Design of Gas Piping Distribution Systems Consisting of Long Pipes Under Any Vacuum Conditions

64th IUVSTA Workshop on Practical Applications and Methods of Gas Dynamics for Vacuum Science and TechnologyLeinsweiler, Germany, 16 – 19/05/2011

Serafeim Misdanitis and Dimitris Valougeorgissemisdan@mie.uth.gr; diva@mie.uth.gr

Department of Mechanical Engineering University of Thessaly, Volos, Greece

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Introduction

Rarefied gas flows in long circular channels have been extensively investigated via linear kinetic theory from the 1960s, implementing various semi-analytical and numerical schemes.

In many applications however, the rarefied gaseous distribution system is consisting not of a single channel but of many channels accordingly combined to form a network.

Such distribution systems are commonly found in several technological fields including

vacuum pumping, metrology, industrial aerosol, porous media and microfluidics.

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Introduction

Computational algorithms dedicated to the design of gas pipe networks (e.g., compressed air, natural gas, etc.) in the viscous regime are well developed, while corresponding tools for the design of gaseous pipe networks operating under any (e.g. low, medium and high) vacuum conditions are very limited.

It is seen that in several applicationsthe gas distribution systems may be represented by complex networks consisting of short and long channels of various cross sections and

the gas flows may be in the whole range of the Knudsen number from the free molecular through the transition up to the viscous regime.

The objective is to compute the mass flow rate and the conductance through each tube and the pressure head at each node of the network, provided that the geometry of each channel of the network is given.

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Formulation of the algorithm

A typical pipe network may be considered as a directed linear graph consisting of a finite number of pipe sections interconnected in a specified configuration.

Each pipe is characterized of its length L, diameter D and some roughness. A point where two or more pipes are joined is known as a junction node or simply as a node.

The closed path uniquely formed by adjacent pipes is a loop, while the open path connecting two fixed-grade nodes is a pseudo-loop. A fixed grade node is a node where a consistent energy grade is maintained (e.g. a constant pressure reservoir).

For a well-defined network with p pipes, n junction nodes, l loops and f fixed-grade nodes the following relation holds:

1p n l f= + + −

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Formulation of the algorithm

Independent of the flow regime, the system of equations describing such a network is consisting of

the mass conservation equations at each node of the network, given by

and the pressure drop equations along each pipe element.

The pressure drop equations may be reduced to a set of the energy balance equations along the closed loops of the network:

When there are fixed-grade nodes in the network then, the system of equations for the mass flow rates is amplified by the energy balance equation around each pseudo-loop connecting two fixed grade nodes:

( ) 0jj i

M Q⎡ ⎤

± − =⎢ ⎥⎣ ⎦∑

( )( ) 0jj k

PΔ⎡ ⎤

± =⎢ ⎥⎣ ⎦∑

, j

refers to the pipes connected to the node i

, j

refers to the pipes making up each k of the l total loops

( )( ) 0jj m

P HΔ Δ⎡ ⎤

± + =⎢ ⎥⎣ ⎦∑ , j

refers to the pipes making up each m of the pseudo

loops

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Formulation of the algorithmIn the present rarefied conditions, the information for pressure drop along a channel is given by:

quantity is the dimensionless flow rate obtained from a data base, which has been developed for this purpose by solving a linearized BGK kinetic equation in a wide range of the Knudsen number, with Maxwell purely diffuse boundary conditions.

A gaseous pipe network in the viscous (or hydrodynamic) regime is still described by the same mass conservation and energy balance equations. The only difference compared to networks operating under rarefied conditions is that the pressure drop along each pipe element is obtained by a corresponding expression based on hydrodynamic principals.

For the purposes of the present work the Darcy–Weisbach equation is implemented:

( ) 01 2 * 34 j j

jjj j

M LP P P

G Dυ

π− = Δ =

*jG

( )2

11 2 2 4

2

8 2lnj jj jj

j j j

M LPP P P fD P Dρ π

⎛ ⎞− = Δ = +⎜ ⎟⎜ ⎟

⎝ ⎠

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Design of a network under vacuum conditions

An iterative process between the system of equations and the pressure drop equations yields the converged values of the unknown quantities. The basic steps of the designed methodology may be summarized as follows:

Assume the pressure head at the nodes of the network. Use the assumed pressure heads to solve the closed system of mass conservation and energy balance equations for the mass flow rates. Use the computed mass flow rates to find, based on the kinetic data, the updated pressure heads at the nodes.

Compare with the previous pressure heads and iterate upon convergence.

The convergence criterion was set equal to 10-9 between the values of the pressure heads computed through the iteration scheme.

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2 3 42M 3M

23M

14Q

6Q2P 3P 4P

26M

29M28M

16M

18M 19M

25M

22M

19

262516

17 1819P

26P25P16P

17P18P

Results of a typical network under vacuum conditions

4 5 6

20 7

21 8

232415 22 9

121314 11 10

5P 6P

4M 5M

20M 21M

30M 31M

37M 38M

11M 10M13M 12M

35M 36M15P

15M 32M

39M14M

24M 6M

7M

8M

9M41M40M

33M34M

27M

17M

42M

1M

8P

9P

10P11P12P13P14P

7P

21P

24P 23P 22P

20P

27P

27

1P

1

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Based on a graphical user interface (GUI) the desired network may be drawn by adding nodes and edges and the corresponding data:

coordinates of nodes length and diameter of pipe elementspressure heads at fixed-grade nodesinformation for the gas and its propertiesthe demands (if any) at the nodes.

Results of a typical network under vacuum conditions

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Results of a typical network under vacuum conditions

To demonstrate the feasibility and the effectiveness of the proposed methodology the sample network described is simulated.

Nodes 1 and 27 refer to two reservoirs, where the pressure is held constant. All tubes are taken to have the same length and diameter, which are equal to L=10m and D=0.1m respectively.

The reference temperature is set to T=290.68K, while the conveying gas is nitrogen (N2).

Two typical simulations are performed. The first one in the viscous (or hydrodynamic) regime, where:

P1=70Pa, P2=60Pa where leakages (or demands) are present Q6=1.4x10-5 and Q14=2.1x10-5 (Tables 1 and 2)

and the second one in the whole range of the Knudsen numberP1=1Pa, P2=1x10-3Pa where there are no leakages present (Tables 3 and 4).

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Results of a typical network under vacuum conditions

Nodenumber

Local Kn

Head Pressure [Pa]

1 9.13(-4) 70.00

2 9.66(-4) 66.12

3 9.97(-4) 64.13

4 1.02(-3) 62.90

5 1.03(-3) 61.98

6 1.05(-3) 61.05

…

…

…

22 1.04(-3) 61.49

23 1.04(-3) 61.66

24 1.04(-3) 61.71

25 1.02(-3) 62.47

26 1.03(-3) 62.12

27 1.07(-3) 60.00

Tubenumber

FromNode

To Node

Mass Flow Rate[kg/s]

Conductance[lt/s]

1 1 2 4.33(-5) 9.06(2)

2 2 3 2.13(-5) 8.67(2)

3 3 4 1.28(-5) 8.46(2)

4 4 5 9.49(-6) 8.32(2)

5 5 6 9.35(-6) 8.20(2)

6 6 7 -4.65(-6) 8.17(2)

… … …

… … …

… … …

37 22 23 -1.81(-6) 8.20(2)

38 22 9 2.05(-6) 8.18(2)

39 13 24 -5.98(-6) 8.18(2)

40 12 23 -3.33(-6) 8.19(2)

41 11 22 -2.68(-6) 8.17(2)

42 10 27 8.32(-6) 8.05(2)

Table 1: Pressure head and Knudsen number at each node of the network.

Table 2: Mass flow rate and Conductance at each tube of the network.

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Results of a typical network under vacuum conditions

Nodenumber

Local Kn

Head Pressure [Pa]

1 6.39(-2) 1.00

2 7.81(-2) 8.18(-1)

3 8.90(-2) 7.18(-1)

4 9.74(-2) 6.56(-1)

5 1.04(-1) 6.16(-1)

6 1.07(-1) 5.96(-1)

…

…

…

22 1.25(-1) 5.11(-1)

23 1.14(-1) 5.59(-1)

24 1.07(-1) 5.96(-1)

25 1.01(-1) 6.31(-1)

26 1.07(-1) 5.96(-1)

27 6.39(1) 1.00(-3)

Tubenumber

FromNode

To Node

Mass Flow Rate[kg/s]

Conductance[lt/s]

1 1 2 4.58(-8) 2.04(1)

2 2 3 2.29(-8) 1.86(1)

3 3 4 1.35(-8) 1.76(1)

4 4 5 8.32(-9) 1.69(1)

5 5 6 4.16(-9) 1.65(1)

6 6 7 4.16(-9) 1.63(1)

… … …

… … …

… … …

37 22 23 -9.36(-9) 1.56(1)

38 22 9 9.36(-9) 1.50(1)

39 13 24 -4.16(-9) 1.63(1)

40 12 23 -5.20(-9) 1.58(1)

41 11 22 -9.36(-9) 1.50(1)

42 10 27 4.58(-8) 1.14(1)

Table 3: Pressure head and Knudsen number at each node of the network.

Table 4: Mass flow rate and Conductance at each tube of the network.

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Concluding remarks

A novel algorithm has been developed for the design of gaseous distribution systems of any complexity consisting of long tubes based on linear kinetic theory.

In the viscous regime, the algorithm provides very good agreement with similar solvers based on hydrodynamic principals.

The results are valid and accurate in the whole range of the Knudsen number.

The algorithm will be extended to include channels of any cross section and more important, channels of short length.

Further development will include optimization of gaseous micro distribution networks in order to be used as an engineering tool in the design of gas network systems.