nonstratified transitional boundaries in horizontal gas—liquid flows

Chemical

Engrneerrng

Science.

ooO9 2509/91 I3.00 t 0.00 0 1991 Pergamon Press plc

Vol. 46, No. I, pp. 1849-1859, 1991.

Printed in Great Britain

ANALYSIS OF STRATIFIED/NONSTRATIFIED TRANSITIONAL BOUNDARIES IN HORIZONTAL GAS-LIQUID FLOWS NEIMA Department

BRAUNER

of Fluid Mechanics

(First received

and DAVID

MOALEM

MARON

and Heat Transfer, Faculty of Engineering, Ramat-Aviv 69978, Tel-Aviv, Israel

17 August

1990; accepted

in revised

University

of Tel-Aviv,

form6 December 1990)

Abstract-Stratified/nonstratified transition in gas-liquid flows has been traditionally tackled via stability analyses, resulting in a transitional boundary which relates mainly to stratified/slug transition. The present study shows that the departure from stratified configuration is associated with a “buffer zone”, confined between the conditions derived from stability analysis (a lower bound) and those obtained by requiring well-posedness of the transient governing equations (an upper bound). These two form the basis for the construction of a complete itratified/nonstratified transitional boundary to the various bounding flow patterns: slug, pseudo-slug and annular. In exploring the relative destabilizing contributions of the two phases along the complete transitional boundary, two zones, a “liquid-controlled” (along the stratified/ slug boundary) and a “gas controlled” (along the stratified/annular boundary), have been identified. It has been found that the liquid dominance decreases with reduction of the tube size or increase of the liquid viscosity.

INTRODUCTION The

extensive

its major

literature

on two-phase

data along the stratified/slug transitional boundary [2-51. Recently, Lin and Hanratty [6] and Andritsos et al. [7] have extended in subsequent studies the classical K-H instability theory for ideal fluids, to account for the various viscous shear stresses due to the mobility of both phases. This “viscid” analysis yields an interfacial disturbance velocity which differs from the liquid velocity, and consequently a nonnegligible destabilizing effect of the liquid phase inertia. The application of the “viscous K-H” instability for the onset of slugging in gas-liquid horizontal flows, shows better agreement with the experimental findings for various liquid phase viscosities. The above and other related studies associated the departure from stratified pattern with a marginal instability boundary and the stability criteria so obtained have been mainly applied for stratified/slug transition. The authors [S-13] recently invoked (in attempting mental

flows

refers

in

part to gas-liquid systems. Indeed, the intensive research activities on a variety of aspects of gas-liquid flows have contributed to a broad understanding of the undertying mechanisms. Of particular interest is the study on the feasibility and transitional boundaries of the possible flow patterns. Amongst the various flow patterns, the stratified flow in horizontal (and slightly inclined) gas-liquid systems is considered as the basic flow configuration, since the relatively large density differential sustains stable stratification for relatively wide ranges of flow rates. Thus, the understanding of the stability of stratified flow is an essential step towards the development of transitional criteria to the various flow patterns which share common boundaries with the stratified configuration. The basic approach in exploring the departure from stratified configuration is undoubtedly a stability analysis. This analysis yields the conditions for the evolution and growth of interfacial perturbations which are considered as precursors to transitions. Early related studies are those which employed classical Kelvin-Helmholtz (K-H) theory for two inviscid layers [l-3]. However, while considering gas-liquid flows, pG/pr. + 1, and assuming that the interfacial disturbance velocity equals the (slower) liquid layer velocity, the liquid destabilizing contribution has been neglected. This results in rather simple Bernoulli-type transitional criteria, whereby the suction forces in the flowing gas phase over the (“stationary”) liquid interfacial disturbance exceeds the restoring gravity forces. Such criteria, required the insertion of empirical constants to match the experi-

to study

the stability

two-phase well

as

and

flows) parallel on

the

transitions

in liquid&liquid

analyses

well-posedness

of

on the

stability

the

(hyperbolic)

as

equations which govern the stratified flow. It has been shown, that the combined conditions for stability and reality of characteristics define a buffer zone which bear a potential for stratified/nonstratified transition and the departure from stratified to other bounding flow patterns is not always located along the marginal stability boundary. It is the purpose of the present study, to establish the entire stratified/nonstratified transitional boundary in gas-liquid flows. This is achieved by exploring the relationships between stability and reality of characteristics for wide ranges of operational condi-

1849

NEIMA BRAUNER

1850

and

tions. While constructing the complete transitional boundary, the relative contributions of the liquid and gas destabilizing terms are explored in order to determine the specific zones for liquid-controlled or gascontrolled stratified/nonstratified transitions. THEORETICAL

ANALYSIS

AND CONCEPTS

In dealing with the various flow patterns and transitions in general liquid-liquid systems, the authors have studied the relationship hetween stability and well-posedness (real characteristics) of the hyperbolic set of the governing equations [9]. Except for nearly equal-density systems (which are dealt with separately [13]), two immiscible liquids of a finite density differential tend to stratify while tlowing together. As such, the stratified flow configuration can be considered as one of the basic and more natural flow patterns. Referring now to the flow of two stratified immiscible layers a, b as described schematically in Fig. 1, and utilizing the average one-dimensional two-fluid transient formulation, the resulting continuity equations for the two layers and combined momentum equation read [9]

DAVID MOALEM MARON

respectively and y,, Y,, are the shape factors which account for a velocity distribution in the two layers. It is worth noting at this point that eqs (l-3) represent a general framework of formulation for the coflow of the two stratified layers. The RHS of the momentum eq. (3), A&, as defined in eq. (4), includes the body forces and shear stresses (t,, TV,ri) which are to be modelled in terms of the flow variables u,, u,,, h according to the particular physical situation under consideration [S, IO]. In searching for the conditions under which the smooth stratified flow is stable the governing transient equations (1 3) are explored along two main routes; stability analysis and conditions for possessing real characteristic roots.

In order to derive the conditions for stability, the linearized forms of eqs (l-3) are perturbed around a smooth fully-developed stratified Aow pattern, utilizing a temporal stability formulation. This yields a dispersion equation which relates the real wave number k to the complex wave velocity, C = u/k: aC2 - 2(h, + ib,)C

+ (i, + id, = 0, C = w/k (64

;(Pb4+;

(PbAbd = 0

&(P,A,) +g tPa4KJ F + P,(l

PbU - Yb)

-

b

(1)

C,,,

= i(b,

+ ibz) & k [(b, t ibZ)* - a(d, + id,)]“’ (W

= 0 with:

r.,~1 2 E ,

ah (aPi, - l?P,,) + (I)b - P.&J cos B z + dX

with (7)

+ (Pb - P&J sin P &(Pib-Pio)=

-g

CT

(4)

d2h/8x2

where uO. ub. h denote here instantaneous local values for the velocities of the phases and lower phase depth,

Fig. 1.

Schematic description of stratified flow configuration.

Analysis of In eq. (7), aAF,,/W, U,, Ud are WilW, u,, ud at steady-state conditions H, U,, U,. Based on eq. (6), the so-called neutral stability conditions are obtained by requiring a zero imaginary part for C, whereby eq. (6b) yields the neutral stable wave number, k,, and the corresponding wave velocity, C,“:

16D;;osB {;$ c[($ - 1) +(y.-l)(l-2~)]+~u:[(~-I) +h

-U(l-2$)]}

(Pb-Pa) + -{

Pb

4 Pb4

COSB >=

dJ-&--

4

0

au,

4,

(84

au, W

or by rearranging

eq. @a): J, + J, = 1 + J,

(9)

with J, =

7r2 16Dgcosp

Reality

of characteristic

roots

The well-posedness of the initial value (hyperbolic) set of equations is ensured provided it possesses real characteristic roots. The test for reality of characteristx roots on the transient equations (l-3) is carried out around the initial condition of a fully-developed stratified solution. The characteristic roots, L of eqs (l-3) are obtained by solving det (T - AX) = 0, where T, X are the matrices of coefficients of the t, x derivatives, respectively of the state variables h, u,, t+,. The condition under which the characteristic roots 11 are real reads &Ub2%&b - 1) + ,%I’%&,

-, A,

PO

“adjustable definitions” for the hydraulic diameters of the two fluids, depending on their relative velocities, has been introduced by the authors for a general twofluid system [S]. This enables to encompass under a common framework, all possible flow regimes of the two phases, laminar-laminar. laminar-turbulent, turbulent-laminar and turbulent-turbulent for wide ranges of density and viscosity ratios. Detailed descriptions of the shear-stress terms utilized for modelling the fully-developed flow and the solution procedure are given elsewhere [S, 91. The steady-state flow variables H, U,, U,, aAF,,/a(H, U,, U,), A,, A,, AL, AL so-obtained are then introduced in eqs (669) for evaluating the stability characteristics.

~ l) -

(rb’%

-

?s,~cz)~

(Pb(9a)

J, =

R2

‘%

Pb

16Dgcos/3

(Pb-

2

1

(9b) r

J”= (Pb-

uk,2

P,b

cm B

(9c)

the superficial In eqs (8) and (9) U,,, U,, represent velocities of the phases (z 4Q/7rDz), and - denotes normalized values (areas by 0’ and lengths by D). Inspection of eqs (8) and (9) indicates that while the structure of either eq. @a) or (9) is invariant under the specific modelling of the wall and interfacial shear stresses, and evolves essentially from the continuity equations and the LHS of the momentum equations, eq. (S’b) for C,, is directly related to the modelling of the various shear-stress terms in the two-fluid momentum equations. In this sense, the form of eqs (8a) and (9) is general and is only indirectly affected by the modelling of shear stresses through the C,, as determined by eq. (8b). Note also that eqs (pa) and (9) may account for the velocity distribution in the two phases by adjustment of the appropriate shape factors, y,, yb. In modelling the shear stresses required in the various AF,, terms in eqs (4) and (7) an approach of

As stated earlier, eqs (8-10) have been derived and discussed with reference to general two-fluid liquid-liquid systems covering wide ranges of physical properties [9]. For the particular cases of twophase gas-liquid flows, po/pb + 1. This may lead to U, = U, $ U, E U,, whereby the (lower) liquid phase may be assumed to be stationary with respect to the faster gas phase, in which case its time and space variations can be ignored. Stability analysis carried out on eq. (2) and reduced eq. (3) yields for the neutral stability conditions U:;[WU.-

V+(y.-l)(l--$)I

=~S[*pgcos~+rrk:] x2 0;

(lla)

(lib) Note that, for turbulent gas flow, y., z tion (1 la) for C,n Q U, becomes similar to ion obtained by simply applying Bernoulli’s in the gas phase flowing over a stationary disturbance (when C,,/U, z 0, is assumed).

1. Equathe criterequation interfacial However,

1852

NEIMA BRAUNER and DAVID MOALEM MARON

Air-W&,

D-9.53

’

1

UGS’ I.Om/s

lh-.Im/s,UGs*t

m/s

cr=70dyne/cm 0-k. I-k , =

-2 &_I$’

Wave Fig. 2. Wave amplification

NumdZr ,

for gas-liquid

system;

as this is not always true, the present analysis includes C,,/U, in eq. (1 la), while eq. (1 lb) provides C,,/U, in terms of the flow conditions. Note that, eq. (1 la) can be obtained directly from the general eq. @a) assuming C,,/U, 4 1 and yb = 1. Similarly, the condition for obtaining real characteristic roots in gas-liquid systems with turbulent gas liquid phase reads (7, = 1) and stationary

It is noteworthy, that it is necessary that both U, 4 U, and AJp, 4 A,/p, hold in order that eq. (12) be a valid approximation of the general eq. (IO). This may not always be the case in gas-liquid two-phase flows, for example at relatively low gas-liquid rates ratio. In two-phase flow problems, the existence of the physical situation assumed, e.g. stratified flow configuration, is not certain under all operational conditions. Therefore, comptex characteristic roots (illposed initial-value problem) may not necessarily indicate an incorrect formulation, but may be attributed to a physical instability of the assumed flow configuration, whereby transition to a different flow pattern may take place. It is therefore of interest to discuss first the overlapping and distinction between the conditions for stability and those for reality of characteristic roots with reference to gas-liquid flows. Relationships between stability and reality OJ characteristic roots The first comparison of interest is that between the general eqs (6) and (10). Inspection of cqs (6) and (7) indicates that in the limit of very short waves, k -+ 03, the condition for stability of smooth stratified flow, reduces to b: - ad, > 0, identical to eq. (10) which constitutes the condition for reality of the character-

IO0

IO’

effect of flow rates and liquid

viscosity.

istic roots. It is noted that in the presence of finite surface tension and with k + cc, eqs (6), (7) and (10) are trivially satisfied, independently of the specific operational conditions. For zero surface tension, however, either of the above two conditions defines a finite range of operational conditions (U,,, U,,) where both stability and well-posedness are assured (for k -+ cc, or any specified k). Another comparison of interest, particularly relevant to turbulent gas-liquid flows, is the range of C,,/U, = C,,/ UG + 0 and C,,/LJ, -+ 1. In this case, the stability condition given by eq. (I 1a) and the corresponding one derived for real characteristic roots, eq. (12), are again identical, independently of the wavelength of the disturbance. A more general insight into the relations between the stability characteristics and the well-posedness of the two-phase gas-liquid stratified flow formulation may be obtained by means of Fig. 2. The variation of the amplification with the wave number (wi = k Im(C) versus k), in terms of (U,,, ULi,,)combinations, is obtained from eqs 16) and (7). For a certain set of (U,,, UL,,,),as depicted by curves (a), there exists no wave number for which the amplification is positive. Thus, for the entire range of k (or wavelengths), all disturbances are expected to decay. On the other hand, for other combinations of (U,,, U,,) as in curves (c), a smooth (stable) interfacial structure is maintained only for k > k,. For 0 < k < k,, or suEiciently long waves, where the stabilizing effect of surface tension becomes small, the disturbances are amplified and hence a wavy interfacial structure develops. Clearly, the range of amplified waves, 0 < k < k,, varies depending on the (cl,,, U,,) combinations as demonstrated in Fig. 2(a). For particular combinations of (U,,, U,,) as represented by curves (b), the amplified range almost diminishes (k, -+ 0 and J, = 0).

Analysis of stratified/nonstratified For each combination of (U,,, U,,), the range for amplified wave numbers is directly obtained by solving eqs (8) and (9) for k,. For the same (U,,, U,,) conditions, eq. (10) is also solved for k = k,, beyond which (k > k,,) the characteristic roots are real. This minimum wave number which ensures real characteristic roots, k,, is also indicated on Fig. 2. In view of Fig. 2, for all k < k,,, an unstable smooth stratified flow is consistently predicted by both stability and reality of characteristics analyses. However, for any k,, -c k -c k,, while the governing eqs (l-3) are wellposed as an initial-value problem, they are still expected to develop a wavy structure. It is to be emphasized that the value of k, is always within the amplified range, k,, < kn. This has been rigorously shown with reference to general two-fluids liquid-liquid systems C9J While Fig. 2(a) relates to air-water system, Fig. 2(b) includes typical trends for the effect of the liquid phase viscosity in general air-viscous liquid twophase flows for specified operational conditions U,, = 1 and U,, = 0.1 m/s. These are identical to those of curve (a) of Fig. 2(a) for air-water system which show no amplified mode. Thus, while for air-water systems, stable stratified flow is predicted for the entire range of wave numbers, unstable wave modes appear as the viscosity of the liquid phase increases. As shown in Fig. 2(b), the range of amplified wave numbers extends and the corresponding nondimensional amplification increases with increase of the liquid viscosity. In parallel, for sufficiently low viscosity (pL < 250 cp, with U,, = 1 and CJLi,,= 0.1 m/s) real characteristic roots are obtained for the entire range of wave numbers. For more viscous liquids, the range of real characteristic roots is reduced and is limited to k > k,, as marked in Fig. 2(b) for ,uL > 260 cp.

STRATlFIEDjNONSTRATIFIED GAS-LIQUID

As demonstrated exists a particular k, --+0, indicating

TRANSITIONS

IN

SYSTEMS

with reference combination

to Fig. 2(a), there

(U,,,,

U,,)

for which

that the range of amplified waves diminishes. In searching for all combinations of (U,,, .!I,,) for which k,, + 0, the so-called zero neutral stability line (ZNS) may be obtained [along which J, in eq. (9) vanishes]. This boundary confines all possible smooth stratified flows. The locus of the curve itself represents the departure from smooth stratified structure. For any operational set (Ua,,,, U,,) outside the k, = 0 boundary, the linear stabiliy analysis predicts exponential growth with time for a finite range of wave numbers, 0 < k 6 k,. The growth of the disturbances in this region may either be damped (due to nonlinear effects) and thus end up with stable wavy stratified flow, or may result in a different flow configuration (due to bridging, for instance.) In parallel to the ZNS boundary, defined by k, = 0, an analogue boundary is built up by searching for all combinations of (U,,, Ub,) which yield real characteristic roots for k,, = 0, by eq. (10). The zero real charac-

transitional

boundaries

18.53

ya'yb'l.

D~ZDScm

--ZRC

botmkq,hrt=O

-2NS

boundary , k, =O

SUPERFICIAL GAS VELOClTY,U&s]

Fig. 3. Zero neutral stability (ZNS) and zero real characteristics (ZRC) boundaries for liquid-liquid and gas-liquid stratified flows (buffer zone shedded).

teristic roots boundary (ZRC) is generally composed of two branches both of which fall [expectedly in view of Fig. 2(a)] away from the ZNS, in the region where no stable smooth stratified flow is expected to exist. The ZNS and ZRC boundaries are demonstrated for comparison in Fig. 3 for typical liquid-liquid (oil-water) and gas-liquid (air-water) systems. It is of importance to indicate some basic points with reference to Fig. 3, which may shed some light on the mechanisms for departure from stratified flow configuration as well as on the effects of physical properties of the two layers: (a) The upper ZRC branch [Fig. 3(a)] corresponds to U, > U,, while along the side branch U, > U,. The two branches approach one another at the region of high velocities of both phases, U,, and U,, the difference of which is just sufficient to balance the gravity term in eqs (10). As these two branches get closer, a double solution is obtained (for either specified U,, or U,,,). Thus, at relatively high velocities, there exists a narrow range of operational conditions for which real characteristic roots are still ensured. The location of this region and its width depend mainly on the density differential. For gas-liquid systems, the ZRC branch along which U, E U, > UG = Uo, corresponds to extremely low gas/liquid rates ratio, and lies outside the

’

1854

NEIMA BRAUNERand DAVID MOALEM MARON

range of practical operation rates, to the left of U,, range in Fig. 3(b). (b) In searching for k, = 0 at specified U,, or Ub,,,, the ZNS boundary may also exhibit multiple solutions. In principle, this is expected due to the nonlinearity of the resulting stability conditions and the discontinuities which evolve from laminar/turbulent flow regime transitions in either of the two phases. This has been widely discussed with reference to general two-fluid liquid-liquid systems in the previous studies [9, lo]. It is, however, interesting to follow here the ZNS boundary for the gas-liquid system in Fig. 3(b). The point S,, for instance, represents limiting conditions for smooth stratified laminar water layer. Increasing either the gas rates, U,, towards S, or U, towards S,, the water layer becomes turbulent at S, or S,. The region between S, and S, or S, corresponds to an unstable flow with laminar liquid layer. The transition to turbulent flow regime at either S, or S3, renders a stable stratified flow again. With further increase of either the gas or liquid rates, point S, is reached were the turbulent smooth liquid layer now attains again a neutral stable situation. Thus, the region lying between the points S&-S, corresponds to a stable turbulent liquid layer. (c) Tke general interpretation of the ZNS line and the ZRC Iine consists in defining three zones; the area below the ZNS boundary, is well understood to be the stable smooth stratified zone. In the buffer zone in between the two baundaries, though amplified interfacial waves exist, the equations which govern the variation of (h, M,, ub) in space and time are still wellposed with respect to all unstable modes. Beyond the ZRC boundary, the unreal characteristic roots imply that the governing equation for the stratified flow configuration cannot accommodate the time and space variations associated with a certain range of amplified wave modes. Thus, while the ZNS boundary may represent a preliminary transition to a wavy interfacial structure, the ZRC boundary, which is well-advanced in the wavy unstable region, represents an upper bound for the existence of a wavy stratified configuration, beyond which another flow pattern prevails. (d) Figure 3(b) indicates that in gas-liquid systems and in the range of U,, P U,, (and W,/U, 9 I), the ZNS line and the ZRC side branch are practically identical implying that stratified/annular transition can be predicted by either stability or reality of characteristic roots. On the other hand, in the range of comparable phases velocities, the ZNS and ZRC criteria deviate from each other. This behavior is typical of gas-liquid systems and is understood in view of eq. (11) with C&/U, = 0 and eq. (12). The inclusion of Fig. 3(a) for liquid-liquid oil-water systems, points to an essential difference in the location of the buffer zone for typical gas-liquid and liquid-liquid systems. As is shown in the figures, in liquid-liquid (oil-water) systems, the ZNS and ZRC lines converge along the upper ZRC branch where U, = U,,,,, $ U, = U,,i,, and deviate in the region U,, > U,, forming there the

buffer zone. Clearly, in the range where the ZNS and the ZRC boundaries are practically identical, either of these can be utilized for predicting stratified/nonstratified flow pattern transitions. (e) In the range where the ZNS and ZRC lines deviate, the buffer region is characterized by the existence of interfacial disturbances, and as such bears a real potential for a flow pattern transition. Whether these disturbances trigger a departure from stratified configuration (due to blockage) depends on the relative thicknesses of layers. For if H/D z 0.5-l. it is likely that the evolution of the interfacial disturbances on the liquid layer will end up in the tube blockage. Thus, when the entry from the smooth into the buffer zone (along the ZNS boundary) is associated with a relatively thick liquid layer, H/D 2 0.5, above fi = 0.5 line as at point S, in Fig. 3(b), the ZNS line which in general represents the transition to a disturbed wavy pattern, will also predict the condition for the development of other flow patterns. On the other hand, when the entry to the buffer zone (along the ZNS boundary) occurs with a relatively thin liquid layer, H/D i 0.5, below fi = 0.5 line as point S, in Fig. 3(b), the ZNS line will predict the development of wavy stratified Row. In the latter case, the transition to other flow pattern is delayed and predicted by the ZRC boundary as long as the relative liquid layer thickness, H/D, remains small in the buffer zone. As the relative liquid layer becomes of the order of the within the buffer region, conduit radius, H/D z 0.5, the disturbed interface may trigger a flow pattern transition, in this case within the buffer zone in the vicinity of H/D 5 0.5. These scenarios are applied to structure a complete transitional boundary for departure from stratified flow which is essentially composed of three sections; along the ZNS line for H/D > OS,along the ZRC line for H/D < 0.5 and along H/D z 0.5 in the buffer region (see sections in Fig. 4 in bold below, which confine the predicted stratified zone, shedded area). Accordingly, one may state that stability analysis may be considered a lower bound for stratified pattern of a relatively thick liquid layer, while reality of characteristics may yield an upper bound for stratified flow of a relatively thin liquid layer.

COMPARISON

WITH EXPERIMENTS

The transitional boundaries and the associated ideas proposed in the preceding section for the departure from stable stratified gas-liquid two-phase flow are now to be tested and discussed in the light of experimental observations. Figure 4 summarizes available experimental data for the departure from stratified flow in air-water systems, and systems of air-viscous liquids of varying viscosity, for 9.53 cm tube diameter [6, 7, 14, 151. Similar comparisons (not included here) are obtained with smaller diameters [16]. Included in Fig. 4 are the calculated curves for ZNS, ZRC and the H/D = 0.5 lines. Inspection of the

Analysis

of

stratified/nonstratified

AIR-LICU!D.

transitional

D=9.53cm

ErpwirrmnloL.~=

--Mbandary

ST--WED

a Ol(l989~

Fig. 4. Stratified/nonstrali~~d

AIR VELoaTY,

transition; comparison

figures indicates that the data associated with thick liquid layer H/D > 0.5 (above and to the left of H/D = 0.5 line), consistently follow the trends and fall closer to the ZNS boundary than to the ZRC curve. In this region of H/D > 0.5, the departure from stratified flow indeed ends up in an intermittent slug flow, implying that the interfacial disturbances which develop on the thick unstable stratified layer trigger a transition. In the region of relatively thin liquid layer, H/D < 0.5, the transitional data follow in principle the ZRC boundary (away from the ZNS curve). This is clearly seen in relatively large diameter systems as in Fig. 4. In large diameters, the initiation of atomization is not associated with simultaneous (intensive) wetting of the upper wall and therefore, the observed line for atomization falls within the still stable stratified configuration. Note that, however the initiation of atomization in small diameter is associated with practically immediate wetting of the upper tube walls and may reasonably be considered as a developing annular pattern. In view of the above, the data for H/D > 0.5 (and along the ZNS boundary) represents a transition from a stable stratified to a slug pattern, whereas the data in region of H/D < 0.5 (along the ZRC boundary) relate to a transition to an annular configuration. In the intermediate range around H/D n, 0.5 and within the buffer zone, the reported flow patterns have been identified as either large wave-stratified flow, pseudopatterns. According to slug, or wavy annular Andritsos and Hanratty [7] and Lin and Hanratty [ 171, the pseudo-slug region is characterized by high amplitude roll waves which intensively wet the upper tube walls, without causing the pressure fluctuations typical of the slug pattern. For large diameters (Fig. 4), the stratified/nonstratified data depart from the ZNS line around H/D z 0.5and climb along the H/D - 0.5line in the buffer region and then follow the ZRC line at H/D < 0.5. The region of pseudoslugs in large diameters, diminishes as large waves do not necessarily intensively wet the upper wall. In small

[lJcs

m

.

Y-SLUG AN-AhWULAR PS-PSELDO SLUG

Slr~tifbd/NmstmWicd 0Atomimtim l

SUERFlClAL

18.55

boundaries

m/s]

with experiments, D = 9.53 cm

diameters, however, the large waves are more likely to wet the upper walls, resulting in a clear region of pseudo-slugs which has been observed within the herein predicted buffer region around H/D z 0.5

CI61.

DISCUSSION

As has been shown in the preceding section, the departure from stable stratified configuration to other bounding flow patterns is associated with a buffer zone as formed in between the ZNS, eqs (8) and (9), and the ZRC, eq. (lo), boundaries. The purpose of the present section is to elucidate some additional basic points regarding the relative contributions included in the equations governing ZNS and ZRC. Basically, eq. (9a) [and eq. (10) as well] includes two destabilizing terms which evolve from the mobility of the two phases and are to be balanced by the gravity and surface-tension stabilizing terms. As already mentioned above, the viscous effects (between the two phases and between the walls and the two layers) directly determine the interfacial neutral wave velocity C,,, eq. (8b), which in turn affects the stability condition, eq. (9a). Thus, the gas (J, = JG) and liquid (J,, = J,) destabilizing terms in eq. (9a) represent the combined effects of inertia [as those that evolve from the LHS of eqs (l-3)] and viscous effects [via C,,, as those that evolve from the RHS of eq. (3)]. This general structure of the stability equation remains unchanged when a different modelling is employed for the various shear-stress terms, resulting in a different interfacial wave velocity. Moreover, even when the viscous effects are completely ignored, resorting to an inviscid K-H instability type of analysis, the structure is still maintained,

velocity

of the resulting

instability

equation

while the corresponding interfacial e.g. for rectangular channel, reads: 1+

5

u-- fi I.

~a U, 1 -

H

(13)

1856

NEIMA BRAUNER and DAVID MOALEM MARON

Indeed, various early studies employed inviscid R-H stability analysis for predicting the stratified/nonstratified transition boundary in gas-liquid twophase flows [l-3]. However, by assuming that pJpb = pG/pL e 1, eq. (13) has been reduced to C,, = U, and the liquid destabilizing term in the stability condition [which is proportional to (C&/U, - 1)] has been neglected. Consequently, the resulting equation [similar to eq. (I la) with C,, = U,,] includes, in fact, only the gas destabilizing term, whereby

&I, 1

- UJ2

> g 2 C

- 1 (1 - fI)D. )

itional data is, as a matter of fact, expected in view of the unjustified assumption C,, = U, [which led to eq. (14)]. A more careful inspection of eq. (13), indicates that even for gas-liquid flows, where theory may still yield PGIPL < 1, the inviscid C,, # CJ,, since (U,/U,) cd/( 1 - H”)] may attain high values, particularly along the stratified/intermittent transition boundary, where the liquid layer is relatively thick and the gas velocity is still higher than the liquid velocity. Moreover, when viscous effects are accounted for, a more realistic expression for the interfacial wave velocity merges, eq. (8b), whereby C,,/U, - 1 # 0 and thus both liquid and gas destabilizing contributions ought to be included along the transitional boundary. As is shown below, the relative contributions of the gas and liquid destabilizing terms along the transitional boundary is to be carefully considered according to the particular physical system (e.g., gas-liquid or liquid-liquid flows) and the range of operational conditions. In order to demonstrate this, Fig. 5 presents the ratios of the interfacial neutral wave velocity to either the average gas or liquid velocities and the corresponding ratio of the liquid destabilizing contri-

(14)

As the remaining gas-phase destabilizing contribution has been found insufficient to balance the gravity term along the experimental stratified/nonstratified transition boundary, various empirical correction factors, K,, have been introduced to match eq. (14) with the data. For instance, Wallis and Dobson [3], Taitel and Dukler [43, Kordyban [2] and Mishima and Ishii [S] enhanced the gas term by introducing K, = 0.5, 1 ~ fi’, 0.49 and 0.74, respectively. The disagreement of the gas destabilizing term with experimental trans-

-

0.9.53

- -‘_ -

cm

- Air-Water Air-Viscous

lO%p

p,_= Icp Liquid

‘\

\ \\

Superficial

Fig. 5. Interfacial

wave celenty

Air

\

\

; : -

Velocity, U&m/s]

and liquid destabilizing contribution = 9.53 cm.

along the ZNS boundary,

D

Analysis

of stratified/nonstratified

bution with respect to the gravity stabilizing term, JL. Clearly. along the ZNS boundary, (k,, J,) + 0, and eq. (9a) reduces to J, + J, = 1. Figure 5 indicates that C,, almost always deviates significantly from the liquid velocity, and thus, C,,/W, - 1 # 0. Correspondingly, the liquid destabilizing effect as represented by J,, certainly cannot be ignored. Moreover, in view of Fig. 4 in the range which corresponds to stratified/intermittent transition, the liquid destabilizing term in fact even dominates, JL > 0.5. It has also to be emphasized that even when C,,/U, z 1 (as in air-water systems, .ur. x 1 cp), the neglect of the liquid relative contribution may be erroneous over most of the stratified/slug transition boundary. This is understandable [in view of Fig. 5(b)], since in this range also C,,/ U, = 1. However, in view of eq. @a), (C,,/U, - 1) and (C,,/UG - 1) are not the only quantities which determine the relative contributions of the liquid and gas destabilizing effects. This is further demonstrated while following the effects of the liquid viscosity: It is shown in Fig. 5 that as the liquid viscosity increases, C,,/UL may significantly deviate from one and attain high values, while simultaneously, C,,/U, decreases. However, the liquid relative destabilizing contribution, J,, although proportional to (C,,/U, - l), decreases with increasing viscosity and thus the range of liquid dominance is reduced. The observation by Hanratty [18] that “the K-H inviscid theory becomes surprisingly more accurate as the liquid viscosity increases”, is indeed well established in Fig. 5. The effects of tube diameter are demonstrated in Fig. 6. In general, both C,,/U, and CJU, increase mildly with increase of the tube diameter and the overall outcome is represented by -I, for varying tube sizes and liquid viscosity. As is shown in the figure, larger diameters affect a wider range for liquidcontrolled transitions, and the effect of tube size becomes more pronounced with increase of the liquid viscosity.

01

Id=

d

IO0

contribution

along

boundaries

1857

The liquid phase dominance along the ZNS boundary, JL, is continuously reduced as the gas rate increases. For sufficiently high gas rates, the contribution of the gas-phase term exceeds that of the liquid phase and J, = 1 - JL z 0.5. Comparisons of Figs 5 and 6 with the transition maps of Fig. 4 indicate that the gas-controlled transition zone is associated generally with E? = H/D < 0.5, under which condition the transition to annular pattern takes place along the ZRC boundary (as explained in the previous section). However, in this range of gas-controlled instability, the ZNS and ZRC boundaries are practically identical in gas-liquid systems and both are predicted by eq. (12) or (lla) with C,JU, + 0. It is to be noted, however, that while C&/U, -0 for high U,,, [Fig. 5(b)], the corresponding ratio for the liquid phase, CJU,, depends on the flow regime in the liquid. For low viscosity liquids (air-water systems), where the liquid phase is turbulent, C,,/CJ, + 1, while for more viscous liquids, where laminar flow prevails, C,,/CJL attains an asymptotic value of % 1.7. To sum up, the complete transitional stratified/nonstratified boundary in gas-liquid flows, is shown to be liquid-controlled, J, + J, along the stratified/slug transition, and gradually becomes gas-controlled, J, 9 J, towards the stratified/annular transition boundary. It is at this point relevant to emphasize that previous studies relating to stratified/slug flow pattern transition, assumed in fact a gas-controlled transition by unjustifiably ignoring totally the liquid contribution [eq. (14)]. As these models ate essentially based on the gas contribution which plays a minor part in the stratified/slug boundary, a reasonable comparison with experiments required the insertion of correction constants, which become greater for lower U,, (greater H/D). The (1 - 2) = K, correction [4], although it yields the larger correction required at lower u GS, is still unphysical, as the other empirical constants; it artificially enhances the gas (negligible) term

102 Id=

IO’

Superficial Fig. 6. Liquid destabilizing

transitional

Air

the ZNS

diameter.

107

100

10’

I6

Velocity,Ucs [m/s] boundary;

effect of liquid viscosity

and tube

+

NEIMA BRAUNER and DAVID MOALEM MARON

1858

along the stratified/slug transition, whereas it is practically ineffective along the stratified/annular boundary where H/D d 1. In view of Figs 5 and 6, the relative contribution of the gas or liquid destabilizing terms depends on a variety of parameters and operational conditions. In the extremes, however, of definite liquid-controlled or gas-controlled zones, empirical constants can be utilized to simplify the general predictive transitional criteria, provided that they are employed in the really dominating contribution.

SUMMARIZING

REMARKS

Previous studies concerned with flow pattern transition in gas-liquid flows, relate mainly to instability criteria (or the simplified Bernoulli criterion) in predicting stratified/nonstratified transitions. Here, however, the departure from stratified configuration to other bounding patterns is shown to be associated rather with a buffer zone, formed in between the ZNS and ZRC boundaries as evolve from stability and well-posedness analyses. The range of stable stratification is thus defined here in view of these two limiting bounds: the lower one obtained from stability analysis (ZNS) corresponds to the evolution of interfacial disturbances, whereas the conditions for reality of the characteristic roots (ZRC) represent an upper bound for the stratified configuration. Whether a departure to other bounding flow patterns can be triggered also by the growing interfacial waves along the ZNS and in the buffer region depends on the relative liquid layer thickness. Consequently, it is proposed that the entire stratified/nonstratified transitional boundary is generally composed of three sections: the ZNS line for H/D > 0.5, H/D - 0.5 in the buffer zone and the ZRC line for H/D < 0.5. Clearly, the choice of H/D = 0.5 comes here instead of a complicated nonlinear stability analysis which is required to determine whether the growing waves in the buffer wavy region will indeed reach the upper tube wall and thus end in a different flow pattern. Note also, that the predicted boundaries are based on fully-developed stratified flow (subjected to infinitesimal disturbances). On the other hand, experimental data taken in finite-length tubes correspond to undeveloped flows which may also be exposed to relatively large disturbances (due to entry and exit effects). In such cases the transitional data may fall outside the buffer region confinements. In exploring the stability boundary along the entire range of operational conditions, liquid-controlled and gas-controlled zones have been identified. It has been shown that in air and water systems the gas-controlled zones relate mainly to the stratified/annular transition, while the stratified/slug transition is essentially liquid-controlled. The range of liquid phase dominance along the transitional boundary increases with decreasing liquid viscosity and increasing tube size. AS such, the stratified/slug transitional data are not to be correlated by the gas destabilizing term only

(modified gas Froude number) as suggested by various earlier studies. It has to be pointed out that the accurate locations of the various predicted boundaries discussed here may require the velocity-profile shape factors and the use of more established friction-factor correlations. However, the shear stress modelling applied here [S] and y. = yb = 1 go to demonstrate that the basic concepts and ideas reasonably predict the general effects and trends of the various physical and geometrical parameters involved in stratified/nonstratified transition. Sensitivity tests carried out on the effect of the interfacial friction factor, which may be amplified due to interfacial wave characteristics show only moderate changes on the transitional boundaries. Similarly, as the gas is always turbulent, y. = 1 and yL = ylb> 1 in the laminar liquid zone have been also tested (and discussed with reference to general liquid-liquid flows [9]). In general, both amplifiedf; and yL > 1, relevant at high U,, and low U,, along the stratified/annular transition may extend the predicted region of feasible stratification. Finally, it is of interest to note that the integrated application of stability and well-posedness considerations are found capable of elucidating also the observed dramatic effects of inclination on the transitional boundaries in inclined flows [16, 193 as well as predicting flow pattern transitions in small conduits

c-w. NOTATION

A

C s” 4 h

H

:, P Re

S U u X

Greek D

Y p v P 0 =

cross-sectional flow area, m* wave celerity, m/s pipe diameter, m friction factor gravity acceleration, m’/s instantaneous layer depth, m layer depth at steady state, m wave number, mm * input volumetric flow rate, m3/s pressure, N/m2 Reynolds number perimeter, m instantaneous axial velocity, m/s axial velocity at steady state, m/s coordinate in the downstream direction, letters

inclination angle shape factor vicosity, kg/m *s kinematic viscosity, m’/s density, m3/s surface tension, N/m shear stress, N/m’

Subscripts a lighter as b bs

fluid superficial, lighter fluid heavier fluid superficial, heavier fluid

m

Analysis

G GS

of stratified/nonstratified

gas gas superficial interfacial

;

liquid

LS n r

liquid, neutral

or imaginary

part

superficial stable

IC

real part real characteristic

s

superficial

roots

Superscripts dimensionless

REFERENCES

E. S. and Ranov, T., 1970, Mechanism of Ill Kordyban, slug formation in horizontal two-phase flow. J. basic Engng 92, 857-864. E. S., 1977, Some characteristics of high PI Kordyban, waves in closed channels approaching KelvinHelmholtz instability. A.S.M.E. J. Fluids Engng 99, 339-346. c31 Wallis, G. B. and Dobson, J. E., 1973, The onset of slugging in horizontal stratified air-water flow. Int. J. Multiphase Flow 1, 173-193. c41 Taitel, Y. and Dukler, A. E., 1976, A model for predicting Bow regime transitions in horizontal and near horizontal gas-liquid flow. A.1.Ch.E. J. 22, 47-55. Mishima, K. and Ishii, M., 1980, Theoretical prediction of onset of horizontal slug flow. Trans. A.S.M.E. .I. Fluids Engng 102, 44-445. T. J., 1986, Prediction of the C61 Lin, P. Y. and Hanratty, initiation of slugs with linear stability theory. Int. J. Multiphase Flow 12, 79-98. N. and Hanratty, T. J., 1987, Interracial c71 Andritsos, instabilities for horizontal gas-liquid flows in pipelines. Int. J. Multiphase Flow 13, 583-603. PI Brauner, N. and Moalem Maron, D., 1989, Two-phase

transitional

boundaries

1859

liquid-liquid stratified flow. Physicochem. Hydra 11, 487-506. c91 Brauner, N. and Moalem Maron, D., 1990, Stability analysis of stratified liquid-liquid horizontal flow. Int. J. Multiphase Flow (submitted). Cl01 Brauner, N. and Moslem Maron, D., 1990, Flow paltern transitions in two phase liquid-liquid horizontal tubes. Int. J. Mulriphase Flow (submitted). N., 1990, Two-phase liquid-liquid annular Cl11 Brauner, flow. Ink J. Multiphase Flow (in press). CI21 Moalem Maron, D., Brauner, N. and Kruka, V. R., 1990, The mechanisms of two phase liquid-liquid viscous core flow. Proc. 6th Miami Int. Symp. Heal and Mass Transfer. Cl31 Brauner, N., 1990, On the relations between two-phase flow under reduced gravity and earth experiments. Int. Comm. Heat Mass Transfer 17, 271-282. T. J., 1989, Cl41 Andritsos, N., Williams, L. and Hanratty, Effect of liquid viscosity on the stratified-slug transition in horizontal pipe flow. Int. .7. Mukiphme Flow 15, 877-892. J. M., Gregory, G. A. and Aziz, K., 1974, A Cl51 Mandhane, Row pattern map for gas-liquid flow in horizontal pipes. Int. J. Multiphase Fknv 1, 537-553. Cl61 Brauner, N. and Moalem Maron, D., 1990, Stability well-posedness and flow pattern transitions in twophase liquid-liquid flows. Proc. 6th. Miami Int. Symp. Heat and Mass Transfer. T. J., 1987, Effect of pipe Cl71 Lin, P. Y. and Hanratty, diameter on the interfacial configurations for air-water flow in horizontal pipes. ht. J. Multiphase Flow 13, 549-563. T. J., 1987, Gas-liquid flow in pipelines. IL181Henratty, Physicochem. Hydra 9, 101-l 14. Cl91 Brauner, N. and Moalem Maron, D., 1990, Analysis of stratified/nonstratified transitional boundaries in inclined gas-liquid flows. Internal report, Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering Tel-Aviv University, Israel. WI Brauner, N. and Moalem Maron, D., 1991. Identification of the range of ‘small diameter conduits’ regarding two-phase flow pattern transitions. Int. Comm. Heat Mass Transfer (submitted).