Slip velocity ratios in an air-water system under steady-state and transient conditions

Chemical Engineering Science, 1967, Vol. 22, pp. 661668.

Pergamon Press Ltd., Oxford.

Printed in Great Britain.

Slip velocity ratios in an air-water system under steady-state and transient conditions G. P. NASSOS~and S. G. BANKOFF Chemical Engineering Department,

Northwestern

University, Evanston, Illinois

(Received 25 August 1966; accepted 4 October 1966)

Abstract-A study was made of slip velocity ratios in air-water flow in a 22 in. vertical pipe under steady-state

periodic conditions.

The steadv-state data for V. vs. Vm fell in a straight line, as suggested by Neal and by Zuber and Findlay. Thk slope correspoids to Co=l.l, which; in conjunction with the rounded void fraction profiles, would indicate very flat velocity profiles. It was found that the time-averagedperiodic V, vs. Vm data also fell on a straight line, of somewhat steeper slope (CO= 1*2), possibly due to a secondorder approximation error.

1. INTRODUCTION PRESFJNT models for the response of a two-phase flow system to a disturbance assume that the instantaneous cross-sectional average slip velocity ratio can be determined from steady-state correlations. The advent of an improved local gas concentration probe [6, 71 has made it possible to obtain information concerning the validity of this assumption. The steady-state data are also of interest in connection with some recent theories of the structure of gas-liquid flow.

2. REVIEWOF LITERATURE NEAL [l] derived expressions for the slip velocity ratio in the bubble and slug flow regimes, noting that slip in two-phase (gas-liquid) flow is a result of the difference in the gravitational forces acting upon the phases, as well as the existence of radial gradients of void fraction and stream velocity. The model introduced two flow parameters: (1) a measure of the resistance to gas motion with respect to the liquid, and (2) a correlation coefficient between the void fraction and the mixture (stream) velocity.

The rise velocity (with respect to the surrounding liquid) of slugs or bubbles large enough to that the tube walls influence their shape was given by 7 Present address: International

where C is a similarity parameter and D is the tube diameter. When the liquid flow rate is not zero, a second component for the gas velocity is introduced, due to the gas being carried along the channel by the surrounding liquid. In fully-developed flow this component is generally greater than the crosssectional average stream velocity because the bubble concentration and the stream velocity are both maximum at the center of the channel. A correlation coefficient may be defined by

COP





cv>

where a is the local void fraction, stream velocity defined by v=av,+(l-a)v,

(1) u is the local

(2)

and the triangular brackets denote the crosssectional averaging operator. In Eq. (2) ug and ur are the local gas and liquid velocities, respectively. The average gas, liquid and mixture velocities are then defined by

Minerals and Chemical Corporation,

661

Libertyville, Illinois.

G. P. NASSOSand S. G.

=-

v,&!L A

An alternative approach (3), which can be employed if the distributional parameter, CO, and the crosssectional average drift velocity, V,,, are both independent of V, for a given flow regime, would be to plot V, vs. V,. If the result is a straight line, the slope and intercept could be identified with C,, and V,,, respectively. It was suggested that a changing slope may be indicative of changing profiles or developing flow. The development of a reliable local gas concentration probe made possible an examination of this postulate, although it should be noted that velocity profiles were not measured. Both Neal and Zuber and Findlay suggested



Q1

v,LJ A=

<(1 -a)o>

and

v,,gQ&

=
A

>

Addition of the two components results in a cross-sectional average gas velocity with respect to the tube wall given by ,,=,,,+c\igD(~)

(3)

An expression similar to Eq. (3) was earlier obtained by GRIFFITHand WALLIS [3] in an analysis of two-phase slug flow. Since flat profiles were obtained, the value of C, in this case was equal to unity. Using the relation

v,=



V,+(l-

BANKOFF

)V,

for the slug flow regime; however, a value of 0.35 was set for C by Zuber and Findlay, while Neal used C= 0.60 to fit data from both the slug flow and bubbly flow regimes. A value of C=O.35 was also used by GRIFFITH [5] for the bubble rise velocity. For the churn-turbulent bubble regime, Zuber and Findlay recommended the following expression :

and Eq. (3), the following expression for the slip velocity ratio was derived : 3. The second factor on the right-hand side of Eq. (4) is the contribution to slip due to the buoyant forces. If this is neglected, Eq. (4) is then identical to that derived earlier (2) with Co = 1/K. A similar expression was later derived by ZUBER and FINDLAY[3], who also extended the analysis to other flow regimes. A more general expression for the second term on the right in Eq. (3) is V, B , defined as the cross-sectional average drift velocity. Expressions for V, were given both for the slug flow and the bubbly churnturbulent regimes. Values for C, can be evaluated from Eq. (1) by integrating the concentration and velocity profiles. Usually, however, these are not both available.

EXPERIMENTAL APPARATUS

The data were obtained from both a forcedcirculation and a natural-circulation loop. The forced-circulation air-water loop, shown schematically in Fig. 1, consisted of a test section with two air-water mixers, followed by a separator and downcomer, all of clear plastic, together with a 100 gal/min pump. The primary air-water mixer produced a steady air-water flow in the test section. Void perturbations were introduced by means of a secondary mixer constructed from drilled Plexiglas discs. Details are given in [7]. The test section was a 22 in. pipe, 84 ft in length, with probe and pressure taps at twelve different locations, as shown in Fig. 1.

662

Slip velocity ratios in an air-water

I.

system under steady-state

AIR WATER SEPARATOR

2. PRESSURE

DROP TAP

6.

AIR WATER MIXER

7.

PERTURBATION

3. PROBE TAP

6. STEADY

4. TEST SECTION

9.

STATE

AIR AIR

SUPPLY SUPPLY

WATER ORIFICE

10. WATER CONTROL VALVE

5. DOWNCOMER

FIG. 1. Forced-circulation

Natural-circulation was achieved by bypassing the 100 gal/min pump. Water coolers were unnecessary in this system, since the water temperature remained essentially constant.

4.

and transient conditions

flow system.

cross-sectional average void fraction in terms of the manometer reading, h:
PROCEDURE

The cross-sectional average void fraction was determined by the pressure drop technique. The total pressure drop in a vertical column of flowing air-water mixture is equal to the sum of a hydrostatic, an acceleration, and a frictional term. At low liquid velocities the momentum and frictional terms can be neglected. The neglected terms contribute oppositely to the hydrostatic term, which means that void fraction values calculated from the pressure drop wilI be slightly lower than the true values. Making use of the fact that p,> >ps, elementary analysis leads to an expression for the

=(P-PJWP&

(7)

An approximate correction for the frictional and momentum pressure drop can be made by noting the pressure drop when the liquid phase is flowing alone. If the frictional pressure drop is not neglected, Eq. (7) becomes = KP-PJd

+AP,lIp,sL

(8)

The frictional pressure drop is related to the liquid velocity by the Fanning equation AP,=4f LVf/2gD

(9)

where f is the Fanning friction factor. For smooth tubes it is given as a function of Reynolds number in the following equation :

663

G. P. N~ssos

and S. G. BANKOFF

f=O*046(NRJ-0~2

5. RESULTS AND DISCUSSION

Using these relations, it is straightforward to derive the more accurate equation:

==?h+ (1- )lS8 hl [

For fully-developed flow in both the slug and bubbly flow regimes, the mixture velocity and void fraction usually increase smoothly from the wall inwards, with no inflection points in the profile. It is convenient, therefore, to approximate these distributions by radial power laws:

1 (10)

For the calculation of the cross-sectional average void fraction, standard 60 in. manometers filled with a fluid of 1.25 sp. gr. were used. Four pressure taps and two manometers were used to assure the reliability of the data taken. If the data from the two manometer systems agreed, this was a good indication that no air was trapped in either system. Local values of the void fraction were determined with an electrical resistivity probe which consisted of an insulated needle with an exposed tip pointing into the flow [6, 71. When a battery and a resistor in series with the probe were connected to ground, a change in voltage was observed, depending upon whether water or air was in contact with the probe. Both air and water flow rates were determined from pressure drop readings across an orifice plate.

+=1-+/m me

(11)

u-a w-l-r*‘/” a,--cr,

(12)

where 0 I m, n s 1, and the subscripts c and w refer to values evaluated at the pipe centerline and wall. Co can be expressed in terms of m, n, and 01, by substituting Eqs. (11) and (12) into Eq. (1). In these tests u ,=O, which leads to (13)

FORCED CIRCULATION NATURAL

Od

CIRCULATION

0

0

0

0/

/

0

0 0

7

/

/

P

4

SLOPE

@

SLOPE =

8

1

0

:

v

0

/ 0”

1

2

3

4

50

1

2

vm (f t/red Fro. 2.

Gas

velocity as a function of mixture velocity. (a) Steady-state data. (b) Time-averaged data.

664

3

4

5

Slip velocity ratios in an air-water

system under steady-state

.oe

.lP

84

86

.10

and transient conditions

qqq 80

0

0

.ol

.lO

.m

.lL

86

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.06

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mm

RUN V-1

91

a .03

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oo

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x4

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RUN v-e
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80

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* 0.066

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mi 20 a

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RUN

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RUN C-6

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FIG.3. Void fraction profiles.

665

.1-p*

i

G. P. N~ssos and S. G. BANKOFF

Thus, as either m or n becomes small, corresponding to flat velocity or void fraction distributions, C,+ 1. For linear velocity and void fraction profiles, C, = 1.5. It can be seen from Fig. (2) that the steady-state data for VBwhen plotted against V,,,, do, in fact, show a linear relationship which, furthermore, appears to be independent of the type of circulation system. The latter point is to be expected, since the local flow distribution should not depend upon the means for imposing the overall pressure difference upon the channel. The criterion used to fit Eq. (12) to the local void fraction data was that the average void fraction calculated from Eq. (12) be equal to the actual average void fraction. This was done by finding the centerline void fraction in terms of the average void fraction. Solving Eq. (12) for < LY > and then rearranging, one obtains ccc= (2n+l)

(14)

when CL,,, is taken as zero (experimental data verify this). Using Eq. (14) and the measured values of LX,and , the void fraction profile data were fitted reasonably well, as seen in Fig. 3. The smallest value of II used to fit the data was ), which corresponds to m = &, based upon Eq. (13), for C,, = 1.1. This would indicate very flat velocity profiles, together with rounded void fraction profiles, which seems reasonable in the churn-turbulent flow regime, or in the transition to slug flow. The intercept in Fig. 2 indicates a value of O-7 ft/sec for V,. Good agreement is shown by evaluating V,, from Eq. (6), which predicts a value of 0.8 ft/sec. If one were to use the expression for slug flow regime, Eq. (5), a value of 0.95 ft/sec would be obtained, when C is taken as O-35. In fact with a pipe size of 2 in., Eqs. (5) and (6) predict the same value for V,, so that with circular channels between 1 and 3 in., either equation is in acceptable agreement with the data. In addition to these data, some measurements were made of and V,, when the flow was subjected to a steady square-wave perturbation of air flow, introduced through the secondary mixer. Time-averaged values of Vse, , and V,,, were

thus obtained from flow and pressure measurements. In general, the calculated value of the timeaverage gas velocity, V*A& *- is expected to be somewhat larger than the true value

( >

V VA sg



as discussed in the Appendix. That this indeed is the case is seen in Fig. 2(b), where the plot has a slightly higher slope and intercept, corresponding to C, = l-2 and Vd = 1*Oft/sec. It is of considerable interest that the data still plot as a straight line, showing that the steady-state theory advanced by Neal and by Zuber can be extended to periodic flows. Acknowledgment-This work was sponsored by Argonne National Laboratory and the Associated Midwest Universities, under the auspices of the Atomic Energy Commission. The facilities were under the general supervision of M. Petrick.

NOTATION

A C CO D

f g h h K L N R@ m,

n

P AP, AP,,

Q r* t vd

666

cross-sectional area similarity for slug flow distributional parameter, Eq. (1) diameter Fanning friction factor gravitational acceleration manometer reading for two-phase mixture manometer reading for liquid system distributional parameter for zero local slip distance between pressure taps Reynolds number power law exponents pressure frictional pressure drop, two-phase mixture frictional pressure drop, liquid system only volumetric flow rate normalized radial position variable time weighted average drift velocity

Slip velocity ratios in an air-water

v, &A,

system under steady-state

gas velocity, cross-sectional average

Subscripts

c g I m

refers refers refers refers refers 0 w refers

superficial gas velocity u local velocity vd local drift velocity

VW

Greek

c( local void fraction p viscosity Il”9 vl* normalized superficial gas velocity and void fraction fluctuations, Eq. (5A) p density C/J slip velocity ratio

and transient conditions

to to to to to to

centerline gas phase liquid phase mixture steady-state conditions wall

Superscripts ,

< >

time fluctuating quantity time average quantity cross-sectional average quantity

REFERENCES [l]

NEAL

L. G., An Analysis of Slip in Gas-Liquid Flow Applicable to the Bubble and Slug Flow Regimes, KR-62 (1963).

[2] BANKOFFS. G., J. Heat Transfer 1960 82 265. [3] ZUBER N. and FINDLAY J. A., The Effects of Non-Uniform Flow and Concentration Distributions and the Effect of the Local Relative Velocity on the Average Volumetric Concentration in Two-Phase Flow, GEAP-4595 (1964). [4] GRIFFITHP. and WALLIS G. B., J. Heat Transfer 1961 83 307 [SJ GRIFFITHP., J. Heat Transfer 1963 86 327 [6] NASKIS G. P., Development of an Electrical Resistivity Probe for Void-Fraction Measurements in Air-Water ANL-6738 (1963). [7] NASSOS G. P., Ph.D. Thesis, Northwestern University, Evanston, Ill. (1965); also ANL-7053 (1965).

R&m&--On a btudie les rapports de la vitesse de glissement dam un courant air-eau, dans une conduite verticale de 70 mm en &at stable et a p&iode stable. Les don&s de Mat stable pour V,:Vm suivaient une ligne droite, comme suggere par Neal et par Zuber et Findlay. La pente correspond a CO= 1,l ce qui, en conjonction avec les profils arrondis de la fraction vide, indiquerait des profils tres plats de la v&cite. On a trouve que les don&es Vg: V,,, periodiques dont on a calcule la moyenne en fonction du temps suivaient egaiement une ligne droite, dans une pente plus raide (CO= 1,2), due peut-etre il une erreur d’approximation de second ordre. Zusammenfassung-Das Gleitgeschwindigkeitsverhahnis in Luft-Wasser Striimungen in einem 2$ Zoll Vertikalrohr unter stationaren periodischen Bedinguugen wurde studiert. Die stationlren werte fur Vo:Vm fielen auf eine Gerade, in Ubereinstimmung mit friiheren Arbeiten von Neal und von Zuber und Findlay. Die Neigung entspricht CO= 1.l was, in Verbindung mit den abgerundeten Leerraumfraktionsprofilen, auf sehr flache Geschwindigkeitsprofi!e hinzudeuten scheint. Es wurde festgestellt, dam die zeitlich gemittelten period&hen V; Vm Werte ebenfalls auf eine Gerade zu liegen kamen, die allerdings, wahrscheinlich infolge eines Nlherungsfehlers zweiter Ordnung, eine stlrkere Neigung (CO= l-2) aufwies.

667

Flow,

0. P. NASSOS and S. G. BANKOFF APPENDIX

If the two-phase flow is periodically perturbed around a steady-state condition, the time-average gas velocity is

where +

(14

## and

In general, however, it is rather difikult to measure this ratio as a function of time. If, on the other hand, a pseudo-average gas velocity is defined by V*A L #-

@A)

this represents a quantity readily amenable to measurement. The approximation error is given by

%?=-

’

are the normalized superficial gas velocity and cross-sectional average void fraction fluctuations, respectively. The sign of the error therefore depends upon the phase difference between q. and (n, - q.), and may, in general, be either positive or negative. To investigate this question more fully, we note that to the first order,

(3A) ?A

Assuming that

V, -V, V,

Nvn‘+

V;

v,

-

=.L+--DV’ <8>

use of the binomial theorem gives sg7

_

(Llyl_~‘+(g_

...)

V#

so that



=

K,’ =

Ysg 3

”

+higher terms (4A)

This can be written, neglecting the third-order correlations, as

This is positive if, as seems reasonable in most cases, the phase lag between the gas velocity and void fraction fluctuations does not exceed 90”.

668