Thermal conduction through a vented support

(1)

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Thermal conduction through a vented support

S. Buhler

To cite this version:

(2)

Classification

Physic-s Abstracts 07.20M 07.20F

Thermal

conduction

through

_a

vented

support

S. Buhler

Institut de

Physique

Nucl£aire, 91406

Orsay

Cedex, France

(Received 15

July

1993, revised 25 October J993,

accepted

J5 Noi>ember 1993)

Abstract. A

simplified

theory of thermal conduction in a support vented with cold vapour is described which allows the construction of convenient graphs for practical

applications.

The _very

same model is

employed

to extend the Wexler correlation [I] beyond its classic limits and

corresponding

diagrams are presented for helium and nitrogen.

1. The vented

support.

Consider a solid

support

vented with cold vapours in a counter-current way with the

following

assumptions

(Fig,

I)

:

.

Temperatures

Tj

and T~

respectively

on each end of the support are in

steady-state

.

Conducting

cross-section A = Constant _;

fl

02

x

°2

_T2 x =

6~

+

d6~

~ = const T₊ dT A dx ~~X

fl

Z= ~ ~

~

~ x = 0

lj

_Ti

hi

lf

_j~ a) b)

(3)

664 JOURNAL DE PHYSIQUE III N° 4

. The average thermal

conductivity

is

given by

A~

where

j

T~

A~

= A dT = Constant T~

Tj

~

.

Specific

heat of the vapour c~ =

Constant _;

If _we _assume _a

perfect

heat

exchange

between the solid conductor and the vent mass flow,

M, we can establish two heat balances

Heat flow

Q(x)

in the wall of the support :

Q=-A.A.jj.

₍₁₎

Perfect heat

exchange

from vapour to wall ;

corresponding

heat balance

a)

on a finite element at :

dQ (x

=

M

_c~

dT

(2)

b) on the total

length

of the support

Q2-Qi

"M.c~.

(T~-Tj).

(3)

From

equations

(I)

to

(3)

we determine

~jj

or its reduced value with Z = AIL :

Outflowing

heat

~jj/Z

from

the support

(at

cryogenic

temperature

Tj)

:

(

Cp

(T2

Ti

~=

₍₅₎

~

jM.c~j

exp -1 ~"~ also the.

Inflowing

heat

Q2/Z

into the support

(at

ambiant

temperature

T~)

jM.c~j

~2

= M ~.

(T~

T~

).

~~~ ~ ~

(6)

Z Z ~

jM.c~j

~~~ Z.A

Likewise we establish the temperature

profile

T(x),

or, with #i = x/L,

expressed

in a

(4)

A

practical

application

for the heat flow at low temperature

Qj/Z

is

represented

in

figure

2a

for different materials of a helium vented support between 4 to 300 K

figure

2b shows the

evolution of its

temperature

profile

under these conditions.

iooo Brass 100 Stainless jo fi

)

_lo Qi

d

~

o-S > Glass ~ o-1 0.2 0.01 0 0,001 0.01 o-I 0 100 200 300

Ml

Z g/s cm

T

a) b)

Fig. 2. Vented supports with cold helium _vapours from 4 to 300K. a) Residual heat flow

Qi/Z

f(filZ)

at 4 K. b)

Temperature

profile T

= f

~,

~

for stainless steel.

L Z

2. Wexler correlation.

Wexler I has

investigated

the boil-off rate of a LHe reservoir, in

particular

when he enhanced

it

artificially

with an immersed electric heater

(Fig.

3a).

This

procedure

allowed him to

distinguish

the radiation and the residual conduction components within the total heat flow

(Fig.

3b)

which is absorbed

through

a continuous

evaporation

of the saturated

cryoliquid

(latent

heat of

vaporization

i~).

. ~ ~ Tot R ~. ~~

~Qll~

a) b)

(5)

666 JOURNAL DE PHYSIQUE III N° 4

Wexler's model, however, is restrictive, since the total

produced

vapour flow

always

has to

vent the

single

neck of his reservoir.

Figure

4a shows the

principle

of a different cryostat with two separate escape lines. The total

boil-off _M~~~ may be

split

into a support vent

flow,

Mv,

and its

complement,

M~,

diverted

through

a

by-pass

such that

The Wexler correlation

QV

"

QSI

+

QR

+

QH

~~~

where

Qsi

=

f (Mv

)

is _now valid for any mass flow distribution : for

li~

=

0 one finds the classic Wexler

configuration

with

Q~

_m

0,

whereas for

M~

_~ 0 one has to extend it to virtual but

negative

(6)

Figure

4b shows a new Wexler

diagram,

now extended to 6 different vent flow modes

explained

in the table below.

Table II. Parameters

definition

for figure

4b.

Mv

Vent _mass flow

MB

By-pass

mass flow

fii~~~ Total mass flow

(produced

boil-ofo

~j~j

Outflowing

heat from vented support

Q~

Radiative heat flow into the

cryoliquid

~j~

Additional heat flow from immersed heater

~j~~~ Total heat flow absorbed from the

cryoliquid

bv

Heat flow evacuated from the vent

only

f~

Latent heat of

vaporization

of the

cryoliquid

3. Calculation of the extended Wexler correlation.

For

general

practice

it is advisable to express mass and heat flows in their reduced

form,

divided

by

Z

=

AIL.

Selection

of

the

fixed

parameters

. Gas nature -

c~,

i~

for

nitrogen,

helium...)

.

Temperatures

Tj,

T~ on each end of the support

.

Supports

material

(stainless

steel, brass.. -

A~).

Calculation

procedure

. choose any value for

Qtot/Z

. calculate

Mtot

Qtot

V

fi

. since

Jitot

=

iiB

₊

iiv

select _any

M~

and

corresponding

Mv

where am

M~

w M~~~

(7)

668 JOURNAL DE

PHYSIQUE

III N° 4

iso

(v/Z

_[W/cml

izo Fluid

Nitrogen

Support

:

Stainless

steel

Temperature

Tl/T2

:

80/300K

90 so 30 ,,." '~'~~~

,.""

. ."

Qh

/Z [W/cull _~,." 30 60 90 120 150 ,,..~ + + + + + + + + + + + + + + °' -~o °.~ °.~ °.~ °.~

0~

_/ Z gIs cmj

Fig.

5. Extended Wexler

diagram

for

nitrogen.

fi~

/Z [W/cmj Fluid : Helium

,

Support

Stainless

steel

Temperature

_:

Tl/r2

_:

4/300K

-3t -26 -21 -Is -11 -6 4

:~

_Qh/Z

_(W/cm] ,? ," ,:. -6

;""

,."'

~?'

-l I ,,.." + +

,:"'

+ + + ~ + 0 o,01

~J2

o-m ~ M~/Z _gIscm] ~~~,,?' ~' -21 ~,,:~ ~ -26 " '~ -31

(8)

As

Qv depends

on the vent mass

flow,

Mv,

we can introduce

kv

=

f(Aiv

as a

secondary

scale in the _same

diagram.

We also _may

proceed

analogously

for

Qv

=

f (M~

),

but the

starting

point

of this third scale will be different for _any

given

cryostat,

depending

on its

specific

QR.

Figures

5 to 7 represent the extended and reduced Wexler

diagrams

for

nitrogen

and helium

in well defined conditions.

z.5

Qv/Z

[W/cmj

Fluid

:Helium

m

Suppon

:

Stainless

steel

Temperature

_:

Tim

_:

4/80K

-3.5 -2.5 -1.5 -0.5 O-b I-b Z-b

Qh

/Z [W/cml ,. .,' ;? ,,." ' _;" -IA ,." :"

,,."'

I

_,r"

+ + + + + + + ++ ++++++++ 0 ,,

o.005 o.ot o.03 o-1

;."

_$i~

/Z _gIscm]

.~

-3.5

Fig.

7. Extended Wexler diagram for helium.

4. Discussion and conclusions.

The

proposed

calculation

procedure

for a vented support should be considered as a first

approach

intended to fix the order of

magnitude.

However the method is convenient and

straight-forward

if one makes use of the

appropriate

diagrams.

We notice that the so-called _« Wexler effect

» (I.e. a moderate

change

of the boil-off rate

resulting

from a substantial increase of the heat QH

dissipated)

is rather

pronounced

for LHe

but far less

perceivable

for LN.

The

validity

of the model used is

certainly

restricted and should be

compared

with some

experimental

data [2].

References

[I] Wexler A.,

Evaporation

rate of

liquid

helium, J App/. Phy.q 22, no. 12 (1951).