Chapter 2 Thermodynamics of Turbochargers

Transcription

1 Chater herodynaics of urbochargers. herodynaic Characteristics Soe essential therodynaic characteristics of gases are needed to know in the turbocharging. hey are usually alied to therodynaics of turbochargers where the charge air and exhaust gas are assued to be coressible erfect gases [ ]. he total teerature t (K) results fro the su of the static s (K) and dynaic teeratures dyn (K). he static teerature is easured at the wall where the gas velocity equals zero due to the viscous boundary layer. t ¼ s þ dyn ¼ s þ c c ð:þ where c is the gas velocity; c is the heat caacity at constant ressure. he total ressure t is calculated fro the isentroic gas equation as follows: j j t t ¼ s ¼ s þ j j M j ð:þ s where s κ c /c v M is the static ressure; is the isentroic exonent of gas; is the Mach nuber of gas (M c/a), a is the sound velocity he secific total enthaly h t results fro the su of the gas secific enthaly and secific kinetic energy of gas. h t ¼ h þ c ð:3þ Sringer International Publishing Switzerland 05 H. Nguyen-Schäfer, Rotordynaics of Autootive urbochargers, Sringer racts in Mechanical Engineering, DOI 0.007/ _

2 herodynaics of urbochargers where the gas secific enthaly h (enthaly er ass unit, J/kg) is defined as hðþ ¼c ð 0 Þþhð 0 Þ ¼ uðþþ q ¼ uðþþr g ð:aþ within ρ is the gas density; 0 is the reference teerature (K); u is the secific internal energy; R g is the gas constant; h( 0 ) 0at 0 0K( 73. C). he secific internal energy u is defined as uðþ uð 0 Þ¼c v ð 0 Þ¼c v D ð:bþ where u( 0 ) 0at 0 0K. hus, u() c v ; h() c ; in K.. Efficiencies of Coressor and urbine he coression rocess in the coressor is a olytroic rocess with increasing entroy due to friction and losses in the coressor. Figure. shows the coression rocess of the intake air fro the state at the coressor inlet (, )to state at the coressor outlet (, ). he coressor efficiency η C is defined as the ratio of the isentroic total enthaly change fro t to st to the olytroic total enthaly change fro t to t. In other words, the coressor needs ore energy in the olytroic rocess (real rocess) than the ossibly inial required energy of the coressor stage in the isentroic rocess (ideal rocess) [, 5, and 6]. h t t Δh ηc Δh s, tt, tt st t t t ½c² st diffuser s Δh,tt Δh s,tt t t ½c² s Fig.. Coression rocess in the coressor stage in h-s diagra

3 . Efficiencies of Coressor and urbine 3 he total total isentroic efficiency of the coressor stage (further called coressor) consisting of the coressor wheel and diffuser is defined as g C ¼ Dh s;tt Dh ;tt ¼ st t t t ð:5þ he total total isentroic efficiency is generally used in the coressor since the kinetic energy of gas in the state could be transfored into the ressure energy in the diffuser. he rocess increases the charge-air ressure. Using therodynaic equations for the isentroic rocess, the coressor efficiency is written in ters of the total ressures and teeratures at the inlet and outlet of the coressor, and the isentroic exonent of the charge air κ a.. g C ¼ j t ð j t t t Þ a ð:6þ he coressor efficiency is deterined by easuring the total ressures and teeratures at the inlet and outlet of the coressor according to Eq. (.6). he axiu total total isentroic efficiency of the coressor η C is norally between 70 and 80 % at the design oint in the coressor erforance a. Analogous to the coressor, the efficiency of turbine results fro the olytroic exansion rocess of the exhaust gas fro the state 3 at the turbine inlet ( 3, 3 ) to state at the turbine outlet (, ), see Fig... he turbine efficiency η is defined as the ratio of the olytroic total enthaly change fro 3t to t to the isentroic total enthaly change fro 3t to s. Physically seaking, the turbine delivers less outut energy due to friction and losses in the olytroic exansion rocess than the ossibly axiu energy given in the isentroic rocess. h 3 a η Δh Δh 3, tt t s, ts Δhs, ts s 3t 3t ½c² 3 3 3t 3t t t a Δ h 3,tt 3 VG Δh s,ts ½c² s Fig.. Exansion rocess in the turbine stage in h-s diagra s

4 herodynaics of urbochargers he total-static isentroic efficiency of the turbine stage (further called turbine) consisting of the turbine wheel and variable turbine geoetry (VG) or waste gate (WG) is defined as [, 3] g ¼ Dh 3;tt Dh s;ts ¼ t 3t s 3t ð:7þ he total-static isentroic efficiency is norally used in the turbine since the kinetic energy of the exhaust gas in the state does not generate any additional ower for the turbine. Alying therodynaic equations to the isentroic rocess, the turbine efficiency is exressed in ters of the total ressure and teerature at the turbine inlet and outlet and the isentroic exonent of the exhaust gas κ g.3. g ¼ t 3t s 3t j j ð Þ g ð:8þ he turbine efficiency is deterined by easuring the total ressure and teerature at the turbine inlet and outlet according to Eq. (.8). he axiu total-static isentroic efficiency of the turbine η is norally between 65 and 70 % at the design oint of the turbine erforance a [5 7]..3 urbocharger Equations he turbocharger consists of the turbine, coressor, and core unit (CHRA: center housing and rotating assebly) including the rotor and bearing syste. Both turbine and coressor wheels are fixed in the rotor shaft that is suorted on the bearing syste of the radial and thrust bearings. he rotating shaft including the coressor wheel, turbine wheel, thrust rings, radial bearings, and seal rings is called the rotor of the turbocharger. Due to exanding the exhaust gas of the engine in the turbine, it generates the turbine ower that deends on the ass flow rate of the exhaust gas through the turbine and the isentroic enthaly dro in the turbine. he effective turbine ower results as P ¼ g _ jdh s j ð:9þ he isentroic enthaly dro in the turbine stage is calculated using therodynaic equations.

5 .3 urbocharger Equations 5 " jdh s j ¼ c ;g 3 j ð j Þ # g 3 ð:0þ Inserting Eq. (.0) into Eq. (.9), one obtains the effective turbine ower in function of the ass flow rate, inlet teerature, and ressure ratio of the turbine. " P ¼ g P ;ideal g _ c ;g 3 j ð j Þ # g 3 ð:þ Due to the friction loss in the bearing syste, the required coressor ower results fro the effective turbine ower and echanical efficiency η. " P C ¼ g P ¼ g g _ c ;g 3 j ð j Þ # g 3 ð:þ Analogously, the required coressor ower is calculated fro the isentroic coressor ower and coressor efficiency. P C ¼ P C;ideal g C _ CDh sc g C ð:3þ where Δh sc is the increase of the isentroic enthaly in the coressor. Using therodynaic equations for an isentroic rocess, the required coressor ower is calculated fro the ass flow rate, inlet teerature, and ressure ratio of the coressor. " P C ¼ _ j Cc ;a ð j Þ # a g C ð:þ Substituting Eqs. (. and.), one obtains the ressure ratio of coressor π C in the first turbocharger equation. C ¼ ¼ ¼ þ c ;g c ;a þ c ;g c ;a _ 3 g _ C C _ 3 g _ C C " #! ð j 3 j ð j Þ g j j ð j Þ j g j Þ a ð Þ a ð:5þ

6 6 herodynaics of urbochargers within the overall efficiency of the turbocharger η C is written as g C ¼ g g g C ð:6þ o achieve the high boost ressure of the charge air, the coressor ressure ratio π C could be iroved according to Eq. (.5) if the overall turbocharger efficiency η C is high, esecially a high echanical efficiency of the bearing syste in low-end torque; the exhaust gas teerature 3 is high due to the large enthaly. Hence, ore turbine ower is generated; the turbine ressure ratio π (turbine exansion ratio) is as high as ossible; the inlet air teerature is as low as ossible; therefore, the charge-air teerature is low leading to a high density of the charge air; the exhaust gas ressure 3 is chosen at an otial ressure in order to coroise between the turbine ower and secific fuel consution; the exhaust gas ass flow rate in the turbine should be large. he first turbocharger Eq. (.5) shows the behavior of the ressure ratio π C ( / ) of the coressor versus the ressure ratio π ( 3 / ) of the turbine. In fact, the diensionless ter δ in the angle brackets < > of Eq. (.5) does not change so uch along the full load curve in the coressor erforance a. herefore, the behavior of the ressure ratios of turbine and coressor can be dislayed at various diensionless araeters δ in Fig..3. he exhaust gas flow in the turbine can be considered as a coressible flow in a nozzle in which the inlet and outlet ressures are 3 and, resectively. Based on the flow equation for coressible fluids in the nozzle, the second turbocharger equation describes the ass flow rate through the turbine in function of the ressure, teerature at the turbine inlet, and the turbine exansion ratio. Fig..3 Behavior of the ressure ratios π C versus π

7 .3 urbocharger Equations 7 sffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j! 3t jg jþ u ð j Þ _ ¼ la t 3t g 3t R g 3t j g ð:7þ where µ is the flow coefficient due friction and flow contraction at the nozzle outlet, A is the throttle cross-sectional area in the turbine wheel. o eliinate the influences of 3t and 3t on the ass flow rate in the turbine shown in Eq. (.7), the so-called corrected ass flow rate is defined as _ ;cor _ ffiffiffiffiffiffi 3t ¼ f ð ;ts Þ 3t sffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j! 3t jg jþ u ð j Þ 3t ¼ la t g R g j g ð:8þ Equation (.8) indicates that the corrected ass flow rate of the turbine is indeendent of the inlet condition of the exhaust gas of 3t and 3t. It deends only on the turbine exansion ratio π,ts. he erforance a of the turbine dislays the corrected ass flow rate over the turbine exansion ratio π,ts at various rotor seeds in Fig... Fro a turbine ressure ratio of aroxiately 3, the ass flow rate has no longer increased, even at higher rotor seeds. In this case, the flow in the turbine becoes a choked flow in which the exhaust gas seed at the throttle area reaches the sonic seed with Mach nuber M. As a result, the exhaust gas ass flow rate through the turbine corresonding to the noinal engine ower ust be saller than the ass flow rate, cor, choke N ax N N π, ts 3 t Fig.. Perforance a of the turbine

8 8 herodynaics of urbochargers at choke. Note that the isentroic turbine efficiency at the choked flow is extreely low (η < 60 %). his low efficiency is unusable in the autootive turbochargers. he echanical efficiency of turbocharger induced by the bearing friction results fro Eq. (.). g ¼ P C P ¼ _ C c ;a ð Þ ð ð:9þ j g C g _ c ;g 3 3 Using Eqs. (. and.9), the efficiency roduct of the echanical and turbine efficiencies is written as g g ¼ j Þ g _ C c ;a ð j j Þ a g C _ c ;g 3 ð j j Þ ð:0þ 3 g he efficiency η η described in Eq. (.0) is a key art in the overall efficiency of the turbocharger. It is deterined by easuring the therodynaic characteristics in Eq. (.0), such as the ass flow rates in the coressor and turbine, teeratures and 3, coressor efficiency η C, ressures at the coressor inlet and outlet and, and as well as ressures at the turbine inlet and outlet 3 and. Figure.5 dislays the efficiency η η versus ressure ratio of turbine π,ts at various rotor seeds. η η 0.7 N C N C,in N C,ax.0 π π, ts 3 t Fig..5 Efficiency η η versus π,ts at various rotor seeds N C

9 .3 urbocharger Equations 9 η η VG osition: : in-flow (0%) : artly oen (30%) 3: artly oen (70%) : fully oen (00%) π Fig..6 Efficiency η η versus π,ts at various VG ositions π, ts 3 t At low rotor seeds corresonding to the low-ressure exansion ratios π, the echanical efficiency is quite sall due to large bearing friction at the low oil teerature in the bearings. Additionally, the turbine efficiency is still sall due to the aerodynaic oerating condition at the low rotor seed. herefore, the resulting efficiency η η reains low. At high rotor seeds, the echanical efficiency of the bearings increases due to the increased oil teerature and therefore less friction in the bearings. As a result, the efficiency η η reaches a axiu value at the turbine ressure ratio at the design oint in the erforance a. As the rotor seed further increases to the axiu seed, the turbine efficiency dros due to the aerodynaic oerating condition at high rotor seeds; the echanical efficiency decreases because the bearing friction rises at high rotor seeds. herefore, the efficiency η η decreases with the rotor seed corresonding to the turbine ressure ratio. However, the efficiency η η deends not only on the turbine ressure ratio but also on the osition of VG (variable turbine geoetry), as shown in Fig..6. Initially, the VG is oen at the in-flow condition for the iniu ass flow rate called 0 % VG (osition ) at the idle condition of the engine. he efficiency η η increases with the turbine ressure ratio because the turbine efficiency becoes larger at high ass flow rates corresonding to a high turbine ressure ratio. he VG begins oening at 30 % VG (osition ). At increasing turbine ressure ratios, the ass flow rate of exhaust gas rises in the turbine leading to the high turbine efficiency. Hence, the efficiency η η increases since the turbine ressure ratio rises to the ressure ratio π at the design oint, as shown in Fig..5. As a result, the efficiency η η is higher than the efficiency at the osition.

10 30 herodynaics of urbochargers π C, tt t t rotor seed N 3 75% DP full load curve 67% 70% P no N ax 65% N k choke line 0 η C 60% N N choke C, cor C t t Fig..7 Coressor erforance a At the osition of 70 % VG (osition 3), the turbine efficiency begins decreasing at high ass flow rates corresonding to large turbine ressure ratios. Finally, the VG is fully oen at 00 % VG (osition ). he ass flow rate significantly increases in the turbine at increasing turbine ressure ratio. herefore, the turbine efficiency reduces with the turbine ressure ratio. As a result, the efficiency η η dros with the turbine ressure ratio. he coressor erforance a in Fig..7 shows the coressor ratio π C versus the corrected ass flow rate. At low rotor seed in the idle condition of about 30 % of the axiu rotor seed N ax, the coressor ressure ratio on the full load curve increases strongly with the ass flow rate to deliver a good transient resonse at low-end torque. At further increasing the rotor seed N to nearly 70 % of N ax, the coressor ressure ratio increases to aroxiately.5 and the engine achieves the axiu torque fro to the design oint DP. Due to the reduced turbine efficiency at the high rotor seeds leading to reducing the overall efficiency of the turbocharger, the coressor ressure ratio decreases at the noinal engine ower P no in the full load curve according to the first turbocharger Eq. (.5).. Resonse ie of urbochargers he resonse tie is an iortant characteristic dealing with turbolag in autootive turbochargers. he turbolag is the delayed tie that the turbocharger needs to reach the axiu engine torque fro the idle.

11 . Resonse ie of urbochargers 3 Fig..8 Resonse tie of a turbocharger e better low-end torque ax LE 90% ax better transient behavior 0 τ 90 t o characterize the turbolag of the turbocharger, the resonse tie τ 90 is defined by the tie interval that is required to reach the low-end torque (LE) LE of 90 % of the axiu engine torque ax in low-end torque LE (see Fig..8). he resonse tie τ 90 results fro the effective turbine ower and the olar ass inertia oent of the rotor. Note that the better the transient behavior of the turbocharger is, the shorter the resonse tie τ 90 is at a given LE. he better the LE is, the higher the LE LE is at a given resonse tie. he angular acceleration of the rotor is calculated fro the turbocharger dynaics equation. g P P C ¼ I h X ð:þ he echanical efficiency of the bearing is calculated as [] g ¼ P fb P ð:þ where P fb is the bearing friction ower. he rotor seed N C of the turbocharger results fro Eqs. (. and.) in N C ¼ X ¼ ðg P P C Þ I h ¼ ðp P fb P C Þ I h ð:3þ Integrating with tie the angular acceleration within the resonse tie τ 90, one obtains the average angular rotor velocity in LE X 90 ¼ Z s 90 0 hðtþdt hs90 ð:þ where h is the average angular acceleration within the resonse tie.

12 3 herodynaics of urbochargers Substituting Eq. (.) into Eq. (.) and eliinating the average angular acceleration, the resonse tie of the turbocharger results in s 90 I X 90 ðg P P C Þ ¼ I N 90 ðg P P C Þ ð:5þ where I is the olar ass inertia oent of the rotor; P is the effective turbine ower; P C is the effective coressor ower; is the echanical efficiency of the bearings. η In order to have a good transient resonse of the turbocharger, it is necessary to kee the resonse tie τ 90 as sall as ossible at a given rotor seed N 90. According to Eq. (.5), since the effective turbine ower P is relatively sall in LE, the resonse tie τ 90 rises leading to a large turbolag of the turbocharger. However, there are soe iroving easures: () he olar ass inertia oent I of the rotor should be as sall as ossible. he turbine wheel lays a key role in the entire olar ass inertia oent of the rotor due to its heavy ass of Inconel 73C. he olar ass inertia oent of the turbine wheel is roortional to its ass and squared wheel diaeter D. Due to * ρd 3, the olar ass inertia oent of the rotor is written as I I ;W / D / qd 5 ð:6þ o reduce the inertia oent of the turbine wheel, there are soe ossibilities as follows: he turbine wheel should be as light as ossible, such as using light titaniu aluinide ial6v (ρ.5 g/c 3 ) instead of heavy Inconel 73C (ρ 7.9 g/ c 3 ); he turbine wheel is scalloed at the back face to reduce its ass. However, the turbine efficiency could be reduced by 3 % due to the unsuitable aerodynaic flow at the scalloed back face leading to a reduction of the effective turbine ower according to Eq. (.); he inflow diaeter D of the turbine wheel should be reduced in order to decrease the olar inertia oent and to increase the turbine efficiency at low rotor seeds leading to the increase of the effective turbine ower. However, the sall turbine wheel has soe negative effects, such as low turbine efficiency at high rotor seeds and sall ass flow rate. As a result, the noinal turbine ower is reduced. () he echanical efficiency η should be increased using airfoil bearings, agnetic bearings, rolling-eleent bearings, or rotating floating ring bearings with two oil

13 . Resonse ie of urbochargers 33 fils. Generally, the rolling-eleent bearings generate less friction ower, esecially in LE. Contrary to the ball bearings, oil-fil bearings induce a little ore friction ower in LE since the oil viscosity in the bearing is high at low rotor seeds. In fact, the ratio of the bearing friction to turbine ower is relatively high in LE due to the sall effective turbine ower and high bearing friction. However, the discreancy between the friction coefficients of the rolling eleent and rotating floating ring bearings is negligibly sall at high rotor seeds. Note that the rollingeleent bearings cost about 0 ties of the oil-fil bearings. herefore, it is recoended to carefully decide which aterial of the turbine wheel should be used, the suitable inflow diaeter of the turbine wheel, either scalloed or unscalloed turbine wheel. Furtherore, which bearing syste should be used in the turbochargers in coroise between the transient resonse and cost of the bearings?.5 urbocharger Matching he atching rocedure of turbochargers based on the engine characteristics, coressor erforance a, turbine erforance a, and first turbocharger equation is discussed in the following section (see Fig..9). he following stes are carried out for the atching rocedure of turbochargers [5 7]: () Fro the engine characteristics (Fig..9, botto left), the oerating field of the turbocharger in the coressor a is deterined at the various oerating oints, such as LE, axiu torque, design oint, and noinal ower of the engine. At any oerating oint, the required ass flow rate of the charge air at the given turbocharger seed results fro the engine ower according to Eqs. (. and.). Additionally, the necessary charge-air density is given by Eq. (.3). he coression ratio without air intercoolers is calculated fro Eq. (.0). In case of using air intercoolers (see Fig..0), the charge-air teerature dros at a lower teerature * leading to the increase of the charge-air density ρ *. he charge-air teerature * after the intercooler is calculated fro the coressed charge-air teerature, coolant inlet teerature c, and intercooler efficiency ε c (usually between 0.6 and 0.8) as ¼ð e c Þ þ e c c \ ð:7þ hus, the charge-air teerature in case of using air intercoolers dros with D ¼ e c ð c Þ\0 ð:8þ

14 3 herodynaics of urbochargers π C Coressor erforance a N π C First turbocharger equation δ ηc C π C (, δ ).0 η C 0.5 0, π ) ( C C N C C, cor C t t ( π ) π 3, cor 3t 3t urbine erforance a 00% VG ( ) 75% VG 50% VG α VG 0% VG Engine characteristics (P e, V cyl, N, n R, ) 0 N C 3 π Fig..9 Matching rocedure of turbochargers coolant c engine coressor intake air charge air cooled charge air intercooler, ρ, Δ c,, ρ * *, *, ρ Fig..0 Coressor using a charge-air intercooler herefore, the charge-air density rises fro ρ to ρ at a sall ressure dro in the intercooler. Using the state equation of a erfect gas, the charge-air density is calculated as

15 .5 urbocharger Matching 35 q ¼ ¼ D c [ q R a R a ð:9þ where Δ c is the ressure dro in the air intercooler. Substituting Eqs. (.9,.7, and.9), one obtains the coression ratio of the coressor in case of using the air intercooler C ¼ ¼ þ D c ¼ q R a þ D c q R a ð e c Þ þ g ð j j Þ a C C ¼ þ e c c þ D c ð:30þ where the charge-air density ρ * is given by the engine requireent and κ a. is the isentroic exonent of the charge air. he charge-air density ρ * is deterined by iteratively solving Eq. (.30) under the given boundary conditions for the engine, coressor, and intercooler. () he oerating oint of the turbocharger is located in the coressor erforance a (Fig..9, to left) at the given ass flow rate and coression ratio of the charge air that have been couted in the ste. he value δ in the diagra of the first turbocharger equation involves the ass flow rates of the charge air and exhaust gas, the teeratures of the exhaust gas and intake air, and the turbocharger efficiency. It results fro the oerating condition of the engine and the guessed/ easured efficiency of the turbocharger. (3) Fro the given coression ratio π C and value δ, the exansion ratio π of the turbine corresonding to the oerating oint is deterined in the diagra of the first turbocharger equation (Fig..9, to right). () Both ass flow rate and exansion ratio of the turbine given in the stes and 3 are used to deterine the oerating oint of the turbocharger in the turbine erforance a (Fig..9, botto right). he oerating oint gives the corresonding VG angular osition of 75 % VG. Note that the ass flow rate of the exhaust gas that is identical with the ass flow rate of the VG turbine, equals the ass flow rates of the charge air and injected fuel in the cylinders based on the air fuel ratio AFR defined in Eq. (.). herefore, the ass flow rate of the exhaust gas is written as AFR þ _ ¼ _ a þ _ f ¼ _ a ð:3þ AFR Using the turboachinery theory, the outflow diaeter D (exducer diaeter) of the coressor wheel and the inflow diaeter D 3 (inducer diaeter) of the turbine wheel are couted at the given effective owers and ass flow rates of the coressor and turbine, echanical efficiency, and turbocharger seed. he atching rocedure of turbochargers is iterated until the guessed values, such as the

16 36 herodynaics of urbochargers efficiencies of the coressor and turbine, efficiency of the air intercooler, etc., are converged. Furtherore, the couted values could be reatched with the easured efficiencies as soon as they are available. References. Aungier, R.H.: urbine Aerodynaics. ASME Press, New York (006). Baines, N.C.: Fundaentals of urbocharging. Concets EI, Inc, Wilder (005) 3. Custy, N.A.: Coressor Aerodynaics. Krieger Publishing Coany, Melbourne (00). Heywood, J.B.: Internal Cobustion Engine Fundaentals. McGraw-Hill, New York (988) 5. Jaikse, D.,Baines, N.C.: Introduction to urboachinery. Concets EI, Inc, Wilder (99) 6. Jaikse, D., et al.: Axial and Radial urbines. Concets EI Inc, Wilder (003) 7. Whitfield, A., Baines, N.C.: Design of Radial urboachines. Pearson Education, Longan Scientific and echnical (990)

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