Rb flow in AWAKEGennady PLYUSHCHEV

(CERN - EPFL)

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

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 0.14°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:

𝑱=𝜶 (𝒏−𝒏𝒔𝒂𝒕 )𝒌𝑻

√𝟐𝝅𝒎𝒌𝑻=𝜶 (𝒏𝟒 √𝟖𝒌𝑻𝝅𝒎

−𝒏𝒔𝒂𝒕

𝟒 √ 𝟖𝒌𝑻𝒘𝒂𝒍𝒍

𝝅𝒎 )𝑛4= 14𝜋 ∫

0

𝜋 /2

∫0

2 𝜋

𝑛𝑉 cos𝜃 sin 𝜃ⅆ𝜑ⅆ𝜃

𝑉=1𝑛0

∫0

∞

𝑉 ⅆ𝑛=4 𝜋( 𝑚2𝜋𝑘𝑇 )

32∫0

∞

𝑒−𝑚𝑉 2

2𝑘𝑇 𝑉 3ⅆ𝑉=√ 8𝑘𝑇𝜋𝑚

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 28°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: 27°C

The simulation shows good agreement with the theoretical curve for infinite volume (for the case with Twall < 28°C and L<0.5m):

- Cylindrical volume r = 0.1m; L = 0.2m

- Base of the cylinder with orifice at 200°C

- Another base and tube is at low temperature 27°C

- Side of the cylinder is also at low temperature 27°C

Results of simulation: 27°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 28°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 (< 28°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 < 28°C

Rb flow on surface of Expansion Volume:

Orifice

Plasma Cell Expansion Volume

Θ

ϕExpansion Volume Wall

r

Normalization for cosine distribution

ⅆΩ

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 0.1°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

28°C;- The transition from hot (200°C) to low (28°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 orificeξ along the conical wall

z

r

ξ

Cone shape: coordinates

The temperature of conical surface as a function of ξ is calculated according the heat equation (the walls are perfectly isolated; the temperature is fixed at the ends).

Heat flux:

𝑇=𝑐1ξ𝑟0

ln(1+ ξ𝑟0 sin𝛼)ξ𝑟0sin𝛼

+𝑐2

1sin𝛼→0

Cone shape: temperature profile

At ξ = 0 the wall temperature is equal to 200°C, at ξ = 0.15m the wall temperature is equal to 28°C r0 = 0.06mα = 60°

The density on axis increases up to 15% due to the conical surface

Additional density

Cone shape: Density on axis