Development of mathematical models for accommodation Manssour FADIL V. Delalande, P. Dolegieviez, B. Rannou SAFERIB Joint Meeting LMU Munich October 1212-13, 2006 - Munich

DEFINITION Accommodation

Capital phenomenon in gas transmission Badly known Very few experimental studies

SPIRAL2

Development of accommodation models Database of the accommodation coefficient

“The accommodation coefficient is defined as the ratio of the energy actually transferred between impinging gas molecules and a surface, and the energy which would be theoretically transferred if the impinging molecules reached complete thermal equilibrium with the surface” A. Roth

2000 °C 20 °C 390 cm

265 cm

137Cs

Without accommodation With accommodation (α α = 0.7)

~ 0.2 % ~ 0.022 %

Models of calculation of the accommodation coefficient

The accommodation coefficient

Model1 (M1) Elastic scattering

Configuration 1 (M1C1)

Configuration 2 (M1C2)

Model2 (M2) Manson formalism

1 Dimension (M2D1)

3 Dimensions (M2D3)

Model 1 : Moderation by elastic scattering

r r VB = 0

r′ mAVA

r mAVA

θ

r′ mBVB

′

VA + / −

T ∝V

ϕ

m A cos(θ ) + / − mB2 − m A2 sin 2 (θ ) = VA m A + mB

2

Hypothesis : 1.

Twall

Discrete calculus

Tgas

Tr

α= Tr −Tgas Twall −Tgas M1C1

2..a.. α constant 2.b. Or non constant

M1C2

Model 2 : Manson formalism Hypothesis : 1. Gas in equilibrium at TG 2. Wall in equilibrium at TS 3. Maxwell-Boltzmann distribution 4. Discrete calculus

α E (TG , TS ) = α E (TG , TS ) =

E f − Ei < E f > − Ei E f − k BTG k BTS − k BTG

TG − TG α E (TG , TS ) = TS − TG

E f = k BTG ∞

∞

1 TG = dEi ∫ dE f E f α MB ( Ei , TG )α R ( E f , Ei , TS ) ∫ kB 0 0 α R ( E f , Ei , TS )

α MB ( Ei , TG ) =

1 e k BTG

Differential coefficient of reflection −

Ei k BTG

Differential coefficient of incidence

General behaviour

α

1,2 1 0,8 0,6

experience M1C1

0,4

M1C2 M2D1

0,2

M2D3 0 0

0,2

0,4

0,6

0,8

1

1,2 Mg/Mw

Calculation /Experiment

1,E+02

1,E+01

1,E+00

1,E-01

1,E-02

M1C1/Exp M1C2/Exp M2D1/Exp M2D3/Exp

Baule approximation

2 .4 µ α(T) ≡ (1+ µ)2 α

1,2 1 0,8 0,6

experience

0,4

M1C2 M2D1

0,2

M2D3 0

Baule 0

0,2

0,4

0,6

0,8

1

1,2 Mg/Mw

SCA code

SCA code

CONCLUSION These calculations are not in disagreement with the experience Good and acceptable approach

Development of experimental database (now more the 130 experiments) Possible improvement of the models (Van der Waals well) Definition of the validity limitations Applications in SPIRAL2