Energy Conversion and Management 51 (2010) 1958–1969

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Energy Conversion and Management

journal homepage: www.elsevier .com/locate /enconman

Impact of two-stage turbocharging architectures on pumping lossesof automotive engines based on an analytical model

J. Galindo, J.R. Serrano *, H. Climent, O. VarnierUniversidad Politécnica de Valencia, CMT – Motores Térmicos, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 June 2009Received in revised form 15 February 2010Accepted 27 February 2010Available online 2 April 2010

Keywords:AutomotiveReciprocating internal combustion enginesTurbochargersTwo-stage turbochargingAnalytical engine modeling

0196-8904/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.enconman.2010.02.028

* Corresponding author. Address: Universidad PolMotores Térmicos. Camino de Vera s/n, 46022 Valenc57; fax: +34 96 387 76 59.

E-mail address: jrserran@mot.upv.es (J.R. Serrano)

Present work presents an analytical study of two-stage turbocharging configuration performance. Theaim of this work is to understand the influence of different two-stage-architecture parameters to opti-mize the use of exhaust manifold gases energy and to aid decision making process. An analytical modelgiving the relationship between global compression ratio and global expansion ratio is developed as afunction of basic engine and turbocharging system parameters. Having an analytical solution, the influ-ence of different variables, such as expansion ratio between HP and LP turbine, intercooler efficiency, tur-bochargers efficiency, cooling fluid temperature and exhaust temperature are studied independently.

Engine simulations with proposed analytical model have been performed to analyze the influence ofthese different parameters on brake thermal efficiency and pumping mean effective pressure. The resultsobtained show the overall performance of the two-stage system for the whole operative range and char-acterize the optimum control of the elements for each operative condition.

The model was also used to compare single-stage and two-stage architectures performance for thesame engine operative conditions. Benefits and limits in terms of breathing capabilities and brake ther-mal efficiency of each type of system have been presented and analyzed.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In automotive applications, combustion engines are required tohave not only a high specific power output but also reduced pollu-tant emissions. For these reasons, current light and medium dutyengines are being highly turbocharged [1], achieving improvedbrake mean effective pressure and enlarged exhaust gases recircu-lation (EGR) available zone [2]. In the recent years, a generalizationof the use of turbocharging in diesel engines has occurred and thesame is happening in petrol engines [3]. The reason is that once thecombustion processes produced a breakthrough with the arrival onnew flexible injection systems and new combustion concepts [4],one of the weakest points for further engine development has beenthe air loop system.

It is a commonplace that the most promising way to achieveEuro 6 and beyond standards is the downsized engine. This tech-nique consists in developing engines with a reduced swept volumebut with the same power output. This has a first output in reducingfuel consumption and lower pollutant emissions, but in the sametime torque at very low speeds is limited and engine transient

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itécnica de Valencia. CMT –ia, Spain. Tel.: +34 96 387 96

.

behavior is worsened. Market-acceptance is always more demand-ing about vehicle dynamics, so as a minimum; performances of thedownsized engine must be maintained not only under steady, butalso in transient conditions.

One way to improve transient performance is to employ a smal-ler turbocharger, since the corresponding turbine is better de-signed for working with low exhaust gases mass flow and haslower thermal and mechanical inertias. Both features allow for fas-ter engine response. However, small size turbochargers limit themaximum engine speed and load. In order to overcome this prob-lem, two turbochargers operation arranged in series or in parallelcan be used. Parallel arrangement of turbochargers eliminates tur-bo-lag issues but both compressors provide the same compressionratio, limited by the maximum compression ratio of the smallestcompressor [5]. However, the high boost pressure required indownsized engines to maintain the power output and the steadystate performance is only obtained if the air compression processis performed in serial configuration [6]. Therefore, two-stage tur-bocharging arises as a practical solution to increase the air chargeinto the cylinders. Besides, higher EGR rates can be achieved whilekeeping reasonable values of air to fuel ratio (AFR) [7].

In a two-stage system, the high-pressure (HP) turbocharger issmaller than the low-pressure (LP) one in order to achieve bettertransient response at low speeds because its reduced inertia com-pared to single-stage system. The low-pressure turbocharger is

Nomenclature

Acronymsbmep break mean effective pressureEGR exhaust gas recirculationHP high pressureimep indicative mean effective pressureLP low pressurefamep friction plus auxiliaries mean effective pressurepmep pumping mean effective pressure

Zc ¼ 1gcðcrÞ

cc�1cc � 1

� �Zt ¼ gt 1� ðerÞ

1�ctct

� �Rc ¼ ðcrÞ

cc�1cc

Rt ¼ ðerÞ1�ctct

Latin symbolsA area (m2)cp specific heat at constant pressure (J/kg K)cr compression ratioer expansion ratioF fuel to air ratioHc heat of combustion (J/kg)K quotient of burn gas specific heat by fresh air specific

heat at constant pressure (cpt/cpc)_m mass flow rate (kg/s)

Oper optimized expansion ratiop pressure (bar)q heat losses (J/kg)R ideal gas constant (J/kg K)

Qc total compression ratioQt total expansion ratio

T temperature (K)u flow velocity (m/s)_W power (J/s)

Greek symbolsc adiabatic exponent (cp/cv)e intercooler efficiencyj pressure loss coefficientq gas density (kg/m3)g efficiencyge brake thermal efficiencygv volumetric efficiencyC temperature including heat losses (K)

Subscripts0 stagnation conditions1 LP compressor inlet conditions2 HP compressor outlet conditions3 HP turbine inlet conditions4 LP outlet conditionsC LP compressor outlet conditionsc compressorH2O cooling fluidIT HP compressor inlet conditionss isentropic processT HP turbine outlet conditions and LP turbine inlet condi-

tionst turbinefa friction plus auxiliaries

J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969 1959

large and optimized for maximum power output operation. Theaddition of an extra intercooler between both compressors is usedto reduce HP compressor inlet temperature and improve overallsystem efficiency. The high-pressure turbine is controlled throughvariable nozzle area or a bypass valve while the low-pressure tur-bine is usually a fixed geometry turbine. Fig. 1 shows a typical two-stage architecture.

Two-stage turbocharging has been studied for heavy and lightduty applications, where a lot of work has been devoted to quantifyperformance improvement and fuel economy that represent this

Fig. 1. Engine scheme.

technology related to the single stage turbocharging. Pflüger [8]presented results of this comparison obtained on a modern 12-l6-cylinder commercial Diesel engine. It showed the huge increaseon boost pressure at low engine speed and on corresponding lowend torque. Besides, maximum torque, rated power output andthe overall system efficiency is improved. Schmitt et al. [9] de-scribed for heavy and light duty applications the engine character-istic curves in the low- and high-pressure compressorcharacteristic maps. As regulated 2-stage charging system is moreflexible and combines benefits of both small and big turbochargers,matching can be optimized for the engine characteristic curves runthrough the center of the compressor characteristic maps at opti-mum efficiencies. In his example, efficiencies of over 70% for HPand LP compressor are reached. Same conclusions are presentedby Choi et al. [10] who quantified for light duty applicationsimprovement of BSFC based on average five operation points of2.2%. The reduction in BSFC for a light duty engine is smaller thanfor a heavy duty engine because vehicle drivability is very impor-tant in these applications, and turbochargers matching are moreoptimized for transient engine operations. This means that is pos-sible to use a smaller turbine with a turbine by-pass for the LPstage, offering packaging and transient operation advantages ow-ing to the lower inertia of the LP unit.

In the literature, different computer simulations with gas dy-namic one dimensional models are presented for engine perfor-mance prediction. These models simulate compressorsperformance using directly manufacturer’s maps with differentinterpolation techniques [11], and turbines performance withmore or less sophisticated models as explained in [12] and [13].With this approach it is not possible to check the influences of dif-ferent variables independently because a compressor or a turbine

1960 J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969

map is needed. As explained in the study presented by Saulnieret al. [6] about the matching of a two-stage turbocharging systemfor a small light duty engine, with these maps the simulationinherits relationships between turbomachines sizes and their effi-ciencies. Therefore, this approach is more adapted in a secondstage of the development process when turbochargers and enginearchitectures are defined. At that time of the design process, theyallow to carry out studies about instantaneous variables [14] diffi-cult to measure in test bench as instantaneous temperature orstudies about the potential of other technologies as Miller Cycleand variable valve timing system when associated to a regulated2-stage turbocharging system [15].

Another approach less time consuming is the simulation of theengine performance with physics-based thermodynamic zero-dimensional models as presented by Gautier et al. [16] or Leeet al. [17]. This approach also needs compressor and turbine mapsto simulate turbocharger performance, so is not convenient tocheck influences of different variables independently. Neverthe-less, as they consider the cycle averaged values of the state vari-ables, capturing the air path dynamics with sufficient accuracy,they are well adapted to develop new control strategies for turbineby-pass actuation. For example, Chasse et al. [18] developed withthis approach a control structure, composed of two cascaded con-trollers and feed forward terms to actuate the wastegate, while Pli-anos et al. [19] fulfilled the same function with a coordinated LQGbased controller adapted to the specific issues of double stageturbocharging.

Unlike the authors mentioned above, who used their models todevelop the two-stage system and its control structure, this workaims to analyze the coupling of the engine and both turbochargersand obtain clear criteria useful for phenomena understanding. Themain objective of this work is the development of an analyticalpre-design model able to determine optimum two-stage architec-tures as a function of basic engine and turbocharging systemparameters. Pre-design capability of such an analytical model ishighly appreciated due to the intrinsic complexity of two-stagearchitectures. This type of models offers an overview of the prob-lem that helps the understanding of complex behavior of systemsand aids decision making process (for example: architecture pre-design purposes like turbochargers size selection, componentsselection and control valves specification) prior to working withmore sophisticated models. Similar analytical model were devel-oped by Zinner [20] for single stage turbocharging but nothing inthe literature appears for two-stage systems with specific downsiz-ing architecture (Fig. 1).

The analytical approach, developed in this work, is very usefulfor two main reasons. By keeping an analytical solution it is possi-ble to check the influence of different variables independently. Asexplained before, this cannot be undertaken with other ap-proaches. By analytical models, simulations have been performedfaster and with reasonable accuracy, which is important whennumerous calculations are needed for having an overview of mul-tivariable problems like two-stage systems for a given downsizedengine application. As application, engine simulations with pro-posed model have been presented and results have been used todefine the specifications for two-stage turbocharging systemscompatible with air requirements and an optimum use of exhaustmanifold gases enthalpy.

The paper is structured as follows. Section 2 is devoted to de-duce in detail the equations that relate compression ratio withexpansion ratio and the analytical solving procedure, while Sec-tion 3 shows the validation with experimental data of the proposedmodel. Section 4 contains the results and discussion concerning:(a) the influence of HP and LP turbines expansion ratio, (b) theinfluence of the inter-cooling system, (c) the influence of exhausttemperature and turbomachines efficiency, and (d) a comparison

between single and two-stage configurations. Finally, conclusionsare presented in Section 5.

2. Governing equations

Flow evolution through a two-stage turbocharging configura-tion and the nomenclature that will be used in the paper for eachcharacteristic point of the system are shown in Figs. 1 and 2,respectively. On one hand the evolution in the LP compressor goesfrom 10 to C0 and the evolution in the LP turbine goes from T0 to 4,and on the other hand the evolution in the HP compressor goesfrom IT0 to 20 and the evolution in the HP turbine goes from 30to T0. The evolution in the intercooler between compressors goesfrom C0 to IT0. The objective is to relate the compression ratio(cr) with the expansion ratio (er) for a given condition in theturbochargers.

The compression ratio of a two-stage configuration is theproduct of the two elemental compression ratios developed bythe HP and LP compressors, neglecting the pressure drop in theintercooler due to internal friction losses. The compression ratiois generally associated with the energy needed by the compres-sor to obtain a specific pressure increase. With respect to the LP(1) and HP (2) compressors, the expressions for compressorspowers [21] are:

_WcLP ¼_mccpcT10

gcLP

pc0

p10

� �cc�1cc

� 1

!¼

_mccpcT10

gcLPcrLPð Þ

cc�1cc � 1

� �¼ _mccpcT10ZcLP ð1Þ

_WcHP ¼_mccpcTIT0

gcHP

p20

pIT0

� �cc�1cc

� 1

!¼

_mccpcTIT0

gcHPcrHPð Þ

cc�1cc � 1

� �¼ _mccpcTIT0ZcHP ð2Þ

Assuming turbulent pressure drop in the intercooler, can bewritten:

pC0 � pIT0¼ j � 1

2qC0u2

C0 ¼ j � 12

_m2c

qC0A2C0

ð3Þ

Therefore pIT0 can be calculated as a function of mass flow, pC0 andTC0:

pIT0¼ pC0 � _m2

cjRTC0

2pC0A2C0

ð4Þ

The temperature downstream the LP compressor (TC0) can be ex-pressed in terms of isentropic efficiency as:

gcLP ¼TC0s � T10

TC0 � T10! TC0 ¼ T10 1þ 1

gcLPc

cc�1cc

rLP � 1� �� �

¼ T10½1þ ZcLP� ð5Þ

The intercooler modifies the fluid thermodynamic properties up-stream the HP compressor. The air mass flow is cooled and the tem-perature decrease depends of the intercooler efficiency (e) and thecooling fluid temperature (TH2O).

e ¼ TC0 � TIT0

TC0 � TH2O! TIT0 ¼ TC0ð1� eÞ þ e � TH2O ð6Þ

Rearranging (2) with (5) and (6) one obtains:

_WcHP ¼ _mccpcZcHPT10 ½1þ ZcLP�ð1� eÞ þ e � TH2O

T10

� �ð7Þ

The total power consumed by HP and LP compressors is calculatedby the addition of (7) and (1) as:

10

20

20s

IT0

C0C0s

p10

p20

pC0

s

h

p30

pT0

30

4

T0s

T0

4s

pIT0

C0Δ p

WcHPWcLPqI

p4

30HP

T0LP

WtLP

WtHP

qHP

qLP

Fig. 2. Thermodynamic evolutions of a two-stage turbochargers configuration.

J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969 1961

_WcTotal ¼ _WcHP þ _WcLP

¼ _mccpcT10 ZcLP þ ZcHP ½1þ ZcLP�ð1� eÞ þ e � TH2O

T10

� �� �ð8Þ

By the same token, the expansion ratio of a two-stage configurationis the product of the two elemental expansion ratios developed bythe HP and LP turbines, and these expansion ratios are generallythe consequence of the energy required by the turbines to produceand transfer the necessary work to the compressors. With respect toHP (9) and LP (10) turbines, the expressions of turbine power are:

_WtHP ¼ _mtcptC30HP gtHP 1� ðerHPÞ1�ctct

� �¼ _mtcptC30HP ZtHP ð9Þ

_WtLP ¼ _mtcptCT0LP gtLP 1� ðerLPÞ1�ctct

� �¼ _mtcptCT0LP ZtLP ð10Þ

where erHP and erLP are the expansion ratio in the HP and LP turbine,respectively defined as:

erHP ¼p30

pT0and erLP ¼

pT0

p4ð11Þ

In addition, C30HP and CT0LP are the turbine inlet temperatures con-sidering heat losses (q) before gas expansion, as Fig. 2 shows.C30HP and CT0LP are defined as:

C30HP ¼ T30 �qHP

cptð12Þ

CT0LP ¼ TT0 �qLP

cptð13Þ

The isentropic efficiency of the HP turbine can be expressed interms of temperature, therefore gas temperature at its outlet canbe written as:

gtHP ¼C30HP � TT0

C30HP � TT0s

! TT0 ¼ C30HP 1� gtHP 1� ðerHPÞ1�ctct

� �h i¼ C30HP ½1� ZtHP� ð14Þ

Total to total efficiency has been used since kinetic energy is furtherrecovered by the LP turbine. Rearranging (10) with (14), one obtains

Eq. (15) where it is clear the effect of heat losses and HP turbineexpansion in the temperature upstream LP turbine (CT0LP).

_WtLP ¼ _mtcptCT0LP ZtLP ¼ _mtcpt TT0 �qLP

cpt

� �ZtLP

¼ _mtcpt T30 �qHP

cpt

� �½1� ZtHP� �

qLP

cpt

� �ZtLP ð15Þ

The total power provided by LP and HP turbines is calculated by theaddition of (15) and (9) as:

_WtTotal ¼ _WtLP þ _WtHP ð16Þ

In the equilibrium, compressor total power and turbine total powershould be equal considering mechanical losses included in the tur-bines efficiency. If both turbochargers (HP and LP) are in the sameshaft we could write that Eq. (16) has to be equal to Eq. (8). Never-theless, this is not the common case. Therefore, we must imposethat the turbine and compressor power is equal individually in eachturbocharger. It means one equation for the HP turbocharger and adifferent equation for the LP turbocharger. Indeed, equalling Eq. (9)to (7) for the HP turbocharger and (15) to Eq. (1) for the LP turbo-charger we could write:

_WtHP¼ _mtcptC30HP ZtHP¼ _mccpcZcHPT10 ½1þZcLP �ð1�eÞþe �TH2O

T10

� �¼ _WcHP

ð17Þ_WtLP¼ _mtcptC30LP ½1�ZtHP �ZtLP¼ _mccpcT10ZcLP¼ _WcLP ð18Þ

where C30LP is defined as:

C30LP ¼ C30HP 1� qLP

C30HP cpt ½1� ZtHP�

� �ð19Þ

and C30LP is equal to C30HP when qLP = 0.Taking into account the fuel to air ratio (F) to relate the burn gas

mass flow with the air mass flow, the following expression can beobtained:

1962 J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969

_mt

_mc¼ ð1þ FÞ ð20Þ

We define K as the quotient between the different specific heat ofburn gas and fresh air:

cpt

cpc¼ K ð21Þ

Replacing the definitions of (20) and (21) in (17) and (18) and rear-ranging both equations one obtains the compression ratio groups asa function of the expansion ratio as follows:

ZcLP ¼ ð1þ FÞK C30LP

T10ZtLP 1� ZtHP½ �

¼ ð1þ FÞK C30HP

T10ZtLP 1� ZtHP �

qLP

C30HP cpt

� �ð22Þ

ZcHP ¼ ð1þ FÞK C30HP

T10

ZtHP

1þ ZcLP½ �ð1� eÞ þ e TH2O

T10

ð23Þ

From Eqs. (22) and (23) is clear that the lower are the heat lossesqHP and qLP the higher would be C30HP and C30LP, respectively. Asa consequence, ZcHP and ZcLP would increase while keeping constantHP and LP turbines expansion ratio (and the rest of parameters).

From Eq. (23) can also be observed the effect of the inter-cool-ing. Indeed, the lower is the cooling fluid temperature for a givenintercooler efficiency the higher would be the compression ratiofor a given expansion ratio (being the rest of parameters constant).The effect of inter-cooling efficiency is clearer if we make a simplehypothesis assuming that cooling fluid temperature is equal toambient temperature (TH2O = T10). Then Eq. (23) can be written asfollows:

ZcHP ¼ ð1þ FÞK C30HP

T10

ZtHP

1þ ZcLPð1� eÞ ð24Þ

Looking at Eq. (24) is clear that the higher is e the higher is ZcHP for agiven expansion ratio.

In order to make evident the effect of the turbochargers isentro-pic efficiency in the equations it is possible to write Z groups as afunction of efficiency and the corresponding expansion and com-pression ratios as appear in Eq. (25).

ZcLP ¼1

gcLPðcrLPÞ

cc�1cc�1

� �¼ 1

gcLPðRcLP � 1Þ

ZcHP ¼1

gcHPðcrHPÞ

cc�1cc � 1

� �¼ 1

gcHPðRcHP � 1Þ

ZtLP ¼ gtLP 1� 1

ðerLPÞct�1ct

!¼ gtLPð1� RtLPÞ

ZtHP ¼ gtHP 1� 1

ðerHPÞct�1ct

!¼ gtHPð1� RtHPÞ ð25Þ

In Eq. (25) RcLP and RcHP groups’ value increase as compression ratiogrows. However RtLP and RtHP groups’ value decrease as expansionratio grows and the minimum value that can achieve is 0. Substitut-ing Eq. (25) in Eqs. (22) and (23) and rearranging one can write:

RcLP ¼ 1þ gcLPgtLPð1þ FÞK C30LP

T10ð1� RtLPÞ½1� gtHPð1� RtHPÞ�

ð26Þ

RcHP ¼ 1þ gcHPgtHPð1þ FÞK C30HP

T10

ð1� RtHPÞ½1þ gtLPð1� RtLPÞ�ð1� eÞ þ e TH2O

T10

ð27Þ

Defining

fLP ¼ gcLPgtLPð1þ FÞK C30LP

T10and

fHP ¼ gcHPgtHPð1þ FÞK C30HP

T10ð28Þ

It can be concluded that the higher are the temperatures ratio andturbochargers efficiency, the higher are the f groups. In addition, itis worth noting that K � 1 and also (1 + F) � 1. Substituting Eq. (28)in Eqs. (26) and (27), the compression ratio for each turbochargercan be expressed as:

RcLP ¼ 1þ fLPð1� RtLPÞ½1� gtHPð1� RtHPÞ� ð29Þ

RcHP ¼ 1þ fHPð1� RtHPÞ

½1þ gtLPð1� RtLPÞ�ð1� eÞ þ e TH2O

T10

ð30Þ

Analysis of a two-stage turbochargers configuration can be de-scribed with two different points of view. In fact, the product ofexpressions (29) by (30) defines the global compression ratio as afunction of the different expansion ratios known in the turbines.So, it can be evaluated the maximum intake manifold pressure fora certain energy and exhaust pressure upstream the HP turbine.But, it is sometimes more convenient in the design process to eval-uate the minimum back pressure cost to obtain a certain compres-sion ratio. In the latter case, expressions (29) and (30) must berewritten to express the expansion ratio in function of compressionratio. For that, expressions (22) and (23) are rearranged as follows:

ZtHP ¼ZcHP

ð1þ FÞKT10

C30HP

½1þ ZcLP �ð1� eÞ þ e � TH2O

T10

� �ð31Þ

ZtLP ¼ZcLPT10

ð1þ FÞKC30LP ½1� ZtHP �

¼ ZcLPT10

Kð1þ FÞC30LP � ZcHPT10C30LPC30HP

½1þ ZcLP �ð1� eÞ þ e TH2 O

T10

� �h i ð32Þ

Taking into account the definition of the Z groups shown in Eq. (25)and the definitions of fLP and fHP from (28), Eqs. (31) and (32) arerearranged as:

RtHP ¼ 1þ ð1� RcHPÞfHP

1þ ðRcLP � 1ÞgcLP

� �ð1� eÞ þ e

TH2O

T10

� �ð33Þ

RtLP ¼ 1þ ð1� RcLPÞfLP 1� gtHP ðRcHP�1Þ

fHP1þ RcLP�1ð Þ

gcLP

� �ð1� eÞ þ e TH2O

T10

h i� � ð34Þ

In this way, the relation between expansion ratios and compressionratios is more complex. The different parameters overlap them-selves and their particular influences are more difficult to extractand to analyse individually with this point of view.

To analyse the performance of this two-stage turbochargingarchitecture in a given engine and to compare different operatingconditions, we can define parameters which allow observing easilyand also quantify at pre-design stage the performance of the wholesystem. The first parameter used is the ratio between the globalcompression ratio and the global expansion ratio (

Qc/Q

t). Whenthis term is greater than the unity, turbocharging generates a netpositive work on the piston during the engine breathing process;when is lower than the unity, the engine has to produce more workto evacuate the exhaust gases of the combustion chamber than getduring breathing fresh one. The second parameter, which is highlyrelated with the previous one, is the pumping mean effective pres-sure (pmep) which quantifies the engine breathing efficiency. It canbe represented as the difference between the average pressure up-stream HP turbine (p30) and the average pressure downstream HPcompressor (p20). Using the definition of the global compressionratio by the global expansion ratio, it can be rearranged to give:

Table 1Basic characteristics of the engine for model validation.

Type of engine Double-stage turbochargedHeavy-duty Diesel engineNumber of cylinders six cylindersInjection system Unit pump injectorCompression ratio 16:1Turbocharger Wastegate HP turbine + LP turbine

Table 2Parameters used for the simulation.

cpt 1209 J/kgKcpc 1001 J/kgKgcHP 0.7 t/tgcLP 0.7 t/tgfa 1gi 0.4gtHP 0.6 t/tgtLP 0.6 t/sgv 0.8e 0.9F 0.062j 0cc 1.402ct 1.311qHP 0 J/kgqLP 0 J/kgHc 42,500,000 J/kgR 287 J/kgKT10 298 KT30 1000 K

J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969 1963

pmep ¼ p30 � p20 ¼ p20Ptp4

Pcp10

� �� 1

� �ð35Þ

Therefore pmep is positive when negative net work is achieved dur-ing gas exchange process. The third parameter used is the brakethermal efficiency (ge). Its definition is usually based on the brakemean effective pressure (bmep) corresponding to the energy trans-ferred from fuel to engine output. The bmep can be divided in threedifferent terms: the indicated mean effective pressure (imep) corre-sponding to fuel transformation efficiency, the pumping meaneffective pressure (pmep) and the friction plus auxiliaries meaneffective pressure (famep). The addition of pmep and famep is gener-ally known as engine mechanical losses. The famep corresponds tothe mechanical losses by friction of the different engine movingparts and the energy consumed by engine auxiliaries such as wateror fuel pumps. The expression of the brake thermal efficiency can bewritten as:

ge ¼imep� famep� pmep

q20 � gv � F � Hcð36Þ

Taking into account the definition of the indicated efficiency (gi)and assuming for simplicity reasons the hypothesis of fa-mep� q20 � gv � F � Hc (mainly at full load operation) to obtain re-

Fig. 3. Comparison of measured and modeled HP and LP expansion ratio as a function oglobal expansion ratio.

sults independently of engine design and engine speed, thefollowing approximate expression is established:

ge ¼ gi �p20

q20 � gv � F � Hc

Ptp4

Pcp10

� �� 1

� �ð37Þ

Engine load and speed for a given engine and specific operating con-ditions could be introduced in the analysis defining the friction plusauxiliaries efficiency (gfa) as:

gfa ¼ 1� famepq20 � gv � F � Hc

ð38Þ

f HP and LP compression ratio, respectively. Comparison of measured and modeled

10 bar9 bar8 bar

7 bar

6 bar

0 2 4 6 8 10LP Expansion Ratio [bar]

0.5

0.7

0.9

1.1

1.3

Glo

bal C

ompr

essi

on R

atio

by

Glo

bal E

xpan

sion

Rat

io

0.37

0.38

0.39

0.4

0.41 Brake Therm

alEfficiency

-2

0

2

4

6

Pum

ping

Mea

n Ef

fect

ive

Pres

sure

[bar

]

5 bar4 bar3 bar2 bar

10 bar9 bar

8 bar

7 bar6 bar5 bar4 bar3 bar2 bar

10 bar9 bar

8 bar7 bar6 bar5 bar4 bar3 bar2 bar

Fig. 4. Global compression ratio by global expansion ratio, brake thermal efficiency and pumping mean effective pressure as a function of LP expansion ratio for differentglobal compression ratios.

1964 J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969

3. Model validation

A validation process was carried out to assess the analyticalmodel accuracy and reliability. Different operating conditions, cor-responding to representative steady state points of the USA anti-pollution directive (US2007), were tested on a two-stage turbo-charging heavy duty engine. The engine used in this validationwas 12 l displacement Diesel engine and its basics characteristicsare shown in Table 1. A more detailed description of the experi-mental installation and measurement methodology is presentedin [7].

A comparison between experimental and simulated results ofexpansion ratio versus compression ratio in the HP and LP turbinesis shown in Fig. 3. Moreover, it can be observed in this figure acomparison between measured and predicted total expansion ratiothrough the turbines. In each case, a good agreement betweencomputed and measured values is obtained with very small differ-ences. These differences can be explained with the inherent exper-imental uncertainties that present the different sensors used in thisstudy. Turbines efficiencies were calculated from compressorworks to include mechanical efficiencies and heat transfer effects.However, small uncertainties in measured inlet and outlet com-pressor temperatures can lead to the small errors encountered inthe results.

Fig. 5. Optimized LP expansion ratio for different compression ratios in function ofthe aftercooler configuration and two design constraints.

4. Results and discussion

At first, the different equations were analyzed with constantparameters as shown in Table 2. Equations were described for

two different points of view. On one hand, the compression ratiois maximized for a given expansion ratio and on the other hand,the expansion ratio is minimized for a given compression ratio.Nowadays, the compression ratio appears like a constraint in theengine design process and a lot of optimization efforts concernthe reduction of the cylinders back pressure. The following resultsgo in the same direction. Equations were solved for different given

J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969 1965

compression ratios and expansion ratios were optimized in the HPand LP turbines to minimize the global expansion ratio.

4.1. HP and LP expansion ratio influence

The distribution of the expansion ratio between the HP and LPturbines is very important in terms of turbocharging performance.Fig. 4 shows the relation between the ratio

Qc/Q

t, the brake ther-mal efficiency and the pumping mean effective pressure as a func-tion of the expansion ratio in the LP turbine, which represents theintermediate pressure between turbines, for different compressionratios. All these performance curves have a maximum (or a mini-mum for the pmep) which corresponds to the minimum cylindersback pressure for this operation condition. The wastegate valveor the VGT mechanism should control the distribution betweenHP and LP turbines to maintain always the best operating conditionshown by maximum values in Fig. 4. With the exposed procedure itis easy to find such a maximum for a given engine operating point,considering that turbochargers are operating within designconditions.

The equations developed before can be solved as a function ofdifferent design objectives: minimum cylinders back pressure ormaximum brake thermal efficiency. Optimized LP expansion ratio(OperLP) obtained to maximize the ratio

Qc/Q

t or the brake thermal

Fig. 6. Global compression ratio by global expansion ratio, brake thermal efficiency,intercooler efficiencies and cooling fluid temperatures.

efficiency are similar for configurations with aftercooler. OperLP canbe adjusted with good agreement (R2 = 99.95%) as a function of thetotal compression ratio with lineal regression as shown in Fig. 5.

The architecture shown in Fig. 1 has an aftercooler with a goodefficiency and low cooling fluid temperature, so there are no differ-ences between LP expansion ratio optimized for each objective.But, if a system without aftercooler or with a low efficiency is con-sidered, slightly differences can be observed in Fig. 5. Theses differ-ences are generated by the imep calculation which take intoaccount a constant volumetric efficiency and constant air fuel ratio.Without a good temperature control downstream the HP compres-sor, the air density entering to the cylinders (and so the engineindicative power) varies substantially depending on the HP com-pression ratio.

4.2. Intercooler performance influence

Efficiency and cooling fluid temperature in the intercooler andthe aftercooler have a strong impact on two-stage turbochargingefficiency. Minimum cylinder back pressure is obtained with thehighest intercooler cooling capacity. But a nonlinearity relation be-tween efficiency and cooling fluid temperature exists. In fact, min-imum back cylinder pressure is more impacted by intercoolerefficiency when cooling fluid temperature is low than when is high.

pumping mean effective pressure and optimized LP expansion ratio for different

1966 J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969

This nonlinear relation can be observed in Fig. 6 where the evolu-tion of the quotient

Qc/Q

t, the brake thermal efficiency and thepumping mean effective pressure are plotted as a function of effi-ciency and cooling fluid temperature for given compression ratios.

When efficiency increases, the optimized expansion ratio distri-bution is modified increasing LP turbocharger work and decreasingHP one to keep minimum cylinder back pressure. In parallel, theoverall system efficiency is improved and the total expansion ratiodecreases for a given compression ratio amplifying the differencesbetween LP and HP expansion ratio. This trend is slightly differentwith the cooling fluid temperature. When the cooling fluid temper-ature decreases, the expansion ratio distribution remains practi-cally constant. The LP turbocharger work reduction observed isonly the consequence of the improved system efficiency and theQ

c/Q

t increase.In this analysis, pressure drop along the intercooler was ne-

glected because it strongly depends of the operating conditions,especially of the air mass and flow density, as shown in Eq. (4)and it is relatively small comparing to compressors pressure ratio(pC0 � pIT0� p20 � p10).

Fig. 7. Calculation of uncertainties for global compression ratio by global expansion ratioefficiencies.

4.3. Exhaust temperature and turbocharger efficiency influence

It can be observed in the compression ratio Eqs. (29) and (30) thatcompressor efficiencies and temperature upstream the HP turbineappear only in the f groups. Turbine efficiencies are present in theterms fLP and fHP but in addition turbine efficiencies appear as factorsof other terms in the compression ratio Eqs. (29) and (30). Beingsmall the terms they multiply, authors have assumed the hypothesisthat their influences can be neglected in the equations and have per-formed a parametric study of all turbocharger efficiencies and HPturbine upstream temperature directly with the f groups.

To validate this hypothesis, calculations were realized varyingcompressor efficiencies and keeping turbine efficiencies constant(gtLP = 0.6 and gtHP = 0.6), and repeated varying turbines efficien-cies and keeping compressors efficiencies constant (gcLP = 0.7 andgcHP = 0.7) to obtain the same values of the fLP and fHP terms. Dif-ferences between both calculations are shown in Fig. 7.

The relative difference obtained for the quotientQ

c/Q

t and thebrake thermal efficiency are kept below 5% in main operating con-ditions. But when turbocharger efficiencies are very small, the er-

, brake thermal efficiency and optimized LP expansion ratio versus LP and HP stage

Fig. 8. Global compression ratio by global expansion ratio, brake thermal efficiency, pumping mean effective pressure and optimized LP expansion ratio for different LP andHP stage efficiencies.

Fig. 9. Global compression ratio by global expansion ratio and brake thermal efficiency differences between single-stage and two-stage architectures versus turbochargerefficiencies.

J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969 1967

1968 J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969

ror increases substantially and exceeds 10%. This operating rangedoes not correspond to the typical operating conditions of a mod-ern engine because is very inefficient working with high compres-sion ratio and very small turbocharger efficiencies. So, errors in thisrange are not taken into account in the study and the hypothesiscan be verified for both parameters.

Concerning the optimized LP expansion ratio, the error is moreimportant and varies from 5% to 20% in the typical engine operat-ing range. The assumption stated before cannot be verified for thisparameter and numerical values cannot be directly processed. Nev-ertheless, curves for both calculations were checked and tenden-cies look very similar as exemplified at Fig. 7 (bottom right) for atotal compression ratio equal to 5 bar. So, trends can be analyzedin the same way to estimate the turbocharger efficiencies influence(or HP turbine upstream temperature) on the optimized LP expan-sion ratio.

Results obtained varying compressor efficiencies and keepingturbine efficiencies constant were used for the parametric studyand are shown in Fig. 8. To have a better representation, figuresare plotted as a function of gcLP � gtLP and gcHP � gtHP, correspondingto the LP turbocharger efficiency and HP turbocharger efficiency,respectively. HP turbine upstream temperature is a proportionalterm in the f groups, so its influence can be analyzed by the sameway. T30 = 1000 K is used as reference in the calculations and tur-bine adiabatic conditions have been imposed in order to simplifyanalysis and conclusions. A multiplication factor can be appliedto the graphics scales to have a visibility of the different resultsas a function of the temperature.

Each operating conditions were calculated with an optimizationprocess to determine the LP expansion ratio which minimizes thecylinder back pressure. Given that compression ratios are limitedto 6 bars to keep operating conditions at least representative of amedium and light-duty Diesel engine. In these cases, the cylinderback pressure is limited to acceptable values.

Fig. 8 shows the evolution of the ratioQ

c/Q

t, the brake thermalefficiency and the pumping mean effective pressure as a functionof turbocharger efficiencies. Bottom plane scales have been reori-ented for surfaces visualization optimization. The higher are theg parameters, the higher are two-stage turbocharger performance.It can be noted that this relation is not linear and the cylinder backpressure increases much more than turbochargers efficiency de-crease. When turbochargers efficiency decrease, brake thermal effi-ciency and pumping mean effective pressure are very impactedand overall system performance falls.

The optimized LP expansion ratio represents work distributionbetween HP turbine and LP turbine. When the efficiency of one tur-bocharger stage is different to the efficiency of the other stage,wastegate regulates the pressure to have higher expansion ratioin the stage with better efficiency. When efficiencies are the same,expansion ratios are quite similar in both stages to minimize backcylinder pressure.

4.4. Comparison between single-stage and two-stage performance

A single-stage turbocharger, which has the same characteristicsthan the two-stage turbocharger studied before but with only theHP stage, was analyzed and results were compared to the two-stage system ones. Both systems have an aftercooler downstreamthe HP compressor to cool the air mass flow entering to thecylinders.

Fig. 9 shows the differences for the ratioQ

c/Q

t and the brakethermal efficiency. Given compression ratios are limited to 4 barto keep acceptable expansion ratio in a single-stage turbocharger.The differences become more and more important as the LP stageefficiency increases and the HP stage efficiency decreases. Withtypical modern engine turbocharger efficiencies (f parameter

assuming T30 = 1000 K and adiabatic conditions in turbine andcompressor), it can be observed the real potential of the two-stagesystem comparing to the single-stage one.

5. Conclusions

An analytical pre-design model able to determine optimumtwo-stage architectures has been presented. Equations giving therelationship between total compression ratio and total expansionratio as a function of basic engine parameters have been first timedeveloped for two-stage systems similarly to well known existingexpressions for single stage turbocharging. This model can be eas-ily solved having analytical solutions and contributes to under-stand the complex interrelations of this type of enginearchitecture. By using non-dimensional or engine size reducedparameters the model results can be easily used for any enginepower requirement.

Using developed equations several studies have been carriedout to analyze different parameters that modify the overall perfor-mance of the system, as an example of such analytical model po-tential. Influences of the expansion degree between LP and HPturbine on the ratio

Qc/Q

t, the brake thermal efficiency and thepumping mean effective pressure were characterized and opti-mum performance for given operating conditions was established.As a result, a linear relation between OperLP and

Qc can be ob-

served. This allows a pre-design of either the wastegate or theVGT opening in the HP turbine; this is one of the main control deci-sions in the two-stage architecture.

In addition, other performed studies are the influence of inter-cooler efficiency, of turbocharger efficiencies, of cooling fluid tem-perature and of exhaust temperature, which were analyzed fordifferent global compression ratios. The main conclusions obtainedhave been that cooling fluid temperature has stronger impact onpumping mean effective pressure than intercooler efficiency. Re-sults show how the pmep varies from �0.6 bar to 0.3 bar modifyingthe cooling fluid temperature from 298 K (obtained with an exter-nal air cooler) to 358 K (obtained with an internal-engine-watercooler), while the pmep varies only from �0.6 bar to �0.3 bardecreasing the intercooler efficiency from 90% to 40% and keepingthe cooling fluid temperature at 298 K. Moreover, a strong influ-ence of turbocharger efficiencies and exhaust temperature onbrake thermal efficiency has been observed. As example, it canbe noted for the operating condition described in Table 2 brakethermal efficiency decrease from 41% to 38% varying compressorefficiencies from 70% to 50% or exhaust temperature from 1000 Kto 700 K.

Finally, the model was applied to compare single-stage andtwo-stage architecture in terms of engine effective efficiency forthe same engine operative conditions. At 2 bar of boost pressure,which corresponds to the typical boost pressure used in light dutyengines, results obtained show two-stage systems provide a differ-ence up to 10% in terms of brake thermal efficiency, especially foroperating conditions where turbocharger efficiencies are low.When boost pressure raises up to 4 bar, which is a common casefor high downsized o medium duty engines, this difference can ex-ceed 100% due to the difficulties for single stage system for achiev-ing high compression ratio with good efficiency.

These analyses and the modeling philosophy associated couldbe expanded to more areas of interaction between HP and LP tur-bochargers, with a deeper study, new conclusions could be ob-tained. In addition, the effect of EGR or turbochargers maps (toaddress efficiency variations in the different operative conditions)could be included in these equations in order to make more com-plex analyses. Authors consider these analyses a necessary tool tounderstand main energy equilibrium factors that affect to HP and

J. Galindo et al. / Energy Conversion and Management 51 (2010) 1958–1969 1969

LP coupling design. Moreover, it provides a very fast and highlycomplementary tool to other approaches such as 1D gas-dynamicmodels.

Acknowledgments

The authors wish to thank the economical support of this workto Spanish Project TRA2007-65433 from Ministerio de Ciencia eInnovación. Olivier Varnier is indebted to the Ministerio de Cienciae Innovación for its support through Grant AP2007-02868.

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