xlcml

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

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

^{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

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~ ⁴He 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}

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)).

The temperature reduction can be calculated by relating the cooling power of the dilution process to the heat capacities of bath phases

{cc.TA(h-Xp'n)+ cclT ÄXp'nldT

=-Öex -81.5~\(Hx)Tz

^(3.16)

v3

^{v~o. Jdt.}

where Qex is an external mixing chamber heat laad.

Wi th ( cf. eq. ( 4. 3) )

dx"." _

n~

v?:.

dl:. - A Vred

(3.17)

where V =(V 3d-V3);v3d, red

equation (3.16) can be transformed to

{ c,T(h-xph)+cdTVl x?h\dT =VfTid{~u

-d1.5(1-+xÎT^2} (3.18).

Vld j dXph n~

Inserting Vred.(l+x)~1, the salution reads

<.91.5T

¹

-~u

^Vred

=-c*{c~h+(cc:~.V3 -c.,)xp\·r~

^(3.19)

n~ V~

J

where -e=l63/cd(v3;v3d)-cc)=11.7

c* is an integration constant.

The temperature reduction T/Tb can be determined, after the phase boundary has covered a distance from hb to he. Hence,

·

^n~d

· T""edj( "

¹

T'!

^2:. Ou~

.,.. (1: _ 0

^12x. ~

h-

O.b1 Xph,e) . d1.5 n₃ 81.5 n₃ h - o.b1 X.ph,'b

(3.20)

where Tb is the starting temperature Te is the final temperature

x ph,b is the starting level of the phase boundary xph,e is the final level of the phase boundary.

Typical Temperature Profiles during a Single Cycle

In this paragraph we will present the calculated results of single-cycle simulations in two different mixing chambers. The processes in a heigh (L=7 cm>D=3 cm) and a flat mixing chamber (L=3 cm<D=6 cm) will be compared. A camparisen between the numerical solutions of the differential equations and the ideal single cycle~

as described in the last paragraph~ is made.

Figure 3.3 presents the single-cycle processes in bath mixing chambers. The temperature profile is drawn every time after the phase boundary has covered a tenth part of the height of the mixing chamber.

In bath processes the dilute-exit tube has diameter 0=1.2 mm and the flow rate is n3=0.6 mmol/s~ resulting in an intrinsic low-temperature limit (eq. (2.41)):

Tmin,i = 5.0 mK

Sinée the phase boundaries initially are at the same relative level (x/h=0.2) and the volumes of both mixing chambers are equal, both single-cycle processes take the same time.

The single cycle in the heigh mixing chamber, figure 3.3.a, starts from a temperature profile with a large temperature gradient in the concentrated phase. Initially, the largest part of the cooling power is used for lowering the concentrated phase temperatures~ and thus the phase boundary temperature remains nearly constant. Only when the mixing chamber nearly has a homogenuous temperature, an overall cooling of the liquid is achieved. The cooling rate then is about the same as predicted by the ideal cooling rate for Óex=O (eq. (3.20), dotted line in figure 3.3).

Besides the negative effect of the initially .large temperature gradient on the lowest temperatures reached~ there is another difficulty for prooving the intrinsic minimum temperature in the mixing chamber under consideration. Because the phase boundary is relatively far away from the dilute-exit tube, a temperature gradient can be present in the diluted phase. As has been stated, the cooling power varies with T²~ the influence of the dilute-exit tube decreases with T- 2. As a

result~ in the heigh mixing chamber a stationary state can be reached,

In document Eindhoven University of Technology MASTER The intrinsic minimum temperature of a 3He circulating dilution refrigerator Keusters, H.W.M.P.M. (pagina 37-44)