s
2
D= 1.6mm
1~---~---~---~
0.1 0.2 0.5 1.0
ri 3 lmmol
Is)figure 2.11. The minimum temperature of the mixing chamber as a function of the 3He 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 (11.4n3T) is supposed~ in the diluted energy flow
• 2
by mass flow only (92.9n3T)
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)
= 8.1T ;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.----~--~----~--~----~--~--~