**3.9-GHz Helium Vessel, Low Temperature Maximum Allowable Working
Pressure**

**EDMS \# D00000**

+-------+-----------+----------------------+------------+------------+
| **    | ## Date   | **Description**      | **         | **Approved |
| Rev** |           |                      | Originated | by**       |
|       |           |                      | by**       |            |
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| None  | 13 March  | Final version for    | Tom        |            |
|       | 2009      | operational          | Peterson   |            |
|       |           | readiness review     |            |            |
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|       |           |                      |            |            |
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|       |           |                      |            |            |
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|       |           |                      |            |            |
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**3.9-GHz Helium Vessel, Low Temperature Maximum Allowable Working
Pressure**

Tom Peterson

First draft 2 January 2009, revised 13 March 2009

[Introduction.]{.underline}

The pressure vessel engineering notes for the 3.9 GHz RF cavity helium
vessels for the tests in Fermilab's Horizontal Test Cryostat (HTC)
established a maximum allowable working pressure (MAWP) in accordance
with Fermilab's ES&H Manual chapter 5031 (following ASME code with some
exceptions) of 2.0 bar both warm and cold \[ref. 1\]. The helium vessels
installed in the 3^rd^ harmonic cryomodule are designated as cavity #5
(IND-102), cavity #3 (IND-112), cavity #7 (IND-114), and cavity #8
(IND-115). Maximum pressure in HTC, even with a cold loss-of-vacuum
accident, would not exceed 2.0 bar, so a higher MAWP for cold operations
was not necessary. ("Cold" means cryogenic temperatures of 80 K or
less.)

However, in FLASH at DESY, the 3^rd^ harmonic cryomodule will be
connected in a system for which low temperature pressures could possibly
exceed 2.0 bar, and for which the 1.3 GHz RF cavity helium vessels are
rated up to 4.0 bar cold (and 2.0 bar warm). These 3^rd^ harmonic helium
vessels must be rated at a pressure consistent with the rest of the
system, a low temperature MAWP of 4.0 bar.

[Scaling room temperature MAWP to low temperatures.]{.underline}

The ASME Pressure Vessel Code (referred to as the "code"), Section VIII,
Div 1, UG-98(c) says, "Maximum allowable working pressure may be
determined for more than one designated operating temperature, using for
each temperature the applicable allowable stress value." \[ref. 2\]
Thus, one may have a higher cold MAWP with a higher cold allowable
stress.

[Stress evaluation.]{.underline}

The pressure vessel engineering notes for the 3^rd^ harmonic cavity
helium vessels included, as the third load case, "Primary + Secondary
stresses evaluation at cryogenic temperature (internal pressure, own
weight of the assembly, thermal contraction)". (See "3^rd^ load case",
pp. 50 -61, in "Pressure Vessel Engineering Note for the 3.9-GHz Helium
Vessel, Cavity #5" \[ref. 1\]. The stresses were evaluated for 2.0 bar
pressure, low temperature material properties, and compared to room
temperature allowable.

Multiplying the predicted stress for 2.0 bar by 2 overestimates the
stress with 4.0 bar pressure, since it is equivalent to multiplying
pressure, gravity, and thermal contraction by 2. Thus, in the concluding
summary table below, 4.0 bar stresses are conservatively estimated as 2
x the 2.0 bar stresses.

[Niobium low temperature allowable stress.]{.underline}

The document, "Engineering note: Summary of Niobium mechanical
properties" \[ref. 3\], contains a thorough review of the reported
properties of high RRR, heat-treated niobium such as in our 3.9 GHz RF
cavities. The conclusion is that a room-temperature allowable stress for
niobium, determined by ASME formulas from very conservatively low
ultimate (115 MPa) and yield (38 MPa) strengths, is 25 MPa \[ref. 3, pg.
13\]. The additional 0.8 derating factor in accordance with Fermilab
rules for an "in-house" pressure vessel results in a 20 MPa allowable
stress at room temperature, which is the value used in our pressure
vessel engineering notes.

The summary of niobium mechanical properties also finds low temperature
yield stresses of 320 MPa or greater and ultimate strengths of 600 MPa
or greater. These result in a cold allowable stress based on tensile
strength of 600/3.5 x 0.8 = 137 MPa \[ref. 3, pg. 13\], over six times
the 20 MPa room-temperature value. Since doubling pressure from 2.0 bar
to 4.0 bar would no more than double the stresses, the increase of low
temperature allowable stress to 137 MPa is more than enough of an
increase in strength to allow a cold MAWP of 4.0 bar.

[Titanium.]{.underline}

The allowable room temperature stress for grade 2 titanium in our
pressure vessel analysis \[ref. 1\] is, including the Fermilab factor of
0.8, 0.8 x 83 MPa = 66.4 MPa, where the 83 MPa comes from the ASME code
allowable stress tables, Tab 1B, Section II, Part D \[ref. 2\].

The only location where doubling of calculated stress would result in a
stress exceeding the room-temperature allowable is in the titanium
bellows.

Low temperature ultimate strength for titanium increases to 872 MPa at
77 K \[ref. 3, Table 4, pg. 10, for grade 2 titanium, welded\] resulting
in a primary allowable stress of 872/3.5 x 0.85 (welded pipe) x 0.8
(taking Fermilab 0.8 factor) = 169 MPa, more than double the
room-temperature allowable.

[Niobium-Titanium.]{.underline}

The room-temperature allowable stress for niobium-titanium alloy
(titanium- 45% niobium), determined by ASME formulas from (Wah-Chang
data) ultimate stress (546 MPa) and yield strength (480 MPa) is 132.6
MPa. The additional 0.8 derating factor in accordance with Fermilab
rules for an "in-house" vessel results in an allowable stress of 106
MPa.

Room-temperature allowable stress is sufficient to allow doubling of all
calculated stresses in the NbTi and still not exceed allowable. So it is
not necessary to invoke a higher low temperature allowable for NbTi.

[Cavity collapse.]{.underline}

The non-linear cavity buckling analysis in the pressure vessel
engineering note \[ref 1, pg. 66\] finds a safety factor of 10 using
room-temperature niobium properties. Deformation which may lead to
buckling is seen at 20 bar external pressure. Thus, even using
room-temperature properties, we have a safety factor of 5 relative to
4.0 bar on the cavity.

[Required test pressure for a 4 bar cold MAWP]{.underline}

From ASME Section VIII, Div 2, Part 8, 8.3.1, the required
room-temperature test pressure for qualifying a vessel for a temperature
other than room-temperature is Pt = 1.15 x MAWP x (St/S) where Pt is
test pressure, S is allowable stress at the operating temperature, and
St is allowable stress at the test temperature. As the tabulated values
below show, our 2.3 bar (33.5 psid) pressure test exceeds the required
test pressures for the 4 bar cold MAWP.

[Conclusion.]{.underline}

+-------------+--------------+-------------+--------------+-----------+
|             | Peak cold    | Peak cold   | Cold         | Test      |
|             | stress       | stress at   | allowable    | pressure  |
|             | location     | 2.0 bar x 2 | stress (MPa) | for 4 bar |
|             |              | (MPa)       |              | cold MAWP |
|             |              |             |              | (bar)     |
+-------------+--------------+-------------+--------------+-----------+
|             |              |             |              |           |
+-------------+--------------+-------------+--------------+-----------+
| Niobium     | Cavity end   | 51.7 x 2 =  | 3xS = 411    | 1.15 x    |
|             | cells region | 103.3       |              | 4.0 x     |
|             |              |             |              | (20/137)  |
|             |              | (Cavity 5   |              | = 0.67    |
|             |              | note Table  |              |           |
|             |              | A18)        |              |           |
+-------------+--------------+-------------+--------------+-----------+
| Titanium    | Bellows      | 115.6 x 2 = | 3xS = 508    | 1.15 x    |
| Gr. 2       | central      | 231.2       |              | 4.0 x     |
|             | convolution  |             |              | (         |
|             |              | (Cavity 5   |              | 66.4/169) |
|             |              | note Table  |              | = 1.8     |
|             |              | A17)        |              |           |
+-------------+--------------+-------------+--------------+-----------+
| NbTi        | NbTi joint,  | 40.9 x 2 =  | 3xS = 318    |           |
|             | Ti shell     | 81.8        |              |           |
|             | weld region  |             | (room        |           |
|             |              | (Cavity 5   | temperature  |           |
|             |              | note Table  | allowable)   |           |
|             |              | A22)        |              |           |
+-------------+--------------+-------------+--------------+-----------+

**Summary table -- worst-case low temperature stresses scaled up to 4
bar internal pressure, compared to cold allowable stress. Cold stress
from the engineering note is multiplied by 2 for a most conservative
estimate of stress with 4.0 bar internal pressure. Note that total
allowable stress is 3 x the primary allowable since secondary stresses
are also included.**

Four bar internal pressure results in low temperature stresses which are
less than the low temperature allowable stress for all three helium
vessel materials: niobium, titanium and NbTi. Thus, the cold MAWP for
these helium vessels may be set at 4.0 bar.

[References]{.underline}

1.  Pressure Vessel Engineering Note for cavity #5 (IND-102). Pressure
    vessel engineering notes for cavity #3 (IND-112), cavity #7
    (IND-114), and cavity #8 (IND-115) are "similar vessels" to IND-102
    per FESHM 5031 and have short engineering notes highlighting the
    differences from IND-102. These engineering notes have been placed
    on a public website at
    <http://ilc-dms.fnal.gov/Workgroups/CryomoduleDocumentation/3rd-Harmonic-Cryomodule-for-DESY/>
    and are also archived with Fermilab's Technical Division safety
    office and Fermilab's ES&H section.

2.  ASME Boiler and Pressure Vessel Code, 2007 edition

3.  "Engineering note: Summary of Niobium mechanical properties," by G.
    Wu, H. Edwards, T. Peterson, dated 12 December 2008. Part of the
    3^rd^ harmonic cryomodule documentation package located online at
    <http://ilc-dms.fnal.gov/Workgroups/CryomoduleDocumentation/3rd-Harmonic-Cryomodule-for-DESY/>
    .