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Original Article

International J of Engine Research2016, Vol. 17(7) 705–712� IMechE 2015Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1468087415609855jer.sagepub.com

Scalable turbocharger performancemaps for dynamic state-based enginemodels

Clay Bell, Daniel Zimmerle, Thomas Bradley, Daniel Olsen andPeter Young

AbstractAdapting turbocharger performance maps to a form suitable for dynamic simulations is challenging for the following rea-sons: (1) the amount of available data is typically limited, (2) data are typically not provided for the entire operating rangeof the compressor and turbine and (3) the performance data are non-linear. To overcome these challenges, curve fits aretypically generated using the performance data individually for each device. The process, however, can take un-economical amounts of effort to implement for a range of compressors and turbines. This article introduces a methodto implement non-dimensional performance maps thereby allowing a range of turbochargers to be modeled from thesame performance data, reducing the effort required to implement models of different sizes. The non-dimensional mapsseek to model the performance of compressor and turbine families in which the geometry of the rotor and housing aresimilar and allow the turbocharger to be scaled for simulation in much the same way used to design customized sizes ofturbochargers. A method to match the non-dimensional compressor map to engine performance targets by selectingthe compressor diameter is presented, as well as a method to match the turbine to the selected compressor.

KeywordsTurbocharger, compressor maps, turbine maps, dynamic modeling, scaling

Date received: 12 June 2015; accepted: 26 August 2015

Introduction

Compressor and turbine performance is typicallyimplemented in dynamic state-based engine modelssuch as those developed in Kao and Moskwa,1

Malkhede et al.2 and Millet et al.3 and used for powersystem studies as in Katrasnik et al.4 and Ibrahimet al.5 as lookup tables or curve fits developed fromperformance maps; however, a significant amount ofeffort is required to convert the data to a form suitablefor simulation. Manufacturers frequently only providedata on the maps around the region of high efficiencywhere the turbocharger is operating at high speeds andpressure ratios near the surge margin. The performanceat low rotor speeds and low pressure ratios, a regionfrequently encountered during simulation at low engineload, must be extrapolated from the available data asillustrated by Moraal and Kolmanovsky6 and discussedin Gamma Technologies.7 Additionally, at compressorsurge, the mass flow rate through the compressor dropssuddenly and can even experience a flow reversal, creat-ing stiff state equations. This region of the map is

difficult to capture using a lookup table which isindexed by pressure ratio and speed parameter; the out-put is undefined for inputs above the surge line and itis difficult to capture the compressor surge well in thefits produced by the methods discussed in Moraal andKolmanovsky.6

Collecting performance data specific to a particularcompressor and turbine and generating curve fits whichaccurately capture the characteristics of the data oftenrequire a significant amount of time and effort. In orderto model several engines ranging in displacement andpower, the process must be repeated for each engine. Ascalable approach to modeling the turbocharger wouldstreamline this process, but has not been presented inthe literature. The approach introduced in this article

Colorado State University, Fort Collins, CO, USA

Corresponding author:

Clay Bell, Colorado State University, 430 N College Ave, Fort Collins,

CO 80524, USA.

Email: [email protected]

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utilizes dimensional analysis techniques, often used inthe design of compressors and turbines, to generate asuitable map once, and then adapt that map for use insimulation across a range of engine sizes. The processreduces the amount of data and effort required tomodel a series of engines.

The contributions of this article are threefold. First,a method is presented to normalize compressor perfor-mance data by the surge line in order to produce tabu-lar data on rectilinear axes which capture the suddenreduction of mass flow at compressor surge and arereadily implemented as a lookup table during simula-tion. Second, following well-established dimensionalanalysis of compressors and turbines, the publishedperformance maps of a given device are converted intothe non-dimensional terms which represent a family ofdevices for which design parameters associated with therotor and housing remain constant. Third, a method isintroduced to select the diameter of the compressorand turbine rotors (the remaining design parameter) inorder to match the non-dimensional performance mapsto an engine for which a state-based model is beingconstructed.

The remainder of this article is divided into threesections. The first briefly reviews methods to fit perfor-mance data and extrapolate into the low rotor speed,low pressure ratio regions of a compressor map. Anapproach to scaling a compressor map using the dimen-sional analysis similar to that presented in Yahya8 toproduce maps for geometrically similar compressors isthen illustrated. The compressor data are normalizedby the surge line to produce a rectangular lookup tablewhich (1) is readily implemented in simulation and (2)captures rapid change in mass flow near the surge line.A similar scaling approach is applied to normalize theturbine map. The second section introduces a methodto select a compressor diameter to match the non-dimensional map to a given engine and to select a tur-bine diameter which will match the compressor torquerequirements. Finally, a set of compressor and turbinemaps are scaled to match several engine sizes in theresults section in order to illustrate the utility of themethod.

Scalable modeling approach

Review of fitting methods

Steady-state compressor mass flow and isentropic effi-ciency data collected at selected rotor speeds and pres-sure ratios are typically available from themanufacturer as a table or map similar to that shownin Figure 1. Given the pressure ratio across the com-pressor and the speed parameter, the data can be usedto determine the mass flow parameter, _mcorr,C =f(Ncorr, (Pd=Pu)), and the isentropic compressor effi-ciency, hc = f(Ncorr, (Pd=Pu)). The speed parameter,Ncorr, and the corrected mass flow are used to account

for temperature and pressure effects of the upstreamair9,10

_mcorr,C =_mC

ffiffiffiffiffiffiTu

p

Puð1Þ

Ncorr =NTCffiffiffiffiffiffiTu

p ð2Þ

The tabular data are not well-suited for simulationsince it (1) is limited to few data points representingnon-linear performance characteristics of the compres-sor and turbine which would introduce significant errorwhen used with linear interpolation and extrapolationtechniques; (2) does not illustrate the performance atlow rotor speed and low pressure ratios, a region fre-quently encountered in simulations during initializationor where the engine is operating across a range of speedand load and (3) includes a region above compressorsurge where the outputs ( _mcorr,C,hc) are undefined forinputs (Ncorr, (Pd=Pu)). For these reasons, curve fits areoften used in simulation models. Four methods of curvefitting compressor data are reviewed and illustrated byMoraal and Kolmanovsky6 in order to extrapolate

Figure 1. Compressor data typically available by request frommanufacturer illustrated as (a) corrected mass flow as a functionof pressure ratio and speed parameter and (b) isentropiccompressor efficiency as a function of speed parameter andmass flow parameter. The compressor shown has a rotordiameter of 189 mm.

706 International J of Engine Research 17(7)



performance to low pressure ratios and low rotorspeeds. The four methods illustrated are as follows:

� The Jensen and Kristensen method11 whichexpresses the dimensionless head parameter and thecompressor efficiency as functions of normalizedflow rate and inlet Mach number. Coefficients forthe expressions are determined using a least-squaresfit to experimental data.

� The Mueller method12 which models the dimension-less head parameter as a quadratic function of nor-malized compressor flow rate.

� The zero-slope line method6 which describes thecompressor flow parameter as a function of pres-sure ratio and the speed parameter. The fit isdivided into a linear region and an exponentialregion by the zero-slope line which connects themaximum mass flow of each speed line.

� A neural network as used by Nelson et al.13 anddetermined to be ill-suited to compressor modelingdue to the large number of required coefficients andthe limited number of data points available to trainthe neural network.

Steady-state turbine mass flow and isentropic effi-ciency data collected at selected rotor speeds and pres-sure ratios are also typically available from themanufacturer as a table or map similar to that shownin Figure 2. Given the expansion ratio across the tur-bine and the speed parameter, the data can be usedto determine the mass flow parameter, _mcorr,T =f(Ncorr, (Pu=Pd)), and the isentropic compressor effi-ciency, hT = f(Ncorr, (Pu=Pd)). Methods to curve fit tur-bine mass flow and efficiency data are also presentedby Moraal and Kolmanovsky.6

All these methods require a significant amount ofexperimental data, as well as effort and judgment toprocess each map. In order to reduce the amount oftime and effort required to implement a turbochargermodel into simulations of various engine scales andapplications, a scalable approach using a ‘‘generic’’ mapis desirable.

Proposed scaling method—compressor

Turbocharger designers use maps in terms of non-dimensional terms to approximate the performance ofnew compressor sizes from similar compressors beforeproducing a prototype.8 The dimensional analysis dis-cussed by Canova et al.14 shows the characteristiccurves for compressors and turbines are dependent onthe working fluid, Reynolds number, and three designparameters: the rotor diameter and trim associated withthe blade design, and the A/R ratio associated with thehousing design. The analysis illustrates how the designparameters influence the characteristic curves describ-ing compressor and turbine mass flow and efficiency,enabling the performance maps to be predicted for agiven set of design parameters.

The method proposed here seeks to use a single setof available performance data to generate a scalablemap for state-based models. Although the non-dimensional data represent a device with two fixeddesign parameters, namely, the trim and A/R ratio, themethod allows a first-cut model to be developed forvarious engine sizes from a single set of device-specificdata.

Starting with manufacturer data for a single com-pressor design, if the diameter of the compressor wheel,Dc, is known, the mass flow parameter and efficiencydata can easily be converted into non-dimensional com-pressor mass flow coefficient, ;C, and Mach numberbased on the rotor tip speed, c0,C

;C = _mcorr,C

ffiffiffiffiRp

D2c

ð3Þ

c0,C =2p

60

� �Ncorr

DcffiffiffiffiffiffiffigRp ð4Þ

Maps implemented in terms of the compressor massflow coefficient and the rotor tip Mach number repre-sent a family of similar compressors with the same trim

Figure 2. Turbine data typically available by request frommanufacturer: (a) corrected mass flow as a function ofexpansion ratio and speed parameter and (b) isentropiccompressor efficiency as a function of expansion ratio and speedparameter. The turbine shown has a rotor diameter of 177 mm.

Bell et al. 707



and A/R ratio, rather than a single design.Manufacturers typically present data in terms of speedparameter and mass flow parameter rather than thesenon-dimensional parameters because the map is pre-sented for a particular selection of rotor(Dc = constant) and it is assumed the compressor willalways handle air near standard temperature and pres-sure (R=Rair, g = gairjSTP).

Figure 3 shows a compressor map in terms of thenon-dimensional parameters developed from data pre-sented in Figure 1. The data were first fit using thezero-slope line method and extrapolated into the lowrotor speed, low pressure ratio region. The map inFigure 4 is generated from the non-dimensional mapby selecting a new diameter, Dc =100mm (smallerthan the original rotor). By selecting a diameter,Dc =200mm (larger than the original diameter), themap in Figure 5 is produced.

From equations (3) and (4), it is easy to see that themass flow rate is proportional to the square of therotor diameter, and the rotor speed is inversely propor-tional to the rotor diameter. By selecting a smaller dia-meter, the rotational speed of the rotor is increased andthe mass flow parameter is reduced as illustrated inFigure 4. By selecting a larger diameter, the rotorspeeds are reduced and the mass flow rate is increasedas shown in Figure 5.

Compressor performance as lookup tables

Lookup tables generally are not used for turbochargerdata because linear interpolation methods result in sig-nificant error, particularly in regions with low resolu-tion data or significant non-linearity.6,11,12 Lookuptables, however, are readily implemented in simulationsand are computationally efficient. By first fitting thedata using one of the methods discussed previously, thecurve fit can be evaluated to create a lookup table at ahigher resolution (using smaller steps in pressure ratioand rotor speed) than the data provided by the

manufacturer. This method reduces the error associ-ated with interpolation within the table data.

At the surge line, the compressor stalls and massflow stops or may reverse flow. The region of the mapabove the surge line is typically not communicated bythe manufacturer, which makes it difficult to implementthe data as lookup tables indexed by pressure ratio androtor speed (either speed parameter or rotor tip Machnumber). The challenge is twofold: First, in the surgeregion, a lookup table would have combinations ofinputs ((Pd=Pu), c0,C) where the outputs (;C,hc) areundefined. This region cannot be avoided duringdynamic simulation due to events such as a suddenchange in speed and/or load and results in undefinedbehavior, where there is no counteracting ‘‘force’’ topush the simulation back onto the map. Second, for alookup table of practical size (or a curve fit of practicalorder), it is difficult to represent the sudden change inoutput parameters near the surge line. Therefore, thesimulation can easily jump over this boundary and getstuck in the undefined area above the surge line.

Figure 3. Compressor map developed from data in Figure 1converted into non-dimensional mass flow coefficient and rotortip Mach number, and extrapolated to low rotor speed andpressure ratio regions.

Figure 5. Compressor map in terms of speed parameter andmass flow parameter developed from non-dimensional mapscaled up to a rotor diameter = 200 mm.

Figure 4. Compressor map in terms of speed parameter andmass flow parameter developed from non-dimensional mapscaled down to a rotor diameter = 100 mm.




The pressure ratio at compressor surge is found fora given rotor speed (Pd=Pu)surge = f(Ncorr), or rotor tipMach number (Pd=Pu)surge= f(c0,C). The pressureratios corresponding to efficiency and mass flow dataat that rotor speed are then normalized by the pressureratio at surge as

Pidx =

Pd

Pu

� �� 1

Pd

Pu

� �surge�1

ð5Þ

By normalizing the pressure ratio by the surge line, arectangular lookup table is produced, indexed by asurge-normalized pressure ratio index, Pidx, instead ofthe pressure ratio, (Pd=Pu). This process effectivelymaps the region below the surge line onto rectilinearaxes as illustrated in Figure 6. Using this index, there isno longer a region above compressor surge in the datawhere the outputs are undefined. The new pressureindex ranges from Pidx =0 where there is no boostacross the compressor, (Pd=Pu)=1, to Pidx =1 at com-pressor surge, (Pd=Pu)= (Pd=Pu)surge, where the flowquickly rolls off to 0. The high-resolution lookup tablecreated by this process can be used for a range of enginesizes.

Proposed scaling method—turbine

An approach similar to that used for the compressordata can be applied to the turbine data. If the diameterof the turbine wheel, DT, is known, the data can easilybe converted into the non-dimensional parameters. Theturbine mass flow coefficient, ;T, and Mach numberbased on the rotor tip speed, c0,T, are calculated as

;T = _mcorr,T

ffiffiffiffiRp

D2T

ð6Þ

c0,T =2p

60

� �Ncorr

DTffiffiffiffiffiffiffigRp ð7Þ

Again, the manufacturers typically present the dataas in Figure 2 in terms of speed parameter,Ncorr =(NTC=


p), and mass flow parameter,

_mcorr,T =(( _mT


p)=Pu), rather than these non-

dimensional parameters, for similar reasons. Figure 7shows the turbine efficiency and non-dimensional massflow coefficient presented in terms of the expansionratio, Pu=Pd, and rotor tip Mach number. The datawere first fit using techniques in Gamma Technologies7

and extrapolated into the low rotor speed, low pressureratio region, and then normalized using equations (6)and (7).

Turbocharger selection using non-dimensional maps

Implementing performance maps in terms of the non-dimensional parameters allows a single set of maps toapproximate the turbocharger performance for a rangeof engine sizes by adjusting the compressor and turbinediameters to match the target brake power and air–fuel(A/F) ratio of each engine. A single set of non-dimensional maps can be utilized in the series of enginemodels instead of repeating the process of finding thedevice-specific compressor and turbine maps, fitting thedata to extrapolate into the region of low pressure andlow rotor speed and converting the maps into a tabulardata format at a higher resolution than provided by themanufacturer. The non-dimensional maps are limitedto the assumption that the trim and A/R ratio in thefamily is fixed. A process to match the non-dimensionalcompressor and turbine maps to a given engine is illu-strated below.

Selecting compressor diameter

The fuel mass flow rate to meet a target engine power,_W, can be calculated as _mF = _W=hThELHV where hTh is

Figure 6. Compressor map normalized by pressure ratio atcompressor surge, resulting in a rectangular lookup table easilyimplemented in a simulation. While rectangular, near the surgeline (Pidx = 1) the map changes rapidly and needs to beimplemented using a sufficient number of data points,interpolated from the original data as described in the text.

Figure 7. Turbine non-dimensional mass flow coefficient andefficiency maps.

Bell et al. 709



the approximate thermal efficiency of the engine andELHV is the lower heating value of the fuel. The airmass flow rate required to operate at a target A/F ratio,RA�F, can then be approximated as

_mair=_WRA�F

hThELHVð8Þ

The intake manifold pressure, Pm, required to meetthis air flow rate in a four-stroke engine can then beapproximated by rearranging the speed-density equa-tions (1) and (2) and assuming an ideal gas as

Pm =120 _mairRTm

hVNVdð9Þ

where Tm is the intake manifold temperature, hV is thevolumetric efficiency of the engine, N is the enginespeed and Vd is the displacement of the engine.

Assuming a small pressure loss across the air filter,Ploss1, and across the aftercooler, Ploss2, the pressureratio across the compressor can be determined as

Pd

Pu

� �Req

=Pm +Ploss2

Patm � Ploss1ð10Þ

where Patm is the atmospheric pressure.Referring to the non-dimensional map, the required

pressure ratio can be found in the region of high effi-ciency and the required compressor mass flow coeffi-cient, ;CReq, can be read as illustrated in Figure 8.

The diameter of the compressor wheel can be calcu-lated by combining equations (1) and (3), substitutingTatm for Tu, (Patm � Ploss1) for Pu and _mair=NParallel for_mC, where NParallel has been introduced to divide the airflow among parallel turbochargers

DC =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_mair

ffiffiffiffiffiffiffiffiffiffiffiffiffiRTamb

p

NParallel;CReq(Patm � Ploss1)

sð11Þ

This diameter will match the normalized compressormap to the given engine. The torque required to drive

the compressor, tC, will be required to match the tur-bine and can be calculated from isentropic relations as

tC =_mcCpTu

NParallelhcvTC

Pd

Pu

� �g�1g

�1 !

ð12Þ

Selecting turbine diameter

The turbine diameter must be approximated to matchthe torque produced by the turbine to that required bythe compressor during operation, tT = tC. The massflow rate through the turbine can be calculated fromthe compressor mass flow and the A/F ratio as_mT = _mc(1+ (1=RA�F)). The isentropic relation forturbine torque is

tT =hT _mTCpTu

NParallelvTC1� Pd

Pu

� �g�1g

!ð13Þ

The lookup tables provide a relationship between themass flow coefficient, efficiency, rotor tip Mach numberand expansion ratio

;T = fPu

Pd,C0,T

� �ð14Þ

hT = fPu

Pd,C0,T

� �ð15Þ

The definition of the rotor tip Mach number and themass flow coefficient provides two more equations withrelation to the diameter

;T =_mT

ffiffiffiffiffiffiffiffiffiRTu

p

PuD2T

ð16Þ

c0,T =vTCDTffiffiffiffiffiffiffiffiffiffiffiffi

gRTu

p ð17Þ

Assuming the temperature of the exhaust gasupstream of the turbine, Tu, is known and the turbineoutlet pressure, Pd, is near ambient, equations (13)–(17)provide a set of five equations and five unknowns (hT,Pu, ;T, c0,T and DT) which are solved for the turbinerotor diameter.

Results—matching non-dimensional mapsto four diesel engines

To illustrate the utility of the non-dimensionalapproach, compressor and turbine wheel diameters areselected to match the non-dimensional maps presentedin Figures 6 and 7 to several engines of various sizes.Table 1 summarizes the power and displacement ofeach engine, and the compressor and turbine diameterscalculated to meet the power requirements.

Steady states corresponding to 20%, 40%, 60%,80% and 100% load were then simulated using eachmodel. The compressor performance was monitoredduring simulation and is plotted on the normalized

Figure 8. Using the required pressure ratio to determine therequired mass flow coefficient to be in the high efficiency regionof the compressor map leaving a surge margin.




compressor maps in Figures 9 and 10. At each load,each engine operates in nearly the same location on thenormalized map. The turbine performance during simu-lation is also presented in Figure 11. Again at each load,each engine operates in nearly the same location on thenormalized turbine map to balance the compressor tor-que. The clusters of simulated points correspond toincreasing loads from left to right in each figure.

The results illustrate that the non-dimensional mapscan be utilized to model the turbocharger compressorand turbine across a range of engine models. Using themethods illustrated in this article, the work flow andeffort required to model engine dynamics in controlstudies where turbocharging plays an important rolecan be reduced by eliminating the need to fit compres-sor and turbine data for each engine individually, butrather match the ‘‘generic’’ family maps to the enginesthrough the selection of wheel diameters. In this way, asingle map can be utilized across a range of studiesinvolving different engines.

Conclusion

Normalizing the pressure ratio in compressor data bythe pressure ratio at compressor surge produces recti-linear axes which are readily implemented as a lookuptable in simulation models. The rectangular surge-normalized map captures the sudden decrease in flowat compressor surge and avoids modeling the surgeregion using empirical fits generated from data in thenormal operating region of the compressor and extra-polated beyond surge.

The non-dimensional maps represent devices wherethe turbocharger design parameters, namely, the A/Rratio and trim, are constant within a family.Implementing turbocharger performance data in state-based engine models using the non-dimensional massflow coefficient and rotor tip Mach number allows theturbine and compressor to be scaled to match the dis-placement and power of the engine. This eliminates theneed for data specific to each turbocharger and reducesthe time and effort required to model a range of enginesizes. For design applications, additional maps repre-senting geometric families of devices could be devel-oped in order to evaluate the influence of trim and A/Rratio on the engine dynamics.

The method presented to scale the compressor andturbine is a useful tool when implementing engine

Figure 9. Compressor operation during simulation overlaid onnon-dimensional compressor map. At each load, each engineoperates at nearly the same point on the map. Clusters ofpoints from left to right correspond to 20%, 40%, 60%, 80% and100% engine load.

Figure 10. Compressor operation during simulation overlaidon surge-normalized compressor map. Each engine operates atnearly the same point on the map. Clusters of points from left toright correspond to 20%, 40%, 60%, 80% and 100% engine load.

Figure 11. Turbine operation during simulation overlaid onnormalized turbine map. Clusters of points from left to rightcorrespond to 20%, 40%, 60%, 80% and 100% engine load.

Table 1. Compressor and turbine diameters calculated tomatch non-dimensional maps to four diesel engines utilized inpower generation.

Engine _W (kW m) Vd (L) DC (m) DT (m)

1 507 15.0 0.082 0.0612 1470 50.3 0.149 0.1123 1975 60.2 0.163 0.1204 2515 77.6 0.186 0.139

Bell et al. 711



models across a range of engine sizes. The method wasused to match non-dimensional compressor and tur-bine performance data generated from a single devicefor a family with a specific A/R ratio and trim to fourdiesel engines ranging from 500kW to 2.5MW. Theresults show that the compressor and turbine operatesimilarly across the range of engine sizes when themaps for the family are scaled to match the enginepower and displacement.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship and/or publica-tion of this article.

Funding

The author(s) received no financial support for theresearch, authorship and/or publication of this article.

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Appendix 1

Notation

c0,C compressor rotor tip Mach number basedon inlet conditions

c0,T turbine rotor tip Mach number based oninlet conditions

Cp specific heat at constant pressure, J/(kg K)Dc compressor wheel diameter, mDT turbine wheel diameter, mELHV lower heating value of fuel, J/kg_mair engine intake air flow rate, kg/s_mcorr,C compressor mass flow parameter, kg K0.5/

(s Pa)_mcorr,T turbine mass flow parameter, kg K0.5/(s

Pa)_mC compressor mass flow rate, kg/s_mT turbine mass flow rate, kg/sN engine speed, r/minNcorr rotor speed parameter, r/min K20.5

NParallel number of turbochargers in parallelNTC turbocharger rotor speed, r/minPatm atmospheric pressure, PaPd downstream pressure, PaPidx surge-normalized pressure ratio indexPloss1 pressure loss across air filter, PaPloss2 pressure loss across aftercooler, PaPm intake manifold pressure, PaPu upstream pressure, PaR specific gas constant, J/(kg K)RA�F air–fuel ratioTatm atmospheric temperature, KTu upstream temperature, KVd engine displacement, m3

_W engine power, J/s

g specific heat ratiohc compressor isentropic efficiencyhT turbine isentropic efficiencyhTh engine thermal efficiencyhV engine volumetric efficiencytC torque required to drive compressor, N mtT torque produced by turbine, N m;C compressor mass flow coefficient;T turbine mass flow coefficientvTC turbocharger rotor speed, rad/s




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