Stratified two-phase superfluid helium flow: II

Cryogenics 31(1997) 739-744 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 001 l-2275/97/$17.00

PII: SOOll-2275(97)00091-X ELSEVIER

Stratified two-phase flow: II* L. Grimaud,

A. Gauthier,

CEA-Grenoble/DRFMC/SBT, Cedex 09, France Received 28 November

superfluid helium

B. Rousset and J.M. Delhaye

CEA-Grenoble/DTP/STI,

17 rue des Martyrs,

7996; revised 7 January

F-38054

7997

This paper presents the physical basis of a one-dimensional two-fluid stratified flow model, adapted for adiabatic or diabatic superfluid helium flow. Closure relationships for wall shear stresses and inter-facial momentum transfer are used to predict the total pressure losses and the void fraction along the line. The numerical resolution is based on a finite difference scheme. The influence of each term in the momentum equations is discussed. Finally, the model has been verified through comparisons with available experimental data’. 0 1997 Elsevier Science Ltd. Keywords: two-phase void fraction

flow; superfluid

helium;

stratified

Nomenclature

Sk

Ak

Vi VIC V,,

D

Dh ? g h L in

M pik

Pk Rek si

Flow area of phase k Tube diameter Helix diameter Hydraulic diameter for phase k Wall friction coefficient of phase k Gravitational acceleration Liquid height Latent heat of vaporization Mass of vapour generated by unit time and unit length of the pipe Total mass flow rate Interfacial pressure Pressure in phase k Reynolds number of phase k Perimeter of the interface

A stratified two-phase flow pattern is characterized by a liquid moving along the bottom of a pipe and a gas moving above it. This flow pattern exists for low gas and liquid flow rates. When the gas velocity is sufficiently low, the interface appears flat and corresponds to a stratified smooth flow. However, when the gas velocity is increased waves are formed at the interface and the flow becomes stratified wavy how. At high gas velocities, atomization occurs at the crest of the waves. The basic equations of two-phase flow can be found in Delhaye’. The application of these equations to a stratified

at the DESY Workshop

model; pressure

drop;

Perimeter of phase k Interfacial velocity Axial mean velocity of phase k Superficial velocity of phase k

Greek letters Void fraction Mean void fraction along the line Tube inclination Quality Mean quality factor along the line Dynamic viscosity of phase k Density of phase k Interfacial shear stress of phase k Wail shear stress of phase k Heat flux densitv

a ; X

x pk

Pk rik rk *

flow of superfluid helium with heat input will be discussed. The equations, both written in the liquid and gas phases, are based on mass, linear momentum and energy balances. The resolution of the system of equations needs closure relationships to express the interfacial and wall shear stresses. It is then possible to calculate the pressure drop and the void fraction knowing the flow geometry, the total mass flow rate, the pressure of the system and the outlet quality.

Flow * Extended manuscript of a presentation on He II Two-Phase Flow

flow

pattern map for superfluid helium

In gas-liquid flow, the two phases can be distributed in the conduit in a wide variety of ways. In the case of horizontal

Cryogenics

1997 Volume

37, Number

11

739

Stratified

two-phase

superfluid

helium flow: L. Grimaud

or slightly inclined pipes, four main flow patterns may arise: stratified, intermittent, annular and bubbly. In the stratified flow pattern, the liquid flows on the bottom of the pipe and the gas over it; both phases are continuous in the axial direction and two sub-patterns may be defined: stratified smooth and stratified wavy. In the intermittent flow pattern, the gas is distributed non-continuously in the axial direction; slugs of liquid are separated by gas plugs and overlap a liquid layer flowing along the bottom of the pipe. In the annular flow pattern, the liquid forms a continuous annular film on the pipe wall; this liquid film surrounds a core of high velocity gas which usually contains entrained drops. In the bubbly flow pattern, the gas phase is distributed as bubbles in an axially continuous liquid phase. The concentration of bubbles is higher near the top of the pipe, but as the liquid rate increases, the bubbles are dispersed more uniformly. Taitel and Dukler3 proposed different physical mechanisms to predict the transitions between the flow patterns. The application of the method of Taitel and Dukler for superfluid helium in a 86 m long, 40 mm inner diameter, 1.4% slope tube is shown in Figure 1. The objective of this map is to determine the flow pattern which could be expected in an experiment performed in our laboratory. It is important to note that these flow pattern transitions do not account for the mass transfer between the phases. However, Taitel and Dukler“ have shown that the effect of boiling on flow in slightly inclined pipes is not drastic. Thus, we expect that these transitions are still valid for superfluid helium in the case of evaporation.

Stratified system

flow geometry

flow equilibrium

r

2

(5)

22-l )]

flow. Geometric

Formulation

2$ - 1 )

S, 4 D(ii-arccos(2~-

,j)

,AD$-(2$1)

(3)

of the balance equations

We consider a stratified flow pattern of two immiscible fluids in a inclined conduit with mass transfer between phases. A flat interface is assumed and the flow is steady under thermodynamic equilibrium all along the line. Furthermore, the longitudinal conduction in the superfluid helium is neglected. The one-dimensional mass, linear momentum and energy equations averaged over the layer thicknesses are given by the following expressions. Mass balance equation

300 3 k

10

2 i

1

E -z 20.1

Stratified

J(

(

3

Figure 2

l-

S, LL Darccos

where + corresponds phase. Linear momentum 0

0.2

0.4

0.6

0.8

1

~ (~kAk~)

740

Flow pattern map (D = 40 mm, slope = - 0.014,

Cryogenics

1997 Volume

37, Number

11

T =

-

$ (A$,)

to the gas phase and - to the liquid

equation

= f r;ZVi- ~kSk - TikSB + pkAkgW

Quality Figure 1 1.9 K)

1 D

for an evaporating

Figure 2 shows an evaporating stratified relationships give the following:

a, ;;i

et al

+ Pi, 2

(7)

Stratified where + corresponds phase.

to the gas phase and - to the liquid

(8)

!P

=

(9)

L

These equations are solved using a finite difference scheme with respect to the position along the line.

Assumptions equations

superfluid

Interfacial

helium

flow: L. Grimaud

et al.

shear stress

The interfacial shear stress T, is expressed in terms of the relative velocity between the phases and an appropriate friction factor J:

Energy equation

)fi

two-phase

for the resolution

Ti

P,(Vg- VI)’

=,i

(18)

2

The coefficient must take into account the presence of waves on the interface which increases the rate of momentum transfer across the interface.

The choice of the friction factors as closure laws Relationships expressing the wall and inter-facial are needed to close the system.

of the

stresses

Wall friction factor The pressure dent:

l

for each phase P, is only gravity depen-

4

Pk= f

ki 0

LPik

+

PkgcosLXhl - Y)l dA

(10)

The longitudinal pressure losses in the liquid and gas phases are equal. The interfacial velocity is taken to be equal to the axial mean velocity of the liquid. Moreover, there is no jump of pressure across the interface and the interfacial shear stresses of the liquid and the gas are equal and opposite:

l

l

The correlations used to describe the wall friction factor are correlations of single phase flow. As the model will be compared to experiments in coiled tubes, correlations for coiled tubes are needed. Several possibilities have been considered and are listed below. l

Srinivasan

et al.5

(19)

Re 5 7 x lo6 and 0.001 5 $5 Vi = V,

(11) l

T,g =

-

Ti,

=

0.135

he,

Anglesea

et al.‘:

(12)

Tl

Pi, = Pi,

(20)

(13)

Shear stress modelling

in stratified

flows 5

2 x 104 5 Re 5 2 x lo5 and 0.005 5 $

5 0.065 hel

Wall shear stress The wall shear stresses 7k are expressed in terms of the phase velocity and appropriate friction factors fk:

l

Ruffel?

o~275Re4~4+ 0.00375

121)

Pk%

?k =fk

~

(14)

2

5 x lo3 5 Re 5 6 x lo5 and 0.005 5 F The friction factorsfk are functions of the Reynolds number of the phases based on the hydraulic diameters defined by:

44 -_=

Dhg b (S,+$) D

hl

CUTD2 (S,+SJ

4A,_(l -ahD*

A

s,

4

(15)

(16)

Interfacial

=

kA-

PkvkDhk pk

(17)

friction factor

The correlations following. l

Re

5 0.021 he,

most often used were established

by the

Taitel and Duklel”:

(22)

fi =fg

Cryogenics

1997 Volume

37, Number

11

741

Stratified

two-phase

for separated . Rosant’:

superfluid

helium flow: L. Grimaud

et al. 550

flow.

T

- Taitel & Dukler

_

k, = 619( 1 - cz) :A(

1 ~ =-210g Jx

V, - V,)’

-++

t -50 i!l

(25)

PO J Pg

where pOrepresents the gas density at 300 K under for stratified air-water flow in tubes.

vgs (m/s)

(27)

friction factors and of Andritsos interfacial friction factor.

1 bar,

Comparison of the different total pressure losses

l

600

- I”

- Srinivasan

t -

Wall friction terms

(28)

Interfacial

friction terms (29)

7iksi

Gravity terms

A&g

K)

l

equation

terms of the total pressure losses

7&Z l

for the

terms of the

different terms of the linear momentum (Equation (7)) are now compared to each other.

l

1000

and Hanratty

The

The different

Using the interfacial friction factor of Andritsos and Hanratty’O, we compare the different wall friction factors with the experiments in the case where maximum adiabatic pressure losses were measured. Using the wall friction factors of Ruffel’, we compare the different interfacial friction factors with the experiments in the case where maximum diabatic pressure losses were measured. As shown in Figures 3 and 4, the best agreement is obtained by using the closure laws of Ruffel for the wall

M(6.3 g/s)-T(1.81

12

8

Figure4 Pressure losses of a diabatic two-phase flow flow with the correlation of RuffeP (M = 6.2 g s-‘, T= 1.81 K)

The choice of friction factors

800

4

(26)

In our case, we used for pOthe density of helium at 1 bar and 300 K.

l

Rosant

2 k 250 %

(24)

Vg,&X=.fg

v,,, LL5

-

(23)

for stratified water-air flow in tubes at high pressure with a void fraction (Y 2 0.9. l Andritsos and Hanratty”:

vg,5

M(6.2 g/s)-T( 1.8 1 K)

l m

(30)

sir@

Acceleration

terms

Anglesea Ruffel

(31)

‘;;i‘ l

e-, 400 %

Terms due to the variation

of the liquid level

ah

Pd& CO@ z

200

l

0

12

8

4

Momentum

(32) transfer due to evaporation

f riz(Vk - Vi)

term (33)

-200 l

vgs (m/s) Figure 3 Pressure losses of an adiabatic two-phase flow with the correlation of Andritsos and Hanratty’O (M = 6.3 g s’, T = 1.81 K)

742

Cryogenics

1997 Volume

37, Number

11

Total pressure loss terms

A!!? k ax

(34)

Stratified I

I

-

two-phase

superfluid

helium

flow: L. Grimaud

et al.

_ Wall gas friction I Interfacial gas friction -

-Gas gravity

1000 T

-Gas acceleration

100 AP theoretical

10 I

1 I

’

, ’/

/’

I

Figure 7 Comparison between measured and calculated pressure losses in the adiabatic case

+-_ _w., 8

4

0

1000 (Pa)

12

vgs Ws) Figure 5 Pressure losses in the gas phase (M = 6.2 g s’, 1.81 K, D = 4 cm, slope = - 1.4%, length = 86.4 m)

The comparison

between

the different

T =

terms

Figures 5 and 6 present the different terms of the pressure losses calculated by the model in the gas phase or in the liquid phase in the case of a diabatic flow. In the gas phase, the main terms are the gas wall friction and the interfacial friction. The acceleration and the the momentum transfer due to evaporation are very similar but their importance is quite small. The gravity term is constant and small since the value of the gas density is very low. For the same reason, the level term is completely negligible and does not appear in Figure 5. _

I

-

Comparison experiments

of the model with the

Wall liquid friction

I -Interfacial -

In the liquid phase, the main terms are the liquid wall friction, the interfacial friction and the gravity of the liquid. The values for these different terms can be very high: between 1000 and 30 000 Pa. The other terms, level and acceleration, are negligible compared to the others. When the superficial gas velocity is low (V,, -=c3 m s-l), the two important terms are the wall friction and the gravity and as they compensate themselves, the total pressure loss is small and almost equal to the interfacial friction term. When the superficial gas velocity is high (V,, > 8 m SC’), the most important terms are the wall and interfacial terms.

-Liquid

Our measurements” are compared to the results given by the model. In the adiabatic case, Figures 7 and 8 show a comparison between the calculated and measured pressure

liquid friction

gravity

Liquid acceleration -Variation

A

of the liquid level

-Total

1.oo

liquid pressure loss , ,/ II

M( 1.5 g/s)-T(1.8

-oM(1.6

g/s)-T(1.93

K) K)

_- l M(3.4 g/s)-T( 1.97 K) A M(3.3 g/s)-T(1.8 K) 0.98 -- n M(6.2 g/s)-T( 1.97 K) __l M(6.3 g/s)-T(1.81

0.92 8

4

12

0.92

vgs (m/s) Figure 6 Pressure losses in the liquid phase (M = 6.2 g s-l, T= 1.81 K, D = 4 cm, slope = - 1.4%, length = 86.4 m)

0.94

0.96 c1 theoretical

Figure8 Comparison between fraction in the adiabatic case

Cryogenics

measured

1997 Volume

1 .oo

0.98 and calculated

37, Number

11

void

743

Stratified

two-phase

superfluid

helium

flow: L. Grimaud

et al. other large projects. This model can be used for the design and sizing of devices which need to be cooled with forced superfluid helium two-phase flow.

Acknowledgements The authors wish to thank the LHC cryogenic team at CERN for their support and interest in this study.

References 1.

w M(6.2 g/s)-T( 1.97 K) 2.

A M(6.2 g/s)-T( 1.9 K)

3.

1000

100 AP theoretical

(Pa)

4.

Figure 9 Comparison between measured and calculated pressure losses in the diabatic case 5.

losses and void fractions. In the diabatic case, the comparison is only made in terms of pressure losses in Figure 9. In every case, the agreement between experimental and calculated points is very good, except for the low values of the quality because of the measurement inaccuracy.

6.

7.

8.

Conclusion 9.

The stratified two-phase flow model presented in this paper gives results in close agreement with the experiments by using correlations established for other fluids and other temperature and pressure ranges. Experiments made on a long line qualify this model for

744

Cryogenics

1997 Volume

37, Number

11

IO.

1I.

Rousset, B., Gauthier, A., Grimaud, L., Bezaguet, A. and Van Weelderen, R., Stratified two-phase superfluid helium flow. Proc. 16th Znt. Cryogenic Engineering Conj, Kitakyushu, Japan, 1996. Delhaye, J.M., Basic equations for two-phase flow modelling. TwoPhase Flow and Heat Transfer in the Power and Process Industries, ed. A.E. Bergles, J.G. Collier, J.M. Delhaye, G.F. Hewitt and F. Mayinger. Hemisphere, 1981, pp. 40-97. Taitel, Y. and Dukler, A.E., A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AZChE J., 1976, 22, 47-55. Taitel, Y. and Dukler, A.E., Flow pattern transitions in gas-liquid systems: measurements and modelling. Multiphase Science and Technology, Vol. 2, ed. G.F. Hewitt, J.M. Delhaye and N. Zuber. Hemisphere, 1986, pp. l-94. Brauner, N. and Maron, D.M., Role of interfacial shear modelling in predicting stability of stratified two-phase flow, Encyclopedia of Fluid Mechanics, Vol. 4, 1995. Srinivasan, P.S., Nandapurkar, S.S. and Holland, F.A., Pressure drop and heat transfer in coils. Chemical Engineering, 1968, 218, CE113-CE119. Anglesea, W.T., Chambers, D.J.B. and Jeffrey, R.C., Measurement of water-steam pressure drop in helical coils at 179 bars. Multiphase Flow Systems Meeting, Glasgow, 1974. Ruffel, A.E., The application of heat and pressure drop data to the design of helical coil once-through boilers. Multiphase Flow Systems Meeting, Glasgow, 1974. Rosant, J.M., Ecoulements diphasiques liquide-gaz en conduite circulaire, Ctude de la configuration stratilike au voisinage de l’horizontale, Thkse de Doctorat 6s Sciences, ENSM Nantes, 1983. Andritsos, N. and Hanratty, T.J., Influence of interfacial waves in stratified gas-liquid flows. AZChE J., 1987, 33, 444-454. Rousset, B., Gauthier, A. and Grimaud, L., Stratified two-phase superfluid helium flow: I. Cryogenics, in press.