Possibility of long-distance heat transport in weightlessness using supercritical fluids

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Possibility of long-distance heat transport in weightlessness using supercritical fluids

Daniel Beysens, Denis Chatain, Vadim Nikolayev, Jalil Ouazzani, Yves Garrabos

To cite this version:

Daniel Beysens, Denis Chatain, Vadim Nikolayev, Jalil Ouazzani, Yves Garrabos. Possibility of long-distance heat transport in weightlessness using supercritical fluids. Physical Review E : Sta- tistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2010, 82 (6), 061126 (11 p.).

�10.1103/PhysRevE.82.061126�. �hal-00551878�

Possibility of long-distance heat transport in weightlessness using supercritical fluids

D. Beysens,

^1,2

D. Chatain,

¹

V. S. Nikolayev,

^1,2

J. Ouazzani,

³

and Y. Garrabos

⁴

1Service des Basses Températures, INAC/CEA, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France

2ESEME, PMMH-ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 5, France

3SARL ArcoFluid, Parc Unitec 1, 4 Allée du Doyen Georges Brus, 33600 Pessac, France

4Institut de Chimie de la Matière Condensée de Bordeaux, UPR 9048, CNRS, 87 Avenue du Dr. Schweitzer, 33608 Pessac Cedex, France 共Received 23 July 2010; published 21 December 2010兲

Heat transport over large distances is classically performed with gravity or capillarity driven heat pipes. We investigate here whether the “piston effect,” a thermalization process that is very efficient in weightlessness in compressible fluids, could also be used to perform long-distance heat transfer. Experiments are performed in a modeling heat pipe共16.5 mm long, 3 mm inner diameter closed cylinder兲, with nearly adiabatic polymethyl- methacrylate walls and two copper base plates. The cell is filled with H₂ near its gas-liquid critical point 共critical temperature: 33 K兲. Weightlessness is achieved by submitting the fluid to a magnetic force that compensates gravity. Initially the fluid is isothermal. Then heat is sent to one of the bases with an electrical resistance. The instantaneous amount of heat transported by the fluid is measured at the other end. The data are analyzed and compared with a two-dimensional numerical simulation that allows an extrapolation to be made to other fluids共e.g., CO₂, with critical temperature of 300 K兲. The major result is concerned with the existence of a very fast response at early times that is only limited by the thermal properties of the cell materials. The yield in terms of ratio, injected or transported heat power, does not exceed 10– 30 % and is limited by the heat capacity of the pipe. These results are valid in a large temperature domain around the critical temperature.

DOI:10.1103/PhysRevE.82.061126 PACS number共s兲: 05.60.⫺k, 05.70.Jk, 44.35.⫹c, 68.03.Cd

I. INTRODUCTION

Heat transport over large distances is classically per- formed by the latent heat transport. Conduction is effective only on short distances because of the slowness of the heat diffusion process. In the 1990s, a new way of the heat trans- fer in very compressible fluids has been evidenced. This phe- nomenon, which is particularly important under weightless- ness, has been called piston effect

关1–3兴. When heating a

fluid confined in a container, a thin hot boundary layer

共HBL兲

of thickness ␦

1

forms close to the heating wall

关4兴.

This layer expands and at the same time compresses adia- batically the rest of the fluid. As a result, a spatially uniform heating of the bulk fluid is achieved in a very short time.

The heat flux is limited by the temperature gradient in the heating wall and in the fluid HBL. If the other side of the sample, located at a distance L from the heating wall, is colder, a cold boundary layer

共CBL兲

of length ␦

2

will simul- taneously form at the corresponding wall

关

see Fig.

1共c兲

be- low兴. In between, the temperature is homogeneous. A heat transport can thus be performed in both HBL and CBL, con- nected by the bulk fluid that acts as a kind of thermal short circuit. While such a “heat pipe” does not work in the sta- tionary regime

共

the piston effect concerns only transient be- havior兲, it could serve to quickly transfer heat.

Studies of transient heat flux in the presence of piston effect have already been performed in

³

He under Earth’s gravity in a flat sample of large aspect ratio

共

thickness of 1 mm and diameter of 57 mm兲 perpendicular to gravity for Rayleigh numbers below and above the convection threshold

关5兴

and in CO

₂

under weightlessness

关6兴. Here, in order to

evaluate whether this process can be used to transport heat on large distances under weightlessness, experiments are per- formed under the conditions of magnetic gravity compensa-

tion in a cylindrical cell with the largest length that preserves magnetic compensation

共aspect ratio of 0.182兲. The cell is

filled with H

₂

near its gas-liquid critical point

共critical tem-

perature T

c

= 33 K兲.

II. EXPERIMENTAL SETUP

Weightlessness conditions are achieved by using magnetic forces to compensate the gravity. The magnetic gravity com- pensation technique is recently reviewed in

关7兴. The HYLDE

installation

共hydrogen levitation device兲

situated at CEA/

Grenoble has been used here. It is described in detail in

关8,9兴.

A. Cell

The fluid is confined in a cylinder of L = 16.5 mm long, 3 mm inner diameter, and 4 mm outer diameter made of poly- methylmetacrylate

共PMMA= Plexiglas, denoted below with

the subscript PA兲 共Fig.

1兲. The cylinder axis coincides with

the magnetic coil axis. PMMA has been chosen because it is transparent and has a low thermal conductivity

共

k

_PA

= 0.125 W m

⁻¹

K

⁻¹

at 33 K兲 and the specific heat of the walls is low

共CPA

= 180 J kg

⁻¹

K

⁻¹

at 33 K兲. With a mass density of 1.15

⫻

10

³

kg m

⁻³

, the thermal diffusivity is D

_PA

= 6.04

⫻

10

⁻⁷

m

²

s

⁻¹

, corresponding to the characteristic diffusion time t

PA

= L

²/

D

PA

= 450 s. The PMMA cylinder is sealed with stainless-steel rings to two parallel electrolytic copper

共thermal conductivity at 33 K is 1130 W m⁻¹

K

⁻¹兲

blocks. They are in thermal contact with the helium bath by thermal conductors. The temperatures of the blocks are mea- sured with Cernox® thermistors. They are made of a com- posite that provides a very weak resistance variation with the magnetic field. Their time constant is on the order of 20 ms.

Stainless steel is a thermal insulator at these low tempera-

1539-3755/2010/82共6兲/061126共11兲 061126-1 ©2010 The American Physical Society

tures

共thermal conductivity is 3.37 W m⁻¹

K

⁻¹兲. The mass of

the upper copper block

共hot part兲, called the “head,” is

m

= 2.37 g. The other block, the “base”

共the coldest part兲, is

kept at constant temperature T

_B

by a closed-loop temperature regulation, the proportional integral differential

共PID兲

device.

The temperature control accuracy is

⫾

1 mK at 33 K and the working temperature range is 15–40 K.

The cell can be filled in situ with pure pressurized H

₂

through a capillary. This capillary is closed by a H

₂

ice plug

共H2

solidification temperature is 14 K兲, whose formation is provided locally by a thermal conductor in contact with the helium bath. The cell is observed by light transmission. Par- allel light is directed with mirrors from outside the anticry- ostat to the sample. The beam is parallel to the axis of the cylindrical cell and is deflected by mirrors to the camera outside the anticryostat. The whole setup is situated in a vacuum chamber.

At room temperature and at equilibrium, H

₂

is actually a mixture of three volumes of orthohydrogen

共parallel spins of

H nuclei, o-H

₂兲

with one volume of parahydrogen

共antipar-

allel spins, p-H

₂兲. This gas is called normal hydrogen,

n-H

₂

. When H

₂

is cooled, the o-H

₂

– p-H

₂

equilibrium is shifted and almost all o-H

₂

gradually transforms to p-H

₂共

more pre- cisely, 96% of p-H

₂

at equilibrium at 30 K兲. The exact criti- cal pressure, temperature, and density of p-H

₂

are T

c

= 32.976 K, p

c

= 1.2928 MPa, and ␳

c

= 31.426 kg m

⁻³ 关8兴.

The cell is filled at critical density ␳

c

by checking that the gas-liquid meniscus does not displace in the cell while ap- proaching T

c

from below. The precision is on the order of

⫾0.5 mm which corresponds to the⫾3% uncertainty of the

critical density. The precision of the determination of T

c

is within 1 mK. This value shifts with the conversion o-H

₂

– p-H

₂

with a time constant of order 50 h

关8兴. Then the

experiments close to T

c

are performed more than 2 days after the beginning of the runs with the value T

c

checked before and after these runs.

B. Measurement procedure

Initially, the base, the head, the fluid, and the cell walls are at the same temperature T

₀

. The head temperature is maintained constant by sending a constant power

共⬃5 mW兲

that compensates the heat losses through the wir- ing

共Fig.1兲. It is important to precisely calibrate the head and

the base temperature sensors. We took benefit of the strong convective flows that form a few millikelvins to T

c

under gravity in the presence of the slightest temperature gradients.

Head

共TH兲

and base

共TB兲

temperatures are assumed to be equal when, under Earth’s gravity, the convection disappears.

Then the magnetic field is switched on and further measure- ments are performed at weightlessness.

The electric power P

_B^full

is supplied to the cell base by the temperature regulation that maintains T

B

constant. P

_B^full 共

t= 0

兲⬇

200 mW when the head and the base are at the same temperature. The power P

_H^full

is supplied to the head. P

_B^full

, P

_H^full

, T

H

, and T

B

are recorded throughout the experiment.

At t = 0, the power P

₀

= 7.5 mW is sent during

⌬t

= 300 s in the head by increasing in a single step the power P

_H^full

. This value corresponds to 1 kW

/

m

²

. It has been cho- sen because

共1兲

it is the smallest value that provides enough precision in the data;

共

2

兲

this power is close to the maximum power that can be extracted by the cell base temperature regulation. The initial temperature T

₀

remains the reference for the base temperature regulation PID

共

i.e., T

_B

= T

₀

within the precision of the PID兲. During the experiment, the tem- perature of the base varies only slightly, on the order of 6 mK

共Fig.2兲. The integration time of the PID is typically 10

s. This value thus corresponds to the temporal resolution of the transient heat flux measurements.

The heat power P

B

transferred by the cell is determined as P

B共t兲

= P

_B^full共t

= 0兲 − P

_B^full共t兲. 共1兲

The power P

B共t兲

is thus limited by the value P

_B^full共t= 0兲.

PBfull

TH

TB

z

Pwfull

TF(z,t) TF(z,t=0)=TB

PHfull

CBL T

He, 2.17K

HBL

TB

HEAD

BASE PMMA 16 mm tube

(a)

TH

H2

(b) (c)

FIG. 1. 共Color online兲The experimental piston effect heat pipe.

共a兲photo;共b兲heat exchange model. Both the head and the base are connected to the helium bath via heat conductors共thin wires兲. The heat powerP_w^fullleaks from the head to the bath共see Sec.III A 2兲. The controlled heat power is supplied to the head共P_H^full兲and to the base共P_B^full兲.共c兲Schematic instantaneous fluid temperature variation provided by the piston effect. The cold 共CBL兲 and hot 共HBL兲 boundary layers are indicated.

-0.5 0 0.5 1 1.5

20 25 30 35 40

0 200 400 600 800 1000

P

B

(mW )

t (s)

P0=7.5 mW

∆t= 300 s

T

H

(K )

V

V

100 mK 100 mK

33.126 33.128 33.130 33.132 33.134 33.136 33.138

0 400 800

t(s) TB(K)

FIG. 2. 共Color online兲Typical measurements of the transmitted heat flux共P_B兲, head temperature共T_H兲, and base temperature共T_B, in the inset兲when a 7.5 mW power is injected into the head during about 300 s 共shown with a bar兲. Solid lines: filled cell, T₀=T_c + 100 mK; dotted lines: empty cell共under vacuum兲.

The measurements were initially carried out in the empty cell to evaluate the heat conduction by the PMMA walls. The cell is then filled and measurements are performed at differ- ent temperatures T

₀

.

C. Measurement results

In Fig.

2

typical evolutions of the transmitted power P

B

and head temperature when the pipe contains H

₂

initially at T

₀

= T

_c

+ 5 mK are shown. The corresponding curves for the empty

共under vacuum兲

cell are shown for comparison. In this latter case, heat is transmitted by the PMMA pipe walls only.

One notes that

共i兲

T

H

is lower when the pipe is filled, which agrees with the higher value of P

B

and

共ii兲

the dynamics of P

_B

at small time is quite different with empty and filled cells.

This is even clearer from Fig.

3

where the typical small time behavior is reported for different temperatures. After the 10 s reaction time of the thermal regulation, P

B

increases linearly.

In contrast, the heat flux in the empty pipe remains close to zero for long time. The difference in behavior highlights two different mechanisms of heat transfer:

共i兲

thermal diffusion in the walls of the empty pipe where the heat flux is zero before the front of heat diffusion reaches the base and

共ii兲

piston effect in the filled pipe. In the latter case, the dynamics

共

and the yield, see below兲 increases when nearing T

c

. The behav- ior of P

B共t兲

at T

₀⬍

T

c共i.e., when both liquid and gas phases

are present in the pipe兲 is similar to that at T

₀⬎

T

c

.

The measured time evolution of the transmitted power P

B

is plotted at different temperatures above and below T

_c

in Fig.

4. One notes that the behavior at

T

₀⬍

T

c

is similar to the one at T

₀⬎

T

c

, even very close to T

c

. The yield defined as Y = P

B/

P

₀

attains at its maximum about 10%. The yield for the filled pipe is always larger than for the empty pipe. At

large times, the pipe walls contribute to the heat transfer, as seen in Fig.

4, where the rise of

P

B

for the filled cell is roughly parallel to the empty cell curve. The maximum heat flux

共PB

M兲

is then always reached at the end of the heating period.

In order to determine what are the relative contributions of the cell walls and the fluid, it is necessary to analyze the process in great detail. This is performed in Sec.

III. It is

found that the wall contribution is much smaller than in the empty cell due to the rapid thermalization of the bulk fluid that reduces the temperature gradient in the pipe walls.

It is possible to infer the “initial” slope dP

B/

dt from Fig.

3. Heat is sent into the cell head at

t = 0 and, after 10–15 s, P

B

grows linearly in time during at least 20–30 s. One thus defines as initial slope the value of dP

B/

dt as measured be- tween 27 and 47 s. During this period, the contribution of the transient conduction by the pipe walls is negligible. The tem- perature dependence of this slope is reported in Fig.

5共a兲

for two experimental runs. The data are similar above and below T

c

. The slope increases when going to T

c

and saturates for

兩T0

− T

c兩ⱕ

0.3 K. The maximum heat flux

共at 300 s兲

P

_B^M共Fig.

4兲

is plotted with respect to temperature in Fig.

5共b兲

for two different runs. Data between both runs show a systematic difference of order 15% that we attribute to the uncertainty of filling of the cell at critical density

共⫾3%兲. They, how-

ever, exhibit the same behavior, similar to dP

B/dt, with an

increase when going to T

c

and saturation for

兩T0

− T

c兩 ⱕ

0.3 K.

-0.1 0 0.1 0.2 0.3 0.4 0.5

33.038 33.04 33.042 33.044 33.046 33.048 33.05

0 20 40 60 80 100

P

B

(mW )

t (s)

T

H

(K )

5 mK

30 mK

TH (5mK) ^{- 0.1 K}

- 1 K

V

0.1 K

3 K

P0=7.5 mW

FIG. 3. 共Color online兲 Small time evolution of the transmitted power 共P_B兲 for several temperature distances T₀−T_c, both above and below the critical point. The head temperature evolution共dotted line兲is also shown to precisely detect the onset of heating. V: empty cell;T₀−T_c= 5 mK共solid line兲, 30 mK共broken line, long dashes兲, 0.1 K共double broken line兲, 3 K共broken line, short dashes兲, −0.1 K 共dotted line兲, and −1 K共dotted-dashed line兲.

0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600

P

0

=7.5 mW

∆ t ≈ 300 s

t (s) P

B

(mW)

5 mK

30 mK - 0.1 K

0.3 K

- 1 K

3 K

V

FIG. 4. 共Color online兲 Measured transient of the transmitted power P_B for several temperatures above and below the critical point. The time interval during which the power is injected is shown with a bar. V: empty pipe; other curves: filled cell;T₀−T_c= 5 mK 共solid line兲, 30 mK共broken line, long dashes兲, 0.1 K共double broken line兲, 3 K 共broken line, short dashes兲, −0.1 K 共dotted line兲, and

−1 K共dotted-dashed line兲.

III. NUMERICAL SIMULATION IN TWO-DIMENSIONS AND ANALYSIS

Several important points influence the choice of a numeri- cal method for the simulation.

共i兲

One-dimensional

共1D兲

numerical simulations are rela- tively easy to perform. However, in the present study, a 1D simulation would not take into account an important effect, the heat exchange with the lateral walls. Lateral walls are heated by the bulk fluid that is quickly thermalized by the piston effect. This wall influence limits the fluid temperature rise and thus the value of the output heat flux. At least a two-dimensional

共2D兲

numerical simulation is necessary to take this influence into account.

共ii兲

The near-critical fluid introduces a nonlinearity in our system. Its simulation thus requires an iterative algorithm.

On each iteration a complete set of temperature values at each node point should be determined. It means that

共i兲

in practice, the equation of state

共EOS兲, i.e., the relation be-

tween fluid temperature T, pressure p, and density ␳ ^{, must be} solved at each iteration, node point, and time step. It involves an enormous total number of EOS treatments. At the same time, the system is driven quite far from the critical point;

the relative temperature distance can be as large as

共T

− T

c兲/

T

c⬃

0.15. This means that

共ii兲

any simple model EOS

共

parametric

关18兴

, van der Waals, linearized

关11兴

, etc.

兲

would not work well. A real EOS for H

₂

needs to be used in the simulation. The available hydrogen EOS

关12,13兴

is the modi- fied Benedict-Webb-Rubin

共MBWR兲

EOS that includes doz- ens of terms. The requirements

共i兲

and

共ii兲

make the compu- tation time prohibitively long when the full hydrodynamic description of the supercritical fluid

共as in关11兴兲

is used.

Recently, some of us

关10兴

suggested a fast calculation method that results in accurate heat transfer calculations wherever the impact of the fluid advection on the heat trans- fer can be neglected. It makes use of the boundary element method

共BEM兲 关14兴

that permits a reduction of the dimen- sionality of the problem: only the boundary surfaces need to be meshed. Here we use the BEM to simulate this problem in microgravity where the fluid convection can be neglected.

A. Problem statement

We want to find the amount of heat transferred to the base from the head of the cell while the temperature of the base T

B

is kept constant. Initially, the temperature of the whole cell is equal to T

₀

.

The cell consist of several parts:

共i兲

cell base

共denoted by

B index兲,

共ii兲

cell head

共denoted by

H index兲,

共iii兲

PMMA tube

共

denoted by PA index

兲

, and

共

iv

兲

fluid

共

denoted by F index兲. These parts need to be modeled separately.

1. Cell base model

The cell base is made of electrolytic copper and is as- sumed isothermal, thanks to its very high thermal conductiv- ity. Its temperature T

B

is controlled by the temperature regu- lation and assumed to be constant

共

=T

₀兲

during each experiment. In reality, the temperature regulation device

共see

above, Sec.

II B兲

has an integration time t

I⯝

10 s that char- acterizes the time over which it averages the actual base temperature data T

B

in order to maintain the base tempera- ture at its reference value T

₀

. We take into account this regu- lation “inertia” indirectly by integrating over t

I

the function P

B共t兲

obtained in the simulation.

The power P

B

transferred to the base is the main output of the simulation to be compared with the experiment. It is a sum of the heat power that arrives from the cell head by the tube walls and by the fluid column,

P

B

= − k

PA

冕

_共S_PA兲

冏 ^⳵ ⳵ ^T n ជ

^PA

冏

B

dS − k

F

冕

_共S_F兲

冏 ^⳵ ⳵ ^T n ជ

^F

冏

B

dS,

共2兲

where k denotes the thermal conductivity and the integration is performed over the cross-section areas of the PMMA tube S

_PA

and of the fluid S

_F共

see Fig.

6兲

; the unit normal vector n ជ is directed inside the base. Since T

B

is assumed to be con- stant in time, the heat power that leaks to the He bath re- mains constant and P

B

given by Eq.

共2兲

corresponds to the measured quantity

共1兲.

0.003 0.004 0.005 0.006 0.007 0.008 0.009

0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0011 0.0012

0.001 0.01 0.1 1 10

dP B/dt(mW/s)

IT₀-T

cI(K)

dY/dt

0.6 0.7 0.8 0.9 1

0.08 0.09 0.1 0.11 0.12 0.13

0.001 0.01 0.1 1 10

P B

M (mW) Y M

IT0-T

cI(K)

(a) (b)

FIG. 5. 共a兲Measured transient of the transmitted heat flux initial risedP_B/dt共see text兲with respect to兩T₀−T_c兩. Run 1: circles; run 2:

squares.T₀⬎T_c: black circles;T₀⬍T_c: open circles. The dotted共broken兲line is the run 1共2兲smoothing ofT₀⬎T_cdata.共b兲Fluid maximum heat transmissionP_B^M共see text兲with respect to兩T₀−T_c兩. Run 1: circles; run 2: circles.T₀⬎T_c: filled circles;T₀⬍T_c: open circles. The dotted 共broken兲line is the run 1共2兲smoothing ofT₀⬎T_cdata.

2. Cell head model

The main part of the head is made of electrolytic copper that can be assumed isothermal, thanks to its very high ther- mal conductivity at this temperature. The other part is made of stainless steel, which conducts heat much worse than cop- per

共ratio of thermal conductivities of steel and copper is

0.003; see Sec.

II A兲

. However, because of its very small thickness, slow temperature evolution, and a good thermal contact with copper, it can also be assumed isothermal.

The power balance of the head can then be written in the form

P

_H^full

= k

_PA

冕

_共S_PA兲

冏 ^⳵ ⳵ ^{T n} ជ

^PA

冏

H

dS + k

_F

冕

_共S_F兲

冏 ^⳵ ⳵ ^{T n} ជ

^F

冏

H

dS + C

_H

dT

_H

dt

+ P

_w^full共TH兲, 共3兲

where P

_H^full

is the controlled heat power supplied to the head.

The first two terms at the right-hand side correspond to the heat power transferred to the tube walls and the fluid, respec- tively. The unit normal vector n ជ is chosen to be directed toward the head. The third term reflects the thermal inertia of the head, the head total heat capacity

共measured in J/K兲

be- ing denoted by C

H

. The power P

_w^full

= P

_w^full共TH兲

is the heat leak due to the copper wire that connects the head to the He bath at temperature T

_He

= 2.17 K. Due to the high thermal diffusivity of the copper, the temperature distribution in the wire can be assumed to follow immediately the temperature of the head. This implies that

P

_w^full共T兲

= S

w

l

w

冕

T_He T

k

_Cu共T兲dT, 共4兲

where S

w

= 0.05 mm

²

and l

w

= 20 cm are the section area and the length of the wire. Since the copper thermal conductivity k

_Cu

does not exhibit a strong temperature dependence be- tween 33 and 43 K

共where the experiments are carried out兲,

P

w共TH兲

can be approximated by the linearized expression

P

_w^full共

T

_H兲 ⯝

P

_w^full共

T

₀兲

+ S

w

l

_w

k

_Cu共

T

₀兲共

T

_H

− T

₀兲

.

共

5

兲

Note that the constant power P

_w^full共T0兲

needs to be supplied to the head both before

共to maintain

T

H

= T

₀兲

and during the experiment

共

as a part of P

_H^full兲

to compensate the heat leaks.

In the following, we will denote P

H共t兲

= P

_H^full共t兲

− P

_w^full共T0兲. It

represents the excess of the supplied power with respect to its value at t = 0. In the above experiment, P

_H

was switched on at t = 0, kept at a constant level P

₀

during the time

⌬t, and

then switched off.

3. PMMA wall model

To estimate correctly the heat exchange between the fluid and the tube, it is important that the heat exchange area 2 ␲ ^R

i

h be reproduced correctly in the simulated 2D configu- ration. The fluid layer is supposed to be sandwiched between two plain walls

共see Fig.6兲

so that the PMMA-fluid contact area is 2dh. This condition defines the depth d = ␲ ^R

i

of the sandwich. The effective fluid layer half-thickness is then l

F

= S

F/

2d = R

i/

2. The effective wall thickness is l

PA

= S

PA/

2d.

The heat flux and temperature are continuous at the PMMA- fluid contact area, where the normal is directed inside the PMMA,

k

_PA

⳵ ^T

PA

⳵ ⁿ ជ ^{= k}

^F

⳵ ^T

F

⳵ ⁿ ជ ^,

T

PA

= T

F

.

共6兲

Since the cell is in vacuum and the radiative heat transfer is negligible because of the low temperature, the zero flux boundary condition can be defined at the external surface of the cell PMMA walls. The isothermal conditions T

PA

= T

H

and T

PA

= T

₀

are defined at the areas of contact PMMA head and PMMA base, respectively. Notice that while T

₀

remains constant, T

H

is a function of time. Its evolution is calculated by using Eqs.

共4兲

and

共5兲.

The heat conduction problem has trivial initial conditions

⳵␸

⳵ ^{t = D}

^PAⵜ2

␸ ^,

␸

兩t=0

= 0,

共7兲

where D

PA

is the thermal diffusivity of the PMMA. It has to be solved for the reduced temperature ␸ ^{= T}

PA共x

ជ ^{,t兲− T}

0

共where

x ជ is the position vector兲 to find the variation of the temperature T

_PA

inside the PMMA. The parameters k

_PA

and D

PA

are assumed to be constant.

4. Fluid model

Due to the symmetry, only half of the cell through a cut along the axis needs to be simulated, the zero radial flux being imposed at the cell axis. The approach

关10兴

was used in the simulations. We summarize it below.

The fluid temperature satisfies the equation

S

P A

d S

F

n n

n

h

2 l

F

l

P A

FIG. 6. Scheme of the cell representation in two dimensions 共fluid layer sandwiched between the PMMA walls兲.

⳵ ^T

F

⳵ ^{t = D}

^Fⵜ2

^T

^F

^{+ g共t兲,}

^共8兲

with

g共t兲 = 冉 ^{1 − c c}

^vp

冊 ^{dT dt ¯ ,}

^共9兲

where T ¯ = T ¯

共

t

兲

is the average fluid temperature

共

not to be confused with space- and time-dependent local fluid tem- perature T

F兲,

c

_v共cp兲

is the specific heat of the fluid at constant volume

共pressure兲, and

D

F

is the fluid thermal diffusivity.

The initial condition for T

F

is

T

_F兩t=0

= T

₀

.

共

10

兲

One can note that the term g

共

t

兲

is only relevant near the critical point where c

pⰇ

c

_v

. Since the cell is closed, the mean fluid density ␳ in the cell remains constant. The average tem- perature T ¯ is defined using the fluid EOS,

⌳共p,

␳ ^{,T ¯}

兲

= 0,

共11兲

where p is the fluid pressure. To obtain realistic results in a wide temperature interval

共in the experiments, the tempera-

ture varied within 20% from T

c兲, we have to use a realistic

EOS. Only the quite complicated NIST MBWR EOS

关12,13兴

is available for parahydrogen. It is unfortunately precise only relatively far from the critical point, typically for T

₀

− T

c

⬎

0.5 K. Discrepancies between simulation and experiments can then be expected close to the critical point because the critical divergence of some parameters will not be properly accounted for in the simulation.

Several methods are available to simulate the thermal be- havior of the cell

共see关5,15,16兴兲. However, with such a com-

plicated EOS, only the approach

关10兴

developed by some of us can be used, the others being prohibitively time consum- ing. The temporal evolution of the average temperature is given directly by the equation first derived in

关3兴, which

replaces the more complicated pressure evolution determina- tion,

dT ¯ dt = 1

␳

^vcv

冕

A

k

F

⳵ ^T

F

⳵ ⁿ ជ ^dA,

^共12兲

where

v=

S

F

h is the total fluid volume. The integration is performed over the whole fluid boundary A. n ជ is the outward normal to A. Instead of the local values for the fluid param- eters

共cv

, c

p

,k

F

, . . .兲 we use everywhere their time-dependent spatially averaged counterparts defined. EOS

共11兲

is used to determine the average density ␳ and the average temperature T ¯ . The fluid thermal conductivity k

F

is determined with a correlation implemented in the NIST

REFPROP

database

关13兴

. Equation

共12兲

has to be complemented with the initial con- dition

T ¯

兩t=0

= T

₀

.

共13兲

5. Reduction of the fluid problem

Two more steps must be performed before the numerical procedure can be applied. First, it is convenient to introduce a new ␺ variable instead of T

F

,

␺

共x

ជ ^{,t兲 = T}

^F共x

ជ ^{,t兲 − T}

0

− 冕

T₀ T¯

冉 ^{1 − c} c

^vp

冊 ^{dT ¯ .}

^共14兲

Equation

共8兲

reduces then to the equation

⳵␺

⳵ ^{t = D}

^Fⵜ2

␺ .

共15兲

One notes that since all the fluid parameters depend on the time-dependent average temperature T ¯ , the fluid thermal dif- fusivity can be written as

D

F

= D

d

f共T ¯

兲 共16兲

and is time dependent. In this equation, the function f is nondimensional and on the order of unity and D

d

is a con- stant. To eliminate this time dependence from D

F

in Eq.

共15兲,

one makes a second step by introducing a new variable ␶ defined by the equations

d ␶ dt = f

共

¯ T

兲

,

␶

兩t=0

= 0.

共17兲

Since ¯ T is a function of t only, the initial value of the prob- lem is fully defined. The substitution of Eqs.

共17兲

into Eq.

共15兲

results in a linear diffusion problem with a constant diffusion coefficient D

d

,

⳵␺

⳵␶ ^{= D}

^dⵜ2

␺ ^,

␺

兩_␶=0

= 0.

共18兲

B. Numerical implementation

A linear 2D heat diffusion problem can be solved by BEM. Below we outline this method for problems

共18兲

that is equivalent to the following integral equation:

D

_d

冕

0

␶

d ␶ ⬘ 冕

A

冋 ^G

^共

^{x ជ − x}

^ជ

^{⬘ , ␶ − ␶ ⬘}

^兲

^q

^共

^x

^ជ

^{⬘ , ␶ ⬘}

^兲

− ␺

共

x

^ជ

⬘ ^, ␶ ⬘

兲

⳵

x⬘

G

共

x ជ ^{− x}

^ជ

⬘ ^, ␶ ⁻ ␶ ⬘

兲

⳵ ⁿ ជ 册 ^d

^x⬘

^A

= 1

2 ␺

共x

ជ ^, ␶

兲, 共19兲

where q共x ជ ^, ␶

兲

= ⳵␺

共x

ជ ^, ␶

兲/

⳵ ⁿ ជ ^{and G} is the Green’s function for

the infinite space,

G共x ជ ^, ␶

兲

= 1 4 ␲␶ ^D

d

exp 冉 ^{− 4}

^兩x

␶ ^{ជ D}

^兩2d

冊 ^.

^共20兲

Equation

共19兲

is solved by breaking the integration contour A into straight boundary elements

共BEs兲. On each of them

␺ and q are assumed to be constant and equal to their values at the node point x ជ

i

in the middle of the ith BE. The time interval

共

0 , ␶

兲

is broken into the intervals

共

␶

f−1

, ␶

f兲

on which

␺ and q are assumed to be constant and equal to ␺

fi

and q

fi

at the ith BE. Equation

共19兲

then reduces to the set of linear algebraic equations,

兺

f=1

F

兺

j=1 N

共

q

_{f j}

G

_ij^Ff

− ␺

f j

H

_ij^Ff兲

= ␺

Fi/

2,

共

21

兲

where i = 1 , . . . ,N. H

ij

and G

ij

are the BEM coefficients,

G

_ij^Ff

= D

_d

冕

_␶^␶_f−1^f

^dt 冕

A_j

G

共

x ជ

i

− x ជ ^, ␶

F

− t

兲

d

_x

A,

H

_ij^Ff

= D

d

冕

␶_f−1

␶f

dt 冕

A_j

⳵ G共x ជ

ⁱ

^{− x} ជ ^, ␶

F

− t兲

⳵ ⁿ ជ

x

d

x

A.

共22兲

Their calculation is discussed in detail in

关14兴. The 2N

vari- ables

兵qFi

, ␺

Fi其 共i

= 1 , . . . , N兲 are related via N more equations that correspond to the boundary conditions given at each of the N BEs. The total number of equations is then 2N and corresponds to the total number of the unknowns. This set of equations should be solved sequentially for each time mo- ment ␶

F

, where F varies from 1 to the maximum required value.

The calculation algorithm needs to be iterative to find T ¯ :

共

1

兲

T ¯

共

t

_F−1兲

is taken as the initial guess for T ¯

共

t

_F兲 共

T

₀

is used for F= 1兲.

共2兲

The fluid parameters are calculated for ¯ T

共tF兲

using the EOS; ␶

F

is calculated with Eqs.

共17兲.

共

3

兲

Coupled problems

共7兲

and

共18兲

are solved by BEM;

the heat flux distribution over the fluid boundary is found.

共4兲

It is used to determine the corrected value of T ¯

共tF兲

with Eq.

共12兲.

共5兲

Steps 2–4 are repeated until the convergence is achieved.

共6兲

Steps 1–5 are repeated for the next time moment.

Since the summation in Eqs.

共21兲

has to be performed over all f = 1 , . . . , F, the described BEM formulation has a disadvantage of an increase of the calculation volume with F. On the other hand, the algorithm is very stable, which permits to choose a relatively large time step with practically no loss in accuracy.

C. Adjustment of the cell parameters

First, one needs to check if the model parameters corre- spond to the actual experimental values. The test consists in comparing the simulation and the experiments in an empty cell of both the evolution of the heat flux and head tempera- ture. This simulation appears to be very sensitive to the pre- cise value of the heat capacity of the head C

H

= m

H

c

H

, where

m

H

and c

H

are the mass and the specific heat of the head. The pieces of the head are made of stainless steel and copper, the specific heats of which at 35 K being close. For simplifica- tion, we will take that of copper

共cH

= 42.6 J/ kg K

关17兴兲. A

good agreement between the calculation and the experiment

共Fig. 7兲

is found for m

H

= 6 g. Taking into account that the copper part weight is 2.4 g and the stainless-steel pieces are about of the same size, this m

H

value is quite reasonable.

D. Some analytical results 1. Small time asymptotics

No heat exchange of the fluid with the tube walls is as- sumed in this section in order to obtain analytical results. At small times, all fluid parameters can be approximated by their values at the initial temperature, i.e., that of the base T

₀

. The temperature profile in the fluid is sketched in Fig.

1共c兲

. It shows three regions: hot and cold diffusion boundary layers

共HBL and CBL兲

that form near the extremities of the cell and a nearly isothermal bulk. The thickness of both HBL and CBL can be estimated with the expression ␦

⬃冑

D

F

t, where D

F

is the fluid thermal diffusivity. This allows the main pa- rameter, i.e., the transferred power to be found in the form

P

B⬇

S

F

k

F共Tbk

− T

B兲/

␦ ^,

共23兲

where T

bk

means the temperature in the bulk of the fluid and k

F

is the fluid thermal conductivity. S

F共=

␲ ^R

i

2兲

is the fluid cross-section area. Unless stated otherwise, T

B

= T

₀

and is constant. Similar to Eq.

共23兲, the transfer to the fluid power

is

P

H⬇

S

F

k

F共TH

− T

bk兲/

␦ ^.

共24兲

The bulk of the fluid remains isothermal, which means that

⌬Tbk

= 0. By making use of Eqs.

共8兲

and

共12兲

one obtains

0 0.2 0.4 0.6 0.8 1

0 1 2 3 4 5 6 7

0 100 200 300 400 500

P

B

(mW ) T

H

-T

0

(K )

t (s)

FIG. 7. Empty pipe: comparison between the simulation共lines兲 and the experiments 共characters兲 for 7.5 mW injected power. T₀

= 33 K. Black line and dots: transmitted powerP_B. Gray line and squares: head temperatureT_H.