The primary characteristic of slug flow () is its inherent intermittence. An observer looking at a fixed position along the axis would see the passage of a sequence of slugs of liquid containing dispersed bubbles, each looking somewhat like a length of bubbly pipe flow, alternating with sections of separated flow within long bubbles (Figure 1). The flow is unsteady, even when the flow rates of gas and liquid, QG and QL, are kept constant at the pipe inlet.
The elementary part of slug flow is a cell, involving the region of a long bubble plus the region of the following liquid slug. The slug of liquid of length LD carrying dispersed bubbles, travels at a velocity V. It overruns a slower moving liquid in the separated film. During stable slug flow liquid is shed from the back of the slug at the same rate that liquid is picked up at the front. As a result the slug length stays constant as it travels along the tube. For horizontal or near horizontal tubes, the liquid shed at the back decelerates under the influence of wall shear and forms a stratified layer. For vertical or near vertical tubes the liquid forms a falling annular film and accelerates as it moves downward. The separated section containing a large gas bubble has a length LS Most of the gas is transported in these large gas bubbles.
As this kind of flow occurs over a wide range of intermediate flow rates of gas and liquid, it is interesting for many industrial processes such as the production of oil and gas, the geothermal production of steam, the boiling and condensation processes, the handling and transport of cryogenic fluids and the emergency cooling of nuclear reactors The existence of slug flow can create problems for the designer or operator. The high momentum of the liquid slugs can create considerable force as they change direction passing through elbows, tees, etc. Furthermore, the low frequencies of slug flow can be in resonance with large piping structures and severe damage can take place. In addition, the intermittent nature of the flow makes it necessary to design liquid separators to accommodate the largest slug length. In contrast, there are numerous practical benefits which can result from operating in the slug flow pattern. Because of the very high liquid velocities, it is usually possible to move larger amounts of liquids in smaller lines than would otherwise be possible in two phase flow. In addition, the high liquid velocities cause very high convective heat and mass transfers resulting in very efficient transport operations.
The entry is organised as follows. In the first part we recall briefly the basic modelling concept. In the second part some of the closure laws quoted in the literature will be evaluated.
Wallis was probably the first to formulate clearly the concept of unit cell (UC) suggested by Nicklin et al. In the past twenty years the model has been perfected by several investigators so that it appears now as the ultimate 1D approach. The concept reposes on the two following assumptions. Firstly, there exists a frame of a given velocity V in which the flow is almost steady. Secondly, in this frame the flow of long bubbles and liquid slugs is fully developed. A review of the scientific literature reveals an abundance of these laws with varying physical validity (see for example the overviews of Taitel and Barnea, 1990; Fabre and Liné, 1992). The weakness of these laws often originates from the narrow range of flow conditions used in their calibration. Their critical role has been discussed by Dukler and Fabre, 1992.
Suppose that we know the specific flow conditions: the pipe size (i.e. its diameter D or its cross-section area A=πD2/4 and its inclination θ, the fluid properties (i.e. the density θk (k = L,G)), the kinematic viscosity νk , the surface tension σ and the volumetric flux of each phase, jk = Qk /A.
A complete model of slug flow should produce at least the following information: the characteristic lengths, LD, LS, and the mean bubble size in the liquid slug, the form of the liquid film (stratified, annular), the characteristic velocities, V, and the mean velocities of both gas and liquid in each part of the cell, the cross-sectional phase fraction in each part of the cell, the mean wall and interfacial shear stresses in each part of the cell and the pressure drop.
The main difficulty in modelling slug flow comes from its chaotic nature. This is suggested by the observation of the succession of bubbles and slugs whose length appears randomly distributed with time (Figure 2). To avoid having to account for the flow randomness, a few assumptions are needed. The initial assumption was to picture the flow as a sequence of cells periodic with both space and time the UC concept was born. However, two weaker assumptions lead to the same model.
The first assumption comes from experimental evidence which is illustrated in Figure 3. The probability density distribution of bubble and slug velocities shows that they are narrowly distributed about their average—in other words they are almost identical. Although this property becomes less evident at high gas or liquid flow rate, this quasi-steady behavior in a moving frame is the key of the success of the UC model. Indeed this property leads to a great simplification since it allows to transform an unsteady problem into a steady one.
The second assumption consists in assuming that the flow is fully developed in each part of the cell. As a consequence, the cross-sectional mean fraction and velocity of each phase do not depend on the longitudinal coordinate inside the long bubbles and the liquid slugs. This assumption is probably stronger than the previous one.
Figure 2. Probability density distribution of bubble and slug lengths. Slug flow in pipe of 5 cm diameter at superficial velocities jG=1.25 m/s and jL=0.97 m/s (Fabre et al., 1993).
Considering steady inlet flow conditions, the mass flow rates of each phase can be specified as:
In order to simplify the discussion the density of both phases will be considered constant so that the "steadiness" dmk/dt is equivalent to djk/dt=0 .
The phase-k is distributed over two elementary regions pictured in Figure 1. The rate of occurrence of the large bubbles β is thus defined by:
where the overbar denotes a time average. Because the flow is supposed steady in some frame of reference this average is likewise interpreted as a space average. β will be further referred to as the rate of intermittence.
Let εk be the fraction of each phase existing over any pipe cross-section: as gas and liquid fill the section, εG+εL=1. Introducing the definitions of the time—or space—average of phase fractions over the cell, the separated region and the dispersed region
leads to the volumetric relations:
The mean phase fraction may be expressed versus the phase fractions in each part of the cell through:
In the frame moving at the velocity V of the cells, the equations of mass and momentum take a simplified form. The coordinate of the moving frame is defined as:
The velocity of phase-k averaged over the pipe cross-section is thus transformed by the change of frame as V - uk . Due to steadiness of the flow in the moving frame, the continuity equation is expressed by:
φk represents the volumetric flux of phase-k entering the long bubble region, and shed from the liquid slug. The time—or space—average of Equation 7 over the cell, the separated region and the dispersed region introduces the following definitions of the mean velocities Uk, UkS, UkD of phase-k:
Equation 8 is the mass balance relating the flux of phase-k entering the long bubble to that entering the liquid slug. In combining Equations 5 and 8 a relation similar to Equation 5 is easily deduced for the mean velocity of phase-k:
The cross-sectional average of the momentum equation for phase-k is:
where p is the mean pressure over the cross-section area of phase-k, τ is the x-component of the stress exerted upon the phase-k by the wall (subscript w) or the interface (subscript i), S is the wetted perimeter and g the gravity. The above equation simplifies in using Equation 7:
with the jump condition
In contrast to the continuity equations which can be simplified by an integration over the different parts of the cell, the momentum equations cannot, unless we have
According to the second assumption, the above equation holds in each part of the cell, so that Equation 11 simplifies and may be averaged over each part of the cell. In the standing frame it yields:
Since the flow is fully developed, the pressure gradient is the same in both phases and must not be distinguished. The mean pressure gradient over the cell results from the mean pressure gradient over each part of the cell weighted by their rate of occurrence:
The pressure gradient involves two contributions the weight of the phases and the wall friction.
It is worth noting that the fully developed flow assumption makes the equation independent of the cell length. Only the intermittence factor b appears. On the other hand, the pressure gradient appears only in Equation 15. Therefore, once the phase fractions and the velocities are determined, the wall friction and the weight of the phases may be calculated. This remark suggests that the problem can be split into two steps. In a first step the phase fractions, RL, RLS, RLD to be determined. The pressure gradient will be determined in a second step.
In order to solve the closure problem for the determination of the phase fractions, the seven independent algebraic equations have been grouped in Table 1 They are nonlinear and present a deficit of four equations with respect to the eleven unknown quantities: RL, RLS , RLD , UL , ULS, ULD, UG, UGS, UGD, β, V. The role of the four closure equations is to restore the missing information. We shall limit the discussion to the most classical method which requires equations for: the velocity V of the large bubble, the void fraction in the large bubble RGS, the void fraction in the liquid slug RGD and the drift velocity in liquid slug UGD-ULD .
Whereas the phase fractions are not coupled to the pressure gradient, the pressure gradient does depend on the phase distribution as shown by Equation 15. Even for horizontal flow in which the weight vanishes, they still have a great influence on the pressure gradient through the intermittence factor β. For the pressure gradient to be calculated, two other closure laws are needed for the shear stress at the wall in the film τkwS and in the slug τLwD .
Different models have been published in the scientific literature. What makes the difference is the choice of the closure laws.
As most of the gas is conveyed by the large bubbles the accurate prediction of their motion and their shape is essential. It is possible to get a crude estimate of the void fraction by assuming that the gas is conveyed at velocity V:
This relation does a fairly good job in some simplified cases. This shows that the phase fractions are primarily sensitive to the long bubble velocity.
Our present knowledge of the motion of long bubbles comes from both theory and a considerable amount of data (see the review of Dukler and Fabre, 1992). Cylindrical bubbles rising in vertical tubes have the shape of a prolate spheroid independent of their length. The shape at the rear depends on whether or not the viscous force is negligible. When negligible, the bubble has a flat back indicating that flow separation and vortex shedding occur. In upward liquid slug flow, the nose may be distorted by the turbulence generated in the wake of the preceding bubble. In downward liquid flow, the structure of the free surface is more complex. The bubble migrates with an asymmetrical shape. Moreover, above some critical liquid flow rate, the bubble is distorted alternately on one side of the tube and then the other.
The shape of the bubble depends upon the pipe inclination. Indeed the experiments in still liquid (Zukoski, 1996) show clearly that the eccentricity increases when the pipe is deviated from the vertical position. When the inclination decreases from 90° to 0°, the cross-sectional area of the film far from the nose departs from a centered annulus to a segment of circle indicating that stratified flow is reached in the liquid film. According to Spedding and Nguyen, 1978, this change in shape occurs between 30° and 40°. When the flow in the film is stratified, the tail has the appearance of a hydraulic jump.
Measured bubble velocities as a function of mixture velocity are shown in Figure 4 for vertical flow, and in Figure 5 for horizontal flow. At high velocity the data are much more scattered. For a more extensive analysis, see the review of Fabre and Liné.
The V(j) relation is linear over certain ranges of mixture velocity j=jG+jL thus supporting the assumption of Nicklin et al. for single bubble motion. The velocity is thus given by:
where C0 and C∞ are two coefficients which remains constant for some range of mixture velocity and fluid properties. This relationship has the peculiarity of separating two physical effects: the mean flow transportation contained in the first term of the r.h.s. and the driving force included in the second term of the r.h.s.
Figure 4. Velocity of long bubbles vs. mixture velocity, θ=0°. D=50mm (Fréchou, 1986); 140 mm ♦, 100 mm ◊, 26 mm , (Martin) ____: Equation 17 with C0=1.2, C∞=0.35
Figure 5. Velocity of long bubbles vs. mixture velocity, θ=0°. D=146 mm (Ferschneider, 1982); 189 mm (Linga); ____: Equation 17 with C0=1.2, C∞=0 ; _ _ _: Equation 17 with C0=1, C∞=0.54
However, secondary effects due to viscosity, surface tension and pipe inclination complicate this law. Indeed the coefficients of Equation 17 take the form:
We know very little about the theoretical expression of the coefficients, except the very nice proofs that
which were given by Dumitrescu, 1943, and Benjamin, 1968, respectively. However, we have a substantial amount of experimental data which illuminates some unexpected features of the bubble motion. These features may be summarized as follows:
C0, C∞ vary monotonously with pipe inclination and surface tension through the Eötvös number and, to a lesser extent, with viscosity through the Reynolds number: these points will not be discussed here.
C0, C∞ may change drastically for some critical values of the dimensionless parameters: in other words the coefficients experience some transitions which deserve to be discussed.
The first transition appears in vertical flow. It is clearly visible in Figure 4 in the vicinity of j=0. For upflow (j>0), Equation 17 fits nicely the experimental data with the recommended values C0=1.2 and C∞=0.35, whereas for downflow (j<0), one must take C0=1and C∞=0.7. Martin observed that the bubble nose experiences a shape transition near j=0, from centered nose in upflow to unstable and non symmetrical nose for downflow. It is likely that the transition occurs when the destabilizing inertia force balances with surface tension force, the transition arising at some critical Weber Number.
The second transition has for a long time been a matter of controversy. It is shown in Figure 5 for horizontal flow. At high enough mixture velocity the long bubble velocity only depends on the mean flow with C0≈1.2 - 1.3 and C∞=0. In contrast, at low enough velocity C0=1 and C∞=0.54 in agreement with theoretical prediction indicating that the bubble should experience a drift in a horizontal pipe containing a still liquid. In fact, this phenomenon may be put in evidence experimentally only in large pipe.
The third transition was put in evidence in vertical flow, although it may also occur in horizontal flow. It can be proven theoretically that the bubble moves faster when the liquid flow is laminar upstream of the bubble nose than when it is turbulent: C0 =0.27 in laminar flow, C0 =1.2 for turbulent flow. This unexpected result suggested to Nicklin et al., 1962, that "the bubble velocity is very nearly the sum of the velocity on the center line above the bubble plus the characteristic velocity in still liquid". The result was confirmed from experiments earned out in a wide range of Reynolds number by Fréchou, 1986 (Figure 6).
The method generally used to determine the holdup in large bubble starts from the assumption that the separated flow region between the nose and the tail is fully developed. The liquid holdup may be known by eliminating the pressure gradient between Equation 13 and 14.
In the foregoing equation the shear stresses at both wall and interface are expressed by single phase relationships, in which the friction factors have to be closed following the method indicated in the entry Stratified Flow. Solving Equation 21 addresses an important issue: the pattern of the interface within the bubble must be known. For vertical flow the liquid forms an annulus, whereas it is stratified in horizontal flow. A transition thus occurs which must be modeled. Very little is known on this problem.
In the recent decade, some experimental data of gas fraction in the liquid slugs have been published. Results obtained with similar flow conditions but different pipe inclinations are illustrated in Figure 7 This presentation has the merit to emphasize that the evolution of the gas fraction with the mixture velocity has the same trend in horizontal and in vertical pipe: this suggests that the same physical process take place and that the same modelling can be used for both cases.
A description of the mechanism of entrainment may be explained as follows. The liquid shed from the rear of a liquid slug flows around the nose of the long bubble to form a stratified or annular film flowing downward. This film enters at a relatively high velocity into the front of the next slug at high relative velocity. As the liquid film enters the slug it entrains some gas. In the mixing zone at the front of the next slug there is a local region of high void fraction, which is clearly observable. In this region of high turbulence level, the mixing process carries some of the bubbles to the front of the slug where they coalesce back into the long bubble.
What is the basic difference between horizontal and vertical flows? Even if the gas fraction is higher in vertical than in horizontal flow, the net flux of entrained gas could be the same provided that the relative bubble velocity is smaller. Since the bubble drift is higher in vertical than in horizontal flow, this could be true. However, V>UGD and we can firmly state that the gas flux is higher in vertical than in horizontal flow. The gas entrainment raises another question. Figure 7 shows that below some mixture velocity there are no bubbles in the slugs. There is some critical velocity difference above which gas is entrained: this is the onset of bubble entrainment. In vertical flow, small bubbles are always generated at the tail of the long ones.
There are a few models in the literature which were developed for predicting the gas fraction in the liquid slugs. We shall not make room for those which are less than satisfactory. These models were developed specifically either for horizontal flow or vertical flow, and the result is quite disappointing when one tries to apply each to the other case. Keeping in mind that the mechanism of entrainment is basically the same whatever the pipe slope, a reliable model should do a good job in both cases.
Figure 7. Gas fraction in liquid slugs. Air-water, D=5cm Vertical flow. : Barnea & Shemer, 1989, : Mao & Dukler, 1991, Horizontal flow. : Andreussi & Bendiksen, 1989; Horizontal flow. ————; Vertical flow. - - -: Equation 22
Andreussi and Bendiksen, 1989, proposed a model which applies satisfactorily to horizontal or slightly inclined flows. It may be demonstrated that the gas fraction is expressed as:
In this equation, the critical mixture velocity and the velocity scale are given by empirical expressions including coefficients chosen for the best fit with experimental data. The onset of entrainment corresponds to a critical mixture velocity which is close to the velocity at the transition pictured in Figure 5.
The drift velocity in liquid slugs can be described by the modified Harmathy equation:
However, this law is not expected to work properly when the viscosity of the liquid is too high.
Another choice is to use a Drift Flux Model for the bubbly region. This model has been proposed from theoretical grounds by Kowe et al., 1988:
in which C1 accounts for the velocity and gas fraction distribution, Cm is the entrained mass coefficient and VB is the rise velocity of bubble in still liquid. For inclined pipe the question has not yet been resolved. As the bubble diameter is needed it has to be closed following the method indicated in the entry Bubble Flow.
It has been shown in previous sections that neither the characteristic length scale L of the cells nor their frequency n are needed to determine both void fraction and pressure gradient. However, there is a practical need for knowing the time or length scales of slug flow.
The mean slug length is one of the characteristic length scales. Since the probability distribution of the velocities are narrowly distributed about their average (Figure 2), the time and length scales are related through:
where , are the mean times of residence of slugs and long bubbles. Introducing the mean time of passage of the cell , the slug frequency may be defined as . Using Equation (2), it can be shown that the mean slug length and the slug frequency are related by:
In the case of horizontal flow, when the superficial gas velocity increases the mean length of the liquid slugs increases and then reaches an asymptotic value lying between 30 to 40D. Up to now, the modeling of slug frequency is not resolved. A more detailed discussion and modelling will be found in the review of Dukler and Fabre.
The UC model concept still appears as modern and robust. It uses at best the 1D balance equations together with two assumptions which are the corner stone of the model, and four closure laws. Concerning the closure laws, it may show that each has a specific influence.
The physical law of bubble velocity supports the model accuracy. There are still some uncertainties which concern mainly the transitions which have a strong influence on the dynamics: loss of symmetry in vertical flow, centered to non-centered nose in horizontal flow, influence of flow regime, annular to stratified regime of the film. The numerous results existing in horizontal and vertical flow must not disguise the lack of result for different inclinations. Much can be learned from numerical experiments of single phase flow around long bubbles.
The two other major issues concern the bubbly part of the liquid slugs: gas fraction and bubble motion. The paths by which gas enters and leaves a slug appear to have been identified. However the models necessary to convert these ideas into general predictive methods have not yet been developed. Careful experiments are needed.
There are no experimental data on the slip velocity between bubbles in the liquid slug and the surrounding liquid. It turns out that none of the existing experiments of bubbly flow are applicable to liquid slugs: it must be realized that in most cases the velocities in liquid slugs are considerably greater than they are in bubbly flow experiments. Numerical experiments on this problem are probably useless since the current turbulence models fail in the presence of a dense cloud of bubbles.
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