improvements of boundary condition models for 1d engine thermodynamic and gas dynamic simulations

IMPROVEMENTS OF BOUNDARY CONDITION MODELS FOR 1D ENGINE THERMODYNAMIC AND GAS DYNAMIC SIMULATIONS

Fernando OrtenziENEA - Italian National Agency for New Technologies, Energy and the Environment

Email: [email protected], phone +39 0630486873

Emiliana VescoRTZ-Soft – Automotive Engineering and Software

ABSTRACT

This paper analyzes the 1D boundary calculations using the Method of Characteristics, in particular the valve outflow and the multi-pipe junction model.

The Benson constant pressure outflow model has been modified imposing the minimum and maximum physical entropy levels during the iterations. A sub iterative cycle has also been made when calculating the Riemann incident and reflected variables in order to improve the convergence especially for lower valve area. Convergence has been reached for all the test also with valve area ratios of 10-6.

A general multi-pipe junction model has been developed; a pressure loss term has been added for ducts with flow away from the junction. Sonic flows has been taken into account and a “equivalent” duct has been built in order to calculate the pressure loss terms. Two tests have been reported in order to show the results when sonic flow has been encountered.

Keywords: Method of Characteristics; pipe boundaries modeling; Outflow; multi-pipe junction.

1. INTRODUCTION

Computer software are widely used in the design and development process of internal combustion engines: time and costs are the advantages of using such software.

The development of thermodynamics and gas dynamics models of manifolds, together with the combustion models are the main interest to build an accurate simulation code capable to calculate the energy and environmental impact of internal combustion engines.

Generally computer software to simulate the engines use 1D codes due to lower computation times than three-dimensional codes.: multi-cylindrical engine with much complex intake ducting together turbo-compressor devices can be simulated requiring very low calculation times with mono-dimensional codes.

Second order conservative schemes are widely used to calculate the 1D gas dynamics within the engine manifolds such Two Step Lax-Wendroff or ;McCormack [1,2,3]. while for pipe boundaries the first order method of characteristics is the most used [1,3,5].

The method of characteristics, with the great contribution of Benson [1] has been used extensively during ’70 and now is used mainly to calculate the gas dynamic at pipe boundaries. Many works have been made to improve this method [5,6,7] and to add new

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boundaries [8,9]: in the present work the outflow [1,10] (flow from cylinder to manifold) and the pipe junction with pressure loss [11] have been analyzed and modified methods have been reported in order to improve convergence (the first) and take into account for chocked flow (the second).

In the first part of the paper the duct models used are described, a second order Two Step Lax-Wendroff scheme has been used together with the TVD flux Limiter in order to limit the oscillations typical of the second order schemes.

In the OutFlow section the Constant Pressure model of Benson [1,10] is analyzed and the modifications done to improve the convergence are reported. Some results are also reported in order to test the accuracy and convergence of the method.

The multi-pipe junction is then reported and the sonic flow control together with an improved “datum” duct are reported. Two examples for a “Y” junction are reported in order to show the behavior of such method in configurations similar to those found in exhaust systems of engines. Finally the conclusions are reported.

2. PIPE BOUNDARIES

To manage the pipe boundaries, a first order model is suggested and the method of characteristics has been used. Derivation of this model is well reported in [1,3,4,] and here it is briefly revised.

The compatibility equations along the characteristics are (figure 1):

Figure 1 Characteristics lines on the space-time plane

; (1)

Along the characteristics ;

; (2)

Along the characteristics .

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Where the values of Δ1 , Δ2 , Δ3 are the source terms:

(3)

(4)

(5)

Defining the dimensionless quantities:

A= aaref ;

U= uaref ;

AA=a A

aref (6)

the Riemann variables can be defined as:

λ=A+ γ−12

⋅U;

β=A− γ−12

⋅U(7)

And the updated values at time step “t+1” are:

λ t+1= λLt +dλ ; β t+1=βL

t +dβ (8)

Where dλ and dβ take account of pipe area variation, friction heat transfer and entropy variation.

At pipe boundaries is common to name the Riemann variables as follows:

λ in=A+ γ−12

⋅U λout=A− γ−12

⋅U(9)

In every boundary problem, the values to calculate are λ in ,

λout and the entropy level AA and all the thermodynamic parameters can

be then calculated.

3. OUTFLOW

In the present work the “Benson Constant Pressure” [1] is revised in order to improve the convergence. This method is reported and discussed in [1] and their limits have been also discussed in [10].

The outflow is quite common and the cylinder exhaust flow is an example (figure 2): when the exhaust valve get opened, the cylinder pressure is much higher than the manifold pressure so the flow if from cylinder to manifold.

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Figure 2example of Outflow (flow from cylinder to manifold)

The governing equations of such boundary conditions are:

Conservation of energy between cylinder and throttle:; Conservation of energy between cylinder and manifold; Isentropic expansion between cylinder and throttle; Continuity equation; Constant pressure (No pressure recovery and then no momentum equation) from throttle to manifold; A Characteristic equation;

the unknown are 6: 3 for the throttle and 3 for the manifold; Benson in [1] solved the equations in terms of Riemann variables.

In the present work some improvements have been made in order to improve the stability of the method: the extreme values of the

entropy levels have set up before the iterative cycle, and a sub cycle has been added in order to improve the calculation of λ in

c and

λout .

The minimum and maximum allowable values of the entropy level Aa

c are when there is no flow or the flow is sonic at the port.

When there is no flow the entropy level is the entropy level of the cylinder:

Aa¿=Aa¿

=Ac⋅( Pref

Pc)

γ−12⋅γ

(10)

For sonic flow the critical sound speed ratio is defined:

Acr=Ac⋅√ 2γ+1 (11)

And then the following relation is valid:

Acr=U cr→λin

c+λout

2=

λout−λ inc

(γ−1 )

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From [1] the pressure ration between Port and cylinder is :

P port

Pc

=φ⋅( 2γ+1 )

γγ−1

The entropy level is defined by

Aac=

λinc+λout

2⋅( Pref

Pport)

γ−12⋅γ

so the maximum entropy level value is:

Aa¿=Acr⋅(P ref

Pc

⋅ 1

φ⋅( 2γ+1 )

γγ−1 )

γ−12⋅γ

(12)

The maximum and minimum allowable values of Aa¿ are then estimated; the following initial values for the unknown

Aa¿ are assumed:

Aa¿=Aa¿ (13)

ΔA ac=

( Aa¿−Aa¿

)2 (14)

And the iteration process can start.

The λ in

c and λout values can be calculated:

λ inc=2⋅λ in

n⋅

Aa¿

Aa¿+ Aa

n

+ λout⋅Aa¿

−Aan

Aa¿+ Aa

n (15)

λout=3−γγ+1

⋅λinc + 2

γ+1⋅√ (γ 2−1 )⋅Ac

2+2⋅(1−γ )⋅λin

2c

(16)

λ inc is function of

λout and Aa

c , but also and λout is function of

λ inc and

Aa¿ . If an sub-iterative cycle is made for the

equations (15) and (16), the two values are now only function of Aa

c , that is the only unknown of the main iterative cycle. The sub

iteration ends when two consecutive values of λout are within allowable value: 1/1000 of the of the main iterative cycle (10-6) and

then 10-9.

With the calculated λ in

c and λout , the following dimensionless speed value can be evaluated:

U =λout− λin

c

( γ−1 )⋅Ac (17)

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Then the coefficient:

C=

( γ−1 )2

⋅U2

(1-(γ−1 )

2⋅U2)

2

(18)

If the squared value of the area ratio is

φ2≤ 4⋅C

γ 2−1 (19)

then the flow at the throat is sonic and the pressure ratio between port and cylinder is defined by:

P port

Pc

=max [φ⋅( 2γ+1 )

γγ −1 ;

φU

⋅( 2γ +1 )

γ +12⋅( γ−1 )⋅(1− (γ−1 )

2⋅U2)]

(20)

In which the term

φ⋅( 2γ +1 )

γγ−1

calculate the value for sonic flow also at the port.

If the flow is subsonic then the pressure ratio is

:

P port

Pc

= 12⋅C

⋅(φ⋅√φ2+4⋅C−φ2 )(21)

The new value of the corrected entropy level can be calculated:

Aa¿=

λinc+λout

2⋅( Pref

Pc

⋅Pc

Pport)

γ−12⋅γ

(22)

If the absolute difference|Aa¿

−Aa¿|>ε

a new Aa¿ have to be calculated:

{ Aa¿< Aa¿

→ Aa¿=Aa¿

+ΔAac¿ ¿¿¿

(23)

ΔA ac=

ΔAac

2 (24)

Equations from (10) to (14) calculate the initial values for the iterative cycle, while those from (15) to (24) are the main iterative cycle

with Aa

c as unknown.

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An example to test the accuracy and the convergence of the outflow model has been done and reported in this section.

A 1 meter duct, with a discretization step of 10 mm has on the right boundary a close end while at left different area ratios have been tested with an external pressure of 500000 Pa and the results are showed after a time of 0.001 seconds. The duct model used in this test is the Two Step Lax-Wendroff with TVD. The thermodynamic initials conditions are: pressure 101325 Pa and temperature 293.15 K. The area ratios tested vary from 1 to 10-6 and convergence has been obtained when the difference between two consecutive values

of Aa

c were below 10-6.

Figure 3Pressure behavior within the duct after 1000 s (left) and number of iterations to convergence at left boundary (right) for different area ratios.

In figure 3, at left the pressure along the duct is depicted and at right the average iterations needed to reach convergence for each area ratio during the calculations. Decreasing the area ratios the pressure at the left end of the manifold decreases and for the lowest values, the pressure does not vary considerably in respect of the initial conditions. The convergence is always obtained and the number of iterations increase decreasing the area ratio and in the figure 3 on the right they vary from 2 for area ratio near 1 to 42 for a value of area ratio of 10-6.

4. MULTI-PIPE JUNCTION

The multi-pipe junction is quite common in internal combustion engines and many works to model such boundary have been made during the years. The first “Constant Pressure” model has been reported by Benson in [1] where the hypothesis that all the pipes connected with the junctions have the same pressure [1]; a new method which uses a different calculation of the entropy level has been developed by Corberan in [7] while the general model that takes account for the pressure drop for flows away from the junction (separating flows) is well reported by Bassett et all. in [11].

In this section the sonic case has been added and a “equivalent datum” manifold has been defined with the following characteristics:

with the equivalent flow as the sum of all the flows (joining flows), with an equivalent stagnation enthalpy, density and speed that are averaged and weighted with the incoming flow-ratio; an equivalent area that is the ratio between the datum flow and the product of datum density and speed.

For a junction with n ducts, the unknowns are 3n: the unknown are 3 for every duct: the corrected incident lambdaλ in

¿c

, the reflected

λout¿

and the entropy level AAc .

For all the pipes with flow towards the junction, the incident lambda λ in

¿c=λin

¿n

and also AAc=A An

so only an unknown has to be

calculated, while for pipes with flows away from the junction (separating flows) the unknowns are 3.

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The method to calculate the boundary conditions at the ends of the pipes connected with a junction is well reported in [1] where a constant pressure for all the pipes model has been used by Benson and [11] where a pressure loss term has been defined for flows away from the junction (separating flows). The method is iterative using a predictor-corrector strategy.

In the present paper the pressure loss model [11] have been improved using a “equivalent” datum duct in case of more than one pipes have the flow towards the junction and taking into account of choked flow if occurred in any of the pipes of the junction.

The first equation, the continuity equation, is used to calculate the sound speed A j

¿

(after the calculation of the pressure loss terms),

then the entropy levels AAc

¿

are calculated. And finally the corrected lambda values λ in

¿c

can be calculated. The above calculation can

be iterated until two following values of A j

¿

are within a little error (10-6).

4.1.ENTROPY LEVEL

The entropy levels are calculated in the same mode as [11] and [7].

For joining flows there is not correction so:

AAcj=A An

j (25)

Where for separating flows, to calculate the corrected entropy level the energy equation is applied.

First the starred dimensionless speed value is calculated:

U j¿= 2

γ−1⋅( λ j

¿−A j¿ )

(26)

Then the stagnation enthalpy is defined:

h0datum

=∑i=1

i=Nj

m¿

i⋅h0i

∑i=1

i=Nj

m¿

i

=∑i=1

i=Nj {m¿ i⋅aref

2

γ−1⋅(A i

¿2+ γ−1

2⋅U i

¿2)}∑i=1

i=Nj

m¿

i(27)

The entropy levels are then:

AAcj=√ h0

datum

aref2

γ−1⋅(A j

¿2+ γ−1

2⋅U j

¿2)(28).

4.2.RIEMANN VARIABLES

The values of corrected λcj

¿

variables are:

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λcj¿ =

λn j

A Acj

+ A j¿¿(1− AAn j

AAc j)

(29)

The equations from 25, that is function of λcj

¿

, calculated in equation (29) can also be iterated in way to improve the convergence of

the model. This iteration from equation (25) to (29) can be terminated when two following values of λcj

¿

have a relative difference less than 10-6.

The calculation is concluded with the equations (29) and (30) with the λ in

j and λout

j together with AAc

j calculated within the iteration process.

λ inj= λc j

¿ ¿ A Ac j(30)

λoutj=2⋅A j

¿⋅A Acj−λc j

¿

(31)

4.3.PRESSURE LOSS TERMS

The calculation of the pressure loss terms starts with finding the direction of the flow. This can be calculated using the flow rate equation:

m¿

j=A j

¿

2γ−1

A A j

¿γ⋅P ref

aref

¿U j¿¿ F j

(32)

The “datum duct” properties can be now evaluated:

the “datum” incoming flow:m¿

datum= ∑i=1

i=N m >0

m¿

i(33);

The equivalent “datum duct” density and speed:

ρdatum=∑i=1

i=N m> 0

m¿

i⋅ρ i

m¿

datum

Udatum=∑i=1

i=N m> 0

m¿

i⋅U i

m¿

datum (34);

and the equivalent datum duct area Fdatum is:

Fdatum=m¿

datum

ρdatum⋅Udatum (35)

Depending on the flow direction, the calculation of Δ j

¿

has a different approach.

Case m¿

>0

When m¿

>0 the flow is towards the junction; a (sonic) critical value of A j

¿

can be calculated and it has the following value:

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Acr j

¿ =λ j

¿c +3−γγ+1

⋅λ j¿c

2 (36)

If Adatum

¿ > Acr j

¿

the flow in the jth pipe is subsonic and the

Δ j¿

is null, else the Δ j

¿

value is the one that give a value of A j¿=Acr

¿j

as follow:

Δ j¿=¿ {Adatum

¿ < Acr j

¿ →Adat¿ −Acr j

¿¿ ¿¿¿¿

(37)

or

Δ j¿=min( Adatum

¿ −Acr j

¿ ,0) (38)

In this way the pressure loss terms take into account of sonic flow condition and does not allow supersonic flow in the ducts with flow towards the junction.

Case m¿

<0

The pressure loss terms are calculated in the equation (39) as an averaged value between all the joining flows:

C j=∑i=1

i=Nj

m¿

i⋅C (i , j )

m¿

datum (39)

Where the index “i” is referred to the ith duct with flow towards the junction and with the index “j” the

jth duct to calculate the Δ j

¿

value.

Two pressure loss coefficients used in this work are [11]:

C ( i , j )=1− 1q j⋅ψ j

⋅cos [ 34⋅( π−θ (i , j ) )]

(40)

Where:

q j=m¿

j

m¿

datum is the mass flow ratio and

ψ j=Fdatum

F j is the area ratio.

A second correlation from [12] can be also used:

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C (i , j )=¿ {θ (i , j )<167→1. 6−1. 6⋅θ (i , j )167

¿¿¿¿(41)

Where θ is the angle between the ducts.

The Δ0 j

¿

, before any check to avoid unreal values is:

Δ0 j¿ =A j

¿⋅{[1+γ⋅C j⋅( 2γ−1 ( λ j c

¿ −A j¿ )

A j¿ )

2

](γ−1)

2

−1}(42).

There are some checks to do in this equation in order to avoid unreal values:

The first control is that the value

1+γ⋅C j⋅( 2γ−1

⋅(λ j¿−A j

¿ )A j

¿ )2

≥0 so

C j must be greater than:

C j≥−1

γ⋅( 2γ−1

⋅( λ j¿−A j

¿ )A j

¿ )2

(43)

The second condition is that the flow must not be supersonic so the Acr j

¿

value is defined

Acr j

¿ =λ j

¿+ γ +13−γ

⋅λ j¿

2 (44)

And the condition is

Adatum¿ −Δ j

¿ <= Acr j

¿

The third condition is the flow cannot become toward the junction (invert the direction):

Adatum¿ −Δ j

¿≥λ j

¿c

The last two conditions can be expressed in the following form:

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Δ j¿=max ( Adatum

¿ −Acr j

¿ ,min ( Adatum¿ −λ j

¿ c , Δ0 j¿ )) (45).

4.4.CONTINUITY EQUATION

In the iterative method the first equation to solve is the continuity equation [11]:

∑j=1

j=N [A j¿

2γ−1

⋅(λ j¿c−A j

¿)¿F j

A Ac

]=0(46)

A “datum” value is defined as a unique duct with the following characteristics:

With a flow rate as the sum of all the flow rates entering the duct; With an equivalent Area; With an equivalent stagnation Enthalpy

If a “datum” value is defined ,the starred dimensionless sound speed of each duct can be expressed in the form:

A j¿=Adatum

¿ −Δ j¿

(47)

WhereΔ j

¿

is the pressure loss term between the datum duct and the jth duct.

The continuity equation has then the following form:

∑j=1

j=N [ ( Adatum¿ −Δ j

¿)2

γ−1

⋅( λ j¿c−( Adatum

¿ −Δ j¿ ))¿

F j

A Ac

]=0(48)

that can be solved in a iterative way with Adatum

¿

as unknown having as input the pressure losses Δ j

¿

, the corrected lambda valuesλ j

¿c

and the entropy levelsAAc in addition to the pipe area

F j and the specific heat ratio . In the following section the pressure loss terms are explained.

4.5.SOLVING METHOD

To solve the multi-pipe junction model a predictor –corrector solver has been used. The unknown of the iterative cycle is the vector of

A j¿

and the initial values of the cycle can be for example those from “Benson” Constant Pressure method and the steps to solve the model are:

1. With equations from (25) to (28) the corrected values of the entropy levels are calculated;

2. with equation (29) the values of λ j

¿c

are calculated;

3. equations from (32) to (45) to calculate the pressure loss terms Δ j

¿

;

4. solve the continuity equation (48) for the unknown Adatum

¿New

;

5. calculate the sound speeds A j

¿

using equation (47);

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6. the steps 1 to 5 are repeated until |Adatum

¿New −Adatum

¿ |<ε.;

7. calculateλ in

j and λout

j from equation (30) and (31).

4.6.EXAMPLES

Below are reported two cases with sonic flows in the joining ducts (the first) and in the separating flows (the second): the results have been compared with those obtained without taking account for sonic flow in way to appreciate the differences.

Example 1

A three pipe “Y” junction have been simulated in the example 1 an example with sonic flow in the joining duct is reported. The pipes and junction details are reported in figure 4.

90°

135°

135°

P=500000 D=0.05

D=0.05P=100000

P=100000

D=0.05

13

2

Figure 4 Overview of the 3 pipe junction of the example 1

Three ducts with equal diameter of 0.05 m and length of 0.5 m are connected with a an angle of 135 degrees for the duct 1 with ducts 2 and 3 and 90 degrees between 2 and 3. At the left end of the duct 1 there is a pressure of 500000 Pa and 593.15 K of temperature and the right boundary of ducts 2 and 3 the pressure is 100000 Pa and 293.15 K while within the ducts the initial conditions are 100000 Pa and 293.15 K of pressure and temperature respectively. The duct model used in this test is the Two Step Lax-Wendroff with TVD and the discretization step is 0.005 m.

The results are showed after 1200 s. In this case there is only one incoming and two outcoming flows and the “datum” manifold is exactly the duct 1. There is symmetry for ducts 2 and 3 so they show the same behavior.

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Figure 5 Pressure and Mach number for example 1 without sonic boundary equations

In figure 5 the values of pressure and Mach number along the ducts are showed the case in which the sonic flow is not taken into account. After 0.5 m, where the junction is located, a supersonic flow can be observed in the duct. the compression wave arriving to the junction is reflected and a expansion wave is produced, in a similar way of the sudden area enlargement. In figure 5 a supersonic flow can be observed and the pressure behavior is reported.

Higher pressures at the boundary of duct 1 have been tested and increasing such value the Mach number grows and for pressure greater than 700000 Pa the calculation crashes. These conditions are possible in the exhaust system of an internal combustion engine.

Figure 6 Pressure and Mach number along the ducts with the improvements of the present work.

In figure 6 the sonic boundary has been added during calculations and as result, the gas speed is limited to be not greater than the sound speed (figure 6 at left) so the duct 1 show a different behaviour than figure 5:the Mach number at junction is limited to 1 and the pressure value is higher than the previous figure. Ducts 2 and 3 are not much influenced by the modifications.

Increasing the Pressure at the left boundary of duct 1 does not reach to a crash during calculations and the method is much efficient in calculating the solution for multi-pipe boundary.

Example 2

In the example 2 a sonic flow in the separating duct is reproduced. The pipes and junction details are reported in figure 7.

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90°

135°

135°

P=100000D=0.025

D=0.05

P=500000

P=500000

D=0.05

13

2

Figure 7 Overview of the 3 pipe junction of the example 2

The “Y” junction is similar to that of the previous example, but rotated and the duct 3 has a diameter of 0.025 instead of 0.05. the initial conditions (temperature and pressure) are the same (100000 Pa and 293.15 K), but the boundary temperature for duct 1 and 2 are 500000 Pa and 593.15 K and the results are showed after 1200 s.

In this case there are 2 incoming and only one outcoming flows. The “datum” manifold has a total flow as the sum of the flows of duct 1 and 2 and a “datum” area is the sum of the two duct’s area.

With these initial conditions the duct1 and 2 see the junction like a sudden contraction and a compression wave is reflected. A higher value of pressure is found in these duct and a supersonic flow is observed for duct 3 as shown in figure 8.

Figure 8 Pressure and Mach number for example 2 without boundary equations

At the distance of 0.5 m, where the junction is located, the pressure and Mach number discontinuity are found. The pressure in the duct 1 and 2 grows from about 3.5 bar to 8.5 near the junction; in the duct 3 the pressure grows from 100000 up to 6 bar. In ducts 1 and 2 the Mach number decreases from the left boundary to the junction from 0.75 to 0.1 and in the duct 3 from 1.1-1.2 to 0.

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Figure 9 Pressure and Mach number for example 2 with the improvements of the present work.

In figure 9, the sonic boundary has been added and the pressure Mach number curve are quite different from the previous figure: the pressure peak in the ducts 1 and 2 is higher while for duct3 is lower. The Mach number at the junction in the duct 3 is limited to 1 and consequently the ach number in duct 3 is always lower than figure 8Figure 8.

This configuration, together with that of example 1 are quite common in internal combustion engines and sonic flows are possible in the exhaust systems of engines and a better accuracy can be achieved if a computer code is able to take into account for shocks in pipe junctions.

5. CONCLUSIONS

In this work the pipe boundaries, the outflow and the multi-pipe boundaries have been analyzed and improved in order to increase the accuracy of the calculation. In the Benson Constant Pressure model, better values of entropy level have been calculated and a

subcycle has been added in order to better calculate the λ in

c and λout values. In this way the convergence is obtained also for very

little area ratios (10-6) and in the tests reported in this paper the number of iterations to reach convergence never exceeds 50.

In The multi-pipe junction, to calculate the pressure drop for flows away from the junction a “datum” duct has been defined. The datum duct has an incoming flow as the sum of all the incoming flows and an average density, speed and area weighted with the flowratio of the ith duct supplying flow to the junction. In unsteady gas flows the gas cannot go to a speed greater than the perturbation that caused its flow: the perturbations have a speed that is equal to the sound speed so the gas cannot go to a speed greater than sound speed. This limitation has been added in the calculation and these modifications to the multi-pipe junction model developed by [11] have been tested and compared on two case tests. The first test analyzed the sonic flow in the incoming flows while the second in the outcoming on a “Y” junction and the results have been showed and discussed.

REFERENCES

1. Rowland S. Benson; edited by J.H. Horlock and D.E. Winterbone. "The thermodynamics and gas dynamics of internal-combustion engines", 1982 Clarendon Press

2. F. Ortenzi, "La Simulazione Dei Motori A 4 Tempi: Le Caratteristiche Di Un Apposito Software",Mototecnica, Anno 19, N°2 - Febbraio 2005

3. Ll. Gascón and J. A. García, "About the TVD property for the flux-corrected transport techniques", Computational Mechanics, 2002, Volume 30, Number 4, Pages 281-285

4. D. E. Winterbone, R. J. Pearson, “Theory of Engine Manifold Design”, Professional Engineering Pub., 2000

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5. J.R. Serrano, F.J. Arnau, P. Piqueras, M.A. Reyes-Belmonte, "Assessment of a methodology to mesh the spatial domain in the proximity of the boundary conditions for one-dimensional gas dynamic calculation", Mathematical and Computer Modelling 54 (2011) 1747–1752

6. T. Katrasnik, "Improved model to determine turbine and compressor boundary conditions with the method of characteristics", International Journal of Mechanical Sciences 48 (2006) 504–516

7. Corberán, J.M., "A new constant pressure junction model for N-branch junctions", 1992, Proc. I. Mech.E., Vol. 206, Part D, pp.117-123

8. J.M. Desantes, J.R. Serrano, F.J. Arnau, P. Piqueras, "Derivation of the method of characteristics for the fluid dynamic solution of flow advection along porous wall channels", Applied Mathematical Modelling, Volume 36, Issue 7, July 2012, Pages 3134–3152

9. A. Onorati, G. Ferrari and G. D’Errico, "Fluid Dynamic Modeling of the Gas Flow with Chemical Specie Transport through the Exhaust Manifold of a Four Cylinder SI Engine", SAE Paper 1999-01-0557

10. P Brejaud, P Higelin, A Charlet and Y Chamaillard, "Development and experimental validation of a new one-dimensional valve boundary condition based on the method of characteristics", Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 2011 225: 1614

11. M. D. Bassett, R. J. Pearson, N. P. Fleming, D. E. Winterbone, "A Multi-Pipe Junction Model for One-Dimensional Gas-Dynamic Simulations", SAE 2003-01-0370

12. G.P. Blair, “Design and Simulation of Four-Stroke Engines”, 1999 SAE International, R-186

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improvements of boundary condition models for 1d engine thermodynamic and gas dynamic simulations

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