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Rb flow in AWAKEGennady PLYUSHCHEV
(CERN - EPFL)
AWAKE plasma cell: overview
β’ Γ4cm, 10m tube
β’ Rb, 200Β°C, 7x1014cm-3
β’ Rarefied regime
Goal: sharp density gradient.
β’ Fast valves are too slow: (10ms x 300m/s = 3m)
Solution: orifices + continuous flow
TheoryMass flow through orifice: , where
Axis density distribution near orifice:
Evaporation rate: , where , and
Stationary flow in long tube: where , and is the viscous slip coefficient.If then
Density profile
Flows in AWAKE
Evaporation area (m2) required to provide the mass flow rate of 1.0mg/s as a function of density and temperature of source:
Calculated flows in order to have 10% density gradient in plasma cell:
400
400
405
405
410
410
415
415
420
420
425
425
430
430
435
435
440
440
445
445
450
450
455
455
460
460
465
465
470
470
475
475
480
480
n, cm-3
n
/ n
, %
T, K
1 2 3 4 5 6 7 8 9 10
x 1020
1
2
3
4
5
6
7
8
9
10
400
410
420
430
440
450
460
470
480
0.140810.14081
0.263630.26363
0.386460.38646
0.509280.50928
0.63211
0.632110.75493
0.75493 0.87776
0.87776 1.0006
1.0006 1.1234
1.1234 1.2462
1.24621.3691
1.36911.4919
1.4919
1.6147
1.6147 1.7375
1.73751.8604
1.8604
1.9832
1.9832
2.106
2.106
2.2288
2.2288 2.35162.4745
n, cm-3
n
/ n
, %
T, K
1 2 3 4 5 6 7 8 9 10
x 1020
1
2
3
4
5
6
7
8
9
10
0.5
1
1.5
2
Source temperatures in AWAKE
Evaporation area: 10cm2
T1 T2
T1
T2 - T1
To set the density gradient with 0.5% accuracy, the temperature of sources should be set with 0.14Β°C accuracy
COMSOL simulation- Qualitative analysis- Continuum flow- What is the structure of
density near orifice inside plasma cell? Is there density maximum near orifice?
n
x
x
n
COMSOL results: Pressure
COMSOL results: Pressure Zoom
There is density overshoot (1-2%) near orifice in front of source
Ends of plasma cell
- At each end of plasma cell there is Volume for Rb to Expand (Expansion Volume).
- The volume has cold wall in order to condensate the Rb and reduce the density
- Thus sharp density gradient created through the orifice
- Simulation to study the Rb flow beyond the plasma cell (Volume for Rb to Expand)
Goal: 1. calculate density on axis2. calculate Rb deposition
Grant estimation: 0.5m available for Expansion Volume
Condensation rate: MaxwellCondensation rate: . This equation obtained from Maxwell distribution:
π±=πΆ (πβππππ )ππ»
βππ πππ»=πΆ (ππ βπππ»π π
βππππ
π β πππ»ππππ
π π )π4= 14π β«
0
π /2
β«0
2 π
ππ cosπ sin πβ πβ π
π=1π0
β«0
β
π β π=4 π( π2πππ )
32β«0
β
πβππ 2
2ππ π 3β π=β 8ππππ
ConCondensation Evaporation
Thus, according to this theory, directed flux on surface will condensate immediately.
Density in infinitely large volume
Empirical formula: Naively (for point source):
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-6
10-5
10-4
10-3
10-2
10-1
100
101
x, m
n /
n 0
For x = 0.5m: 2.5x10-5. Which corresponds to 1.75x1010cm-3, for nominal density of 7x1014cm-3
Saturation pressure / density
-20 -10 0 10 20 30 40 5010
7
108
109
1010
1011
1012
T, C
n, c
m-3
Solid phase: T<39CLiquid phase: T>39C
Saturation density of to 1.75x1010cm-3 corresponds to 28Β°C. Thus it is logical to keep the Expansion Volume under this temperature.
In this case, the pressure in Expansion Volume will be determined mainly by the theoretical limit for infinite volume and not by the saturation density due to evaporation from the Expansion Volume surface.
Results of simulation: 27Β°C
The simulation shows good agreement with the theoretical curve for infinite volume (for the case with Twall < 28Β°C and L<0.5m):
- Cylindrical volume r = 0.1m; L = 0.2m
- Base of the cylinder with orifice at 200Β°C
- Another base and tube is at low temperature 27Β°C
- Side of the cylinder is also at low temperature 27Β°C
Results of simulation: 27Β°C
In order to have density profile close to the theoretical limit for infinite volume: Rb flow should not hit the surface with temperature higher than 28Β°C (for Expansion Volume of 50cm). Thus all Rb which hits walls will be condensed. The transition from hot to cold temperature should be on lateral wall parallel to Rb flow:
Rb
Cold walls (< 28Β°C)
Plasma Cell
Transition from hot to cold temperature should be on this surface (parallel to Rb flow).
The Expansion Volume should be long enough to trap most of the Rb. For 50cm long chamber, the escaping Rb:
Guidelines for Expansion Volume
The shape and lateral size of Expansion Volume is not crucial for density profile on axis.
Rb deposition for Twall < 28Β°C
Rb flow on surface of Expansion Volume:
Orifice
Plasma Cell Expansion Volume
Ξ
ΟExpansion Volume Wall
r
Normalization for cosine distribution
β Ξ©
For cylindrical Expansion Volume (r = 0.2m; L = 0.5m) the Rb layer per 2weeks:
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
r, m
Rb
laye
r, c
m/(
2wee
ks)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
0.1
0.15
0.2
x, m
Rb
laye
r, c
m/(
2wee
ks)
Perfect Expansion Volume shape
For Sphere D=50cm, after 14day (24 hours per day) the Rb layer would be less than 1mm:
Rb flow on surface of Expansion Volume:
From this equation we can derive the Expansion Volume shape, for which the Rb flow is constant for any point on surface of Expansion Volume. For cosine distribution this is Sphere:
Rb
Plasma Cell
Conclusions
Plasma Cell:- The density profiles and flows inside the plasma cell was calculated analytically, using
the long tube approximation;- The on axis density has overshoot (~2%) near the orifice in front of the source;- The temperature of Rb sources should be controlled with better than 0.1Β°C precision;
Expansion Volume:- Even in infinitely large volume the minimum of density is limited by vacuum flow
propagation;- For the 50cm long Expansion Volume, the temperature of walls should be less than
28Β°C;- The transition from hot (200Β°C) to low (28Β°C) temperature should be on lateral wall of
Expansion Volume which is parallel to Rb flow;- The Expansion Volume should be long enough to capture most of the Rb;- The shape and lateral size of Expansion Volume is not crucial for density profile on axis;- For homogeneous Rb deposition the Expansion Volume should have the spherical shape.
Expansion Volume with cone shape
z axis (along cell axis, starts at the end of orifice)r axis is perpendicular to orificeΞΎ along the conical wall
z
r
ΞΎ
Cone shape: coordinates
The temperature of conical surface as a function of ΞΎ is calculated according the heat equation (the walls are perfectly isolated; the temperature is fixed at the ends).
Heat flux:
π=π1ΞΎπ0
ln(1+ ΞΎπ0 sinπΌ)ΞΎπ0sinπΌ
+π2
1sinπΌβ0
Cone shape: temperature profile
At ΞΎ = 0 the wall temperature is equal to 200Β°C, at ΞΎ = 0.15m the wall temperature is equal to 28Β°C r0 = 0.06mΞ± = 60Β°
The density on axis increases up to 15% due to the conical surface
Additional density
Cone shape: Density on axis