GeMer, D.. I d . Eng. Chem., Fundam., 10, 534 (1971). Kase, S..J. Appl. Polym. Sci., 18, 3279 (1974). Matovich. M. A,, Pearson, J. R. A,. Ind. Eng. Chem., Fundam., 8, 512 (1969). Miller, J. C., S.P.E. Trans., 3, 134 (1963). Nickell, R . E., Tanner, R. I., Caswell. E., J. FIMMech., 65, 189 (1974). Pearson. J. R. A,, Matovich. M. A,. Ind. Eng. Chem., Fundam., 8, 605 (1969). Pearson, J. R. A,, Shah, Y. T., Trans. SOC.Rheol., 16, 519 (1972). Pearson, J. R. A., Shah, Y. T.. Ind. Eng. Chem., Fundam., 13, 134 (1974).
Shah, Y. T., Pearson, J. R. A.. Ind. Eng. Chem., Fundam.. 11, 150 (1972). Trouton, F. T.. Proc. Roy. SOC.Ser. A, 77, 426 (1906). Weinberger, C. B., Ph.D. Dissertation, University of Michigan, 1970 Weinberger, C. B., Goddard, J. D., Int. J. Multiphase Now, 1, 465 (1974).
Received for reuiew January 20, 1975 Accepted June 30,1975
A Model for Gas-Liquid Slug Flow in Horizontal and Near Horizontal Tubes Abraham E. Dukler’ and Martin G. Hubbard Chemical Engineering Department, Universify of Houston, Houston, Texas 77004
A model is presented which permits the prediction in detail of the unsteady hydrodynamic behavior of gas-liquid slug flow. The model is based on the observation that a fast moving slug overruns a slow moving liquid film accelerating it to full slug velocity in a mixing eddy located at the front of the slug. A new film is shed behind the slug which decelerates with time. Mixing in the slug takes place first due to the mixing eddy and then due to the usual diffusion due to turbulence. The model predicts slug fluid velocity, velocity of propagation of the nose of the slug, film velocity as a function of time and distance, length of the slug, film region behind the slug, and mixing eddy and shape of the surface of the film region. Agreement with experimental data is good.
Introduction Gas-liquid flow in conduits is a more complex phenomenon than single-phase flow primarily because the spacial distribution of the two phases is unknown and difficult to specify quantitatively. A variety of such distributions have been qualitatively described by numerous investigators (Hewitt and Hall Taylor, 1970; Hoogendoorn and Welling, 1965; Kosterin, 1949). Hubbard and Dukler (1966) suggested that these many observed patterns really represented the superposition of only three basic distributions: separated, intermittent or slug, and distributed flows. A still unresolved problem is the prediction of the particular combination to be expected given the flow rates, fluid properties, conduit size, and inclination. A useful empirical correlation has been proposed (Baker, 1954). Intermittent or slug flow exists over a wide range of flow rates for moderate pipe sizes in a horizontal configuration. Such a flow pattern is inherently unsteady with large time variation of the mass flow rate, pressure, and velocity at any cross section normal to the tube axis. This is so even when the gas and liquid flow to the system is steady. As a result, processes of heat and mass transfer are also unsteady with substantial fluctuations in temperature and concentration. This poses special and difficult problems for the designer. I t is the purpose of this paper to present a systematic model for the hydrodynamics of slug flow from which the time varying behavior can be predicted. The model presented here is based partly on the work of Hubbard (1965) which was presented by Hubbard and Dukler (1968) but unpublished. That version has been substantially revised to eliminate certain empirical aspects and certain simplifications made in the original work which have now been found to be unnecessary. Some Related Research A number of early investigations (Govier and Omer, 1962; Hoogendoorn, 1959; Martinelli and Nelson, 1948)
measured pressure drop and in some cases average holdup under conditions where slug flow was observed to exist. A careful study of these data and the experimental techniques reveal serious limitations in the results. In many instances a portion of the pressure measuring system was not in fully developed slug flow. In some cases the slug length (estimated by methods to be reported here) exceeded the distance between pressure taps. In most of these studies only air space existed between pressure taps at times. In all cases pressure drop was measured using highly damped manometers to smooth out the fluctuations. Furthermore, these studies provide no information on the characteristics of slug flow such as frequencies, spacing, slug length velocities, etc. The earliest attempt to study the details of slug flow were by Kordyban (1961). He proposed a simple model where slugs move at the velocity of the gas and “skate” or slide over the top of a substrate film without interaction or mixing between the slug and film. Experimental data were taken in 6 ft long tubes of 0.315 and 0.420-in. i.d. with pressure taps separated by 1 ft. Thus the validity of the data is seriously in question. Based on their concept, the authors develop an expression for pressure drop but this is in poor agreement with their data. In fact the classical Martinelli correlation is shown to be a better predictor of the data than their own correlation. In 1963 Kordyban and Ranov (1963) reported new experimental data from a 10-ft long pipe of 1.25-in. diameter. They generate slugs by forcing the two-phase mixture through a riser before entering the test section. If this riser had not been used it is doubtful if they would have been able to observe slug flow, especially a t the higher flow rates in their work. In this instance the pressure measuring technique produced data that were more meaningful. However, comparison with the earlier Kordyban model leads the authors themselves to suggest that the model is not adequate. The beginnings of an attempt to understand the details Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
337
LEVEL
DROPS
SLUG JUST PASSES OUT OF VIEW TO THE RIGHT LEVEL DROPS
LEVEL REBUILDS AND WAVE NEARLY BRIDGES PIPE
POSITION (OR TIME)
Figure 1. The physical model for slug flow. BRIDGING OF PIPE BY LIQUID
of slug flow can probably be found in the work of Nicklin et al. (1962), who attempted to modify expressions for the slug velocity which was developed theoretically for a vertical system to the horizontal case. These ideas were generalized somewhat by Hughmark (1965), who was concerned with the heat transfer problem and by Neal (1963), who examined some of the limited data of Richardson (1958) on slug velocity. This was further examined by Marrucci (1966). Oliver and Wright (1964) reported on heat transfer during slug flow in a test section of 0.25-in. diameter and 4 f t long. They made no measurements of the detailed characteristics of the flow since their concern was the heat transfer process. Slug flow in these small sizes is a very special situation which is equivalent to bubble flow with Taylor type bubbles having bullet shaped noses separated by liquid plugs moving in laminar flow. There is little or no asymmetry and the similarity with slug flows in larger diameter tubes is small. Greskovich and Shrier (1972) applied the model presented by Hubbard and Dukler (1968) to horizontal systems in pipes of 1.5 to 6.07-in. diameter but used independent relationships for slug frequency and holdup. In general the results were shown to predict pressure drop with good reliability. In a subsequent paper (Bonnecaze et al., 1971) they modified the preliminary model to account for tube inclination. New relations for slug velocity were prepared based on a potential flow model but the relative importance of gravity and friction forces was not clearly demonstrated. Despite these untenable assumptions, agreement with some field data taken by Esso on a 10,000-ft section of 6-in. line was good. Vermeulen and Ryan (1971), following some aspects of Hubbard's model, compared their own data taken in a 0.5in. x 18-ft test section with their simplified form of the model. Their data included, in addition to the horizontal flow configuration, some results at positive and negative inclinations of 7'. Agreement was good but the model did not account for film acceleration and the data itself was used to estimate this quantity. Since such acceleration is over 50% of the whole, the compromise involved is clear. No comparison was presented between experiment and the simplified theory for many of the details of the flow. Application of two-phase flow in capillary tubing could be of industrial importance in a few specialized instances. Here the flow is laminar, the gas bubbles are almost symmetrically located in the pipe, liquid film exists at all points on the wall and the analysis is subject to a near rigorous development. The papers of interest are those by Cox (1964), Bretherton (1961), Goldsmith and Mason (1963), Suo and Griffith (19641, and Suo (1968). Most of these papers are directed toward determining the rate of propagation of the gas bubble out of the tube but they also consider the velocity distribution and residence time of the gas phase. 338
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
SLUG FORMATION
0 SLUG SWEEPS UP LIOUID
LEVEL DROPS
Figure 2. The process of slug formation.
A Description of the Condition of Slug Flow, Its Initiation and Dissipation The process of slug flow is a highly complex unsteady phenomenon. An understanding of the flow mechanism has been developed from extensive visualization studies including short exposure time still photographs, motion picture sequences at several speeds, and dye tracer work. The study was carried out in a 1.5-in. i.d. horizontal pipe, 65 f t long. The mechanism proposed was substantiated by measurements of wall pressure using fast response pressure transducers located flush with the wall. The results are summarized below. Visual Observations. Refer to Figure 1 for a sketch of an idealized slug which has been fully established and to Figure 2 for a sketch of the process of slug formation in horizontal pipes. The picture of slug flow which emerges from these observational studies is as follows. 1. Liquid and gas flow concurrently into a pipe. Near the entrance the liquid flows as a stratified phase with the gas passing above. At gas and liquid velocities under which slug flow takes place, the liquid layer decelerates as it moves along the pipe. As a result, its level increases, approaching the top of the pipe. At the same time, waves appear on the liquid surface. Eventually the sum of the rising liquid level plus the wave height is sufficient to bridge the pipe momentarily blocking the gas flow. (See Figure 2A, B, and C). 2. As soon as the bridging occurs, the liquid in the bridge is accelerated to the gas velocity. The liquid appears to be accelerated uniformly across its cross-section, thereby acting as a scoop, picking up all the slow moving liquid in the film ahead of it and accelerating it to slug velocity. By this mechanism, the fast moving liquid builds its volume and becomes a slug. (See Figure 2D.) A fully formed slug is shown in Figure 1. 3. As the slug is formed and moves down the pipe, liquid is shed uniformly from its back and forms a film with a free surface. This liquid in the film decelerates rapidly from the slug velocity to a much lower velocity as controlled by the wall and interfacial shear. (See zone l f in Figure 1.) 4. Once a slug is formed as it travels down the pipe it first sweeps up all the excess liquid which had entered the pipe since the last slug was formed. From that point on it picks up liquid film which has been shed from the preceding slug. Since the slug is now picking up liquid at the same rate that it is shed, its length stabilizes. 5 . The slug has a higher kinetic energy than that of the
I
4
k O . l S /
Vs= 5.0 ft./sec.
Figure 3. Pressure-time trace for slug flow.
liquid film. Thus, the film penetrates a distance into the slug before it is finally assimilated a t the slug velocity. This over-running phenomenon creates an eddy at the front of the slug which is essentially a mixing vortex. The distance of penetration constitutes the length of the mixing eddy. In this mixing zone gas is entrapped due to the violent mixing operation. (See zone 1, in Figure 1.) 6. As the gas rate and consequently the slug velocity increase, the degree of aeration of the slug increases. Ultimately the gas forms a continuous phase through the slug. When this occurs the slug begins bypassing some of the gas. At this point the slug no longer maintains a competent bridge to block the gas flow so the character of the flow changes. This point is the beginning of “blow-through’’ and the start of the annular flow regime. Qualitative Confirmation of the Model. Figure 1 shows an idealized pressure profile through a slug unit viewed a t an instant in time and based on the mechanism presented above. A sharp rise in pressure takes place across the mixing zone associated with the force necessary to accelerate the slow moving liquid in the film ahead of the slug to the velocity of the slug. There follows a linear change in pressure due to gradient shear in the body of the slug and equivalent to that which would take place in full pipe flow with no slip between the distributed gas and liquid. In the film zone the pressure is essentially constant since the pressure drop is small compared to that in the liquid slug. Consider a wall pressure transducer located a t point, T , along the test section. This point is selected so that there is never more than one slug between T and the discharge location. Then the pressure measured a t T is that due to the pressure drop across one slug. When the slug passes out of the pipe so that vapor space exists between T and the exit (before another slug arrives) the pressure drops to essentially that at the exit. As a slug moves across the station a t T the profile of pressure through the slug can be recorded. Figures 3-5 are time traces obtained using air-water in a 1.5-in. horizontal pipe with the pressure transducer located 8.5 f t upstream of the discharge. Three different slug flow conditions appear. Figure 3 represents a low flow rate where the film and slug velocities approach each other and thus where film acceleration and its associated pressure drop are small. Figure 5 represents a very high slug velocity and a low film velocity so that a large pressure drop is required to accelerate the film. In each trace the time during which a slug passes over the measuring station is indicated by a bar as determined from a separate conductivity probe. These figures show the principal contributions to the pressure pulse associated with a slug. The zone, labeled 1, is the pressure rise due to the increase in hydrostatic pressure on the arrival of the slug over the measuring station. Zone 2 is an additional abrupt rise in pressure a t the front of the slug associated with the acceleration of the liquid film in front of the slug. Zone 3 is a gradual linear pressure rise due to the frictional pressure gradient across the back of the slug behind the mixing zone. The pressure trace as measured a t T also displays changes in pressure when the slug drops out of the pipe and also reflects changes in the static pressure
Figure 4. Pressure-time trace for slug flow.
r vs= 174 f t
/sec
Figure 5. Pressure-time trace for slug flow.
of the entire system. In particular this is evident in Figure 4 after the slug passes and these variations can be explained in terms of coupling between the slugs (Hubbard, 1965). Note that as the slug velocity increases there is a corresponding increase in the fraction of the total pressure drop across a slug which is due to acceleration of the liquid film. In these experiments independent measurements were made of slug and film velocity as well as of film flow rate. It was thus possible to calculate the pressure drop due to acceleration and that due to friction. These were shown to agree very well with the measurements made from several hundred traces of the type shown in these figures. Thus there has been demonstrated the reasonableness of this two-zone model which requires a pickup of the film in a mixing zone, the existence of a “full pipe flow” region behind the mixing vortex and a process of shedding of the film behind the slug to form a film region. In the discussion which follows, the process of slug flow is treated as a fully deterministic one. In fact, experiments make it evident that slug flow has certain random features. In particular, the time and space between slugs does vary and so does the pressure drop across slugs, the film velocity, and other features. There has been no satisfactory treatment of this randomness. However, the data show that the probability distributions are narrow. In the work that follows each quantity can be interpreted as the mean of a set of values which distribute about the mean.
The Hydrodynamic Model Pressure Drop across a Slug. Figure 1 shows that there are two contributions to the pressure drop across a slug. The first, APa, is the pressure drop that results from the acceleration of the slow moving liquid film to slug velocity. The second, APf, is the pressure drop required to overcome wall shear in the back section of the slug. The total pressure drop across a slug is thus AP, = A P ,
-k
APi
(1)
Pressure drop in the gas phase above the liquid film is negligible. Acceleration Contribution. A slug that ha5 stabilized in length can be considered as a body receiving and losing mass a t equal rates. The velocity of the liquid in the film just before pickup is lower than that in the slug and a force is therefore necessary to accelerate this liquid to slug velocity. This force manifests itself as a pressure drop given by
where x is the rate at which mass is picked up by the slug, Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
339
-+ “fe-
I
t
I
AB C
Figure 6. The pickup process at the front of a slug.
Figure 7. The shedding process at the back of a slug.
V, is the mean velocity of fluid in the slug, and Vf, is the mean velocity of fluid in the film in front of a slug. This pressure drop due to acceleration takes place over a mixing eddy at the front of the slug which penetrates a distance, l, into the body of the slug. Frictional Contribution. Behind the mixing eddy in the body of the slug pressure drop takes place due to wall friction. For the calculation of this term, the similarity analysis for two-phase frictional pressure drop developed by Dukler et al. (1964) is applied. Within this part of the slug the two phases are homogeneously mixed with negligible slip. Under this condition, the recommended pressure drop equation becomes
The translational velocity, Vt, must also satisfy the following relationship
The similarity analysis showed that for “non-slip” conditions f, could be correlated as a unique function of Re, when this parameter is defined in the following manner and that when the liquid holdup exceeded 0.7 ( R , > 0.7) this correlation is identical to ones for single phase flow.
translational mean fluid velocity at = velocity in the slug nose the slug
apparent velocity adding fluid at the slug nose
+ gained by X
vt
VS
P L ARS
Comparing this equation with eq 7 indicates that the correct definition of the mean slug velocity is
Thus, V, can be calculated from a knowledge of the input volumetric flow rates and the pipe area and is independent of the distribution of liquid between the film and the slug. Equation I can now be written as
v,
=
v, +
X ~
PLARS
It is convenient to define a term C In order to calculate pressure drop across a slug using eq 2, 3, and 4 the following quantities must be determined: liquid pickup and shedding rate, x ; film velocity, Vf,; average fluid velocity in the slug, V,; slug holdup, R,; slug length 1,; and length of acceleration section, 1., Slug Velocity. The scooping model requires that two characteristic slug velocities be defined. V, represents the mean velocity of the fluid in a slug relative to the pipe wall. The observed rate of advance of the slug, Vt, is the sum of V, and the rate of buildup a t the front due to film pickup. As discussed above, the slug length remains constant because the amount of liquid shed from the rear is identically equal to the amount which adds to the front. To develop a relationship between these two velocities, consider the overall balance between liquid entering the pipe and that which leaves with a slug unit of length, l,, as it passes out of the pipe. The mass of liquid in a slug unit is (lsRs IfRf)ApL and the time for the slug unit to pass out of the pipe is l / u s . However, as the slug unit passes out of the pipe, the slug which follows overruns and captures part of the liquid in the film. This liquid does not move out with the slug unit as calculated above. The mass of liquid picked up by a second slug during the time the first slug unit passes out of the pipe is xl,/Vt. Since u, = Vt/lu, the material balance becomes W X = (Z,R, + IfRf)v, -(5) APL PLA Similarly developed, the gas phase material balance is
+
Adding eq 5 and 6 and simplifying yields for the slug translational velocity
340
Ind. Eng. Chem.. Fundam., Vol. 14, No. 4, 1975
which is the ratio of the rate of shedding to the rate of flow in the slug.
v,
= (1
+
C)V,
(11) In order to use eq 9 for calculating Vt it is necessary to develop independent expressions for the pickup or shedding rate, x , and liquid holdup in a slug, R,. The mechanism for pickup and shedding are quite different. This provides for independent relationships between the variables. The Pickup Process. Refer to Figure 6. Consider a slug whose outline is designated by the solid line and whose front is located at the plant A-A a t a specified instant in time. In the interval A t , the front of the slug shown dotted moves to plane C-C. The film, formerly located at AA, moves only to BB because of its lesser velocity. The amount of film shown cross-hatched is picked up and mixed into the slug. Thus the mass rate of pickup is
x = p L A R f , ( V , - vfJ
(12) The Shedding Process. Consider the region of the slug behind the mixing zone. According to the model the flow in this region is equivalent to fully established pipe flow and, as seen in eq 3, the pressure drop is calculated as if a pipe flow turbulent velocity distribution existed there. The mechanism for shedding can now be understood by reference to Figure 7 . The average velocity of the liquid in the slug is distributed radially from a value of zero at the wall to a value above V, a t the center. There is one specific radial location where the local velocity, u , equals the average velocity, V,. Designate this radial position as r p . At values of r < r p the fluid moves faster than V,. Thus it advances in the direction of flow with respect to the motion of the slug. But in the region r > r p (see cross-hatched area) the fluid moves slower than the average fluid in the slug. Thus it eventually is shed from the rear of the slug. This rate of shedding is the mass rate of flow in the slug across any
plane drawn normal to the pipe axis less that which flows in the area r < rp.
x = R,pLV,A
-s,”
2avpLudr
u* ’
’*’ - -
( y = R - r)
n
(13)
There exist well established relationships for the velocity distribution for turbulent pipe flow in terms of the dimensionless parameters, u f and y + where u+ = -U‘ Y +
.30
(14)
20
50 100 ~ e ,iIO-?
500
Figure 8. Relationship between C and slug Reynolds number.
U
(15) *E.-
Written in terms of these variables the equation for C becomes
-
dxt 1 2
3
Figure 9. Definition of control volume in the film. where y is the value of y + a t the pipe centerline.
A well accepted equation for the central region of the pipe, y + > 30, in terms of these variables is
u+ = A
+
1
- In y + K
(18)
where K is the von Karman constant. There exist some differences in the literature concerning the values to be assigned to A and K . For this work we use A = 5.75, K = 0.38. Note that a t u = V, (u+),, =
while cy
the film velocity, Vf, and the film holdup, Rf, as a function of position from the rear of the slug (or of time since slug passage). Refer to Figure 9 and consider the principle of momentum conservation as applied to the control volume of fluid which exist between plane 1 and 2 separated by dxf. The pressure in the vapor space is constant snd independent of x f as shown by experiment. Thus a balance in the xf direction is
In this equation the momentum associated with the gas phase is neglected since the density is small compared to the liquid and its change in velocity across the control volume is small. P represents the average hydrostatic pressure acting on the liquid over the liquid area which exists in the plane normal to xf. This, of course, depends on the value of the pressure in the gas phase, P,, and the distance from the surface to the center of pressure in the liquid.
= K [ ~ - A ]
There is now sufficient information to perform the integration of eq 16. Using smooth friction factors for smooth pipes all terms in the equation can be calculated once Re is specified. The integrations, which are straightforward, were carried out over the range 30,000 IRe, I400,000 and the resulting values of C are presented in Figure 8. I t is seen that, in accord with experiment, the value of C varies only slightly over a wide range of Reynolds numbers. The integration of eq 16 is cumbersome to carry out but the relation between Re, and C can be approximated by observing the near loglinear relationship shown in Figure 8. For purpose of computation the following equation is recommended C = 0.021 In (Re,) + 0.022 (21) Hydrodynamics of the Film. In developing the equation for the pickup rate a new parameter was introduced, Rf,,the fractional pipe area occupied by film just before pickup by a slug. As seen from eq 2 it is also necessary to find Vf,, the film velocity a t this same point. Both of these quantities depend on the process of deceleration in the film behind the slug after shedding takes place. In order to arrive a t these quantities, the momentum balance is applied to the film. By making the application to a differential length of film, it will be possible to derive expressions for
= P , ipLgL 50 gC
(22)
where .$ is the ratio of the distance from the surface to the center of pressure in the liquid to the diameter. F , is the force a t the pipe wall due to friction and this force tends to retard the flow. F , is the force due to gravity acting in the xf direction. With /3 the angle between the horizontal and the axis (for up flow this angle is positive), then
F , = pLg4 AR, (sin p ) dx gc
(23)
The subscripts 1 and 2 designate the planes to which the bracketed term applies. If plane 1 and 2 are allowed to approach each other then in the limit the momentum equation becomes
gc
7wpw
PLA
- g L R f sin p
(24)
T, is the wall shear stress and P , is the perimeter wetted by liquid film. Note that Vf designates a film velocity averaged over its cross-sectional area, (RfA). In order to integrate this equation, it is necessary to develop expressions for T,, P,, E and a relationship between Rf and Vf.
Ind. Eng. Chem., Fundam., Vol. 14. No. 4, 1975
341
The Wetted Perimeter, P,: Let b' be the angle which subtends the liquid in the film as shown in Figure 11. Then by definition OD Pw = 2
3
Figure 10. Flow relative to the stagnation point, S.
Thus Re, =
2nBR
Re, 0
(34)
The Distance to Center of Pressure on the Liquid, 4. The force due to pressure is obtained by an integration of the pressure over the liquid crossectional area and this result is [ - R , 71~ 02 c o s 8 + 2 ./2 sin3 2 3
Figure 11. Definition of the angle, 8.
Relationship between Vf and Rf.The profile of the film behind a slug appears as shown by the solid curve in Figure 9. The curvature a t the top of the pipe is due to surface tension. If surface tension were zero, then the curve would be as shown by the dotted extension to point S. In fact, the forces due to surface tension are easily shown to be small compared to the other forces acting and the shape of the film at the back of the slug is assumed to end a t the point designated as S. Since the slug moves without a change in length, S must move along the pipe a t the same velocity as the front of the slug, namely Vt. Define a velocity, w, measured relative to the velocity of propagation of this stagnation point, S. Then at any point Wf
(33)
= Vf -
(2 5)
vt
But the force in terms of the distance to center of pressure is
Equating these two expressions and solving for E gives
4
1 8 = -- cos 2
2
+
1
- sin3
3nR,
Note that Rf and b' are uniquely related by
R, =
B-sint,
(36) 2n Substituting the equation for Vf in terms of Rf (28), and the equations for T, (30), P , (34), and E ( 3 5 ) into eq 24 gives, after considerable transformation and simplification
Let the coordinate system translate to the right a t this velocity, Vt. The point, S, is then stationary and flow then takes place relative to S as shown in Figure 10. A material balance between flow across plane 3 and any other plane drawn normal to the film flow is WfRiPLA = W ~ R ~ P L A (26) From eq 11 and 25
In this equation, Fr is the Froude number
v,
-
v,
=
cv,
= -w,
(2 7)
Substituting (25) and (27) into (26) and recognizing from (11)that Vt = (1 C)V, provides the needed relationship between Vf and Rf.
+
V, = V,
[1 - C
R1)]
= SVs
(38) and $ is the dimensionless distance measured from the stagnation point, S.
(28)
Wall Shear Stress, T ~ In. the film region the wall shear due to liquid flow is estimated by assuming pseudo parallel flows.
$ =
1, -
D
Xf
(39)
The variables are separated and integrated to give (4 Oa)
(29)
where Substituting for Vf from eq 28 gives W(R,) = (30) The friction factor for the film is evaluated from smooth tube f(Re) data with the Reynolds number of the film based on the hydraulic diameter
[:
C2Rs2 - Ri2 Fr
8 R, s i n 5
+
8 sin2 -
1 - cos 8 f f B 2O- + R Sin P n Fr
;
- - cos-
(4Ob)
(31)
Re, =
nDR B P
342
Re,
W
Ind. Eng. Chem.. Fundam., Vol. 14. No. 4, 1975
(32)
Note that W is a function of Rf as given by eq 40b and varies along the film. At xf = 0, just before pickup of the next slug, J. = lf/D and R f = Rf,. At point S, x = lf, J. = 0, and Rf = R,. The profile for Rf vs. xf can be obtained from
sRs
W(Rf)dRf =
1, D
Xt
(41)
Rf
The integration of eq 40a must be accomplished numerically and requires a value of lf. The procedure starts by evaluating W at the upper limit and adding increments of hRf until a value of Rf is produced for which the integral equals lf/D. This value of Rf = Rf,. The Length of the Slug, I,, and the Film Region, If. The length of a slug unit is
Velocity of the Gas Phase. Consider a coordinate system moving with the velocity, Vt. Now examine velocities of the gas phase relative to this moving coordinate system as shown in Figure 10 for the liquid phase. In order to satisfy continuity requirements
(42) With some rearranging this equation becomes
and that of the film region is (43)
(45)
The length of the slug can be calculated from a material balance on the liquid. The rate of liquid flow into the pipe is WL. Consider a plane normal t o the flow a t some position downstream where fully developed slug flow exists. We proceed by calculating the mass of liquid crossing the plane in (a) the time it takes for the slug to pass, T,, and (b) the time it takes for the film to pass, Tf. The sum of these two quantities is then divided by the time for passage of one slug unit, 1/uS.
Note that for liquid slugs that are free of entrained gas, R, = 1.0. Then the gas velocity equals the velocity of propagation of the slug, Vt, at all positions behind the slug. When R, is not unity, then the gas velocity varies slightly with location. At the point of pickup, where Rf = Rf,, the gas velocity is
M , = mass c a r r i e d in the slug =
While the gas phase velocity has no influence on the pressure drop it is important to the modelling of heat and mass transfer operations for slug flow. Length of the Mixing Eddy, I,. The depth of penetration of the liquid film into the slug appears to depend on the relative velocity between slug and film. This observation derived on physical grounds is verified from studies of high speed films and still photographs. The relative velocity is (V, - Vfe). A simple and effective estimate can be obtained using the “velocity head” concept. This velocity head is defined as
JOT’
; I I ,
V$RspLdt
V&RgL9
= vfiRsp,Ts
1
vt
Similarly for the film
Jo VfARfpLdt Tf
,!A!
=
(46)
VH = PL(’S
- ‘fJ2 2sLgc
From eq 28, VfRf = V,[Rf
where SL is the specific weight of the liquid. Resistance to flow in unusual geometries has been shown to be related to the velocity head as is the trajectory distance of h fluid released from a nozzle. A convenient measure of each of these quantities is the “number of velocity heads.” From the observations mentioned above a simple correlation emerges
- C(R, - Rf)]
1, = 0.3VH
Solving for 1, and rearranging gives
(47)
_ _wL __
[ R f e ( l + C) - CR,]Z 1“ = PLA v, %{R, - [Rf,(l + C) - CR,]Z}
vt
where
=
.fo’
R f ( l + C) - CR R f e ( l + C) - CR:
(7)
The relationship between R f and xf can be calculated from eq 41. Such calculations indicate the Z approaches 1.0 for all flow conditions. This confirms what is observed in experiment, that the film drops quickly to near its value of Rf, a t a short distance behind the slug and then continues to decrease only slightly. With Z set equal to 1.0
A Summary of Equations, Variables, and Procedure for Solution. In Table I the equations are listed in sequence and the new variables which are introduced in each are identified. Note that the variables appearing in the equation set are 16 in number, namely: AP,,APa, APf, x, Vs, Vfe, Rs, L, if, L, Res, WL, WG,Vt, Rfe, us. The physical , MG) and the tube size and orientation properties ( p ~p, ~ ML, (D, A, p) are, of course, known. As shown, the number of independent equations is 12. Thus four variables must be specified to effect a solution. W Land WGare input data. It has been shown (see section on “Description of Slug Flow”) that the slug frequency is controlled by conditions near the entrance and is not coupled to the behavior of a slug. At this time no independent relationship exists for R,. Thus the independent variables which are required as input to the model are WL, WG,us, R,. Despite the complex appearance of the equation set the Ind. Eng. Chem.. Fundam.. Vol. 14, No. 4. 1975
343
Table I. A Summary of Equations and Variables SEOUENCE NUMBER
Table 11. Computational Sequence
EPUATION NO. I N TEXT
EPUATION
I
I
2
2
3
3
P L . P O , PLI D
w
~
WG, t RS.
vs CALCULITE
C4LCULATE
Vs
C4LCUL4TE R e s Ea 4
4
4
S
8
8
10
= 0.021 In Re.
T
+
0.022
C
'L'
'G
*NUMERICALLY INTEGRATE Ea A F R O M R , TO A V A L U E OF R t e WHICH M A K E S
21
P LRSAVS 8
v,
=
+ ClVS
v, +*A'
(I
9
1NTEGR4L
9. I1
2
1 ,/ D
4-7
40A
C4LCULATE
1
NOTE
V I L U E S OF R f . V f . V s . X i , C4N BE OBTAINED
B Y STORING THE INTERM E D l 4 T E V b L U E S OF
408
THE lNTEGR4L 4S THE C A L C U L I T I O N PROCEEDS.
8:
I
-
C
[
v
]
20
34
36
IO
44
".
2 -1,
I1
PI'
I2
l.'o.lslvs-vf,,~~
"s
43
47
'L
approach to solution is quite straightforward and is accomplished readily by computer, given the required four input variables. In fact, only one calculation loop is needed due to the coupling between eq 40 and 49. The block diagram of Table I1 details the calculational procedure.
Comparison of Model Prediction with Experiment An experimental study of slug flow for an air-water system has been carried out in a horizontal smooth glass tube 1.5-in. i.d. and 65-ft total length. The test section length was made up of 8.5-ft pipe lengths joined in Plexiglas blocks which were carefully machined to match the i.d. with no discontinuities. Fast response pressure transducers were located in each block and connected to the flow channel through small drill holes kept full of water. A tee type entrance section was used with water introduced on the run and air on the tee. Extensive experimental studies with various entrance sections demonstrated that slug frequency and other fluid characteristics are independent of the configuration. The following variables were directly measured over a wide range of liquid and gas rates, V,, Vt, Vf,, U sl,,, l,, R,, us. Tabulated data may be found in Hubbard (1965) along with detailed discussion of experiment techniques. The measurements clearly demonstrated the stochastic nature of the slug flow phenomenon. All the quantities above except for frequency displayed a range of values. Because of this fact, repeated measurements were made over the passage of many slugs a t each gas-liquid rate pair. Sample mean values were calculated from 25-100 observa344
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
tions. Standard deviations associated with these measurements are as follows: Vt = 8%; V, = 5%; Vf, = 22%; 1, = 35%; hp, = 20%. Part of this deviation is due to the randomness of the slug formation process with the result that each slug is somewhat different in length with different associated film velocities. In part, the deviation is due to error in measurement. Measurement error is estimated to be: V, and Vt = 5%;Vf, = 15%; 1, = 25%; AP, = 15%. As indicated above, test of the model requires as input data values of the frequency, us and slug holdup, R,. Experimentally measured frequencies as obtained from slug count are shown in Figure 12. In a future paper the slug initiation phenomena will be explored and a model will be presented for predicting these data. In the meantime, if the model presented here is to be applied to other conditions, the correlation of Grescovich and Shrier (1972) is recommended. Values of the slug holdup, R,, were measured using an impact probe especially designed to follow the transient as a slug passed. A water filled, Ih-in. tube formed in the shape of the impact tube of a pitot was placed at the centerline with the opening facing upstream. This tube was connected to a sealed water reservoir located outside of the test section into which was fitted a pressure transducer. A second transducer was located at the wall of the test section a t the plane of the mouth of the impact tube. Stagnation and static pressure were recorded continuously during slug flow. When the slug passed across the tube a sharp rise of momentum flux and impact pressure was observed. Since the fluid velocity could be calculated and the liquid and gas moved without slip, it was a straightforward matter to determine p , and R , using the pitot tube equations. Results of these experiments appear in Figure 13. It should be noted that the detector was positioned a t the axis and radial gradients in air concentration exists. In addition, considerable fluctuation in impact pressure with time was observed making it difficult to determine the impact pressure characteristic of that slug. For these reasons considerable scat-
c
20
I
I
I
L 4.
15
.
1413
-
I:! r
Y L.C"
0
01
03
02
04
WG ilb/sec)
2
3
Figure 12. Slug frequency data of Hubbard (1965)
4
B
9
MEASURED
Vr
5
6
7
IO
I1
12 13
15
14
16
17
Figure 14. Comparison of theory with experiment: V,. 10
R, 075
-
>
a a
FLOW RATE OF
20
Y 05
0
10
20
i
WATER IN A 1.5" Q HORIZONTAL TUBE
4 Y)
30
v,
Figure 13. Hubhard data for slug holdup.
CALCULATED
Vt
Figure 15. Comparison of theory with experiment: VT. ter was observed and this is clearly evident in the data. Fortunately, the predicted results from the model are not especially senSitive to the value of R, used as input. Theory vs. Experiment: V,. Values of V, were measured by removing the end of the tube and allowing the slugs to flow out the end and then undergo a free fall trajectory to the floor. Knowing the centerline height of the test section and measuring the horizontal distance from the end of the test section to the point where floor contact was made, the velocity of the slug as it left the tube could be calculated from well established trajectory equations. A comparison of measured values of V, with values calculated from the model, eq 8, appears in Figure 14 where agreement is seen to be excellent. Theory vs. Experiment: Vt. Electrical contact probes were located 8.5 f t apart in the last two Plexiglas blocks of the test section. These probes were introduced into the top of the block and penetrated only Y 4 - h into the test section. The circuits were such that when the front of a slug passed, the conducting water closed a circuit which started a timer. When the front of the same slug reached the next contact probe the timer was stopped. In this way the rate of advance of the front of the slug, or Vt, was measured. Equation 21 was used to calculate C as predicted from the model and Vt was then calculated from eq 15. A comparison of predictions of the model and experiment appear in Figure 15. Again agreement is seen to be excellent. Theory vs. Experiment: Vfe.Experimental values of Vf, were obtained by observing the trajectory of the film as it flowed out of the end of the pipe in free fall in the manner similar to that described above for V,. It was observed that the horizontal distance reached by the film decreased
,--. $19 W
w 0 3
$
0 4
FLOW RATE OF WATER IN A 1 5 ' D HORIZONTAL TUBE
0 2
0 3 31 I b / $ e c
Y
I
0 0
0 2
0 4 CALCULATED
06
(?)
OB
10
Figure 16. Comparison of theory with experiment: Vf,.
with time since the last slug passed and this is in accord with the prediction of the model. This distance was recorded just before the next slug left the pipe and values of Vf, were calculated. Values of VfJV, can be predicted from the model using eq 28 with Rf, calculated from eq 40a. A comparison of calculated and measured values of this ratio appear in Figure 16. There appears to exist a small systematic error with measured values being 10-15% lower than calculated ones. In these measurements air resistance was not considered in the trajectory equation and as a result the values of Vf, will be somewhat higher than those indicated here. Considering the fact that there exists some controversy on correct relaInd. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
345
I
Conclusion A model has been presented which can predict the detailed structure of slug flow, given flow rates, fluid properties, tube geometry and inclinations, and measured or predicted values of slug frequency and slug liquid holdup. In a future paper prediction of v, and R , will be discussed.
SOLID LINES INDICATE PREDICTION OF MODEL
WATER IN A I 5 " D HORIZONTAL TUBE
0 0
5
IO
20
IS
25
30
35
", Figure 17. Comparison of theory with experiment:slug length.
nP
0 U Y
a VI
2 n m J
0
20
10
30
", Figure 18. Comparison of theory with experiment: U s .
tionships to predict wall shear for open channel flow, this is very good agreement indeed. Theory vs. Experiment: Slug Length 1,. Slug length was measured using a single electrical contact probe described for the determination of Vt. In this case a counter was started when the nose of the slug contacted the probe and was stopped when the contact was broken as the back of the slug passed over the probe contact. Using this time interval and the measured value of Vt it was possible to calculate an experimental value of l s . Because of the highly aereated nature of the front of the slug an accurate determination of arrival time was difficult. Furthermore, randomness in the system is particularly evident in slug length. A comparison of predicted slug length from the model (solid lines) and the measurements (data points) appears in Figure 17. Despite difficulties in measurement the agreement is within experimental error and the correct trend is observed. Theory vs. Experiment: AP,. For each flow condition pressure traces were recorded similar to those of Figures 4-6 and from these AP, was measured for 25-50 successive slugs and an average calculated. AP,,from the model is predicted eq 1, 2, and 4 as indicated in Table 11. A comparison of experiment with the prediction of the model appears in Figure 18. There is a weak dependence of predicted values on the liquid rate through the appearance of WL in eq 42. For almost all conditions agreement between prediction and experience is within 20%. Considering that these difficult transient measurements are within this accuracy the agreement seems to be very good indeed. 346
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
Notation A = cross-sectional area of the pipe A f = cross-sectional area of the film C = ratio of rate of mass pickup to rate of mass flow in the slug (see eq 10) D = pipe diameter f s = friction factor for the slug f f = friction factor for the film F , = force acting on an element of film in the flow direction due to gravity F , = force at the pipe wall acting on an element of film in the flow direction due to friction g, = conversion factor gL = acceleration of gravity 12 = Von Karman constant 1, = length of slug l f = length of film 1, = length of slug unit lm = length of mixing eddy P = average hydrostatic pressure acting on film cross-sectional area P, = perimeter wetted by the film AP, = pressure drop due to acceleration across slug APf = pressure drop due to friction across slug U s= pressure drop across slug r = radial coordinate Rf = fraction of pipe flow area occupied by film R f , = fraction of pipe flow area occupied by film just before pickup R, = fraction liquid holdup in the slug Ref = Reynolds number of film Re, = Reynolds number of slug SL = specific weight of the liquid Tf = time for passage of a film T , = time for passage of a slug u = local velocity u* = friction velocity u+ = dimensionless local velocity (See eq 14) Vf = average velocity of fluid in the film Vf, = average velocity of fluid in the film just prior to pickup by the next slug V, = average velocity of fluid in the slug Vt = average translational velocity of the nose of the slug W , = gas mass flow rate WL = liquid mass flow rate W = grouping of variables (see eq 40b) y = distance coordinate measured from the wall y+ = dimensionless y (see eq 14) 1: = rate of mass pickup by slug a = defined by eq 20 /3 = angle between pipe axis and the horizontal y = value of y c at y = R 0 = angle which subtends the liquid film (see Figure 11) p~ = viscosity of gas p~ = viscosity of liquid V, = slug frequency 5 = ratio of the distance from the film surface to the center of pressure to the tube diameter p~ = density of gas p~ = density of liquid w = velocity measured relative to Vt (see eq 2 5 ) Literature Cited Baker, O., OilGas J., 185 (July 1954). Bonnecaze. R. H., Erskine, W., Grescovich, E. J., AlChEJ., 17, 1109 (1971). Bretherton.F. P.. J. NuidMech.. 10. 166 (1961). Cox, B. G., J. FluidMech , 20, 13 (1964). Dukler, A. E.,Wicks, M., Cleveland, R. G., AlChE J., 10, 44 (1964).
Goldsmith, H. L., Mason, S.G.. J. ColbidSci., 18, 237 (1963). Govier, G. W., Omer, M. M., Can. J. Chern. Eng., 40, 93 (1962). Greskovich. E. J.. Shrier, A. L., lnd. Eng. Chern.. Process Des. Dev., 11, 317 (1972). Hewitt, G. F., Hall Taylor, N.S. "Annular Two Phase Flow" Pergamon Press, 1970. Hoogendoorn, C. J., Chern. Eng. Sci., 9, 205 (1959). Hoogendoorn, C. J., Welling, W. A. "Symposium on Two Phase Flows," E x 6 ter, 1965. Hubbard, M. G.. Ph.D. Thesis, University of Houston, 1965. Hubbard. M. G., Dukler, A. E., "Proc. 1966 Heat Transfer 8 Fluid Mech. Inst.," M. A. Saad and J. A. Miller, Ed., Stanford University Press, 1966. Hubbard, M. G., Dukler, A. E., paper presented at the AlChE National Meeting, Tampa, Fla.. 1968. Hughmark, G. A,, Chern. Eng. Sci., 20, 1007 (1965). Johnson, H. A,, Trans. ASME, 77, 1257 (1955). Kosterin. S. I., lzv. Ak. Nauk SSSR, Otd. Tekh. Nauk, No. 12, 24 (1949).
Kordyban, E. S., Trans. ASME, 83, 613 (1961). Kordyban, E. S..Ranov, R . R., "ASME Mukiphase Flow Symposium," p 1, 1963. Marruci, G., Chern. Eng. Sci.. 21, 718 (1966). Martinelli, R. C., Nelson, B. D., Trans A S K , 70, (1948). Neal, L. G.. "An Analyses of Slip in Gas-Liquid Flow" Report of lnstitutt for Atomenergi, Kjeller Res. Est. (1963). Nicklin, D. J., Wilkes, J. O., Davidson, J. F., Trans. lnst. Chern. Eng., 40, 61 ( 1962). Oliver, D. R., Wright, S.I., Brit. Chern. Eng., 9, 540 (1964). Richardson, B.,Argonne National Laboratory Report ANL-5949 (1958). Suo, M.. Trans. ASME, J. Basic Eng., 90, 140 (1968). Suo, M., Griffith, P., Trans. ASME, J. BasicEng., 86, 576 (1964). Vermuellen, L. R., Ryan, J. T., Can. J. Chern. Eng., 49, 195 (1971).
Received for review February 5,1975 Accepted June 5,1975
Nonideality of Binary Adsorbed Mixtures of Benzene and Freon-I 1 on Highly Graphitized Carbon at 298.15 K Earle D. Sloan, Jr. and J. C. Mullins' Chemical Engineering Department, Clernson University, Clernson, Sooth Carolina 2963 1
Experimentaladsorption isotherms of binary gas mixtures of benzene and Freon-11 are presented for comparison with predictions by the ideal solution theory and by a two-dimensionalvan der Waals equation of state. The homogeneous carbon black, Sterling MTFF-D-7 (310OOC) with a surface area of 9.6 m2/g, was the adsorbent. A commercial electrobalance was used to measure both mixture isotherms at 298.15 K below 10 Torr and pure component isotherms at 273.15 K and 298.15 K at pressures up to 125 Torr. A chromatographic technique was used to correct the mixing rule for the energy parameter crii The calculated isotherms, using the corrected mixing parameter for the van der Waals equation, are shown to agree with the experimental isotherms. A small departure from the ideal adsorbed solution theory was found.
Introduction Many theories of adsorption such as the early BET theory (Brunauer et al., 1938) and a more recent theory by Lee and O'Connell (1972) assume sitewise homogeneity for the adsorbent. In spite of this basic assumption there is a paucity of data for adsorption equilibria of mixtures on either sitewise homogeneous adsorbents or adsorbents with a homogeneous field. The basic difficulty is that the composition of the adsorbed phase, which is frequently less than a monomolecular layer on a relatively small (e.g., 10 m2/g) surface area, must be determined. Friederich and Mullins (1972) obtained data for ideal mixtures of similar molecules using an equilibrium calculation method suggested by Van Ness (1969). The purpose of this paper is to present a different method for determining homogeneous absorbent mixture equilibria and to extend the mixture data available to include components which exhibit nonideality in the monomolecular layer region. Adsorption Equilibrium Relations Two equilibrium relations have been used here in the measurement of mixture equilibria on homogeneous adsorbents: a modification of the Gibbs-Duhem relation for the adsorbed phase and a relation for determining infinite dilution activity coefficients by a chromatographic technique. Gibbs Adsorption Isotherm. The Gibbs-Duhem relation for a two dimensional adsorbed film, restricted to constant temperature, is known as the Gibbs adsorption isotherm (Van Ness, 1969)
N
A n
- -dd.rr
2 xi dpia = 0 i=l
+
(constant T )
(1)
The chemical potential of species i in the ideal gas mixture is given by
pig(^, p , y i ) = G i o ( T ) + RT In P y l
( 2)
Since a t equilibrium the chemical potential of component i is equal in the adsorbed and gas phases, the differential of eq 2 may be substituted into eq 1 to obtain a useful relation.
--A
nRT
d n + d l n P + N
( x i d In y i ) = 0 (constant T ) (3) i.1
If eq 3 is restricted to constant gas phase composition, a means for calculating spreading pressure results P
=
n d(ln P) (constant T and y i )
(4)
In eq 4 the number of moles adsorbed may be determined by the mass adsorbed and an average molecular mass
i=1
Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
347
