A model of turbocharger radial turbines appropriate to be used in zero- and one-dimensional gas...

Energy Conversion and Management 49 (2008) 3729–3745

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Energy Conversion and Management

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A model of turbocharger radial turbines appropriate to be used in zero- andone-dimensional gas dynamics codes for internal combustion engines modelling

J.R. Serrano a,*, F.J. Arnau a, V. Dolz a, A. Tiseira a, C. Cervelló b

a CMT-Motores Térmicos, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spainb Consellerı́a de Cultura, Educación y Deporte, Generalitat Valenciana, Spain

a r t i c l e i n f o

Article history:Received 6 March 2007Received in revised form 26 November 2007Accepted 29 June 2008Available online 23 August 2008

Keywords:Internal combustion enginesTurbochargersRadial turbinesEngine modellingIntake/exhaust processes

0196-8904/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.enconman.2008.06.031

* Corresponding author. Tel.: +34 96 387 96 57; faxE-mail address: [email protected] (J.R. Serrano)

a b s t r a c t

The paper presents a model of fixed and variable geometry turbines. The aim of this model is to providean efficient boundary condition to model turbocharged internal combustion engines with zero- and one-dimensional gas dynamic codes.

The model is based from its very conception on the measured characteristics of the turbine. Neverthe-less, it is capable of extrapolating operating conditions that differ from those included in the turbinemaps, since the engines usually work within these zones.

The presented model has been implemented in a one-dimensional gas dynamic code and has beenused to calculate unsteady operating conditions for several turbines. The results obtained have been com-pared with success against pressure–time histories measured upstream and downstream of the turbineduring on-engine operation.

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1. Introduction

Turbocharging increases the power of internal combustion en-gines and reduces specific fuel consumption [1]. However, theapplication of this technique poses coupling problems betweenthe engine and the turbocharger. For example, at low engine speedwith small mass flow rate, a turbine with high expansion ratio (i.e.,with a small effective section of the exhaust gas passage) is neces-sary to supply the power needed to meet the compressor require-ments. However, for high-speed engine operating points, a turbinewith a larger effective area would be enough to supply the powerrequired by the compressor. Therefore, a single turbine might failto adapt correctly to all the working conditions of an engine. Tosolve this problem variable geometry turbines, capable of alteringthe effective area of the gas flow passage, can be used. A solutioncommonly used is to vary the angle of inclination of the statorguide blades, thus changing the effective flow area. Variable geom-etry turbines of this kind are referred to in this study as VGT. An-other possibility is to change the width of the gas flow passageby relocating the stator guide blades, with a constant angle, alongan axis parallel to the rotor shaft. Variable geometry turbines ofthis second type are referred to in this study as angle fixed turbine(AFT).

The uptake of variable geometry turbines by engineering firmshas been a slow process, as they are difficult to manufacture at a

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reasonable cost, and due to problems in achieving the requiredreliability of the variable geometry mechanism. However, manyof current high-speed direct-injection Diesel engines and recentlydeveloped downsized petrol engines are being equipped with var-iable geometry turbines. The reason lies in the wide operatingrange of such engines, in which this type of turbine allows (withthe appropriate control) to improve the transient response of theengine and to reduce the pumping losses in steady operation [2].Therefore, smoke emissions, NOx emissions (when combined withEGR) and specific fuel consumption are reduced in comparisonwith fixed geometry turbines.

Zero- and one-dimensional models are able to reproduce theglobal engine behaviour with reasonable computational costs [3–6]; therefore, in this context the correct physical modelling of thevariable geometry turbine provides a powerful tool for the designof the necessary matching between turbocharger and engine plusthe required control strategies. On the one hand, the modellingof the turbine must take into account the fluid-dynamic behaviourof the gas, that is, the boundary conditions to be set at the exhaustmanifold end. This is necessary in order to guarantee that the dy-namic interaction between the cylinders and the turbine, as wellas the flow evolution downstream of the turbine and along the restof the exhaust system, are correctly computed. On the other hand,the modelling of the turbine must take into account the energyconversion and the irreversibilities generated in the process [7];that is the production of mechanical energy from the gas expansionthrough the turbine stator and rotor. This energy will be availableto the compressor, and a balance between the energy produced by

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Nomenclature

AcronymsAFT angle fixed turbineEGR exhaust gas recirculationFGT fixed geometry turbineVGT variable geometry turbine with moving stator blades

Latin symbolsa speed of sound (m/s)A amplitude of pressure wave (bar)c gas velocity (m/s)cp specific heat at constant pressure (J/kg K)cv specific heat at constant volume (J/kg K)D diameter (m)Disp displacement (m)h specific enthalpy (J/kg)M Mach number_m mass flow rate (kg/s)

N turbocharger speed (rps)_m� corrected mass flow rate: _m

ffiffiffiffiffiffiffiT00p

=p00 ðkg=s K0:5MPaÞN* corrected turbocharger speed: N=

ffiffiffiffiffiffiffiT00p

(rps/K0.5)n polytropic indexg polytropic coefficient of the expansion in the rotork polytropic coefficient of the expansion in the statorp, Pr pressure (Pa)R ideal gas constant (J/kg K)R reaction degreerexp ratio of expansion in the turbineS cross section area (m2)T temperature (K)TQ torque (Nm)u blade tip velocity (m/s)_W work transfer rate (J/s)

Greek symbolsc adiabatic exponent (cp/cv)a stator blades angle (�)g efficiency- angular velocity (rad/s)x Relative velocity (m/s)

Subscripts and superscripts0 stagnation conditions (also indicates inlet turbine con-

ditions)1 conditions between turbine stator and rotor2 turbine outlet conditionsa axialeff effectiveg polytropic coefficient of the expansion in the rotori incident pressure wavek polytropic coefficient of the expansion in the statorlimit value for which the stator blades direct the flow tangen-

tial to the rotorn nut of the wheelr radial (also indicates reflected pressure wave)rot rotorR relative conditions (also indicates reflected pressure

wave)s isentropic processst statort, T transmitted pressure waveTs, T/s total to static conditions

3730 J.R. Serrano et al. / Energy Conversion and Management 49 (2008) 3729–3745

the turbine and the energy consumed by the compressor must beperformed. Following this, the operating point can then be ob-tained and, in turn, the flow boundary conditions at the engineinlet.

Presumably, the easiest way to attain the proposed objectives isthe introduction of the turbine characteristic curves, as suggestedby Benson [8] and found in the literature [9]. However, these sim-ulations require a wide range of previous measurements in orderto characterise the turbine. Additionally, only quite recently tur-bine manufacturers have attempted to test them under pulsatingflow conditions, as those found in real engine operation, and thusit will usually be necessary to relate the characteristic curves ofthe turbine under steady flow to its behaviour when coupled tothe engine. Several authors have studied this difference in turbinebehaviour under steady flow and pulsating flow conditions overthe last few years [10–13]. Moreover, imposing the turbine charac-teristic curves always implies the need to interpolate and excludesany option to extrapolate, in addition to assume a totally quasi-steady behaviour in the turbine, thus making it impossible to takeinto account mass accumulation during unsteady operation. Final-ly, the use of interpolation functions has the drawback of increas-ing the calculation time.

The simplest model developed for a radial turbine was that pro-posed for a fixed geometry unit by Watson and Janota [14]. Centralto this model is the representation of the turbine as a nozzle lo-cated at the exhaust manifold outlet, which reproduces the pres-sure drop across the turbine for a specific mass flow rate. In thecase of radial turbines with high reaction degrees, in which expan-sion is produced in two steps, critical flow conditions are reached

for an expansion rate of approximately 3, whereas a nozzle reachesshock conditions with an expansion rate of approximately 1.89.Therefore, this model must consider additional solutions forshocked flow conditions. A further drawback of this model is thatthe effective section of the nozzle is assumed to be constant,whereas it should be a function of the expansion produced, in or-der to represent accurately the fluid-dynamic behaviour of a radialturbine.

An alternative to this initial and basic model is that described byPayri et al. [15] and by Winterbone [16], in which the same ideali-sation of the turbine as a simple nozzle is considered, but settingthe effective area so that a given mass flow produces half the pres-sure drop generated in the turbine. Thus, the problem of criticalexpansion and that of the non-constant nozzle area can be solved.However, the nozzle outlet pressure corresponds now to the pres-sure between the stator and the rotor. This has to be established asa function of the pressure at the turbine outlet, which is assumedto be constant. This situation prevents the calculation of the pres-sure–time histories at the turbine outlet and the calculation of thepossible effect of this variable upon the behaviour of the turbine.

Hribernik et al. [17] proposed a more complete model for dou-ble entry turbines, based on pipe junctions, in which variable sec-tion nozzles are introduced to simulate the flow expansion insidethe stator, whereas a predictive model is used to describe the rotor.The pressure–time histories calculated at the cylinder outlet in asix-cylinder engine matches the experimental results. However,results at the rotor outlet were not presented.

In 1991 and further in 1996, Payri et al. [18,19] presented amodel based on two nozzles in series, separated by an intermedi-

J.R. Serrano et al. / Energy Conversion and Management 49 (2008) 3729–3745 3731

ate reservoir with the same volume than the turbine. A rather sim-ilar model concerning the inclusion of an intermediate volume up-stream of the rotor was also proposed by Baines et al. [20]. Themodel proposed by Payri et al. [19] is sketched in the diagramsof Fig. 1. This geometry combined the advantages of the first mod-els mentioned above and solved the problem of high expansion ra-tios and the calculation of the instantaneous pressure downstreamof the turbine. Moreover, the model allowed for mass accumula-tion in the volume and therefore the consideration of unsteadyphenomena. The main hypotheses of this model were that thebehaviour of the turbine was quasi-steady throughout the nozzlessimulating the turbine (from both the fluid dynamics and the ther-modynamic point of view) and the pressure drop was the sameacross the stator than across the rotor. The first hypothesis is com-mon practice in zero-dimensional models that calculate internalcombustion engines and is common practice to solve the boundaryconditions used in one-dimensional and gas dynamics codes alsoused to calculate internal combustion engines. The second hypoth-esis facilitates the calculation of the effective areas of the nozzlesrepresenting the turbine stator and rotor. Moreover, the secondhypothesis is quite representative of what occurs in a turbine witha reaction degree (R) of 0.5. This is the case in turbines withoutguide blades in the stator and with radial blades in the rotor[14]. These turbines are normally used in automotive applications.

In order to calculate the effective area of the nozzles, Payri et al.[19] assumed that the mass flow through each one (Fig. 1) was thesame as that passing through the turbine, as indicated in (1) and(2):

_m �ffiffiffiffiffiffiffiffiffiffiffiffiffiffifficRT in0

ppin0

¼ Seff � Fpout

pin0

� �ð1Þ

Fpout

pin0

� �¼ c � pout

pin0

� �1=c

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

c� 1� 1� pout

pin0

� �ðc�1Þ=c" #vuut ð2Þ

where Seff is the effective area of the nozzle equivalent to the statoror the rotor, and ‘‘in” and ‘‘out” represent the inlet and outlet con-ditions, respectively. In spite of the flow inlet conditions beingknown, i.e. mass flow rate, pressure and temperature at the turbineinlet, it was necessary to define the pressure drop across the statorin order to determine the effective area of the equivalent nozzle.Subsequently, when the turbine outlet conditions of the flow wereknown, the pressure drop produced in the rotor had to be also de-fined in order to calculate the effective area of the nozzle represent-ing it. Payri et al. [19] calculated the value of the pressure at thevaneless space (p1 at Fig. 1) imposing the condition of an equal

Fig. 1. Thermodynamic process of the flow traversing a radial

expansion ratio across the stator and the rotor, as shown in (3). Thisis consistent with the second hypothesis of their model:

p1

p00¼ 1

2� 1þ p2

p00

� �ð3Þ

This model was also applied to fixed geometry radial turbineswith two inlets (twin-scroll). In this case, three nozzles repre-sented the turbine. Two nozzles were located at the entrance ofthe volume and reproduced the pressure drop across the stator.While the third nozzle, as in the previous case, reproduces thepressure drop across the rotor. The model was validated for a sin-gle-inlet turbine with a waste gate in a four-cylinder-in-line en-gine, and for a twin-scroll turbine in a six-cylinder in-line engine.In addition, this model allowed easy access to the correct time his-tories of all the variables that describe the behaviour of the turbine[19].

In 1996, Chen et al. [21] published the modelling of a mixedflow turbine under pulsating flow conditions. This was a develop-ment of the model for a fixed geometry turbine under steady andpulsating conditions presented by Chen and Winterbone [22].The proposed model simulated the spiral part of the casing as aconvergent tube of a certain length, as a function of the geometryof the turbine to be modelled, from the opening of the volute up toan azimuth angle of 180�. The volume of the casing could be main-tained by using an appropriate definition for the flow area of theduct. This hypothesis assigned a length to the volute, which is avery important issue in non-steady flow studies. The rotor wassimulated using a quasi-steady model, as justified by the fact thatthe Strouhal number, which describes the relative importance ofthe unsteady flow, was much smaller than unity for the rotor con-sidered in the study. Finally, the authors added several loss models,which provided a balanced improvement of the steady flow simu-lations under conditions far from those used in the design. The pro-posed model was applied to four cases of turbine operation underpulsating flow conditions, confirming its ability to predict theinstantaneous mass flow. In addition, an increase in the flowcapacity under pulsating flow conditions is demonstrated, whichwould not have been predictable with a model based on interpola-tions in the turbine map, according to the authors.

Macek et al. [23] presented another relatively recent model forradial turbines. The model described the passage of the flowthrough the turbine as an initial acceleration, at the opening ofthe volute, followed by flow acceleration at the turbine volute.Afterwards, the absolute velocity was transformed into relativevelocity, taking into account the rotor velocity and the incidencelosses. Subsequently, the relative expansion under the influence

turbine and geometry diagram of the turbine model [19].


of a centrifugal power was simulated and the relative velocity inrelation to the fixed coordinate system was once again trans-formed, taking into account the losses at the outlet. Another notedfeature was the diffuser at the outlet of the turbine, which was rep-resented by a flow deceleration. This model, in comparison withother existing models, included the transformation to relative con-ditions. Although many adjustment parameters are needed, themodel offered an interesting line of research.

Only a few articles are found in the literature referring to theone-dimensional modelling of radial turbines with variable geom-etry. An example is the one presented by Kessel et al. [24] in whicha turbine model was designed in order to obtain data to train aneural network aiming to simulate the behaviour of a variablegeometry turbine. The model was based on a series of thermody-namic transitions, which represent the processes occurring insidea radial turbine. These steps were followed by a special treatmentfor nozzle geometry, vaneless spaces and rotor inlet outside the gi-ven design conditions in order to fit experimental data from theturbine testing. The neural network was used to find the pressuredrop between two given points, by considering a parameter thatdescribes local efficiency. Following this, the corresponding tem-perature drop and other variables, such as increases in enthalpyand entropy, were calculated. In 1999, Nasser and Playfoot [25]presented a model for a radial turbine with moving blades. Themass flow rate through the VGT was calculated using the nozzleEq. (1) and by taking the cross section of the throat area of the sta-tor blades as the effective area. As such, there is no distinction be-tween the geometrical and the effective sections. This model issimilar to that presented by Macek et al. [23] for fixed geometryturbines.

The thesis of present paper is that, before variable geometryturbines spread, it has been commonplace to use the R = 0.5hypothesis to model the behaviour of radial turbines used in turbo-chargers; as they were mostly designed without guide blades inthe stator and with radial blades in the rotor, thereby being consis-tent with that hypothesis [14]. When modelling variable geometryturbines, with guide blades in the stator, such a hypothesis cannotbe used at every operative condition. Nevertheless, if an alternativemethod were found to calculate the pressure drop across the tur-bine stator and rotor (instead of assuming R = 0.5), the modelsbased on mass accumulation in an intermediate volume (Fig. 1)would be a good starting point to calculate the behaviour of a var-iable geometry turbine.

Therefore, this paper proposes a model for calculating R and thepressure drop through the stator and rotor. The aim is to extend, upto variable geometry turbines, the validity of models based on avolume between two nozzles in series (Fig. 1). In this way, oncethe pressure drop will be established for each nozzle, it is possibleto calculate the effective area and to determine the permeabilityacross the stator and rotor of the variable geometry turbines.

An important objective of the proposed model is that the onlyinputs will be geometrical parameters, the corrected mass flowand speed ( _m� and N*) and the total to static expansion ratio, which

Fig. 2. Velocity triangles of a radial tu

are available in the turbine maps normally provided by turbo-charger manufacturers.

2. Calculation of reaction degree

Fig. 2 shows the diagrams of the velocity triangles at the rotorinlet and outlet of a radial turbine and the nomenclature that willbe used in the paper for the velocity vectors. The calculation of thereaction degree of variable geometry turbine is demonstrated asfollow:

The definition of R is usually based on the energy transferred,that is, the ratio between the energy transferred due to the pres-sure change in the rotor and the total variation of energy:

R ¼ h1 � h2

h00 � h20¼ ðh10 � c2

1=2Þ � ðh20 � c22=2Þ

h00 � h20ð4Þ

Since no work is developed at the stator h00 = h01. If the fluid canbe regarded as an ideal gas, it can be assumed that

h10 � h20 ¼ cp � ðT10 � T20Þ ð5Þ

In addition, the energy transfer in the rotor can be representedas the product of the torque by the angular velocity, which isknown as the Euler equation [14]:

_W ¼ - � TQ ¼ _m � ðu1ch1 � u2ch2Þ ¼ _m � cp � ðT10 � T20Þ ð6Þ

From the previous equation, it is readily obtained that

T10 � T20 ¼ðu1ch1 � u2ch2Þ

cpð7Þ

If as design hypotheses swirl at the exit is neglectedðch2 ¼ 0! c2

2 ¼ c2a2Þ and it is assumed that the radial velocity at

the stator outlet (i.e., at the rotor inlet) is equal to the axial velocityat the rotor exit (cr1 = ca2 = c1 � cosa1), from Eqs. (4)–(6) one has

R ¼ 1� c21 � c2

2

2u1ch1¼ 1� c2

h1 þ c2r1 � c2

a2

2u1ch1ð8Þ

Taking into account the aforementioned conditions, and thevelocity triangles, (Fig. 2), this can be rearranged to give

R ¼ 1� ch1

2u1¼ 1� c1 � sin a1

2u1¼ 1�

ca2 � sin a1cos a1

2u1¼ 1� tan a1

2� ca2

u1ð9Þ

Here, a1 is the gas entry angle to the rotor, which will be deter-mined by the stator guide blades. In the case of a rotor with radialblades (b1 = 0) and without guide blades to direct the flow, thench1 = u1 (Fig. 2). Therefore, it is observed from Eq. (9) that R is 0.5.This is the case of most of the fixed geometry turbines withoutguide blades.

It is shown that from the definition of R and using two commonhypotheses at design conditions [14] (radial component of thevelocity at rotor inlet equal to axial component at rotor outletand no swirl at turbine outlet), R can be expressed as shown in(10):

rbine at the rotor inlet and outlet.


R ¼ 1� tan a1

2� ca2

u1ð10Þ

Considering that mass flow rate at the rotor outlet is a functionof the geometric area at its exit (S2) and the gas conditions at theturbine outlet (10) can be rewritten as

R ¼ 1� tan a1

2 � u1�

_mRT2

S2p2ð11Þ

The velocity u1 and the rotor exit area S2 can be written as

u1 ¼ pND1 ð12Þ

S2 ¼p4ðD2

2 � D2nÞ ð13Þ

where D1 and D2 are the external and internal diameters of the tur-bine rotor, respectively (see Fig. 1) and Dn is the rotor nut diameter.Substituting (12) and (13) into (11) gives

R ¼ 1� tan a1

2pD1N� 4 � _m

pðD22 � D2

nÞ�RT2

p2ð14Þ

On the other hand, if cp is assumed constant, the isentropicbehaviour of the variable geometry turbine can be expressed interms of temperatures as

gTs ¼T00 � T20

T00 � T2sð15Þ

Taking into account the previous stated hypothesis of no swirlat the turbine outlet and assuming that c0 � c2, the followingapproximate expression can be established.

gTs ¼T00 � T20

T00 � T2s� T0 � T2

T00 � T2sð16Þ

It is worth noting that the approximation of (16) (based on bothhypothesis stated in previous paragraph) is quite consistent. More-over, if the low values of gas kinetic energy at turbine inlet andoutlet sections are compared with the gas enthalpy values. Follow-ing, solving (16) for the exit gas temperature and considering that

T2s ¼ T00 �p2

p00

� �ðc�1=cÞ

ð17Þ

the value of T2 is obtained as a function of the turbine inletconditions:

T2 ¼ T00T0

T00� gTs 1� p2

p00

� �ðc�1=cÞ !" #

¼ T00 � f 0p2

p00;

T0

T00;gTs

� �

ð18Þ

When taking into account (18) (obtained from (15)), R can beexpressed as

R ¼ 1� 2p2D1ðD2

2 � D2nÞ�

_m � tan a1

N�R � T00 � f 0 p2

p00; T0

T00;gTs

� �p2

ð19Þ

The relationship between the turbine inlet temperature (T0) andthe inlet stagnation temperature (T00) has to be also obtained fromthe corrected variables, which are available in turbine maps. In-deed, considering that

T00

T0¼ 1þ c� 1

2M2

0 ð20Þ

The Mach number for the turbine inlet conditions can be rewrit-ten as

M0 ¼c0

a0¼

_mRT0

p0S0� 1ffiffiffiffiffiffiffiffiffiffiffiffi

cRT0p ¼

_m�

S0

ffiffiffiffiR

c

sT00

T0

� � cþ12ðc�1Þ

ð21Þ

By substituting this value into (20), the following is obtained:

T00

T0¼ 1þRðc� 1Þ

2c_m�

S0

� �2 T00

T0

� �cþ1c�1

ð22Þ

An iterative process can solve Eq. (22), giving an initial value of1 to the temperatures ratios, (22) easily converge to a value slightlyhigher than one. Introducing this result in Eq. (19) and rewriting itas a function of corrected variables yields

R ¼ 1� 2 �Rp2D1ðD2

2 � D2nÞ�

_m� � tan a1

N�� f p2

p00; _m�; S0;gTs

� �

fp2

p00; _m�; S0;gTs

� �¼ p2

p00

� ��1 T00

T0

� ��1

� gTs 1� p2

p00

� �c�1=c !" #

ð23Þ

Therefore, once some measurable geometric parameters of theturbine (a1, D1, D2, Dn and S0) and the turbine map for each positionof the variable geometry mechanism are known, it is possible tocalculate R for each operating point.

Nevertheless, in the case of a VGT (Fig. 3), it is not always pos-sible to use the Eqs. (4)–(9) to calculate R. Indeed, the pictures atthe top of Fig. 3 show that there will be an intermediate positionbetween ‘‘VGT open” and ‘‘VGT closed” for which the chord of eachblade is tangential to the turbine rotor. For values of a1 (Fig. 2)higher than this intermediate position the flow exiting the statoris no longer directed towards the rotor but to the intermediate vol-ume between the stator and the rotor; like the ‘‘VGT closed” pic-ture shows in Fig. 3. Thus, the direction of the rotor inlet velocityis not dependent upon the angle of the stator blades (a1) but uponthe angle of the rotor blades (b1). Therefore, the turbine can be con-sidered without guide blades in the stator. Taking into account thatradial rotor blades are generally used and the remaining compo-nents of the design hypothesis, R can be supposed equal to 0.5,as it has been demonstrated from Eqs. (4)–(9).

In order to apply the previous model, several tests were carriedout on a VGT turbine (Table 1). From the data obtained during thetests R was calculated using (23). The tests consisted on measuringthe performance maps for the VGT using a specific test rig for tur-bochargers. This is fully described in [26,27]. Measurements werecarried out for seven constant positions of the VGT, and for eachposition, several operating speeds and expansion ratios weretested. The specifications of the transducers used in the turbo-charger testing are shown in Table 2.

Since the angle of the blades in the VGT stator is directly relatedto the movement of the rack from the variable geometry mecha-nism, the blades angle can be clearly determined once the rack po-sition is established. An angle of the stator blades of 42�corresponds to the open VGT and an angle of the stator blades of86� corresponds to the closed VGT. The results obtained are shownat the bottom of Fig. 3, where it can be observed that R increaseswhen the VGT opens. In addition, Fig. 3 shows that R tends to 0.5when the blades angle tends to a1 � 68�. This is the a1 value atwhich the axis of each blade is tangential to the turbine rotorexternal circumference.

Additionally, an AFT, whose blades always have a constant a1

angle (Fig. 4) was tested coupled to an engine on a test bench(whose scheme is shown in Fig. 11). This turbine consists of a mov-ing rod to which the stator blades are attached and a vacuumpump which controls the position of the rod. More characteristicsof this variable geometry turbine with fixed angle are shown inTable 1. In this case, for a given opening and turbine speed, onlyone point was measured, and then R was calculated from Eq.(14) by using the average values of pressure and temperature mea-sured downstream the turbine on engine tests. Information aboutthe precision and range of the transducers used in the tests are

Fig. 3. Different blade positions of a VGT and R obtained for the different measured points.

Table 1Characteristics of the turbochargers used to validate the model

Comp.VGT VGT Comp.AFT AFT Comp.FGT FGT

Inlet diameter (mm) 39 33 41 40 95 58Outlet diameter (mm) 38 38 35 50 63 66Number of rotor blades 6 9 6 11 7 12Number of stator blades – 11 – 11 – –A/R 0.42 0.61 0.42 0.62 0.77 0.83

Table 2Characteristics of the sensor used in the experimental measurements

Pressure Temperature Mass flow

Type Piezo-resistive

Piezo-electric

Thermo-couple

Thermo-resistance

Hot-wire

Hot-wire

Model Kistler4045 A5

Kistler7031

K type Pt100 Sensyflow

Range 0–5(bar)

0–5(bar)

273–1533(K)

73–663(K)

0–720(kg/h)

80–2400(kg/h)

Precision 0.1% 0.7% 0.3% 0.033%(at 273 K)

1.5% 1.5%

Fig. 4. Diagram and view of an AFT. Variation of R versus the displacement of the turbine shaft and versus the turbine mass flow rate.



shown in Table 2. The results obtained are shown at the bottom ofFig. 4. They clearly show that R is dependent on the motion of theturbine axis. In contrast to what occurs in the VGT, R increaseswhen the turbine closes. The cause of this behaviour can be theparticular way in which the AFT reduces the stator effective section(AFT_Disp = 0 corresponds with the drawing showed at top ofFig. 4). It can be presumed that a lamination process is producedin the stator when the AFT is closed and therefore R is increased.

Both Figs. 3 and 4 show also a dependency with other turbinevariables; for example, R generally decreases when the exhaustgas flow increases. The results plotted in Fig. 3 show this trendfor each position and each turbine speed. The exception is the caseof VGT closed, where R = 0.5 has been imposed as previously ex-plained. It is worth noting that when the AFT is closed (AFT_Disp = 0) R remains also virtually constant, since the flow rate inthe AFT is almost constant for the different expansion ratios con-sidered (Fig. 4). Fig. 4 also shows that R varies between 0.25 and0.1 when the AFT is open (AFT_Disp = 10). Consequently, thechange of relative speed in the rotor should be lower when R de-creases and this would account for the very low efficiency that isusually observed for a completely open AFT.

3. Calculation of pressure at the stator outlet

Once R has been calculated as a function of the corrected tur-bine variables and of some easily measurable geometric parame-ters, it is necessary to establish its relation with the intermediatepressure between turbine stator and rotor in order to apply thechosen model of two nozzles plus an intermediate volume.

If R is defined according to Eq. (4), and considering the gas to bea perfect gas, one may write

T1

T0¼ T00

T0

T2

T00þ R 1� T20

T00

� �� ð24Þ

where the ratio T00/T0 can be calculated using (22); the ratio T2/T00

can be calculated using (18) and the ratio T20/T00 can be calculatedfrom gTs definition like (25) shows

T20

T00¼ 1� gTs 1� p2

p00

� �c�1c

" #ð25Þ

Substituting (25) and (18) in (24), the following is obtained:

T1

T0¼ 1þ T00

T0ðR� 1ÞgTs 1� p2

p00

� �c�1c

" #ð26Þ

If the thermodynamic process of the gas traversing the turbineis known, it is possible to relate the intermediate pressure to thetemperature and (with the help of (26)) to R, as (27) shows

p1

p0¼ 1þ T00


p00

� �c�1c

" # !k=k�1

ð27Þ

where k is the polytropic exponent that defines process in the sta-tor. In the following paragraphs, the thermodynamic processes inthe turbine will be discussed in order to calculate the k coefficient.

Taking into account that in any thermodynamic process of aflow traversing the turbine, it may start from certain initial condi-tions (p0,T0) and it may achieve certain final conditions (p2,T2), insuch a way that p0 > p2 and T0 > T2. Therefore, the polytropic expo-nent of the process takes values between 1, for the extreme case inwhich T0 = T2, and �1.33 in the case of isentropic process for ex-haust gases. Assuming that the process undergone by the gas inthe turbine is adiabatic but irreversible, it is physically impossiblefor the polytropic exponent to take values below 1, since thatwould imply a temperature increase. Likewise, any value above

1.33 would imply a decrease of entropy in the final state whencompared with the initial state.

If the gas traversing the turbine undergoes a polytropic processwith a constant polytropic index n, Eq. (28) can be proposed to cal-culate the polytropic index n, after taking logarithmics andrearranging:

p2

p0¼ T2

T0

� �n=n�1

) nn� 1

¼ln p2

p00þ c

c�1 ln T00T0

ln T2T00þ ln T00

T0

ð28Þ

where n is a function of corrected variables from the turbine mapsby combining (28) with (18) and (22), respectively.

However, the polytropic process through the stator is generallyquite different from the process through the rotor. Therefore, itseems convenient to assume the hypothesis that the process acrossthe stator and the process across the rotor have different (but con-stant) polytropic exponents. Thus, for the total process from theturbine inlet to the turbine outlet, it can be established that

p2

p0¼ p2

p1� p1

p0¼ T2

T1

� �g=g�1

� T1

T0

� �k=k�1

¼ T2

T0

� �n=n�1

ð29Þ

Since in the turbine maps there are not available data of pres-sure and temperature at the vaneless space the polytropic expo-nents (k and g) cannot be directly obtained. However, it ispossible to relate them taking logarithmics and rearranging (29),as shown in (30):

kk� 1

¼ gg � 1

þ nn� 1

� gg � 1

� � ln T2T00þ ln T00

T0

ln T1T0

ð30Þ

and further by substituting (28) in (30) as shown in (31):

gg � 1

¼k

k�1

ln T1

T0� ln p2

p00þ c

c�1 ln T00T0

h iln T1

T0� ln T2

T00þ ln T00

T0

h i ð31Þ

where the ratio of logarithms can be calculated as a function of cor-rected variables from the turbine maps by using (18), (22) and (26),respectively. Nevertheless, there are still two unknown values in(31), k and g. Therefore, it is necessary to consider an additionalhypothesis. The behaviour of the VGT and the AFT under differentoperative conditions are going to be analysed in the following para-graphs in order to stablish the additional hypothesis.

With respect to the VGT, the operative conditions with the low-est isentropic efficiency are when the VGT is closed or fully open. Inthe case of the VGT described in Table 1, Fig. 5 shows the isentropicefficiency versus VGT opening for several turbocharger speeds. Fig.6 shows the thermodynamic evolutions in such opening condi-tions. On the one hand, in the left part of Fig. 6, the hypothesis thatR is 0.5 when the VGT is closed has been used. In this case, the tur-bine efficiency is lower than at intermedium openings and thethermodynamic process should be similar to the k–g processshown in the left part of Fig. 6. That is, when the VGT is closedand the passage area of the stator is very small, the irreversibilitiescaused by friction are very significant and the entropy increment isgoing to be higher in the stator than in the rotor. Being the objec-tive to calculate the pressure between stator and rotor, a satisfac-tory hypothesis would be to assume the polytropic process in thestator with lower slope than the process in the rotor. Therefore,the polytropic process in the stator will have a polytropic exponentbetween n and 1. An equivalent but more accurate statement willbe that the polytropic exponent in the rotor (g) will be limited be-tween n and c. On the other hand, for a VGT fully open also the tur-bine isentropic efficiency is lower than at intermedium openings;due to there is not and optimum angle of incidence at the stator in-let. Therefore, the thermodynamic process is going to be as the k–gprocess shown in the right part of Fig. 6. This means again with

0.35

0.45

0.55

0.65

0.75

0.85

0 20 40 60 80 100

VGT opening (%)

IsentropicEfficiency

120 rps/ K 110 rps/ K 100 rps/ K90 rps/ K 80 rps/ K 70 rps/ K60 rps/ K 50 rps/ K 40 rps/ K

y = 2.9159x - 2.8182R2 = 0.9829

0.35

0.45

0.55

0.65

0.75

0.85

1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24

"n" polytropic exponent

Isentropicefficiency

√

√

√ √

√

√ √

√

√

Fig. 5. Relation between isentropic efficiency, VGT opening and polytropic exponent n.

p0

p2

0

2s

2

hs

hr

ht

1n1γ1k-g

1s

n( +1)/2

k-gh (J/kg)

s (J/kgK)

p1n p1

p1γ

p0

p2

0

2s

2

hs

hrht

1n 1γ1k-g 1s

n( +1)/2

k-g h (J/kg)

s (J/kgK)

p1n

p1

p1γ

Υ Υ

Δ

Δ

ΔΔ

ΔΔ

Fig. 6. h–s diagrams for low efficiency conditions. VGT closed (left) with R = 0.5, and VGT 100% open (right) with R > 0.5.



higher increase of entropy in the stator than in the rotor; andtherefore, with a polytropic exponent between n and c in the rotorprocess. In this second case R is higher than 0.5 (Fig. 3), which im-plies a low level of expansion at the stator.

In the case of a variable geometry turbine of the AFT type, whenit is fully closed or fully open (and turbine isentropic efficiency isvery small), the same hypothesis previously exposed for the VGTcan be assumed. In the AFT case, it has to be considered that Rtakes values in accordance with those showed in Fig. 4, insteadof Fig. 3.

Finally, at intermedium openings of a VGT, the incidence anglein the stator blades is closer to the optimum and this is why theefficiency is higher (Fig. 5). Therefore, the thermodynamic processshould be as the k–g process shown in Fig. 7, with lower increase ofentropy in the stator than in the rotor. Consequently, the statorpolytropic exponent (k) is in these cases always between n and c.Similar hypothesis can be done for AFT maximum efficiencyopenings.

In summary, based on previous paragraphs analysis and in or-der to calculate the k polytropic exponent (necessary to solve(27)) it will be assumed the following hypotheses:

1. On the one hand, if the turbine efficiency is high enough thepolytropic exponent of the process in the stator is between cand n (c > k > n).

2. On the other hand, if the turbine efficiency is not high enough,the process in the rotor is the one between c and n (c > g > n).

The polytropic exponent n will be used to discriminate betweenboth situations, due to the linear relation between n and turbineefficiency (as Fig. 5 shows). Indeed, if n polytropic exponent ishigher than the average between isothermal and isentropic process(n > (c + 1)/2) the turbine efficiency will be considered high enoughto assume the first hypothesis (c > k > n) and viceversa. These situ-ations are also exemplified at Fig. 6 and at Fig. 7. In both cases n iscalculated using (28).

Once the boundaries for k and g have been established, they willbe calculated as a pondered addition of these limits. This is shownin (32) and (33). In the case of (32), the weights used to ponderedare chosen to impose that the higher is the isentropic efficiency(i.e. the lower is the distance between n and c) the closer is the kpolytropic exponent to c. In the case of (33), the weights used to

p0

p2

0

2s

2

hs

hr

ht

h (J/kg)

s (J/kgK)

1s

p1s p1

p1n

1k-g 1n

n( +1)/2

k-g

h

Υ

Υ

ΔΔ

Δ

Δ

Fig. 7. h–s diagrams for high efficiency conditions. VGT 20% ope

pondered are chosen to impose that the lower is the isentropic effi-ciency (i.e. the lower is the distance between n and the unity) thecloser is the g polytropic exponent to c. Once g is calculated, kcan be obtained from (30).

ifcþ 1

26 n < c! k ¼

n n� cþ12

� ��1þ cðc� nÞ�1

n� cþ12

� ��1þ ðc� nÞ�1

ð32Þ

if 1 < n 6cþ 1

2! g ¼

cðn� 1Þ�1 þ n cþ12 � n

� ��1

ðn� 1Þ�1 þ cþ12 � n

� ��1 ð33Þ

4. Synthesis of the proposed VGT model

In summary, the calculation of the intermediate pressure maybe formally expressed as

p1

p0¼ 1þ T00


p00

� �c�1c

" # !k=k�1

ð34Þ

where

R ¼ 0:5 if a1 > alimit

R ¼ 1� 2�Rp2D1ðD2

2�D2nÞ� _m��tan a1

N� � f p2p00; _m�; S0;gTs

� �if a1 6 alimit

(

ð35Þ

and k is calculated at any turbine position as a function of the n (28)polytropic exponent by using (32) or (33). If (33) is used, remainingk polytropic exponents can be further calculated using (30).

Once the pressure drop across the stator (p1/p0) is calculated,the effective areas of the nozzles equivalent to the stator and therotor can be obtained from the nozzle Eq. (1). For the stator, onehas

Seff st ¼ _mT �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiRcT00

pp00

� 1c� p00

p1

� �1=c

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

c� 11� p1

p00

� �ðc�1Þ=c" #vuut

0@

1A�1

ð36Þ

and for the rotor

( +1)/2

p0

p2

0

2s

2

hs

hr

t

h (J/kg)

s (J/kgK)

1s

p1s p1

p1n

nk-g

1k-g1n

Υ

Υ

Δ

Δ

n (left) with R = 0.5 and VGT 60% open (right) with R > 0.5.


Seff rot ¼ _mT �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiRcT10

pp10

� 1c�

p10R

p2

� �1=c

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

c� 11� p2

p10R

!ðc�1Þ=c24

35

vuuut0B@

1CA�1

ð37Þ

where p10r is the relative stagnation conditions at the rotor inlet.Considering the relation between pressure and temperature in

an isentropic expansion can be easily demonstrated Eq. (38):

p10R

p2¼ p1

p2� T10r

T1

� �c=c�1

ð38Þ

The relationship between stator inlet temperature (T1) and therelative stagnation temperature (T10R) has to be obtained fromthe corrected variables, which are available in turbine maps. In-deed, considering that

T10R ¼ 1þ c� 1M2

1Rð39Þ

T1 2

Fig. 8. Correlations obtained for the effective areas of the nozzles equivalent to the AFT and VGT stators.

Fig. 9. Correlations obtained for the effective areas of the nozzles equivalent to the AFT and VGT rotors.

Considering the nomenclature of Fig. 2, the Mach number forthe relative conditions at the turbine inlet (M1R) can be rewrittenas

M1R ¼x1

a1ð40Þ

In addition, from (4)–(9) was demonstrated that with radialblades in the rotor (b1 = 0) the following identities can be obtainedcr1 = x1 = c0 (Fig. 2). Therefore, Eq. (40) may be rewritten as

M1R ¼x1

a1¼ c0

a1¼

_m�

S0

ffiffiffiffiR

c

sT00

T0

� � cþ12ðc�1Þ

! ffiffiffiffiffiT0

T1

sð41Þ

Considering Eqs. (41), (39) can be rewritten as

T10R

T1¼ 1þ Rðc� 1Þ

2c_m�

S0

� �2 T00

T0

� �cþ1c�1

" #� T0

T1ð42Þ

Considering (22), Eq. (42) can be rewritten as

T10R

T1¼ 1þ T00

T0� 1

� �� T0

T1ð43Þ

Fig. 10. Effective areas of the nozzles equivalent to stator and rotor for an FGT.


Considering Eqs. (43), (38) can be rewritten as

p10R

p2¼ p1

p2� 1þ T00

T0� 1

� �� T0

T1

� �c=c�1

ð44Þ

In addition, considering (45) and substituting in (44), (46) isobtained:

p1

p2¼ p1

p0� p00

p2� T00

T0

� � c1�c

ð45Þ

p10R

p2¼ p1

p0� p00

p2� T0

T00

� � cc�1

� 1þ T00

T0� 1

� �� T0

T1

� � cc�1

ð46Þ

Eq. (46) shows that the relative total to static expansion ratio(p10R/p2) can be calculated as a function of total to static expansionratio (p00/p2) and corrected variables (considering also (22) and(26)). It is worth noting that usually T0/T00 is close to one, therefore,(46) shows that p10R � p1.

The calculation of the effective areas has been carried out forthree single entry turbines: a VGT, an AFT turbine and a fixedgeometry turbine without guide stator blades (FGT). Their charac-teristics are also shown in Table 1.

Fig. 8 shows in the case of an AFT the relationship between sta-tor effective area and stator displacement and in the case of a VGTthe relationship between stator effective area and stator blades an-gle. In both cases, the effective area clearly increases when the tur-bine trends to open. Fig. 9 shows that it is possible to correlate theeffective area of the nozzle equivalent to the turbine rotor (37) as afunction of the flow rate and the operation speed both correctedusing the gas conditions at the rotor inlet. Fig. 9 shows that for aspecific turbocharger speed, this effective area increases whenthe gas mass flow (and therefore the expansion ratio) increases.In addition, for a specific flow rate, the effective area of the equiv-alent nozzle at the rotor decreases when the speed increases. Thiscan be accounted for by considering that, as the speed increases,the centrifugal forces produced when the rotor turns also increaseand the passage of the exhaust gas is obstructed.

Obviously, the model developed is also useful for fixed geome-try turbines with or without guide stator blades. The effective areasof the nozzles equivalent to the stator and the rotor of the FGTwithout guide stator blades were calculated assuming that R is

0.5 (see equations between (4)–(9)) and using the data from theturbine maps provided by the turbocharger manufacturer. The re-sults obtained are plotted in Fig. 10, where it can be observed theexcellent level of correlation obtained with respect to the turbineoperative variables. Fig. 10 shows that the stator effective area(36) increases, albeit slightly, when the flow rate is increased andthe corrected operating speed is decreased. In addition, Fig. 8showed that in spite of the variable geometry mechanism positionexplains most of the stator effective area variation; it should de-pend on mass flow rate too. Conversely, the effective area of thenozzle equivalent to the rotor correlates linearly with correctedmass flow rate and expansion ratio.

5. Comparison between measured and modelled data

Once the turbines were characterised, the correlations obtainedfor the nozzle effective area representing the stator and the rotorwere implemented in a global gas dynamic code for engine model-ling developed at CMT-Motores Térmicos TM and called wave actionmodel (WAM). WAM is one-dimensional, non-homoentropic andunsteady; more details can be found in the works referred from[28–36]. WAM supplies the inputs required by the obtained corre-lations in order to calculate the effective areas.

The turbine model has been validated using tests conducted inengine test bed, so that the turbines were coupled to compressionignition engines. The AFT is part of the turbocharging group of a2.2 L displacement engine, the VGT is part of a 1.9 L engine andthe FGT is coupled to a 10.8 L engine. The features of these enginesare shown in Table 3.

The engines have been installed in test benches with all theequipment and instrumentation necessary to control their perfor-mance and measure their operational variables. In each test, themost significant parameters related to the operative conditions ofthe turbine have been measured; such as the pressure and temper-ature at the inlet and outlet of the turbine, the flow rate passingthrough the engine, the operating speed of the turbo and the posi-tions of the VGT and AFT stators. Information about the precisionand range of the transducers used in the tests are shown in Table2. The results obtained from the turbine model (within theWAM) have been compared to the measurements for checking if

Table 3Characteristics of the engines used to validate the model

Engine with VGT Engine with AFT Engine with FGT

Type of injection Direct Direct DirectNumber of

cylinders4 4 6

Displacement 1.9 L 2.2 L 10.8 LCompression

ratio18.3:1 18:1 16:1

Rated power 88 kW/4000 rpm 98 kW/4000 rpm 340 kW/1800 rpmRated torque 300 Nm/2000 rpm 314 Nm/2000 rpm 2200 Nm/1200 rpmRated speed 4500 rpm 4500 rpm 1800 rpmBore 80 mm 85 mm 123 mmStroke 93 mm 96 mm 152 mmConnecting

rod length139 mm 152 mm 225 mm


it reproduces properly the fluid-dynamic behaviour of the threedifferent radial turbines.

Fig. 11 shows a scheme of the 2.2 L Diesel engine at which wastested the AFT. At each measurement position, a straight duct wasdesigned (diameter and length) according to inlet and outlet com-pressor cross section area to ensure an essentially one-dimensionalflow, without any complex features originated by singularities. Inaddition, was taken into account that the position of any of thetransducers did not coincide with, or lie close to, any of the pres-sure nodes associated with the standing-wave pattern established

AFT TUR

-10.0

-5.0

0.0

5.0

10.0

1.00 1.20 1.40 1.60 1.80 2.00 2.20Pr

Erro

r %

Error_CD Stator (%) Error_

Fig. 11. Scheme of engine experimental test and

[37]. Arrays of three ‘‘Kistler 7031” piezoelectric transducers withwater cooled adaptors were used at each measurement position(upstream and downstream of compressors), with a distance be-tween two consecutive sensors of 0.05 m, in order to get a suitablecompromise between the assumption of linear propagation be-tween transducers and measurement precision [38,39]. Transducersignals were calibrated at each test case, the precision reported bythe pressure transducer manufacturer is 0.7% of the full scale.Simultaneously, instantaneous pressure versus time and versuscrank angle was measured. Using a Yokogawa high frequency(maximum 100 kHz) acquisition system, 40,000 records (threeinstantaneous pressure upstream and three downstream for eachmeasurement) were acquired in time with an acquisition fre-quency of 20 kHz and 14,400 measurements were made versuscrank angle, each 0.5 crank angle degrees, comprising 10 enginecycles. Beam-forming techniques were used for wave decomposi-tion [38] in order to obtain the pressure wave components fromthe measured pressure–time histories. In the engine scheme ofFig. 11 are also shown the most important elements than modifythe wave dynamics of the system around turbine and compressor.It is well known that the wave dynamics of the system where tur-bine or compressor are located influence their performance [40].More details about the described experimental technique andother tests performed with the same arrangement can be foundin [38,41]. The mass flow through the AFT was calculated byimposing to the turbine model the instantaneous pressure tracesmeasured upstream and downstream the AFT. The mass flow er-

BINE

2.40 2.60 2.80 3.00 3.20 3.40 3.60 (T/s)

CD Rotor (%) Error_Mass Flow (%)

error in mass flow prediction for the AFT.

EXCITATION RESPONSE

Ai

At

Ar

Ar2

AT

-0.02

-0.01

0

0.01

0.02

0 120 240 360 480 600 720

Crank Angle (º)Pr

essu

re (b

ar)

AR

-0.1 4

-0.0 6

0.02

0. 1

0 120 240 360 480 600 720

Crank Angle (º)

Pres

sure

(bar

)

AR

100

125

150

175

200

0 200 400 600 800 1000Fc. (Hz)

dBl

AT

100

120

140

160

0 200 400 600 800 1000Fc. (Hz)

dBl

PUPSTREAM

2.2

2.4

2.6

2.8

3

0 120 240 360 480 600 720Crank Angle (º)

Pres

sure

(bar

)

PDOWNSTREAM

1. 05

1. 07

1. 09

1. 11

1. 13

0 120 240 360 480 600 720Crank Angle (º)

Pres

sure

(bar

)

Modelled Measured Fig. 12. Wave decomposition in time and frequency domain for measured and modelled signals. Comparison with an AFT. 10% opening and 120,000 rpm.


rors obtained for all the points, measured at different speed at AFTpositions, were always lower than 5%, as Fig. 11 shows. The linesshowed in Fig. 11 join the points that correspond to the sameAFT opening. In an equivalent way, Fig. 11 shows (also in the formof errors) how much should be reduced with respect to the unity agiven stator and rotor coefficients of discharge (CD), which multi-ply the effective areas provided by the turbine model, in order toprovide zero mass flow error when imposing the expansion ratio.As expected, also in this case the errors are mainly below 5%.

Fig. 12 shows an example of the instantaneous results obtainedfor the case of the AFT at 10% opening and 120,000 rpm. The inci-

dent pressure pulse (Ai) obtained from the pressure measured up-stream of the turbine and the reflected pressure pulse (Ar2)obtained from the pressure measured downstream of the turbineare the excitations used as input variables for the model. In Fig.12 reflected (AR) and transmitted (AT) pulses, obtained also frompressure wave decomposition (from PUPSTREAM and PDOWNSTREAM)have been compared with the corresponding modelled results inthe time and the frequency domains (without considering theaverage value). In addition, Fig. 12 shows the comparison of themeasured pressure signal upstream and downstream of the AFT,including also the average value. The flow lamination through

Piezo-resistive transducer

temperature

mass flow rate

Piezo-resistive transducer

temperature

mass flow rate

VGT TURBINE

-10.0

-5.0

0.0

5.0

10.0

1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80Pr (T/s)

Erro

r %

Error_CD Stator (%) Error_CD Rotor (%) Error_Mass Flow (%)

FGT TURBINE

-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.0

1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80Pr (T/s)

Erro

r %

Error_CD Stator (%) Error_CD Rotor (%) Error_Mass Flow (%)

Fig. 13. Schemes of engines used for testing and errors of mass flow prediction for the VGT and FGT.


the AFT at 10% opening is evidenced by the very small amplitude ofthe pressure waves downstream the turbine (PDOWNSTREAM and AT)compared with those upstream turbine (PUPSTREAM and AR). Indeed,this is consistent with the hypotheses previously discussed in Sec-tion 3.

It is worth noting that the good prediction obtained in the fre-quency domain up to quite high frequencies (600 Hz) especially inthe reflected pressure wave (AR). These good results, also obtainedin many other operative conditions, qualify the turbine model forbeing used in noise prediction tasks.

Fig. 14. Comparison of measured and modelled pressures in a VGT for different blade angles and operating points.

1500 rpm

1

2

3

4

5

6

7

0 180 360 540 720Crank angle (º)

Pres

sure

[bar

]

1800 rpm

1

2

3

4

5

6

7

0 180 360 540 720Crank Angle (º)

Pres

sure

[bar

]

1200 rpm

1

2

3

4

5

0 180 360 540 720Crank Angle (º)

Pres

sure

[bar

]

800 rpm

1

1.5

2

2.5

3

0 180 360 540 720Crank Angle (º)

Pres

sure

[bar

]

Experimental Turbine InletTurbine Outlet Compressor Outlet

Fig. 15. Comparison of measured and modelled pressures in an FGT for different operating points.



The validation of the model for the VGT and the FGT was per-formed without modifying either the intake or the exhaust mani-fold of the used engines. The engines schemes and the location ofthe transducers are shown in Fig. 13. Information about the preci-sion and range of the transducers used in the tests are shown inTable 2. In addition, Fig. 13 shows a validation of the mean valuesprovided by the turbine model through the plotting of the errorsobtained when calculating the average mass flow through theVGT and through the FGT. Fig. 13 shows that all the errors are be-low ±5%, except for quite low expansion ratios at a certain openingof the VGT. As in the case of Fig. 11, the lines join the points withthe same VGT opening and the error of the discharge coefficients(which would be necessaries to obtained zero mass flow error)are generally of the same order but opposite sign than the massflow errors.

The measured-modelled comparison for the VGT results wascarried out using steady tests of the 1.9 L engine described above.Fig. 14 shows an example of the results obtained for different an-gles of VGT stator blades, different turbine speeds and different en-gine operative conditions. Fig. 14 shows the comparison betweenthe modelled and measured pressure–time histories at the turbineinlet, at the turbine outlet and also at the compressor outlet. Fig. 14shows a good agreement between the experimental values andthose obtained from the modelling.

Finally, in the case of the FGT, Fig. 15 shows the comparison be-tween measured and modelled pressure histories for four full loadconditions at different engine speeds (800, 1200, 1500 and1800 rpm) of the 10.8 L displacement engine. These results covera wide operating range for a heavy-duty engine, and allow observ-ing the good agreement achieved between the measured and mod-elled values. Fig. 15 shows that the model is able to reproduce eventhe asymmetric behaviour of the instantaneous pressure caused bythe divided exhaust manifold typical of truck engines; due to theinstantaneous piezo-resistive transducer was placed in one of thebranches of the exhaust manifold.

6. Conclusions

A new model for variable geometry turbines suitable to be usedin either zero or one-dimensional engine codes has been devel-oped. This model is a natural evolution of a previous one for fixedgeometry turbines in which the turbine is represented by two idealnozzles, which reproduce the pressure drops across the stator andthe rotor, and at an intermediate cavity, which reproduces massaccumulation in the system.

The structure of the new model presented is the following.Firstly, the reaction degree R of the variable geometry turbine un-der the desired performance conditions is calculated. A methodol-ogy has been proposed in which geometrical parameters and thecorrected variables, obtained from the maps supplied by the man-ufacturer, are used as the only inputs for R calculation. Once R isestablished, it is possible to calculate the pressure drops producedat the stator and at the rotor taking into account: the turbine effi-ciency (represented by n exponent) the type and the position of thestator blades. Once these pressure drops are known, it is possible tocalculate the effective areas of the equivalent nozzles representingthe turbine stator and rotor.

Concerning the VGT and the AFT, it was concluded that R has ahigh level of correlation with the opening and with the flow ratepassing through the turbine. In addition, the effective area of thestator correlates with the position of the turbine, and that of the ro-tor with the corrected mass flow rate, the corrected speed or theexpansion ratio. Thus, it was shown that the effective area of thenozzle equivalent to the rotor, for a specific rotating speed, in-creases as the gas flow or the expansion ratio increases. However,

for a given flow rate, the effective area of the nozzle equivalent tothe rotor diminishes when rotating speed increases. This is causedby the centrifugal forces produced when the turbine rotor turnsand by the obstruction imposed to the passage of exhaust gases.

To validate the turbine model, it has been introduced into a glo-bal engine model based on a one-dimensional gas dynamic code.Results were compared to tests performed on an engine bench,and it was concluded that for both VGT and AFT, this model is ableto reproduce the fluid-dynamic behaviour of the turbine with goodaccuracy in both time and frequency domains.

Finally, the model was applied to a FGT using the maps suppliedby the manufacturer as the only input data, as stated in the projectobjectives. As a result, it has been possible to achieve, over a wideoperative range, a highly precise reproduction of the turbine fluiddynamics coupled to an internal combustion engine.

Acknowledgements

The authors would like to thank Dr. Antonio Torregrosa for hishelpful suggestions and Mr. Daniel del Valle for the equationssupervision.

The authors thank Renault SA and Generalitat Valenciana(Grant GV06/057-20060547) for the material and financial supportof this study.

References

[1] Ugur K. Effect of turbocharging system on the performance of a natural gasengine. Energy Convers Manage 2004;46:11–32.

[2] Galindo J, Serrano JR, Vera F, Cervelló C, Lejeune M. Relevance of valve overlapfor meeting Euro 5 soot emissions requirements during load transient processin HD Diesel engines. Int J Vehicle Des 2006;41:343–67.

[3] Lee SJ, Lee KS, Song SH, Chun KM. Low pressure loop EGR system analysis usingsimulation and experimental investigation in heavy-duty Diesel engine. Int JAutomotive Technol 2006;7(6):659–66.

[4] Ugur K. Study on the design of inlet and exhaust system of a stationary internalcombustion engine. Energy Convers Manage 2005;46:2258–87.

[5] Rakopoulos CD, Giakoumis EG, Rakopoulos DC. Cylinder wall temperatureeffects on the transient performance of a turbocharged Diesel engine. EnergyConvers Manage 2005;45:2627–38.

[6] Cho B, Vaughan ND. Dynamic simulation model of a hybrid powertrain andcontroller using co-simulation – Part I: Powertrain modelling. Int J AutomotiveTechnol 2006;7(4):459–68.

[7] Rakopoulos CD, Giakoumis G. Development of cumulative and availability ratebalances in a multi-cylinder turbocharged indirect injection diesel engine.Energy Convers Manage 1997;38(4):341–69.

[8] Benson RS. The thermodynamics and gas dynamics of internal-combustionengines, vol. I. Oxford: Oxford University Press; 1982.

[9] Moraal P, Kolmanovsky I. Turbocharger modeling for automotive controlapplications. SAE Technical Paper 1999-01-0908; 1999.

[10] Karamanis N, Martinez-Botas R. Mixed-flow turbines for automotiveturbochargers: steady and unsteady performance. Int J Engine Res2002;3(3):127–38.

[11] Chen H, Hakeem I, Martinez-Botas R. Modelling of a turbocharger turbineunder pulsating inlet conditions. Proc Inst Mech Eng 1996;210:397–407.

[12] Iwasaki M, Nobuyuki I, Marutani Y, Kitazawa T. Comparison of turbochargerperformance between steady flow and pulsating flow on engines. SAETechnical Paper 940839; 1994.

[13] Winterbone DE, Nikpour B, Frost H. A contribution to understanding turbineperformance in pulsating flow. Proc Inst Mech Eng 1991;C433/011:19–30.

[14] Watson N, Janota S. Turbocharging the internal combustionengine. London: McMillan Publishers Ltd.; 1982, ISBN 0-333-24290-4.

[15] Payri F, Desantes JM, Boada J. Prediction method for the operating conditionsof a turbocharged Diesel engine. In: Proceedings of the motor symposium’86,Prague, vol. 2; 1986. p. 8–16.

[16] Winterbone DE. The theory of wave action approaches applied to reciprocatingengines. In: Weaving JH, editor. Internal combustion engineering: science &technology. London: Elsevier Applied Science; 1990. p. 445–500.

[17] Hribernik A, Dobovisek Z, Cernej A. Application of rotor characteristics for one-dimensional turbine modelling. Proc Inst Mech Eng 1994;C484(034):239–49.

[18] Payri F, Benajes J, Jullien J, Duan Q. Non-steady flow behaviour of superchargerturbine. In: Proceedings of the third EAEC international conference,Strasbourg; 1991. p. 347–51.

[19] Payri F, Benajes J, Reyes M. Modelling of supercharger turbines in internal-combustion engines. Int J Mech Sci 1996;8–9:853–69.

[20] Baines NC, Hajilouy-Benisi A, Yeo JH. The pulse flow performance andmodelling of radial inflow turbines. Proc Inst Mech Eng1994;C484(006):209–18.


[21] Chen H, Hakeem I, Martinez-Botas R. Modelling of a turbocharger turbineunder pulsating inlet conditions. Proc Inst Mech Eng 1996;210:397–407.

[22] Chen H, Winterbone DE. A method to predict performance of vaneless radialturbine under steady and unsteady flow conditions. In: Proceedings ofImechE international conference on turbochargers and turbocharging; 1990.p. 13–22.

[23] Macek J, Vávra J, Vı́tek O. 1-D model of radial turbocharger turbine calibratedby experiments. SAE Technical Paper 2002-01-0377; 2002.

[24] Kessel JA, Schaffnit J, Schmidt M. Modelling and real-time simulation of aturbocharger with variable turbine geometry (VTG). SAE Technical Paper980770; 1998.

[25] Nasser SH, Playfoot BB. A turbocharger selection computer model. SAETechnical Paper 1999-01-0559; 1999.

[26] Luján JM, Serrano JR, Bermúdez V, Cervello C. Test bench for turbochargergroups characterization. SAE Technical Paper, 2002-01-0163; 2002.

[27] Galindo J, Serrano JR, Guardiola C, Cervelló C. Surge limit definition in a specifictest bench for the characterisation of automotive turbochargers. Exp ThermFluid Sc 2006;30:449–62.

[28] Payri F, Benajes J, Galindo J. One-dimensional fluid-dynamic model forcatalytic converters in automotive engines. SAE Technical Paper 950785; 1995.

[29] Benajes J, Reyes E, Luján JM. Modelling study of the scavenging process in aturbocharged Diesel engine with modified valve operation. Proc IMechE PartC: J Mech Eng Sci 1996;210.

[30] Benajes J, Reyes E, Galindo J, Peidró J. Predesign model for intake manifolds ininternal combustion engines. SAE Technical Paper 970055; 1997.

[31] Benajes J, Reyes E, Bermúdez V, Serrano JR. Predesign criteria for exhaustmanifolds in IC automotive engines. SAE Technical Paper 980783; 1998.

[32] Payri F, Reyes E, Serrano JR. A model for load transients of turbocharged Dieselengines. 1999 SAE Trans–J Engines 2000;108(3):363–75.

[33] Benajes J, Luján JM, Serrano JR. Predictive modelling study of the transient loadresponse in a heavy-duty turbocharged Diesel engine. SAE Technical Paper2000-01-0583; 2000.

[34] Payri F, Reyes E, Galindo J. Analysis and modelling of the fluid-dynamic effectsin branched exhaust junctions of ICE. Int J Gas Turbine Power: Trans ASME2001;123(1):197–203.

[35] Payri F, Benajes J, Galindo J, Serrano JR. Modelling of turbocharged Dieselengines in transient operation. Part 2: Wave action models for calculating thetransient operation in a high speed direct injection engine. Proc IMechE Part D,D06501 2002;216:479–93.

[36] Galindo J, Luján JM, Serrano JR, Dolz V, Guilain S. Design of an exhaustmanifold to improve transient performance of a high-speed turbochargedDiesel engine. Exp Therm Fluid Sci 2004;28:863–75.

[37] Dowling AP, Ffowcs Williams JE. Sound and sources of sound. NewYork: Wiley; 1983.

[38] Piñero G, Vergara L, Desantes JM, Broatch A. Estimation of velocity fluctuationin internal combustion engine exhaust systems through beam formingtechniques. Meas Sci Technol 2000;11:1585–95.

[39] Seybert AF. Two-sensor methods for the measurement of sound intensity andacoustic properties in ducts. J Acoust Soc Am 1988;6:2233–9.

[40] Epstein AH, Ffowcs Willians JE, Greitzer EM. Active suppression ofaerodynamic instabilities in turbomachinery. J Propul 1989;5(2):204–11.

[41] Torregrosa AJ, Serrano JR, Soltani S, Dopazo JA. Experiments on wavetransmission and reflection by turbochargers in engine operating conditions.SAE Paper 2006-01-0022; 2006.

A model of turbocharger radial turbines appropriate to be used in zero- and one-dimensional gas...

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