Rb flow in AWAKE
Gennady PLYUSHCHEV
(independent scientist
aka
gentleman scientist)
AWAKE plasma cell: overview
•
•
•
Ø4cm, 10m tube
Rb, 200°C, 7x1014cm-3
Rarefied regime
Goal: sharp density gradient.
•
Fast valves are too slow:
(10ms x 300m/s = 3m)
Solution: orifices + continuous flow
Theory
𝜋𝑚𝑘𝑇
,
2
Mass flow through orifice: 𝑀 = 𝑊𝑟 2 𝑛
𝑛
where 𝑊 ≅ 1.3
1
Axis density distribution near orifice: 𝑛 = 2 −
0
Evaporation rate: 𝐽 =
𝛼(𝑝−𝑝𝑠𝑎𝑡 )
,
2𝜋𝑚𝑘𝑇
where 𝑎 ≅ 1.0, and log10
Stationary flow in long tube: 𝑝 + 𝑝𝑎 =
where 𝑝𝑎 =
4𝜇𝜎𝑝
𝑎
2𝑘𝑇
,
𝑚
(𝑥/2𝑟)
2 (𝑥/2𝑟)2 +0.25
𝑝𝑠𝑎𝑡
101325
𝑥
4040
𝑇
(𝑝0 + 𝑝𝑎 )2 + 𝐿 (𝑝1 + 𝑝𝑎 )2 −(𝑝0 + 𝑝𝑎 )2
and 𝜎𝑝 is the viscous slip coefficient.
𝑥
= 4.312 −
If 𝑝1 − 𝑝0 ≪ 2𝑝0 then 𝑝 = 𝑝0 + 𝐿 (𝑝1 − 𝑝0 )
Density profile
Flows in AWAKE
Calculated flows in order to
have 10% density gradient
in plasma cell:
Evaporation area (m2) required to
provide the mass flow rate of 1.0mg/s
as a function of density and
temperature of source:
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
n
x
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.
- 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 axis
2. calculate Rb deposition
Density in infinitely large volume:
𝑛
1
Empirical formula: 𝑛 = 2 −
0
(𝑥/2𝑟)
2 (𝑥/2𝑟)2 +0.25
𝑛
1 𝐴
Naively (for point source): 𝑛 = 2 𝑥 2
0
Condensation rate limitation
How condensation works?
p
Flow from orifice. In
equilibrium, it is equal
to condensation rate
Condensation rate: 𝐽 =
1
𝛼(𝑝−𝑝𝑠𝑎𝑡 )
,
2𝜋𝑚𝑘𝑇
thus: 𝑝 − 𝑝𝑠𝑎𝑡 = 𝛼 𝐽 2𝜋𝑚𝑘𝑇
Condensation condition: the
pressure near the wall is higher
then saturation pressure
𝑛 − 𝑛𝑠𝑎𝑡
1 𝑀 2𝜋𝑚
=
𝛼 𝑚 𝑘𝑇
Minimum possible density near the wall is limited by condensation rate.
Even if the wall temperature is 0K, the pressure near wall ≠ 0.
Example of minimum possible density: 0.896mg/s; 39°C; L = 0.2m; r = 0.1m
Cylinder side
Cylinder base
Simulation of plasma cell end
-
Cylindrical volume r = 0.1m; L = 0.2m
Base of the cylinder with orifice at 200°C
Another base is at low temperature
Side of the cylinder at temperature which
goes linearly from high to low
- Boundary condition:
If 𝑛 > 𝑛𝑠𝑎𝑡 then condensation
If 𝑛 < 𝑛𝑠𝑎𝑡 then evaporation
thus we underestimate the density by
approximately 1011cm-3
Simulations performed:
- 3 temperature used: 39°C; 70°C; 1K
- Standard and smaller diameter (0.1m
instead of 0.2m)
1.
Temperature decrease linearly
2.
3.
39°C
200°C
0.1m
1K
4.
0.18m
70°C
200°C
0.
200°C
39°C
200°C
200°C
r=0.05m
39°C
Results of simulation: -272°C
0.
1K
200°C
0.18m
This simulation was performed to
compare calculated values with
theoretical ones.
Results of simulation: -272°C
Results of simulation: 2D density
39°C
70°C
39°C
.05m
Results of simulation: axis density
39°C
70°C
39°C
.05m
Results of simulation: Rb on side walls
39°C
70°C
39°C
.05m
Results of simulation: Rb on side walls
39°C
70°C
39°C
.05m
Results of simulation: Rb on base wall
39°C
39°C
70°C
.05m
Results of simulation: Rb on base wall
39°C
70°C
39°C
.05m
Conclusions
- 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 minimum density is limited by condensation rate (1012cm-3 for r=0.1m; L=0.2m). To
decrease this minimum large volume should be used (or developed surface!);
- Even in infinitely large volume the minimum of density is limited by vacuum flow
propagation;
- The Volume for Rb to Expand should be as large as possible;
- The side walls should be kept at low temperature.