Beam Pipe Vacuum
An estimate of the pressure in the proposed small diameter beam pipe for STAR based on outgassing of the pipe and it's conductance.
Simplified model of the beam pipe:
The proposed beam pipe is a small diameter pipe centered on the interaction region with larger pipes on either end, extending to the pumps on either side of the STAR hall. By symmetry, the flow in the center is equal in both directions, so for this estimate the beam pipe is assumed closed at the mid plane and the pressure is calculated for one side. For ease of calculation it is assumed that the outgassing source in the small diameter section is, rather than distributed, located at the midpoint in this half section. It is assumed that the transition from the small diameter pipe to the larger pipe is abrupt. Likewise, the outgassing surface in the large diameter section is treated as a point source at the midpoint of this larger section.
Half length of the small diameter pipe
Diameter of the small beam pipe
Length of the large diameter pipe
Diameter of the large diameter pipe
Surface area of the small pipe contributing to outgassing.
Surface area of the large pipe contributing to outgassing
Aluminum outgassing rate after 20 h bake at 100 Deg C
[1]
Amount of gas outgassing from the small pipe
Amount of gas outgassing from the large pipe
Temperature in the STAR hall
Assume worst case that water is the outgassing material. In reality, after baking it will be a lighter gas with higher velocity.
Aperture area of the small diameter pipe.
Aperture area of the large diameter pipe.
Obtain conductance values using the expression:
[2]
where
is the thermal gas velocity
is the probability that a gas molecule entering the pipe passes out the other side. The value depends on the length to diameter ratio and is given in ref. [3]
is the area of the pipe aperture
[3]
Conductance used for the small pipe
[3]
Conductance for the large pipe, full length
To get conductance for half of the long pipe, to be used for pressure contribution from outgassing of the large pipe we have:
To find the pressure from the center section outgassing, add the series conductances:
The conductance added correctly with the Haefer method gives a larger value [4], but for a simple worst case estimate we use this simpler approach.
Pressure from outgassing of the central small pipe
Pressure from outgassing of the large beam pipe section
Estimated total pressure in the central region in excess of the pump pressure. This value will be less for a better estimate of the conduction, but the biggest uncertainty will be the outgas rate after baking.
Note, the biggest contribution is from the larger beam pipe. This could be improved with a second expansion of the pipe once it is clear of the STAR detectors.
Check with simple expression for conductance:
OK very close to same.
References:
John F. O'Hanlon, "A User's Guide to Vacuum Technology" John Wiley and Sons, Inc, 2nd Edition, 1989.
The following items refer to the book referenced above.
1. Appendix C.1 (p.444) As shown in the table there is some variation in this number, with one case about 10 times worse than the number we have chosen and another 4 times better.
2. eq.3.25 (p.34)
3. Table 3.1 (p.36)
4. section 3.4 Molecular Flow (p.46-48)
