RAREFIED GAS DYNAMICS AND ITS APPLICATIONS TO VACUUM ...

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RAREFIED GAS DYNAMICS 

AND ITS APPLICATIONS 

TO VACUUM TECHNOLOGY 

Felix Sharipov 

Departamento de Física 

Universidade Federal do Paraná 

Curitiba, Brazil 

http://fisica.ufpr.br/sharipov 

Felix Sharipov – p. 1

Typical problems 

isothermal flows, Poiseuille flow 

P1 

✲ P2 P1 > P2 




P1 

✲ P2 P1 > P2 

˙M mass flow rate? 




P1 

✲ P2 P1 > P2 


Q heat flow rate? 




P1 

✲ P2 P1 > P2 



density distribution? 




P1 

✲ P2 P1 > P2 



density distribution? 

over the whole range of Kn 



non-isothermal flows, thermal creep 

T1 

✲ T2 T1 < T2 




T1 

✲ T2 T1 < T2 





T1 

✲ T2 T1 < T2 






T1 

✲ T2 T1 < T2 



density (or pressure) distribution? 




T1 

✲ T2 T1 < T2 



density (or pressure) distribution? 

over the whole range of Kn 



Thermomolecular pressure difference 

P1, T1 

P2, T2 




P1, T1 

P2, T2 

˙M = 0 no mass flow 




P1, T1 

P2, T2 


What is the pressure ratio? 




P1, T1 

P2, T2 



P2 

P1 

= 

T2 

T1 

γ 




P1, T1 

P2, T2 



P2 

P1 

= 

T2 

T1 

γ 

0 ≤ γ ≤ 0.5 


Knudsen number 

Kn = 

molecular mean free path 

characteristic size 



Kn = 



Kn≫1 Free molecular regime. 

Every particle moves independently on each other 



Kn = 





Kn≪1 Hydrodynamic regime. 

Continuum mechanics equations are solved 



Kn = 





Kn≪1 Hydrodynamic regime. 

Continuum mechanics equations are solved 

Kn∼ 1 Transition regime. 

Kinetic Boltzmann equation is solved 

or DSMC method is applied 


Rarefaction parameter 

equivalent mean free path 

ℓ = µvm 

P 




µ - viscosity 

ℓ = µvm 

P 





ℓ = µvm 

P 

vm = 2kT/m most probable molecular vel. 





ℓ = µvm 

P 

vm = 2kT/m most probable molecular vel. 

P - pressure 


Boltzmann equation 

f(t, r, v) - velocity distribution function 



f(t, r, v) - velocity distribution function 

n(t, r) = f(t, r, v)dv - density 

u(t, r) = 1 

n 

P (t, r) = m 

3 

T (t, r) = m 

3nk 

q(t, r) = m 

2 

vf(t, r, v)dv - bulk velocity 

V 2 f(t, r, v)dv - pressure 

V 2 f(t, r, v)dv - temperature 

V 2 Vf(t, r, v)dv - heat flux vector 

V = v − u 



∂f 

∂t 

+ v · ∂f 

∂r 

= Q(ff∗) 



Q(ff∗) = 

∂f 

∂t 

 

+ v · ∂f 

∂r 

= Q(ff∗) 

(f ′ f ′ ∗ − ff∗) |v − v∗|bdb dε dv∗ 

v ′ and v∗ ′ - pre-collision molecular velocities 

v and v∗ - post-collision molecular velocities 



Discrete velocity method: 

v1, v2, ... , vN, 



Discrete velocity method: 

v1, v2, ... , vN, 

The BE is split into N differential eqs. coupled via 

the collisions integral 


Kinetic equations 

Till now, a numerical solution of the exact Boltz.Eq. 

requires great computational efforts 





BGK model 

Q(ff∗) = ν f M − f 





BGK model 

S model 

Q(ff∗) = ν 

 

f M 

Q(ff∗) = ν f M − f 

 

1 + 

2m(q · V) 

15n(kT ) 2 

mV 2 

2kT 

 

5 

− 

2 

 

− f 





BGK model 

S model 

Q(ff∗) = ν 

 

f M 

f M = n 

Q(ff∗) = ν f M − f 

 

1 + 

m 

2πkT 

2m(q · V) 

15n(kT ) 2 

3/2 

exp 

mV 2 

2kT 

 

5 

− 

2 

 

 

m(v − u)2 

− 

2kT 

ν = P/µ - frequency of intermolecular collisions 

 

− f 


Viscous slip coefficient 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

✻ 

y 

. 

.. 

..... ........ ..... ...... 

.. 

. ..... . . 

✻ 

. 

. 

. 

. 

. 

. 

✻ ✻ 

. 

✲ 

uy 

x 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

✻ 

y 

. 

.. 

..... ........ ..... ...... 

.. 

. ..... . . 

✻ 

✻ ✻ 

The most used formula 

. 

. 

. 

. 

. 

. 

. 

uy = 

✲ 

α -accommodation coefficient 

λ - mean free path 

uy 

x 

2 − α 

α λduy 

dux 



All disagreements with experiments are 

eliminated by fitting α: 

while in reality 

0.1 ≤ α ≤ 2, 

0.9 ≤ α ≤ 1, 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

✻ 

y 

. 

.. 

..... ........ ..... ...... 

.. 

. ..... . . 

✻ 

. 

. 

. 

. 

. 

. 

✻ ✻ 

. 

✲ 

uy 

x 



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

✻ 

y 

. 

.. 

..... ........ ..... ...... 

.. 

. ..... . . 

✻ 

. 

. 

. 

. 

. 

. 

✻ ✻ 

. 

✲ 

uy 

x 

uy = σPℓ duy 

dx 

σP - viscous slip coefficient 

ℓ equivalent mean free path 

at x = 0 



Diffuse scattering 

σP = √ π/2 = 0.886 estimation by Maxwell 




σP = √ π/2 = 0.886 estimation by Maxwell 

σP = 1.016 solution of BGK model 

σP = 1.018 solution of S model 

σP = 0.985 solution of Boltzmann Eq. 



Non-diffuse scattering 

Estimation by Maxwell 

σP = 0.886 

2 − α 

α 



Non-diffuse scattering 

Estimation by Maxwell 

σP = 0.886 

2 − α 

α 

S model with CL bound.cond., (Sharipov-2003) 

2 − αt 

σP = 1.018 

αt 

1 − αt 

− 0.264 

σP is sensitive to the gas-surface interaction 

αt 


Gas-surface interaction law 

Experiment by Porodnov et al. (1974) 

technical (contaminated) surface 

gas He Ne Ar Kr Xe H2 N2 CO2 

αt 0.88 0.85 0.92 1.0 1.0 0.95 0.91 0.99 


Gas-surface interaction law 

Experiment by Porodnov et al. (1974) 

technical (contaminated) surface 

gas He Ne Ar Kr Xe H2 N2 CO2 

αt 0.88 0.85 0.92 1.0 1.0 0.95 0.91 0.99 

For a technical surface αt is very close to unity for the most 

of gases 


Thermal slip coefficient 


uy = σT 

µ 

ϱ 

d ln T 

dy 

at x = 0 

σT = 0.75 estimation by Maxwell 

σT = 1.175 solution of S model 

σT = 1.01 solution of Boltzmann Eq. 


Temperature jump coefficient 


Tg = Tw + ζTℓ dT 

dx 

ζT = 1.662 estimation by Maxwell 

ζT = 1.954 solution of S model 


Flow through a tube 

˙M = πa2 P 

vm 

 

−GP 

a 

P 

dP 

dx 

+ GT 

a 

T 

GP = GP (δ) GT = GT (δ) 

δ = a 

ℓ 

dT 

dx 

 



Free molecular regime δ = 0 

GP = 8 

3 √ π , GT = 1 

2 GP 

Hydrodynamic regime δ → ∞ 

GP = δ 

4 + σP, GT = σT δ 



Transitional regime GP 

GP 

3.5 

3 

2.5 

2 

1.5 

BGK 

1 

0.01 0.1 1 10 

δ 

Cercignani et al. (1966) 




GP 

3.5 

3 

2.5 

2 

1.5 

BGK 

S 

1 

0.01 0.1 1 10 

δ 


Sharipov, 1996 




GP 

3.5 

3 

2.5 

2 

1.5 

BGK 

S 

BE 

1 

0.01 0.1 1 10 

δ 



Loyalka & Hamoodi (1991) 




GP 

3.5 

3 

2.5 

2 

1.5 

BGK 

S 

BE 

1 

0.01 0.1 1 10 

δ 



Loyalka & Hamoodi (1991) 

BGK and S model 

provide reliable 

results for isothermal 

flows 



Transitional regime GT 

GT 

0.8 

0.6 

0.4 

0.2 

BGK 

0 

0.1 1 10 

δ 

Loyalka (1994) 




GT 

0.8 

0.6 

0.4 

0.2 

BGK S 

0 

0.1 1 10 

δ 






GT 

0.8 

0.6 

0.4 

0.2 

BGK 

S 

BE 

0 

0.1 1 10 

δ 



Loyalka & Hickey, 1991 




GT 

0.8 

0.6 

0.4 

0.2 

BGK 

S 

BE 

0 

0.1 1 10 

δ 



Loyalka & Hickey, 1991 

BGK does not, 

while S provides 

reliable results for 

non-isothermal flows 



Numerical data on GP and GT can be found in 

Sharipov & Seleznev, Data on Internal Rarefied 

gas Flows J. Phys. Chem. Ref. Data 27, 657-706 

(1998) 

Numerical calculations of ˙ M can be carried out 

on-line 

http://fisica.ufpr.br/sharipov 


Direct Simulation Monte Carlo 

M particles are considered simultaneously 

M ∼ 10 7 − 10 8 


Direct Simulation Monte Carlo 

M particles are considered simultaneously 

M ∼ 10 7 − 10 8 

• Free motion of particles 

• Interaction with solid surface, Elimination and 

Generation of particles 

• Simulation of collisions 

• Calculation of macroscopic quantities 


Orifice flow 

⑦ 

W = ˙ M 

, 

˙M0 

P0 

˙ M0 = 

✲ 

P1 

√ πa 2 

vm 

P0 


Orifice flow into vacuum P1 = 0 

W 

1.6 

1.5 

1.4 

1.3 

1.2 

1.1 

1 

DSMC 

0.9 

0.01 0.1 1 10 100 1000 

δ 



W 

1.6 

1.5 

1.4 

1.3 

1.2 

1.1 

1 

DSMC 

Liepman, 1961 

0.9 

0.01 0.1 1 10 100 1000 

δ 



W 

1.6 

1.5 

1.4 

1.3 

1.2 

1.1 

1 

DSMC 

Liepman, 1961 

Barashkin, 1977 

0.9 

0.01 0.1 1 10 100 1000 

δ 



W 

1.6 

1.5 

1.4 

1.3 

1.2 

1.1 

1 

DSMC 

Liepman, 1961 


Fujimoto & Usami, 1984 

0.9 

0.01 0.1 1 10 100 1000 

δ 



W 

1.6 

1.5 

1.4 

1.3 

1.2 

1.1 

DSMC 

Liepman, 1961 


1 

Fujimoto & Usami, 1984 

Jitschin, 1999 

0.9 

0.01 0.1 1 10 100 1000 

δ 


Orifice flow at P1 > 0 

W 

1.6 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

P1/P0=0 

P1/P0=0.1 

P1/P0=0.5 

P1/P0=0.9 

Sreekanth, 1965 

Porodnov et al., 1974 

0.1 1 10 100 1000 

δ 



Flowfield at P0/P1 = 100 and δ = 1000 

8 1.0000 

4 

0 

0 4 8 12 16 

0.3162 

0.1000 

0.0316 

0.0100 ϱ/ϱ0 

8 1.0000 

4 

0 

0 4 8 12 16 

0.6687 

0.4472 

0.2991 

0.2000 T/T0 

8 8.0000 

4 

0 

0 4 8 12 16 

2.8284 

1.0000 

0.3536 

0.1250 Mach 

density 

temperature 

number 



Flowfield at P0/P1 = 10 and δ = 1000 

8 1.0000 

4 

0 

0 4 8 12 16 

0.5623 

0.3162 

0.1778 

0.1000 ϱ/ϱ0 

8 1.0000 

4 

0 

0 4 8 12 16 

0.6687 

0.4472 

0.2991 

0.2000 T/T0 

8 5.0000 

4 

0 

0 4 8 12 16 

1.5811 

0.5000 

0.1581 

0.0500 Mach 

density 

temperature 

number 


Holweck pump 

✻ 

gas flow 

η 

✻ 

✲ 

ξ 

U 

✲ 

Fore vacuum 

✏ 

✏✶ ❇ 

A ❇ 

✏✶ 

A ❇ 

. α ... 

... . 

✛ 

✲ 

✏ 

✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏ ✏✏ W 

High vacuum 

✻ 

H 

❄ 

Scheme of pump 


Holweck pump 

✻ 

gas flow 

η 

✻ 

✲ 

ξ 

U 

✲ 

Fore vacuum 

✏ 

✏✶ ❇ 

A ❇ 

✏✶ 

A ❇ 

. α ... 

... . 

✛ 

✲ 

✏ 

✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏✏✏ ✏✏✏✏✏✏ ✏✏ d 

b 

y 

W 

High vacuum 

✻ 

✲x 

z✠ 

 

✛ ✲✛ 

✲ 

a 

✻ 

H 

❄ 

✻ 

❄ ✻ 

h 

❄ 

Scheme of pump 

Scheme of groove 


Holweck pump. First stage 

Four problems are solved 

(P ) 

• Poiseuille flow in x direction G x (δ) 

(P ) 

• Poiseuille flow in z direction G z (δ) 

• Couette flow in x direction G (C) 

x (δ) 

• Couette flow in z direction G (C) 

z (δ) 

✛ ✲✛ 

y 

✻ 

✲x 

z✠ 

 

✲ 

d b 

✻a 

❄ ✻ 

It takes long CPU time 

h 

❄ 


Holweck pump. Second stage 

S - pumping speed 

ℓ dP 

dη 

= sin α 

U 

vm 

It takes short CPU time 

Gη = S 

vmℓ 2 

P cos α[G(C) z − ℓz 

ℓ G(C) x ] − GηPh 

(P ) 

G z sin2 α + ℓz ) 

ℓ 

G(P x cos2 α 


Holweck pump. Second stage 

It allows us easily to change many parameters 

such as: 

• groove inclination 

• fore vacuum and high vacuum pressures 

• angular velocity of rotating cylinder, 

• species of gas 

• temperature of the gas 


Holweck pump. Results 

Limit compression pressure ratio 

1000 

K0 

100 

10 

1 

Ar 

N2 

1 10 100 

δf 


Holweck pump. Results 

Dimensional pumping speed 

0.3 

Gη 

0.2 

0.1 

0 

N2 

Ar 

0.1 1 10 100 

δh 


Recent results 

Slip and jump boundary conditions 

• Velocity slip coefficients of single gas for 

different gas-surafce interaction laws 






• Velocity slip coefficients for gaseous mixtures 







• Temperature jump coefficient of single gas 

for different gas-surafce interaction laws 







• Temperature jump coefficient of single gas 

for different gas-surafce interaction laws 

• Temperature jump coefficient for gaseous 

mixtures 



over the whole range of the gas rarefaction 

• Single gas flows through long tubes and 

channels 





channels 

• Mixture gas flows through long tubes and 

channels 





channels 


channels 

• Gas flow through orifices and slits 





channels 


channels 


• Couette flow of a single gas 





channels 


channels 



• Couette flow of mixtures 





channels 


channels 



• Couette flow of mixtures 

• Modelling of vacuum pumps 


Numerical programs 

Some calculations of flow rate through tubes, 

channels and orifices can be carried out in 

dimensional quantities on line 

http://fisica.ufpr.br/sharipov/ 


Numerical programs 

Some calculations of flow rate through tubes, 

channels and orifices can be carried out in 

dimensional quantities on line 

http://fisica.ufpr.br/sharipov/ 

Thank you for your attention

RAREFIED GAS DYNAMICS AND ITS APPLICATIONS TO VACUUM ...

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