Summary of Rb loss simulations
The goal: assess different techniques
for optimization density gradient at
the ends of plasma cell
Parameters
•
•
•
•
•
•
Temperature: T = 200⁰C = 473K
Density: n = 7x1020m-3
Dimensions: inner diameter: 4cm; length: 10m
Knudsen number ~0.1 (rarefaction parameter ~10)
Mean free path ~3.6mm
Pressure: 𝑝 = 𝑛𝑘𝑇 = 6.5Pa = 0.065mbar = 0.049torr
• Thermal velocity: 𝑣 =
2𝑘𝑇
𝑚
= 301m/s
• Rubidium atom mass: 1.443x10-25kg
• Rubidium atom diameter: 496-606pm
Fast valve, No orifice (iris)
What is depletion length?
Fast valve: 10ms
𝑓𝑛 𝑥, 𝑡 =
𝑛0
𝑥 𝑚
1 + erf −
2
𝑡 2𝑘𝑇
50cm at 1-2ms
𝛾−1
2 𝛾−1
𝑛
2
2
𝛾 𝛾−1 𝑚 𝑥
=
−
𝑛0
𝛾
𝛾 + 1 2 𝛾 + 1 2𝑘𝑇 𝑡
1D Theory (FM+C) + 1D DSMC Petrenko A,
https://indico.cern.ch/event/357090/contri
bution/2/material/slides/2.pdf
50cm at 3-4ms
Kersevan R,
https://indico.cern.ch/event/328455/con
tribution/11/material/slides/1.pdf
120000
100000
80000
50cm at 2-3ms
60000
50cm at 3ms
40000
20000
15cm at 0.67ms
0
2D DSMC Petrenko A,
https://indico.cern.ch/event/357090/
contribution/2/material/slides/2.pdf
0
0.2
0.4
0.6
3D DSMC Plyushchev G
0.8
1
Shakhov EM, Non-stationary rarefied gas
flow into vacuum from a circular pipe
closed at one end
Electron trapping
=> Length of the transition region should not exceed 10-15cm
Lotov KV, http://arxiv.org/pdf/1408.4448v1.pdf
Very simple estimation of outflow
through orifice
• Particles escapes from orifice (continuum
regime) with speed of sound: 𝑐 =
𝛾
2
2𝑘𝑇
𝑚
=
275m/sec.
• Number of particles escaped per second:
𝑛𝜋𝑎2 𝑐 = 1.51x1019/sec => 2.18mg/sec
• In reality, it is half-density should be used =>
1.09mg/sec
• Total mass of Rb in 10m@4cm = 1.27mg
Less simple estimation of outflow
through orifice
• Mass outflow for infinitely large volume to vacuum
(Sharipov F., Rarefied gas flow through a thin orifice):
𝜋𝑚
2
𝑀 = 𝑊𝑎 𝑛𝑘𝑇
2𝑘𝑇
Mass
of Rb
Accounts for rarefaction,
for molecular regime = 1
Orifice
radius
Pressure
• Rarefaction: 1.23 => W=1.15 => M=0.77mg/sec
• For continuum: W=1.51 => M=1.01mg/sec
• Total mass of Rb in 10m@4cm = 1.27mg
Summary of leak rate values
•
•
•
•
Molecular flow theory: 0.67mg/sec
Rarefied flow: 0.77mg/sec
Continuum flow: 1.01mg/sec
Simple continuum estimation: 1.09mg/sec
• Simulation: 0.52mg/sec
Possible explanation of error: we don’t have
infinitely large volume
Analytical tails
• When orifice small compared to inner diameter of plasma
cell, the physics, near orifice, is similar to gas flow through
a thin orifice
• Gas density on axis of orifice (Danilatos G., Direct
simulation Monte Carlo study of orifice flow):
𝑛
𝑛0
1
2
= −
𝑥
2
2𝑎
𝑥
2
2𝑎 +0.25
• => ramp length is order of magnitude of orifice diameter.
• => gas density on axis symmetric with respect to orifice
plane
• This equation could be used for plasma wakefield
simulation to see the influence of this profile on electron
trapping
3D DSMC simulation Double Orifice no
Source: boundary conditions
50cm
4cm
4cm
1cm
Symmetry
wall
The idea :
some time
20cm
Thermal wall
Fast valve
To have large volume between both orifice to drive outflow to gain
3D DSMC simulation Double Orifice no
Source: density profile (1e7 particles)
Density profile integrated over last 4ms (25.9-29.9ms) in order to increase statistics:
10000
8000
6000
4000
2000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Red line: theory for infinitely large volume
Blue lines: orifice 1 and 2
0.7
0.8
3D DSMC simulation Double Orifice no
Source: density profile (1e7 particles)
14000
14000
0.00ms
12000
14000
5.31ms
12000
10000
10000
10000
8000
8000
8000
6000
6000
6000
4000
4000
4000
2000
2000
2000
0
0
0
0.1
0.2
0.3
0.4
0.5
14000
0.6
0.7
0
0.8
0
0.1
0.2
0.3
0.4
0.5
14000
0.66ms
12000
19.9ms
12000
0.6
0.7
0.8
10000
10000
8000
8000
8000
6000
6000
6000
4000
4000
4000
2000
2000
2000
0
0
0.1
0.2
0.3
0.4
0.5
14000
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
10000
10000
8000
8000
6000
6000
6000
4000
4000
4000
2000
2000
2000
0
0.2
0.3
0.4
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0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.6
0.7
0.8
29.9ms
12000
8000
0.1
0.5
25.2ms
0
10000
0
0.4
14000
15.3ms
12000
0
0.3
0
0
14000
1.33ms
12000
0.2
12000
10000
0
0.1
14000
9.96ms
12000
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Approximately 15cm at 30ms
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Density in plasma cell, a.u.
3D DSMC simulation Double Orifice no
Source: density profile (1e7 particles)
Time, sec
3D DSMC simulation Single Orifice
with Source: boundary conditions
DSMC: hard sphere model
50cm
2cm
4cm
1cm
Symmetry
wall
Thermal wall
Source
(constant flux)
20cm
2cm
3D DSMC simulation Single Orifice
with Source: results
Final data
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Convergence reached at around 0.5sec (with plasma cell tube initially filled)
Simulation inflow (=outflow): 0.52mg/sec. This equivalent to 45g/day.
If orifice will be open only 3 seconds each 30 seconds: 4.5g/day.
3D DSMC simulation Single Orifice
with Source: density profile
Density profile in the center of plasma cell (inside r=4mm)
4000
3500
3000
2500
2000
1500
1000
500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
3D DSMC simulation Double Orifice
with Source: boundary conditions
50cm
2cm
2cm
4cm
1cm
Symmetry
wall
Thermal wall
Source
(constant flux)
The idea of second orifice:
10cm
2cm
1. Prevent any possible vortex creation
2. Both orifices are symmetrically placed with respect to
source tube => the symmetry simplifies the understanding of problem (in case with low
collisions between particles, it could be considered as superposition of source and two
orifices with plasma cell with orifice at the end
3D DSMC simulation Double Orifice
with Source: results
Final data
0.04
0.02
Results are very similar to the simulation with single
orifice: leak rate:
0.52mg/sec
convergence:
0.5sec
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
5000
4000
3000
2000
1000
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
3D DSMC simulation Double Orifice
with Source: Flow
3D DSMC simulation Double Orifice
with Source (constant density): results
8x1020m-3
9x1020m-3
7x1020m-3
10x1020m-3
Conclusions:
1. In this case (stationary case) the density near the source is higher that the density in
plasma cell.
2. Thus the injection tube (between Rb source and plasma cell) should be very short and
should have large diameter in order to have smaller density gradient difference.
3. The injection tube should be as close as possible to orifice
Challenge 1
• For Rb vapor, pressure
depends on temperature
• Pressure near the source
should be higher, then in
plasma cell.
• => source should be as
close as possible to
plasma cell (and to iris)
• Can our source provide
this constant flux
(~0.77mg/sec)?
Steck, D.A., Rubidium 85 D Line Data
Evaporation rate
Surface, m2
Density, 1020m-3
Flux = 0.77mg/sec
Temperature, K
𝐽=𝛼
𝑛𝑘𝑇 − 𝑝𝑒𝑞
2𝜋𝑚𝑘𝑇
𝑝𝑒𝑞 = 101325 × 10
4040
4.312− 𝑇
Pound G.M., Selected Values of Evaporation and Condensation Coefficients for Simple Substances
Possible practical application
10m
Valve 2
Valve 1
Source
(constant flux)
Questions:
1. If valve 1 and 2 close, what is the time to fill this volume with Rb source from one end?
(Initially plasma cell is empty!) (for particular geometry it is > 5sec)
2. If valve 2 is closed and valve 1 is open, what is the time to reach the equilibrium?
(Initially plasma cell is filled with Rb!) (for particular geometry it is > 8sec, see next slide)
Total mass of Rb in 10m@4cm = 1.27mg. Flow through orifice = 0.52 - 0.77mg/sec
Possible closed loop Rb vapor system:
70 °C
200 °C oil tank
R ~ 10 cm
A valve which is
normally closed
and opened to
let beams pass.
It’s not necessary
for this valve to
be leak tight
and fast.
The amount of liquid Rb in the reservoir can
be limited to ~10 cm3 (15 g). The main
question is how much liquid Rb will stick to
70 °C walls before it starts to flow down to
the reservoir? Let’s assume the 70 °C surface
is ~ πR2 = 3.14*(10 cm)2 = 300 cm2 and Rb
layer is 1 mm thick => V = 300 cm2 * 0.1 cm =
30 cm3 => The total mass of Rb is likely to be
below 100 g.
At 70 °C equilibrium vapor pressure of Rb
is 2000 times lower than at 200 °C.
Rb flow is 0.5 mg/sec = 100 g / 2 days =>
Another option may be a cycled operation –
70 °C tank will be heated up once a day or so.
Rb in this system is in the closed loop
because it’s either in a liquid or in a vapor
form.
Rb
190 °C
70 °C
Challenges: 2
• If we going to have orifice system (or source
system) from both ends of plasma cell => the
both sources should be perfectly aligned (to
avoid density ramp)