the pist mti, ad V is the residual vlume betwee the pist ad the spl (whe the pist is at the tp f its bre,)

4 6 Aalysis f pressure wave dyamics i fuel rail system Q p, P c Rail V r, P r A c )( P Q A m P f )( A f A Q f Figure 3 Schematic f a ACR ijectr. The mti f the pist/pluger system ca be mdeled as a mass-sprig system, m&& + K = PA Pf Af F (4) where m is the cmbied mass f the pist ad pluger, K is the itesifier sprig rate, P f is the fuel pressure ad A f is the pluger area. The fuel pressure ca be mdeled as Ad f Qdt f Pf = B V A f f (5) where Q f is the fuel flw ad V f is the vlume f fuel i the lwer secti at the start f ijecti.

5 It. Jl. f Multiphysics Vlume Number The fuel flw ca be mdeled as Q f = A P ρ f (6) where A is the zzle rifice flw area. The ijectr mdel is the slved with the system f equatis -6 usig Matlab. 3.. RAI MODE The wave mechaics i the rail ca be mdeled with the wave equati u = c t u. The Matlab fucti PDEPE slves elliptic ad parablic partial differetial equatis (PDE s.) The wave equati hwever is a hyperblic PDE. PDEPE ca be used t slve the wave equati by applyig reducti f rder ad the slvig a system f PDE s. Substitutig u v = t (7) it the wave equati gives v t = c u The pressure i the fluid is (8) P=B u (9) The mius sig i Eq 9 idicates that tesi is egative pressure ad cmpressi is psitive. Differetiatig Eq 7 with respect t gives v = u = u = P t t t B. Multiplyig bth sides by -B gives P = B v t () Multiplyig bth sides f Eq 8 by -B ad substitutig Eq 9 it Eq 8 gives = B v P c t ()

6 8 Aalysis f pressure wave dyamics i fuel rail system Q p 3 = / 3 Q Figure 4 Schematic f rail gemetry. The system f equatis ad ca the be slved give iitial ad budary cditis fr pressure ad velcity. A schematic represetati f the rail is shw i Figure 4. The ijectrs are cected at equally spaced psitis alg the rail crrespdig t the egie cylider lcatis i the blck ad head. Oly e ijectr psiti is shw because ly e ijectr delivers fuel at a give pit i time. The rail is divided it three sectis, each f a differet legth. The PDEPE mdule i Matlab ca slve a system f equatis fr all three sectis, simultaeusly. This ca be de by makig the equatis -dimesial i the -crdiate, by dividig each i equatis ad by their respective legths, where the -dimesial crdiate, X = = = 3 3 [, pg ] The crdiate X the varies betwee ad fr all three systems. The system f equatis is the: B v = P = c t X ad P v B t X B v c = P = t X ad P v B t X B 3 v3 3 = P c t = X ad 3 P3 v3 B t X () The system eeds t be cupled at the tw cmm budaries, = ad 3 =. The pressure is equal bth sides f these budaries s that, i -dimesial frm, P (,t) = P (,t) ad P (,t) = P 3(,t) The flw acrss these budaries must be equivalet t the flw frm the pump ad the flw t the ijectr. The flw ca be divided by the crss-sectial area f the rail t give the velcity. The velcity budary cditis are the v = v (,t) v (,t) ad v p = v 3(,t) v (,t), as shw belw i Figure 5.

7 It. Jl. f Multiphysics Vlume Number Q p v p = v 3(, t) v (, t) v (, t) v 3(, t) v (, t) v (, t) Q v = v (, t) v (, t) Figure 5 Velcity budary cditis. The remaiig tw budary cditis are zer velcity at = ad 3 = /. This is equivalet t fiig the ed pits f the rail i a fied-fied system, ad ca be mdeled i PDEPE as v (,t) = ad v 3(,t) =. The rail mdel is the slved i the Matlab. The physical gemetry f the rail is a meter lg tube with a prt i the middle (at 5 mm) fr the pump supply, tw prts 5 mm frm each ed (# ad #6 ijectrs,) ad the remaiig ijectrs (# - #5) 5 mm apart. The rail vlume is.7 ad the crss-sectial area is 7 cm. 3.. RESUTS The results frm ruig the ijectr simulati uder Peak Trque peratig Cditis are shw i Figure 6. The upper pae i Figure 6 is the Rail Pressure, the middle pae is the ijecti pressure, ad the lwer pae is the Rate f Ijecti (ROI). The rate f ijecti is simply the istataeus flw rate f fuel it the egie. Tw f the simplificatis that were used i the ijectr mdel are. Istataeus mti f the ctrl valve spl, ad. Needle (zzle valve) always pes. Mdelig the mti f the spl requires aalysis f the magetic ad the dyamics, ad takes the frm f a -ff -liearity. Mdelig the eedle mti als requires additial dyamic aalysis ad is als -liear. Bth f these effects were mitted i rder t simplify the mdel. Fuel ijectrs typically als have a additial check valve betwee the pumpig uit ad the eedle the Reverse Flw Check (RFC.) The purpse f the RFC is t prevet reverse flw f ehaust gases, it the ijectr. This ca ccur at the Ed f Ijecti (EOI,) after the ctrl valve is clsed ad while the eedle is still pe. The mti f the RFC als is -liear. Asimplified represetati f the ijectr ROI is shw i Figure 7. The EOI is typically a eact reversal f the start f ijecti (SOI.) The SOI ad EOI are mdeled i the rail simulati as tt tt Q = Q e ad Q Q e ( ) si ss ( ) = / t c d / c ei ss (3) where t c is the time cstat ad t d is the time (time durati f the ijecti.) Equatis 3 are used t tur the ijecti i the Budary Cditis. The scillatis that ccur at the begiig f the ROI i figure 6 are typical f real systems. These scillatis als typically ccur i the EOI. The scillatis culd be prduced i the mdel by addig a decayed sie fucti t the equati. This effect, ad a mdel f the RFC were als mitted i rder t simplify the mdel.

8 3 Aalysis f pressure wave dyamics i fuel rail system Rail pressure (MPa) Ijecti pressure (MPa) Rate f ijecti (mm3/ms) Time (ms) Figure 6 Ijectr.m utput at Peak Trque. Figure 7 ROI (mm 3 /ms) Time (ms) Simplified represetati f ROI..5 I Matlab M-file e ca calculate the pressure ad velcity distributi i the rail frm a sigle ijecti evet. The utput is i the frm f surface maps (pressure as a fucti f time ad rail psiti) ad -y plts f pressure as a fucti f time. The surface plts give a verall view f the wave prpagati i the rail. A surface plt f the respse frm Ijectr firig uder Peak Trque perati is shw i Figure 8. The respse is a result f bumps i the surface prpagatig utward frm the pump lcati, ad dips i the surface prpagatig utward frm the ijectr lcati, at the wave speed

9 It. Jl. f Multiphysics Vlume Number Cylider ijecti at Peak Trque 6 Pressure (MPa) Rail psiti (mm) Figure 8 Rail respse due t Ijectr at Peak Trque. 9 Pressure at ijectr 5 due t cylider ijecti at PT 8 Pressure (MPa) Figure Time (ms) Effect f cylider cylider 5 at PT. B 7. GPa c = = = 44m/ s ρ 8. gm / cc The bjective f this study is t evaluate the effect f the respse successive ijectrs firig rder is , respectively. Therefre, we eed t eamie the respse due t cylider at the lcati f cylider 5. This is shw i Figure 9. The simulati was ru

10 3 Aalysis f pressure wave dyamics i fuel rail system.4 Pressure at ijectr 5 due cylider ijecti at HS. 9.8 Pressure (MPa) Time (ms) Figure Effect f cylider cylider 5 at HS. fr 4 ms f ijecti, ad a additial 6 ms after EOI. Durig the ijecti evet, the pressure decreases because the ijectr flw rate (ut flw) is greater tha the pump flw rate (i flw.) After EOI, the pressure recvers because the flw t the ijectr has stpped. If the simulati were ctiued fr ather ~6 ms, the pressure wuld fully recver t the ctrl pressure (i this case 8 MPa,) after which the pump ctrl wuld hld the pressure cstat (with a certai amut f steady state errr.) The egie speed at PT is rpm. The time betwee ijectis is therefre mi rev 6 S ms t = revs 3ij mi 6 7 ms ij S =. / The eact timig f the et ijecti hwever is depedat the eact peratig cditi. The time it takes t mve frm a valley t a peak i Figure 9 is apprimately.3 ms. The timig at rpm is 6.4 ms per ijecti. S a differece i egie speed f rpm wuld make the differece betwee startig the et ijecti at 8.3 vs 7.7 MPa. Therefre, the amplitude f the wave i the recver prti f Figure 9 is the measure f variability the system. The respses due t Cylider at HS ad Egie Idle are i Figures ad. Similar data ca be pltted fr the remaiig cyliders. Table is a summary f the aalysis cditis, where P c is the ctrl pressure (MPa,) v f is the fuel delivery (mm 3,) ROI is the rate f ijecti (mm 3 /ms,) speed is the egie speed (rpm,) v is the velcity f il t the ijectr (m/s,) v p is the velcity f il frm the pump (m/s) ad t d is the ijecti durati (ms.) Table is a summary f the results. Oly data fr cyliders 3 are pltted because, by symmetry, cyliders 4 6 wuld prduce the same result. PDEPE requires e iitial cditi fr each f the si PDE s i the system. Fr the velcities, the iitial velcity was set t zer, ad the pressure was set t P c. I a real system hwever the iitial state wuld deped the fial state f the previus ijecti cycle. The

11 It. Jl. f Multiphysics Vlume Number Pressure at ijectr 5 due t cylider ijecti at Idle Pressure (MPa) Time (ms) Figure Effect f cylider cylider 5 at Idle. Table Aalysis cditis State P c v f ROI Speed v v p t d Idle PT HS Table Variati i pressure durig pumpig recver State Cylider Cylider Cylider 3 Idle HS PT ly way t iput a pressure ad velcity distributi it PDEPE is with aalytical fuctis (fuctis f ad t.) This is t practical t d fr several cycles. Hwever it ca be de fr e ijecti, t determie the effect. A graph f the pressure (slid curve) alg the legth f the rail after cylider 4 fired is shw i Figure. The dashed curve is a aalytical epressi that apprimates the pressure. Figure 3 is a similar set f curves represetig the velcity distributis. These fuctis are calculated i the file iputs.ls. Iputtig these epressis it the iitial cditi i PDEPE prduced the results shw i Figure 4 ad 5. Figure 4 is a surface plt f the pressure vs. rail psiti ad time. Figure5 is a plt f the pressure drp durig ijecti, ad the recver after EOI.

12 34 Aalysis f pressure wave dyamics i fuel rail system 9 Aalytical represetati f rail pressure at ed f cylider 4 ijecti appr actual Pressure (MPa) Figure Pressure iput t ijectr. Rail psiti (mm). Aalytical represetati f rail velcity at ed f cylider 4 ijecti. appr actual Velcity (m/s) Figure 3 Velcity iput t ijectr. Rail psiti (mm) Bth plts shw a clear ifluece frm the previus cycle with additial wave effects evidet i the surface plt ad icreased variability i the P vs. t plt (frm ±.35 MPa t ±.5 MPa.) The variati i fuel delivered by the ijectr, due t the variati i pressure, ca be determied frm the ijectr mdel by itegratig the ROI trace ver the ijecti durati. Iputtig the mi ad ma prjected pressure at the ed f the recvery it the ijectr mdel prduced the results summarized i Table 3. The largest variati i fuel delivery ccurred at PT with ±.5 mm3. HS ad Idle were ±.75 ad ±.75 mm3. The HS ad Idle pits shuld be acceptable fr egie perati.

13 It. Jl. f Multiphysics Vlume Number Aalytical represetati f rail velcity at ed f cylider 4 ijecti. appr actual Velcity (m/s) Rail psiti (mm) Figure 4 Pressure waves due t ijectr (with IC s). 9 Pressure at ijectr 5 due t cylider ijecti at PT 8 Pressure (MPa) Time (ms) Figure 5 Effect f cylider cylider 5 (with IC s). The PT variability des typically perate that high, but eeds t be lwered i rder t achieve future emissis stadards COMPARISON TO ANAYTICA SOUTION The mdel f the rail is a fied-fied system with velcity cstraits at the ijectr ad pump cectis. A eact aalytical mdel f the rail, with ijectr ad pump iputs, wuld be difficult t prduce because it requires simultaeus sluti f a system f PDE s. Hwever, it is pssible t cmpare the umeric sluti fr tw simplified mdels f the rail: a fied-fied ad a fied-free mdel.

14 36 Aalysis f pressure wave dyamics i fuel rail system Table 3 Variati i fuel delivery due t pressure Peak Trque HS Idle P(MPa) v(mm 3 ) P(MPa) v(mm 3 ) P(MPa) v(mm 3 ) It ca be shw that the sluti t a fied-free system is P=4P cs cs ( ) t cs ω = [4, pg 433] The sluti methd is preseted here i a slightly differet frm tha [4.] A geeral sluti f the wave equati ca be epressed as: ω u A c B ω c ( t, ) = si + cs C si t D cs ω + ω t With e ed f the rail fied, ad the ther ed free, the budary cditis are u ad u (, t) = = Substitutig the first cditi it the geeral sluti gives u(, t) = B( Csiωt+ Dcs ωt) = B= Substitutig the secd cditi it the geeral sluti gives u ( t, ) ( t, ) = ω ω ω ω A cs ( Csiωt+ Dcsωt) cs,,,... c c c c k = = k = 35 The atural frequecies are the ( ) c ω =, = 3,,... The geeral sluti ca the be rewritte i the frm v ω u( t, ) = siωt+ u csωt si ω c ( )

15 It. Jl. f Multiphysics Vlume Number u (, ) Slpe = P /B Figure 6 Saw tth wave iitial cditi. where u ad v are the displacemet ad velcity at time zer. The velcity prfile is the v( t, ) = u( t, ) = ( vcs t ω & ω uωsi ωt)si c The iitial cditis are v p = ad u = B (, ) (, ) The velcity at time zer is &u ω = v c = (, ) si v = The geeral sluti ca the be rewritte i the frm ω u( t, ) = usi c csω t Fr the saw tth wave shw i Figure 6, the iitial cditi u P = B (, ) Ca be epaded thrugh Furier series represetati as P = B b si = p

16 38 Aalysis f pressure wave dyamics i fuel rail system where b Frm itegrati by parts, d si = cs cs d = si cs Oe ca fid, b P B d P B = si = si d P = B The iitial cditi ca the be represeted as P = B P B The verall sluti is the sum f all pssible slutis which ca be epressed as u( t, ) usi ( = ) cs t ω = The iitial cditi is the P u(, ) usi ( ) = cs = si B = These tw series d t have the same frm ad therefre the u cstat cat be evaluated. If hwever, istead f usig the saw tth wave f Figure 6, we use the triagle wave f Figure 7, we btai the fllwig: = u = b si (, ) = si cs P P = cs B = cs B cs si =

17 It. Jl. f Multiphysics Vlume Number u (, ) P /B Figure 7 Triagle wave iitial cditi. where the Furier cefficiet is: b The Furier cefficiet the reduces t b Rearragig the cefficiet equati gives b P B = si d P B P + + B si d P B d P = si + ( ) si B d P B d P B = si si d P B The first tw itegrals ca be slved with Itegrati by Parts: = d si cs cs d = cs + cs d This reduces t = d si si cs si d

18 The Furier cefficiet is the This reduces t The iitial cditi ca the be epressed as The eve terms f the series, =, 4, 6..., are equal t zer. Epadig the ifiite series the gives Re-epressig the series s that eve terms are -zer, gives Therefre, = = cs si ( ) si si = = = cs si ( ) si si si si si si si si = = + si si u P B P B (, ) si si 8 8 = = = si si = b P B = 8 si b P B = si cs P B si cs + = 4 P B P B cs cs + si cs P B + si cs cs P B cs 4 Aalysis f pressure wave dyamics i fuel rail system

19 It. Jl. f Multiphysics Vlume Number Figure 8 Respse f a Fied-Free system. The iitial cditi ca the be epressed as u (, ) 8 P = B = cs si ( ) Ad the geeral sluti ca be epressed as u ( t, ) 8 P = B = cs si ( ) csω t Fially, the pressure ca the be epressed as P B u = P t = 4 cs cs ( ) csω = A surface plt f the utput frm the Matlab simulati is shw i Figure 8. The large egative velcity is a result f the free ed havig applied resistace. This pressure respse wuld ly be pssible with vibrati f a steel bar, because the pressure i a liquid wuld drp belw vapr pressure ad the stp decreasig. A -d plt f the results frm the Matlab simulati, verlaid with the results frm the aalytical epressi fr the fied-free system is shw i Figure 9. The aalytical sluti was carried ut t the first terms i the ifiite series. The sluti was calculated i. ms time steps ver the legth f the rail segmet. The plt shws gd agreemet betwee the aalytic ad umeric results. The aalytic sluti is calculated i the file Fied-Free.ls.

20 4 Aalysis f pressure wave dyamics i fuel rail system 3 Pressure i a "fied-free" fluid clum Slid lie = Aalytical sluti Dashed lie = Numeric sluti.6 ms.4 ms. ms. ms Pressure (MPa).8 ms. ms. ms.4 ms Rail psiti (m) Figure 9 Cmparis f aalytical ad umeric results. The sluti t the fied-fied system is smewhat simpler. As with the fied-free system, a geeral sluti f the wave equati ca be epressed as: ω u A c B ω c ( t, ) = si + cs C si t D cs ω + ω t ( ) With bth eds f the rail fied, the budary cditis are u (,t) = ad u (,t) =. Substitutig the first cditi it the geeral sluti gives u(, t) = B( Csiωt+ Dcs ωt) = B = Substitutig the secd cditi it the geeral sluti gives u( t, ) = ω A c si C si t D cs ( ω + ω t )= ω ω si = =,,,... c c 3 ω = c

21 It. Jl. f Multiphysics Vlume Number P (, ) P P (, ) = P + / Figure Fied-Fied iitial cditi. The first iitial cditi is v (,) = The velcity at time zer is &u ω = ACω C = c = (, ) si The geeral sluti ca the be rewritte i the frm ω u( t, ) = usi c csω t The pressure ca the be epressed as P=B u B u c = ω cs ω c csω t where B is the Bulk Mdulus f the fluid. The ttal sluti is the the sum f all pssible slutis, B p = ωu c = The cstat u ca be evaluated frm the remaiig iitial cditi. Figure is a graph f a simple iitial cditi that wrks well fr demstrati f the sluti. Fr a eve Furier series, f( ) = a + acs where a = ω cs c csω t = ad a f ( ) d f = ( ) cs d

22 44 Aalysis f pressure wave dyamics i fuel rail system 9. Cmparis f aalytical t umeric sluti f the wave equati fr a fied-fied system with liear iitial pressure distributi Slid lies = Aalytical sluti. Dashed lies = Numeric sluti Pressure (MPa) Time (ms) Figure Fied-fied system utput. Fr the fucti i Figure, the ffset cstat is a P d = + P = + = P + P = + Ad the series cstat is a = P + cs d P = si P = si + si + = si + cs d cs si d

23 It. Jl. f Multiphysics Vlume Number Which reduces t a = ( ) The iitial pressure is the ( cs ) P = P + + (, ) cs cs = Ad the ttal sluti is the superpsiti f the ffset (a ) ad the ifiite series ( ) P( t, ) = P + + cs cs csω t = ( ) ( ) ( ) Figure is a cmparis f the aalytic results t the umeric results. Agai i this case the aalytic ad umeric results shw gd agreemet. The aalytical sluti was carried ut fr the first fur terms f the ifiite series. The results fr this system are calculated i the file Fied Fied.ls. 4. CONCUSIONS. The pressure waves mve thrugh the rail at the wave speed. The actual wave speed is 44 m/s. This ca be verified frm Figure 9 where the wave traverses the rail ( m) i appr.7 ms. The velcity fr the wave wuld the be m v = = = 49 m/ S t 7. ms. The disturbaces i the pressure distributi, due t ijectr ad pump flw, prpagate back ad frth iside f the rail, reflectig ff f the eds. 3. The pressure scillatis are larger tha what is see i a real physical system. This is prbably due t the missi f dampig effects i the rail. 4. A iitial pressure distributi i the rail superimpses t the pressure distributi that is geerated by the et ijecti. 5. The pressure variati is t sufficiet t cause emissis prblems at Egie Idle r HS, but are sufficiet t cause prblems at Peak Trque. RECOMENDATIONS. Use utput frm the mdel t geerate pressure disturbaces due t ijecti ad pump flw, ad apply these sub-slutis as iitial cditis t the d Alembert sluti. It may the be pssible t pseud-aalytically slve the system, ctiuusly frm e ijecti t the et.. Repeat the aalysis with dampig built it the mdel. 3. Add -liearity fr the spl, eedle ad RFC. 4. Cuple the ijectr ad rail mdels t give a mre accurate ROI budary cditi.

24 46 Aalysis f pressure wave dyamics i fuel rail system REFERENCES [] Advaced Egieerig Mathematics, 6th Editi, Erwi Kreyszig, Jh Wiley ad Ss, New Yrk, C988. []. Zhg, et al., Effect f Cycle t Cycle Variati i the Ijecti Pressure i a Cmm Rail Diesel Ijecti System Egie Perfrmace, SAE paper [3] Qiag Hu, et al., Predicti f Pressure Fluctuatis Iside a Autmtive Fuel Rail System, SAE paper [4] Vibrati Fr Egieers, Adrew D. Dimargas, Pretice Hall, Eglewd Cliffs, New Jersey C99.