Eirini Koutantou

Supervisor: Prof. D. Valougeorgis

Holweck pumpmodeling

Department of Mechanical Engineering,University of Thessaly

Presentation contents:

1) Introduction

2) Statement of the problem

3) Computational scheme

4) Results and discussion

5) Concluding remarks

2

3

Vacuum: the pressure of the gas is much lower than the one of its

environment

Pump:device that is used

to move fluids

Vacuum pump:movement of gas

molecules due to flow induced by a vacuum

system

Was invented in:1650

by:Otto von Guericke

General Terminology:

4

• Pressure:

• Ideal gas equation:

• Mean free path:

• Reynolds number:

• Knudsen number:

mfp:

Fp

A

B

mRTpV Nk T

M

2 2

1 1

2 2n d n d

where: 8kTm

Reud

v

1

2Kn

d

Absolute vacuum:density of

molecules=0

5

Vacuum Terminology:

• Mass flow:

• Pumping speed:

• Pump throughput:

• Conductance/Conductivity:

• Compression ratio:

mM

t [kg/h, g/s]

dVS

dt

[m3/s, m3/h]

pV

V mRTQ p

t tMpV

dVQ S p p

dt[Pa m3/s =W]

pVQC

p

in row:1/Ctot = 1/C1 + 1/C2

parallel:Ctot = C1 + C2 + …

20

1

pK

pP1: inlet pressureP2: outlet pressure

6

vacuum

(mfp range)

rough vacuum:

mfp << 10-4 m

medium vacuum:

10-4 m - 10-1 m

high vacuum:

10-1 m - 103 m

ultra high vacuum:

mfp >> 103 m

vacuum

(pressure range)

rough vacuum:

105 Pa - 100 Pa

fine vacuum:

100 Pa - 10-1 Pa

high vacuum:

10-1 Pa - 10-5 Pa

ultra high vacuum (UHV):

10-5 Pa - 10-10 Pa

extreme high vacuum(XHV):

10-10 Pa - 10-12 Pa

Definition of vacuum ranges:

Gas flow regimes:

Kn > 0.5:

Free Molecular

-Equation of Boltzmann(without the collision term)

-ultra, extreme highvacuum

0.01 < Kn < 0.5:

Transition regime

-Equation of Boltzmann(empirical approaches)

- fine, medium vacuum

Kn << 0.01:

Viscous or continuum flow(laminar or turbulent)

-Described by the equations NS- rough vacuum

7

fluid displacedby a space and is

forwarded to another

gases are removed by extracting them in the atmosphere

change of the kinetic state of

the moving fluid

cause condensation or chemical

trapping of gas

Pump tree:

Gas transfer: Positive displacement

8

• Diaphragm pump:

- Well known forenvironmental reasons

- low maintenance cost

- noiseless

Rotary pumps

• Roots pump:

- design principle was discovered:in 1848 by Isaiah Davies

- implemented in practice:Francis and Philander Roots

- in vacuum science: only since 1954

Reciprocating pumps

9

Gas transfer: Kinetic

- 1913 :Gaede - molecular

- 1957 :Dr.W.Becker - turbomolecular

• (Turbo) molecular pump:

Entrapment pumps

• Cryopump:

- concentration on cold surface

- profitable for some gases

Drag pumps

10

Examples!

AUDI

MERCEDESBMW

MEDICAL APPLICATIONAEROSPACE APPLICATION

Applications:

- Refrigeration systems

- Food industry

- Laboratory experimentation

- Mechanical vacuum

- Medicine

- Aerospace industry

- Formula 1 and

Automotive industries

Holweck pump:

11

Invented by:Fernand Holweck

Constructed by:Charles Beaudouin

Molecular pump:- Outer cylinder with grooves, spiral form- Inner cylinder with

smooth surface

The rotation of the smooth cylinder causes

the gas flow

Fernand Holweck

(1890-1941)

3D problem

12

Simulation: much computational effort

Neglect: end effects and the curvature of the geometry(total effect = 0.05 )

4 independent problems: 2D flowin grooved channel

region of solution:

Geometry:

13

H : distance between plates

W x D : groove cross section

W : groove width

D : groove depth

L : period

Isothermal walls:

Τ=Τ0

Characteristic length:

Η

Boundaries of flow domain:

- Inlet: (x΄= -L/2)

- Outlet: (x΄= L/2)

- Top wall: (y΄= Η)

- Bottom wall: (y΄=-D)

General description of individual problems:

14

1. Longitudinal Couette flow

2. Longitudinal Poiseuille flow

3. Transversal Couette flow

4. Transversal Poiseuille flow( , )i

f fQ f f

t i

Boltzmann equation:

BGK model:

( )M

i

f fv f f

t i

23

[ ( , )]2

2 ( , )( , )

2 ( , )

i i

B

m u i t

k T i tM

B

mf n i t e

k T i t

Maxwell distribution function:

Steady state flow:

Taylor expansion:( )M

i

fv f f

i

0

0

n n

n0

0

T T

T

20

0 0

31

2 2

M i iuf f

RT RTwhere:

Polar system coordinates:

2 2

x yc c

1tany

x

c

ccos sinx y

dc c

x y x y ds

Linear differentiation of distribution function

15

Longitudinal Couette: Longitudinal Poiseuille:

Fluid flow: in direction z’

Cause of flow: moving wallin direction z’

Cause of flow: pressure gradientin direction z’

0,0, ,zu u x y

0 01o

Uf f h

u0 1o

U

u

Linearization0 1f f hXp z Xp 1Xp

xx

H

yy

H0

xxc

u0

y

ycu 0

zzc

u

0

0

Pv 0

0 0

P H

u02ou RT

Non dimensionalvariables

'

0

zz

uu

U

0

0

u

U 0

ou

U

'

0

zz

uu

u Xp Xp Xp

reducedBGK equationsafter projection

x y zc c ux y

where:21

, , , , , , , zc

x y x y z z zx y c c h x y c c c c e dc

1

2x y zc c u

x y

Macroscopic velocity:

1616

Fluid flow: in direction x’

Cause of flow: moving wallin direction x’

Cause of flow: pressure gradientin direction x’

Linearization

xx

H

yy

H0

xxc

u0

y

ycu 0

zzc

u

0

0

Pv 0

0 0

P H

u02ou RT

Non dimensionalvariables

( , ), ( , ),0x yu u x y u x yTransversal Couette: Transversal Poiseuille:

0 01o

Uf f h

u0 1o

U

u

0 1f f hXp x Xp 1Xp

0

0

u

U 0

ou

U Xp Xp

where:

'

0

xx

uu

U

'

0

y

y

uu

U

'

0

xx

uu

u Xp

'

0

y

y

uu

u Xp

2 1 2 cos sinx yu ux y

2 1 2 cos sin cosx yu ux y

2x yc c

x y

21, , , , , , , zc

x y x y z zx y c c h x y c c c e dc

221 1

, , , , , , ,2

zc

x y x y z z zx y c c h x y c c c c e dcand

reducedBGK equationsafter projection

Macroscopic velocity:

Macroscopic quantities:

17

Longitudinal flows:

22

0 0

1,zu x y e d d

22

0 0

1, sinyzP x y e d d

1

0

2 ,2

z

LG u y dy

H

/2

/2

2( ,1)

L H

yz

L H

HCd P x dx

L

Transversal flows:

22

0 0

1,x y e d d

22

2

0 0

1 2, 1

3x y e d d

22

2

0 0

1, cosxu x y e d d

22

2

0 0

1, sinyu x y e d d

22

3

0 0

1, sin cosxyP x y e d d

/2

/2

2,1

L

xyL

HCd P x dx

L

1

02 ,

2x

LG u y dy

H

Density deviation:

Temperature deviation:

Macroscopic velocity:

Stress tensor:

Flow rate:

Drag coefficient:

18

Boundary conditions:

Couette

Poiseuille

eq

wf f

2

02

32

02

wu

RTeq ww

nf e

RT

, , , , , ,2 2

L Ly y

H HInlet – Outlet: Periodic

Interface gas-wall: Maxwell - diffusion

0 0ncStationary walls:

Moving wall: 2 zc 0yc

Stationary walls:

2 coswn 0yc

Longitudinal

Transversal Stationary walls:

Moving wall:

0 0nc

0 0nc

wnStationary walls: 0nc

where

0

0

Longitudinal

Transversal

where nw is defined by the no-penetration condition: 0u n

• Discretization:- Physical space [ (x,y) or (x,z)] : (i,j)

where i=1,2,…,I and j=1,2,…,J

- Molecular velocity space (μm,θn) : (ζm , θn) where 0 < ζm < ∞ and 0 < θn < 2π

m=1,2,…,M and n=1,2,…N

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Discrete Velocity Method

DVMSet consists of:

Μ × Ν discrete velocities(16 × 50 × 4)

3200

• Discretized kinetic equations:(e.g. transversal Couette flow)

, ,

, , , 2

, , , , , 1 2 cos sini j i j

i j m n

m i j m n i j i j m m x n y n

du u

ds

, , , ,

, , ,2

i j m n i j

m i j m n

d

ds

Set ofalgebraic equations:

2 × Μ × Νequations/node

Algorithm:

20

Parameters:

δ μm θn Ny_cha

D (D/H) W (W/H) L (L/H)

Couette: U0 / Poiseuille: Χp

• Grid format :Channel and Cavity

• Grid reverse:

Scan of grid:

1st 2nd

3rd4th

end of scanning

• Geometries:

21

L = 2:

L = 2.5:

L = 3:

• Rarefaction parameter:

δ 0 10-3 10-² 10-¹ 1 10 100

Total runs:

18 geometries 7 δ

=

126

• Results:

Mass flow rate

Drag coefficient

Macroscopic velocities

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Transversal Poiseuille flow:

Knudsen minimum: δ=1

Normalization of results:

F. SharipovL=3 , W=1.5 , D=0.5

δ

L W D 0 10-3 10-² 10-1 1 10 100

2 0.5 0.5 3,211 3,157 2,815 1,976 1,511 2,766 15,544

2 0.5 1 3,212 3,158 2,814 1,975 1,509 2,752 15,433

2 1 0.5 3,149 3,096 2,760 1,945 1,534 2,996 17,228

2 1 1 3,159 3,106 2,769 1,953 1,539 2,984 16,625

2 1.5 0.5 3,159 3,107 2,777 1,983 1,651 3,523 20,918

2 1.5 1 3,159 3,107 2,775 1,978 1,641 3,514 20,856

2.5 0.5 0.5 3,227 3,173 2,829 1,988 1,516 2,754 15,450

2.5 0.5 1 3,227 3,173 2,828 1,986 1,513 2,740 15,340

2.5 1 0.5 3,183 3,129 2,789 1,961 1,532 2,927 16,742

2.5 1 1 3,191 3,137 2,796 1,968 1,535 2,914 16,625

2.5 1.5 0.5 3,171 3,119 1,219 1,241 1,617 3,312 19,412

2.5 1.5 1 3,173 3,120 2,785 1,980 1,610 3,301 19,333

3 0.5 0.5 3,239 3,184 2,839 1,996 1,520 2,746 15,387

3 0.5 1 3,239 3,184 2,838 1,993 1,517 2,732 15,277

3 1 0.5 3,202 3,148 2,806 1,974 1,530 2,880 16,428

3 1 1 3,208 3,154 2,811 1,978 1,533 2,867 16,313

3 1.5 0.5 3,193 3,139 2,801 1,985 1,594 3,171 18,496

3 1.5 1 3,196 3,142 2,802 1,984 1,588 3,160 18,408

0.01 0.10 1.00 10.00

0.5

1.0

1.5

2.0

2.5

Mass flow rate:

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Transversal Poiseuille flow:

Normalization of results:

F. SharipovL=3 , W=1.5 , D=0.5

δ

L W D 0 10-3 10-² 10-1 1 10 100

2 0.5 0.5 0,470 0,470 0,471 0,476 0,488 0,488 0,393

2 0.5 1 0,471 0,471 0,472 0,477 0,489 0,488 0,393

2 1 0.5 0,436 0,436 0,438 0,449 0,483 0,499 0,403

2 1 1 0,443 0,443 0,445 0,456 0,488 0,500 0,400

2 1.5 0.5 0,414 0,414 0,417 0,436 0,493 0,522 0,420

2 1.5 1 0,415 0,415 0,418 0,436 0,493 0,522 0,420

2.5 0.5 0.5 0,476 0,476 0,477 0,480 0,490 0,487 0,392

2.5 0.5 1 0,477 0,477 0,478 0,481 0,491 0,487 0,392

2.5 1 0.5 0,449 0,449 0,450 0,459 0,486 0,496 0,399

2.5 1 1 0,455 0,455 0,456 0,464 0,489 0,497 0,400

2.5 1.5 0.5 0,431 0,431 0,434 0,449 0,494 0,513 0,413

2.5 1.5 1 0,432 0,320 0,434 0,449 0,494 0,513 0,412

3 0.5 0.5 0,480 0,480 0,481 0,484 0,491 0,487 0,392

3 0.5 1 0,481 0,481 0,481 0,484 0,492 0,487 0,392

3 1 0.5 0,457 0,457 0,459 0,466 0,488 0,494 0,398

3 1 1 0,462 0,462 0,464 0,470 0,491 0,495 0,398

3 1.5 0.5 0,442 0,443 0,445 0,457 0,495 0,508 0,408

3 1.5 1 0,443 0,443 0,445 0,457 0,494 0,508 0,408

0.010 0.100 1.000 10.000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Drag coefficient:

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channel inlet:

Longitudinal Couette:

Transversal Couette:

L=3, W=1, D=1

L=3, W=1, D=1

Macroscopic velocities

channel middle:

L=3, W=1, D=1

L=3, W=1, D=1

cavity start:

L=3, W=1, D=1

L=3, W=1, D=1

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channel inlet:

Macroscopic velocities

channel middle:cavity start:

Longitudinal Poiseuille:

Transversal Poiseuille:

L=3, W=1, D=1L=3, W=1, D=1

L=3, W=1, D=1 L=3, W=1, D=1

L=3, W=1, D=1

L=3, W=1, D=1

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Longitudinal Couette:

Longitudinal Poiseuille:

Macroscopic velocities

velocity contours:

L=2 , W=1 , D=1

δ=0.1

L=2 , W=1 , D=1

δ=1

L=2 , W=1 , D=1

δ=10

L=2 , W=1 , D=1

δ=0.1

L=2 , W=1 , D=1

δ=1

L=2 , W=1 , D=1

δ=10

27

Transversal Couette:

Transversal Poiseuille:

Macroscopic velocities

velocity streamlines:

L=2 , W=1 , D=1

δ=0.1

L=2 , W=1 , D=1

δ=1

L=2 , W=1 , D=1

δ=10

L=2 , W=1 , D=1

δ=0.1

L=2 , W=1 , D=1

δ=1

L=2 , W=1 , D=1

δ=10

• Four different flow configurations have been examined:1. Longitudinal Couette flow

2. Longitudinal Poiseuille flow

3. Transversal Couette flow

4. Transversal Poiseuille flow

• Results have been obtained in the whole range of Knudsen number and for various values of the geometrical parameters: L/H , W/H , D/H.

• Synthesizing these results in a proper manner designed parameters such as pumping speed and throughput can be obtained.

• Optimization of the Holweck pump will follow soon!!!

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Thank you for your attention !!!