A Reduced Complexity Model for the Compressor Power of an ......compressor isentropic efficiency...

10
Tao Zeng Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824 e-mail: [email protected] Devesh Upadhyay Ford Motor Company, Dearborn, MI 48124 e-mail: [email protected] Guoming Zhu Fellow ASME Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824 e-mail: [email protected] A Reduced Complexity Model for the Compressor Power of an Automotive Turbocharger Control-oriented models for automotive turbocharger (TC) compressors typically describe the compressor power assuming an isentropic thermodynamic process with fixed isentropic and mechanical efficiencies for power transmission between the turbine and the compressor. Although these simplifications make the control-oriented model tracta- ble, they also introduce additional errors due to unmodeled dynamics. This is especially true for map-based approaches since the manufacture-provided maps tend to be sparse and often incomplete at the operational boundaries, especially at operational conditions with low mass flow rate and low speed. Extrapolation scheme is often used when the com- pressor is operated outside the mapped regions, which introduces additional errors. Fur- thermore, the manufacture-provided compressor maps, based on steady-flow bench tests, could be quite different from those under pulsating engine flow. In this paper, a physics- based model of compressor power is developed using Euler equations for turbomachi- nery, where the mass flow rate and the compressor rotational speed are used as model inputs. Two new coefficients, speed and power coefficients, are defined. As a result, this makes it possible to directly estimate the compressor power over the entire compressor operational range based on a single analytic relationship. The proposed modeling approach is validated against test data from standard TC flow bench tests, standard supercharger tests, steady-state, and certain transient engine dynamometer tests. Model validation results show that the proposed model has acceptable accuracy for model- based control design and also reduces the dimension of the parameter space typically needed to model compressor dynamics. [DOI: 10.1115/1.4039285] 1 Introduction It is common for plant models, used in the air path control of turbocharged diesel engines, to assume that the ideal power con- sumed by the compressor is defined by an isentropic thermody- namic process. The actual power is then derived from either a compressor isentropic efficiency map [13] or an empirically fit- ted isentropic efficiency map [46]. A common alternative approach is to define the compressor power as a first-order dynam- ics using an ad hoc time constant with the turbine power as the input [7,8]. On the other hand, map-based compressor power mod- els are relied on the overall turbocharger (TC) system efficiency maps as a function of the vane positions [9] applied to the calcu- lated turbine power. Empirically fitted compressor efficiencies are typically second- or third-order polynomials of the blade speed ratio. The polynomial coefficients are often dependent on the shaft speed [5,6,10,11] and are identified against the populated regions of the manufacturer-supplied maps. These polynomial models are, therefore, also subject to extrapolation when used under opera- tional conditions outside the manufacturer-supplied test points. Note that the typical manufacturer-supplied compressor perform- ance map is based on hot-gas flow-bench test data under steady flow conditions. Hence, actual maps could deviate from the manufacture-supplied ones under pulsating flow when the com- pressor is coupled to an internal combustion engine [1,2,12,13]. Another issue, often encountered when operating at light load conditions, is the sparsity of the manufacturer-supplied compres- sor performance maps in these operational areas, indicated in Fig. 1 for a sample turbocharged engine. When operating the com- pressor outside the mapped region, extrapolation under some smoothness constraint becomes necessary. Several investigations into extrapolation based on map extension are reported in open lit- erature; see Refs. [14] and [15]. The traditional actual compressor power is computed from the isentropic efficiency in the following equation: _ W c ¼ 1 g is c _ m c T in c air p P out P in c air 1 c air 1 ! (1) This compressor power model ( _ W c ) relies on a number of meas- ured inputs: the compressor inlet temperature (T in ), the mass flow rate ( _ m c ), and the inlet and outlet pressures (P in and P out ), Fig. 1 Operating range deficit between mapped and desired engine operating range Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT, AND CONTROL. Manuscript received October 11, 2016; final manuscript received January 17, 2018; published online March 27, 2018. Assoc. Editor: Azim Eskandarian. Journal of Dynamic Systems, Measurement, and Control JUNE 2018, Vol. 140 / 061018-1 Copyright V C 2018 by ASME Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 05/30/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Transcript of A Reduced Complexity Model for the Compressor Power of an ......compressor isentropic efficiency...

Page 1: A Reduced Complexity Model for the Compressor Power of an ......compressor isentropic efficiency map [1–3] or an empirically fit-ted isentropic efficiency map [4–6]. A common

Tao ZengDepartment of Mechanical Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: [email protected]

Devesh UpadhyayFord Motor Company,

Dearborn, MI 48124

e-mail: [email protected]

Guoming ZhuFellow ASME

Department of Mechanical Engineering,

Michigan State University,

East Lansing, MI 48824

e-mail: [email protected]

A Reduced Complexity Modelfor the Compressor Power ofan Automotive TurbochargerControl-oriented models for automotive turbocharger (TC) compressors typicallydescribe the compressor power assuming an isentropic thermodynamic process with fixedisentropic and mechanical efficiencies for power transmission between the turbine andthe compressor. Although these simplifications make the control-oriented model tracta-ble, they also introduce additional errors due to unmodeled dynamics. This is especiallytrue for map-based approaches since the manufacture-provided maps tend to be sparseand often incomplete at the operational boundaries, especially at operational conditionswith low mass flow rate and low speed. Extrapolation scheme is often used when the com-pressor is operated outside the mapped regions, which introduces additional errors. Fur-thermore, the manufacture-provided compressor maps, based on steady-flow bench tests,could be quite different from those under pulsating engine flow. In this paper, a physics-based model of compressor power is developed using Euler equations for turbomachi-nery, where the mass flow rate and the compressor rotational speed are used as modelinputs. Two new coefficients, speed and power coefficients, are defined. As a result, thismakes it possible to directly estimate the compressor power over the entire compressoroperational range based on a single analytic relationship. The proposed modelingapproach is validated against test data from standard TC flow bench tests, standardsupercharger tests, steady-state, and certain transient engine dynamometer tests. Modelvalidation results show that the proposed model has acceptable accuracy for model-based control design and also reduces the dimension of the parameter space typicallyneeded to model compressor dynamics. [DOI: 10.1115/1.4039285]

1 Introduction

It is common for plant models, used in the air path control ofturbocharged diesel engines, to assume that the ideal power con-sumed by the compressor is defined by an isentropic thermody-namic process. The actual power is then derived from either acompressor isentropic efficiency map [1–3] or an empirically fit-ted isentropic efficiency map [4–6]. A common alternativeapproach is to define the compressor power as a first-order dynam-ics using an ad hoc time constant with the turbine power as theinput [7,8]. On the other hand, map-based compressor power mod-els are relied on the overall turbocharger (TC) system efficiencymaps as a function of the vane positions [9] applied to the calcu-lated turbine power. Empirically fitted compressor efficiencies aretypically second- or third-order polynomials of the blade speedratio. The polynomial coefficients are often dependent on the shaftspeed [5,6,10,11] and are identified against the populated regionsof the manufacturer-supplied maps. These polynomial models are,therefore, also subject to extrapolation when used under opera-tional conditions outside the manufacturer-supplied test points.Note that the typical manufacturer-supplied compressor perform-ance map is based on hot-gas flow-bench test data under steadyflow conditions. Hence, actual maps could deviate from themanufacture-supplied ones under pulsating flow when the com-pressor is coupled to an internal combustion engine [1,2,12,13].Another issue, often encountered when operating at light loadconditions, is the sparsity of the manufacturer-supplied compres-sor performance maps in these operational areas, indicated inFig. 1 for a sample turbocharged engine. When operating the com-pressor outside the mapped region, extrapolation under somesmoothness constraint becomes necessary. Several investigations

into extrapolation based on map extension are reported in open lit-erature; see Refs. [14] and [15].

The traditional actual compressor power is computed from theisentropic efficiency in the following equation:

_Wc ¼1

gisc

_mcTincairp

Pout

Pin

� �cair�1

cair

� 1

!(1)

This compressor power model ( _Wc) relies on a number of meas-ured inputs: the compressor inlet temperature (Tin), the mass flowrate ( _mc), and the inlet and outlet pressures (Pin and Pout),

Fig. 1 Operating range deficit between mapped and desiredengine operating range

Contributed by the Dynamic Systems Division of ASME for publication in theJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedOctober 11, 2016; final manuscript received January 17, 2018; published onlineMarch 27, 2018. Assoc. Editor: Azim Eskandarian.

Journal of Dynamic Systems, Measurement, and Control JUNE 2018, Vol. 140 / 061018-1Copyright VC 2018 by ASME

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Page 2: A Reduced Complexity Model for the Compressor Power of an ......compressor isentropic efficiency map [1–3] or an empirically fit-ted isentropic efficiency map [4–6]. A common

respectively. Additionally, the air properties, such as specific heat(cair

p ) and isentropic index (cair), are assumed to be fixed. The isen-tropic efficiency (gis

c ) used in Eq. (1) is either an empirical modelor is available from mapping data-based flow bench tests. In areview of the existing literature, there are various empirical mod-els used to describe the isentropic efficiency. In Ref. [6], the effi-ciency is expressed as a third-order polynomial function of thenondimensional mass flow rate,uc, that is a function of the massflow rate and the blade tip speed. The polynomial coefficients ai

are three individual functions of inlet Mach number. In Ref. [16],the efficiency is modeled using an elliptical fit based on mass flowrate and pressure ratio. This model depends on the maximum effi-ciency in terms of corrected mass flow rate and compressor pres-sure ratio. In Ref. [17], the efficiency model from [16] is furthermodified for pressure ratio variations. Physics-based high fidelitycompressor model was presented in Ref. [18] without any discus-sion of reduced complexity. In Ref. [19], compressor work isdefined as a polynomial function of corrected mass flow rate andthe model is further fitted with experimental data. Table 1 summa-rizes these approaches and presents results from a previousperformance assessment [22] of these models inside themanufacture-supplied maps. However, it has been well estab-lished that these models do not necessarily behave well underextrapolation, especially under light load conditions [11]. In thispaper, an alternative approach is investigated such that the modelparameters are derived from compressor physics, and therefore,have clear physical interpretation. Further, a reduced-order modelis proposed. Prediction capability of the proposed reduced-ordermodel is investigated and is shown to be adequate for controldesign. Additionally, this model also allows smooth extension tooperational conditions beyond the typically mapped operatoinalrange. In the approach adopted in this study, the Euler equationsfor turbomachinery are used for developing a model for predictingcompressor power. The proposed model uses the compressor massflow rate and the compressor angular velocity as inputs. Two newparameters, the speed and power coefficients, are defined. It isfound that using a quadratic analytic function to model these twocoefficients is adequate to characterize the compressor performanceover the entire operational range. The model is validated using hotgas flow bench test data, steady-state engine dynamometer test dataas well as certain transient simulation and test data. The validationresults indicate that the proposed reduced-order model is suitablefor both steady-state and certain transient operations under realisticpulsating exhaust flow conditions. Note that even though the Eulerequation is only applicable for steady-state, it turns out that the pro-posed Euler equation based model is capable of providing accurateprediction under certain transient operational conditions.

The rest of the paper is organized as follows: Section 2 dis-cusses the development of the proposed compressor model basedon the Euler equations with different slip factors. Section 3 pro-vides results from model identification based on flow bench tests.Section 4 discusses results from model validation against transientdata from GT Power simulations as well as from engine dyna-mometer tests. Section 5 concludes the paper.

2 Compressor Power Modeling

2.1 Compressor Power Model Based on the EulerEquations. The proposed model is derived, in part, using Eulerequations of turbomachinery. In dealing with the Euler equations, onemust rely on the compressor geometry and velocity triangles associ-ated with the gas flow at the impeller inlet and outlet. For complete-ness, figures of the compressor geometry and velocity triangle arereproduced from Refs. [1] and [23]. A typical centrifugal compres-sor geometry layout and the velocity triangles of the gas flow at thecompressor impeller inlet and outlet are shown in Figs. 2 and 3.

The Euler equation provides the energy transfer to the fluid as aproduct of the angular velocity and torque shown in the belowequation:

_Wc ¼ xs ¼ x _mcðvoutrout � vinrinÞ ¼ _mcðU2Ch2 � U1Ch1Þ (2)

For a centrifugal impeller, it is assumed that the air enters theimpeller eye in the axial direction so that the initial angularmomentum of the air at the inlet of the compressor can beassumed to be zero (U1Ch1 � 0) [1,23]. The ideal compressorpower equation can then be reduced to

_Wc no swirl ¼ _mcU2Ch2 (3)

or equivalently

_Wc no swirl ¼ _mcxR2Ch2 (4)

where U2 ¼ xR2. Under nominal operation, the flow exiting theblades will deviate from the ideal blade back-sweep angle b andexit at some angle b0. This deviation from the ideal is referred toas slip. Under the influence of slip, the corrected absolute flowvelocity can be expressed as

C0h2 ¼ U2 � Cr2 tan b02 (5)

where Cr2 is the impeller outlet flow radial velocity. The ratio ofCh2 to C0h2 is defined as slip factor r

r ¼ Ch2

C0h2

(6)

The slip factor depends on a number of factors such as the num-ber of impeller blades, the passage geometry, the ratio of impellereye tip to impeller exit diameters, the mass flow rate, and the com-pressor speed [23]. Introducing the slip factor in Eq. (4) and theexpression for C0h2, the compressor power without inlet swirl iswritten as

_Wc no swirl ¼ _mcxR2C0h2r ¼ _mcxR2ðU2 � Cr2 tan b02Þr (7)

Using the conservation of mass flow rate yields

_mc ¼ Cr2q2A2 ¼ Cr1q1A1 (8)

Further assuming Cr2 � Cr1 (uniform radial flow with no radialflow losses) leads to ðCr2 ¼ ð _mc=q2A2ÞÞ � ðCr1 ¼ ð _mc=q1A1ÞÞ.Note that due to required simplification, it is reasonable to assumethat the speed change in the radial flow direction is small due tosmall flow losses in the compressor and the error resulting fromthis simplification can be absorbed indirectly via model parametercalibrations. Substituting for Cr2 ¼ ð _mc=q1A1Þ in Eq. (7), thecompressor power can be reformulated as

_WC no swirl ¼ rR22 _mcx

2 � rR2 tan b02q1A1

!x _m2

c (9)

In order to account for flow and “windage” losses, the actualrequired input work must be greater than the theoretical value nec-essary to achieve the target flow rate [24]. To account for this, apower loss factor w is used to modify the ideal power and a fric-tion loss power term _Wf is introduced into power model to definethe actual, loss-compensated, power required to achieve a desiredmass flow rate as

_Wc ¼ w _Wc no swirl þ _Wf (10)

In this study, the factor w is assumed to be a design parameterand is therefore assumed fixed for a given compressor design.Ideally, this parameter must be established experimentally oridentified through standard techniques (this work). In order todefine the friction loss term, the loss models proposed in

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Table 1 Comparison among different efficiency modeling approaches (assuming constant inlet condition)

Reference Compressor efficiency model[Number of fitting coefficients]

{number of inputs}[R2]{ error meandeviation (EMD)}

Jensen et al. [6,20] SgC ¼ a1u2c þ a2uc þ a3; ai ¼

ai1 þ ai2M

ai3 �Mi ¼ 1; 2; 3 where M is inlet Mach number [9]{2} [0.971] {6.1}

Guzzella–Amstutz [16]gc ¼ gc;max � vTQv vT ¼ ½ucorr � ucorr;gmax

;pc � pc;max� Q ¼a11 a12

a21 a22

� �ucorr;gmax

and pc;max are the corrected mass

flow parameter ucorr and the compression ratio pc corresponding to the maximum efficiency gc;max

[4]{3} [0.927]{ 10.9}

Andersson [17]gc ¼ gc;max � vTQv; vT ¼ ½ucorr � ucorr;gmax

1þffiffiffiffiffiffiffiffiffiffiffiffiffipc � 1p

� pc;max�; Q ¼ a11 a12

a21 a22

� �[4]{3} [0.790]{22.2}

Canova et al. [19]

gc ¼GcCpTinðp

k�1k

c �1ÞðPref

Pin

ÞffiffiffiffiffiffiffiTref

Tin

rWcorr

Wcorr ¼ A1 þ A2ucorr þ A3u2corr þ A4u3

corr where Aiði ¼ 1; 2; 3; 4Þ are

linear functions of corrected speed xcorr ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTref=Tin

p[4]{3} [0.947]{4.9}

Sieros et al.: simple linear [21] Y ¼ a1 þ a2X; Y ¼ ðpc þ gcÞ2 X ¼ ðpc þ gc þ 1=pcÞðpc þ gcÞ [2]{3} [0.919]{11.6}

Sieros et al.: generalizedlinear 1 [21]

gc ¼ A1 þ A2pc þ A3p2c A1 ¼ a1 þ a2xnor þ a3x2

corr A2 ¼ a4 þ a5xnor; A3 ¼ a6 [6]{3} [0.978]{5.3}

Sieros et al.: generalizedlinear 2 [21]

gc ¼ A1 þ A2pc þ A3 logðpcÞ SS A1 ¼ a1; A2 ¼ a2 þ a3xnor; A3 ¼ a4 þ a5xnor [5]{2} [0.980]{4.8}

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Page 4: A Reduced Complexity Model for the Compressor Power of an ......compressor isentropic efficiency map [1–3] or an empirically fit-ted isentropic efficiency map [4–6]. A common

Refs. [23] and [25] are used here. The friction loss is modeled as asum of the losses over the impeller and the diffuser and is definedas cubic functions of the mass flow rate

_Wf ¼ kf _m3c ¼ ðkfi þ kfdÞ _m3

c (11)

where kfi and kfd are friction coefficients for the impeller and dif-fuser, respectively. Hence, the actual compressor power needed toachieve a desired mass flow rate can be written in its expandedform with all the loss modifiers as

_Wc ¼ wr R22 _mcx

2 � wr R2 tan b02q1A1

x _m2c þ kf _m3

c (12)

Two new parameters, the power coefficient CPower and the speedcoefficient CSpeed, are defined as follows:

CPower ¼_W c

_m3c

(13)

CSpeed ¼x_mc

(14)

Using this notation, the compressor power in Eq. (12) can berearranged in terms of the power and speed coefficients as follows:

CPower ¼ wr R22 CSpeedð Þ2 �

wr R2 tan b02q1A1

CSpeedð Þ þ kf (15)

It is clear from Eq. (15) that Cpower varies not only with com-pressor operating condition (such as Cspeed) but also with com-pressor designs parameters. Different compressor designs couldalso impact the power law (15) through compressor specificgeometry, power loss factor, and slip factor. A generalized power

coefficient model can, however, be written for a specific compres-sor design. Since for a given compressor design, the parameters w,kf , R2 tan b02, and A1 are constant. This allows the power coeffi-cient to be expressed compactly as a function of the operatingparameters, speed coefficient, and the slip factor for a given oper-ating condition i as

CPower i ¼ f ðCSpeed i; riÞ; i ¼ 1; 2;…; n (16)

One of the goals of this work was to derive a reduced orderpower model and investigate its prediction capability. The powerand speed coefficients in conjunction with an appropriate slip fac-tor model may be such a candidate model. It may be possible tofurther reduce the model order by making the power coefficient afunction of the speed coefficient only as indicated in the belowequation:

CPower ¼ f ðCSpeedÞ (17)

It is clear that in order to achieve the form shown in Eq. (17),the slip factor must be defined either as a parameter fixed bydesign or defined in terms of the speed and/or power coefficient tomaintain the homogeneity of the compressor power model withrespect to Cpower and Cspeed. This leads us into an investigation ofslip factor models.

2.2 Investigation of Slip Factor Models for Flows OverCentrifugal Compressor. Several slip factors models for centrif-ugal compressors are readily available in the open literature[21,22,23,26–29]. The slip factor typically depends on the com-pressor design parameters as well as the operating conditions,such as compressor rotational speed and mass flow rate. In orderto preserve the impact of flow variations on slip, the slip factormodels, as proposed in Refs. [26], [28], and [30], are investigatedin this work

slip1 : r¼1þ _m tanb02q1A1xR2

�0:5� 1�e�2p

Z cosb02� �

0<r<1ð Þ Reffstrupð Þ

(18)

slip 2 : r ¼ 1� p=Zð ÞxR2 cos b02

xR2 �_m

q1A1

tan b02

0 < r < 1ð Þ Stodolað Þ

(19)

slip 3 : r ¼ 1� a_m

xR32

0 < r < 1ð Þ Stahlerð Þ (20)

where a is a design parameter in Eq. (20) related to the flow exitangle and is a constant for a given impeller design, and Z is thenumber of impeller blades. Integrating each of these models intothe compressor power model (15), a generalized compressorpower model can be established with the following structure:

_Wc

_m3c

¼ e1

x_mc

� �2

� e2

x_mc

� �þ e) CPower

¼ e1 CSpeedð Þ2 � e2 CSpeedð Þ þ e3 (21)

where the coefficients e1, e2, and e3 depend on the selected slipfactor model and are defined in Table 2. The model structure inEq. (21) is referred to as the generalized compressor power modelin the rest of this paper. Note that the three coefficients in Eq. (21)take constant values and are fixed by compressor design under theassumption of constant or slowly varying inlet conditions. Oncethe relationship between the power and speed coefficients hasbeen identified, the compressor power for a given operating point

Fig. 2 Compressor geometry

Fig. 3 Velocity triangles of a centrifugal compressor at therotor inlet and outlet

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defined the pair h _mc;xi, which can be established as in the belowequation:

_Wc ¼ _m3c e1

x_mc

� �2

� e2

x_mc

� �þ e3

!(22)

This approach shows that the compressor operation can be rep-resented via an analytic quadratic function rather. The proposedmodel has only two inputs (the compressor mass flow rate and TCrotational speed) and three parameters to be identified. In practicalapplications, while the mass flow rate is typically available as ameasurement, the TC speed is not a standard measurement. In theabsence of a TC speed measurement, the observed value of theTC speed via an appropriate observer design may be used. Alter-nately, TC speed can also be obtained by solving the turbochargerrotor dynamic differential equations as in Refs. [3], [7], [8], and[31], and the compressor mass flow rate can be calculated usingan observer based on shaft speed and pressure ratio [6]. In thispaper, the compressor power model is validated under the formerassumption that two parameters, mass flow rate and TC rotationalspeed, are available as measurements.

2.3 Compatibility With Corrected Mass Flow Rates. Cor-rected compressor mass flow rates _mcorrect ¼ _mð

ffiffiffiffiffiT1

p=P1Þ and cor-

rected TC angular velocity xcorrect ¼ x=ffiffiffiffiffiT1

pare typically used in

compressor performance maps for nonstandard inlet conditions. Inthis case, T1 and P1 are nondimensional, where T1 ¼ ðTinlet=TrefÞand P1 ¼ ðpinlet=prefÞ. It was natural to investigate the structure ofthe power coefficient based compressor power model when usingthese corrected terms. It is found that the power coefficient withthe corrected variables has the same structure as before and isscaled by the modifying term

ffiffiffiffiffiT1

p=P1 and 1=

ffiffiffiffiffiT1

pas shown

below. To show this, the corrected power coefficient is defined asCpower correct ¼ ð _Wc=ð _mcorrectÞ3Þ; and the corrected speed coeffi-cient can be defined as Cspeed correct ¼ ðxcorrect= _mcorrectÞ. Introduc-ing these terms directly into the power coefficient model as inEq. (23), the proposed model with corrected power coefficient isshown in Eq. (24)

_Wc

_m3¼ e1 x _mð Þ2 � e2 x _mð Þ þ e3

_Wc ¼1ffiffiffiffiffi

T1

p=P1

� �3

_mffiffiffiffiffiT1

p=P1

� �3e1

T1

P1

� �2 x=ffiffiffiffiffiT1

p

_mffiffiffiffiffiT1

p=P1

!20@

�e2

T1

P1

� �2 x=ffiffiffiffiffiT1

p

_mffiffiffiffiffiT1

p=P1

!þ e3

!

_Wc ¼1ffiffiffiTp

=P

!3

_mcorrectð Þ3 e1

T1

P1

� �2 xcorrect

_mcorrect

� �2

�e2

T1

P1

� �2 xcorrect

_mcorrect

� �þ e3

!(23)

_Wc

_mcorrectð Þ3¼ 1ffiffiffiffiffi

T1

p=P1

� �3

e1

T1

P1

� �2 xcorrect

_mcorrect

� �2

�e2

T1

P1

� �2 xcorrect

_mcorrect

� �þ e3

!

Cpower correct ¼1ffiffiffiffiffi

T1

p=P1

� �3

e1

T1

P1

� �2

Cspeed correctð Þ2

�e2

T1

P1

� �2

Cspeed correctð Þ þ e3

!(24)

Equation (24) means that the model needs to be calibrated byreference temperature and pressure when compressor operatesunder nonstandard inlet conditions. In the rest of this paper, allderivations and results are based on the model (22) without thecorrected parameters, since all the measurements in this paper areunder the standard inlet conditions at sea level.

3 Model Identification

3.1 Model Identification Using Standard Hot Gas FlowBench Test Data. Test data from standard hot gas flow benchtests were used to identify the three model parameters, e1; e2; ande3; for the generalized compressor model as in Eq. (21) for threedifferent compressor designs, compressors 1, 2, and 3. The inletconditions of compressors are assumed fixed in line with standardtest protocol [32]. The minimum test speed for compressors 1 and2 was 30k rpm, while the lowest speed for compressor 3 was 46krpm. Model parameters for the power coefficient model as inEq. (21) are identified using a least squares optimization based ona model error cost function as in the below equation:

J ¼Xn

i¼1

_Wmodel

c ðiÞ � _Wmes

c ðiÞh i 2

(25)

In Eq. (25), n¼ 322 is the number of steady-state compressor

operating test points for the hot gas flow bench tests. _Wmodel

c is the

calculated compressor power using Eq. (22); and _Wmes

c is thestandard compressor power calculated from measured inputs asEq. (26) (see Ref. [23])

_Wmes

c ¼ _mmesc cpðTmes

out � Tmesin Þ (26)

Since the geometric design parameters R2, A1 and b02 for com-pressor 1 were available. Compressor 1 was selected as a candi-date for identifying the parameters, w, kf , and a, for all three slipmodels using the model structure (15). Note that the parameter ais relevant only for the slip model-3. The values of the identifiedparameters are shown in Table 3.

It is interesting to note that identified value of the power lossfactor w is greater than unity, indicating that the compressorpower necessary to achieve the desired flow rate must be largerthan the ideal power calculated in Eq. (4) in order to account for

Table 2 Model coefficients for various slip models

Slip model 1 Slip model 2 Slip model 3

e1 wR22ð0:5þ 0:5e

�2pZ cos b

02 Þ wR2

2ð1� ðp=ZÞcos b0

2Þ wR22

e2 wR2 tan b0

2

q1A1

ð0:5e�2p

Z cos b02 � 0:5Þ

wR2 tan b

0

2

q1A1

wa

R2

þ tan b0

2R2

q1A1

!

e3

kf � wtan b

0

2

q1A1

!2 kf wtan b

0

2a

q1A1R2

þ kf

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losses. This is due to assuming that Cp is constant and Cr2 equal toCr1. The order of magnitude of the identified parameter a (Slipmodel-3) agrees with the result in Ref. [28]. Given the value forthe friction coefficient kf , the maximum friction loss power identi-fied was around 2.38 kW over the entire compressor operatingrange. Modeling errors are investigated using four error metrics:the coefficient of determination R2, EMD, and PEBð65%Þ andPEBð610%Þ that are the percentage of total data points within theerror bound 65% and 610%, respectively. The error evaluationparameters used in these calculations are defined in the belowequations:

error ið Þ ¼_W

model

c ið Þ � _Wmes

c ið Þ_W

mes

c ið Þ(27)

EMD ¼ error�meanðerrorÞn

(28)

PEB 6k%ð Þ ¼m

n;m ¼

Xn

i¼1

Nk ið Þ; where jerror Nkð Þj � k% (29)

where n is the total sample number. Based on model fitting error,as reported in Table 3, the slip model-3 has the best fit (least error)for compressor 1. Since compressor design parameters were notavailable for compressors 2 and 3, we could only identify the threelumped parameters e1, e2, and e3 for the generalized power model(21) for these two compressor designs. The identified values forthe coefficients of the generalized power coefficient are shown forall three compressor designs in Table 3.

Results from the fitted models for the three different compres-sor designs are shown in Figure 4(b). The log–log plot is shown inFig. 4(c) that each compressor design has a unique characteristiccurve. This is expected and reflects the design differences amongthe different compressors. In fact, such plots allow a quick andeasy comparison of different compressor designs. As an example,it is clear that for a given mass flow rate and TC shaft speed, thecompressor power follows the trend, Power1> Power2>Power3,indicating that design-1 perhaps have a larger loss relative to theother designs. Also, note that it is clear that compressor 3 mayoffer the widest operating range of the three designs considered inthis study. The log10-scale plots offer new insights. It is easy tosee that the log10-scale representation is a linear transformation ofquadratic curves; that is, a power law relationship exists betweenthe power and speed coefficients. The log model is shown in thebelow equation:

log10ðCPowerÞ ¼ p log10ðCSpeedÞ þ q (30)

Table 3 Identified model coefficients for three model variants and three compressor design variants

w kf a R2 EMDa PEBb (65%) PEB (610%)

Model (21)compressor 1

Slip model 1 1.295 19,040 NA 0.968 4.42 58.9% 83.52%Slip model 2 1.323 6374 NA 0.968 4.56 66.44% 84.00%Slip model 3 1.351 11,390 1.334 0.989 3.24 79.15% 92.90%

e1 e2 e3 R2 EMD PEB (65%) PEB (610%)

Model (22) Compressor 1 3.52� 10�8 0.0123 1752 0.992 3.24 79.15% 92.90%Compressor 2 2.98� 10�8 0.0162 3547 0.995 3.73 68.31% 90.22%Compressor 3 1.68� 10�8 0.0086 1987 0.996 2.61 81.34% 91.69%

p q NA R2 EMD PEB (65%) PEB (610%)

Model (30) Compressor 1 2.439 �10.25 NA 0.994 5.97 70.07% 86.11%Compressor 2 2.423 �10.31 NA 0.989 3.78 58.01% 85.83%Compressor 3 2.262 �9.551 NA 0.996 2.37 86.63% 94.61%

aEMD: error mean deviation.bPEB: percentage of data points within error bound.

Fig. 4 Identification results for the generalized compressor-power model for compressors 1, 2, and 3: (a) flow bench testrange for three compressors, (b) model identification under lin-ear scale, and (c) model identification under log scale

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In Eq. (30), p is the slope of each characteristic line and q is they-intercept for each design. From Eq. (30), we get the power lawmodel, derived from a log10 plot as

CPower ¼ 10qðCSpeedÞp !_Wc

_m3c

¼ 10q x_mc

� �p

(31)

The p and q values for the power-law plots were also identifiedand are noted in Table 3. The power law representation implies ascaling invariance, and therefore, provides a good measure of thesensitivity of power coefficient to changes in the speed coefficient.One important advantage of this is that Eq. (31) can, theoretically, beobtained from only two compressor operating conditions that span thehigh and low load operational conditions. This is in line with the two-point fitting method used for determining the power law exponent.Hence, this method may significantly reduce the experimental burdenduring the early stages of compressor development.

In summary, model fitting results confirm that the generalizedcompressor power model proposed in this work is able to repro-duce the compressor behavior. Proposed models only need two orthree parameters to be identified. These results also demonstratethat for a centrifugal compressor, the operating characteristics canbe reduced to an analytic quadratic function in linear scale and astraight line in logarithmic scale. A summary of the relative errorsof prediction, for the various compressor designs and the modelvariants, is shown in Fig. 5. It is clear that prediction error is wellcontained within a 65% relative error band.

3.2 Model Identification With “Supercharger StandardTest.” Supercharging standard tests (SST) [33] are performedwith an electric motor driving the compressor as opposed to theturbine. Given the wider speed control of electric motor, it is pos-sible to perform more extremely light load (low-speed and lowmass flow rates) tests relative to standard hot-gas flow bench tests.Three different compressor designs, compressors 4, 5, and 6, weretested under this test protocol. The lowest compressor angularvelocity achieved was 20k rpm compared to 30k rpm in the flowbench tests. The minimum mass flow rate was 0.01 kg/s in thecompressor dynamometer, compared to 0.02 kg/s in the flowbench test.

Following the same procedure as for the previous set of com-pressors, we identified the three lumped parameters ei for the gen-eralized power coefficient model as in Eq. (22). The identifiedvalues are indicated in Table 4. The behavior of the fitted model iscompared against SST data in Fig. 6. Results indicate, as before,

that the proposed model is able to reproduce the observed data.The log scale plots also reproduce a similar power law behaviorobserved for the previous set of compressors. The parameters forthe power law model are included in Table 4.

4 Model Validation

4.1 Model Validation Based on Steady-State EngineDynamometer Test Data. The previously identified model forCpower ¼ f ðCspeedÞ was validated against a more realistic data set

Fig. 5 Modeling error (27) for compressor power models

Table 4 Fitted model coefficients identified through super-charging test data

Model (22) e1 e2 e3 R2

Compressor 4 1.928� 10�6 0.0618 858.1 0.995Compressor 5 1.965� 10�6 0.0618 858.1 0.996Compressor 6 1.967� 10�6 0.0564 718.3 0.995

Model (29) p q NA R2

Compressor 4 2.379 �10.09 NA 0.995Compressor 5 2.378 �10.07 NA 0.989Compressor 6 2.378 �10.06 NA 0.996

Fig. 6 Identification results for the generalized compressor-power model for compressors 4, 5, and 6: (a) supercharger testranges for three compressors, (b) model identification underlinear scale, and (c) model identification under log scale

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from engine dynamometer steady-state tests. These tests wereconducted on a heavy duty diesel engine. The turbocharger on thisengine used a compressor equivalent to the design variant com-pressor 1. Test details can be found from previously publishedworks [34] and [35]. The steady-state test covers the entire engineoperating range (185 testing points). The compressor operatingrange is defined by a mass flow rate range between 0.01 and0.45 kg/s and compressor speed range between 5.9k and 109krpm. Note that these tests achieve lighter operational conditionsrelative to the flow bench tests used for model identification forwhich the TC speed was limited, at the low end to, 30k rpm forthe hot-gas tests, and 20k rpm for the SST tests. The engine opera-tional range drives compressor operation beyond the flow-benchdata and provides an opportunity to test the range extension prop-erties of the proposed model. In Fig. 7(a), we show the engine testgrid overlaid on the compressor mapping points (from hot-gasflow-bench tests). It is clear that the engine operation at light loadconditions forces the compressor to operate in regions not coveredby the flow-bench data.

The Cpower model, previously identified for compressor 1, wasused to reproduce the power-coefficient versus speed-coefficientrelationship for the engine test data. In Fig. 7(a), the predictedCpower is compared against the calculated values for Cpower forboth the flow-bench tests as well as the engine steady-state tests.

Recall that the calculated values for Cpower, are based on Eq. (26)and use measured inputs. Since the engine data set extends tolower load conditions (TC speed< 30k rpm), the characteristiccurves for the engine data are plotted in two sets to span opera-tional regimes above and below the 30k rpm TC speed threshold.This was done primarily to assess the quality of the model predic-tion for light conditions that extend beyond the standard mappingdomain. It is clear from Figs. 7(b) and 7(c), that, while the modeladequately replicates the calculated values for Cpower for TCspeeds> 30k rpm, there is a significant error under operationalconditions of TC speeds< 30k rpm, where the calculated valuesfor Cpower lose monotonicity and appear to be quite random. Thisobviously anomalous behavior may have sources other than flowirregularities from extremely low flows.

To further understand the error sources, a sensitivity analysiswas performed on the power calculations. The error in the meas-ured power-coefficient is amplified by artificially low mass flowrates reported by the production mass flow sensor. Mass flow ratesensors are known to suffer from nonlinearity and loss of accuracyat low flow rates as observed for these tests. This is confirmedfrom the model behavior for the SST tests that used laboratorygrade sensors and for which the Cpower model was able to repro-duce the measured values quite adequately. An additional sourceof error may be attributed to a positive bias in the temperaturedownstream of the compressor due to heat transfer from thehot-end (turbine). This effect of heat transfer on the calculatedcompressor under light load operating conditions was also verifiedexperimentally by Serrano et al. [36] and Chesse et al. [37]. How-ever, this effect may not completely describe the error in the Cpo-

wer model observed here since the successive low load testconditions would result in a progressively cooler turbine housingleading to reduced heat transfer effects. In summary, since theerror in the predicted power is less than 0.23 kW in light loadrange and the model showed good predictive capability for theSST experiments for light load operation, it is safe to project thatthe proposed model is adequate for light load extrapolation.

Fig. 7 Comparison of Cpower model performance against calcu-lated values based on flow bench data and engine steady-statedynamometer test data for compressor 1: (a) engine test rangeand flow bench test range, (b) comparison under linear scale,and (c) comparison under log scale

Fig. 8 Model validation over US06 GT-Power transientsimulation

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4.2 Model Validation for US06 Transient Cycle Based onGT-Power Cycle Simulations. Next, the model behavior is veri-fied against transient data from GT-Power simulations for a US06cycle. The TC speed varied between 14k and 105k rpm during thetest. The model selected for evaluation was based on compressor1 design and slip model-3. Model prediction for Cpower is com-pared against calculated values as before. Figure 8 shows that themodel prediction for Cpower agrees quite well with the expectedvalues as calculated values over the test cycle.

4.3 Model Validation Over Federal Test Procedure CycleBased on Engine Dynamometer Tests. Model prediction capa-bility over an federal test procedure (FTP) test cycle with enginedynamometer data was investigated. The model candidate used isbased on compressor 1 design and slip model-3. Since the calcu-lated Cpower (for comparison) relies on the compressor down-stream temperature, we assessed the signal behavior of therelevant sensor. This is because the location of the postcompressor

temperature sensor, in the test engine configuration, was expectedto introduce measurement error from slow dynamics and delay aswell as heat transfer losses. In Fig. 9, it is clear that during thisload step, the mass flow rate and turbocharger speed has a muchfaster response compared to the temperature sensor. In order toaddress this, we corrected the measured temperature signal usinga lead-lag filter. The lead-lag filter was designed to match theexpected temperature profile for the same load step by invertingthe compressor power model (26) shown in the below equation:

_mcp Tout � Tinð Þ ¼ _m3f x _mð Þ ) T̂out ¼_m3f x _mð Þ

_mcpþ Tin (32)

The corrected temperature, downstream of the compressor, iscompared against the measured signal in Fig. 10. The same filterwas then applied to the measured temperature signal for the fullcycle and the corrected temperature was used to evaluate themeasured value of the compressor power using (26). The cor-rected measured temperature signals are shown in Fig. 10. Thepredicted compressor power was established from the predictedCpower and known mass flow rate. The predicted and calculatedvalues are compared in Fig. 10. The filter is seen to converge at160 s, after which, it is clear that the predicted compressor poweris capable of reproducing the actual compressor reasonably well.With this confidence, it is reasonable to claim that using the pro-posed method the compressor power can be predicted reasonablywell based on measurements of compressor mass flow rate andTC speed.

5 Conclusion

A compressor power model, based on the Euler turbomachineryequations with realistic assumptions, was developed. Two newperformance coefficients, the power and speed coefficients, wereproposed as an alternative to multiple performance maps. The pro-posed correlation between Cpower and Cspeed is especially useful indefining the compressor power necessary for achieving a desiredcompressor mass flow rate. This compressor power demand canthen be translated into a variable geometry turbo vane position or

Fig. 9 Normalized measurements for mass flow rate, TCspeed, and compressor downstream temperature for a loadstep

Fig. 10 Model validation against transient engine test data for a FTP 75 cycle

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an assist demand for assisted boosting systems. This relationshipcan also be easily used to compare compressor design variantswith respect to performance and range. The model is validatedagainst data sets from standard turbocharger flow bench tests,steady-state engine dynamometer tests as well as transient enginesimulations and test. Validation results indicate that the proposedmodel provides accurate compressor power prediction over abroad range of compressor operating conditions and provides foran easy and reliable extrapolation for operating conditions outsidethe standard mapping domain. Further, the proposed modelreduces the dimensionality of the parameter space typically neces-sary for such applications. The reduced order, reduced complexitymodel is especially useful for the control applications. Futurework will focus on improving prediction accuracy in the face ofmeasurement noise as previously discussed. Model-based controldesign based upon proposed model will be investigated.

Nomenclature

A ¼ geometric area (m2)_m ¼ mass flow rate (kg/s)P ¼ pressure (N/m2)< ¼ universal gas constant (J kg�1 K�1)T ¼ temperature (K)

W ¼ work (J)c ¼ isentropic indexq ¼ gas density (kg/m3)s ¼ torque (N�m)x ¼ compressor angular velocity (rad/s)

Subscripts

c ¼ compressorin ¼ represent the compressor upstream or inlet condition

out ¼ represent the compressor downstream or outlet condition

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