J D Fuerst

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A v Fermi National Accelerator Laboratory 
TM-1676 
An Investigation of Thermally Driven Acoustical 
Oscillations in Helium Systems • 
J. D. Fuerst 
Fermi National Accelerator Laboratory 
P.O. Box500 
Batavia, Illinois 60510 
August 1990 
* Published in the proceedings of the Low Temperature Engineering and Cryogenics Conference 1990, 
Southampton, England, July 17-19, 1990. 
0 Operated by Universities Research Association Inc. under contract with the United States Department of Energy 
Low Tempera.ture Engineering a.nd Cryogenics Conference and Exhibition, July 17-19, 1990 
AN INVESTIGATION OF THERMALLY DRIVEN ACOUSTICAL OSCILLATIONS IN 
HELIUM SYSTEMS 
J.D. Fuerst 
Fermi National Accelerator Laboratory* 
Batavia, Illinois USA 
Introduction 
Ab:itract 
The phenomenon of thermal-acoustic oscillation is seen to arise 
spontaneously in gas columns subjected to steep temperature gradients, 
particularly in tubes connecting liquid helium reservoirs with the ambient 
environment. This is often the arrangement for installed cryogenic 
instrumentation and is accompanied by undesirably large heat transfer rates 
to the cold region. Experimental data are collected and matched to 
theoretical predictions of oscillatory behavior; these results are in good 
agreement with the analytical model and with previously collected data. 
The present experiment places the open ends of oscillating tubes of various 
lengths and cro88 sections in communication with flowing helium in the 
subcooled, 2-phase, or superheated state while the other ends are 
maintained at some controlled, elevated temperature. Assorted cold end 
conditions are achieved through adjustments to the Fermilab Tevatron 
satellite test refrigerator to which the test cryostat is connected. The 
warm, closed ends of the tubes are maintained by isothermal baths of 
liquid nitrogen, ice water, and boiling water. The method is contrasted to 
previous arrangements whereby tubes are run from room temperature into 
or adjacent to a stagnant pool of liquid helium. Additionally 1 the effect of 
pulsations in the flowing helium stream is explored through operation of 
the refrigerator's wet and dry expanders during data collection. These 
data confirm the theory to which they were compared and support its use 
in the design of cryogenic sensing lines for avoidance of thermoacoustic 
oscillation. 
Under certain conditions columns of gas may be induced to oscillate in an undamped 
manner. Typically this behavior occurs within a length of tube, either open at both ends 
or closed at one end and with frequencies associated with acoustic phenomena. The former 
case establishes what is known as the Rijke oscillation, where heat added to a grid placed 
in the lower half of a vertical pipe drives thermoa.coustic oscillations when gas flows up 
through the pipe. Details concerning this effect are supplied by Feldman [lj. The latter 
case prescribes the Sondhauss effect (after Sondhauss, who in 1850 conducted tests with 
heated glass tubes [21), where heat added to the closed end of a pipe causes gas in the pipe 
to oscillate. In the world of cryogenics this effect is often known as Taconis oscillation, 
after K.W. Taconis who witnessed and described the phenomenon during experiments with 
solutions of 3He in 4He in 194g [3]. The generic term "thermoacoustic oscillation" is also 
used in cryogenics to describe this behavior, as it primarily occurs in liquid helium systems 
where tubes open to the liquid helium environment extend outside the insulation system and 
expose their clos•d ends to the ambient. Chief among the problems associated with this 
behavior is the enormous increase in heat flux from warm end to cold end which can be 
1000 times as great as that explained by conduction down the tube itself [4j. Many 
references are available detailing the history of the phenomenon [1,2,3,4,5,6,7j. 
* Research sponsored by Universities Research Association Inc. under contract with the 
U.S. Department of Energy 
2 
Kramers undertook the first mathematical exploration of the stability curve for 
thermoacoustic oscillations in his paper of 1949 [8]; results were in poor agreement with the 
experimental evidence of the day. This work was extended by Nikolaus Rott in a 1969 
paper [5], resulting in a •uccessful theoretical model mapping out with selected dimensionl~ss 
parameters a stability curve within which oscillations a.re predicted and external to which 
oscillations are nonexistant. Parameters include the temperature ratio (warm end/cold end), 
length ratio (warm length/cold length, each assumed constant and separate~ by a 
discontinuous jump), and ratio of tube diameter t.o Stokes. boundary layer th1cknes~. 
Numerical calculations are performed later by Rott m [91 which completely map out th1S 
curve. It is discovered that there are in fact two branches of the curve: a lower branch 
where the boundary layer thickness is large relative the the tube diameter (obeying the 
"fully viscous" assumption) and an upper branch where the boundary layer thickness is 
large at the cold end but small (normal boundary layer approximations) at the warm end 
[9]. It turns out that most of the upper branch is inaccessible in practice because the 
temperature ratios are excessively high. 
Conclusions supplied by Rott in [9] state that from an oscillation production 
standpoint, the optimum position of the jump in temperature is at the center of the half-
open tube and little effect is seen for increases in length ratios up to 2:1. Also, he predicts 
the minimum temperature ratio capable of supporting oscillations to be about six. This 
value was verified by Hoffman, Lienert, and Quack [10] and by the present investigation. 
Rott determined that the failure of the Kramers theory lies in the nature of helium's 
thermophysical properties. While Kramers' theory produced asymptotes of the stability 
curve, they lie at virtually infinite temperature ratios for helium (see reference [9) for a 
more complete discussion). Using Rott's theory, one sees that in addition to altermg the 
geometry of the oscillating system to eliminate oscillations, one can either heat or cool the 
closed end to reach stable portions of the stability curve. Following Haycock [2], it is seen 
that heating the closed end will increase the kinematic viscosity and control acoustic 
damping while cooling the closed end decreases the thermal potential below that capable of 
sustaining oscillations - the lower branch of the curve is reached. 
Theoretical treatments of similar problems include Liburdy's analysis of an annular 
half-open tube [11], modeling the fact that often times thermoacoustical oscillations may be 
damped by inserting a small rod into the oscillating line. The analysis follows Rott closely. 
This work shows that stability is increa.!ed by a decrease in annular gap; this is effected by 
a significant upward adjustment in the lower branch of the stability curve and a somewhat 
more modest downward shift in the upper branch. Additional work by Liburdy [12] 
investigates the stability question for laminar steady flow of cryogens. The analysis proceeds 
in similar fashion to that of the half-open tube case. The effects of flow are shown to 
reduce the instability envelope by shifting the upper branch downward as flowrate increases. 
Not much change in the lower branch is recorded. 
Experjmenta.l Analysis 
Recently, experimental work has centered around quantitative verification of the theories 
of Rott. For example, Gorbachev, et al [13] used a 4 millimeter inner diameter tube with 
the open end at 4.2K and the closed end at ambient conditions to take data. Warm and 
cold lengths of tube were altered by adding coils at the respective locations. For these 
data, "warm" is defined as T>150K while "cold" means T<l50K. Hoffman, Lienert, and 
Qua.ck 1101 took data specifically designed to verify the Rott stability curve; results were 
qualitative y correct but shifted. Yazaki et al [14] also experimentally verified the Rott 
curve but used a slightly different technique. Instead of using a half-open tube, the 
oscillator consist•d of a U-shaped tube with both open ends at ambient and the bend in the 
"U" existing at the midpoint and located at the cold region, providing great flexibility in 
selection of operating conditions. 
The current apparatus makes use of a warm isothermal bath and helium bleed flow to 
enforce the assumption in the Rott theory of constant warm and cold temperatures 
sepa.rated by a discontinuity as shown in Figure 1. Thermometry is provided at appropriate 
locations to verify these conditions. Tests are conducted usirtg half-open tubes with a 
diaphragm-type pressure transducer mounted at the warm end such that minimal a'dditional 
warm volume extraneous to the tube is created. Helium is contained in a flowing line 
contiguous with the refrigeration system. The system used is one of many satellite 
refrigerators operated in support of the Tevatron superconducting dipole array at Fermi 
3 
National Accelerator Laboratory [15]. The refrigerator provides a ncminal 600 watts of 
refrigeration capacity at 4.5K with about one gram per second of 4.5K liquefier load. 
Adjustment of the refrigerator operating conditions allows . data to :,. taken at a variety. of 
cold end temperatures and pressures. Tubes of three different diameters are tested, with 
modifications to the length ratios accomplished by adding lengths of warm tubing. A map 
of the stability curve is generated by varying tube diameter, tube length ratio, and fluid 
flow parameters (see results next section). Also, the effects of pulsations in the cold helium 
stream were evaluated by operating the refrigerator's reciprocating expansion engines during 
data collection {16/· The oscillating sensor tubes are of 3.18 mm O.D, 6.35 mm O.D, and 
12.7 mm O.D., al with 0.89 = wall thickness. Cold lengths are about 500 mm. 
By varying the pressure and temperature in the flowing line through the test cryostat, 
a map of the stability/instability region was generated. A major disappointment was the 
inability to verify both branches of the stability curve, as done by Hoffman et al. This 
would require use of a higher temperature warm bath and may be attempted in the future. 
Oscillations were clearly defined and there was rarely doubt as to their presence or absence. 
Operation of the test refrigerator was smooth enough to allow extended periods of steady 
state, during which measurements were taken. Three graphs of the stability curve were 
produced, each with a different length ratio (0.3, 1.0, 2.0). These graphs are shown in 
figures 2, 3, and 4. Superimposed on the data are the theoretical predictions of Rott for 
each case (solid lines). To fix nomenclature, ; = length ratio (warm to cold), a = 
temperature ratio (warm to cold), and A = ratio of tube diameter to Stokes boundary layer 
th.ickness = (d
0
/2)[e.,/(!,_v,)]'" where d0 = tube I.D., A, = sound speed of gas at T,0", v, = 
kinematic viscosity at '!'"""' and 1, = cold length. 
Specific to the ; = 0.3 graph Figure 2, one sees some agreement between the 
theoretical prediction of the stability curve and the data. Without the ability to precisely 
control fluid properties (see Yazaki, et al), e.n exact demarce.tion between oscillating and 
non-oscillating regions is difficult to produce. Rather, the author has attempted to generate 
enough data to evenly cover the field with cloaely spaced data points, allowing the border 
between oscillation and non-oscillation to manifest itself. As the graph demonstrates, it is 
possible to state that at least qualitatively, the Rott theory is correct for S = 0.3. 
Osdllation frequencies varied between 25 and 125 Hz. The S = 1.0 gre.ph Figure 3, 
contains many more data points and hence ought better to define the ste.bility curve. 
However, much of the data is clumped together around certain more ree.dily accessible 
parameter values. Nevertheless, agreement between the date. and the Rott theory for > = 
1.0 io better than the ; "' 0.3 case. Frequencies were in the 16 to 100 Hz range, with the 
majority of the data oscillating at about 25 Hz. Fine.Uy, the ; = 2.0 graph Figure 4, 
shows the best (albeit still clumpy) distribution of data and comes closest to defining the 
inner portion of the stability curve. Frequencies ranged in the 12 to 90 Hz range, with 
most points showing frequencies of about 17 Hz. All three graphs show pronounced 
intermingling of the oscillation/no oscillation points in the upper right corners of the plots. 
Data in this region all share the characteriotics of high tempe1·ature ratio and high A. This 
combination is created almost exclusively in the 12.7 mm dia. tube due to the role of 
diameter in A. It is possible that thermoacoustic oscillations are less likely in tubes of 
relatively large I.D.; data from other sources do not contain sensing lines as large as the 
largest used in this experiment. 
No particular effects were recorded in view of the fact that the low temperature end of 
the sensing tubes wu maintained via a flowing helium stream rather than a stagnant pool -
the results are in reasonable agreement with experiments done using the latter method. 
Operation of the refrigerator's expansion engine superimposed a low frequency "jump' 1 
corresponding to engine speed (2-6 Hz) on the sensing lines' pressure traces. Oscillations of 
thermoacoustic frequencies were never observed to initiate or dissipate with the 
commencement or cessation of engine operation. However, on rare occasions, oscillations 
were seen to arise in an otherwise steady state situation when a check valve in the 
refrigerator distribution piping began to chatter. Rough estimates of chatter frequency 
correspond to thermoacoustic frequencies. The conditions under which these oscillations were 
induced generally did not fall inside the stability boundary (the oscillating region). When 
the check valve ceased to chatter, the oscillations dise.ppeared. It is possible to conclude 
that oscillations may be induced in a sensing line by introducing pressure fluctuations of 
suitable frequency at the cold end even if conditions point to o. stable configuration on the 
Rott stability curve. This feature is worth further investigation. 
4 
Conclusions 
Thermoacoustic oscillations interfere with the efficient operation of a cryogenic helium 
system, potentially increasing the system heat leak by a factor of 1000. Unless the 
phenomenon is understood, the likelihood of suffering these effects in a cryogenic system is 
high due to the almost universal requirement for instrumentation lines (pressure, 
temperature, flow, etc.) which must run from the cold regions of the system (where they are 
open to the process) out to the ambient (where they are connected to transducers, etc.). 
The best theory explaining and predicting the occurrence of thermoacoustical oscillation is 
that of Nikolaus Rott. This theory describes the phenomenon in terms of a "stability 
curve" defined by the dimensionless parameters ~' a, and A. For a given geometry, cold 
end condition, and ambient temperature, it is possible to predict whether or not oscillations 
will occur. Looked at from the other direction, one can use the Rott stability curve to 
design instrumentation lines that will not oscillate for given ambient temperatures and cold 
end conditions. 
Results confirm the lower branch of the Rott stability curve as well as the nose of the 
curve. Unfortunately, no data wa.s produced to pin down the upper branch of the curve. 
More effort is required in this area. Low frequency pulsations in the flowing helium stream 
were seen to have no effect on the presence or absence of oscillations; rather, the oscillation 
(or lack thereof) merely "rode" the lower frequency pulsations. However, oscillations were 
induced by the onset of chatter in a check valve somewhere in the system (of frequencies 
similar to thermoacoustic). These oscillations vanished when the chatter quit. The 
possibility of such an event must be taken into account when designing cryogenic sensing 
lines. Finally, oscillation in lines of relatively large diamete1 is less predictable, perhaps 
playing a less· dominant role in the system behavior compared to other effects such as 
convection. 
Acknowledgements 
Jay Theilacker of Accelerator Division/cryogenic systems is to be thanked for his advice 
and support. The technicians of Accelerator Division/Cryogenic Systems were instrumental 
in the construction a.nd installation of the experimental apparatus. 
References 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
Feldman, K.T., JR., Review of the literature on Rijke' thermoacoustic phenomena, L 
Sound and Vibration, 7: 83 (1968). 
Ha.ycock, R. H., Techniques for eliminating therma1~acoustical oscillation in cryogenic 
instrumentation, Cryogenically Cooled Sensor Technology, 245: 143 {1980). 
Taconis, K. W. et al, Measurements concerning the vapour-liquid equilibrium of solutions 
of 3He in 'He below 2.lQK, Physica, 15: 733 (11149). 
Bannister, J.D., Spontaneous pressure oscillations in tubes connecting liquid helium 
reservoirs to 300K environments, in: "Liquid Helium Technology", Pergamon Press, 
New York (1966) p. 127. 
Rott, N., Damped and thermally driven acoustic oscillations in wide and narrow tubes, 
J, of Applied Math and Physics, 20: 230 (1969). 
Lord Rayleigh, "Theory of Sound", Vol. II (Macmillan 181l6). 
Clement, J.R. and Gaffney, J., Thermal oscillations in low temperature apparatus, in: 
"Advances in Cryogenic Engineering", Vol. 1, Plenum Press Inc., New York (1960) p. 
302. 
Kramers, H.A., Vibrations of a gas column, Physica, 15: 971 (L949l. 
Rott, N., Thermally driven aco'1stic oscillations. Part II: Stability imits for helium, L 
of App!ied Math and Physics, 24: 54 (lll73). 
Hoffman, T. et al, Experiments on thermally driven gas oscillations, Cryogenics, 13: 490 
[1973). 
Liburdy, J.A., Thermal acoustic oscillation in annular flow of low temperature helium, 
American Society of Mechanical Engineers (paper) n82-HT-12 (1982). 
Liburdy, J.A. and Wofford, J.L., Stability criterion for the occurrence of thermally 
induced oscillations in steady laminar flow, American Society of Mechanjcal Engineers 
(paper) n79-HT-74 (1979). 
5 
13. Gorbachev, S.P. et al, Simple method for elimination of thermoacoustic oscillations in 
cryogenic tubes, Instruments and Experimental Techniques, 29: 258 (1986). 
14. Yazaki, T. et al, Stability limit for thermally driven acoustic oscillation, Cryogenics, 19: 
393 (1979). 
15. Rode, C.H., Tevatron cryogenic system, in "12th Inter. Conf. of High Energy Acee!.", 
1983. 
16. Peterson, T.J., Fermilab 1s satellite refrigerator expansion enginesi in: HAdvances in 
Cryogenic Engineering", Vol. 29, Plenum Press, New York (1984), p.443. 
10 
CUD 
LENGTH 
LHE 
TEMP. 
TEMP. 
PRESS. 
INSULATING VACUUM 
Figure 1: ·Experimental Apparatus 
0 
0 0 
0 
0 
0 
0 
• • • . "· .... 
0 
o.P 
0 
0 
Co 
0 
0 
0 
• 0 • 0 • 
• &. • 6b 
oc9 oc9 0 
0 0 
0 
0 
0 
0 0 0 
Oo 
0 0 0 
0 
• oscillation• 
o no oscHl&tions 
2-1-~~~~~~~~~~~~~~~~......i-
2 10 100 1000 
Figure 2: Data for length ratio > = 0.3 
10 
0 
0 
0 
••• •• 
0 
0 
6 
·~ .... ·~ •• -~ 0 CW! • • 0 
•• ' ... 0. •• :• etc. 
• • 0 
0 0 
0 
• oscillation11 
o no oscillations 
2.+-................... ,__ ...... -.-....... ...-...----....... ...+-
2 10 100 1000 
Figure 3: Data for length ratio > = 1.0 
100+-...._ ................. ~ ..... ----~--.1~.-.~--............ ~ 
10 
0 
0 
2 10 
~., .. . .... -,,. .. ~ 
• • • • •• 
• • 0 
0 
Oo 
0 
0o 8 
0 0 
oO % 0 
o o• 
0 
0 
• 0 
0 
0 
o o""i, 
0 
0 /1 
0 o 8 
<P 
'i>"o 
~ 0 
0 
0 
0 
0 
• oseillalions 
o no oscillation• 
JOO 1000 
Figure 4: Data for length ratio > = 2.0 


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