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Restriction Model Independent Method for Non-IsentropicOutflow Valve Boundary Problem Resolution

Felipe CastilloRenault, France

Emmanuel Witrant, Luc DugardGipsa Lab, Grenoble France

Vincent TalonRenault, France

Copyright c© 2012 Society of Automotive Engineers, Inc.

ABSTRACT

To meet the new engine regulations, increasinglysophisticated engine alternative combustion modeshave been developed in order to achieve simultaneouslythe emission regulations and the required enginedrivability. However, these new approaches requiremore complex, reliable and precise control systems andtechnologies. The 0-D model based control systemshave proved to be successful in many applications,but as the complexity of the engines increases, theirlimitations start to affect the engine control performance.One of these limitations is their inability to modelmass transport time. 1-D modeling allows some of the0-D models limitations to be overcome, which is themotivation of this work.In this paper, two quasi-steady outflow boundarymodels are developed: One based on the isentropiccontraction and another based on a momentumconservation approach. Both are compared with theresults of computational fluid dynamics (CFD) 3-Dsimulations. Then, an innovative method for solvingthe outflow boundary problem taking into account theentropy correction at the boundary for a 1-D unsteadygas flow modeling is presented. This method permitsthe boundary problem to be solved independentlyof the restriction model, which is typically capturedin the resolution method. It means that a physicalrestriction model can be modified without needing tochange the boundary resolution method. A Newton-Raphson algorithm is used with a modified Method ofCharacteristics (MOC) scheme to solve the boundaryproblem along with an extrapolation for the initializationof the scheme, which reduces the calculation timean increases the solution accuracy. Additionally, the

performance of the proposed method is comparedwith the scheme presented in the literature and themethod for solving the unsteady state is validated usinga GT-Power model as reference.

INTRODUCTION

In the recent years, Diesel engine emissions regulationshave become stricter while preserving the performanceof the engine. Although significant improvements havebeen made over the past years, there are still manychallenges that need to be overcome in order to meetthe future emission regulations. The introduction ofsophisticated alternative combustions modes such ashomogeneous charge compression ignition (HCCI), lowtemperature combustion (LTC) and premixed controlledcompression ignition (PCCI) offer a great potential toreduce the engine emission levels [1] [2] [3]. However,these new modes require different fueling strategiesand in-cylinder conditions, thus creating the need formore complex, reliable and precise control systems andtechnologies.

Dual-loop exhaust gas recirculation (EGR) with bothhigh and low-pressure recirculations is one of thenew strategies proposed to achieve the appropriateconditions to implement multiple combustion modes [4].However, ensuring the adequate in-cylinder conditionsis still a very difficult task, as the introduction of theEGR brings many control challenges due to the lackof EGR flow rates and mass fraction measurements.An efficient control of the in-cylinder combustion andengine-out emissions not only involves the total in-cylinder EGR amount, but also the ratio between the

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high pressure EGR (HP-EGR) and the low pressureEGR (LP-EGR). Indeed, this ratio is crucial as thegas temperatures and compositions are significantlyaffected. The HP-EGR gas is helpful to stabilize thecombustion at low load since its temperature is high.The LP-EGR reduces the engine-out NOx emissionwithout excessive smoke as it is filtered by the particlefilter. Controlling the air fractions in the intake manifoldis an efficient approach to control the in-cylinder EGRamount [5] [6]. For engines with dual EGR systems,the air fraction upstream of the compressor indicatesthe LP-EGR rate and the air fraction in the intakemanifold indicates the total EGR rate. Therefore, if theair fractions in each section are controlled well, then theHP and LP-EGR can be well controlled.

Nevertheless, controlling the air mass fraction is adifficult task, as direct measurement of the air fractionis not available on the production engines and asthe dynamics of the admission air-path can be highlycomplex. One of the actual problems to control theair mass fraction in the intake manifold is the EGRmass transport time. This phenomenon is muchmore significant in the LP-EGR as the distance thatthe gas has to travel is much longer that the oneassociated with HP-EGR. Indeed, this phenomenoncauses a degradation of the overall engine emissionperformance during the strong transients. Severalair mass fraction/EGR rate estimation methods havebeen proposed in the literature to overcome someof the actual limitations [7] [8] [6]. However, most ofthese estimation techniques are based on 0-D enginemodeling, which does not permit to take into accountthe mass transport time.

In this paper, a 1-D aerodynamics modeling platformis developed in order to provide an accurate whitebox model to synthesize and validate control lawsand estimators that are able to take into account themass transport time. The study detailed in this paperis focused on the cylinder intake valve model. Themathematical modeling of 1-D unsteady gas flow in apipe system is based on Euler’s equations. In orderto solve the cylinder inlet valve boundary condition,a specific resolution of Euler’s equations need tobe implemented. In this case, due to its satisfyingrobustness and accuracy, a method of characteristicsmodified to take into account the non-homentropic flowsthrough the intake valves is implemented.

Our paper is organized as follows. First, the equationsand hypotheses introduced to build intake valvemodels are described. A classical quasi-steady one-dimensional outflow model is developed along withan alternative momentum-based quasi-steady outflowmodel. Both models are compared with CFD 3-D

simulation results.

Then, a modified MOC is proposed to develop aninnovative non-homentropic outflow boundary resolutionmethodology that allows to implement different outflowmodels without modifying the boundary resolutionscheme. Additionally, a Newton-Raphson algorithm isintroduced in the scheme, along with an extrapolationto initialize the resolution method. It reduces thecalculation time and increases the accuracy.

Finally, the performance of the proposed scheme iscompared with traditional approaches and an unsteadyvalidation of the scheme is done using GT-Power asreference.

MOTIVATION

A simple example illustrates the problematic thatactually exists in the control of the air mass fractionin the intake manifold when a sudden change of theLP-EGR rate occurs. The system shown in Figure1consists in a tube with restriction on both ends. Atthe left end, there is known a pressure pin, a knowntemperature Tin and a known air mass fraction Fin.Assume that the system is in steady state at time 0.Then, a sudden change of Fin (from 0 to 1) is introducedat time 0 at the left end and the response is simulatedusing a 0-D and a 1-D model. The results obtained arepresented in Figure 2.

L=2m

PinTinFin

D=5cm

Figure 1: System used to carry out the 0-D and 1-Dapproaches comparison

As depicted in Figure 2, there is an important differencebetween the response obtained with the 0-D model andthe one with the 1-D model. For example, the 0-D modelover-estimates the air mass fraction between 0 and 0.1seconds (transient phase), which would compromisethe engine NOx emission performance in that timeinterval. The 1-D model response presents a morerealistic behavior of the air mass fraction during thetransient response, as it takes into account the masstransport time. This example illustrates the potentialof using 1-D models to synthesize control laws andobservers that are efficient during the transients.

QUASI-STEADY RESTRICTED OUTFLOW MODELS

In this section, the equations and assumptions fromliterature outflow models are derived along with a

2

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Mass F

raction

Mass Fraction vs. Time

0-D

1-D

Figure 2: Right tube end air mass fraction of the 0-D and 1-Dmodels in response to a unit step input air mass fraction.

model based on the momentum equation instead ofthe isentropic contraction. Figure 3 depicts an outflowrestricted boundary where three quasi-stationaryplanes have been defined. The plane 0 represents thestagnation state, the plane 1 is located just before therestriction and the plane 2 is located in the restrictionthroat. The gas exits the pipe by the restriction throatas a jet of cross-sectional area C2. There are sixunknown quantities: the pressure p, the velocity u andthe density ρ at planes 1 and 2. Hence, six equationsare needed in order to solve the boundary problem. Atthe time when these models were developed, desktopcomputers memory did not allow to pre-calculatesolutions into data-maps. Indeed, Newton-Raphsonalgorithms were used to solve the boundary problem,introducing an iterative problem at each time step. Themain issue of this approach is that the convergencealgorithm had to be modified for each specific model.Pre-processed data-maps at the interface between thequasi-steady model and the 1-D Lax-Wendroff are thusadvantageous. Therefore, all the models developed inthis section are put into data-maps.

Most approaches that are used to solve the outflowrestriction boundary problem assume to have anisentropic contraction between plane 1 and 2 (e.g seeBenson’s proposal) [9] [10]. However, there has beenother propositions such as [11], where a polytropicconstant found through a data-map is proposed inorder to create a non-hometropic approach, allowingto use the same formulation as the homentropic case.Consequently, the same overall boundary problemresolution process can be used. In this section, anadditional method to create data-maps inspired from amomentum approach is presented and then compared

Plane 1 Plane 2

u1

1

p1

u2

2

p2

Figure 3: Outflow Restriction Schema

with the traditional isentropic contraction approachand 3-D CFD steady state results. The purposeof this section is to illustrate some physical outflowrestriction models that can be obtained and the interestof generating a unique boundary resolution method.

OUTFLOW MODEL USING THE ISENTROPICCONTRACTION EQUATION

The following assumptions are made in order to createthe model:

A-1 under subsonic conditions, the back-pressure pbat plane 0 is equal to the pressure at the throatmeaning pb = p2 (no allowance is made for thepressure recovery);

A-2 the state is quasi-steady over the three planes;

A-3 the contraction is isentropic between planes 1 and2.

To start the development of this classical data-mapbuilding method, three basic equations are used:energy conservation, mass conservation and isentropiccontraction (A-3). These equations write respectivelyas follows [10]:

a2tot = a2

1 +γ − 1

2u2

1 = a22 +

γ − 1

2u2

2 (1)

p1

p2= Φ

u2

u1

(a1

a2

)2

(2)

p1

p2=

(a1

a2

) 2γγ−1

(3)

3

where a is the sound speed, u is the particle speed,

atot =√a2 + γ−1

2 u2 is the total sound speed, γ is

the specific heat ratio and Φ = C2

C1the sectional ratio.

Combining (2) and (3) gives:

((a1

a2

)2) γγ−1

= Φu2

u1

(a1

a2

)2

(4)

The energy equation (1) can be written as:

(a1

a2

)2

=

(atota2

)2

− γ − 1

2

(u1

a2

)2

(5)

Replacing (5) in (4) and defining the non-dimensionalspeeds as A = a

atotand U = u

atot.

U1 = ΦU2

[(1

A2

)2

− γ − 1

2

(U1

A2

)2] −1γ−1

(6)

This equation provides a static relationship between thespeed at the throat U2 and the speed at the boundaryU1. A2 can be written in terms of U2 and A1 in terms ofU1 using the energy conservation equation (1). With (3),a relationship between U1 and the pressure ratio p1

p2is

found, which is what is finally captured in the data-map.However, this is only valid for sub-sonic flows and acomplementary analysis has to be performed for thesonic flow case.

Under sonic conditions the particle speed equalsto the sound speed, which implies from (1) thatU2 = A2 =

√2

γ+1 . Using this result in (6), the criticalnon-dimensional particle speed U1cr is found as:

U1cr = Φ

√2

γ + 1

[(γ + 1

2

)− γ2 − 1

4(U1cr)

2

] −1γ−1

(7)

It is important to notice that U1 under sonic conditions isindependent on the pressure ratio p1

pb. It only depends

on the area ratio Φ = C2

C1. The critical pressure ratio pcr,

at which the flow becomes sonic can be expressed as:

pcr =

(γ + 1

2

) γγ−1

(A1cr)2γγ−1 (8)

where A1cr is found using the energy conservation and(7). (8) allows to determine whether the flow is sonic. Atthis point, all the required information to build the data-map is available. In order to obtain the data-map, thefollowing procedure is proposed:

1. setting a range of U2 equals to[0,√

2γ+1

](Subsonic range), A2 is found using (1);

2. now that U2 and A2 are generated, (6) solvednumerically can be employed to find U1. To findA1 use the energy equation once again;

3. Using 3, the values of p1p0

can be found for thesubsonic range with the assumption A-1;

4. to include the sonic range in the data map, usethe critical pressure ratio pcr. If p1

p0> pcr(Φ), set

U1 = U1cr(Φ) using 7.

OUTFLOW MODEL USING A MOMENTUM BASEDEQUATION

In this subsection, a momentum-based outflow data-map is developed. Figure 4 presents a control volumedescription to formulate the momentum equation at theboundary.

Figure 4: Proposed pressure distribution for the momentumapproach

A control volume is once again created between Planes1 and 2. However, this time there is a particular quantitydistribution on Plane 2 that allows to formulate themomentum equation. There are three sections definedin Plane 2: two sections right in front of the restrictionwalls and one facing the restriction throat (see Figure4). Some assumptions are made on Plane 2 in order todefine all the quantities at each section.

H-1 u2 normal to the restriction wall at Plane 2 is zero.

H-2 The pressure and speed on Plane 2 right next tothe throat are p2 and u2 respectively.

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H-3 The pressure right next to the restriction on plane 2is equal to p1.

All the information required to build the model is nowavailable. The momentum conservation in the virtualcontrol volume shown in Figure 4 implies that:

p1C1 − p2C2 − p1(C1 − C2) = ρ2u22C2 − ρ1u

21C1 (9)

To start with the data-map generation procedure, thesound speed equation a =

√γpρ and (9) are combined

to obtain:

p2

p1=

1 + γ(u2

a2

)2

1 + γΦ

(u1

a1

)2

(10)

The goal is to express U1 as a function of U2 and A2,similarly to the previous approach. Combining the massconservation with (10) gives:

Φu2

u1

a21

a22

=

1 + γ(u2

a2

)2

1 + γΦ

(u1

a1

)2

(11)

<=> Φu2

u1

a21

a22

+ γu2

a22

u1 = 1 + γ

(u2

a2

)2

Using the energy conservation (5) and factorizing toobtain a quadratic function of U1 gives:

(γ − Φ

γ − 1

2

)U2

1 −(A2

2

U2+ γU2

)U1 + Φ = 0 (12)

Equations (10) and (12) permit to create the data-mapusing this momentum-based approach. As done in theliterature approach, it is necessary to analyze the sonicflow case. This case in characterized by A2 = U2 =√

2γ+1 , which provides U1 under sonic conditions as:

(γ − Φ

γ − 1

2

)U2

1cr −√

2(γ + 1)U1cr + Φ = 0 (13)

And the pressure ratio under sonic flow is expressed as:

p1cr =

1 + γ

1 + γΦ

(U1cr

A1cr

)2

(14)

Similarly to the isentropic data-map procedure, U1 undersonic conditions does not depend on the pressure ratiop2p1

but only on Φ. The procedure to generate the data-map is presented as follows:

1. setting a range of U2 equals to[0,√

2γ+1

](subsonic range), then find A2 using (1);

2. from U2 and A2 , (12) can be employed to find U1

and A1 is given by the energy equation;

3. using (10), the subsonic values of p1p0

are found(from the assumption A-1);

4. to include the sonic range in the data-map, use thecritique pressure ratio pcr: If p1p0 > pcr(Φ) then setU1 = U1cr(Φ) from (13).

OUTFLOW MODELS COMPARISON

Figure 5 shows the isentropic and the momentumapproach data-maps plotted together in order toevaluate their differences. Additionally, the resultsobtained using steady-state CFD 3-D simulation(Fluent) are shown with the purpose of illustratingthe predictability of each model. Figure 5 shows howthe non-dimensional flow velocities are systematicallyhigher in the model using isentropic contraction thanin the momentum-based development. This behavioroccurs because the isentropic contraction approachdoes not take into account the increase of entropythrough the restriction, which allows for greater flowspeeds.

Figure 5: Steady flow results comparison between outflowboundary restriction models

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When comparing the models with the CFD 3-Dsimulation results, it can be seen that both modelsoverestimate the non-dimensional particle speed, whichis expected as friction (among other phenomena)has been neglected in order to simplify the problem.Nevertheless, the momentum-based model is morepredictive than the isentropic model and comparableto the 3-D references for small area ratios Φ < 0.3. Inorder to visualize the difference of predictability of themodels, the discharge coefficients (ratio between CFDresults and the model results) that fit with the CFDsimulation are calculated and presented in Table 1.

Area Ratio Momentum Model Isentropic Model0.2 0.9798 0.78410.4 0.9414 0.78030.6 0.9229 0.81890.8 0.9320 0.8871

Table 1. Discharge coefficients for both models with respectto the CFD results

The results shown un Table 1 illustrate the interest ofimplementing a boundary resolution method for themomentum approach as better predictability is obtainedwith this approach. The traditional resolution method(Appendix I), based on the isentropic contractionbetween plane 1 and 2, only works under the isentropiccontraction assumption. This creates the need forsearching an alternative resolution method for theother physical approaches such as the momentum-based approach presented in this section or CFD orexperimental results if available. In the next section, asolution for this limitation is presented.

OUTFLOW BOUNDARY PROBLEM RESOLUTION

In the previous section, outflow valve equations havebeen solved and the results have been captured intodata-maps. The next step is to create an interactionbetween these data-maps and the in-pipe resolutionscheme (two-step Lax Wendroff + Total VariationDiminishing (TVD)). The method of characteristics(MOC) has proved to be a versatile method to createthis interaction [9] [10] [12]. The MOC resolutionscheme is based on Euler’s equations (mass, energyand momentum conservation) expressed in thehometropic formulation. However, the intake valvedoes not necessarily satisfy this assumption and theMOC consequently has to be modified in order to treatthe non-homentropic case. In the previous section,different approaches to model the outflow boundarywere presented. Nevertheless, it is important to recallthat the literature outflow boundary resolution methodsare based on the isentropic contraction between planes1 and 2 of Figure 3. This motivates the search for aresolution method that is able to solve the boundaryproblem independently of the model taken for theoutflow boundary. For example, in this paper, the

momentum-based model developed cannot be useddirectly with the literature resolution methods, recalledin Appendix I. A polytropic coefficient has been usedto integrate a momentum-based approach with thetraditional outflow boundary resolution method [11].

In this section, a resolution method able to solvethe boundary problem, independently on the chosenquasi-steady outflow modeling approach, is presented.Moreover, CFD or experimental results presented in theform of a data-map can be solved. Additionally, someadvantages derived from the method are exposed, suchas faster convergence time, greater accuracy, soundspeed and pressure reference free method and entropycorrection at the boundary.

PROPOSED METHOD

Figure 6 shows the scheme of the MOC approach usedin this study. The index D represents a quantity at theboundary and the index L represents the interpolatedpoint between D and D − 1 where the trajectory u +a goes at time n. The index S is associated to thetrajectory u at time n.

n+1

nL S

u+a

u

DD-1D-2

Boundary

XS

XLX

t

Figure 6: Characteristics Outflow Restriction Boundary

In the trajectories u + a and u, the following conditionsare respected (called Riemann invariants) forhomentropic flow:

λ = an+1D +

γ − 1

2un+1D = anL +

γ − 1

2unL (15)

sn+1D = snS (16)

However, the intake valve flow does not respect thehomentropic assumption and the MOC has to bemodified to take into account the difference of entropyacross the valve. The changes of entropy due to heattransfer and friction at the boundaries are neglectedas the change over only one finite element is small

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in comparison with the change along the whole tube.On the other hand, the variation of entropy level canbe significant, that’s why the Riemann invariant for theoutflow case is modified as follows [10]:

λLC = λL +An+1D

An+1AD −AnALAn+1AD

= An+1D −

(γ − 1

2

)Un+1D

(17)

where AA is the entropy level defined as:(p

pref

) γ−12γ

=A

AA(18)

Entropy correction

As previously shown, a modification of the MOC is donein order to take into account the entropy variation atthe boundary. A treatment of (17) is performed withthe purpose of suppressing the need for defining thetraditionally used pressure and sound speed referencesand allowing the entropy correction to be written directlyin terms of the available quantities at time n. Equation(17) can be written as:

un+1D =

2

γ − 1

(λLC − an+1

D

)(19)

Which is expressed in terms of entropy levels as:

un+1D =

2

γ − 1

(λnL + an+1

D

(1 − AnAL

An+1AD

)− an+1

D

)(20)

Replacing in (19) the entropy levels by 18 gives:

un+1D =

2

γ − 1

λnL − anL

(pn+1D

pnL

) γ−12γ

(21)

This equation gives a direct relationship between pD anduD as λL, aL and pL are known at time n.

Resolution method

In this part, the proposed resolution method is explainedin details. As depicted in Figure 6, there is a secondtrajectory that takes into account the path along theparticle speed u. Along this trajectory, the entropy level

does not change as the heat losses and friction havebeen neglected for one finite element. Equation (16)represents the information obtained at the boundaryfrom the trajectory u. Knowing that under isentropicconditions

p

ργ= constant (22)

and replacing ρ by the sound speed equation ρ = γpa2 the

following is obtained:

p(1−γ)a2γ

γγ= constant (23)

The value of (22) at the point S is denoted as Ss andcalculated by a linear interpolation [10]. Equation (16)can be reformulated using the value of Ss and quantitiespreviously defined:

aD =

√γS

1γs p

γ−12γ

D (24)

This equation relates directly the pressure with thesound speed at the boundary using the informationcoming from the trajectory u. Equation (21) alsorelates directly the pressure and the particle speedat the boundary. Therefore, knowing the pressuremeans solving the whole boundary problem due tothe information coming from the Riemann invariants.However, in order to determine the value of thepressure at the boundary, the outflow boundary modelsdeveloped in the previous section have to be taken intoaccount.

As the interest of this study is to establish a resolutionmethod independently of the chosen outflow model, therelation between the non-dimensional particle speedand the pressure ratio is described by a data-map, suchas the ones developed in the previous section, insteadof an analytical equation. Hence, the outflow modelscan be expressed as:

PR = datamap(UD,Φ) (25)

where PR is the pressure ratio pDpb

, pb being the knownback pressure at the stagnation state. Note that thereis an algebraic loop in this formulation because thepressure is required to calculate the particle speed andthe particle speed has to be known in order to calculate

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the pressure. A numerical procedure is thus needed tosolve the boundary problem.

To start the iteration process, it is necessary to setan initial pressure at the boundary. This initial valueis essential as the convergence and the speed of theiterative algorithm depend on that value. In this method,the solution of the Lax-Wendroff scheme is used toinitialize the pressure so the initial value is as closeas possible to the solution minimizing the amount ofiterations and the calculation time. Hence, the initialpressure at the boundary at each time step is set as:

pitD = 2pD−1 − pD−2 (26)

Where pitD is the initial pressure for the iterative algorithmat the boundary. Equation (26) is a linear extrapolationof the pressure at the boundary using the two closestfinite elements in the tube. This extrapolation can bevery close to the solution, overall during the steady-state stages, avoiding the use of the iterative algorithmwhich decreases the calculation time.

Now that an initial pressure has been set, (21) and (24)are used to determine the values of uD and aD. Allthe information at the boundary has been found, butthe coherence with respect to the outflow model as tobe imposed. The non-dimensional speeds have to becalculated in order to compute (25) and find out whetherthe solution is consistent or not.

The knowledge of UD and the area ratio Φ allowsthe pressure ratio to be computed with (25). As theback pressure is known, the pressure at the boundaryobtained with the outflow model is determined whenmultiplying pb by the pressure ratio previously obtained.The comparison between this pressure and the initialpressure is the criterion that determines whether pitDis a solution or not for the boundary problem. Theconvergence criterion can be expressed as follows:

f = pitD − datamap(U1,Φ) ∗ pb < ε (27)

where epsilon sets the convergence accuracy. Anotherimportant advantage of this method appears in (27)as ε can be defined in terms of pressure. Therefore,the convergence criterion can be easily established.However, nothing guaranties that the extrapolation usedin the initial pitD satisfies the criterion of (27). That is whya iterative procedure has to be implemented until (27)is satisfied. The Newton-Raphson method has proved

to improve the performance of the inflow boundaryresolution method [12] and has also been successfullyused for solving the outflow boundary resolution [11].Hence, it is proposed in this study to upgrade the valueof pitD at every iteration loop.

pit+1D = pitD − f it

dfit

dp

(28)

(28) describes the proposal of the NR algorithm forthe outflow boundary problem. The calculation ofthe derivative dfit

dp creates the need for generating asecond data-map containing dPR

dUDand the computation

of the remaining parts of the analytic derivative, whichis inconvenient in terms of simplicity and calculationtime. The numerical computation of the derivative isthus more favorable for this application. The Newton-Raphson algorithm proposed for the boundary solutionis described as follows:

pit+1D = pitD −

f(pitD)

f(pitD+∆p)−f(pitD)∆p

(29)

When ∆p in (29) is small, a better approximation to theanalytic Newton-Raphson algorithm is obtained. Thisiteration procedure is only necessary under subsonicconditions. When sonic flow at Plane 2 is reached, thevalue of UDcr is constant independently of the pressureratio pD

pb, which allows to determine UDcr and ADcr

directly using equations (7) and (13) for the isentropicand momentum approaches respectively. The solutionof these equations is stored in a data-map of thefollowing form: UDcr = datamap(Φ). For sonic flow, (21)and (24) hold, therefore:

λLatot

=γ − 1

2UD +

aL

pγ−12γ

L

√γS

1γs

AD (30)

and

atot =

√γS

1γs p

γ−12γ

D

AD(31)

Combining (30) and (31) gives:

8

pD =

λLADcr(ADcr + γ−1

2 UDcr)√

γS1γs

2γγ−1

(32)

Using this procedure, there is no need for using aniterative algorithm for the sonic flow. At this point, allthe variables under subsonic and sonic flow have beendetermined and the resolution method is completed.

Method implementation

To implement the subsonic method, follow the steps:

1. calculate λnL and Sns ;

2. initialize pitD = 2pn+1D−1 − pn+1

D−2. This is the linearextrapolation using the two closest nodes insidethe pipe where pn+1

D−1 and pn+1D−2 are obtained by

the Lax-Wendroff scheme,

3. use (24) to calculate aitD and (21) to compute uitD

4. calculate the total sound speed as shown in (??)and compute the non-dimensional speed U itD andAitD;

5. compute pDpb

= datamap(U itD ,Φ) using any outflowmodel (data-map);

6. compute (27). If f < ε then return pn+1D = pitD,

an+1D = aitD and un+1

D = uitD. Otherwise, use (29) toupdate pitD and return to the step 3;

To implement the sonic method follow the steps below:

1. calculate λnL and Sns ;

2. find UDcr with UDcr = datamap(Φ), (see (7)and (13)) and calculate ADcr using the energyconservation.

3. calculate pn+1D using (32);

4. use equation (24) to calculate an+1D and (21) to

compute un+1D .

OUTFLOW METHOD CONVERGENCE PERFORMANCE

It is now proposed to study the convergenceperformance of the scheme introduced in this paper byperforming a comparison with the approaches found inthe literature (see Appendix I). The convergence resultsregarding a transient test of an intake valve are detailed.As the criterion of convergence of Benson’s approach

(33) is different from the criterion of the schemeproposed in this study (27), a more general criterionis used in order to perform an objective comparison.The parameter tolerance defined by TolX = ∆Xit

Xit isused as a convergence criterion for both schemes. Thistolerance is set to 10−5 for the test presented in thissection.

Figure 7: Comparison of number of iterations required by theliterature scheme and the proposed scheme. Convergencecriterion: TolX=10-5

Figure 8: Accuracy obtained by the schemes

Figures 7 and 8 show the results obtained. As canbe seen in Figure 7, the proposed resolution schemeprovides a quicker convergence than the literaturescheme: approximatively 5 times faster for a toleranceof 10−5. For smaller tolerances, this iteration numberratio between the schemes can increase significantly:for example for a TolX = 10−6 the proposed schemeis 8 times faster. Figure 8 shows the final TolX of thesolution obtained for each scheme. It is logical thatall the values are under 10−5 as it is the convergencecriterion. However, the proposed method presents asystematically smaller criterion magnitude than the oneusing the literature’s scheme. This behavior is dueto the quadratic convergence of the Newton-Raphsonmethod.

9

UNSTEADY FLOW SIMULATION VALIDATION ANDRESULTS

It is important to validate the performance of the outflowboundary resolution method under unsteady flow withrespect to some reference. GT-Power is used asreference for the unsteady flow validation, due to itsversatility. The validation is done using the schematicpresented in Figure 9.

L=2m

PinTin D=5cmout

Figure 9: Scheme used for unsteady flow validation of theoutflow boundary resolution method

The tube’s left end, which has no restriction, is suddenlyopened introducing a shock wave into the tube. pinand Tin are set greater than the tube initial conditionsinside the tube in order to generate a shock wave andtest the entropy correction at the boundary, respectively.The pressure, speed and temperature are captured inthe middle of the tube every time step for the validation.This experiment is reproduced in GT-Power and in aMatlab Simulink platform where the resolution schemepresented in this study has been implemented. Theresults are presented in Figures 10 and 11. Theconditions for the test are: an initial tube pressure,speed and temperature of 105[Pa], 0[m/s] and 300[K]respectively and pin = 1.2 ∗ 105[Pa] and Tin = 500[K].

0 0.05 0.11

1.05

1.1

1.15

1.2

1.25

Time[s]

Pre

ssur

e[10

0*kP

a]

Pressure vs. Time for Phi=0.2

0 0.05 0.1-20

0

20

40

60

Time[s]

Spe

ed[m

/s]

Speed vs. Time for Phi=0.2

0 0.05 0.1300

350

400

450

500

Time[s]

Tem

pera

ture

[K]

Temperature vs. Time for Phi=0.2

0 0.05 0.11

1.05

1.1

1.15

1.2

1.25

Time[s]

Pre

ssur

e[10

0*kP

a]

Pressure vs. Time for Phi=0.4

0 0.05 0.1-20

0

20

40

60

80

Time[s]

Spe

ed[m

/s]

Speed vs. Time for Phi=0.4

0 0.05 0.1300

350

400

450

500

Time[s]

Tem

pera

ture

[K]

Temperature vs. Time for Phi=0.4

GT-PowerMethod

Figure 10: Scheme used for unsteady flow validation ofthe outflow boundary resolution method for Φout = 0.2 andΦout = 0.4

As it can be seen in the simulation results (Figures 10and 11), the responses obtained with the proposedresolution method match the results obtained withGT-Power. Even when some differences betweenthe results are appreciated due to the temperaturedispersion in the GT-Power simulation, the results aresatisfactory because this phenomenon is due to the

0 0.05 0.11

1.05

1.1

1.15

1.2

1.25

Time[s]

Pre

ssur

e[10

0*kP

a]

Pressure vs. Time for Phi=0.6

0 0.05 0.10

20

40

60

80

100

Time[s]

Spe

ed[m

/s]

Speed vs. Time for Phi=0.6

0 0.05 0.1300

350

400

450

500

Time[s]

Tem

pera

ture

[K]

Temperature vs. Time for Phi=0.6

0 0.05 0.11

1.05

1.1

1.15

1.2

1.25

Time[s]

Pre

ssur

e[10

0*kP

a]

Pressure vs. Time for Phi=0.8

0 0.05 0.1-50

0

50

100

150

Time[s]

Spe

ed[m

/s]

Speed vs. Time for Phi=0.8

0 0.05 0.1300

350

400

450

500

Time[s]

Tem

pera

ture

[K]

Temperature vs. Time for Phi=0.8

GT-PowerMethod

Figure 11: Scheme used for unsteady flow validation of theoutflow boundary resolution for Φout = 0.6 and Φout = 0.8method

GT-Power numerical scheme implemented in the tube,not in the boundary resolution method.

CONCLUSIONS

In this paper, the outflow boundary encountered at theoutlet end of a tube has been studied. Two differentquasi-steady outlet models have been explained indetail, and then compared with results from a 3-D CFDsimulation. The momentum-based model exhibits morepredictability than the traditional quasi-steady modelbased on the isentropic contraction assumption. Thisfact motivates the search of a more general boundaryresolution method, able to solve the boundary problemfor any quasi-steady outlet model that can be put intoa data-map of the form described during this study.A modified Method of Characteristics has been usedto create the interaction between the outlet modelsand the tube numerical scheme, always consistentwith a compressible flow. The proposed resolutionmethod allows to integrate a quasi-steady outlet modelwith the wave action scheme, diminishing the amountof iterations at each time step and increasing theaccuracy of the response. Thid is due to the Newton-Raphson algorithm implemented of the iteration loopand the extrapolation used in the initialization of theboundary resolution method. The resolution schemehas been validated with GT-Power as reference, whichhas demonstrated the effectiveness of the proposedapproach.

DEFINITIONS, ACRONYMS AND ABBREVIATIONS

CFD:Computational fluid dynamicsEGR:Exhaust gas recirculationHCCI:Homogeneous charge compression ignitionHP-EGR:High pressure exhaust gas recirculationLP-EGR:Low pressure exhaust gas recirculationLTC:Low temperature combustionMOC:Method of Characteristics

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NR:Newton RaphonPCCI:Premixed charge compression ignitionQSS:Quasi steady stateTVD:Time variation diminishing

REFERENCES

[1] K.Akihama, Y.Takatory, K.Inagaki, S.Sasaki, andA.Dean. Mechanism of the smokeless richdiesel combustion by reducing temperature. SAEtransactions-Journal of Engines, 110 Paper 2001-01-0655, 2001.

[2] M.Alriksson and I.Denbrantt. Low temperaturecombustion in a heavy duty diesel engine usinghigh levels of EGR. Proceeding of the SAEconference, Paper 2006-01-0075, 2006.

[3] T.Ryan and A.Matheaus. Fuel requirements forHCCI engine operation. SAE transactions-Journalof Fuels Lubricants, 112,1143-1152, 2003.

[4] A.Hribernik. The potential of the high and low-pressure exhaust gas recirculation. Proceeding ofthe SAE conference, Paper 2002-04-0029, 2002.

[5] M.Ammann, N.Fekete, L. Guzella, andA.Glattfelder. Model-based control of the VGTand EGR in a turbocharged common-rail dieselengine: theory and passanger car implementation.Proceeding of the SAE conference, Paper 2003-01-0357, 2003.

[6] J.Chauvin, G.Corde, and N.Petit. Constrainedmotion planning for the airpath of a diesel HCCIengine. Proceedings of the 45th IEEE conferenceon decision and control, 3589-3596, 2006.

[7] J.Wang. Air fraction estimation for multiplecombustion mode diesel engines with dual-loopEGR systems. Control Engine Practice 16, 1479-1468, 2008.

[8] I.Kolmanovski, J.Sun, and M.Druzhinina. Chargecontrol for direct injection spark ignition engineswith EGR. Proceedings of the 45th IEEEconference on decision and control, 34-38, 2000.

[9] R. Benson. The Themodynamics and GasDynamics if Internal-Combustion Engines, volumeVolume I. Clarenton Press-Oxford, 1982.

[10] DE. Winterbone and RJ. Pearson. Theory ofEngine Manifld Design: Wave Action Methods forIC Engines. Society of Automotive Engineers. Inc,2000.

[11] G.Martin. 0-D -1-D Modeling of the air path ofICE Engines for control purposes. PhD thesis,Universite d’Orleans, 2009.

[12] G. Martin, P. Brejaud, and A. Charlet P. Higelin.Pressure ratio based method for non-isentropicinflow valve boundary condition resolution.Proceeding of the SAE conference, Paper2010-01-1052, 2010.

APPENDIX I

CLASSICAL NON-HOMETROPIC OUTFLOWBOUNDARY RESOLUTION METHOD

Subsonic flow

Benson [9] introduced a strategy to solve thehomentropic outflow boundary problem based onthe isentropic contraction assumption. The methodcan be modified to solve the non-homentropic casesby the introduction of the ”starred Riemann variables”.In this section, only the basic procedure is presented.Benson’s resolution method for sub-sonic flow is basedon the solution of:

f (A) =(A

4γ−1 − Φ2

)(λ−A)

2 − γ − 1

2

(A2 − 1

)Φ2 = 0

(33)

at every time step. The remaining quantities at theboundary are found using the following equations

U =2

γ − 1(λ−A) (34)

And

A =

(p

pref

) γ−12γ

(35)

Different solution methods for 33 have been proposed.Benson proposed the following method:

1. the algorithm is initialized at An = λ+12 as the

solution is in the range [1, λ];

2. the initial step is defined as ∆An = λ−14

3. (33) is evaluated using An:if f (A) < 0 then set An+1 = An − ∆An else setAn+1 = An + ∆An;

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4. if f (A) < ε then return A,else set ∆An+1 = ∆An

2 and An = An+1. Go backto step 3.

Martin proposed to use a Newton-Raphson algorithmto solve equation 33 instead of the iterative methodof Benson [11]. His method introduces a polytropiccoefficient to model the non-homentropic cases that isobtained by data-maps. However, the method does notimprove significantly the calculation time and it doesnot take into account the change of entropy level at theboundary. The procedure is described as follows:

1. the algorithm is initialized at An = λ+12 as the

solution is in the range [1, λ];

2. using the energy equation 1 = A2n + γ−1

2 U2n, the

value of Un is determined;

3. the polytropic coefficient is found using a data-map of κ versus λ/a0, where a0 = λ

An+ γ−12 Un

;

4. now that κ, An and λ are known, (33) is evaluated.If f (A) < ε the solution has converged. Else theNewton-Raphson algorithm is introduced in orderto update the value ofAn. It is run until the solutionis converged.

Sonic flow

When the flow in the throat is choked there is a sonicflow. As seen in the previous section, the modelequations change as well as the resolution methoddoes when a sonic flow occurs. Benson proposed asolution method based on the solution of the followingequation (the same approach is done in [11]):

f

(A

At

)cr

= Φ2 −

[γ + 1

γ − 1−(

2

γ − 1

)(A

At

)2

cr

](A

At

) 4γ−1

cr

(36)

This equation cannot be solved analytically, thereforeanother numerical method has to be introduced.

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