Vacuum I

1. Vacuum I G. Franchetti CAS - Bilbao G. Franchetti

2. Index Introduction to Vacuum Vacuum and the Beam Flow Regimes Creating Vacuum G. Franchetti

3. Vacuum in accelerators All beam dynamics has the purpose of controlling a charged beam particle Structure of magnets and RF structure have the purpose of creating a proper guiding (E,B) structure However in an accelerator are present a jungle of unwanted particles which creates a damaging background for beam operation G. Franchetti

4. Characteristic of Vacuum Vacuum is considered as a Gas which is characterized by Macroscopic Quantities Pressure Temperature Density Aim: Microscopic Quantities Composition Minimize the interaction of beam with vacuum gas Kinetic Theory G. Franchetti

5. Typical numbers R.J. Reid G. Franchetti

6. Particle – Wall collision m The velocity after wall collision depends on the particle-wall interaction v2 v1 Momentum is propagated to the wall interaction G. Franchetti

7. Pressure SI Units: m other units: G. Franchetti

8. Particle – Particle interaction In a gas large number of collisions v1’ m m Most likely process P-P collision v1 For elastic collisions: 1) Energy conservation 2) Momentum conservation Interaction m v2 v2’ m Temperature G. Franchetti

9. Typical numbers Air at T= 20o C 20% O2  MO = 8x2 g-mole = 8x2/NA = 2.65 x 10-23 g 80% N2 MN = 7x2 g-mole = 7x2/NA = 2.32 x 10-23 g Therefore Molecules run fast ! But the average velocity is G. Franchetti

10. Velocity Distribution When a gas is at equilibrium the distribution of the velocity follows the Maxwell-Boltzmann distribution max velocity ave. velocity G. Franchetti

11. Equation of state An ideal gas in a container of volume V satisfies the equation of state For a gas in equilibrium SI units P  Pressure [Pa] V  Volume in [m3] n  moles [1] T  absolute temperature [K] R0 = 8.314 [Nm/(mole K)] universal constant of gas kB= 1.38x10-23 [JK-1] Boltzmann constant NA = 6.022x1023 [1] Avogadro’s Number G. Franchetti

12. Mean free path Vacuum Gas Beam free path = path of a particle between two collisions G. Franchetti

13. Mean free path A Effective area covered by the rest atoms Cross-section Density of atoms Transverse area v v # particles at l + Δl that did not collide N(l) = particle at l Particles that did not collide G. Franchetti

14. Mean free path How many particles collide between and ? Therefore dN particles travelled a distance l and then collided Probability that a particle travel l and then collide is In a gas Mean free path G. Franchetti

15. Example Air at T = 200 C and P = 1 atm From equation of state atom/m3 Diameter of molecule of air  Mean free path G. Franchetti

16. N2 at T = 200 C, d = 3.15 10-10 m G. Franchetti

17. Vacuum and Beam Vacuum Gas Beam G. Franchetti

18. Beam lifetime After a beam particle collides with a gas atom, it gets ionized and lost because of the wrong charge state with respect to the machine’s optics Beam of particles going through a vacuum gas survives according to As beam particle have a velocity v0, then From the equation of state Beam lifetime sets the vacuum constrain Beam lifetime G. Franchetti

19. Example It is important to know the cross-section of the interaction beam-vacuum Of what is formed the vacuum ? Does it depends on the energy ? Example LHC H2 at 7 TeV at T = 50 K Nuclear scattering cross section at 7 TeV for different gases and the corresponding densities and equivalent pressures for a 100 hours beam lifetime for LHC Design Report G. Franchetti

20. O. Gröbner, CAS 2007 G. Franchetti

21. Impingement rate For a Maxwell-Boltzmann distribution Wall or surface J  units: [#/( s m2)] and G. Franchetti

22. More on collision mean free path mean free path G. Franchetti

23. Knudsen Number Flow Regimes Continuous Regime Particles collide mainly among them the fluid is continuous Molecular Regime Transitional Regime particles collide mainly with walls This is a regime in between continuous and molecular SI units: [1] G. Franchetti

24. The Throughput Vessel with a gas at The quantity Number of particles is proportional to the number of particles is called throughput SI units: [Pa m3/s] G. Franchetti

25. Conductance In absence of adsorption and desorption processes the throughput does not change 1 2 T = constant equal SI units: [m3/s] Conductance G. Franchetti

26. Composition (Series) C2 C1 Pu1 Pd1=Pu2 Pd2 equal throughput T = constant Conductance of 1+2 G. Franchetti

27. Composition (Parallel) C1 Vessel 2 Vessel 1 Conductance of 1+2 Pu Pd C2 T = constant G. Franchetti

28. Molecular Flow Kn > 0.5 P < 1.3 x 10-3 mbar D = 0.1 m Regime dominated by Particle – Wall interaction Molecular regime m Classical Mechanics m v2 v1 ? v2 v1 ? ? G. Franchetti

29. Cosine-Law A particle after a collision with wall will loose the memory of the initial direction After a collision a particle has the same velocity |v| The probability that particles emerge in a certain direction follow the cosine law surface = solid angle G. Franchetti

30. Consequences Concept of Transmission probability 1 2 A gas particle can return back! In the molecular regime two fluxes in opposite direction coexist N1 = particle per second through 1 N2 = particle per second through 2 particle flux = G. Franchetti

31. Conductance of an Aperture Gas current Volumetric flow Downstream Upstream Conductance Throughput G. Franchetti

32. Conductance of a long tube long tube = no end effect d stationary state L  0 make α  ∞ which is wrong: This result is valid for L long The dependence of 1/L is consistent with the rule of composition of conductance G. Franchetti

33. Continuum flow regime Kn < 0.01 P > 6.5 x 10-2 mbar V = 0.1 m Roughly more than 100 collision among particles before wall collision Local perturbation of propagate through a continuum medium Collision among particles create the Viscosity Collision with walls cancel the velocity Max. velocity zero velocity at the walls G. Franchetti

34. Viscous/Laminar regime Reynold Number Laminar Regime ρ = density of gas [Kg/m3] v = average velocity [m/s] Dh= hydraulic diameter [m] Turbulent Regime A = cross sectional area B = perimeter η= fluid viscosity [Pa-s] M. Lesiuer, Turbulence in Fluids, 4th Edition, Ed. Springer G. Franchetti

35. Reynold Number as function of Throughput For air (N2) at T=20C η = 1.75 x 10-5 Pa-s with Therefore the transition to a turbulent flow takes place at the throughput of For a pipe of d = 25 mm At P = 1 atm this threshold corresponds to v = 0.389 m/s [mbar l /s] [mm] G. Franchetti

36. Laminar Regime Fluid is incompressible: true also for gas when Ma < 0.2 Fluid is fully developed Motion is laminar  Re < 2000 Velocity at Walls is zero Fully developed flow entry length velocity distribution at injection in the pipe velocity distribution of a fully developed flow G. Franchetti

37. Conductance in Laminar Regime long tube = no end effect D Pu Pd L Conductance: where The conductance now depends on the pressure! G. Franchetti

38. Conductance in Turbulent Regime long tube = no end effect The throughput becomes a complicated relation (derived from the Darcy-Weisbach formula) Darcy friction factor, dependent from the Reynold number D Pu Pd L G. Franchetti

39. Sources of vacuum degradation G. Franchetti

40. Evaporation/Condensation At equilibrium condensation PE = saturate vapor pressure Example: water vapor pressure evaporation G. Franchetti

41. Outgassing The outgassing is the passage of gas from the wall of the Vessel or Pipe to the vacuum Mean stay time Θ= fraction of sites occupied “Throughput” due to outgassing Ns = A x 3x1015 A = surface in cm2 P. Chiggiato, CAS 2007 G. Franchetti

42. Leaks QL < 10-6 mbar l /s QL < 10-5mbar l /s QL < 10-4mbar l /s very tight tight leaks Small Channels High vacuum system Diameter hole Leaks rate 0.01 mm 10-2 mbar-l/s 10-10 m 10-12 mbar-l/s K. Zapfe, CAS 2007 317 years for leakage of 1 cm3 G. Franchetti

43. Pumps and Gas flow Ideal Pump particles that enter in this chamber never returns back !! Pumping speed S0 [m3/s] S0 is Here Example N2 at T=20C but Take D=0.1m S0 = 0.92 m3/s G. Franchetti

44. Pumps and Conductance Pumping speed of ideal pump S0 Pumping speed S [m3/s] d conductance C As the throughput is preserved Example: if the pipe is long l=100d m3/s G. Franchetti

45. Pumping Process -- Pump-down Time -- Ultimate Pressure hole Pump throughput due to evaporation throughput due to outgassing throughput due to leaks effective throughput G. Franchetti

46. Pumping Process -- Pump-down Time -- Ultimate Pressure Therefore Ultimate pressure  equilibrium between sources of throughput and pumps throughput P/P0 1 t [s] Pump down time  G. Franchetti

47. Multistage pumps First stage pump down Second stage P/P0 1 t [s] G. Franchetti

48. Creating the Vacuum: Pumps Examples of vacuum in some accelerators LHC  10-10 – 10-11 mbar (LHC Design Report) FAIR HEBT  10-9 mbar SIS100  10-12 mbar A. Kraemer EPAC2006, TUPCH175 J.Y. Tang ICA01, TUAP062 G. Franchetti

49. Positive Displacement Pumps Principle: A volume of gas is displaced out of the Vessel Piston Pumps Out Vessel Rotary Pumps Liquid ring pumps volume V Pout Dry vacuum pump Pin A compression of the volume V is always necessary to bring the pressure from Pin to a value larger then Pout G. Franchetti

50. Piston Pumps Vessel Po Pi Po Minimum volume Vmin G. Franchetti