CONSIDERATIONS ON MIXING CHAMBERS

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CONSIDERATIONS ON MIXING CHAMBERS

A. de Waele, G. Coops, H. Gijsman

To cite this version:

JOURNAL DE PHYSIQUE

Colloque

C6,

supplhent au

no

8,

Tome 39, aolit 1978, page

_C6-1150

CONSIDERATIONS ON MIXING CHAMBERS

A.T.A.M. de Waele, G.M. Coops and H.M. Gijsman,

Eindhoven University o f Teechno logy, Eindhoven, The Netherlands.

R6sum6.- Nous d i s c u t o n s quelques p r o p r i b t 6 s d'un systsme d e b o f t e s P mslange. Le concept d'une b o f t e B mElange continu e s t i n t r o d u i t .

A b s t r a c t . - Some o f t h e experimental p r o p e r t i e s of a m u l t i p l e mixing chamber a r e d i s c u s s e d . The concept of a continuous mixing chamber i s introduced.

DISCRETE M I X I N G CHAMBERS.- I n a d i s c r e t e mixing chamber a r e l a t i v e l y l a r g e p o r t i o n of t h e incoming 3 ~ e i s d i l u t e d . The s i m p l e s t combination of d i s - c r e t e mixing chambers i s a double mixing chamber ( f i g u r e l ) , where t h e 3 ~ e i s d i l u t e d i n two s t e p s . The i n t e r a c t i o n between t h e two mixing chambers and t h e r e s t of t h e d i l u t i o n r e f r i g e r a t o r i s q u i t e complicated. I n t h i s paper we w i l l d i s c u s s some of t h e e s s e n t i a l p r o p e r t i e s of t h e s t a t i o n a r y s t a t e . I t can be d e s c r i b e d w i t h t h e following e q u a t i o n s / 1 , 2 / : 2

6i

+

12

it

T: = 96

nl

T? + 12 ( i t

-

i l ) T1, (1) The i n d i c e s 1 and 2 r e f e r t o MC1 o r MC2 r e s p e c t i v e l y

6

i s a h e a t i n g power; V. i s t h e molar volume of 3 ~ e i n t h e d i l u t e phase;

no

g i v e s t h e v i s c o s i t y by

11 = and II T~ i s t h e temperature-dependent term i n t h e osmotic p r e s s u r e along t h e phase sepa- r a t i o n curve.

The o u t l e t tube of MC2 must a l s o s a t i s f y t h e u s u a l c o n d i t i o n of s m a l l v i s c o u s h e a t i n g / 2 / :

4

Z2 << a T2/(00 A2 V:

1.

( 6 )

S i n c e T1 > T2, a = 54 ~ / m o l ~ ~ and X V = 43 ~ / m o l ~ '

0 0 .

c o n d i t i o n (5) i s a u t o m a t i c a l l y met when t h e tube

i s designed t o g e n e r a t e - n e g l i g i b l e v i s c o u s h e a t i n g .

THE DOUBLE M I X I N G CHAMBER AT HIGH TEMPERATURES.-

_.

.

W e w i l l assume

Q1

= Qp = 0 i n t h i s paragraph and i n t r o d u c e a dimensionless parameter A d e f i n e d by A = 5 1 2 Z 1 n o V o n t ~ ( 7 n o ~ ; ) .

I n a s i n g l e mixing chamber t h e c o o l i n g power i s g i v e n by

+

12

A

T ? = 96

it

T:

.

t 1 ( 4 ) P u t t i n g

%

=

+

i t can be d e r i v e d f o r given 2 2 Ti, t h a t

A

T~ =

il

T1 + (\- "1) T2

.

Since

F

m

0

5

n1

5

nt i t f o l l o w s t h a t T1 and T2cannot be

b o t h s m a l l e r t h a n Tm, i n agreement w i t h our expe- r i m e n t s / l / .

THE INFLUENCE OF Z,

.-

When Z 2 = 0 , e q u a t i o n (3)

2 2

shows t h a t T1 > T 2

.

I f f u r t h e r m o r e = 0 and T2 = Tm, e q u a t i o n s ( l ) , (2) and (4) g i v e

_-

%=

2 2

= 96

il (T1

-

T 2 ) , assuming t h a t Ti i s t h e same i n both systems. Hence t h e c o o l i n g power of MC2 a t a c e r t a i n temperature i s l a r g e r than t h e c o o l i n g power of a s i n g l e mixing chamber a t t h e same tempe- r a t u r e .

When Z2 cannot be n e g l e c t e d , t h e c o o l i n g power of MC2 i s reduced and may be even s m a l l e r

t h a n t h e c o o l i n g power of a s i n g l e mixing chamber. Therefore Z2 should be s o s m a l l t h a t t h e second term i n t h e r i g h t hand s i d e of e q u a t i o n ( 3 ) can b e n e g l e c t e d :

2 2 '

22 <<

no

V. T2 Tl!(non2 V:) (5)

F i g . 1 : Schematic drawing of a double mixing cham- b e r .

The v a l u e of A = 1 when both f a c t o r s i n t h e r i g h t 2 hand s i d e of e q u a t i o n (2) a r e zero. Hence 8 T;= T1 and n l =

it.

A t t h e corresponding T. v a l u e ( f o r g i v e n n t and Z1) Ah ( s e e f i g u r e 1) has a maximum. The system must be designed i n such a way t h a t t h e phase boundaries a r e i n s i d e t h e mixing chambers a t t h i s p o i n t .

When A < 1 t h e r e i s no 3 ~ e flow through

MC2 and T1

#

2.8 T2. T h i s u n r e a l i s t i c r e s u l t i s a consequence of t h e f a c t t h a t we took 4 2 e x a c t l y e q u a l t o zero. When 42 i s f i n i t e , b u t s m a l l , t h e r e i s a small flow through MC2.

A s a r e s u l t of t h e ~ ~ - d e ~ e n d e n c e of t h e osmotic p r e s s u r e , Ah c a n i n p r i n c i p l e be v e r y l a r g e a t h i g h temperatures. I f one of t h e phase bounda- r i e s would be d r i v e n o u t of i t s mixing chamber, t h e system w i l l n o t o p e r a t e p r o p e r l y . F o r t u n a t e l y such a s i t u a t i o n does n o t occur. A t h i g h v a l u e s of Ti ( r e s u l t i n g i n A << I ) i t f o l l o w s from e q u a t i o n

(3) t h a t T1 % T2. Hence Ah i s s m a l l : t h e double mixing chamber behaves a s a s i n g l e mixing chamber.

When A

-

> 1 , which i s t h e v a l u e of main experimental i n t e r e s t , a v a l u e of

G1

5

At

i s found 2 and T1 = 8 T:. DEPENDENCE ON Z1.- The c u r v e s r e p r e s e n t i n g t h e Tp- 21 dependences ( f o r f i x e d o t h e r e x t e r n a l parameters) a r e f a i r l y f l a t i n neighbourhood of t h e minimum. Furthermore t h e dependence of t h e optimum Z1 ( g i - v i n g t h e minimum T2 f o r g i v e n Q2) on 4 2 i s s m a l l . Hence t h e o p e r a t i o n of t h e system i s n o t g r e a t l y a f f e c t e d by t h e p a r t i c u l a r choice of Z1. This i s a g e n e r a l f e a t u r e of systems of t h i s kind and i t i s

i n agreement w i t h experiments. I t f a c i l i t a t e s t h e c o n s t r u c t i o n of e.g. double o r t r i p l e mixing cham- b e r s c o n s i d e r a b l y .

THE CONTINUOUS M I X I N G CHAMBER.-In d i s c r e t e mixing chambers t h e temperature of a flow of 3 ~ e . i n t h e c o n c e n t r a t e d phase i s lowered by d i l u t i n g a por- t i o n of t h e o r i g i n a l 3 ~ e flow. For a s u b s t a n t i a l temperature r e d u c t i o n t h i s p o r t i o n i s l a r g e . I t i s p r e f e r a b l e t o d i l u t e t h e 3 ~ e i n s m a l l q u a n t i t i e s i n a l a r g e n u m b e r (>10) of mixing chambers. I n t h i s c a s e t h e mixing i s p r a c t i c a l l y r e v e r s i b l e . A sche- m a t i c drawing of such an assembly of mixing cham- b e r s , c a l l e d a "continuous mixing chamber", i s g i v e n i n f i g u r e 2. From t h e e n t h a l p y b a l a n c e of an element can be d e r i v e d t h a t d ~ / d i . I n t e g r a t i o n

y i e l d s T. = where r i s t h e t o t a l I n

p o r t i o n d i l u t e d i n t h e a r r a y of mixing chambers. For r = 0.5 i t f o l l o w s t h a t T _{i n} = 1 1 Tout. When

Tin = 2 Tout only 18 % of t h e 5He h a s t o be diluted.

F i g . 2 : Schematic diagram of a continuous mixing chamber.

I n p r a c t i c e t h e performance of t h i s system

is degraded by t h e f i n i t e h e a t c o n d u c t i v i t y i n t h e v e r t i c a l d i r e c t i o n . Taking t h i s e f f e c t i n t o account

t h e r e l a t i o n between t h e temperatures becomes :

where S is t h e t o t a l a r e a of t h e phase boundaries, R i s t h e d i s t a n c e between t h e phase boundary and t h e main 3 ~ e f l o w , and t h e h e a t c o n d u c t i v i t y of t h e c o n c e n t r a t e d phase i s g i v e n by K

IT. For a good

e f f i c i e n c y B should be much s m a l l e r t h a n 1. When

r = 0.18, R = 3 an, = 50 pmol/s, T = 15 mK and

B

= 0.1 t h e a r e a i s 3 cm2.

Between t h e elements of a continuous mixing chamber, p a r t i t i o n s must be i n s t a l l e d . They pre- vent i n t e r n a l c i r c u l a t i o n s t h a t would degrade t h e performance i n such a way t h e system behaves a s one d i s c r e t e mixing chamber r a t h e r than a s a con- t i n u o u s one.

The performance of continuous mixing cham- b e r s w i l l b e i n v e s t i g a t e d i n our l a b o r a t o r y i n t h e n e a r f u t u r e .

CONCLUSIONS.- A m u l t i p l e mixing chamber i s a con- v e n i e n t t o o l f o r extending t h e temperature range of a d i l u t i o n r e f r i g e r a t o r . I n a d d i t i o n t o t h a t t h e c o o l i n g power i s i n c r e a s e d a t lower tempera- t u r e s . A t h i g h temperatures t h e c o o l i n g power i s t h e same a s of a s i n g l e mixing chamber.

ACKNOWLEDGEMENTS.- This work was supported by the " S t i c h t i n g Fundamenteel Onderzoek der Materie, P.O.M.".

References

/ l / D e Waele, A.T.A.M., Reekers, A.B. and Gijsman, H . M . , Proc. ICEC 6 , GrenoBle 1976, 112

/ 2 / I b i d ; Proc. 2nd I n t . Symp. on Quantum F l u i d s and S o l i d s , S a n i b e l I s l a n d , F l a . , USA (19771,