The Development of a High Cooling and Low Ultimate Temperature Three
Stage Superfluid Stirling Refrigerator
by
Carolyn L. Phillips
B.S., Mathematics
Massachusetts Institute of Technology, 1999
SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SEPTEMBER 2001
@2001 Carolyn L. Phillips. All rights reserved.
The author hereby grants to MIT permission to reproduce
and to distribute publicly paper and electronic
copies of this thesis document in whole or in part.
MASSACHUSETTS INSTIEUT
OF TECHNOLOGY
DEC 10 2001
LIBRA
JIM
,t
BA1~tR
Signature of Author:
V
I
Department afechanical Engineering
August 31, 2001
Certified by:__
John G. Brisson
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by:.
Ain Sonin
Professor of Mechanical Engineering
Chairman, Committee for Graduate Students
The Development of High Cooling Power and Low Ultimate Temperature
Three Stage Superfluid Stirling Refrigerators
by
Carolyn L. Phillips
Submitted to the Department of Mechanical Engineering
On August 31, 2001 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
The superfluid Stirling refrigerator (SSR) is a Stirling cycle refrigerator which provides
cooling to below 2 K by using a liquid 3He- 4He as a working fluid. In 1990, Kotsubo and
Swift demonstrated the first SSR and by 1999, Patel and Brisson (Patel) had developed
an experimental prototype capable of reaching a low temperature of 248 mK using two
Stirling refrigerator stages. The goal of this thesis was also to develop a deeper
understanding of the SSR and the technical issues involved in its operation and also to
further develop the SSR built by Patel.
This thesis is divided into four parts. In the first part, technical developments to the SSR
are discussed. Also the details of the three-stage SSR developed for this work are
presented. In the second part, a two-stage SSR with larger recuperators are operated to
see whether new ultimate temperature and cooling powers could be achieved. Operating
from a high temperature of 1.05 K and with a 3.0% SHe-4He mixtures, this SSR achieved
a low temperature of 329 mK and delivered net cooling powers of 1 mW at 606 mK, 500
pW at 408 mK. Next this thesis describes the operation of the first three-stage superfluid
Stirling refrigerator. Unfortunately, due to experimental difficulties, the merits of the
three-stage SSR were not demonstrated and further work is still required. The lowest
ultimate temperature reached by three-stage SSR was 338 mK from a high temperature of
1.07 K. The third part of this thesis sought to ascertain the heat dissipation due to the
flexing and relaxing of the bellows in the SSR. The dissipation of two types of bellows
was measured at 1.4 K and the measurements were used to project dissipation rates for a
family of similar bellows. In the fourth part of this thesis, a numerical analysis was
developed to predict the distribution of 3 He particles in the SSR during operation. This
analysis confirmed that the third stage of the SSR requires a working fluid separate from
the first and second stage of the SSR in order to reach temperatures lower than 200 mK.
Thesis Supervisor: John G. Brisson
Title: Professor of Mechanical Engineering
2
Contents
10
1. Introduction
1.1. T he Stirling C ycle...............................................................
12
1.2. Properties of 3He_ He mixtures................................................
14
1.3. Two phase region and the SSR................................................
17
1.4. H istory of the SSR ...............................................................
22
27
2. Experimental Apparatus
2.1. Description of the three-stage SSR...........................................
27
2.2. Development of the Patel and Brisson SSR.................................
31
2.2.1.
H eat Exchangers.........................................................
2.3. Heat leak in the third-stage fill lines..........................................
32
34
Third Stage V alves......................................................
35
2.3.2. Void Space Analysis...................................................
36
2.3.1.
41
3. Experimental Results of SSR
3.1. Review of the Experimental Results of the SSR...........................
41
3.2. Two Stage SSR with Large Upper and Lower Recuperator...............
44
3.2.1.
Description of Two-Stage SSR........................................
3.2.2. Experimental procedure and results...................................
44
47
3.3. Three-Stage SSR with two Large Recuperators and one Small
R ecuperator.......................................................................
52
D escription of SSR ......................................................
52
3.3.1.
3.3.2. Experimental procedure and results...................................
57
60
4. Metal Bellows Dissipation
4.1. Experimental Apparatus........................................................
3
62
4.2. Procedure........................................................................
63
4.3. Development of Bellows Dissipation Model...............................
64
4.4. Bellows Dissipation in the SSR..............................................
67
4.5. Temperature Limits on using Bellows Expander.........................
69
72
5. Theoretical Development
5.1. The Schmidt Model............................................................
72
5.2. Sub-Kelvin Stirling Refrigerators...........................................
76
5.3. Limitations of the 3He-4He Working Fluid................................
76
Concentration.........................
77
5.4. Phonon and Roton Effects on
3He
5.5. H eat E xchangers..............................................................
88
91
6. Summary and Conclusions
93
Bibliography
4
List of Figures
1.1
The States of the Stirling Refrigerator Cycle...................................
13
1.2
A Sketch of the Phase Diagram of the 3He-4 He fluid. ........................
15
1.3
The compression and expansion of 3He in a piston bypassed by superfluid
4 H e ....................................................................................
1.4
A single phase Stirling refrigerator using 3He in a superfluid 4He
background ..........................................................................
mixture in a piston in the two phase region. ........................
1.5
3He-4He
1.6
A depiction of a Stirling cycle whose low temperature piston is operating
partially in the two phase region. ................................................
1.7
17
18
19
The states of the 3He-4He mixture in the low and high temperature piston
mapped onto a sketch of the phase diagram. ...................................
1.8
A representation of a single stage machine consisting of two back-to-back
1.9
180 degree out of phase SSR's exchanging heat through a recuperator.....
1.10
A diagram of the two stage back-to-back SSR design with plastic
recuperators........................................................................
........
2.1
A schematic of the three-stage superfluid Stirling refrigerator.
2.2
The arrangement of alternate layers of Kapton templates within the
2.3
16
20
23
25
28
recuperator to form a counterflow heat exchanger. ...........................
30
A diagram of the plastic heat exchanger ........................................
33
5
2.4
A schematic of the dual superfluid tight valves. ..............................
35
2.5
An illustration of 3He particles flushed into a cold volume..................
37
3.1
A schematic diagram of a two-stage SSR. ....................................
45
3.2
Data for a 1.1 cm compressor stroke and a 1.08 cm expander stroke. (a)
Cooling power versus cold piston temperature for cycle times of 15, 27,
and 40 seconds. (b) Intermediate Piston temperature versus cold piston
tem perature for the data given in (a). .............................................
3.3
49
Data for cooling power versus cold piston temperature for a 15 second
cycle period, 1.1 cm compressor stroke, and a range of expander strokes... 50
3.4
Data for cooling power versus cold piston temperature for a 23 second
cycle period, 1.1 cm compressor stroke, and a range of expander strokes... 50
3.5
Data for cooling power versus cold piston temperature for a 27 second
cycle period, 1.1 cm compressor stroke, and a range of expander strokes... 51
3.6
Data for cooling power versus cold piston temperature for a 40 second
cycle period, 1.1 cm compressor stroke, and a range of expander strokes... 51
3.7
A schematic of the three-stage SSR.............................................
3.8
Calibration curves for resistance versus temperature of the carbon resistor
thermometer mounted on the SSR's hot platform..............................
3.9
59
Data for the cold piston temperature versus second intermediate compressor
piston for different two-stage cycle periods.....................................
4.1
56
Data for the cold piston temperature versus third-stage cycle period for
different two-stage cycle periods.................................................
3.10
54
59
A schematic of the experimental apparatus used to measure bellows
dissipation ..........................................................................
61
4.2
A plot of the measured bellows dissipation as a function of stroke length ...64
4.3
A annular disk model of bellows...............................................
4.4
The weld bead which is modeled as the source of all dissipation. Dissipation
occurs as the angle between the annular disks changes.......................
4.5
64
65
Energy dissipated per unit bead length per cycle versus normalized stroke.. 65
6
4.6
Non-dimensional energy versus non-dimensional stroke as per Eq. 4.3 in
the text. The fit curve is Eq. 4.4 in the text....................................
66
4.7
Fraction of losses in the SSR due to bellows dissipation....................
68
4.8
(a) Predicted cooling power minus bellows dissipation of a third-stage
expander versus cold piston temperature. (b) The fraction the bellows
dissipation makes up of the cooling power. These data points were
generated using the Schmidt model to predict the cooling powers and
Eq. 4.4 to model the bellows dissipation.........................................
69
4.9
A schematic of the idealized expander with valves...........................
69
4.10
Projected cooling power of non-ideal expander (a) and cooling power ratio
of non-ideal expander to ideal expander (b) versus temperature............ 70
5.1
An equation summary of the Schmidt model for a single stage Stirling
cy cle ...................................................................................
5.2
73
The equation summary of the Schmidt model expanded to model a
two-stage Stirling device. A two-stage Stirling device consists of three
74
pistons connected by two recuperators............................................
5.3
The equations summary for a Stirling cycle with an indefinite number of
75
stages chained together..............................................................
5.4
The temperature distribution for a Stirling device is plotted versus position.
The expansion and compression space are assumed to be isothermal. Three
different temperature distributions for the regenerator are shown............
5.5
80
The 3He concentration of the cold, intermediate, and hot piston volume of
a two-stage SSR plotted versus temperature. The black triangles correspond
to the SSR with cold regenerators, the black diamonds correspond to the SSR
with hot regenerators, and the squares correspond to the SSR with linear
regenerators.........................................................................
5.6
The predicted minimum and maximum 3He concentrations of the hot,
intermediate, and cold piston volumes of four different models of the 3He
distribution in the SSR are plotted versus temperature.
The squares
correspond to a model that includes both regenerator volumes and 4He
7
82
fountain pressure. The * correspond to a model that includes fountain
pressure but omits the regenerator volumes. The triangles correspond to a
model that omits fountain pressure but includes regenerator volumes. The
circles correspond to a model that omit both fountain pressure and
regenerator volum es................................................................
5.7
83
A plot of the 3He expected distributions versus temperature for three SSR's
with different total clearance volumes. The triangles correspond to an SSR
whose clearance volume is 27% of its total volume. The * correspond to the
same SSR with half that clearance volume (clearance volume is 15.6% of the
total volume). The black squares correspond to the same SSR with no
85
clearance volum e....................................................................
5.8
The minimum and maximum
3He
concentrations plotted versus temperature
for an SSR whose hot piston is at 1.05 K, intermediate piston at 0.5 K, and
85
cold piston at 0.2 K and which was loaded with 3% mixture................
5.9
The maximum and minimum concentrations of 3He are plotted for the cold
piston volume and hot piston volume of a single stage SSR whose cold and
hot pistons are at 0.1 K and 0.3 K. The squares correspond to the SSR
loaded with a 3% 3He mixture and large recuperators (6.05 cm 2). The
triangles correspond to an SSR loaded with 3% 3He mixture with small
recuperators (1.5 cm2). The diamonds correspond to an SSR loaded with
1.5% 3He mixture and large regenerators (6.05 cm2)........................
86
5.10
Curves of constant 3He osmotic pressure......................................
87
5.11
Maximum and minimum 3He concentrations for the hot and cold piston of a
single stage SSR. The square correspond to an SSR modeled using the
numerical Boltzmann model loaded with a 3% mixture. The diamonds
correspond to an SSR modeled using the numerical Boltzmann model loaded
with a 1.5% mixture. The * corresponds to an SSR modeled using the
interpolated data model loaded with a 3% mixture. The black triangles
correspond to an SSR using the interpolated data model loaded with a 1.5%
m ixture.............................................................................
5.12
The change in 3He moles in three regenerators over the cycle...............
8
. 88
89
5.13
A graph of the change in
3He
moles in the hot piston and hot regenerator in
the two-stage SSR over the cycle.................................................
5.14
A graph of the change in 3 He moles in the cold piston and cold regenerator
in the two-stage SSR over the cycle.............................................
5.15
89
90
A graph of the change in 3 He moles in the cold piston and the regenerator in
the single-stage SSR over the cycle..............................................
9
90
List of Tables
3.1 Single Stage SSR 's...................................................................
42
3.2 Tw o-Stage SSR 's.....................................................................
43
3.3 A comparison of the performances of Patel's different SSR's..................
44
3.4 Displacement volumes and clearance spaces for different expander strokes... 48
4.1 Manufacturer's specifications for bellows.........................................
62
5.1 The specifications for the SSR modeled unless otherwise specified.............
81
9
Chapter 1
Introduction
This thesis is a continuation of research in the development of the superfluid Stirling
refrigerator. The superfluid Stirling refrigerator (SSR) is a Stirling cycle refrigerator that
uses 3He He as the working fluid to cool to sub-Kelvin temperatures. The basic
components of a single stage Stirling refrigerator are a hot (compressor) piston and a cold
(expander) piston connected by a regenerator. The cyclic compression and expansion of
the ideal gas within these pistons pumps heat from the cold temperature reservoir to the
high temperature reservoir. For temperatures below 1 K, the 3He component of the 3 He4 He
mixture behaves as an ideal gas in an inert background of superfluid 4He. Superleak
bypasses in each piston allow the superfluid 4He component to flow freely through the
pistons while the 3 He is expanded and compressed within the piston cylinders.
Kotsubo and Swift demonstrated the first single stage SSR in 1990 [1,2]. In 1992,
Brisson and Swift further developed and improved the single stage SSR performance by
using a recuperative SSR design [3-6]. In the latter design, two refrigerators are operated
180 degrees out of phase with each other and a counterflow heat exchanger is used as the
regenerator. This SSR design is more practical since there is a dearth of low temperature
materials that can provide the high heat capacity necessary for an efficient regenerator
matrix. This first recuperator was made of CuNi tubes. Brisson and Swift achieved a
low temperature of 296 mK while operating from a high temperature of 1.05 K. Using
10
the same machine, Watanabe, Swift, and Brisson later reached 168 mK while operating
from a compressor temperature of 383 mK [7].
In 1997 a new larger single stage SSR was built by Patel and Brisson which used a
counterflow heat exchanger recuperator manufactured from plastic. Plastic was selected
to improve the SSR performance by mitigating the low temperature effect of Kapitza
resistance to heat transfer and to thus improve the low temperature heat exchanger
performance. Using a small plastic heat exchanger, they achieved a low temperature of
344 mK from a high temperature of 1.0 K [8]. Using a large plastic heat exchanger, they
achieved a low temperature of 291 mK [9].
Patel and Brisson also developed a two-stage SSR and using a large upper plastic
recuperator and a medium sized lower recuperator, the SSR achieved a low temperature
of 248 mK [10]. They suggest that a larger low temperature recuperator would reduce
the ultimate temperature achieved by this machine.
The work presented here further develops Patel's two stage SSR and develops a threestage SSR. The goal of this thesis was also to develop a deeper understanding of the SSR
and the technical issues involved in its operation.
This thesis is divided into four parts. The first is a discussion of the cryostat originally
designed by Patel and Brisson and modified in this work. The modification of this
cryostat includes the development of reliable epoxy-copper joints in the recuperator
flanges to replace the often unreliable indium seals used by Patel and Brisson. Another
modification was the superfluid tight valves. The valves closing off the fill lines to the
third stage needed to be superfluid tight in case the fill lines needed to be evacuated to
thermally isolate the low temperature SSR. Finally the use of void volume in the thirdstage fill line to the SSR to insure thermal isolation between the I K and the low
temperature SSR is discussed.
In the second section, the experimental results of the single and two-stage SSR as
reported by Patel and Brisson are reviewed and followed by new two-stage experimental
results of the SSR and three-stage SSR results.
The third section of the thesis discusses the measurements of the dissipative effects in
the flexure of metal bellows within the SSR. The dissipation of the bellows in the SSR
and how this limits the performance of the SSR has always been an unknown. The tests
11
measure the dissipation of two different size bellows for several stroke lengths. The
experimental results are compared to measurements made by Brisson and Swift [12].
The results are made non-dimensional and used to predict the performance of other
bellows of similar construction.
The fourth part discusses the theoretical and numerical models developed to
understand the spatial distribution of 3 He atoms in the SSR during operation. These
models confirm the necessity for the third stage of the SSR to have a working fluid
separate from the first and second stage in order to reach low temperatures of 100 mK
without a phase separation of the
3He-4He
mixture. These models also show the effect of
different variables in the SSR design on 3He atom distribution in the machine.
This first chapter provides a background for the work done in this thesis. It includes a
discussion of the mechanics of a Stirling refrigerator, the physical properties of 3He- 4 He
working fluid, and the history of the use of this working fluid in Stirling refrigerators.
1.1 The Stirling Cycle
The Stirling engine was first patented by Robert Stirling in 1816. The initial application
of the Stirling cycle was as a Stirling engine, a machine that transformed a thermal
energy into mechanical energy, however the same cycle can be used to do the reverse,
transform work into heat flow. The basic concept of the Stirling Refrigerator is that by
the expansion and compression of an ideal gas, heat can be pumped from a low
temperature reservoir to a high temperature reservoir. The three important components
of the Stirling cycle is the compressor, regenerator and expander. Figure 1.1 shows the
operation of the Stirling Refrigerator. The expansion space is thermally linked to a low
temperature reservoir (not shown in Fig. 1.1). The compression space is thermally linked
to a high temperature reservoir (not shown in Fig. 1.1). At the beginning of the cycle the
working fluid (an ideal gas) is in the expansion space and the compression space is
assumed here to have zero volume. In the first part of the cycle, the working
12
Low
Temperature
Reservoir,TL
High
Temperature
Expansion Space
I Compression space
Regenerator
Reservoir,TH
I
I
~
K;
(a)
Isothermal Expansion
.*
QI
(b)
Fluid displaced to the right. Heat absorbed from Eggenerator
(c)
f
777212
i17zzzi~
zzzi
Z21Z11[
.1
X
+
h
Q
Isothermal Compression
(d)
Fldid displaced to the left. Heat rejected to Regererator
(e)
D
L.J
Cool Regenerator
Warm Regenerator
Figure 1.1: The states of the Stirling refrigerator cycle. From (a) to (b) the gas in the expansion
space is expanded, absorbing heat from a low temperature reservoir. From (b) to (c) the gas is
displaced through the regenerator space, absorbing the energy stored there. From (c) to (d) the
gas is compressed, rejecting heat to a high temperature reservoir. From (d) to (e) the gas is
displaced back through the regenerator space, rejecting energy to the regenerator. In our ideal
Stirling recuperator, the temperature distribution is time independent.
13
fluid in the expansion space is expanded at constant temperature, transferring heat from
the low temperature reservoir to the working fluid. In the next part of the cycle the two
pistons are moved in tandem so that the working fluid is displaced through the
regenerator at constant pressure into the compression space. As the working fluid moves
through the regenerator, it absorbs heat from the regenerator so that it enters the
compression space at the temperature of the hot reservoir. It is assumed at this point that
the expansion space has zero volume. The working fluid is then compressed at a constant
temperature, transferring heat to the high temperature reservoir. The working fluid is
then returned at constant pressure back through the regenerator into the expansion space,
rejecting heat to the regenerator on the way so that it enters the expansion space at the
temperature of the low temperature reservoir. Again it is assumed that the cycle ends
with the compression space at zero volume.
1.2 Properties of 3He- 4He mixtures
Unique properties of helium at very low temperatures allow helium to be used as a
working fluid in a variety of low temperature engineering applications. Helium-4
undergoes a phase transition at 2.17 K. Above this temperature the liquid 4He behaves
as a viscous fluid. Below this temperature the 4He is a superfluid, a fluid that flows
without viscosity.
Helium-3, on the other hand, does not go through the superfluid transition until the
temperature of approximately 2-3 mK is reached [13]. For the purpose of this work 3He
always remains a normal (viscous) fluid.
3He_ He mixtures exhibit both superfluid and viscous effects. The two substance
properties of 3He- 4He mixtures are dependent upon both the concentration of the 3 He and
the temperature of the mixture. The superleak bypassed piston mechanism of
manipulating a 3He-4He mixture works only for a temperatures below 2.17 K and low
3He
concentrations. In Fig. 1.3 a phase diagram of the 3 He- 4He mixture shows the
change of properties of the mixture with change in temperature and concentration. The
diagram has three major regions. At low temperatures and high concentrations, the
mixture exists at two phase in a 3He rich liquid and a 4He rich liquid. At high
14
2.2
Viscous Homogeneous
liquid mixture
Superfluid Homogeneous
liquid mixture
E
1.0-
Two Phase Region
00
(Pure 'He)
0.0637
0.5
1.0
3
x, concentration of He
(Pure
3He)
Figure 1.2: A sketch of the phase diagram of the 3He- 4He fluid. The three major regions are the
superfluid homogeneous mixture region, the viscous homogeneous mixture region, and the two phase
region. The dotted lines represent lines of constant 4He chemical potential. The grey portion of the
figure represents the region where 3He acquires the properties of a Fermi gas.
concentrations and high temperature, the helium mixture is a homogeneous viscous fluid.
Finally at lower 3He concentrations at temperatures below 2.17 K, the mixture is a
homogeneous superfluid. To first approximation, the 3He component in this phase
behaves as an ideal Boltzmann gas. The 3He component has viscosity whereas the 4He
component of this phase is superfluid. The lambda line in Fig. 1.2 delineates the
boundary between the homogeneous superfluid mixture, the homogenous viscous
mixture.
The grey region in Fig 1.2 is where the 3He behavior changes from that of a
Boltzmann gas to that of a Fermi gas. The 3He component can still be compressed and
expanded in this region using superleak bypassed pistons. The effect of operating an
SSR in this region has never been thoroughly examined.
The dotted lines in Fig 1.2 correspond to lines of constant 4He chemical potential. Due
to the high mobility of the superfluid 4He in the 3He_ He mixture, the mixture does not
sustain a gradient in the 4He chemical potential. Temperature of the 3He- 4He mixture
15
Motion
Motion
4H
ee
Superleak 'lowL
.
3H
(a)
eAH,
(b)
4
Figure 1.3: From (a) to (b) the piston is moved upwards. The superfluid He can move freely through
3
the superleak and is unaffected by the piston movement. The He particles cannot move through the
superleak, so the concentration of the 3He-4He mixture is reduced. The 3He component of the mixture
is effectively expanded like an ideal gas.
varies throughout the internal volume of the SSR; and consequently the concentration of
the 3He-4He mixture in each of these volumes will varies to maintain a constant chemical
potential throughout the SSR volume. Hence the temperature and concentrations of the
fluid particles in the SSR all lie on a line of constant chemical potential.
Figure 1.3 shows a conceptual drawing of how the 3He can be expanded (or
compressed) by a piston bypassed by a superleak. The volumes "behind" the pistons are
filled with 4He. The superfluid 4He can flow freely between the two volumes. Moving
the piston effectively raises or lowers the concentration of 3He-4He mixture in the lower
volume.
Figure 1.4 shows a conceptual drawing for an SSR with a 3He-4He mixture as a
working fluid. Pistons like the ones shown in Fig. 1.3 are connected by a regenerator. In
operating this SSR, the pistons are moved in the same way as for a standard Stirling
refrigerator. It should be noted that the pistons used in a real SSR do not have sliding
seals. Bellows are used to compress and expand the working fluid
16
IX
,4 He
4He
Figure 1.4: A single phase Stirling refrigerator using 3He in a superfluid 4He background
1.3 Two Phase Region and the SSR
The superfluid Stirling refrigerator does not operate in the two phase region shown in
Fig. 1.2. In this region, the mixture separates into a dense dilute 3He-4 He mixture and a
less dense concentrated
3He
phase. Figure 1.5 shows three "snapshots" of a piston as it
"compresses" and "expands" a helium mixture in the two phase region. Using the second
law, we will argue that when the low temperature piston operates in the two phase region
for the entire cycle, there is no cooling of the low temperature reservoir.
In Fig. 1.5 there is a pure 3He slug floating on top of a dilute phase 3 He- 4He mixture.
We assume this that
3He
slug is sufficiently sized to neither disappear nor to entirely fill
the low temperature piston during the proposed expansion and compression of the low
temperature piston. The 3He-4He mixture in the low temperature piston is connected to a
constant pressure reservoir of superfluid 4He through a superleak. Compression of the
mixture causes the 3 He slug to increase and expansion of the piston causes the
3He
slug to
decrease in size. Assuming the temperature and pressure are fixed throughout the
process, the concentration of the dilute 3He-4He mixture doesn't change. We will assume
that the low temperature piston is attached to an ideal regenerator so that the working
17
3
'-He'dilute
phase
3He diute phase
Pu re4He/
Pure 4He
3
Hee
Pure 4
H
motion
\ Control
motion
Volume
(c)
(b)
(a)
Figure 1.5: The 3He He mixture in the piston in (a) is in the two phase region. When the fluid is
"compressed" (b), the 3He layer grows and when the fluid is "expanded" (c), the 3He layer decreases.
The dilute 3He phase is the same concentration in (a), (b) and (c).
fluid that enters or exits the expander is always at the same temperature. We apply the
second law to the control volume (shown in Fig. 1.5a) around this low temperature piston
over a cycle,
IdSc=
+ jsdN
- csdNOUt + Sgen
where Scv is the total entropy of the control volume,
(1.1)
Q is
the heat transfer into the control
volume, dNin and dNout is the molar flow of mixture into or out of the control volume, s is
the entropy per unit mole carried by that flow, T is the temperature of the control
surface, which by assumption is the same as that of the low temperature reservoir, the
fluid in the system, and the fluid entering and exiting the system, and Sgen is the entropy
produced in the control volume during the cycle. Since the initial and final states are the
same, the net change in entropy over a cycle is zero. The entropy of the mixture can be
expressed as a function of the temperature T and concentration x of the mixture [14].
The fluid exiting the control volume is at the same temperature and concentration of the
fluid exiting the control volume so sin (x, T)= sou (x, T).
The net change in mass in the
system is zero so the net molar flow in equals the net molar flow out, so the entropy
entering and exiting the system in Eq. 1 cancel out. Thus, the equation for the heat
transfer reduces to
18
Q = -TSgen
(1.2)
.
Since T and Sgen are both positive numbers, Q must be less than or equal to zero
indicating that the compression/expansion process does not lead to cooling of the
reservoir when the piston contains a two phase fluid. Note that if our regenerator were
not ideal, the temperature of the entering fluid would have been higher than the
temperature of the exiting fluid and thus also the entropy of the fluid entering the system
is higher than that of the fluid exiting. A SSR equipped with a non-ideal regenerator
would therefore dump more heat into the low temperature reservoir than the ideal
regenerator case discussed above.
4
He
In thermal contact with
Expansion
Volume
3He
Superleak
In thermal contact with
High Temperature
Clearance
Volume
Clearance
Volume
LowTemperature
Mass
U
3He- 4 He
e.
Regenerator
fluid displaced through Regenerator
I
Compression
I Volume
Reservoir
0h)
isothermal compression
I)
h)
I Ifluid displaced through Regenerator
(3 h)
(31
(approximately)
isothermal expansion
(41)14
(40
Q
V(
4 h)
II
Figure 1.6: A de iction of a Stirling Cycle whose low temperature piston is operating partially in the two
phase region. A He slug develops in the low temperature piston during isothermal compression. This
causes significant heat rejection into the low temperature mass during compression. The 3He slug is
dissipated during isothermal expansion. The states of the 3He-4He mixture in the low and high temperature
pistons are labeled on the left and right respectively. In Fig. 1.7, these states are mapped on a two phase
diagram
19
We have demonstrated that if the expander of the SSR is operating in the two phase
region for its full cycle, it cannot continue to cool its low temperature reservoir.
Having
shown that a SSR will not cool if the expander piston is operated entirely in the twophase region, we now wish to consider how the cooling power of the SSR will change as
the state of the fluid in the SSR expander approaches and crosses into the two-phase
region. We wish to show that an SSR that operates in the two phase region at some part
of its cycle, will eventually operate in the two phase region for all its cycle. We will
make this argument by considering an SSR with discrete, not sinusoidal, piston motions.
Figure 1.6 depicts the stages of the cycle of an SSR with clearance spaces (volumes not
swept by the pistons) in its low and high temperature pistons. As the high temp piston
"compresses" the mixture in the SSR (the process between Fig 1.6.1 to Fig 1.6.2) the
concentration in the SSR increases and the heat of compression is largely rejected to the
The compression process increases the concentration of
high temperature reservoir.
1.0-
0 h4h
(20,
-
3
h)
(
-A
(h,
3
h)
3)
superfluid
homogeneous mixture
.ool
E
(21,31)
(41)
"stable"
x, 3He Concentration
Figure 1.7: The states of the 3He-4He mixture in the low and high temperature piston are mapped onto a
sketch of the phase diagram. The high temperature reservoir holds the high temperature at a constant
temperature. The low temperature mass is assumed to drop in temperature as it is cooled. State 2 and 3 are
4
3
marked by a A, 0, o for successive cycles. The dark line labeled "stable" depicts the states of the He- He
mixture in the low temperature piston when operating in the two phase region reduces the cooling power of
4
3
the low temperature piston to zero. There is no concentration change in the He- He mixture in high
3
is
in
the two phase region.
temperature piston when the He-4He mixture in the low temperature piston
20
the homogeneous fluid in the low temperature clearance volume until the fluid is in the
saturated dilute mixture state. With further compression, the mixture in the low
temperature clearance volume phase separates, rejecting the latent heat of the phase
change to the low temperature mass. We assume the SSR in Fig 1.6 is operating such
that the 3He rich slug goes back into the dilute state during the expansion stroke. The
states of the 3He-4He mixture in the low and high temperature piston are shown plotted
on a phase diagram in Fig 1.7.
The upper left curve in Fig 1.7 represents the states of
the 3He-4He mixture in the high temperature piston. All the states of the 3He-4He mixture
in the high temperature piston are at the same temperature since the piston is connected to
a high temperature reservoir. Alternatively, the states of the 3He-4He mixture in the low
temperature piston experience a temperature drop as the low temperature mass is cooled.
For the small temperature drop in the low temperature mass we consider in Fig 1.7, we
model the concentration of the 3He- 4He mixture as not changing between cycles in each
piston.
As the 3 He- 4He mixture in the low temperature piston enters the two phase region, heat
is rejected to the low temperature mass instead of the high temperature reservoir. The low
temperature mass is not cooled until the 3He-4He mixture in the low temperature piston
emerges from the two phase region. Because each new state 1I is at the same
concentration of the previous state 11, state 11 of the 3He-4He mixture in the low
temperature piston moves into the two phase region. In an ideal SSR with no external
head loads or internal dissipations, the 3 He- 4He mixture in the low temperature piston
would eventually run up and down a stable line touching the two phase boundary at one
end.
If we accounted for the change in concentration of the 3He-4He mixture in the low
temperature piston between cycles, state 11 would converge even faster to the two phase
boundary, since the concentration of the mixture in the low temperature piston increases
as the low temperature mass gets colder. The new state 11 would be closer to the two
phase boundary than the previous state 11.
This analysis demonstrates that once the low temperature piston has crossed into two
phase region, the SSR's ability to cool is quickly disabled. There are other issues such as
the compromised heat rejection on the high temperature platform, and the high entropy
21
generation incurred in compressing and expanding the system with a 3He slug that
actually make the performance of the two phase working fluid SSR even worse than
implied.
1.4 History of the SSR
The first SSR was a single stage device built by Kotsubo and Swift [1] which used a
configuration not unlike Fig. 1.4. The regenerator used in this design was an array of 30
CuNi capillary tubes 200 micron in diameter 38 cm in length jacketed in a 3He bath. The
3
He bath was chosen as the thermal matrix material for the regenerator because below I
Kelvin
3He
is one of the few materials with significant heat capacity. However, the low
thermal conductivity of the 3He made it a flawed regenerator material. The SSR had to
be operated at very low frequencies, 240 seconds per cycle, to allow time for thermal
diffusion into and out of the regenerator's 3He bath. Another undesirable characteristic of
this SSR design is that over time, the operation of the SSR was compromised by a slow
diffusion of the 3He atoms across the superleaks. Using 3He_4He mixture concentration
of 12% and using a high temperature reservoir of 1.2 K, temperatures around 0.6 K were
achieved [2].
The second SSR improved on the first SSR by replacing the
3He
bath regenerator with a
recuperator design. Two superfluid Stirling refrigerators are placed back to back, their
compressors and expanders are connected, respectively, by superleaks as shown in Fig
1.8.
3He-4He
mixture fills both sides of the expander and compressor pistons. The
regenerator consisted of a counterflow heat exchanger made of 238 CuNi capillary tubes
250 microns in diameter silver soldered in a hexagonally closed pack array with
alternating rows corresponding to each half of the SSR. This SSR was an improvement
on the previous design for several reasons. The superfluid 4He reservoirs in the Kotsubo
and Swift design is eliminated, each half of the SSR acts as the 4He reservoir for the
other. The problem of the slow diffusion of the 3 He atoms through the superleaks
disappears since if the two SSR's have been properly loaded, the average concentration
on one side of the superleak is the same as that on the other side. In fact, the performance
of the two SSR's should improve over time since if there is any initial 3 He mass
22
_
imm N
-
___
-
__
-
- = %- -11 11 1- -
-
~
3
A
Figure 1.8: A representation of a single stage machine consisting of two back-to-back 180 degree out of
phase SSR's exchanging heat through a recuperator.
imbalance between the two SSR's, the slow diffusion of 3He across the superleaks will
tend to resolve the imbalance. Also the slow diffusive heat transfer into the 3He bath in
the recuperator of the Kotsubo and Swift design is replaced by a more rapid convective
heat transfer between the two mass flows in the recuperator of this design. This faster
heat transfer mechanism allowed the SSR to be operated at the higher frequency of 20
seconds per cycle. This SSR used a 3He-4He mixture concentration of 6.6%, a piston
3
volume displacement of 0.8 cm 3 and a volume of 7 cm per SSR half. With a He
evaporation refrigerator providing a high temperature reservoir of 1.05 Kelvin, this
refrigerator was able to achieve temperatures of 296 mK and net cooling powers of 930
gW at 700 mK and 140 [tW at 500 mK. The same SSR was operated by Watanabe,
Swift, and Brisson and achieved a temperature of 168 mK while the high temperature
piston was held at 387 mK by a 3 He evaporation refrigerator [7,14].
Further improvements of the SSR design involved first a more careful examination of
the material used to build the SSR. When designing for performance below I Kelvin,
new material properties become important. Designing for efficient heat transfer between
two different materials requires considering the Kapitza thermal boundary resistance for
those two materials. Kapitza thermal boundary resistance is defined by
Rk = ATQ
where AT is the temperature drop across the interface of two different materials and Q is
the heat transfer rate per unit area. At low temperatures this thermal resistance can
dominate the heat transfer between two materials. For reasons beyond the scope of this
text, the Kapitza thermal boundary resistances between helium and plastics are much
23
-
-
__ - __
bwwk
smaller than that between helium and metals. The next development in superfluid
Stirling refrigeration, therefore, was the replacement of the CuNi capillary tubing heat
exchanger with a heat exchanger made out of plastic.
A single stage SSR was built by Patel and Brisson that used a heat exchanger
constructed out of a Kapton-epoxy composite material. The heat exchanger consisted of
alternating layers of 127 pm thick and 25.4 gm thick Kapton glued together with Stycast
1266. Each 127 ptm layer had five passages 2.38 mm in width and 20 cm long. Small,
medium, and large heat exchangers with a total of 50, 100, and 200 flow passages
respectively were constructed by stacking ten, twenty, and forty 127 gm thick layers
respectively. The small, medium, and large recuperators had 1.5 cm 3, 2.4 cm 3 , 6.05, cm 3
of recuperative volume per SSR half respectively [16, 17].
This new single stage SSR was a much larger machine than the previously built
SSR's. With a large plastic heat exchanger, this SSR had a total volume of 48.3 cm 3 per
SSR half and high and low temperature piston volume displacements of 17.7 cm 3 and 9.4
cm 3 respectively. Using a 3 He- 4 He mixture concentration of 3%, this SSR achieved a
temperature of 291 mK from a high temperature reservoir of 1.0 K, and a net cooling
power of 3705 pW at 750 mK, 977 pW at 500 mK and 409 RW at 400mK. The cooling
power of this new SSR was a great improvement over the previous SSR's by a factor of 7
at 500 mK and by a factor of 4 at 750 mK [9, 17]
All SSR's constructed until this point were single stage SSR's, meaning one
compressor, one expander. Patel and Brisson hoped to improve on previous performance
by building a two-stage SSR. This new SSR, depicted in Fig. 1.9, consist of high,
intermediate, and low temperature pistons. The intermediate and low temperature pistons
were rigidly connected together so that these pistons moved together. One plastic heat
exchanger connects the high and intermediate temperature pistons and another plastic
heat exchanger connects the intermediate and low temperature pistons. The best low
temperature performance of this new design was achieved while using a large heat
exchanger between the high and intermediate temperature pistons and a medium size heat
exchanger between the intermediate and low temperature pistons. The total volume of
24
High
I
t.
4
He Evaporation
Refrigerator
Temperature
Pistons
i(belows)
( l w
Plastic
Heat
Constant Temperature
Heat Exchanger
Exchan ger
(recup erator)
(copper)
Pla
a ic
Intermediate
Temperature
He
Exc hanger
(re cuperator)
Pistons
(bellows)
UUM
.M~
Low
,\Temperature
e---Pistons
(bellows)
Constant Temperature
Heat Exchanger
(copper)
One of three sets of cage bars
Figure 1.9: A diagram of the two stage back-to-back SSR design with plastic recuperator.
this SSR was 64.5 cm' per SSR half. The high, intermediate, and low temperature piston
volume displacements were 17.7 cm 3 , 9.4 cm 3 , and 5.3 cm 3 respectively. Using a 3%
concentration
3He-4He
mixture, this SSR achieved a temperature of 248 mK [10, 17].
The best cooling power performance of this new design was achieved while using a
large heat exchanger between both sets of pistons. The volume of the SSR in this
configuration was 68 cm 3 per SSR half. Although it only achieved a low temperature of
307 mK, the SSR achieved cooling powers of 1 mW at 626 mK, 500 pW at 452 mK and
100 jW at 344 mK [11, 17].
Although the new two-stage SSR achieved a new ultimate low temperature, Patel and
Brisson believed that the full potential of this cryostat had not been demonstrated.
Several experimental problems arose during the operation of this machine prevented it
25
_--_-__'__-___1._- _-- - 1 I --
-__'
from running at full capacity. The SSR is thermally insulated from a 4.2 K liquid 4He
bath by a vacuum. A superfluid 4He leak from the cryostat to the vacuum can, therefore,
put a significant additional heat load on the SSR.
The first goal of this work was to resolve the experimental issues of the two-stage SSR
built by Patel and Brisson. The second goal was to attempt to improve on previous SSR
performances by adding a third SSR stage to the two-stage machine. The details of the
machine built are described in Chapter 2. The experimental results of operating both the
two and three stage SSR's are presented in Chapter 3.
In addition there are other important technical issues particular to cryostats using
moving parts below 1 Kelvin. Chapter 4 presents the result of an experiment to ascertain
the heat dissipation due to the flexing and relaxing of the bellows. An understanding of
how much heat load on the SSR is due to its moving parts will help us understand the
ultimate limitations of this type of refrigeration device.
Due to chemical properties of the 3He-4He mixture, the third SSR stage needed to use a
separate working fluid from the rest of the SSR. The analytical and numerical analysis
used to explore this issue is presented in Chapter 5.
26
Chapter 2
Experimental Apparatus
2.1 Description of the three-stage SSR
Figure 2.1 shows a schematic of the three-stage SSR. This refrigerator uses the
counterflow recuperative configuration, so it has back-to-back SSR's operating 180
degrees out of phase with each other and counterflow recuperators as the regenerators.
This SSR consists of four isothermal platforms; the hot, first, and second intermediate
and cold platforms are connected by Kapton recuperators are shown in Fig. 2.1. The
compressor pistons on the hot platform are held at approximately 1 Kelvin by a 4 He
evaporation refrigerator. The third stage of the SSR is thermally connected to the second
stage of the SSR on the second intermediate platform, but third stage is filled separately
and the
3He-4He
mixture cannot flow between the third stage and the second stage.
The hot, first intermediate and second intermediate, and the cold platform of the SSR
are made of solid blocks of OFHC copper on which the pistons are mounted. The pistons
are made with edge welded stainless steel bellows, which have convolutions that nest into
one another to minimize void volume. The effective areas of the pistons based on the
manufacturer's specifications [18] are 17.74 cm 2 for the hot platform pistons, 13.61 cm 2
for the first intermediate platform pistons, 7.68 cm 2 for the second intermediate platform
(second stage), 23.42 cm 2 for the second intermediate platform (third stage) and 3.16 cm 2
27
Stan fne
low temperature valves
void spaces
Out
4
in
He Expa nsion
Refrigera
tor
Hot Platform
First
tage
1of 3 Kapton-Epoxy
Composite
Heat Exchangers
First
jIntermiediate
Platform
Ifill
3rd Stage SSR
econd
tage
lines
low temperature valves
Second
Intermediate
Platform
Third
Stage
1 of 5 Bellows Sets
Cold Platform
Copper
U
Kapton-Epoxy
U
Vycor Glass (superleak)
Moving Part
D
3
He-4 He Mixture
Figure 2.1: A schematic of the three-stage superfluid Stirling refrigerator.
for the cold platform. The hot platform pistons are rigidly connected together and driven
sinusoidally using a push rod from a room temperature drive. The first-intermediateplatform and second-intermediate-platform second-stage pistons are similarly connected
and driven together using a common push rod. The second-intermediate-platform thirdstage pistons and cold platform pistons are also respectively rigidly connected and are
28
independently driven by two push rods actuated from room temperature. The hot
platform temperature is pinned at approximately 1.0 K by a 4He evaporation refrigerator.
Within each piston platform, there are superleaks made from porous Vycor glass, which
allow the superfluid
3He-4He
to flow freely between the halves of the SSR during
operation. In the hot platform, the superleaks are three Vycor cylinders 6.03 cylinders in
length with diameters of 1.39 cm, 1.35 cm, and 0.72 cm. In the first intermediate
platform, the superleaks are three Vycor cylinders 10.63 cm in length with diameters of
0.74 cm. In the second intermediate platform (second stage), the superleaks are three
Vycor cylinders 15.16 cm in length with diameters of 0.74 cm. On the second
intermediate platform (third stage) the superleaks are four Vycor cylinders 13.09 cm in
length with diameters of 0.74 cm. On the cold platform, the superleaks are three Vycor
cylinders 12.06 cm in length with diameters of 0.73 cm. The total volume of the Vycor
glass in the first and second stage of the SSR is 53.2 cm 3 . The total volume of the Vycor
glass in the third stage is 21.9 cm 3 . Since 28% of the Vycor glass is void space [19], the
glass contributes 15 cm 3 to the 3 He- 4He mixture volume of the first and second stage of
the SSR and 6.1 cm 3 to the 3He- 4He mixture volume of the third stage. The 3He that
diffuses into this volume does not participate in the operation of SSR.
Within each piston platform, there are also isothermal heat exchangers made from
nested OFHC copper cylinders press fit into the piston platforms. A 76 pm gap exists
between the inner walls of an outer cylinder and outer walls of an inner cylinder. At the
top of each cylinder is a flow distributor 0.635 mm deep and 0.317 cm wide around the
cylinder circumference. Each half of the hot piston platform contains one cylinder 2.14
cm in length with a diameter of 3.80 cm, which provides a total heat transfer area of
65.94 cm2 . Each half of the first intermediate platform contains two cylinders that
provide a total heat transfer area of 209.70 cm 2 . The first cylinder is 3.97 cm in length
with a 4.11 cm diameter while the second cylinder is 4.88 cm in length with a 3.52 cm
diameter. Each half of the second intermediate (first and second stage) platform contains
four cylinders 6.07 cm in length providing a total heat transfer area of 276.6 cm 2 . The
diameters of the cylinders are 4.43 cm, 3.91 cm, 3.39 cm, and 2.88 cm respectively. Each
half of the second intermediate (third stage) platform contains four cylinders 5.02 cm in
length providing a total heat transfer area of 257.4 cm 2 . The diameters of the four
29
25 /pmKapton
I
127 pm Kapton
Flow through recuperator
of SSR half #1
A
o
Flow through recuperator
of SSR half #2
:>
Figure 2.2: The arrangement of alternate layers of Kapton templates within the recuperator to
form a counterflow heat exchanger. The flow of opposite halves of the SSR alternate between
successive layers of the 127 gm Kapton sheets.
cylinders are 4.94 cm, 4.43 cm, 3.79 cm, and 3.16 cm respectively. Each half of the cold
piston platform has seven cylinders 4.77 cm in length providing a total heat transfer area
of 379.5 cm2 . The diameters of the seven cylinders are 5.83 cm, 5.32 cm, 4.82 cm, 4.30
cm, 3.80 cm, 3.29 cm, and 2.78 cm respectively.
The recuperators, shown in Fig. 2.2, used in this SSR are of a plastic design type
constructed and built by Patel [16]. The recuperative portion consists of alternating
layers of 127 km Kapton film and 25.4 ptm Kapton film [25] glued together using Stycast
1266 [20]. Each 127 pm layer has five passages 2.38 mm in width and 20 cm in length.
The recuperators from the hot platform to the first intermediate and from the first
intermediate to the second intermediate platforms (first and second stages of the SSR) are
large recuperators with forty 127 gm layers and thirty-nine 25.4 pm layers. Each large
recuperator has a total volume of 20.9 cm 3, of which 12.1 cm 3 (6.05 cm 3 per SSR half) is
dedicated to recuperative heat transfer. The recuperator from the second intermediate to
the cold platform is a small recuperator with ten 127 gm layers and nine 25.4 gm layers.
The total volume of the small recuperator is 11.10 cm 3, of which 3.02 cm 3 (1.51 cm 3 per
SSR half) was dedicated to recuperative heat transfer.
30
As can be seen in Fig. 2.1, a total of four fill lines run from room temperature to the
SSR. The two fill lines to the two halves of the first and second stage SSR are designed
to be sealed at low temperature by valves mounted on the hot platform. These valves are
actuated manually from room temperature and prevent the 3He-4He from moving up and
down the fill capillaries during operation of the SSR.
The two fill lines into each of the SSR halves of the third stage are sealed at low
temperature by valves mounted on the 300 mK platform. The valves are also actuated
manually from room temperature and also act to prevent the 3He-4He mixture from
moving up and down the fill capillaries during operation of the SSR. These valves were
designed and tested to be superfluid tight in case it became necessary to pump out the fill
lines to prevent heat loss. As the third stage SSR is thermally isolated from the
4He
evaporation refrigerator (the hot platform), the fill lines are wrapped around the 4He
evaporation refrigerator so that the third stage's
3He_
He working fluid can be liquified
during loading.
Calibrated ruthenium oxide [21] and germanium [22] and carbon resistor thermometers
mounted on the outside of the piston platform are used to monitor the temperature. The
uncertainty of our temperature measurements are + 0.67 mK at 1.0 K and + 1.02 mK at
350 mK. Cooling powers are measured by monitoring the voltage across and current
through a heater made of wound manganin wire. The uncertainty of our cooling power
measurements is ± 2 pW.
The total volumes of the SSR's are given minus the void spaces and the volume of the
fill lines because these volumes are inactive during the operation of the SSR. The total
volume of the first and second stage is 136 cm 3 . The total volume of the third stage SSR
is 103.3 cm 3.
2.2 Development of the Patel and Brisson SSR
The first and second stage of the SSR were designed and built by Ashok Patel. The
major sections of the third stage were also designed and built by Patel but finished and
assembled in this work. The following sections will delineate three alterations to Patel's
design. The first section is an alteration made to plastic recuperators. The second and
31
third sections are design choices made in the third stage to prevent a thermal linkage
between the third stage and the first and second stage of the SSR.
2.2.1 Heat Exchangers
The plastic heat exchangers consist of a copper header silver soldered to stainless steel
tubes that then sealed to the plastic body of heat exchanger using epoxy. The header
consists of two machined pieces of OFHC copper, as shown in Fig 2.3, that were in turn
sealed together with an indium o-ring. The outer piece of the metal header is designed to
mate with specific copper heat exchangers on the SSR itself and by breaking the indium
seal and replacing the outer piece, the heat exchanger could be put into different positions
in the SSR.
The problem with this design was that the procedure of silver soldering the stainless
steel tubes to the inner copper header piece caused the copper to anneal. The
compressive forces required to make the indium seal between inner and outer header
piece would plastically deform the inner head piece and, over time, compromise the
indium seal. Fixing this problem required breaking the seal, filing the copper surfaces
flat, and remaking the indium seal.
This unreliable sealing method was replaced with an improved method developed
specifically for these joints. All of the indium seals between the inner and outer header
pieces were replaced by a Stycast 1266 epoxy seal. To prevent cracking in the epoxy, the
epoxy is allowed to cure under no compressive force. When mounted on the SSR the
epoxy is put into compression by the through bolts in the flange. This compressed epoxy
joint remained superfluid tight over repeated thermal cycles.
The drawback to using the epoxy seal is if a crack develops in the seal, the copper
flanges must be broken apart, carefully cleaned and resealed. Performing this procedure
causes the heat exchanger to endure physical stresses and risks damage to the rest of the
heat exchanger. Also, the ability to change flanges between the different heat exchangers
is lost and thus move different heat exchangers to different positions.
32
3E
0
LI
.
E-
--
AW
Seal
Kapton-Epoxy Heat Exchanger
To Pistons
Outer Metal
Header Piece
(OFHC copper)
-
-
Inner Metal
Header Piece
(OFHC copper)
Silver Solder Joint
To Pistons
Epoxy Joint
Stainless Steel
Tubes
Figure 2.3: A diagram of the plastic counterflow heat exchanger.
33
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I
shaft to room
temperature
where actuated
t S5
LII
Brass
Stainless Steel
Threads
Bearing
Bellows soldered
with lead-tin to Bras S
and Steel
cage bar s-
-
Fill Lin es
stainless steel tip
polished to 1 /. m finis h
4
to SSR
sharp edge machined
in brass
Figure 2.4: A schematic of the dual superfluid tight valves.
2.3.1 Third-Stage Valves
The first method discussed to prevent the superfluid 4He from putting a heat load on
the third stage was evacuating the fill lines completely after the SSR has been filled. In
order to prevent the superfluid 4He in the third stage from flowing in and out of the
evacuated fill lines during expansion and compression, the valves between the pistons
and the fill lines need to be superfluid tight. Fig. 2.4 is a schematic diagram of the dual
valves that were designed, constructed, and successfully tested to be superfluid tight.
The valve set was designed to allow two flow lines to be closed by the rotation of a
shaft, which is actuated at room temperature. The important components of the valve set
are the stainless steel valve stem that has been highly polished to a 1 micron finish in a
conical region and the brass seat machined with a sharp corner edge [23]. The seal is
35
created by the contact between the polished section of the valve stem and the sharp edge
of the brass seat. The valve stem is pressed into the brass seat by rotating the actuating
shaft. Bellows soldered with lead tin solder to the valve stem base and brass seat seal the
body of the valve.
The bearing referred to in Fig. 2.4 is a simple stainless steel sleeve over the rotating
shaft. Unlike the design this valve set was based on [23], it does not require a special
bearing between the actuating shaft and the valve stem to prevent a torque from being
placed on the valve stem. This because the two valve stems are attached to a brass plate
that slides up and down on a set of cage bars, inhibiting any rotational movement of the
valve stems.
This valve set also includes a positive withdrawal system. The brass plate which
controls the opening and closing of the valve stems can be pushed (valves closed) or
pulled (valves opened) by the rotating shaft.
This valve set was tested by filling it with superfluid 4He at 1.2 Kelvin and found to be
superfluid tight.
2.3.2 Void Space Analysis
The alternative method of preventing the thermal link between the 4He expansion
refrigerator and the second intermediate platform was to design void spaces thermally
connected to the hot platform. The theory behind adding void spaces to the fill lines and
the analysis to choose their volume follows.
As discussed earlier, the high thermal conductivity of pure 4He is due to the unimpeded
motion of the superfluid and the normal fluid counterflow. A superfluid moves to the hot
end of the container where it is converted to a normal fluid particle by absorbing thermal
energy from the hot end. The normal fluid particle is then displaced away from the hot
end by other superfluid particles towards the cold end. At the cold end, the normal
particle (a phonon-roton excitation) is converted back to a superfluid particle by rejecting
thermal energy to the low temperature end of the container. There is little interaction
between the superfluid and the normal particle as they flow past each other.
36
3
He- 4 He mixture
Vhot
O0 0 0
000
0
0
00
0
0
00
0
0
0
0
0
00~
00 0El00
0000
0
000000
'hot
0 00
00
0
0
0
0
0
D
0
0
0
0 000
00
0 00
Thot
0
Vcold
1o0
00
000
Thot
Phonon-roton particles
0 0 3He
T
cold
particles
Figure 2.5: On the left, a container at a hot temperature is loaded with 3He-4He mixture. On the
right, the volume on the bottom has been brought to a cold temperature. Few phonon-roton particles
exist at this temperature. The 3He particles are flushed into this cold volume and are just sufficient to
balance the pressure of phonon-roton "particles" at high temperature. Note that the 3He particles are
conserved while the phonon-roton particles are not.
In 3He-4He mixtures, however, the 3He component of the mixture is viscously locked
to the viscous normal component of the 4He, the phonon and roton excitations in the 4He
component of the mixture. If a 3He- 4He mixture is placed in a container with a
temperature gradient the superfluid counterflow currents will begin to flow. The normal
fluid particles will carry 3He particles to the low temperature end of the container until
the concentration of the 3He component is high enough to counter and stop the flow of
the normal component (and hence, the superfluid component flow too.)
If the temperature gradient in the container is too large or the initial 3He concentration
of the 3He_ He mixture is too low a condition known as "heat flush" will occur in the
mixture. Heat flush occurs when all the 3He is flushed to one end of the container leaving
a portion of the container filled with a region of "pure" 4He. The pure region is a region
of high thermal conductivity in the fluid. This condition can be prevented from occurring
in the fill capillary by choosing the geometry shown in Fig 2.5.
The distribution of 3 He in the capillary tube can be determined using the phonon-roton
gas model [24] for the normal component of the 4He and the Boltzmann gas model for the
3He
component of the mixture. The phonon-roton gas of excitations in the 4He
37
component is conceptually similar to Planck's Black body gas of photons. The pressure
of the phonon-roton gas corresponds to the so-called fountain pressure of the 4He which
obey the empirically derived relation
(2.1)
3 2
pf (T) = p4 (BT 4 + AT / e(-A1 /T) +Ce(-A2/ T)
where fit parameters A, B, C, A1 ,A2 have values of 23.2 J/mole K3 /2, 6.75x10- 3 J/mole K4,
500 J/mole, 8.65K, and 15.7 K respectively. Note that pf depends only on temperature.
In steady state, the phonon-roton gas component and the 3He ideal gas component of
the mixture must be in mechanical equilibrium. The total pressure of the "gases" at the
hot end of the cylinder must equal to total pressure at the cold end of the cylinder,
[n 3 RT + pf (T)]hot = [n 3 RT + pj (T)Icold
(2.2)
where n3 is the molar density of the 3He in the mixture and R is the universal gas
constant.
A conservative model used to ensure that the
3He
never gets flushed completely out of
the "I Kelvin" volume into the capillary is to model the system as two volumes, one at
1.4 K and the other at 0.3 K, connected by a passage of negligible volume between them.
The 0.3 K volume corresponds to the fluid volume of the experimental apparatus between
the "I Kelvin" volume (the "void space" in Fig. 2.1) to the valve seat of the low
temperature third-stage valves. The volume is loaded with a 3He-4He mixture that has
average concentration of xi. We wish to find the minimum hot to cold volume ratio such
that the pressure of the "gases" at the hot end equals the pressure of the "gases" at the
cold end.
The fountain pressure is negligible in the cold volume, since at 0.3 K the 4He
is nearly exclusively superfluid. Equation 2.2 is satisfied when the phonon-roton gas and
the 3He gas are in mechanical equilibrium. If the hot and cold temperatures are too large
there maybe only negative solutions for the hot density
(nA)hot.
Physically this
corresponds to all the 3He being forced to the cold end of the apparatus and high thermal
conductivity between the high temperature region where the fluid is essentially pure 4He.
In this case Eq. 2.2 is not satisfied. The smallest high temperature volume for a given
initial concentration, a given Thot, and a given T,.Id that avoids the high thermal load of
the heat flush effect is determined when all the 3He particles in the system are flushed
into the cold volume and, in addition, the 3He pressure in the cold volume is equal to the
38
fountain pressure of the phonon-roton gas in the hot volume. In this case Eq. 2.2 is still
valid, becoming,
n3,coldRTOjl
= pf (Thot).
(2.3)
We approximate the molar volume of 3He, v 3, as v 4 0/xi, where v4 0 is the molar volume
of pure 4 He. Since all of 3He particles initially loaded into the container are now only in
the cold volume,
(n3,cold )Vc = N3tot = VH +VC
V3
(n3,cold) = 4
C
-
(2.4)
VH +VC
+
VH
(2.5)
Xi
Substituting Eq. 2.5 into Eq. 2.3 and then solving for VH
(2.6)
RT
VH +VCXiPf(Thot)
=2.7)
VH =VC Pf1Th;V4
x;RTe
At T = 1.4 K, Eq. 2.1 gives the value Pf = 4074 Pa for the fountain pressure.
Substituting this value and xi = 1.5%, the lowest concentration to which the SSR has ever
been filled, into Eq. 2.7, a minimum ratio of the hot volume to the cold volume is
determined
Vhot
=~
VC(4074
o
Pa)(27 x10-6 m
NmoK )(0.3K)
.94V
(2.8)
(0.015)(8.314
The cold volume within the low temperature SSR consists of the volume inside in
closed low temperature valve and the volume of the capillaries from the 4He evaporation
refrigerator to the low temperature valves. This volume was estimated to be
approximately 1 cm 3 per fill line. An extra void volume of 3 cm 3 per fill line was added
and thermally pinned to the 4He evaporation refrigerator. Based on the conservative
model used, this extra hot volume should be more than sufficient to avoid the
thermomechanical thermal conductivity of the 4 He. We note that this analysis does
require the third stage valves to seal at least tight enough to prevent the flow of 3He
39
particles through them or the cold volume could be a hundred times larger than calculated
here.
40
Chapter 3
Experimental Results of SSR
3.1 Review of the Experimental Results of the SSR
This section will review the performance of the SSR operated by Patel and Brisson
(Patel). Patel gathered data on five separately configured SSR's, two single stage SSR's,
and three two-stage SSR's [8-11,17]. The first single stage SSR was operated using a
small recuperator (1.5 cm 3 recuperative volume per half). The second single stage SSR
was operated using a large recuperator (6.05 cm 3 recuperative volume per half). The
volumes of the hot and cold pistons and the recuperators of these two SSR's can be found
in Table 3.1. The first two-stage was operated using a small upper recuperator and
medium lower recuperator (2.4 cm 3 recuperative volume per half). The second twostage SSR was operated using a large upper recuperator and a medium lower recuperator.
The third two stage SSR was operated using a large upper and lower recuperator. The
volumes of the hot, cold, and intermediate pistons and the recuperators for these two twostage SSR's can be found in Table 3.2. In Table 3.3 the performance of these four SSR's
are compared.
As can be seen from the results in Table 3.3, the results of the two-stage SSR were
disappointing. The SSR with the small upper and the medium lower recuperators was
believed to have recuperators with too little surface area. The second and third two-stage
SSR performances were compromised by technical problems. The two-stage SSR with
41
the large and medium recuperator developed a leak from the SSR volume to the
insulating vacuum. This caused an extra heat load on the SSR and eventually caused the
SSR to warm up . Thus very little data could be collected. In the two large recuperator
SSR test a mechanical interference prevented the pistons from being centered properly .
The resulted in a mass imbalance between the two halves of the SSR and imperfect
recuperation in the heat exchangers.
Table 3.1: Single Stage SSR's
Single-stage SSR
Volumes per SSR half
Hot Piston:
(1.0cm stroke)
Swept Volume
17.74 cm
Clearance Volume
12.04 cm 3
Small Recuperator Volume
1.5 cm
Heat transfer surface area
238 cm2
Large Recuperator Volume
6.05 cm 3
Heat transfer surface area
952 cm 2
Swept Volume
13.34 cm
Clearance Volume
9.83 cm 3
Kapton Recuperator:
Cold Piston:
(0.98 cm stroke)
(0.69 cm stroke)
-Swept
Volume..........9.39
Clearance Volume
Volume Total:
cm~
11.94 cm 3
Small Recuperator SSR
87.56 cm 3
Large Recuperator SSR
96.6 cm
42
Table 3.2: Two-Stage SSR's
Single-stage SSR
Volumes per SSR half
Hot Piston:
(1.0cm stroke)
Swept Volume
17.74 cm
Clearance Volume
12.04 cm 3
Small Recuperator Volume
1.5 cm3
Heat transfer surface area
238 cm 2
Kapton Recuperator:
Lr
Recuperator Volume
6.
.....
Heat transfer surface area
952 cm 2
Swept Volume
13.61 cm 3
Clearance Volume
11.85 cm 3
Swept Volume
9.39 cm
Clearance Volume
13.96 cm 3
Med Recuperator Volume
2.4 cm3
Heat transfer surface area
476 cm 2
Large Recuperator Volume
6.05 cm~.....
Heat transfer surface area
952 cm 2
Swept Volume
7.68 cm
Clearance Volume
12.66 cm 3
Swept Volume
5.30 cm3
Clearance Volume
13.85 cm 3
Intermediate Piston:
(1.0 cm stroke)
(0.69 cm stroke)
-
.
.
.
Kapton Recuperator:
Cold Piston:
-(1.0
-(0.69
cm stroke)-
cm stroke)
Volume Total:
Small (Upper) and
Medium (Lower)
120 cm 3
Recuperator SSR
Large (Upper) and
Medium (Lower)
129cm 3
Recuperator SSR
Large (Upper) and
Large (Lower)
136 cm 3
Recuperator SSR
43
.
.
.
Table 3.3: A comparison of the performances of Patel's different SSR's
Single stage SSR
Single stage SSR
Two stage SSR
Two stage SSR
Two stage SSR
Small recuperator
Large recuperator
Small recuperator
Large recuperator
Large recuperator
Med recuperator
Med recuperator
Large recuperator
1.5 % Mixture
3 % Mixture
Cooling
T
Cooling
T
Power
(mK)
3% Mixture
Power
(mK)
(j W)
(piW)
Cooling
T
3 % Mixture
T
Cooling
Power
(mK)
(piW)
344
0
291
0
282
0
400
97
400
409
361
500
358
500
977
750
1856
750
3705
3 % Mixture
T
Power
(mK)
248
(piW)
0
Cooling
Power
(mK)
(piW)
307
0
100
344
100
485
500
452
500
617
1000
626
1000
3.2 Two Stage SSR with Large Upper and Lower Recuperator
This section describes the operation of the two stage machine investigated in this work.
This two-stage SSR with large upper and lower recuperators is identical to the SSR
described in Table 3.1 (except for a slightly larger hot piston stroke) and to the second
SSR operated by Patel. Both large recuperators had 6.05 cm3 of recuperative volume per
SSR half. The SSR was operated with a 3%
3He-4He
mixture and a 1.1 cm compressor
stroke (17.74 cm 3 volume displacement) and a 1.08, 0.96, 0.92, 0.78, and 0.61 cm
expander stroke. Using the 1.08 cm stroke, from a high temperature at 1.05 K, this SSR
achieved a low temperature of 329 mK and delivered a net cooling power of 1 mW at 606
mK, 500 gW at 408 mK.
3.2.1 Description of the Two-Stage SSR
Figure 3.1 shows a schematic diagram of the two stage SSR. This refrigerator uses the
counterflow recuperative configuration, so it has two SSR's operating 180 degrees out of
phase with each other with counterflow recuperators as the regenerators. This SSR
44
1 of 3 pairs of bellows
4
He Evaporation
Refrigerator
Hot Platform
LJ:AW.
Constant Temperature
Heat Exchanger
(copper)
Plastic
Heat
Exchanger
(recuperator)
I
a
W&
-
'Wa 41WWN
Intermediate
Platform
.
0
[
Plastic
Heat
Exchanger
(recuperator)
3 He- 4
He m ixture resides
inside bell )ws and passages
Cold Platform
Constant Temperature
Heat Exchanger
(copper)
One of three sets of cage bars
D
Copper
U
Kapton-Epoxy
E
Vycor Glass (superleak)
Moving Part
Figure 3.1: A schematic diagram of a two-stage SSR. The internal volume of
the SSR is filled with 3He- 4He working fluid.
consists of three isothermal platforms; the hot (compressor), intermediate (expander),
and cold (expander) platforms are connected by Kapton recuperators are shown in Fig.
45
3.1. The compressor pistons on the hot platform are held at approximately 1 Kelvin by a
4He
evaporation refrigerator.
The hot, intermediate, and the cold platforms of the SSR are made of solid blocks of
OFHC copper on which the pistons are mounted. The pistons are made with edge welded
stainless steel bellows, which have convolutions that nest into one another to minimize
void volume. The effective areas of the pistons based on the manufacturer's
2
specifications [18] are 17.74 cm2 for the hot platform pistons, 13.61 cm for the
intermediate platform pistons, 7.68 cm 2 for the for the cold platform. The hot platform
pistons are rigidly connected together and driven sinusoidally using a push rod from a
room temperature drive. The intermediate and cold platform pistons are similarly
connected and driven together using a common push rod.
Within each piston platform, there are superleaks made from porous Vycor glass,
which allow the superfluid 4He component to flow freely between the halves of the SSR
during operation. In the hot platform, the superleaks are three Vycor cylinders 6.03
cylinders in length with diameters of 1.39 cm, 1.35 cm, and 0.72 cm. In the intermediate
platform, the superleaks are three Vycor cylinders 10.63 cm in length with diameters of
0.74 cm. In the cold platform, the superleaks are three Vycor cylinders 15.16 cm in
length with diameters of 0.74 cm. The total volume of the Vycor glass in the two-stage
of the SSR is 53.2 cm 3 . Since 28% of the Vycor glass is void space [19], the glass
contributes 15 cm 3 to the 3He-4He mixture volume two-stage SSR. The
3He
that diffuses
into this volume does not participate in the operation of SSR.
Within each piston platform, there are also isothermal heat exchangers made from
nested OFHC copper cylinders press fit into the piston platforms. A 76 pm gap exists
between the inner walls of an outer cylinder and outer walls of an inner cylinder. At the
top of each cylinder is a flow distributor 0.635 mm deep and 0.317 cm wide around the
cylinder circumference. Each half of the hot piston platform contains one cylinder 2.14
cm in length with a diameter of 3.80 cm, which provides a total heat transfer area of
65.94 cm2 . Each half of the intermediate platform contains two cylinders that provide a
total heat transfer area of 209.70 cm 2 . The first cylinder is 3.97 cm in length with a 4.11
cm diameter while the second cylinder is 4.88 cm in length with a 3.52 cm diameter.
Each half of the cold platform contains four cylinders 6.07 cm in length providing a total
46
heat transfer area of 276.6 cm 2 . The diameters of the cylinders are 4.43 cm, 3.91 cm,
3.39 cm, and 2.88 cm, respectively.
The recuperators used in this SSR are of a plastic design type. The recuperative
portion consists of alternating layers of 127 ptm Kapton film and 25.4 gm Kapton film
[25] glued together using Stycast 1266 [20]. Each 127 gm layer has five passages 2.38
mm in width and 20 cm in length. The recuperators from the hot platform to the
intermediate and from the intermediate to the cold platforms are large recuperators with
forty 127 pm layers and thirty-nine 25.4 pm layers. Each small recuperator has a total
volume of 10.7 cm 3, of which 3.0 cm 3 (1.5 cm 3 per SSR half) is dedicated to recuperative
heat transfer. Each medium recuperator has a total volume of 12.5 cm 3, of which 4.8 cm 3
(2.4 cm 3 per SSR half) is dedicated to recuperative heat transfer. Each large recuperator
has a total volume of 20.9 cm 3 , of which 12.1 cm 3 (6.05 cm 3 per SSR half) is dedicated to
recuperative heat transfer.
Calibrated ruthenium oxide [21] and germanium [22] thermometers mounted on the
outside of the piston platform are used to monitor the temperature. The precision of our
temperature measurements are + 0.67 mK at 1.0 K and + 1.02 mK at 350 mK. Cooling
powers are measured by monitoring the voltage across and current through a heater made
of wound manganin wire. The precision of our cooling power measurements is + 2 pW.
The total volumes of the SSR's is given minus the void spaces and the volume of the
fill lines because these volumes are inactive during the operation of the SSR. The total
volume of the first and second stage is 136 cm 3.
3.2.2 Experimental procedure and results
The SSR was prepared for operation by first cooling the refrigerator to 1.0 K, then
centering the pistons on each platform to ensure equal volumes of working fluid in each
SSR half, and finally filling the refrigerator with a 3.0% SHe-4He mixture. The fill lines
to the two-stage SSR were then closed and the SSR was operated at various speeds using
a hot piston stroke of 1.1 cm (19.5 cm 2 volume displacement and 11.6 cm 2 of clearance
space) and intermediate/cold pistons strokes of 1.08 cm, 0.96 cm, 0.91 cm, 0.78 cm and
0.61 cm. The associated volume displacements and clearance volumes for each of these
47
Table 3.4: Displacement volumes and clearance volumes for different expander strokes
Expander
Volume Displacement
Clearance Volume
Int. Piston
Cold Piston
Stroke
Int. Piston
Cold Piston
1.08 cm
14.7cm 3
8.29 cm
3
11.3 cm 3
12.4 cm 3
0.96 cm
13.1 cm 3
7.37 cm
3
12.0 cm 3
12.8 cm 3
0.91 cm
12.4 cm 3
6.99 cm
3
12.5 cm 3
13.0 cm 3
0.78 cm
10.6 cm 3
5.99 cm
3
13.3 cm 3
13.5 cm 3
0.61 cm
8.30 cm 3
4.68 cm 3
14.5 cm 3
14.2 cm
3
expander strokes are tabulated in Table 3.4. The cooling power of each expander stroke
and cycle period was measured by measuring the temperature of the cold and
intermediate piston temperature while supplying a constant heat load to the cold piston
platform.
The temperature was measured by recording the minimum temperature of the platform
during the cycle. This is slightly different method than the method used by Patel and
Brisson, who averaged the maximum and minimum temperature of the platform during
the cycle. But since typical values of peak to peak temperature difference during a cycle
are 6 mK, 7 mK, and 8 mK for the cold, intermediate, and hot piston temperatures, the
difference between Patel's measurements and the measurements in this work would be
approximately 3 mK.
Figure 3.2 - 3.6 provide the data obtained with this two-stage SSR while operating
from a high temperature of 1.06 K + 10 mK. Figure 3.2 shows the performance of this
SSR operating with a 1.1 cm compressor stroke and a 1.08 expander stroke using a 15,
27, and 40 second cycle period. In Fig 3.3 - 3.6, each figure compares the different
cooling power of different expander stroke lengths at a constant cycle period. Each
figure corresponds to a 1.1 cm compressor stroke and a 3% 3He-4He mixture. This data
shows that with the exception of the 0.78 cm stroke, the cooling power of the SSR
decreased as the stroke length decreased. The data for the 0.78 cm stroke was the last
data gathered over a two week period, so this data's deviation from the rest of the data is
probably due to an extra heat load caused by the leak into the SSR vacuum insulation.
The performance of this two-stage SSR did not significantly improve on the previous
performances of two-stage SSR's. The cooling power of this SSR matches well and even
48
slightly exceeds the cooling power the other two-stage SSR with larger recuperators. But
the low temperature performance of this SSR was disappointing. A very slow superleak
into the vacuum can was found to be putting an extra heat load on the SSR after two
weeks of operation. Also, this SSR had an extra source of mechanical dissipation. The
intermediate and cold pistons are rigidly connected together so that they can be
sinusoidally driven at the same phase angle. Each piston is connected to a brass plate
with mechanical screws. The two brass plates are connected by a piece of Teflon. The
threads of the screws to the brass plate to the intermediate piston were found to have
partially stripped during the operation of the SSR and the brass plate was no longer
firmly attached to the pistons.
0.6 ,'
I0 00
a;
00
()
F-
0
0.5 -
C
0
00
(L
0)
E
CD
6
---
4
0
2' M)
yce tim e
A 15 ss
27 s
-40
s
A 15s
a)
-2 27 s
---
a)
40 s
C
0
0.3
0.4
0.5
0.6
0.7
0.4
0.8
0.3
Cold Piston Temperature (K)
0.4
0.5
0.6
1
0.7
1
0.8
0.9
Cold Piston Temperature (K)
(b)
(a)
Figure 3.2: Data for a 1.1 cm compressor stroke and a 1.08 cm expander stroke. (a) Cooling
power versus cold piston temperature for cycle times of 15, 27, and 40 seconds. (b) Intermediate
piston temperature versus cold piston temperature for the data given in (a).
49
1200 1000 800
Expander Strokes
A-1.08 cm
-X-0.96 cm
--- 0.92 cm
-00.7 8 cmr
-00.61 cm
-
Zi,
600 0)
0
0)
400
-
200 -
00.3
0.5
0.4
0.6
0.7
0.8
0.9
Cold Piston Temperature (K)
Figure 3.3: Data for cooling power versus cold piston temperature for a
period, 1.1 cm compressor stroke, and a range of expander strokes.
15 second
cycle
1200 ,
1000
-
800 0
600
-
400
-
Expander Strokes
0
0-
-X-0.96 cm
-O--0.92 cm
-0.78 cm
0)
200 -
-<>-0.61
cm
00.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Cold Piston Temperature (K)
Figure 3.4: Data for cooling power versus cold piston temperature for a 23 second cycle period,
1.1 cm compressor stroke, and a range of expander strokes.
50
1200 1000 -
(D
800 8Expander
600
Strokes
1.08 cm
-X-0.96 cm
0)
0
400 -
-0-0.92 cm
-0--0.78 cm
200-
- -.
cm
0
0.3
0.5
0.9
0.7
1.3
1.1
Cold Piston Temperature (K)
Figure 3.5: Data for cooling power versus cold piston temperature for a 27 second cycle period, 1.1
cm compressor stroke, and a range of expander strokes.
12001000800.
Expander Strokes
600>M
---
1.08 cm
400
-X-0.96 cm
-0-0.92 cm
200
-0-0.78 cm
-*-0.61 cm
0
0.3
0.5
0.9
0.7
1.1
1.3
Figure 3.6: Data for cooling power versus cold piston temperature for a 40 second cycle
period, 1.1 cm compressor stroke, and a range of expander strokes.
51
3.3 Three-Stage SSR with two Large Recuperators and one Small
Recuperator
This section describes the first operation three-stage SSR and reports on its preliminary
experimental performance. This three-stage SSR has a total internal volume of 239 cm 2
and uses two Kapton heat exchangers having 12.1 cm 3 devoted to recuperative heat
transfer and one Kapton heat exchanger having 3.0 cm 3 devoted to recuperative heat
transfer. Unfortunately, a superleak into the vacuum insulation space around the SSR
allowed very few performance data points to be collected. This SSR achieved a cold
piston temperature of 338 mK and second intermediate piston temperature of 458 mK
while operating from a high temperature of 1.07 Kelvin with a 3%
3He-4He
mixture. The
performance of this three-stage SSR was disappointing in that it did not improve on the
two-stage SSR performance. However, these results were obtained despite a major
external heat load on the SSR, a possible 3He mass imbalance, and an undersized
recuperator volume in the third-stage SSR. When these issues are resolved, the
performance of the three-stage SSR is expected to improve significantly.
3.3.1 Description of the SSR
Figure 3.7 shows a schematic of the three-stage SSR. This refrigerator uses the
counterflow recuperator configuration, so it has two SSR's operating 180 degrees out of
phase with each other and counterflow recuperators as the regenerators. This SSR
consists of four isothermal platforms; the hot, first intermediate, second intermediate, and
cold platforms are connected by Kapton recuperators as shown in Fig. 3.7. The
compressor pistons on the hot platform are held at approximately 1 Kelvin by a 4He
evaporation refrigerator. The third stage of the SSR is thermally connected to the second
stage of the SSR on the second intermediate platform, but third stage is filled separately
and the 3 He- 4He mixture cannot flow between the third stage and the second stage.
The hot, first intermediate, second intermediate, and the cold platform of the SSR are
made of solid blocks of OFHC copper on which the pistons are mounted. The pistons are
made with edge welded stainless steel bellows, which have convolutions that nest into
52
one another to minimize void volume. The effective areas of the pistons based on the
manufacturer's specifications [18] are 17.74 cm 2 for the hot platform pistons, 13.61 cm 2
for the first intermediate platform pistons, 7.68 cm 2 for the second intermediate platform
(second stage), 23.42 cm 2 for the second intermediate platform (third stage) and 3.16 cm 2
for the cold platform. The hot platform pistons are rigidly connected together and driven
sinusoidally using a push rod from a room temperature drive. The first intermediate and
second-stage pistons are similarly connected and driven together using a common push
rod. The second-intermediate third-stage pistons are rigidly connected so as to move
together and are driven by a push rod from room temperature. The cold platform pistons
are similarly connected and driven. The hot platform temperature is pinned at
approximately 1.0 K by a 4He evaporation refrigerator.
Within each piston platform, there are superleaks made from porous Vycor glass, which
allow the superfluid 3He- 4He to flow freely between the halves of the SSR during
operation. In the hot platform, the superleaks are three Vycor cylinders 6.03 cylinders in
length with diameters of 1.39 cm, 1.35 cm, and 0.72 cm. In the first intermediate
platform, the superleaks are three Vycor cylinders 10.63 cm in length with diameters of
0.74 cm. In the second intermediate platform second-stage, the superleaks are three
Vycor cylinders 15.16 cm in length with diameters of 0.74 cm. On the second
intermediate platform third-stage the superleaks are four Vycor cylinders 13.09 cm in
length with diameters of 0.74 cm. On the cold platform, the superleaks are three Vycor
cylinders 12.06 cm in length with diameters of 0.73 cm. The total volume of the Vycor
glass in the first and second stage of the SSR is 53.2 cm 3 . The total volume of the Vycor
glass in the third stage is 21.9 cm 3 . Since 28% of the Vycor glass is void space [19], the
glass contributes 15 cm 3 to the 3He-4He mixture volume of the first and second stage of
the SSR and 6.1 cm 3 to the 3 He- 4He mixture volume of the third stage. The 3 He that
diffuses into this volume does not participate in the operation of SSR.
Within each piston platform, there are also isothermal heat exchangers made from
nested OFHC copper cylinders press fit into the piston platforms. A 76 pm gap exists
between the inner walls of an outer cylinder and outer walls of an inner cylinder. At the
top of each cylinder is a flow distributor 0.635 mm deep and 0.317 cm wide around the
cylinder circumference. Each half of the hot piston platform contains one cylinder
53
IStae
fillne
low temperature valves
void spaces
4 He Expansion
Refrigerator
Out In
Hot PI atform
II
-j.UU I
1of 3 Kapton-Epoxy
Composite
Heat Exchangers
- U
Second
Stage
---
First
Stage
First
Intermiediate
Platform
3rd Stage SSR
fill lines
"'UL
low temperature
valves
Second
Intermediate
Platform
D
U
U
I
Third
Stage
1 of 5 Bellows Sets
Copper
Kapton-Epoxy
Vycor Glass (superleak)
Cold Platform
~..
I
Moving Part
Figure 3.7: A schematic of the three-stage SSR The internal volume
of the SSR is filled with 3He- 4He working fluid
2.14 cm in length with a diameter of 3.80 cm, which provides a total heat transfer area of
of the first intermediate platform contains two cylinders that
65.94 CM2. Each half
22
provide a total heat transfer area of 209.70 cm2 . The first cylinder is 3.97 cm in length
with a 4.11 cm diameter while the second cylinder is 4.88 cm in length with a 3.52 cm
diameter. Each half of the second intermediate (second stage) platform contains four
54
cylinders 6.07 cm in length providing a total heat transfer area of 276.6 cm 2. The
diameters of the cylinders are 4.43 cm, 3.91 cm, 3.39 cm, and 2.88 cm, respectively.
Each half of the second intermediate (third stage) platform contains four cylinders 5.02
cm in length providing a total heat transfer area of 257.4 cm 2 . The diameters of the four
cylinders are 4.94 cm, 4.43 cm, 3.79 cm, and 3.16 cm, respectively. Each half of the cold
piston platform has seven cylinders 4.77 cm in length providing a total heat transfer area
of 379.5 cm2 . The diameters of the seven cylinders are 5.83 cm, 5.32 cm, 4.82 cm, 4.30
cm, 3.80 cm, 3.29 cm, and 2.78 cm, respectively.
The recuperators used in this SSR are of a plastic design type. The recuperative
portion consists of alternating layers of 127 [tm Kapton film and 25.4 gm Kapton film
[25] glued together using Stycast 1266 [20]. Each 127 pm layer has five passages 2.38
mm in width and 20 cm in length. The recuperators from the hot platform to the first
intermediate and from the first intermediate to the second intermediate platforms (the first
and second stages of the SSR) are large recuperators with forty 127 m layers and thirtynine 25.4 gm layers. Each large recuperator has a total volume of 20.9 cm 3 , of which
12.1 cm 3 (6.05 cm 3 per SSR half) is dedicated to recuperative heat transfer. The
recuperator from the second intermediate to the cold platform is a small recuperator with
ten 127 gm layers and nine 25.4 tm layers. The total volume of the small recuperator is
11.10 cm 3, of which 3.02 cm 3 (1.51 cm 3 per SSR half) was dedicated to recuperative heat
transfer.
Calibrated ruthenium oxide [21], germanium [22], and carbon resistor thermometers
mounted on the outside of the piston platform are used to monitor the temperature. The
precision of the ruthenium oxide and germanium thermometers, which were used on the
first and second intermediate and the cold platform, temperature measurements are +
0.67 mK at 1.0 K and + 1.02 mK at 350 mK.
The hot platform's temperature was measured using a carbon resistor thermometer.
We had some concern on the temperature measurement provided by the carbon resistor.
The resistance of these carbon thermometers are known to drift with thermal cycling.
This thermometer was calibrated initially in August 1992 and calibrated a second time
against a germanium thermometer in March 1998. It was used in this work in August
2001. Since it was originally calibrated in August 1992, the carbon thermometer has
55
been cycled from room temperature to 1 K approximately 15 times. Figure 3.8 shows the
calibration curve of the resistance of the thermometer generated in August 1992 and
March 1998. Between the two calibrations, the resistance of the carbon resistor
thermometer dropped 6.74 Q on average, or 1.12 Q per year. These two calibrations
were used to project a calibration for August 2001. This calibration is still thought to
overestimate the temperature of the hot piston since the majority of the carbon resistor
thermometer's thermal cycling occurred between March 1998 and August 2001. The
difference in temperature measured between the March 1998 calibration curve and
projected curve is approximately 90 mK. The March 1998 calibration curve measured
the temperature of the hot piston as 1.15 K + 10 mK during operation of the SSR. The
projected calibration curve measured the temperature of the hot piston as 1.07 + 10 mK.
Cooling powers are measured by monitoring the voltage across and current through a
heater made of wound manganin wire. The precision of our cooling power measurements
is + 2 Rw.
The total volumes of the SSR's are given minus the void spaces and the volume of the
fill lines because these volumes are inactive during the operation of the SSR. The total
volume of the first and second stage is 136 cm 3. The total volume of the third stage SSR
is 103.3 cm 3 .
20D
-+-12MNth
U)195-
1999
Ag-20
cco 185
6 185
c'J 175-
165 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
Figure 3.8: Calibration curves for resistance versus temperature of the carbon resistor
thermometer mounted on the SSR's hot platform.
56
3.3.2 Experimental procedure and results
The SSR was prepared for operation by first cooling the refrigerator to 1.0 K, then
centering the pistons on each platform to ensure equal volumes of working fluid in each
SSR half. The first and second stage were filled first with a 3.0 % 3He-4He mixture. For
this SSR, the fill lines to the two-stage SSR were not able to be closed. The low
temperature valves to the two-stage SSR had been removed because of a leaky
subcomponent. The two-stage SSR was test operated with a hot piston stroke of 1.03 cm
(18.27 cm 3 volume displacement and 12.3 cm 3 clearance volume) and first/second
intermediate piston stroke of 0.96 cm (13.1 cm 3 and 7.37 cm 3 volume displaced, 12.0 cm 3
and 12.8 cm 3 clearance volume).
The operation of the two-stage was paused while the third stage was filled with a 3.0 %
3He-4He
mixture. The fill lines to the third stage of the SSR were then closed and the
SSR was operated a hot piston stroke of 1.03 cm (18.27 cm 3 volume displacement and
12.3 cm 3 clearance volume) and (second stage) first/second intermediate piston stroke of
0.96 cm (13.1 cm 3 and 7.37 cm 3 volume displaced, 12.0 cm 3 and 12.8 cm 3 clearance
volume). The third stage was operated with a second-intermediate third-stage piston
stroke of 1.03 cm (24.1 cm 3 volume displaced, 14.98 cm 3 clearance space)and a cold
piston stroke of 1.03 cm (3.25 cm 3 volume displaced, 6.67 cm 3 clearance space).
The cold piston temperature was measured for combinations of two-stage cycle periods
and third stage cycle periods. The temperature was measured by recording the minimum
temperature of the platform during the cycle. In Fig 3.9, the cold piston temperature is
plotted against the cycle period of the cold piston for different cycle periods of the twostage SSR. Effective operation of the third stage SSR required it to have a cycle period
approximately ten times longer than the two-stage SSR. In Fig 3.10 the cold piston
temperature and intermediate temperature are plotted for each of the two-stage cycle
periods.
The slow third-stage cycle period and high cold to second intermediate piston
temperature ratio indicate that the third-stage recuperator is undersized.
The data collected provides far from a complete map of the third stage behavior.
Unfortunately a superleak into the vacuum insulation caused the SSR to warm up.
57
Attempts were made to operate despite this leak by warming up the SSR to 4 K and
pumping on the vacuum, but eventually the leak was too large for this approach to work.
A variable level of the leak in the vacuum can and therefore variable level of external
heat load on the SSR may explain why the behavior of the SSR changed erratically day to
day. Without an external heat load on the SSR, it might be found that even longer cycle
periods are necessary to reach lower temperatures.
It is also possible that there was a 3He mass imbalance in the SSR causing imperfect
recuperation. Four out of the five sets of pistons were centered manually, without the
assistance of the electric "switches" that help locate the piston top and bottom piston
position. For one piston, the second intermediate third-stage compressor, the electronics
that measured the position of the piston inside the cryostat failed and a less reliable
position gauge at room temperature was used to position the piston and measure the
stroke.
It is also possible that the heat rejection in the hot piston was compromised because the
fill lines to the two-stage SSR could not be closed. Closing off the fill lines prevents the
3He
from flowing in and out of them during compression and expansion. However, the
SSR was still able to operate with them open due to the small (but undetermined) flow
rate through these lines. The fill lines, which have an inner diameter of 0.1 mm and have
a length of approximately a meter between the hot piston platform and the platform at 4.2
K.
Operating with these three experimental problems, the three-stage SSR was only able to
match the temperature of the single stage small recuperator SSR in Table 3.3. However,
resolving these technical issues and increasing the recuperative volume of the third stage
of the SSR ought to result in marked improvement over these preliminary results.
58
0.38 Two-stage cycle time
0.375 -
---
0.37 0.365
11.5 s
-- l]- 13 s
-
-- A--15 s
E 0.36 0.355 -
d)
_0
0.35 0.345 0.34 0.335
40
50
60
70
80
90
100
110
120
Third Stage Cycle Period (s)
Figure 3.9: Data for the cold piston temperature versus third stage cycle period for
different two-stage cycle periods.
0.38 0.375 0.37 0.365 -
E
U)
C
0
Two-stage cycle time
---11.5 s
-D--13 s
-A-
15s
0.36 0.355 0.35 0.345 0.34 -
0.335 -0.44
0.46
0.48
0.5
0.52
0.54
0.56
Second Intermediate Piston Temperature (K)
Figure 3.10: Data for the cold piston temperature versus second intermediate
compressor piston for different two-stage cycle periods.
59
Chapter 4
Metal Bellows Dissipation
The compression and expansion of the working fluid within the Stirling cycle is produced
by the motion of pistons. Traditional sliding seal pistons would not be adequate in a subKelvin superfluid Stirling refrigerator because the superfluid 4He would leak through the
sliding seal. In addition, the friction of the seal sliding against a surface would produce
an unacceptable heat load on the system. Therefore, pistons sealed with bellows seals are
used in these designs, specifically, edge welded bellows. Unfortunately, the action of
flexing and relaxing bellows is also a source of mechanical dissipation. This issue was
considered by Brisson and Swift (B&S), who made measurements for small displacement
dissipation of the bellows used in their SSR [12]. However, the piston strokes and
bellows sizes used in this cryostat are well outside B&S's measured range. Here we
describe more extensive measurements of bellows dissipation than those of B&S. It was
found that the losses due to bellows flexure accounted for less than 1 % of overall Stirling
refrigerator losses for single stage SSR's operating around 0.35 K with a high
temperature platform at 1.0 K. Using a numerical model, it was also found that bellows
flexure losses could account for as much as 10% of the total available cooling power of
an SSR operating at 100 mK with a high temperature platform at 300 mK.
60
t
4
.1
Sinusoidal
motion
He eviaporation
refri gerator
First
Platforn
Refrigerator
platform
T-1.4 K
4 He
nTK
Copper strips
Drive shaft
Brass bars
(weak thermal
links)
f-
2
Bellows
Type
60050-1
Bellows test
Second
Platforn
splatform 1
(stationary)
LZ
Drive shaft
mounting
hardware
Coppe r
Cage
Brass
bars
Teflon drive shaft
Drive shaft
mounting
hardware
4--
Third
Platform
Bellows test
platform 2
(stationary)
Bellows
Type
60035-2
-A
Hi
I
Figure 4.1: A schematic of the experimental apparatus used to measure bellows
dissipation
61
4.1 Experimental Apparatus
Figure 4.1 shows a schematic of the experimental apparatus, created by modifying a twostage SSR. The first platform of the SSR, the refrigerator platform, consists of a 4 He
evaporation refrigerator that was maintained at a temperature of 1.4 K. The platforms are
suspended in a vacuum space and are structurally supported using Kapton/epoxy
composite stantions (not shown in Fig 4.1). The bellows interior are vented to the
vacuum space. The second and third platforms were connected to the first platform
independently by a large brass rod providing a weak thermal link. The brass rod from the
second platform was 2.5 cm in diameter and 50 cm long. The brass rod from the third
platform was 1.9 cm in diameter and 25 cm long. One end of each brass rod was
thermally connected to the first platform using a flexible stack of six OFHC copper sheets
to avoid thermally induced stresses in the apparatus as it cooled down. Each copper sheet
was approximately 1.5 cm by 4 cm by 0.13 mm thick.
Mounted on each of the second and third platforms is a pair of bellows. The pair of
bellows on each platform consists of two bellows rigidly connected together by "cage
bars" that pass through oversize holes in the platform. There is no contact between the
cage bars and the walls of the oversized holes. As a result of the cage bars, one half of
the bellows pair on a platform is flexed 180 degrees out of phase with the other half. A
thin-walled stainless-steel drive shaft is connected to bellows pair on the second platform
and is sinusoidally driven at room temperature. In turn, the bellows pair on the third
platform are connected to the bellows pair on the second by a 2.2 cm long, 0.95 cm
diameter Teflon rod. The two pairs of bellows are thus driven in tandem by the drive
shaft.
The bellows used are of the welded bellows type made by Senior Flexonics [18]. All
Table 4.1: Manufacturer's (Senior Flexonics) specifications for the bellows discussed in this work
Bellows
Maximum
Spring Rate
Convolutions,
Stroke (cm)
(N/cm) k
N
Smax
Number of
OD (cm) ID (cm)
60030-1
10
2.62
1.40
1.68
44
60035-1
24
3.81
2.46
1.48
19.5
60050-1
16
4.80
3.53
2.18
26
62
bellows used in the SSR have contoured diaphragms that nest into each other to minimize
the clearance volumes in the compressed bellows. The bellows are made from 347
stainless steel. The bellows' stainless steel flanges are soft soldered onto brass flanges
that are, in turn, bolted to the copper platform. The bellows on the second platform and
third platform are 60050-1 and 60035-2 edge welded bellows respectively. The
manufacturer's specifications for these bellows appear in Table 4.1.
Mounted on each platform is a calibrated resistance thermometer and a heater, depicted
in Fig 4.1 as rectangles with symbol "T" and "H", respectively.
4.2 Procedure
The 4 He evaporation refrigerator was maintained at a temperature of 1.4 K. The
second and third platforms were cooled to 1.4 Kelvin. The two pairs of bellows mounted
on the second and third platforms were then driven sinusoidally with a stroke of 0.98 cm
and a period of 11.1 seconds. The system was left operating for several hours to allow
the temperatures to come to steady state. When the temperatures of the platforms
stabilized, the temperatures of the platforms were noted and the bellows drive was shut
off. Then the platforms were electrically heated by using the appropriate platform heater.
The power to the heater was adjusted until the temperature of the platform matched the
steady state "bellows flex" temperature. Then the voltage across and the current through
the heater were measured for both platforms. The power electrically dissipated into the
platform, which is equal to the mechanical dissipation in the bellows, is the product of the
voltage and the current in the heater.
The procedure was then repeated for strokes of 0.38, 0.45, 0.66, 0.81, and 1.03 cm with
strokes of period 5.89, 6.01, 5.96, 6.1, and 9.5, seconds respectively.
Figure. 4.2 shows the data collected in this test plus the data reported by B&S.
63
60050-1
350 300 250 -
0
C
200 150
-
60035-2
)
0
100
60030-1
50 -
00.2
0.4
0.6
Stroke (cm)
0.8
1.0
Figure 4.2: A plot of the measured bellows dissipation as a function of stroke length.
Each curve is labeled with the specific bellows type. The measurements for type
60050-1 and 60035-1 were collected in the experiment described in this chapter. The
measurements for type 60030-1 were collected by Brisson and Swift.
4.3 Development of Bellows Dissipation Model
The bellows can be simply modeled as a stack of annular disks alternately welded in their
inner and outer radii as shown in Fig. 4.3.
welds
Figure 4.3: A annular disk model of bellows. The annular disks are alternately
welded in their inner and outer edges.
64
Weld
bead
Figure 4.4: The weld bead which is modeled as the source of all dissipation.
Dissipation occurs as the angle between the annular disks changes.
In the actual bellows, the diaphragms of the bellows are not rings and flat but are shaped
to improve the bellows performance. In our model, all the bellows dissipation is
assumed to occur in the bead of the weld. Furthermore, the energy dissipation per unit
length of the weld bead depends only on the variation of the angle 0 that the bellows
makes with the horizontal as shown in Fig 4.4. The variation of the angle AO, can be
estimated from the number convolutions in the bellows N (two disks per convolution),
the stroke length S, and the radial width of the annular diaphragm L the outer bellows
0.8 60050-1
60030-1
0.6 Fit
C
0.4
60035-2
cL
0)
a)
0.2
C
0.02
0.04
0.06
Stroke/NL
0.08
0.10
Figure 4.5: Energy dissipated per unit bead length per cycle versus normalized stroke as per Eq. 4.1.
65
radius minus the inner bellows radius. Assuming that the angle 0 is small our
assumptions can be put in the form,
(4.1)
S
f(AO
E=
NC(D,, +Di )
NL)
where Edi,, is the energy dissipated by the bellows per cycle, D" is the outer bellows
diameter, Di is the inner bellows diameter and f is a function yet to be determined.
Our data and the data from Brisson and Swift are plotted in Fig. 4.5. The data for
bellows type 60030-1 and 60035-2 agrees well. Bellows type 60050-1, however,
performed much better in comparison. A fit to the type 60030-1 and 60035-2 data can be
used as an upper bound for bellows dissipation. The exponential fit shown in Fig. 4.6 to
the type 60030-1 an 60035-2 dissipation values is
Edss
Ediss
N
( D ,+
=A
e
(4.2)
N L4
D)
where A is 0.010 pJ/cm cycle and
has value 62. Unfortunately this model is not easily
Fit
-3
10
g~60050-1
.0
6-
C.
4-
60035-2
60030-1
20)
C
'i 010
C
8-
(
6-
E
4-
0
z
2-
0
C:
0)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Non-dimensional stroke
Figure 4.6: Non-dimensional energy versus non-dimensional stroke as per Eq. 4.3 in the text.
The fit curve is Eq. 4.4 in the text.
66
generalized and the data necessary to refine this model are unavailable from the
manufacturer.
The dissipation data can be non-dimensionalized using parameters published by the
manufacturer, Senior Flexonics. Table 4.1 contains the manufacturer's values for the
bellows spring rate, k, and the maximum recommended stroke, Smax. Using these values
to non-dimensionalize the energy dissipation and the stroke we find
Ediss
2=
G
S(4.3)
(ma)
2kSmax
where G is a function to be determined. The data is plotted on the basis suggested by Eq.
4.3 in Fig. 4.6. An exponential fit to this data is provided by
diss
C
S
(4.4)
_
Y2kSmax
where C has a value of 2.5x10-5 and y has a value of 5.8. This fit overestimates the
dissipation for the small strokes reported by Brisson and Swift. The large stroke data
points for a specific bellows are underestimated by this curve. Assuming that there are
no other relevant dimensionless groups, Eq. 4.4 can be used as an estimator for the
dissipation in the Senior Flexonics OTS type bellows. (Senior Flexonics specifics 16
basic types in this product line with the outside diameters that vary from 0.95 to 40.6
cm.)
4.4 Bellows Dissipation in the SSR
There are significant loss mechanisms in the SSR. Patel and Brisson assert that the losses
in the single stage SSR are primarily due to recuperator losses [9]. However the
magnitude of the contribution of the bellows flexure in comparison to the overall losses
have never been determined. Using the model for the bellows losses developed above
and comparing them to the cooling power data and the theoretical phonon-roton cooling
power curve present by Patel and Brisson for their single stage recuperator this
magnitude was determined. This was done by using our data to generate a bellows
67
dissipation rate using their 0.69 stroke and the fact that they use a 60050-1 bellows for
their expander stage. The total loss is calculated by taking the difference between the
actual cooling power and the predicted phonon-roton model predictions. The ratio of
these results as a function of the expander temperature is shown in Fig. 4.7. Figure 4.7
demonstrates that throughout the low temperature range of their measurements, the
bellows losses account for less than a percent of the overall losses in their SSR.
However, the cooling power of the SSR drops off at lower temperatures. The third
stage of the SSR described in Chapter 2 utilizes bellows of type 60030-2. We wished to
project how the bellows dissipation will effect the third stage of the SSR. A simple
Schmidt model was used to predict the cooling powers of the cold SSR when the cold
platform is a third of the temperature of the hot platform for a variety of temperatures.
Bellows dissipation was modeled using Eq. 4.4. At each set of temperatures the stroke
size that maximized cooling power minus bellows dissipation was found. The results of
this numerical analysis can be seen in Fig. 4.8a and Fig 4.8b. When the SSR hot platform
is at 300 mK and the cold platform is at 100 nK, dissipation in the bellows is 10 % of the
total cooling power of the cold piston. Although bellows dissipation is, relatively
speaking, a larger heat load on the SSR at lower temperatures, Fig. 4.8a and Fig. 4.8b
suggest that cooling can still be achieved.
6-
5-
a)
0
0
3
a)
a)
(D
0
a)
0
0~
0.4
0.5
0.6
0.7
Temperature (K)
Figure 4.7: Fraction of losses in the SSR due to bellows dissipation.
68
0.8
0.3-
0
0.2
-
0.250~
(L
.5
0
0
0.2
-
> 0.15
-
0.1
-
0.05
-
00
0
a.
0
E
0
0
0.15 -
0.1 -
0
0.05
I
I
0.1
0.2
-1
------7
0.3
0
0
Temperature of Cold Platform (K)
0.3
0.2
0.1
Temperature of Cold Piston (K)
(b)
(a)
Figure 4.8: (a) Predicted cooling power minus bellows dissipation of a third-stage expander versus
cold piston temperature. (b) The fraction the bellows dissipation makes up of the cooling power. These
data points were generated using the Schmidt model to predict the cooling powers and Eq. 4.4 to model
the bellows dissipation.
4.5 Temperature Limits on using Bellows Expanders
A more theoretical model was developed to determine the effect of bellows dissipation
versus cooling powers at different temperatures. For this analysis, a highly idealized
model of an isothermal expander with non-dissipative valves was considered. The
==O
OM=M3
Figure 4.9: A schematic of the idealized expander with valves.
69
dissipation due to the flexing of the metal bellows as described by Eq. 4.4 is assumed to
be the only losses in the model. Losses due to pressure oscillations in the bellows were
ignored.
The working fluid, 3%
3He_
He mixture, enters the expander. The expander is
bypassed by a superleak to a large 4He reservoir so that the expander acts only on the 3He
component of the mixture which we will assume behaves as an ideal gas throughout the
expansion process. The molar volume of the 3 He is assumed to be of the form
Vmolar
=
27.58X10-6X m 3/mole where x is the molar concentration of 3He. The working fluid is
assumed to enter and exit the expander at the temperature of the expander. The clearance
volume in the expander for a given bellows is determined by assuming there is a 0.125
mm space in each convolution of the bellows.
The analysis requires finding the optimal valve timing and stroke for both the nondissipative and dissipative cases. The calculated optimal cooling power of the expander
are plotted in Fig 4.1 Ga as a function of temperature. Fig 4.1Gb shows the ratio of the
cooling power using non-dissipative bellows to those with the dissipation that follow Eq.
4.4. These analyses suggest that, ideally, these bellows can be used to provide cooling to
temperatures below 100 mK using 3 He- 4 He mixtures. It should be noted that the Fermi
60050-1
a)
b)
60050-1
0.8
-2
-
60035-1
10
60035-1
C_
C
10
-3
60030-1
0
0.6
-
0.4
-
0.2
-
Cci
0
0:
0.
C
C
1o
-4
60030-1
0)
0
CL
10
10
C0
3
10
-5
-6-
0.0 2
34
567
2
3
5
2
0.1
Refrigeration temperature (K)
3
4 56
7
2
3
0.1
Refrigeration temperature (K)
Figure 4.10: Projected cooling power of non-ideal expander (a) and cooling power ratio of non-ideal
expander to ideal expander (b) versus temperature.
70
4 5
temperature for the 3He in a 3% mixture is about 200 mK; and hence, the use of the
Boltzmann ideal gas model is questionable for temperatures below 50 mK.
71
Chapter 5
Theoretical Development
In the following chapter an analytical and numerical model of the superfluid Stirling
refrigerator using a modified Schmidt model that takes into account the phonon and roton
excitations is discussed. Additionally, this chapter explores the reasons for and the
implications of the addition of a third stage to an experimental apparatus.
5.1 The Schmidt Model
Gustav Schmidt developed in 1871 a simple mathematical model of the Stirling cycle.
In the Schmidt model, the working fluid is assumed to be an ideal gas. The discrete
compressive and expansive motions of the pistons described earlier in the ideal cycle are
replaced with sinusoidal motions of the pistons. The sinusoidal motion of the compressor
and expander pistons are ninety degrees out of phase. The Schmidt model also accounts
for a clearance volume, a volume not swept by the piston motion, in both the
compression and expansion spaces and for a non-zero regenerator volume. Temperature
in the regenerator is modeled as varying linearly with position in the direction of flow
and being uniform normal to the direction of flow. The average temperature of the
regenerator is therefore the log-mean temperature difference of the low and high
temperature [26].
72
Compression
Space
Expansion
Space
Qc
W CWh
0
+ I
Qh
Recuperator
XI
II
T
T
IV
V
Vi
'V
Vc
V, =V
Vh
+ Vs [1 + cos(8 +
Q= Wc =
7FVswcPmeansino
Vswh
v
A)]/2
Vh = VcIh+ Vswh [1 + cos(
((1-b b2 -i)
Qh = Wh
(=
efficiency of refrigerator
c
Pressure Relationships
P=
+ VSh/Th)
2
2
(1/2)((VSWC/Tc) +2(V,,,/Tc)(VSh/Th) (VSwhh) )1/
NR
s(1+bcos(8 +
)
2
Pmean =
N
s = VSWC/(2Tc) + Vcic/Tc + Vr1n(Th/Tc)/(Th-Tc)
_QC
7 7
tan-' (V,,csin(cx)/Tc)/(Vswccos(x )/Tc
b
TVswhPmeanSin(l-)
Variables 0, c, s, and b are defined as
W =WC+Wh
)]/2
W
Pmax
+ Vwh/(2 Th) + VcIh/Th
b = c/s
NR
Pminrnns(1NR
+b)
Variables:
V, ,
cold (expansion) volume
Vs , cold swept volume
Vcf , cold clearance volume
Vh ,
hot (compression) volume
0,
(A,
TC,
drive angle
phase angle (-T/2)
cold temperature
Th,
hot swept volume
Vh , hot clearance volume
N,
R,
hot temperature
moles of gas
universal gas constant
Vswh,
Vr,
recuperator volume
Figure 5.1: An equation summary of the Schmidt model for a single-stage Stirling cycle
The summary of the equations of the Schmidt Model is contained in Fig 5.1 [26]. The
volumes of the cold and hot piston vary as a function of the crank angle, 9, which is a
function of time 9 = ot. It is assumed that there is no pressure drop across the
recuperator so the pressure of the system is to be spatially uniform. On the other hand, as
the hot volume, cold volume, and total volume of the Stirling device changes during the
cycle, the pressure changes with it in time.
The Schmidt model can be expanded to handle Stirling cycles with more than one
stage. Figure 5.2, provides an equation summary for a generalized two-stage Stirling
73
W,
W2
W
3
sw1
N1"
Qi
VrA
VrB
V2 = Vsw 2 [1+ cos(O + cx )]/2 +Vd2
V1 = V5 [1+ cos0]/2 +V 1
V3 = Vsw 3[1+ cos(O - 7) /2 +Vc
13
b
Q2 = W 2 = 7TVsw 2 Pmeanin( -(x)
Q1 = WI= TVswPmneansin(j3) (41b2_
Q3 = W3 =
nTVsw 3 Pmeansin(fl-7) ( (1-b2)_1)
Pressure Relationships
variables s, f3, c, and b are defined as
W= W1 + W2 + W3
s
for a refrigerator where
piston 1 is a compressor and
piston 2 and 3 are expanders
77~
1 +
V
Vsw2
d2
sw 3
7-++
27
NR
s(1+bcos() + (3))
Vd3
7
SNR
Pmean
VrAln(Tl/T 2)
+
Q3 + Q2
W
=
tan
Q3
Regenerator B
Kegeneraior iA
t-
+
VrBln(T 2/T 3)
(T2-13)
(Vsw 2 /T2)sinx + (V,, 3/T3)sin7
(Vs w53 f 3 )cos7 + (Vsw /T2)cos +(Vsw /T
2
1 1)
+
vsw
c
P
+2
+
cos(7)
2
2 cs
rn
Pmi
c
sNR
NR
s(1 -b)
NR
rnns(1 +b)
os(7-)
27s7_)
b = c/s
Variables
VrA, volume of regenerator A
VrB, volume of regenerator B
0, drive angle
cx, phase angle of piston 2
y, phase angle of piston 3
N, moles of gas
R, universal gas constant
Vi, total volume of piston i
Vwi, volume swept by piston i
Vci1 , piston i clearance volume
Tj, temperature of piston i
Q, heat transfer to piston i
Wi, work transfer from piston i
Figure 5.2: The equation summary of the Schmidt model expanded to model a two-stage Stirling
device. A two-stage Stirling device consists of three pistons connected by two recuperators.
device. Piston 2 and 3 in Fig 5.2 are generalized to have separate phase angles with
respect to the drive angle. In the two-stage SSR described in Chapter 2, the phase angle
of the intermediate and cold pistons are the same. In Fig. 5.3, the Schmidt model has
74
W1
W,
W2
n
10 2,
VswI
Vsw
V
01
I
V
LI
-------
V
wz!
A
Regenerator 1
VrR
'
Vn
-
Qn
Regenerator (n-1)
Vr,(n-)
egenerator 2
Vr 2
variables s, 0, c, and b are defined as
Volume per Piston
Pressure Relationships
Vi= Vi[1+ cos(O + (i)]/2 +Vc, 1
s=
V - + V
s~~k
=
+ (L1)(Vrkln(Tk/T(k+1))
1-- (Tk-T(k+1)
Heat absorbed and Work Performed per Piston
Qi=W
.
= TrVw iPmeansin(#-cx)
Pmean
n
b
(Vswi/Ti)sinc\ i
=
S=tan-'
p =NR
s(1+bcos(O + 0))
max
bNR
n
(V wi/Ti)cosni
n
n
I
NR
Pmin = s(1+b)
V11/2
E Z(-i
i=1 j=1
Tj
V5*'I cos(nXi-a )
Tj
I
b = c/s
Variables
Vi, total volume of piston i
V5 , volume swept by piston i
Vc,,, piston i clearance volume
Tj, temperature of piston i
Qi, heat transfer to piston i
Wi, work transfer from piston i
Vrk, volume of regenerator k
0, drive angle
(xi, phase angle of piston i (a, =
N, moles of Gas
R, universal gas constant
0, by convention)
Figure 5.3: The equations summary for a Stirling cycle with an indefinite number of stages chained
together
been expanded to handle a Stirling cycle for a device with an indefinite number of stages
connected in a chain.
75
5.2 Sub-Kelvin Superfluid Stirling Refrigerators
As previously discussed in the introduction, a low concentration
3He-4He
mixture can
be used as the working fluid in a Stirling cycle below 2.17 K. Below this temperature,
the 4 He becomes a superfluid and has no viscosity. The superfluid 4He can flow freely
through superleaks. Thus, the normal viscous 3He particles in a low concentration 3He4He
mixture can be expanded or compressed by a piston bypassed by a superleak to a
reservoir of pure 4He. A low temperature Stirling cycle can be constructed out of two
such pistons connected by a regenerator.
With a few minor modification the Schmidt model can be applied to the low
temperature Stirling refrigerator model. For the application of the Schmidt model to this
system, we will assume our regenerator is perfect. The regenerator is assumed to have
time independent temperature distribution. As in the Schmidt model, temperature is
modeled as varying linearly with position in the direction of flow and being uniform
normal to the direction of flow. Also, viscous interactions are assumed to be negligible
so there is no pressure drop across the regenerator and the spatially uniform pressure
assumption used in the Schmidt model still applies.
We assume an SSR of volume Vtotal is loaded with
3He-4He
mixture at 3He molar
concentration of xi. During the operation of the SSR, the number of moles of 3He in the
total volume is conserved and is equal to N 3 = xiVtotal/vm where vm is the molar volume of
the solution. The number of moles of 3 He, N3 , is treated as the number of moles of the
gas in the Stirling device in the equations shown in Fig 5.1 (or Fig 5.2 or Fig 5.3).
5.3 Limitations of 3He- 4He Working Fluid
The cooling power of the SSR is affected by the changes in the distribution of the 3 He as
the SSR cools down. As the expansion space of the SSR cools, the average density of
the "ideal gas" increases in the chamber. Conversely, since the total number of 3He
atoms is conserved, the average density (concentration) of the 3He component in the
compressor must decrease as the expander cools.
76
Using the Schmidt model above, minimum concentrations of 3He in the SSR can be
determined for any set of operating conditions by solving for the 3He concentration of
the hot piston at the maximum pressure. Likewise, the minimum concentrations of 3He in
the SSR can be determined for any set of operating conditions by solving for the 3 He
concentration of the cold piston at the maximum pressure. The results are
Xirn=
V,_,__x__
Vlotal i
Ths(1+ b)
and
5.4 Phonon and Roton Effects on
Xx.
x
3He
-
V, ,x.
Vtotal X
51
Ts(I - b)
(5.1)
Concentration
So far this model assumes that the only active particles undergoing the Stirling cycle are
the 3He atoms. We have assumed that the superfluid 4He has acted solely as a nonparticipating background substance during the cycle. This is not quite true.
Below 2.17 Kelvin, He can be modeled as a two fluids, a superfluid with no viscosity,
and a normal viscous fluid. The normal fluid consists of phonon and roton excitations in
the 4He that behave roughly as viscous fluid of gas particles (whose number is not
conserved) and have an associated pressure called the fountain pressure. The fraction of
the whole each of the two components make up is dependent upon the temperature of the
4He.
At 2.17 K, the 4He is almost exclusively normal fluid. Below 1 K, 4He is almost
exclusively superfluid.
Since the fountain pressure of the 4 He is higher on the hot end and lower on the cold
end of the SSR, it tends to flush the 3He atoms towards the cold end of the SSR. The
effect of the 4He fountain pressure is to increase the concentration of the 3He in the
expander while decreasing the concentration of
3He
in the compressor. Thus, the
maximum and minimum prediction of the concentrations of the
3He
Schmidt model tends
to be lower than the actual maximum concentration and higher than the actual minimum
concentration. Thus, a more accurate model needs to account for the participation of the
normal viscous component of the 4He.
Like the previous analyses we will assume that the system has reached a cyclic steady
state and that the average temperature of each platform is steady. We can think of the by
77
the SSR as a volume whose spatial temperature distribution is known. The liquid 3He4He
is modeled as a incompressible fluid and the
3He
is assumed to behave
thermodynamically as a monatomic Boltzmann gas. We assume that there are sufficient
3He
atoms in the total volume that, for the given temperature distribution, the chemical
potential of the 4He is constant.
A simple form for the chemical potential of the 4He in a 3He- 4 He solution is [14]
44 = Pvm - xRT - pf (T)vm
(5.2)
where P is the total pressure of the 3He- 4He liquid, x is the molar concentration of 3 He, vm
is the molar volume of 4He and pf is the 4He fountain pressure. An empirical expression
for the fountain pressure is,
(5.3)
pf (T)= [BT 4 +AT 3/2-Al IT +CeA2 IT I/vM
where A, B, C, A,, A2 have values of 23.2 J/mole-K3 /2 , 6.75x10- 3 J/mole-K 4, 500 J/mole
8.65 K, and 15.7 K, respectively [24].
Equation. 5.2 can be rewritten as
(5.4)
-(14 + Pvm) = Pf (T)vm + xRT.
Since the pressure of the system is spatially constant the left side of the equation is both
spatially constant and constant relative to temperature for the fluid in the SSR. Since we
have said that the spatial temperature distribution of system is known, for a given mass of
3He
atoms in the SSR, the spatial distribution of the 3He atoms is determined by Eq. 5.4.
The number of moles of 3He in the system is determined by integrating the 3He molar
concentration over the total volume,
N
3
dV.
=
(5.5)
VM )
Thus if the temperature distribution and the number of moles of 3 He atoms in the total
volume is known, the distribution of the 3He atoms in the system can be determined,
although not necessarily in closed form.
The simple method for accounting for the effects of the fountain pressure of the 4He in
a SSR is to neglect the regenerator volume and assume the piston volumes are isothermal
78
[25]. This allows the 3He atom distribution to be determined explicitly. The above set
equations can be reduced to
Pf (Tc)vm + xcRTc = Pf (Th)vm +xbRTh
(5.6)
N 3 = _c Vc+jj
(5.7)
Vh
where Vc is the volume of the cold piston, T, is the temperature of the cold piston, Vh is
the volume of the hot piston, and Th, is the temperature of the hot volume. Equation 5.6 is
a statement that the sum of the partial pressures of the phonon-roton gas and the 3He ideal
gas must be uniform throughout the SSR volume. Equation 5.7 is a statement of the
conservation of the mass of the 3He. The concentrations of the cold and hot volumes, xc
and Xh can be solved for explicitly. If the regenerator volume is a small fraction of the
SSR total volume these equations can be used to calculate the 3He distribution in the
refrigerator. However, for larger regenerators, these equations will overestimate the
concentration of 3 He atoms in the cold end. In the SSR with the two large regenerators
described in chapter 2, the volume of the regenerators made up approximately 9% of the
SSR volume. This SSR had large clearance volumes. If the clearance volumes in this
SSR were reduced, the regenerator volume could be as much as a third of the whole
volume. Therefore, a fine tuning of the model of 3He distribution in the SSR should
include the regenerator volume. Accounting for this volume requires changing from an
analytical approach to a numerical one.
A numerical model was developed based upon the specifications of the experimental
SSR described in Chapter 2 in order to determine the distribution of
3He
atoms in the
SSR during different stages of operation. The cold piston and hot piston are assumed to
be isothermal. The temperature distribution in the regenerator is assumed to be
monotonic function of position in the direction of flow, uniform normal to the direction
of flow, and time independent. Figure 5.4 shows a sketch of the temperature function of
a single stage SSR. The regenerator is shown with three different possible temperature
functions, a hot, linear, and cold function. In a regenerator with a hot or cold temperature
function, the temperature of the fluid does not change linearly through the regenerator.
The temperature of the fluid in the regenerator is biased towards the cold or warm end.
However, in this model, the fluid still enters or exits each piston at its temperature.
79
Expansion
Space
Compression
Space
Regenerator
I
I
I
2;KA;
i
i
I
I
SVI
I
V
C
r
'V
regenerator with a
hot' temperature function
TC
I
I
h
-
h
regenerator with a
-
"c5id"TI& perture functibn
Position
Figure 5.4 The temperature distribution for a Stirling device is plotted versus position. The
expansion and compression space are assumed to be isothermal. Three different temperature
distributions for the regenerator are shown.
Equation 5.5 is rewritten as a function x(T),
x(T) = C-p
(5.8)
Tv
RT
where C is a constant substituted for -p4-Pvm. N3, the total number of moles of 3He in the
SSR is now
N3 =
-
Vc )+ (
(5.9)
hVh )+ N3,reg
where N3,reg is the number of moles of 3He in the regenerator and approximated by
(5.10)
N3,reg = A c(L)nX
i=I
1M
where A is5 the total flow channels cross sectional area of the regenerator, L is the length
of the regenerator, and n is the number of finite volumes the regenerator is divided into in
the numerical model. The numerical model solves for the value of constant C such that
Eq. 5.9 is satisfied for a predetermined value of N3.
80
The above set of equations models a single-stage SSR but can easily be modified to
model a two-stage SSR by changing Eq 5.9 to
N3 =
xC V
-Vc
+xh
)+(
Vh+x
Vh
+
' Vi
(5.11)
+ N3,reg,cold + N3,reg,hot
where xi and Vi are the 3He concentration and volume of the intermediate stage
respectively.
N3,reg,coId
and N3,reg,hot represent the number of moles in the regenerator
between the cold and intermediate volumes and the intermediate and hot volumes,
respectively, and are defined analogously to Eq. 5.10.
The value of the volumes of the cold, intermediate, and hot platforms changes over the
cycle; therefore to fully explore the distribution of 3He in the SSR over a full cycle, the
value of C needs to be determined for different points in the cycle.
A series of numerical simulations were performed using the approach discussed to
determine what variables in the SSR most affect the 3He distribution in the SSR. Unless
otherwise specified, the SSR numerically simulated is the two-stage SSR described in
In order to solve the system of equations, the described temperature function of the
regenerator needs to be known. A regenerator with a "hot" temperature function, like the
one sketched in Fig 5.4. will store more 3He atoms than a regenerator with a "cold"
temperature function. In an operating SSR, this temperature function oscillates during
the cycle and even determining its exact average value is numerically difficult. In order
to explore the effect of the regenerator temperature function on the distribution of 3 He
particles in the SSR, three temperature functions for each of the two regenerators in the
Table 5.1: The specifications for the SSR modeled unless otherwise specified.
Two-stage SSR (Two large recuperators)
Single Stage SSR (One large
Initial 3He- 4He concentration: 3%
recuperator)
Piston Strokes:
1 cm
Piston Strokes:
Recuperator Volume: 6.05 cm per Recuperator
Hot Piston
Intermediate
1 cm
Recuperator Volume: 6.05 cm 3
3
Cold Piston
Hot Piston
Cold Piston
Piston
temperature
1.05 K
0.5 K
0.3K
0.3K
0.1K
Vswept
17.74 cm 3
13.61 cm 3
7.68 cm2
23.42 cm 3
3.16 cm3
Vciearance
12.04 cm 3
11.85 cm 3
12.66 cm
14.98 cm 3
6.67 cm 3
81
two-stage SSR were considered. The hot and cold temperature functions are of the form
(5.12)
T(x)=Tj l+A[(h 1f)-]l
A
where x is the position in the direction of the flow such that x = 0 corresponds to the low
temperature end of the regenerator and x = 1 corresponds to the high temperature end of
the regenerator. For the regenerator between the hot piston (Th
=
1.05 K) and
intermediate piston (Ti = 0.5 K), the values 1.15 and 90 were substituted for A to generate
a hot and cold temperature function, respectively. For the regenerator between the
intermediate piston (Th = 0.5 K) and the cold piston (TI = 0.3 K), the values 0.7 and 90
were substituted for A to generate a hot and cold temperature function, respectively. The
functional form of Eq. 5.12 somewhat arbitrary but it is reminiscent of the sort of
temperature functions found in counterflow heat exchangers with a heat capacity
imbalance.
0 8-
0.4 -
0.2 -0-
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
3He concentration
Figure 5.5: The 3He concentration of the cold, intermediate, and hot piston volume of a two-stage SSR plotted
versus temperature. The black triangles correspond to the SSR with cold regenerators, the black diamonds
correspond to the SSR with hot regenerators, and the squares correspond to the SSR with linear regenerators. The
top points correspond to the hot piston volume, the middle points correspond to the intermediate piston volume, and
the bottom points correspond to the cold piston volume. The six data points per SSR correspond to the minimum and
maximum 3He concentration in each piston in the cycle. The dotted lines correspond to lines of constant 4He
chemical potential. The solid line shows the boundary of the two phase region.
82
In the SSR with two large regenerators, where the total regenerator volume still only
makes up 9% of the total SSR volume, the effect of the regenerator temperature function
on determining the 3He distribution in the SSR is small. Figure 5.5 shows the
3He
concentration of the hot, intermediate, and cold piston volumes of the SSR. The black
diamonds, black triangles, and squares represent an SSR with two hot, cold, and linear
regenerator temperature functions, such as was described in Fig. 5.4. The two data points
per piston volume per SSR correspond to the minimum and maximum 3He concentration
during the operating cycle. Constant g 4 curves are shown as dotted lines on this figure.
The data points for the minimum or maximum
3He
concentration for an SSR lie on
constant [4 curves. The data points shown for each SSR thus outline all of the constant
g4
curves that the 3He- 4He mixture in the SSR operates on during the cycle. The solid
black lines delineates the edge of the two phase region. These three SSR's operate well
within the superfluid homogeneous mixture region. The difference between an SSR with
cold, hot, and linear regenerators is small.
0.8 -
: 0.6
-
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
3He concentration
Figure 5.6: The predicted minimum and maximum 3He concentrations of the hot, intermediate,
and cold piston volumes of four different models of the 3He distribution in the SSR are plotted
versus temperature.
The squares correspond to a model that includes both regenerator volumes
and 4He fountain pressure. The * correspond to a model that includes fountain pressure but
omits
the regenerator volumes. The triangles correspond to a model that omits fountain pressure but
includes regenerator volumes. The circles correspond to a model that omit both fountain pressure
and regenerator volumes.
83
For an SSR with the same spatial temperature distribution, piston volumes, and initial
3He
concentration mixture, Fig. 5.6 contrasts the following different models for 3 He
distribution within the SSR: a model that includes both fountain pressure and regenerator
volumes, a model that omits fountain pressure but includes regenerator volumes, a model
that includes fountain pressure but omits regenerator volumes, and a model that omits
both fountain pressure and regenerator volumes. As can be seen, for an SSR operating
below 1.1 K and whose regenerators make up approximately 9% of its total volume, a
simple analytical model neglecting both the regenerator volumes and the effect of the 4He
fountain pressure can still accurately predict the 3He distribution within the SSR volume.
On the other hand, a careful measurement of the clearance volumes in the SSR is
crucial to determining the expected 3He distribution. Despite design efforts to minimize
clearance volumes in an SSR, clearance volumes still make up a significant percent of the
total volume. The clearance volume of the SSR modeled in Fig 5.5 and Fig 5.6 is 27% of
the total volume of the SSR. The clearance volumes often have convoluted geometries,
as they include the isothermal heat exchanger volume, so that an accurate measurement
of these volumes can be difficult. Figure 5.7 shows the 3He distribution of the same SSR
with different fractions of clearance volume.
Figure 5.7 shows the 3He distribution in the SSR for three different initial load
concentrations of 3 He-4 He mixture. This figure demonstrates that the SSR could be
loaded with a mixture as rich as 5% and reach 300 mK without crossing into the two
phase region during operation.
In Fig 5.8 the minimum and maximum 3He concentration of the SSR loaded with 3%
mixture is plotted if the hot piston is 1.05 K, the intermediate piston is 0.5 K, and the cold
piston is 0.2 K. The maximum cold piston 3He concentration comes close to the
boundary of the two phase region. This suggests this SSR is temperature limited to
approximately 0.2 K by the two phase region.
84
In order, therefore, for lower
1
L
\AX\3
0.8
2D
0.6
E
0.4
'
0.2
0
0.02
0.06
0.04
3
0.1
0.08
He concentration
Figure 5.7: A plot of the 3 He expected distributions versus temperature for three SSR's with
different total clearance volumes. The triangles correspond to an SSR whose clearance volume is
27% of its total volume. The * correspond to the same SSR with half that clearance volume
(clearance volume is 15.6% of the total volume). The black squares correspond to the same SSR
with no clearance volume. Note that the difference in scale between Fig 5.5 and Fig 5.6.
08
E 0.4-
.2
00
00
.5
00
00
.8
00
3He concentration
Figure 5.8: The minimum and maximum 3He concentrations plotted versus temperature for an SSR
whose hot piston is at 1.05 K, intermediate piston at 0.5 K, and cold piston at 0.2 K and which was
loaded with 3% mixture.
temperatures than 0.2 K to be reached, a third stage of the refrigerator needs to have a
working fluid separate from that of the first two stages. This allows the third stage to
operate on a different family of p4 lines and avoid the two phase region. Figure 5.9
shows the minimum and maximum 3He concentrations for the cold and hot piston
85
0.35
0.3 -
\f
0&\
0.25
0.2
0.15
0.1
*
V.0
A
0.05
0
0
0.01
0.02
0.03
0.04
3
0.05
0.06
0.07
0.08
0.09
He concentration
Figure 5.9: The maximum and minimum concentrations of 3 He are plotted for the cold piston volume
and hot piston volume of a single stage SSR whose cold and hot pistons are at 0.1 K and 0.3 K. The
squares correspond to the SSR loaded with a 3% 3He mixture and large recuperators (6.05 cm 2 ). The
triangles correspond to an SSR loaded with 3% 3He mixture with small recuperators (1.5 cm 2 ). The
diamonds correspond to an SSR loaded with 1.5% 3 He mixture and large regenerators (6.05 cm 2 ).
volumes of a separate third stage SSR. The SSR is a single stage SSR that has the
specification described in Table 5.1. The assumed parameters for three SSR's are shown,
a single stage SSR with a large regenerator loaded with 3% mixture, a single stage SSR
with a large regenerator loaded with 1.5% mixture, and a single stage SSR with a small
regenerator loaded with a 3% mixture. As can be seen in Fig. 5.9, this single stage SSR
is unable to reach 100 mK using a 3% mixture without operating in the two phase region.
The 3He has been modeled so far as an ideal Boltzmann gas. However at lower
temperatures, its behavior changes to that of an ideal Fermi gas and the distribution of the
3He
atoms within the system can no longer be derived using the Boltzmann ideal gas
laws. We wished to determine how large an effect this behavior change has on 3He
distribution in the third stage of the SSR. From 0.3 K to 0.1 K the fountain pressure of
4He
is in the range of 2 Pa to 0.02 Pa (as compared to 540 Pa at 1.05 K) so the we can
neglect the fountain pressure in this temperature range. Eq. 5.4 now reduces to
(5.12)
-(94 +Pvm)=xRT.
86
The osmotic pressure of 3He in solution is H 3 =
RT,
Vm
where vm is the molar density
of 4He. So Eq. 5.12 implies that curves of constant p4 correspond to curves of constant
3He
osmotic pressure.
The new model we will develop to account for the transition of 3He gas to a Fermi gas
will use experimentally determined data of the properties of 3 He in this temperature
range. The Radebaugh tables 3He- He properties include a table of 3 He osmotic pressures
versus temperature for constant concentrations of 3He [27]. This data was interpolated to
generate curves of 3He concentration versus temperature along curves of constant
osmotic 3He pressure. Examples of these lines are shown in Fig 5.10.
Assuming the same temperature function for the piston volumes and regenerator as the
SSR modeled in Fig. 5.9, a new numerical model determined the
3He
constant osmotic
pressure curve such that Eq. 5.10 was satisfied for a given loaded 3 He concentration. The
data points generated by this method include the effect of the 3He atoms changing from a
1.4 -
1.2
-
1-
0.8
CU
E
FT
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0. 3
3
He Concentration
Figure 5.10: Curves of constant 3He osmotic pressure. For temperatures where the 4 He fountain
pressure is negligible, these curves correspond to lines of constant 4He chemical potential in the
superfluid homogeneous mixture region. The dotted line corresponds to the boundary of the two
phase region. In the two phase region lines of constant 4He chemical potential and 3He osmotic
pressure diverge.
87
0.35
A
0.3
13
o9
hE&13
0.25
0.2
L 0.15
E
CD
0.1 -A
A
AK
X1
0.05
0
0
0.01
0.02
0.03
0.04
3 He
0.05
0.06
0.07
0.08
0.09
concentration
Figure 5.11: Maximum and minimum 3He concentrations for the hot and cold piston of a single
stage SSR. The square correspond to an SSR modeled using the numerical Boltzmann model loaded
with a 3% mixture. The diamonds correspond to an SSR modeled using the numerical Boltzmann
model loaded with a 1.5% mixture. The * corresponds to an SSR modeled using the interpolated
data model loaded with a 3% mixture. The black triangles correspond to an SSR using the
interpolated data model loaded with a 1.5% mixture The dotted lines correspond to lines of constant
4
He chemical potential. The solid line shows the boundary of the two phase region.
Boltzmann gas to a Fermi gas since these points are based on the Radebaugh tables. The
data points for the maximum and minimum 3He concentration of the hot and cold piston
volume derived from an interpolation of real data are compared to the numerical
Boltzmann model data in Fig 5.11. The difference between the model based on the
Radebaugh tables and the numerical data was small for the 3% mixture but more
significant for the 1.5% mixture. Notice that the squares corresponding to the Boltzmann
model no longer run parallel to the constant g4 curves in Fig 5.11.
5.5 Heat Exchangers
The numerical analysis used to model the state of 3He concentration in the SSR for a
given temperature function allows an interesting perspective on how the 3He flows in the
SSR during operation. Figure 5.12 shows how the regenerators in the two-stage SSR
and single-stage SSR described in Table 5.1 store
88
3He
atoms during the cycle. Figure
5.13, 5.14, and 5.15 all show how the storage of 3He atoms in the regenerators compare
to the mass flows out of the piston volumes.
0.00250.002 0.0015 0.001 0.0005
0
-0.0005
-0.001 -0.0015 -0.002-0.0025
0
Figure 5.12: The change in 3He moles in three regenerators over the cycle. The solid line
corresponds to the regenerator between the hot and intermediate piston volumes on the twostage SSR. The dark broken line corresponds to the regenerator between the intermediate
and cold piston in the two-stage SSR. The light broken line corresponds to regenerator
between the hot and cold piston of the singe-stage SSR.
0.006-
0.004-
.0.002-
CD
0
0
'
2
3
4
..--
6
-0.002 -
-0.004-
e
Figure 5.13: A graph of the change in 3He moles in two volumes in the two-stage SSR over the cycle. The
solid line corresponds to the regenerator between the hot piston volume and the intermediate piston volume.
The broken line corresponds to the mass flow out of the hot piston volume.
89
0.004
-
0.002
N
N
0)
N
2
5
6
CV)
-0.002
-0.004-
-0.006 -
0
Figure 5.14: A graph of the change in 3He moles in two volumes in the two-stage SSR over the cycle.
The solid line corresponds to the regenerator between the intermediate piston volume and the cold
piston volume. The broken line corresponds to the mass flow out of the cold piston volume during
the cycle.
0.003
-
0.002
-
0.001
-
,'7
0
0
,-'1
V
4
2
5
6
-0.001 ........................ ..................
-0.002
-
-0.003 -
0
Figure 5.15: A graph of the change in 3He moles in two volumes in the single-stage SSR over the cycle.
The solid line corresponds to the regenerator between the cold piston volume and the hot piston volume.
The broken line corresponds to the mass flow out of the cold piston volume during the cycle.
90
Chapter 6
Summary and Conclusions
The goal of this thesis to was to improve the performance of the SSR initially built by
Ashok Patel and to develop a better understanding of the SSR. This was accomplished in
the following ways:
1.
A two stage SSR with large recuperators was operated in an attempt to improve on
the low temperature performance of the SSR. Unfortunately, due to unusually high
heat loads induced by a mechanical linkage failure and a high residual gas
concentration in the vacuum can (due to a small superleak in the SSR), the
performance of the two-stage SSR did not improve over that of the previous work.
2. A three stage SSR was built and operated. This was the first successful operation of a
three stage SSR. Further experimental work is required to fully evaluate the three
stage SSR.
3. The dissipation in metal bellows was measured at 1.2 K. The energy dissipated by
these bellows for strokes between 0 and 1 cm was found to vary between 0 to 500
pJ/cycle. These measurements showed the losses due to bellows flexure to account
91
for less than 1% of refrigerator losses for a single stage SSR. A characteristic
equation was developed to estimate the energy dissipation rate for the entire family of
Senior Flexonic bellows.
4. A simple numerical model was developed to understand the 3He particle distribution
in the SSR during operation. An accurate model of the 3He concentrations in the SSR
allows optimal SSR design and choice of initial loading concentrations for to prevent
the 3He-4He phase separation in the cold end of the SSR. The variance between this
numerical model and the simpler analytical models in predicting the 3He particle
distribution within a two-stage SSR was shown for a SSR with the dimensions of the
experimental two-stage SSR used in this work. The small variance between the
numerical and analytical models suggests that for SSR's of relatively small total
recuperator volume and hot platform temperatures below 1.1 K, the error incurred by
ignoring recuperator volumes and 4He fountain pressure is minimal. Alternatively,
since in even best designs, clearance volumes make up a significant proportion of the
internal volume, an accurate measure of this volume is necessary. This model also
demonstrates the storage of 3He particles in the regenerator as compared to the flow
of 3He through the regenerator.
Further evaluating the three stage SSR would require the following steps. The valves to
the hot platform need to be repaired. Various electrical issues that compromised the
accurate centering and positioning of the pistons need to be fixed. Most importantly, two
new Kapton-Epoxy composite heat exchanger need to be built, one to replace the heat
exchanger between the intermediate and cold platform of the two stage SSR which is
suspected of having developed a large superleak, and a large heat exchanger for the third
stage of the SSR so that it can be fully evaluated.
92
Bibliography
[I] V. Kotsubo and G. W. Swift. Superfluid Stirling refrigerator: A new method of
cooling below I Kelvin. In Proceeding of Sixth InternationalCryocoolers
Conference, volume 2, pages 59-70, Bethesda Maryland, 1991. David Taylor
Research Laboratory.
[2] V. Kotsubo and G. W. Swift. Superfluid Stirling-Cycle refrigeration below I Kelvin.
Journalof Low Temperature Physics, 83:217-224, 1991.
[3] J. G. Brisson, V. Kotsubo, and G. W. Swift. The superfluid Stirling refrigerator, a
new method of cooling below 0.5 Kelvin. PhysicaB, 194-196:45-46, 1994.
[4] J. G. Brisson and G. W. Swift. A recuperative superfluid Stirling refrigerator.
Advances in Cryogenic Engineering, 39B:1393, 1994.
[5] J. G. Brisson and G. W. Swift. Measurements and modeling of a recuperator for a
superfluid Stirling refrigerator. Cryogenics, 31:971-982, 1994.
[6] J. G. Brisson and G. W. Swift. High temperature cooling power of the superfluid
Stirling refrigerator. Journalof Low Temperature Physics, 98(3/4):141-157, 1995.
93