Two-phase flow in a vertical tube

ChemicalEngineer&g Science, 1963, Vat 18, pp. 4146. Pergamon Press Ltd., Oxford. Printed in Great Britain.

Two-phase flow in a vertical tube E. Q. BASHFORTH,J. B. P. FRASER,I-L P. HUTCHSON and R. M. NEDDERMAN Department of Chemical Engineering, Pembroke Street, Cambridge (Received 3 August 1962) I

I

Abstract-In a certain range of gas velocities it is possible to hold a quantity of liquid as a fihn on the walls of a vertical tube, there being no net movement of the liquid. Pressure drop and hold-up measurements have been made in this regime using air and water. A theoretical relationship between the two has been-derived and has been checked experimentally.

INTRODUCTION

THE boundaries between the various regimes of co-current two-phase flow have been extensively studied, see, for example, G~VIER et al. [l]. Counter-current flow has received less attention. NICKLIN and DAVIDSON [2] have studied the flooding of annular counter-current flow. No data, however, are available for conditions in the flooded regime. This is likely to become important if the suggestion of KAFAROV[3], that distillation columns should be run in the flooded regime, the so-called “emulsion region”, is taken up. The condition of no net liquid flow, i.e. the critical condition between co-current and counter-current flow is of particular interest. If a quantity of liquid is-injected into a vertical tube, up which air is being blown, three different modes of behaviour can be observed, depending on the air velocity: (a) At low air velocities the liquid falls down the tube under gravity. (b) At high air velocities the liquid is expelled from the top of the tube. (c) Over a considerable range of air velocities the liquid remains in the form of a film of clearly defined length on the tube wall. In a perfectly smooth tube there would be nothing to fix the position of this Iilrn but it is found in practice that any irregularity in the tube is sufficient to locate the film. This regime is the limiting case of flooding when the liquid flow rate is zero. At air velocities only slightly above the. minimum to maintain this regime, the flow in the film is very agitated and the system looks much like that called

“churn flow”t in co-current systems. As the air velocity is increased the tim becomes thinner and smoother. At the upper limit of the regime the tim has small ripples but none of the large waves that are found at the lower limit. It is much more like an annular flow, though “Helmholz waves” seem to be absent. This work is confined to a study of this limiting case of two-phase flow, it being considered more worthwhile to get extensive data for a limited regime than to produce overall correlations. Pressure drop and hold-up measurements have been obtained and the limits of stability of this regime have been determined. EXPERIMENTAL

The experiments were carried out in a perspex tube 6 ft long and 1 in. diameter open to atmosphere at the top. The air was supplied from a 9 p.s.i.g. main through a rotameter and a bell-shaped entry to the bottom of the tube. During a run the air flow rate was set at some predetermined value, and .a known volume of water was injected into the lower end of the tube from a burette, via a specially constructed entry. This is shown in Fig. 1. When the water was injected it settled down into a film of clearly defined length in about 5 set, the rest of the tube being virtually dry. After this the Ghu contracted by about O-2 cm/set as water was lost by entrainment. At the lowest air velocity at t We use the phrase “churn flow” with its wider meaning of very agitated flow and not its restricted meaning of tlow with bridging of the liquid. 41

E. Q. BASHFORTH, J. B. P. FRASIZ,H. P. HUTCHLWN and R. M. NEDDERMAN

-il

~

II

Iin

,

Scale

J

Inlet ports

i/i

/

“D Bras< washer Wooden

block

150

100

Film length,L,

i 10

cm

FIG. 2.

The slopes of these graphs were obtained by the method of least squares. The former gave H the hold-up per unit length of tube and the latter dAp/dL the rate of increase of pressure at the bottom of the tube due to increase in film length. By taking the slopes of these graphs, H and dAp/dL were not subject to any fixed errors such as non-uniformity of the ends of the ftlm, or entry and exit effects. These merely alter the intercept on the axis. FIG. 1.

which measurements were taken (30.8 ft/sec) there was slight loss of liquid from the bottom of the tube. Below this velocity, dumping suddenly became appreciable. We, therefore, deduced that 31 ft/sec is the lower limit of our regime. It will be seen later that the results obtained at 30.8 and 30.9 ft/sec do not fit our correlations whereas those obtained at 32 ft/sec and above do. The upper limit of the regime was an air velocity of 41 ft/sec. Above this the film moved continuously up the tube. Various volumes of water Q were injected into the tube and the resulting pressure drops Ap and Hm lengths L were measured. The pressure drop was measured between the entry device and the top of the tube. Both Q and Ap were plotted against L and the results were closely linear, see Figs. 2 and 3 for a typical run.

THEORY

The pressure drop and net force on the film per unit length can be expressed in terms of the mean shear stress of the air on the film of water (z,), of the water on the tube wall (tW) and the air on the dry parts of the tube wall (zo). Since increase in

Film length, L, FIG.

42

3.

cm

Two-phaseflow in a vertical tube. the length of the film is at the expense of decrease in the length of dry tube

where we have substituted for zh its value according to the well known Blasius Equation: rg = +pAUi x O-079 Re-‘14

where D is the diameter of the tube, and the net force on the liquid film per unit length is F = nD(+

zw)

provided the film thickness is small compared with the tube diameter. For zero net flow this gives rise to the condition: where pw is the density of water and g the acceleration due to gravity. FRIEDMANand MILLER [4] have analysed the. results of many experiments on films with ripples. They found that even when the ripples were large the mean film thicknesses were the same as those predicted assuming laminar flow in a smooth film. We have, therefore, related the pressure drop and hold-up measurements by an analysis assuming the film to be smooth and laminar and the velocity profile to be parabolic, as in Fig. 4, i.e. u = a + by + cy2 where u is the velocity of the water at a distance y from the wall and a, b and c are constants. Since there can be no slip at the wall a = 0, and hence the flow rate q is given by 2 3 4

where pA, Us and Re are the density, superficial velocity and Reynolds Number of the air. We have here an expression relating our pressure drop and hold-up measurements. We can, therefore, check our theory against experiment. Had it been possible to measure the pressure gradient the simpler expression -3aD’ pwHg=Fdz

dp

could have been used, where dp/dz is the pressure gradient within the wet part of the tube and must not be confused with dAp/dL which is the change in pressure drop due to a change in film length. The equation zF = -22, is of interest. It reflects the circulation of the water in the film, upwards near the air/water interface and downwards near the tube wall. As a result of this, two-thirds of the fihn is supported by the air and one-third by the wall. The net force on the tube wall is downwards, i.e. against the air flow. The interfacial skin friction factor C,, is defined as

‘IF --- tp:‘Uf C,, IS plotted against

=!T+y

Re = PA”sD where m is the mean film thickness. But q = 0 therefore b = ( -2cm/3).

PA

in Fig. 5.

= p(b + 2cm) = -2pb

DISCUSSION

+-; where pw is the viscosity of the water. Hence pwgH = $

and hence from (1) 37rD2dAp pwgH =--

8

dL

Dz,

3nD + 4 pAUf x O-079 Re-‘l4

(2)

The values of H and dAp/dL obtained from the slopes of the graphs of the experimental results are presented in Table 1. H’, the “theoretical” hold-up and Cfp are calculated from equations (2) and (3) and are also presented. H is plotted against H’ in Fig. 6, and it can be seen that the agreement is good except for the two points obtained at velocities of 30.8 and 30.9 ft/sec, i.e. on the limit of the regime. The measured holdup is generally about 10 per cent below the theoretical. This difference is less than might be expected 43

E. Q. BASHFORTH, J. B. P. FRASER,H. P. HUTCHLWNand R. M. NEDDERMAN

I

0.6, Interface

E S E

cc 0.4

3 k. c

Ca

r,

C, 2 I0

t

0 0.5 0.3

/

”

;:y

i

* Y 0

0.1

0.2

0.3

Predicted

0.4

0.5

hold-up,

H’,

0.6

0~7

0.8

cm3/cm

FIG. 6. FIG. 4.

H = 1.02 x lo9 Re-“’

in view of the many assumptions made in the theory. The plot of C,, against Re using logarithmic coordinates (Fig. 5) is linear except again for the two points on the limit of the regime. The results are best correlated by the equation C, = 0.90 x lo’* Re-4’5

(4)

This equation must be considered to be completely empirical. The logarithmic plot of H against Re (Fig. 7) is also linear suggesting that

(5)

This too is empirical and it will be realized that both (4) and (5) cannot be exact if equation (2) is to hold. The figure of 31 ft/sec as the lower limit of the regime is consistent with the results of NICKLIN and DAVIDSON [2]. If their results for the critical air velocity to cause flooding are extrapolated to zero liquid flow rate, a value of 31 ft/sec is obtained. Our regime is, therefore, the limiting case of flooding, or churn, flow with zero net liquid flow. We, therefore, feel justified in presenting our holdup results as a continuation of NICIUIN and DAVIDSON’S as in Fig. 8. The value of 31 ft/sec is

0.1

u

‘f ;

0.0:

::

s .4

2

9

0.0;

:: ‘=

2 6 I

0.0;

Air

2.2

2.0

I.8 Reynolds FIQ.

number,

Re.

x IO4

Air Reynolds number, FIG.

5.

44

7.

Rc

x 104

Two-phase flow in a vertical tube Table 1.

not considered

us (ft/sec)

H (cm*/cm)

dAp/dL (cm HsO/cm)

30.8 30.9 32.3 32.3 34.1 36.3 37.4 37.4 38.7 41.2 41.2 41.2

0.499 0519 0462 0.456 0.393 0.360 0.331 O-327 0290 2.260 0272 0.269

0.106 0.101 0.0615 00607 00474 00445 0.0323 0.0390 00330 0.0263 0.0245 0.0282

to

depends

be

0.1237 ml180 @0678 00685 00507 0+428 00350 0.0358 00296 0.0227 0.0218 0.0240

17,000 17,008 17,900 17,908 18,980 19,800 20,700 20,700 21,400 22,700 22,700 22,700

Acknowledgements-The experimental work was carried out (by E.Q.B. and J.B.P.F.) as part of the requirements of the Chemical Engineering Tripos at Cambridge University.

4 g m P

u Y .?

CfF D F H H’

I.00 I

,: I

e” Re u, E c1A PW

1

0.96

PA

PW 7D

TF

/

I

IO

15

20

25

Superficial

air

velocity,

us,

5

30

35

40

45 TW

FIG. 8.

ft/Sec

AP

,

O-852 O-818 0518 0.512 0427 0.399 0.354 0362 0.320 0.276 0.263 0291

0.895 0.891 0903 0904 0.918 0.924 0.931 0931 0.939 0945 0943 0.944

a b1 Constants

Y

Exiropolotion

e

NOTATION

a fundamental

on

0.98

H'

ems/cm

constant. It the nature of the gas, the liquid and the tube. Our experiments were carried out in the tube originally used by NICKLIN and DAVIDSON, which may account in some part for the excellence of the agreement. doubtless

CfF

Re

Acceleration due to gravity Mean film thickness Pressure Volumetric flow rate of liquid per unit of circumference Velocity in the liquid Distance from the tube wall Distance along the tube Interfacial friction factor defined by equation (3) Diameter of tube Total force on a unit length of film Volume of liquid held up in unit length of tube Value of H predicted by equation (2) Length of flhn Volume of water injected into tube Air Reynolds Number Superficial air velocity Voidage fraction Viscosity of air Viscosity of water Density of air Density of water Mean shear stress exerted by the air on the dry parts of the tube Mean shear stress exerted by the air on the liquid film Mean shear stress exerted by the liquid on the tube Wall Difference in pressure between top and bottom of the tube

GOVIER,G. W., RADFO~D, B. A. and DUNN, J. S. C., Canad. J. Chem Engng. 1957 35 58. NICKLXN,D. and DAVIDSON,J. F. (In press). KAFAILOV,V. V., Int. Symp. Distill. Brighton p. 116. Inst. of Gem. Eng. London 1960. FRIEDMANS. J. and MILLER, C. O., Industr. Engng. Gem. 1941 33 885. 45

E. Q. BASHPORTH, J. P. B. FRASER,H. P. HUTCHI~~Nand R. M. NEDDERMAN R&nn~Il eat possible, avec certaines vitesses de gaz d’entrainer une quantitt de liquide comme un film sur les parois dun tube vertical sans avoir de mouvement net du liquide. Des mesures de la pcrte de charge et de la retention du liquide ont et& faites avec Pair et l’eau. Une relation theorique entre les deux a et& Ctablie et v&ifiee exp&imentalement. Znsanunenfaasung-In einem senkrecht stehenden Robr wird durch ein nach oben strijmendes Gas ein Fliissigkeitsfdm an der Rohrwand im Gleichgewicht gehalten. Es wird eine theoretische Beziehung zwischen Druckabfall des Gases und zuriickgehaltener Fltissigkeitsmenge abgeleitet und ftir das System Luft-Wasser mit experimentellen Daten verghchen.

46