This document summarizes the results of computational modeling of gas flow in a Holweck pump. The pump geometry was modeled as 2D grooved channels with varying dimensions and rarefaction parameters. Discrete velocity methods were used to solve the Boltzmann equation for the gas distribution function. Mass flow rates were computed for 126 combinations of channel length, width, depth and rarefaction parameter. The results showed a Knudsen minimum in the mass flow rate around a rarefaction parameter of 1, consistent with theory. Normalizing the results matched published data, validating the computational approach.

1. Eirini Koutantou
Supervisor: Prof. D. Valougeorgis
Holweck pump
modeling
Department of Mechanical Engineering,
University of Thessaly

2. Presentation contents:
1) Introduction
2) Statement of the problem
3) Computational scheme
4) Results and discussion
5) Concluding remarks
2

3. 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

4. General Terminology:
4
• Pressure:
• Ideal gas equation:
• Mean free path:
• Reynolds number:
• Knudsen number:
mfp:
F
p
A
B
mRT
pV Nk T
M
2 2
1 1
2 2n d n d
where: 8kT
m
Re
ud
v
1
2
Kn
d
Absolute vacuum:
density of
molecules=0

5. 5
Vacuum Terminology:
• Mass flow:
• Pumping speed:
• Pump throughput:
• Conductance/Conductivity:
• Compression ratio:
m
M
t
 [kg/h, g/s]
dV
S
dt
[m3/s, m3/h]
pV
V mRT
Q p
t tM
pV
dV
Q S p p
dt
[Pa m3/s =W]
pVQ
C
p
in row:
1/Ctot = 1/C1 + 1/C2
parallel:
Ctot = C1 + C2 + …
2
0
1
p
K
p
P1: inlet pressure
P2: outlet pressure

6. 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 high
vacuum
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. 7
fluid displaced
by 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:

8. Gas transfer: Positive displacement
8
• Diaphragm pump:
- Well known for
environmental 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. 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. 10
Examples!
AUDI
MERCEDESBMW
MEDICAL APPLICATIONAEROSPACE APPLICATION
Applications:
- Refrigeration systems
- Food industry
- Laboratory experimentation
- Mechanical vacuum
- Medicine
- Aerospace industry
- Formula 1 and
Automotive industries

11. 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)

12. 3D problem
12
Simulation: much computational effort
Neglect: end effects and the curvature
of the geometry
(total effect = 0.05 )
4 independent problems: 2D flow
in grooved channel
region of solution:

13. 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)

14. General description of
individual problems:
14
1. Longitudinal Couette flow
2. Longitudinal Poiseuille flow
3. Transversal Couette flow
4. Transversal Poiseuille flow( , )i
f f
Q f f
t i
Boltzmann equation:
BGK model:
( )M
i
f f
v f f
t i
23
[ ( , )]
2
2 ( , )
( , )
2 ( , )
i i
B
m u i t
k T i tM
B
m
f n i t e
k T i t
Maxwell distribution function:
Steady state flow:
Taylor expansion:( )M
i
f
v f f
i
0
0
n n
n
0
0
T T
T
2
0
0 0
3
1
2 2
M i iu
f f
RT RT
where:
Polar system coordinates:
2 2
x yc c
1
tan y
x
c
c
cos sinx y
d
c c
x y x y ds
Linear differentiation of distribution function

15. 15
Longitudinal Couette: Longitudinal Poiseuille:
Fluid flow: in direction z’
Cause of flow: moving wall
in direction z’
Cause of flow: pressure gradient
in direction z’
0,0, ,zu u x y
0 0
1
o
U
f f h
u
0
1
o
U
u
Linearization0
1f f hXp z Xp 1Xp
x
x
H
y
y
H 0
x
xc
u 0
y
yc
u 0
z
zc
u
0
0
P
v 0
0 0
P H
u 02ou RT
Non dimensional
variables
'
0
z
z
u
u
U
0
0
u
U 0
ou
U
'
0
z
z
u
u
u Xp Xp Xp
reduced
BGK equations
after projection
x y zc c u
x y
where:
21
, , , , , , , zc
x y x y z z zx y c c h x y c c c c e dc
1
2
x y zc c u
x y
Macroscopic velocity:

16. 1616
Fluid flow: in direction x’
Cause of flow: moving wall
in direction x’
Cause of flow: pressure gradient
in direction x’
Linearization
x
x
H
y
y
H 0
x
xc
u 0
y
yc
u 0
z
zc
u
0
0
P
v 0
0 0
P H
u 02ou RT
Non dimensional
variables
( , ), ( , ),0x yu u x y u x yTransversal Couette: Transversal Poiseuille:
0 0
1
o
U
f f h
u
0
1
o
U
u
0
1f f hXp x Xp 1Xp
0
0
u
U 0
ou
U Xp Xp
where:
'
0
x
x
u
u
U
'
0
y
y
u
u
U
'
0
x
x
u
u
u Xp
'
0
y
y
u
u
u Xp
2
1 2 cos sinx yu u
x y
2
1 2 cos sin cosx yu u
x y
2
x yc c
x y
21
, , , , , , , zc
x y x y z zx y c c h x y c c c e dc
2
21 1
, , , , , , ,
2
zc
x y x y z z zx y c c h x y c c c c e dcand
reduced
BGK equations
after projection
Macroscopic velocity:

17. 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
L
G u y dy
H
/2
/2
2
( ,1)
L H
yz
L H
H
Cd P x dx
L

Transversal flows:
2
2
0 0
1
,x y e d d
2
2
2
0 0
1 2
, 1
3
x y e d d
2
2
2
0 0
1
, cosxu x y e d d
2
2
2
0 0
1
, sinyu x y e d d
2
2
3
0 0
1
, sin cosxyP x y e d d
/2
/2
2
,1
L
xyL
H
Cd P x dx
L

1
0
2 ,
2
x
L
G u y dy
H
Density deviation:
Temperature deviation:
Macroscopic velocity:
Stress tensor:
Flow rate:
Drag coefficient:

18. 18
Boundary conditions:
Couette
Poiseuille
eq
wf f
2
02
3
2
02
wu
RTeq w
w
n
f e
RT
, , , , , ,
2 2
L L
y y
H H
Inlet – 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

19. • 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
19
Discrete Velocity Method
DVM
Set 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
d
u u
ds
, , , ,
, , ,
2
i j m n i j
m i j m n
d
ds
Set of
algebraic equations:
2 × Μ × Ν
equations/node

20. 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
3rd
4th
end of scanning

21. • 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

22. 22
Transversal Poiseuille flow:
Knudsen minimum: δ=1
Normalization of results:
F. Sharipov
L=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:

23. 23
Transversal Poiseuille flow:
Normalization of results:
F. Sharipov
L=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:

24. 24
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

25. 25
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

26. 26
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. 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

28. • 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!!!
28

29. 29
Thank you for your attention !!!