Post on 19-Jul-2015
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
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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
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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
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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:
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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:
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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
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[ ( , )]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:
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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:
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Boundary conditions:
Couette
Poiseuille
eq
wf f
2
02
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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:
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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:
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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
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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 !!!