Volume 55A, number 4
15 December 1975
PHYSICS LETTERS
ON THE MINIMUM TEMPERATURE ATTAINABLE IN DILUTiON REFRIGERATORS J.A. GEURST, F.A. STAAS and W. Van HAERINGEN Philips Research Laboratories, Eindhoven, The Netherlands
Received 30 October 1975 Minimum temperatures attainable in dilution refrigerators of both the double circulation and conventional types are calculated. They are shown to be very sensitive to the way circulation is achieved. For a double circulation refrigerator a model is proposed in which the concentration at injection differs from the equilibrium concentration.
Recently dilution refrigerators have been successfully operated in which besides the 3He circulation as in the conventional dilution refrigerator also 4He is circulated [1, 2]. The 4He circulation is achieved by injecting 4He through a superleak in an additional mixing chamber situated on top of a conventional one. For an earlier version of a dilution refrigerator in which only 4He is circulated see ref. [3] Double circulation has the advantage of allowing for a larger cooling power as a result of enhanced 3He circulation without meeting Kapitza-resistance problems (see ref. [1]). It turns out experimentally, however, that this cooling power together with the minimum temperature attainable depend critically on the way 4He is injected. We analyse here the processes which, under certain unfavourable circumstances, limit the cooling power of a dilution refrigerator with double circula.
tion. It will then appear that there is a close connection with similar processes in a conventional dilution refrigerator. Finally it wifi be shown how the occurrence of these limiting processes can be avoided as much as possible. A schematic drawing of the 4He upper mixing chamber is injected through is shown in fig. 1. At A1 pure 3He enters and a dia superleak, while concentrated 4He leaves the mixing chamlutedatsolution of ~He in ber A 2, the connection with the lower mixing chamber. These last two flows are distinguishable because of the 3He phasesolution. separation between the concentrated and Dilute solutions will be characterdilute ised by the molar ratio c = n3/n4. It is assumed that in steady operation the mixing chamber has a uniform pressure p and a uniform temperature T, the latter assumption being justified theenergy high heat conductivity 3He.by The balance for the of the concentrated
—
A
: L—
A 1
2
I
~ ~
— _~
ur:
dl e
e
p 1. Schematic drawing
Fig. of the upper mixing chamber in a double circulation dilution refrigerator.
region enclosed by the broken line in fig. I reads:
‘~3~~‘c h~(C~)} +n 4{p4(c1) h4(Ce)} + Q = 0. (1) A situation is envisaged in which the effects of kinetic energies and gravitational potential differences can be neglected. In the equation c1 is the molar ratio at A1 and Ce the equilibrium value of the molar ratio at A2. The possibility of such a difference in concentration between A1flow andrates A2 isofa 3He crucial our analysis. The molar andpoint 4He in denoted by n 3 and ñ 4 fl3hc respectively, are connected by of n3/n4 = Ce. In eq. (1) is the enthalpy flow rate the incoming 3He at A concentrated 2, ~3h3(Ce) is the 3He partial enat A thalpy flow rate of the outgoing diluted 2, ñ4p4(c1) ing 4He atisAthe free enthalpy flow rate of the incom1 (due to entropy filtering in the superleak the enthalpy reduces to the free enthalpy) and ñ4h4(C~) 4Heisatthe A partial enthalpy flow rate of the outgoing 2. The external heat load is denoted by Q. Since of h4 4He, = P4 we + Ts4 being entropy canwith writes4(1) as the partial molar
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fi3{hc_hd(Ce)} +fl
4{114(Ci)_/24(Ce)}+Q=0.
Sd(Ce)}
fl4~{p4(Ci)p4(Ce)}.
(3)
Here Sd =s3 +(l/c)s4. The left hand side of(3) constitutes the net entropy flux out of the mixing chamber resulting from convection and conduction of heat. Because of the entropy balance it follows that the right hand side of (3) being equal to ~~5/T represents the entropy production in the interior of that region. The dissipation can take place by irreversible mixing at phase separation byfriction mutualoccurs frictionwhen or by a combination ofboundaries, both. Mutual 4He flows at a supercritical velocity. It is instructive to look at the entropy production more closely. On account of the Gibbs-Duhem relation + (a~ 4/ac)~,~ = 04He we may write for the passed entropy production a per mole a —{p~(c~) — p 4(C0)} ~ T (4) 3(c0) /~c J~ r d r d T J C T J C P3’ ~z3(c1) Plotting~3(c1) p 3(c) as a function of the molar ratio C as in fig. 2, we see that according to (4) the entropy —
— -~--
—
—
—
252
c,
(2)
The osmotic enthalphy hd is defined by hd = h3 +(1/c)Ts4 [e.g.4].Notethatn3h~(c~)+n4p4(c~) = 1~h(Ce),where h is tht total molar enthalpy of the dilute solution and ii = + fl4 Eq. (2) may be put in a form familiar from the analysis of a conventional dilution refrigerator viz., fl3 ~1h~— hd(Ce)} + Q~= 0, where Qtot = Q ~ with = n~ {jz4(c1) — p4(c~)}.The power Q~5is directly proportional to the osmotic pressure difference between A1 and A2 and may be called a pseudo heat load.4He It isand clearly delivered by the pump differs from zero when the injecting molar the pure ratio c 1 at A1 differs from the equilibrium ratio 4He may be somolar large and! Ce A2. In or at mixing offact 3Heinjection and 4He of may be so hampered that at the place of injection A 1 the equilibrium concentration is not reached. . It will now be shown that the power Q~delivered by the pump is equal to the heat dissipated in the intenor of the upper mixing chamber. Since p~ ji3(Ce), eq. (2) can be rewritten as —
15 December 1975
0.02
1 ° -~
004
C. 006
_____—
-1 ‘2
~ ~‘
Fig. 2. Thermodynamic potential difference p3(c)
ratio C
—
Mc versus
= fl3/n4 at 70 mK.
production is equal to l/T times the area hatched in the figure. Note that at T = 0 the graph will coincide with the corrected p-curve for a Fermi-Dirac gas [5]. The hatched area, however, is the sum of the two areas denoted by I and II respectively. Accordingly the entropy production may be split additively into two contributions a1 and a11 given respectively by ~
41 {p~—p3(c1)}dc=~{p~—p3(c1)}c1(5) 0
and Ce
~H
=
1 c’ 1’ j 1~i~p3(c)} dc. —
(6)
C1
The entropy production (per mole 4He quantity injected) a~ thatis the takes place when c 3He 1 moles of soluleave the concentrated solution and enter a dilute tion of 3He in 4He with a molar ratio of c 1. A model can be proposed according to which this happens in 4He is injected into the neighbourhood of A1 where the mixing chamber. After theregions processofofthe dilutionupper has taken place, drops or larger dilute solution may be thought to move through the upregions willa on per mixing chamber from3He A1 these to A2. Meeting counterfiow of concentrated their way from A1 to A2 enlarge their molar ratio from c1 to the equilibrium value Ce. The entropy 4He produced during this last process is, per mole of passed, given by a11. 3He is the p0The driving force in the dilution of
Volume 55A, number 4
PHYSICS LETTERS
tential difference p~ p3(c). In the case of an approximately reversible dilution this potential difference is 3He very close to zero. However, when the supply of is hampered at the phase separation surface or when the solvent 4He flows with supercritical velocity, the potential difference ~ p 3(c) might become appreciable. At the phase separation surface there will exist a relation between3He thefrom localthe potential difference anddiconcentrated to the the mass flow In of the usual theory of the thermodynamlute solution. ics of irreversible processes this relation is assumed to be linear. The kinetic coefficient of the linear relation —
15 December 1975
the influence of gravitational potential differences and kinetic should be reconsidered. At zero molar ratio theenergies minimum temperature is most unfavourable and lies just above 100 mK. This is indeed observed experimentally [7].
—
will enter the expressions (5) and (6) for the entropy productions a~and OIL. It is therefore of importance in the analysis of dilution refrigerators to have some idea of the magnitude of the kinetic coefficient. However, no data concerning the coefficient are available at this moment. Let us now return to eq. (2). The minimum temperature attainable in a dilution refrigerator with double circulation for which the molar ratio at the place of injection is given by c1 is obtained from eq. (2) using by putting thenumerical external data heat for loadthe Q thermodynamequal to zero. By known ic functions of helium [6] this minimum temperature can be calculated. The numerical results are plotted in fig. 3 as curve a. The value of the molar ratio c 1 will 4He is injected depend on way in which pure through thethe superleak into the mixing chamber. It is seen from the figure that minimum temperatures of the order of 10 mK already result from a slight deviation of the molar ratio c 1 from its equilibrium value Ce. Irreversible dilution should therefore behowever, suppressed as much as possible. At low temperatures,
100 —
b
I 0
°
002
004
006
a—.--— 1
Fig. 3. Minimum temperature attalnable in,a double circuiation refrigerator (curve a) and in a conventional dilution refngerator (curve b) asa function of the molar ratio ci.
The minimum temperature attainable in a dilution refrigerator of the conventional type may be subject to similar limitations to those in for a double circulation refrigerator. The energy balance the mixing chamber of a conventional dilution refrigerator in the case where the molar ratio of the dilute solution leaving the mixing chamber is equal to c 1 reads 133
h C —hd(cl) +Q=0.
7
Eq. (7) can be rewritten as an entropy balance equation, viz,, ~ 1 —n3 {s~— Sd(Cl)} -~=n3 ~{p~ j.z3(c1)}. (8) —
—
Here the left hand side represents the net entropy outflux. The entropy production in the mixing chamber is therefore given by the right hand side. maythe result 4He Itinto mixfromchamber an “uncontrolled of tube with the ing through theinjection” connection still. The uncontrolled injection can be caused by convectional instabilities in the tube [8] The minimum .
temperature attainable at a Qgiven obtamed from (7) by putting = 0. value Curveofbc1inisfig. 3 shows the resulting values (inversion curve, see ref. [5]). It is seen that in the neighbourhood of the equilibrium value Ce both curves a and b practically coincide. Returning to the double circulation refrigerator we 4He is in dihave first to avoid that the injected pure rect contact with the concentrated 3He. Otherwise the maximum value of the entropy production results. It is desirable to have at the injecting surface of the superleak a molar ratio as near to the equilibrium value Ce as possible. This can be achieved by giving the injecting surface of the superleak a large area and by situating it at the bottom of a cup which contains the dilute solution (see fig. 1). Another way to obtain a large phase separation surface is to let the connection tube with the lower mixing chamber end at a higher point in the upper mixing chamber than the injecting superleak [2] It will be clear that a large phase separation surface area is also essential for a conventional dilution .
re rigerator. 253
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References [1] F.A. Staas, A.P. Sevenijns and H.C.M. van der Waerden, Phys. Lett. 53A (1975) 327. [2] G. Frossati, G. Schumacher and D. Thoulouze, Proc. 14th Intern. Coaf. on Low Temp. Phys., 1975, vol. 4, p. 13— 16, [3] K.W. Taconis et al., Physica 56 (1971) 168. [4] C. Ebner and D.O. Edwards, Physics Reports 2 (1971) 77.
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[5] D.O. Edwards, D.F. Brewer, P. Seligman and M. Yaqub, Phys. Rev. Lett. 15 (1965) 773. [6] R. Radebaugh, Nat. Bus. Stand. Techn. Note 362, Washington, 1967. [7] F.A. Staas and H.C.M. Van Den Waerden, Proc. 14th Intern. Conf. on Low Temp. Phys. 1975, vol.4, p. 17—20. [8] J.C. Wheatley, R.E. Rapp and R.T. Johnson, J. Low Temp. Phys. 4 (1971) 1.