MASTER

The intrinsic minimum temperature of a 3He circulating dilution refrigerator

Keusters, H.W.M.P.M.

Award date:

1985

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THE INTRINSIC MINIMUM TEMPERATURE OF A 3He CIRCULATING

DILUTION REFRIGERATOR

H.W.M.P.M. Keusters

Qï8 ^{4} 47~

*ÓJ *

I ### I

afstudeerhoogleraar: prof. dr. H.M. Gijsman dagelijkse begeleiding: ir. C.A.M. Castelijns

dr. A.T.A.M. de Waele

Containing report of the work performed during a two-months
stay at the Institut für Festkörperforschung, Kernforschungs-
anlage, Jülich, ^{We~t }Germany.

supervision: dr. R.M. Mueller

mixtures showed that the mutual friction between the helium isotapes plays a dominant role. Starting from the new empirical relations we derived some consequences for the design of

dilution refrigerators, for which up to now the mutual friction between the 3He and the 4He superfluid was neglected. In particular attention was paid to the temperature distribution in a tube, when thermal conduction of the mixture cannot be neglected. Furthermore, we obtained an expression for the intrinsic minimum temperature of 3He circulating dilution refrigerators, which is principally different from the limit, as predicted by the model ignoring mutual friction.

The minimum temperature may be measured by determination of the low-temperature limit during single-cycle cooling. For this purpose we designed a flat mixing chamber providing homogenuous temperature profiles inside the mixing chamber. The first

experimental results indicate the validity of the new expression for the minimum temperature, at least for those tube dimensions on which our empirical relations were based.

Uit recente metingen van de stromingseigenschappen van He- He mengsels blijkt dat de wederkerige wrijving tussen de helium

isotopen bij stroming van 3He door 4He een dominante rol speelt.

Tot dusver werd aangenomen dat er geen wrijving tussen 3He en het superfluïde 4He bestaat. Uitgaande van nieuwe empirische relaties zijn in dit verslag enkele consequenties afgeleid voor het ontwerp van mengkoelers. In het bijzonder is aandacht besteed aan het temperatuursprofiel in een buis bij zeer lage temperaturen, waar de warmtegeleiding in het mengsel niet kan worden verwaarloosd.

Er is een nieuwe uitdrukking afgeleid voor de intrinsieke minimum temperatuur van een 3He circulerende mengkoeler. Deze wijkt principieel af van de limiet, zoals voorspeld door het model waarin wrijving wordt verwaarloosd.

Experimentele toetsing van de minimum temperatuur is mogelijk door het bepalen van de lage-temperatuur limiet van een, niet continue, single-cycle koeling. Een voor dit doel geschikte mengkamer werd ontworpen, die zo plat mogelijk was teneinde temperatuursgradienten in de mengkamer te vermijden. De eerste meetresultaten wijzen op de juistheid van de nieuw berekende

minimum temperatuur voor die buizen waarvoor de empirische relaties werden opgesteld.

Contents

List of Frequently Used Symbols 1

Some Quantities Used in this Report 3

1. Introduet i on 4

2. Hydrodynamics of 3He-4He Mixtures 6

2.1 The Single Mixing Chamber 6

2.2 The Mechanical Vacuum Model 9

2.3 New Experimental Results 13

2.4 Temperature Profile in the Dilute-Exit Tube 16 2.5 The Intrinsic Low-Temperature Limit 25 3. Temperature Distributions inside the Mixing Chamber 29 3.1 Temperature Profile during Continuous Operatien 29 3.2 Temperature Profile during Single-Cycle Operatien 33

4. The Single-Cycle Mixing Chamber 42

4.1 The Mixing Chamber -42

4.2 Measuring Devices 44

5. Results 51

5.1 Temperature Profile during Continuous Operatien 51 5.2 Temperature Reduction during Single-Cycle Operatien 53

5.3 The Minimum Temperature 53

5.4 Flow Rate Measurements 60

6. Conclusions and Suggestions 62

7. Heat Capacity of a 3He-4He Mixture 64

7.1 The 3He Effective Mass 64

· 7.2 Experimental Set-up 69

7.3 Experimental Results 73

References 78

Appendix: Calculation of the Temperatures during Single Cycle 80

Nawoord 86

LIST OF FREQUENTLY USED SYMBOLS subscripts: c: concentrated phase

d: diluted phase me: mixing chamber

ph: phase boundary 3: 3He

4: 4He symbol

A surface area

C'JF heat capacity of the Fermi gas

Cvm heat capacity of the ^{3}He-^{4}He mixture
c3d,c

3c heat capacity of ^{3}He

D

### .

diameter of the dilute-exit tubeE energy flow

E0 sealing energy flow H magnetic field

osmotic enthalpy per mol ^{3}He in the diluted
ph a se

H_{3}c,HJd rnalar enthalpy of ^{3}He

h height of the mixing chamber

j3

3He flow rate density

L length of the dilute-exit tube

1 distance from the entrance of the dilute- exit tube

sealing length sealing length n flow rate p pressure Q heat flow

~ex ~s

### s

external heat load heat input

rnalar entropy T temperature

TF Fermi temperature

defined in eq.

(2.29)

(2.14) (2.29)

(2.24)

symbol

minimum temperature

intrinsic minimum temperature sealing temperature

sealing temperature sealing temperature

~ reduced temperature in the dilute-exit tube b time

tsc sealing time

### v,v

_{3}

### ,v

3d rnalar volumex a 3He concentratien

~ height in the mixing chamber Zm flow resistance

y

experimentally determined constant

~ experimentally determined constant

~ reduced heat capacity of the mixture reduced energy flow

Fermi energy viscosity of 3He reduced temperature viscosity of 3

He reduced tube length

reduced length in the dilute-exit tube chemical potential

reduced height in the mixing chamber reduced time

defined in eq.

(3.13) (2.14) (2.29)

(2.13) & (2.29)

(3 .13)

(2.18) (2.18) (7.10) ( 2. 28 )-

(3.13)

(2.13)

### &

(2.29) (3.13)(3.13}

SOME QUANTITIES USED IN THIS REPORT C3c=ccT

c3d=cdT os 2

H3 =hrT +hx(x-x0)

-1 -2 , cc=22 .8 J .mol . K

-1 -2 , cd=104.3 J.mol .K

-1 -2 , hr=84.1 J.mol .K , hx=17.6 J.mol -1 , x0=o.066

-1 -2 , hc=11.4 J.mol .K

-1 -2 , hd=92.9 J.mol .K

-6 3 -1 V3=36.8.10 m .mol

-6 3 3 -1

### v

_{3}

^{d=430.10 }

^{m .(mol }

^{He) }

, xT=0.506 K -2

-4 -1 -1 Kc=3.3.10 J.s .m

-4 -1 -1

Kd=3.0.10 J.s .m

-8 2

nd=5.10 Pa.s.K

At temperatures below 5 K, helium is interesting to investigate because it shows an extraordinary and unique behaviour.

First of all, at saturated vapour pressure, helium is a liquid down to zero temperature. Furthermore, bath stable isotapes (3He and 4He) can be in a superfluid state. Although 3

He and 4

He have a
practically identical chemical behaviour, large differences in the
physical properties occur. These differences are caused bij a different
quanturn behaviour; 4He (spin=O) is a boson, whereas 3He (spin=~) ^{is a }
fermion. Below 2.17 K, 4He shows Bose-Einstein condensation and is
superfluid.

The helium isotapes can farm a stable mixture: the 3He-4 He mixture. This property is used for cooling to the millikelvin region

in a so called dilution refrigerator. In the year 1978 the group

"Kryogene Technieken" of the Eindhoven University of Technology completed a dilution refrigerator having a maximum 3

He circulation rate of about 2.5 mmol/s.

In this machine the flow properties of 3He-4

He mixtures were measured. The current model on these flow properties is the so-called Mechanical Vacuum model"(MV-model), which is basedon the assumption that there is no mutual friction between the 3He atoms and the 4

He superfluid. Experimental results, however, created doubts with respect to the validity of this model. An intensive investigation of the flow properties of 3He rnaving through 4He in tubes was started. This resulted in a set of empirical relations describing the distributions of the pressure, flow rate density, 3He concentratien and temperature, as functions of the external conditions at temperatures below 150 mK.

These relations are important not only for a fundamental

understanding of the 3He- 4He flow properties at low temperatures, but also for the design of dilution refrigerators.

In chapter 2 the temperature distribution in the dilute-exit tube is calculated, according to the Mechanical Vacuum model and according to our new insights. The temperature profile in the exhaust tube

determines the minimum temperature of the mixing chamber. An expression for this minimum temperature is derived according to both models.

We wanted to investigate the validity of the calculated

minimum temperature experimentally during a so called single cycle.

For this, it was necessary to know the temperature distribution inside the mixing chamber (chapter 3). The design of a new mixing chamber and some additional features are discussed in chapter 4. Experimental results are given in chapter 5; chapter 6 contains a discussion of the experiments.

In dilute solutions.the 3He atoms are an interesting example of a Fermi gas. Theproperties are described by the phenomenological theory of Landau (section 7.1). Application of the theory requires the knowledge of the effective mass m* of the 3

He quasiparticles. This value can be obtained from the microscopie theory or from experiments. Concerning the latter, we mention specific heat measurements, measurements of the secend-sound velocity and the osmotic pressure. In chapter 7 of this report some preliminar experiments on the measurement of the specific heat of a 3He-4

He mixture are described. From this kind of experiment, in principle a value of m* can be determined. The werk was performed during a two-months visit at the Low-Temperature group of the

Kernforschungsanlage in Jülich, West Germany.

Unless stated otherwise, in this report use is made of SI-units.

This chapter deals with the hydrodynamicsof 3He moving through superfluid 4He, according to both the Mechanical Vacuum model and to our new insights. First a short description of the dilution refrigerator with a single mixing chamber is given (section 2.1).

Assuming no mutual friction between the two He-isotopes, some

consequences ~f the 3He flow through tubes will be derived (section 2.2). After presenting the empirical relations insection 2.3, the consequences of these relations will be considered too (section 2.4).

In the last sectien a comparison between the refrigerator's intrinsic minimum temperature, following from both the models will be made.

2.1 The Single Mixing Chamber

In figure 2.1 the phase diagram of 3He-4He mixtures at saturated vapour pressure is presented. The diagram consists of three regions.

1.~

### Tl KJ 1

### .5

### x

*figure 2.1. The phase diagram of Ziquid **3**He-**4*

*He mixtures *
*at saturated vapour pressure. The hatched part of the *
*diagram denotes the region in which the measurements took *
*pZace. * I: *4He is *~r.maZ; II: ^{4}*He is superfZuid; * III:

*phase separation region. *

In the first region bath components are normal; in the secend the 4He in the mixture is superfluid; a mixture with a composition represented by a point in region III is unstable, and it will split in a part having a high 3He concentratien (concentrated phase) and a part with a low concentratien of 3He (diluted phase), with

compositions given by the boundaries of region III. The hatched part of the diagram denotes the region in which the measurements described in this report took place.

The hydrodynamics of 3He rnaving through 4He is studied in a 3 He circulating dilution refrigerator. Figure 2.2.a shows the low-temperature part of it. For a more complete description of dilution refrigerators we refer to Lounasmaa, 1974.

Figure 2.2.b gives a schematic view on the parts of the refrigerator near the mixing chamber. A concentrated stream of liquid 3

He, precaoled by the heat exchangers, enters the mixing chamber. In this mixing

chamber the concentrated phase floats on top of the diluted phase.

Inside the mixing chamber cooling occurs by 3He crossing the phase boundary.

Normally, it is supposed that the mixing chamber has a homogenuous temperature. In this report temperature gradients in the concentrated phase will be considered, while the diluted phase initially has a constant temperature equal to the phase boundary temperature T

0h. From energy conservation it fellows that the total heat laad on the mixing chamber

### (Ó )

has to equal the difference between the osmotic enthalpy per mol^{3}

### H~

of the diluted phase and the incoming concentrated phase respectively (Kuerten, 1984):where Ti is the temperature of the i ncomi ng concentrated 3He stream

n3 is the 3He rnalar flow rate.

(2.1}

The cooling power (Qph)' generated at the phase boundary, can be calculated as the difference between the osmotic enthalpies at the phase boundary:

still pumping line

~__.__ -.J...--4---1

still \

*v--_ *concentrated entrance

mixinq chamber concentrated phase

dilute phase

a

*figure 2.2.a Schematic drawing *
*of the Zow-terrrperature part *
*of a **3**He circuZating diZution *
*refrigerator. *

*figure 2.2.b Scheme of the *
*mixing chamber. The fZow *

*resistance * *Zm has Zength L and *
*diameter *D.

tube

exchangers

b

(2.2).

The diluted 3He stream leaves the mixing chamber through the 3He exhaust tube. In the experiments on the 3He flow properties, a cylindrical flow resistance Zm is installed at the entrance of this tube. Over this flow resistance, we investigated the temperature rise, the 3He concentratien drop and the pressure fall, as functions of the flow rate and the dimensions of the tube.

2.2 The Mechanical Vacuum Model

Figure 2.3 schematically shows the dilute-exit channel of the mixing chamber and the physical parameters involved.

**0 ** ^{l }

^{l }

**Tm ** . ^{Te }

^{Te }

**Pm **

o~

**0 ** _{Pe }

_{Pe }

**n3---;. ** .

**Xm ** ^{xe }

^{xe }

**L **

*figure 2.3. Schematic representation of the dilute-exit *
*tube of the mixing chamber . *

. In general, the flow channel is of a cylindrical shape (tube), of which the length L is an order of magnitude larger than the diameter

D (L»D).

The main assumption of the Mechanical Vacuum model is that the 3He moves through the 4He superfluid without mutual friction. This implies that the 4He chemical potential ~

### 4

^{d }is constant in the liquid

(2 .3).

With equation (2.3) temperature distributions in the mixture will be calculated. Supposing that only the 3He component is flowing, conservation of energy in this tube imposes

where

### Q

is the total heat flow in the tube H~^{5 }is the osmotic enthalpy per mol 3

He in the diluted phase, given by

where ~

### 3

^{d }is the molar chemical potential of 3 He in the mixture

Hence,

is the contribution of 3

He to the total entropy of the mixture per mol 3

He.

Gibbs-Duhems' relation yields

where x is the concentratien of 3

He in the mixture Vd is the molar volume of the mixture

Sd is the rnalar entropy of the mixture p is the pressure.

(2.4)

(2.5)

( 2.6}.

Since the temperature is so low that the contribution in the entropy of the normal 4

He can be neglected, whereas the superfluid
4He carries no entropy, we may put Sd=x.s3d. Furthermore, from (2.3)
it fellows d~4^{d=O. }Equation (2.4) can be transformed to

(2.8}

### ö

^{is }

where

### v

3d is the rnalar volume of the 3He in the mixture (V3d=Vd/x).given

### Q

^{= }

by

tt D^{1 }

### - - ·

4

Ket . d.T

### T

dlwhere Kd is the thermal conductivity coefficient of the mixture (Kd=3.10-4 Wm-1).

(2.9)

We use the low-temperature approximation for the entropy since the 3 He concentratien is nearly equal to the 3He concentratien at zero

temperature (x=x0)
5~a. ^{= } ^{cel }

### T

-1 -2 with cd=104.3 J.mol .K ,

(2.10)

and the relation determining the pressure gradient in the flow resistance, assuming Poisseuille flow

dp= -t'ld.

### n

_{3}

### Vd.

^{12}

### a

^{dl }

T~ ~ rr D^{4 } ^{(2.11) }

where nd/T2

is the dynamic viscosity of the mixture

-8 2

at temperature T (nd=5.10 Pa.s.K ).

The equations (2.8-11) result in a well-known differential equation for T(l), containing terms due to enthalpy flow, viseaus loss and heat conduction respectively, which add up to zero (Wheatley, 1968):

(2.12).

It is convenient to introduce dimensionless parameters t and ^{À }
(temperature and length respectively), defined by

t

### =

r;r0 and À

### =

^{1;1}

_{0 }(2.13),

where _{5.53.10-}^{4 }_{. }d ^{.1/~ }

(2.14.a)

and ^{d813 }

n~ (2.14.b).

Equation (2.12) then simplifies to

dt.2._

### -'-~(_:_.ei\:)=

^{0 }

ctÀ. \.^{4 } d.À. t dÀ.

(2.15).

The solutions of this differential equation give the reduced t(À) temperature distributions according to the MV-model. The equation is solved bij Van Haeringen, 1980. In figure 2.4 sets of solutions t( À) are presented for various boundary condit i ons. f·1ore extended considerations of these solutions will be given in sectien 2.4.

### t

### 1

### 5 10 15

*figure 2.4. The * *t(À) * *dependenee in the diZute-exit tube. *

*The set of curves wiZZ be disaussed in seation 2.4. *

*(Van Haeringen, 1980). *

For temperatures t8

>>1 (T>1.5T0) the term due to heat conduction may be neglected. Ignoring this term, equations (2.15) yields

- 0

This equation has the solution:

or,

where tm=Tm/T0 is the temperature of the mixing chamber in reduced farm

C* is an integration constant.

2.3. New Experimental Results

(2.16).

(2.17)

Recently experimental results have been reported about the flow properties of 3He through the superfluid 4He following from measurements with the dilution refrigerator at the Eindhoven

University of Technology (de Waele et al., 1983; Castelijns et al., 1985). As stated befare these experimental results are in contradiction with some basic consequences of the MV-model. New empirical relations have been found for the distributions of temperature, pressure and 3He concentratien in relation to the flow rate and the external conditions.

·In this section a summary is given.

The 3

He concentratien difference (öx) between the terminals of a flow resistance Zm was found to be dependent on the cross-sectional area of Zm and not on the shape of this cross-section, according to

(2.18)

where j3 is the flow rate density ^{(j}3=n3^{/(rrD}
2/4))

y is an experimentally determined constant (y=(35 ~ 3).10-9 (SI))

a is an experimentally determined constant (a=2.8 ~ 0.4)

L is the length of the flow resistance.

From measuring the pressure difference over the flow resistance due to the 3He flow, it resulted that

p

### =

constant (2.19).The observations suggest that not a Poiseuille profile, but a so called plug flow profile is present in the tube. The picture we have of this sart of flow is a homogenuous velocity of the 3He atoms over a tube cross section, probably except from the region with width v

along the tubes• wall {figure 2.5).

### 0

*figure 2.5. Assumed veZoaity *
*distribution of the * *3*

*He *
*atoms in the **fZow *

*resistance. *

We do not know the order of magnitude of the quantity v, apart from an estimation on the maximum and minimum value. A maximum value can be estimated from an analysis of equation (2.18).

Since *v *reduces the effective diameter Deff of the tube, part of
the uncertainty in the experimentally determined constants a and y

(eq. (2.18)), could bedue toa decrease of Deff' From such an analysis, we estimated the maximum v-value to be of order 100 ~m (cf. Keltjens, 1983). From measurements of the viscosity of 3He-4

He mixtures the
minimum value of *v *was found to be 1 ~m ( Castel ijns et al., 1985).

As far as temperatures are concerned, a relation between the temperature and the 3He concentratien at the terminals of the flow resistance has been found:

(2.20) where a=0.20 ~ 0.01 K2, independent of L, Dar

### n

_{3. }

Combination of eqs. (2.18) and (2.20) leads to

### T

_{ct }

^{1 }

*= *

^{T}

^{m }

^{2 }

^{+ }

^{a }

### Y

^{L . }

### J3

^{o<. }^{(2.21). }

From this equation a ~(T^{2})-l-relation can be concluded. This is
contrary to the MV-model, predicting a ~(T^{4})-l-relationship

(cf. equation (2.17)).

Using the newly found relationships, an experimental expression for the low-temperature osmotic enthalpy of the diluted 3

He-4
He
mixture can be derived. This expression corresponds to that, derived
from thermadynamie calculations on the ^{3}He-4He mixture (Kuerten et al.,
1985):

(2.22) -2 -1

where hr=84.1 J.K .mol
h x= 17 . 6 ^{J . }mo 1 . -1

2.4 Temperature Profile in the Dilute-Exit Tube

In this section the temperature distribution inside the dilute- exit tube will be derived according to our new insights. The derivation is quite analoguous to that in sectien 2.2.

The system under investigation is depicted in figure 2.6. It schematically shows the exhaust tube of the mixing chamber.

**0 ** ^{l }

^{l }

### I ) I

n

### 3

Hj0.1...~...-i ----:--;_..-~~### 3

^{~H> }

Q(Q)~ **f---Q(l) **

**lmxm **

*figure 2. 6. * *Schematic representation of the dil u te-exit-*
*tube. * Ós *represents the heat suppZy at *ls.

Inside the tube, energy is transported by thermal conduction (Q) and enthalpy flow n

### 3

^{H~s. }The main differences with the derivation according to the MV-model are the absence of viseaus losses and the pressence of mutual friction between the 3He and 4He. If there is no external heat supply, the law of conservation of energy reads

(2.23) where E is a constant energy flow, determined by the boundary

conditions of the system.

The system under investigation can be generalized by including a heat influx Qs at a certain point 1

5 in the tube. This heat input can be bath positive and negative; the first may represent the heating caused by the incoming concentrated 3

He by means of a heat exchanger,

whereas a negative heat input represents cooling at the point ls.

### ó

_{5 }as a function of 1 can be expressed:

(2.24) where o is the well-know o-function.

By insertion of Qs(l) and the expression for the osmotic enthalpy (equation (2.22)), the energy conservation law (2.23) at a certain point 1 in the tube yields (cf. figure 2.6):

l

n?.(hTT^{2}+ hx(lf.-x_{0}^{))_ }Tt

### D,_.

^{Ket }

### (dT) _ jO./L)dl

^{= }

4 T eH l _{0 }

### n3(

hT### T~

^{+ }

### hx

(x.""-### Xo))_ TC rf. Kc.t (dT)

4 T eH ^{l:O }

(2.25).

From equations (2.23,25) it is seen that

### Ë

equals the energy flow at the entrance of the tube.Equation (2.25) is only valid if H~s and Kd have the same

values, both inside the tube and in the bulk mixture being macroscopically at rest. Evidence for H~s to have this same value, was gained from an experiment in which the temperature inside the dilute-exit tube was measured (Kramer, 1985).

Another experiment, in which the viscosity of the 3He-4He mixture inside the tube was measured (Faessen, 1985), showed that the fluid in which the 3He is moving has the same viscosity as the bulk mixture.

Since viscosity and heat conductivity are related to the same physical phenomena, it may be concluded that on first order the heat conductivity coefficient Kd has the same value in the flowing mixture also.

Re 1 a ti on ( 2.18) detenni nes the concentra ti on drop .6.x=xm -x, and

therefore it is convenient to rearrange the differential equation (2.25):

l

· 1. n l)^{2 } Kc:4 ciT

*J · * · '

n_{3 ( }hT T + hx (x-xm))_ - · - · - _ Q!l(t)dl =

### E

4 T dl

0

(2.26) where

### Ê

^{'= }

^{n3 }h,..

### T~

-^{TC }Dl.

^{l<d. }

### (dT)

4 T *eH * l:.O

By inserting the experimentally detennined relation (2.18}, the last equation reacts

### ri3(hrTl -hx ~l.f')

^{-ïtD2 Kd }

^{dT- ( }### Q~(L)dl *= ( *

~ 4

### T

dl*0**) *

( 2. 27).

It is convenient to introduce dimensionless parameters t, À,
qx(À) and ^{E, } defined by

I

reduced temperature t=T/T 0

reduced length (2.28)

I , I

reduced energy flow E=E /E_{0}.
By choosing

(2.29)

equation (2.27) reduces to a simple ferm:

À

12 d\:

### J

~ - À -

### .2.._ - 9.

^{(À) }d À=

*é.*

2 t dÀ. ~ (2.30).

This differential equation turns out to held also fora 3He-4He counterflow tube, as applied in superfluid injection methods; this analoguous problem was solved by Van Haeringen, 1980.

First, we will consider solutions of the differential equation, while leaving a heat input qs(À) out of consideration. Afterwards this heat lead can be introduced quite easily.

The differential equation can be solved after transformation to a more convenient ferm:

The salution of the homogenuous equation yields

_1 . ex p

### (X"+

^{2 }

^{À }

### i.)

### t2

0where t

0 is the reduced temperature at the entrance of the tube.

A particular salution of the differential equation reads

À

- ' = e.xp(A.^{2}+lÀ€)j

### e.xp(-C.u.

^{1}

### +2J-lé)) dJ-A

~~ 0

The explicit t(\)-relationship fellows from the combination of the homogenuous and the particular solution respectively:

2 ( exp l-(À.^{4}4-l..À.~))

t À,\:0 ,f.}: - - - - : - - - - '/l:.! - j.\czxp(-( . .u. ... 2}-lf.))cl.fl-

(2.31).

*o *

In figure 2.7, for various values of ^{E, } some of the t(À) dependences
are drawn for t 0=1 and t0=2 respectively.

t

4~-~~--n----

### 0

\

\

### 1 3

*figure 2. 7. Temperature *

*proftles in the the dilute-*
*exi t tuhe for various *
*values of *E. *Profiles *
*for two mixing chamber *
*temperatures * t

0=1 *and *

t0=2 *are shown. *

In order to gain insight in the general temperature behaviour

inside the tube, it is necessary to transferm eq. (2.31) so that universal lines are obtained. This is achieved by reducing the number of parameters on which t depends from three to two by a translation along the À-axis:

(2.32).

The differential equation transfarms to:

l~

### _

^{l}

^{1 }

^{-}

^{I }dt ( t \

^{I ( }L )

- *. t\ * . - . -

### =

t. - a o '### =

*E.*~o

l. \:. d *!...' *

(2.33).

The salution of (2.33), which is analoguous to the original differential equation, yields

### ta.(À,e.') =

^{e.xp }

### (-(À'~•lA'E.'))

1/l:.~· .. o

### 0

^{J~~xp (-}

^{(fA-2.+ }l"U.€'))

^{cJ..}-'-}(2.34).

We can choose tÀ'=O

*ai *

will by varying a(t0); for further use it is
conven i ent to choose tÀ , =0=2/ *l1r.*

Equation (2.34) expresses the general temperature dependenee in
the tube as a function of À^{1 } and e'. In figure 2.8 curves for various
values of e' are shown. The solutions very much resemble these depicted
in figure 2.4, following from the MV-model.

; In the figure two lines tm=l.S and te=2.5 have been drawn for showing the various possible curve segments between the reduced mixing chamber temperature (tm) and the reduced exit temperature te. Initially, most curve segments fa 11 ow the 1 i ne with e '=0, denoted as the· asymptoti c curve. These curve segments all have the point A, representing the

entrance of the tube in À'-space, in common. The curves starting from A can be divided into three categories.

First, we consicter the curve segment, which follows the asymptotic curve continuously (A-B). The corresponding physical state is the so- called free flow: e'=O, in figure 2.8. There only is one reduced tube length which corresponds to a free flow between temperatures tm and te.

This tu~e has reduced length Af' corresponding to the phycial length Lf=Af.l O.

Secondly, there is a set of temperature profiles with a sudden rise in temperature near the end of the tube (A-C). These curves correspond to a heat supply at the end of the tube, for instanee by means of a heat exchanger. The rise of temperature is rather streng

localized within a distance ~À=l from the end of the tube.

The third category of curve segments (A-E) represents the rather rare situation that there is cooling at the end of the tube. Again, the influence of this cooling is noticed within ~À=l from the end of the tube only.

**t **

**curve ** **1 ** **2 ** 3 **4 ** **5 ** **6 ** **7 ** **8 ** **9 **

e:' **-10 ****-2 t-10-4 1-10-6 **_{H0- 8 0 } 10-8 **10-6 10-4 10-2 **

**4 **

**1 ** **2 ** **3 **

**3 **

**te· ** **2.5 ** **2 **

**te·**

**1 **

*figure 2.8. GeneraZized temperature profiles in the diZute-*
*exit tube for various vaZues of *e:'. *Two Zines * **tm=1.5 ***and *
*te=2.5 **have been drawn in order to show the various *
*possibZe curve segments between the two temperatures. *

*The curve * *t*^{2}*=2.À' **is indiaated **(-·-).A **fuZZ expZanation *
*of the figure is given in the text. *

In conclusion, in a tube with length A>l there is a free flow, problably except from a region 6À=l from the end of the tube. It now can be understood, for what reason the line (F-G) starts from a different À•-value. This curve segment corresponds toa tube length A<l, causing the influence of heating at the end of the tube to be noticable at the entrance of the tube, in the mixing chamber.

We saw that an expression for the free flow can be found by entering ~·=o in the general solution of the differential equation

(2.34) with tÀ•=o=2/ln. For high reduced temperatures we can simplify this expression. De 1 'Hopital •s proposition:

lim f(x)/g(x) = lim f'(x)/g'(x)

in case lim f(x) = lim g(x) = 0 and lim g'(x)F 0, applied to equation (2.34) yields

### x ...

lim^{oo }

-l. À'. ex p (-À' ^{2}^{) }
-exp(-X^{1})

*= * ^{(2.35). }

Equation (2.35) corresponds to the experimentally determined relation (2.21). From figure 2.8 it can be concluded that the deviations from the experimental relation are of importance for t<3.

Next, we will make an extension to the system, which at the same time may serve as an illustration of the use of the foregoing in

designing dilution refrigerators. Both the influences of the heat input Qs and of a varyi ng of tube di mens i ons wi l1 be considered. The system present under investigation is depicted in figure 2.9. It schematically shows the dilute-exit tube, consisting of two tubes with different

diameter and a heat exchanger between them. The heat exchanger, positioned at ls, supplies a heat Qs, originating form the warm concentrated 3He. The boundary conditions are such that the mixing chamber and the high-temperature bath have definite values tm and te respectively; the À-scale is indicated.

For constituting the temperature profile inside this tube system, under the given boundary conditions, we notice that both tubes are

**tm **

I

**0 **

**A, **

**4 ** ~I

### J

**Ä2 **

### I ^{te }

^{te }I I 1 I I I I

À~

**12 **

*figure 2.9. DiZute-exit channel consisting of two tubes *
*with different diameters**3 * *connected by a heat exchanger *

### .

*supplying an amount of heat *Q

5•

I

longer than 1

0. As a result, the temperature profiles in the first parts of bath tubes correspond to free flow. Since all temperatures are t>3, the free flow is represented by t2=2À. The influence from the higher temperature parts is noticed in the last parts of length

~À=l only.

Figure 2.10 gives the temperature distribution in bath tubes for several values of the supplied heat.

Qs determines the difference in slopes of the temperature

profiles on bath sides of the heat exchanger. In physical terms, the heat input balances the heat fluxes from the heat exchanger into the tubes:

l l

*j * ^{Q~(l) } ^{d.l _ } *J *

^{Ka }

### (-1 **(dT) _ **

**· o ****o ****T ** ~'

### dl

L1lsI

### (olT) \

^{cil }

**A,_ ** dl

### L+l~)

^{(2.36). }

For eperating the dilution refrigerator in an efficient way, with temperatures tm and te' the heat input from the heat exchanger is

restricted to limited values. First, the heat input always is positive, because in a stationary state the incoming concentrated 3He stream has to be cooled.

The maximum heat input for the system under investigation is
indicated in figure 2.10. For a heat input larger than Qs ,ma *x *the

.

### . .

. .. . ...### -

^{... }

### .

### 10 ^{. . }

^{. }

### .

:1

**Os ma **

*I*

**t **

**8 ** **O<às<Os,max **

**6 **

**4 **

**2 **

**0 ** **2 ** **4 ** **6 ** **a ** ^{10 } ^{J2 }

^{10 }

^{J2 }

*figure 2.10. Temperature profiles in the extended dilute-*
*exit tube system; profiles for various values of *Q

### .

5

*are shown. The dotted curve represents the unstable *

### . .

*situation in which *Qs>Qs,max·

dotted line in the figure would be obtained, resulting in a negative dt/dÀ-value near the end of the tube. Since there would be a heat laad on the exit bath from the cold side of the refrigerating system, the system tends to a stationary state at a higher temperature.

Next, we will discuss some features for determining suitable

dimensions for the dilute-exit channel, which can be used in designing dilution refrigerators. The tube under consideration, connects the mixing chamber with temperature tm and the coldest heat exchanger with temperature te. While designing a dilution refrigerator the average werking temperatures of the mixing chamber and the heat exchangers have to be estimated. Usually, the ratio between the two temperatures

is 2.8 (for instance: Lounasmee, 1974). However, in order to give a graphic illustration, we will consicter an example in which tm=1.5 and te=2.5 respectively (see figure 2.8).

As stated, it is unfavourable to chose the tube-length langer than indicated by the free flow. If the tube length is too long (A-E, figure 2.8), a heat load from the cold diluted side to the concentrated side would follow. As a result, the temperatures tm and te will rise according to the constant ratio te/tm, and the boundary conditions can not be satisfied.

In order to operate the refrigerator as efficient as possible, the tube length should be chosen short. However, a too short tube lengthwilllead toa surplus heat load on the mixing chamber (curve segment F-G in figure 2.8); by choosing the reduced tube length A>1, this always is avoided.

In conclusion, the tube connecting the mixing lowést heat exchanger should have a reduced length a tube length

(

### ~

^{2 }

### )'·9

### L >

1lt.1~

### nJ

chamber and the

I

1 0 ^{~ }1 , 1 ead i ng to

(2.37).

In order to operate the dilution refrigerator as efficient as possible, the set of parameters L, D and

### n

_{3 }have to be optimized according to eq. (2 .37).

2.5 The Intrinsic Low-Temeperature Limit

In this section the minimum temperature of the mixing chamber lmin during single cycle is calculated. During a single cycle the

temperature can be lowered because na relative warm concentrated 3He has to be cooled. However, the temperature is restricted, because of heat loads from the dilute-exit tube and from external he~t loads respectively. During a single cycle lasting long enough, a stationary state will be reached at temperature Tmin·

Three factors determine Tmin:

i) The cooling power obtained by the dilution of the 3

He. From equation

(2.2), H~s and the relation for the phase separation on the diluted si de:

x_{5 }(T)

### =

x_{0 }+ xT T

^{2 }where xr=0.506 K , -2

it can be derived that the cooling power equals:

*( hr *+ hx

### x

_{1 -}h,) n?>T'

### =

81. rs### n

1\2

(cf. equation (2.2)).

ii) The heat conduction from the dilute-exit tube on the mixing chamber (=(nD2

/4).(Kd/T).(dT/dl)_{1}=0), where (dT/dl)l=O is the temperature
gradient at the entrance of the dilute-exit tube .

### .

iii) External heat loads: Qex·

i) to iii) are combined by the energy conservation law:

### ( hT

+### hx. Xi-he_) n3T;in =

^{it }

### \)~

^{Kó }

### (c;T)

^{+ }

### Qex

*i, * T dl l::.O

(2.38)

- The temperatrue gradient dT/dl is calculated from the temperature profiles in the sectien 2.4. Analoguously, eq. (2.38) is reduced by the quantities (2.28):

(2.39)

By applying the energy conservation law to the tube (cf. eq. (2.25)):

\:.2. - À -*2_ * eH:. =

### l:~i"

^{-}

^{_1_ }

### (oH:\

2. t dÀ. 2 ^{tmin } d'i}À:.O

(2.40)'

we obtain an equation, which is analoguous to the differential equations

~erived in the last chapter:

2.

### dt

### l

_{2 }

^{-À-}

_{- · -}

_{t }

^{1 }

_{dÀ. }

### =

_{é. }

^{(2.41) }

where é.

### = 9.

_{e.x }

^{-t-}

^{l~in }

^{( hx. }

^{XT }

### -~)

2

### hT

hTSince we know the solutions of this differential equation for

various values of ~:::, tmin can be determined: equation (2.41) can be

compared with equation *(2.33) and solutions can be derived from figure *

I

2.8. After multiplication with r0, tmin determines the minimum temperature of the mixing chamber during single-cycle operation.

In figure 2.11 Tmin is given as a function of

### n

3, for various values of the external heat load Qex' The diameter of the exit tube is chosen 1.6 mm.'

10~---~---

**Tm in ** **lmKJ **

**s **

**2 **

**D= 1.6mm **

1~---~---~---~

**0.1 ** **0.2 ** **0.5 ** ^{1.0 }

^{1.0 }

**ri 3 lmmol **

^{Is) }

*figure 2.11. The minimum temperature of the mixing chamber as *
*a function of the * ^{3}*He flow rate. Curves for various vaZues *
*of the externaZ heat Zaad *

*Q *

*are shown.*

*ex *

The curve Qex=O represents the fundamental intrinsic low-temperature limit, Tmin,i' of the mixing chamber with this specific dilute-exit tube.

Sine~ Óex is neglected, the cooling power balances the heat flow from the dilute-exhaust tube only.

The right-hand side of equation (2.41) is small ((hxxT-hc)/2hr==0.02), and it can be shown that

(2.42).

Analoguous calculations based upon the MV-model have been carried out befare (Wheatley, 1968; Van Haeringen et al., 1979).

Taking viseaus losses into account, while ignoring mutual friction, the calculations resulted in an intrinsic minimum temperature TM~n . :

m1 , 1

1"1\in ~.· = 3.8b . \0 _

### T MV -4(1 ~

### , D''3

^{(2.43). }

Again, this intrinsic minimum temperature is valid for single-cycle operation, taking into account the influence of the dilute-exit tube only.

The main difference between bath minimum temperatures is that in our calculations Tmin,i depends on the flow rate density, whereas the MV-model prediets a diameter dependenee only.

It is emphasized that the derivation of the intrinsic minimum temperature is largely independent of the temperature at the high temperature end of the tube. As a result, the minimum temperature of the mixing chamber can be investigated without specifying the temperature in the higher temperature partsof the refrigerator in great detail.

The final goal of our experiments will be the proof of the calculated intrinsic minimum temperature.

The new calculated T . . can be decreased at will by decreasing

m1 n, 1

the flow rate density. In practice, two difficulties will show up, prohibiting that very low temperatures are reached. At first, by decreasing n3/d2 to very low values, 1~ will become very large

(cf. eq. (2.28)). This would lead to unrealisticly long tubes between the mixing chamber and the lewest heat exchanger.

A secend prohibition will be the presence of a non negligible external mixing chamber heat laad. From figure 2.8 the influence of such a heat laad (Óex) on the minimum temperature attained, can be

seen. At low flow rates (low Tmin,i!) the influence of Óex will be steadily more dominant, and for sufficiently low flow rates it completely determines the minimum temperature reached.

CHAPTER 3: TEMPERATURE DISTRIBUTIONS INSIDE THE MIXING CHAMBER

In this chapter the temperature evolution inside the mixing chamber during single-cycle eperation is calculated. Final purpose of our investigations is the measurement of the intrinsic low- temperature limit. For this, an appropriate mixing chamber had to be designed.

During a single cycle the concentrated 3He stream is prevented from entering the mixing chamber. Because no concentrated 3He has to be caoled anymore, the cooling power is used for further lowering the temperature of the mixing chamber.

In section 3.1 the temperature profile during continuous operatien is calculated. Starting from this profile, the progress of temperatures during the single cycle is calculated in section 3.2.

3.1 Temperature Profile during Continuous Operatien

In the continuous mode of operation, the diluted 3He, exhausted

from the mixing chamber is supplied again at the concentrated side tsection 2.1). In a stationary state, the phase boundary is in a steady position.

The phase boundary temperature Tph is an order of magnitude higher than the intrinsic minimum temperature. Since the heat flow from the dilute- exit tube varies with T-2 and the cooling power Qph rises with r2, in the continuous mode the cooling power is some orders of magnitude larger than the heat flow mentioned. As a result, we consider the temperature profile in the diluted phase homogenuous, disregarding the influence of heat conduction from the dilute-exit tube.

Figure 3.1 gives the scheme of the mixing chamber on which our calculations are based. The system under investigation is considered 1-dimensional; the height x is calculated from the bottorn of the mixing chamber. For our calculations we assume that there is no external

mixing chamber heat laad, but only a heat laad from the dilute-exit tube.

### x

### 0

*figure 3. **1. * *Diagram of the mixing chamber. *

*In the concentrated phase energy flow by *

• • *2 *

*conduction * *(QJ * *and convection * *(11.4n**3**T) *
*is *supposed~ *in the diluted energy flow *

• *2 *

*by mass flow only * *(92.9n**3**T) *

The energy conservation law applied to a slice ~x (figure 3.1) yields:

(3.1) where T is the temperature at the position of ~x

### Ó

is the heat conduction in the concentrated phase.In the stationary state, the energy conservation law applied to an infinitesimal slice dx in the concentrated phase yields:

(3.2) where A is the area of the mixing chamber

Kc/T is the thermal conductivity of the concentrated
3He (Kc=3.3 10-^{4 }W.m-1).

Fora slice dx on top of the phase boundary equation (3.1) can be transformed to

=

### A

^{Kc. }

### (dT)

T dx x ~X"h

(3.3) where xph is the height of the phase boundary.

With the salution of eqs. (3.2) and (3.3), and the assumption that the temperature in the diluted phase is homogenuous, the temperature profile in the mixing chamber yields

• 0~ x< x ^{p'n }

(3.4)

### T

^{2 }

### (x)

= 8.1### T ;k ^{x }

^{L }

^{<x } ^{~ }

^{h }

{ x-x • P"- -

1 + *1.1 * exp ^{-2 } A

### •.l"

^{92.<3 }

^{r\~ T~h) }

where h is the height of the mixing chamber.

The temperature profile is determined by the flow rate density n3/A, the phase boundary temperature Tph and the level of the phase boundary inside the mixing chamber xph' In figure 3.2 some temperature profiles inside a mixing chamber of height 7 cm are shown, for various values of

### n

3/A, Tph and xph'Until now, in calculations on single-cycle processes (for instanee Peshkov, 1970), a homogenuous temperature all over the mixing chamber was assumed. From our calculations, the concentrated phase appears

to have a higher temperature, corresponding to a larger enthalpy content.

This enthalpy content is the lowest for low flow rate densities and low

80.----~--~----~--~----~--~--~

**lmKJ ** **T **

**60 **

**40 **

**20 **

**0 **

**dil. ** **con. **

**1 ** **2 ** **4 ** **6 **

**xlcml **

*figure 3.2. * *Temperature profiles in a mixing chamber *

cJ~' *length 7 cm. Profiles for phase boundary temperatures *
*15 and 25 mK" * *and * *for **flow * *rate densities **Q): *

### n

_{3/A=O. }

^{8 }*and *

*®: *

*0.04 mmol/s.m*

*2*

*are shown. *

**7 **

phase boundary temperatures. As will be seen, this fact is of great importance in designing a mixing chamber suitable for investigation of the intrinsic low-temperature limit during single-cycle operation)

3.2 Temperature Profile during Single-Cycle Operatien Energy Conservation

During a single cycle the concentrated 3He is stopped from entering the cold part of the dilution refrigerator. However, it can not be avoided that a certain amount of 3

He already present in the entrance line and heat exchangers enters the mixing chamber. For our calculations, this will be leftout of consideration, on which we will return later.

- During single-cycle eperation there is no flow of 3He in the concentrated phase anymore. The phase boundary moves towards the top of the mixing chamber as a result of the dilution of 3

He.

The model on which our calculations are based, in principle is the same as we used for calculating the temperature profile during continuous eperation (see figure 3.1). However, because single cycle is an instationary process, the energy conservation law in general time-dependent ferm has to be used

### C oT

^{+ }

### u3 ~T

^{= }

### Vm i..(~. aT)

### àt *ox * ^{àx }

^{àx }

^{T }

^{óx }

where

### u

_{3 }is the velocity of 3He atoms Vm is the molar volume

K/T is the heat conductivity C is the heat capacfty.

(3.5)

In agreement with figure 3.1 the system is treated 1-dimensional.

The mixing chamber has such large diameter and 3

He veloeities are so small that, besides the viseaus heating, we can ignore the effect of dissipation by mutual friction. Furthermore, the heat capactties of the mixing chamber and the measuring devices installed in it are neglected.

In the concentrated phase (u

3=0!) the general energy conservation law (3.5) is transformed by substitution of the concentrated phase quantities:

### e.c. T

^{àT }

### =

^{V?J }

### {~

èl.T _### ~

(### àT)

^{1}

^{\ }

*ot. *

^{T }

^{àx' }*T' öx * l

where eeT is the molar heat capacity of 3He

-1 -1

(cc=22.8 J.mol .K ).

(3.6)

The differential equation in the diluted phase can be constituted analoguously. However, a difficulty occurs in defining the 3He flow rate. During single-cycle operation the external flow rate differs from the rate in which 3He is diluted, because the volume of the diluted phase inside the mixing chamber increases. In this report we set

### n

_{3 }

for the amount of 3He exhausted from the mixing chamber. As a result, the.amount of 3He diluted per second during single cycle equals

(1+x)ri3.

The velocity of the 3He atoms in the diluted phase is determined:

( 3. 7)

The general energy conservation law in the diluted phase then yields:

### c T J oT- n~

v?.d.### àT} = ^{V }

^{d }

### f

^{Kd.. }

### ozT -

\(ei (### ö1 \

^{2}

### l (

^{3. 8) }

ct

### l

*à*l: A

*à*

### x

^{?> }

### l

^{T }

^{à }

### x

^{4 }

^{\}

^{2 }

^{èx }

^{Î } J

^{Î }

where cdT is the heat capacity of the diluted phase per

3 . -1 -2

mol He (cd=104.3 J.mol .K ).

buring a single cycle~ ^{4}He enters the mixing chamber through the
dilute-exit tube, reoccupying the volume of the exhausted 3He.

Since the 4

He atoms carry no energy, there is no influence on the energy balance (3.8).

Tagether with an initial condition and two boundary equations for each differential equation, we now have a complete set of equations for calculating the evolution of temperatures during single-cycle operation. As the initial condition we use the continuous temperature dîstribution eq. (3.4). Next, we will consider the 4 boundary conditions concerned.

i) Since no 3He enters the mixing chamber, the heat flux at the top of the mixing chamber is zero:

(dT

### /dx)

^{= }

^{0 },'(:"" ^{( 3. 9). }

ii) At the phase boundary, in principle, a temperature difference between the two phases may be caused by the Kapitza resistance. In our calculations such a difference will be neglected, hence

(3.10) where Tph,c is the temperature in the concentrated phase just

above the phase boundary

Tph,d is the temperature in the diluted phase just below the phase boundary.

iii) Another boundary condition, valid at the phase boundary, expresses that the cooling power generated balances the heat conduction fluxes from both the concentrated and the diluted phases:

### ats n~(Hx)r;h: A{~(uT) _Ket (oT) \

^{(3.11). }

Tph *OX * ^{XUph } Tpn àx xt Xph}

iv) At the bottorn of the mixing chamber the heat conduction from the dilute-exit tube has to be accounted for. In order to determine the amount of heat conducted, we use tbe calculations of the temperature distribution inside the dilute-exit tube (section 2.4). The boundary condition can be formed by equalization of the heat flow into the mixing chamber and the heat conduction from the dilute-exit tube:

*A ( àT) _ -*

^{tt }

*D' ( oT) *

*XX *

^{X::.O-}### 4 hl

^{L:O }

^{(3.12). }

Numerical Salution

For solving the differenttal equations it is convenient to introduce dimensionless parameters:

reduced temperature B=T/Tsc with

reduced time reduced height in

-r=t/t with

SC

the mixing chamber ~=x/h.

(3.13)

The sealing quantities have a physical meaning. tsc is the time approximately needed for a single-cycle process starting in a mixing chamber completely filled with concentrated 3He. e2 equals the Péclet number determining the ratio at which energy is transported by

conveetien and by conduction respectively. With eqs. (3.13) the differential equations (3.6) and (3.8) are transformed to

and

### v g. *à9 _ , * *o*

^{1}*g. _ ' * (à9)z.

^{+ }

^{g. } ^{octt }

^{octt }

*t *

^{èrt -}

^{tf } ^{?J~}

^{2 }

^{&}

^{2 }

^{óf · } ^{~ }

(3.14),

(3.15).

Since there is na analytical salution found, the differential
equations were solved numerically. The salution methad used ^{i~ }the
explicit Euler methad (see for instance: Dahlquist et al., 1974;

Walsh, 1973). Appendix A contains all information on the numerical salution of the differential equations system. Attention is paid to linking the two differential equations at the rnaving ohase boundary.

Temperature Reduction during Single-Cycle Operatien under Optimum Circumstances

By means of a simple thermadynamie calculation the maximum

'temperature reduction during a single-cycle processcan be calculated.

Apart from being instructive the calculation can serve the analysis of single-cycle processes. The derivation is analoguous to that given by Peshkov, 1970.

Our calculations start from an ideal mixing chamber model:

- there is an homogenuous temperature all over the mixing chamber.

- there is na heat flow from the dilute-exit tube; the tube has such a large diameter that any influence can be neglected (cf. (2.42)).