pipeline simulation by the method of characteristics for...

The 11th International Fluid Power Conference, 11. IFK, March 19-21, 2018, Aachen, Germany

Figure 1: The pump design of URACA GmbH&Co.KG

Pumps of this type typically consist of a drive unit and a pumped media unit. To prevent mixing of the lubricating liquid with the pumped medium, these two units are completely separated. The drive unit assembly is used to transfer the energy of the connected drive. The rotation is converted into an oscillating plunger motion. The plungers then transfer energy to the pumped medium, moving it from the suction space to the high-pressure space.

Switching between the space environment suction and discharge is performed by the valve block.

Figure 2: Pump valves (URACA GmbH&Co.KG)

Figure 2 shows the phases of operation of the plunger and valves. (1) shows the front position of the plunger when both valves are closed. From this position, the suction process (2) begins. Reducing the pressure in the plunger chamber leads to the opening of the suction valve, which enables the medium to enter the working chamber. At position (3), the suction process is complete. The pressure in the working chamber is equal to the pressure in the suction space, and the suction valve is closed. During the discharge stroke (4), the pressure in the working chamber is increased. The discharge valve opens and lets the medium pass from the working chamber to the pressure space.


Pipeline simulation by the method of characteristics for calculating the pressure pulsation of a high-pressure water plunger pump

Dr.-Ing.(Rus) Maxim Andreev*, Dipl.-Ing. Uwe Grätz* and Dipl.-Ing. (FH) Achim Lamparter**

ESI ITI GmbH, Schweriner Str. 1, 01067 Dresden, Germany* URACA GmbH & Co. KG, Sirchinger Str. 15, 72574 Bad Urach, Germany**

E-Mail: [email protected], [email protected], [email protected]

The article describes ways to adapt the method of characteristics to solving the problem of pressure pulsation calculation of a high-pressure plunger water pump considering a complex pipeline network using a CAE software “SimulationX”. The objective of this adaptation is to increase the stability of the numerical solution and reduce the calculation time. To verify the accuracy of the simulation, the pressure pulsations were compared with pulsations in various parts of a real complex pipeline. As a result, a compromise between accuracy and speed of calculations was achieved, which improves the process of pump development.

Keywords: Fluid power networks, digitalization, connectivity, communication Target audience: Industrial Hydraulics, Simulation, Design Process

1 Introduction

Every engineer involved in simulating technical systems faces the task of selecting appropriate modelling assumptions. The current state of computer technology and software allows modelling systems of almost any complexity with a very high degree of accuracy, achieving a perfect match with experiment. However, the solution cannot always be obtained quickly; at times, the calculation time can be up to weeks or even months, especially when it comes to optimization problems.

This problem is especially relevant for engineers involved in the simulation of hydraulic systems. The methods of Computational Fluid Dynamics (CFD) and methods of 1D-simulation based on Ordinary Differential Equations (ODE) can be applied. CFD achieves high computational accuracy but requiring a long computational time. In the case of 1D-Simulation, strong assumptions must be introduced, but it allows you to quickly obtain a result.

Thus, any mathematical model is always a compromise between accuracy and calculation time, and the art of simulation consists of choosing the optimal path to achieve this compromise. This article describes the process of achieving this compromise in solving the problem of pressure pulsation calculation in a high-pressure plunger water pump while considering the effect of a complex pipeline network.

2 Plunger pump as a source of pulsations

The object of study – a system with integrated three plunger high-pressure water pump is shown in Figure 1.

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As can be seen, the theoretical volumetric pump flow is essentially non-linear and contains discontinuities. This fact makes the analysis of pressure fluctuations in the pipeline in the frequency domain problematic and makes the calculation in the time domain preferable.

3 One-dimensional model of wave propagation in pipelines and methods of numerical solution

To simulate a pressure pipeline, the following assumptions are made:

• the pipeline is filled with liquid

• the cross section of the pipeline is constant and does not vary with pressure

• for calculations, the velocity averaged over the cross section of the pipe is used and the influence of the nonuniformity of the velocity distribution is neglected

• heat exchange between liquid, pipe walls and the environment is not considered

The mathematical model under such assumptions can be based on the one-dimensional (𝑥𝑥 - length coordinate) Euler equations:

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 + 𝜕𝜕(𝜕𝜕𝜌𝜌)

𝜕𝜕𝑥𝑥 = 0

𝜕𝜕(𝜕𝜕𝜌𝜌)𝜕𝜕𝜕𝜕 + 𝜕𝜕(𝜕𝜕𝜌𝜌2 + 𝑝𝑝)

𝜕𝜕𝑥𝑥 = 𝜕𝜕 (𝐹𝐹 + 𝐺𝐺)

where 𝜕𝜕 – fluid density, 𝜌𝜌 – flow velocity, 𝑝𝑝 – pressure, 𝐹𝐹 – friction loss, 𝐺𝐺 - gravity pressure drop.

For convenience of further calculations, the equations are rewritten in primitive variables 𝑝𝑝 and 𝜌𝜌 /2/:

𝜕𝜕𝑝𝑝𝜕𝜕𝜕𝜕 + 𝜌𝜌 𝜕𝜕𝑝𝑝

𝜕𝜕𝑥𝑥 + 𝜕𝜕 ∙ 𝑐𝑐2 𝜕𝜕𝜌𝜌𝜕𝜕𝑥𝑥 = 0

𝜕𝜕𝜌𝜌𝜕𝜕𝜕𝜕 + 1

𝜕𝜕𝜕𝜕𝑝𝑝𝜕𝜕𝑥𝑥 + 𝜌𝜌 𝜕𝜕𝜌𝜌

𝜕𝜕𝑥𝑥 = 𝐹𝐹 + 𝐺𝐺

Where 𝑐𝑐 – sound speed.

In the case of a liquid, as a rule, the expression 𝜌𝜌 ≪ 𝑐𝑐 holds. In this case, the equations take the final form:

𝜕𝜕𝑝𝑝𝜕𝜕𝜕𝜕 + 𝜕𝜕 ∙ 𝑐𝑐2 𝜕𝜕𝜌𝜌

𝜕𝜕𝑥𝑥 = 0

𝜕𝜕𝜌𝜌𝜕𝜕𝜕𝜕 + 1

𝜕𝜕𝜕𝜕𝑝𝑝𝜕𝜕𝑥𝑥 = 𝐹𝐹 + 𝐺𝐺

(1)

Differential equations in partial derivatives (1) must be supplemented by algebraic equations that describe the properties of a fluid. Density is calculated as a function of pressure 𝑝𝑝, temperature 𝑇𝑇 and the volume fraction of undissolved gas 𝛼𝛼𝑈𝑈.

𝜕𝜕 = 𝜕𝜕(𝑝𝑝, 𝑇𝑇, 𝛼𝛼𝑈𝑈)

Similarly, the calculation of the equivalent bulk modulus 𝐸𝐸𝑒𝑒 is needed for the calculation of 𝑐𝑐. Finally, 𝑐𝑐 is calculated with the equation:

𝑐𝑐 = √𝐸𝐸𝑒𝑒𝜕𝜕

The equivalent bulk modulus 𝐸𝐸𝑒𝑒 is a measure of the equivalent fluid compressibility (including the wall elasticity).


The following factors have a significant effect on the flow pulsation of the pump \1\:

• a finite number of plungers

• nonlinearity due to the kinematics of the mechanism

• delays in the valve actuation due to the compressibility of the medium.

Figure 3 shows an example of the ideal volume flow of a three-plunger pump as a function of crankshaft angle.

Figure 3: Volume flow of each plunger (Q1-3) and total volume flow (Qsum) as a function of crankshaft angle (without delays in the valve actuation)

In this case, there is no delay in opening the valve, which is typical for a pump without external load. The pulsation of the volume flow is exclusively due to a finite number of pistons and the nonlinearity of the kinematics of the mechanism.

When the load pressure is increased, there is no volume flow up to a certain angle of rotation of the crankshaft until the pressure valve opens. An example of the calculation of the flow for this case is shown in Figure 4.

Figure 4: Volume flow of each plunger (Q1-3) and total volume flow (Qsum) as a function of crankshaft angle (with delays in the valve actuation)

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At first, it is necessary to transform the equations into suitable ordinary differential equations, referred to as compatibility equations on characteristics /4/:

• compatibility equation on C+ characteristics:

𝑑𝑑𝑑𝑑 + 1𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 − (𝐹𝐹 + 𝐺𝐺)𝑑𝑑𝑑𝑑 = 0

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = +𝜌𝜌

• compatibility equation on C− characteristics:

𝑑𝑑𝑑𝑑 − 1𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑 − (𝐹𝐹 + 𝐺𝐺)𝑑𝑑𝑑𝑑 = 0

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = −𝜌𝜌

Along curves C+ and C− the set of partial differential equations become ordinary differential equations. These characteristics are shown in Figure 7.

Figure 7: Lines of the characteristics

Characteristics cross the line 𝑑𝑑𝑗𝑗 at the points 𝜌𝜌 and 𝑓𝑓. Pressure 𝑑𝑑𝑐𝑐, 𝑑𝑑𝑓𝑓 and velocity 𝑑𝑑𝑐𝑐, 𝑑𝑑𝑓𝑓 at points 𝜌𝜌 and 𝑓𝑓 are determined by linear interpolation.

Figure 8: Interpolation of the values at the points c and f.


Thus, the problem reduces to solving hyperbolic systems of partial differential equations (1). Among the many available methods to solve this problem, three main approaches are highlighted:

Discretization of the equation for the finite control volume. In this case, the pipe volume is divided into a finite number of volumes with the pressure and velocity calculated at the centres and boundaries respectively. Partial derivatives along the length of the tube are transformed into finite differences. Thus the system of equations becomes a system of ordinary differential equations. The advantage of this approach is the ease of integration into an ordinary differential equations (ODE) solver. There are no additional restrictions on the time step. The disadvantage is the numerical oscillations that arise when calculating the propagation of shock waves /3/, making this approach unsuitable for solving this problem.

The method of characteristics is based on the idea that the independent variables 𝑡𝑡 (time) and 𝑥𝑥 (coordinate) are related by the equation of characteristics. Along these characteristics the PDE becomes an ordinary differential equation. Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. The disadvantage of this approach is the restriction of the time step by the Courant–Friedrichs–Lewy condition. Nevertheless, this method allows for solving the problem of shock wave propagation in pipelines, if there is not much difference in the density and speed of sound in neighbouring cells /3/.

The methods based on Godunov‘s scheme are based on the solution of the Riemann problem for each inter-cell boundary. This scheme allows for solving the problem of shock wave propagation of almost any complexity, but it is usually more computationally time-consuming /3/.

As a compromise between the simplicity and accuracy of the solution, the method of characteristics was chosen.

4 Description of the simplified method of characteristics

At time 𝑡𝑡𝑗𝑗 the values of the pressure 𝑝𝑝𝑖𝑖,𝑗𝑗 and velocity 𝑣𝑣𝑖𝑖,𝑗𝑗 at points 𝑥𝑥𝑖𝑖 are known.

Figure 5: General case of the initial conditions

The values of these parameters for time 𝑡𝑡𝑗𝑗+1 = 𝑡𝑡𝑗𝑗 + Δ𝑡𝑡 need to be calculated.

Figure 6: Pipe at the time step 𝑡𝑡𝑗𝑗+1

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Figure 9: Connection of the results of the solver of the method of characteristics (blue) and ODE-Solver (red)

In case of variable density and speed of sound, the equations take the form /3/:

(𝑣𝑣𝑖𝑖,𝑗𝑗+1 − 𝑣𝑣𝑐𝑐) +1

𝜌𝜌�̅�𝑐 ∙ 𝑐𝑐�̅�𝑐(𝑝𝑝𝑖𝑖,𝑗𝑗+1 − 𝑝𝑝𝑐𝑐) − (𝐹𝐹𝑐𝑐 + 𝐺𝐺)Δ𝑡𝑡 = 0

(𝑣𝑣𝑖𝑖,𝑗𝑗+1 − 𝑣𝑣𝑓𝑓) −1

𝜌𝜌𝑓𝑓̅̅ ̅ ∙ 𝑐𝑐�̅�𝑓(𝑝𝑝𝑖𝑖,𝑗𝑗+1 − 𝑝𝑝𝑓𝑓) − (𝐹𝐹𝑓𝑓 + 𝐺𝐺)Δ𝑡𝑡 = 0

where:

𝜌𝜌𝑐𝑐(𝑓𝑓)̅̅ ̅̅ ̅̅ =𝜌𝜌𝑐𝑐(𝑓𝑓),𝑗𝑗+1 + 𝜌𝜌𝑐𝑐(𝑓𝑓)

2

𝑐𝑐𝑐𝑐(𝑓𝑓)̅̅ ̅̅ ̅̅ =𝑐𝑐𝑐𝑐(𝑓𝑓),𝑗𝑗+1 + 𝑐𝑐𝑐𝑐(𝑓𝑓)

2

In these cases, the system of equations can be solved only by iterative methods. The following assumptions can significantly speed up the calculations:

𝜌𝜌𝑐𝑐(𝑓𝑓)̅̅ ̅̅ ̅̅ = 𝜌𝜌𝑀𝑀, 𝑐𝑐𝑐𝑐(𝑓𝑓)̅̅ ̅̅ ̅̅ = 𝑐𝑐𝑀𝑀, 𝜈𝜈𝑐𝑐(𝑓𝑓)̅̅ ̅̅ ̅̅ = 𝜈𝜈𝑀𝑀

where 𝜌𝜌𝑀𝑀, 𝑐𝑐𝑀𝑀, 𝜈𝜈𝑀𝑀 are average values of density, velocity of sound, and viscosity calculated at each time step using the fluid properties:

𝜌𝜌𝑀𝑀 = 𝜌𝜌(𝑝𝑝𝑀𝑀, 𝑇𝑇𝑀𝑀, 𝛼𝛼𝑈𝑈𝑀𝑀)

𝑐𝑐𝑀𝑀 = 𝑐𝑐(𝑝𝑝𝑀𝑀, 𝑇𝑇𝑀𝑀, 𝛼𝛼𝑈𝑈𝑀𝑀)

𝜈𝜈𝑀𝑀 = 𝜈𝜈(𝑝𝑝𝑀𝑀, 𝑇𝑇𝑀𝑀, 𝛼𝛼𝑈𝑈𝑀𝑀)

where 𝑝𝑝𝑀𝑀, 𝑇𝑇𝑀𝑀, 𝛼𝛼𝑈𝑈𝑀𝑀 are average values of the pressure, temperature and gas fraction.

𝑝𝑝𝑀𝑀 =∑ 𝑝𝑝𝑖𝑖𝑛𝑛+1𝑖𝑖=1𝑛𝑛 + 1

𝑇𝑇𝑀𝑀 = 𝑇𝑇𝐴𝐴 + 𝑇𝑇𝐵𝐵2

𝛼𝛼𝑈𝑈𝑀𝑀 = 𝛼𝛼𝑈𝑈𝐴𝐴 + 𝛼𝛼𝑈𝑈𝐴𝐴2

Due to the conversion of the sound velocity at the end of each step, a new calculation of the time step Δ𝑡𝑡 is performed.


Now, the difference scheme can be written:

(𝑣𝑣𝑖𝑖,𝑗𝑗+1 − 𝑣𝑣𝑐𝑐) + 1𝜌𝜌𝜌𝜌 (𝑝𝑝𝑖𝑖,𝑗𝑗+1 − 𝑝𝑝𝑐𝑐) − (𝐹𝐹𝑐𝑐 + 𝐺𝐺)Δ𝑡𝑡 = 0

(𝑣𝑣𝑖𝑖,𝑗𝑗+1 − 𝑣𝑣𝑓𝑓) − 1𝜌𝜌𝜌𝜌 (𝑝𝑝𝑖𝑖,𝑗𝑗+1 − 𝑝𝑝𝑓𝑓) − (𝐹𝐹𝑓𝑓 + 𝐺𝐺)Δ𝑡𝑡 = 0

where 𝐹𝐹𝑐𝑐 and 𝐹𝐹𝑓𝑓 are the Friction Losses values calculated for states at points 𝜌𝜌 and 𝑓𝑓.

The time step size is calculated by considering the Courant-Friedrich-Levy Number (or Courant Number) 𝑘𝑘𝐶𝐶𝐶𝐶𝐶𝐶:

Δ𝑡𝑡 = 𝑘𝑘𝐶𝐶𝐶𝐶𝐶𝐶Δ𝑥𝑥𝑐𝑐 .

For a stable numerical solution, choose the time step such that the 𝑘𝑘𝐶𝐶𝐶𝐶𝐶𝐶 < 1 /3/, /4/.

These equations can be solved without the use of iterative methods for all grid points except the first and last. At these points, the pressure and velocity are calculated by taking the following boundary conditions into account:

• the left end of the pipe:

(𝑣𝑣𝐴𝐴 − 𝑣𝑣𝐴𝐴𝑓𝑓) − 1𝜌𝜌𝜌𝜌 (𝑝𝑝𝐴𝐴 − 𝑝𝑝𝐴𝐴𝑓𝑓) − (𝐹𝐹𝐴𝐴𝑓𝑓 + 𝐺𝐺)Δ𝑡𝑡 = 0

• the right end of the pipe:

(𝑣𝑣𝐵𝐵 − 𝑣𝑣𝐵𝐵𝑐𝑐) + 1𝜌𝜌𝜌𝜌 (𝑝𝑝𝐵𝐵 − 𝑝𝑝𝐵𝐵𝑐𝑐) − (𝐹𝐹𝐵𝐵𝑐𝑐 + 𝐺𝐺)Δ𝑡𝑡 = 0

Pressure and velocity are calculated with a discrete time step. If the time step of ODE-solver is smaller than the time step of the local solver of method of characteristics, the result will be in the form of steps. For smoothing the result, the pressures 𝑝𝑝𝐴𝐴𝑓𝑓, 𝑝𝑝𝐵𝐵𝑐𝑐 and velocities 𝑣𝑣𝐴𝐴𝑓𝑓, 𝑣𝑣𝐵𝐵𝑐𝑐 are calculated as follows:

𝑑𝑑𝑑𝑑𝑡𝑡 (𝑢𝑢𝐴𝐴𝑓𝑓) =

𝑢𝑢{1}𝑓𝑓,𝑗𝑗+1 − 𝑢𝑢{1}𝑓𝑓,𝑗𝑗Δ𝑡𝑡

𝑑𝑑𝑑𝑑𝑡𝑡 (𝑢𝑢𝐵𝐵𝑐𝑐) =

𝑢𝑢{𝑛𝑛+1}𝑐𝑐,𝑗𝑗+1 − 𝑢𝑢{𝑛𝑛+1}𝑐𝑐,𝑗𝑗Δ𝑡𝑡

Where 𝑢𝑢 = 𝑝𝑝 ∨ 𝑣𝑣, 𝑢𝑢{𝑖𝑖}𝑓𝑓(𝑐𝑐),𝑗𝑗 – pressure or velocity in the point 𝜌𝜌(𝑓𝑓), obtained from the construction positive (or negative) characteristics from point 𝑥𝑥𝑖𝑖 at time 𝑡𝑡𝑗𝑗.

The Friction Losses 𝐹𝐹𝐴𝐴𝑓𝑓 and 𝐹𝐹𝐵𝐵𝑐𝑐 are calculated as:

𝐹𝐹𝐴𝐴𝑓𝑓 = 𝑓𝑓(𝑣𝑣𝐴𝐴𝑓𝑓)

𝐹𝐹𝐵𝐵𝑐𝑐 = 𝑓𝑓(𝑣𝑣𝐵𝐵𝑐𝑐)

These equations must be solved together with equations describing the processes in the devices connected to the ends of the pipe.

Figure 9 shows an example of the calculation of the pressure at the boundary of the pipe.

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Figure 12: Comparison of simulation results with measurement results.

Figure 13: Comparison of simulation results with measurement results.

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5 Simulation results

To validate the method described above, a mathematical model of the test bench of URACA GmbH & Co. KG was created /5/. The test bench consists of a three-plunger water pump driven by an electric motor and a simple pipeline (see Figure 10).

Figure 10: Scheme of the test bench

At the beginning of the pipeline, a pressure sensor pd1 is installed. The second pressure sensor pd2 is installed at 21,68 m from the sensor pd1. The total length of the pipeline is 30,32 m. The purpose of the valve is to set the offset pressure.

The mathematical model of the test bench, modelled in the SimulationX 3.8, is shown in the Figure 11.

Figure 11: Simulation model of the test bench

In the plunger pump model, the following factors were considered /5/:

• kinematics of the mechanism;

• valve static characteristics;

• compressibility of liquid in the working chamber.

The pipeline is simulated using two pipe models with the built-in characteristic method. Local resistance is not considered. The local time step in the method of characteristics is ≈1.6 ms, 𝑘𝑘𝐶𝐶𝐶𝐶𝐶𝐶 = 0,95.

Comparisons of simulation results with measurement results are shown in Figure 12 and Figure 13.

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/2/ Randall J. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, United Kingdom, 2002.

/3/ Beck, M., Modellierung und Simulation der Wellenbewegung in kavitierenden Hydraulikleitungen, Univ. Stuttgart, Germany, 2003.

/4/ Popov D.N., Panaiotti S.S., Ryabinin M.V., Gidromehanika: uchebnik dlya vuzov. Second Edition. BMSTU, Moscow, 2002 (in Russian).

/5/ Beck, F., Durchführen von detaillierten Pulsationsmessungen an einem Hockdruckpumpenprüfstand zur Absicherung und Weiterentwicklung einer numerischen Pulsationsberechnung, Bachelorarbeit, Dualen Hochschule Baden-Württemberg, Stuttgart, Germany, 2016


The comparison with the data shows a good correlation of the fundamental harmonics and characteristic peaks. At the same time, a significantly less damping in the simulation model can be observed. This can be explained by the absence of the hydraulic resistances of the knees in the model and also a possible influence of non-stationary friction.

It should be noted that a very simple model from the point of view of creation and parametrization was used to obtain these results. The real-time factor in the calculation, depending on the solver used, is between 2 and 3, which makes this model very convenient for rapid preliminary assessment of the level of pressure fluctuations in the pipeline.

6 Summary and Conclusion

1. A high level of pulsations in the volume flow of a three-plunger pump makes it necessary to have a method for predicting the level of pressure ripples considering the pipeline.

2. The nonlinear nature of pulsations makes analysis in the time domain preferable.

3. To solve the problem of wave propagation in the pipeline, the method of characteristics was used with a local variable time step built into the ODE-solver.

4. Comparison with the results of the experiments showed the feasibility of using this model for preliminary calculations of pressure pulsations in the pipeline.

5. This model is effectively used by URACA GmbH&Co.KG to predict mechanical vibrations considering wave processes in the pipeline and to design resonance pulsation dampers.

Nomenclature

Variable Description Unit

𝜌𝜌 Fluid Density [kg/m³]

𝑣𝑣 Flow Velocity [m/s]

𝑝𝑝 Pressure [bar]

𝑥𝑥 Length Coordinate [m]

𝑡𝑡 Time [s]

𝑇𝑇 Temperature [K]

𝐹𝐹 Friction Loss [m/s²]

𝐺𝐺 Gravity Pressure Drop Factor [m/s²]

𝑐𝑐 Sound Speed [m/s]

𝛼𝛼𝑈𝑈 Volume Fraction of Undissolved Gas [%]

𝐸𝐸𝑒𝑒 Equivalent Bulk Modulus [bar]

𝑘𝑘𝐶𝐶𝐶𝐶𝐶𝐶 Courant-Friedrich-Levy Number (Courant Number) [-]

References

/1/ URACA GmbH & Co.KG, Volumenstrompulsation in Abhängigkeit vom volumetrischen Wirkungsgrad, URACA GmbH & Co.KG, Bad Urach, Germany, 2007.

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pipeline simulation by the method of characteristics for...

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