Znd. Eng. Chem. Res. 1994,33, 1208-1221
1208
SEPARATIONS A Unified Model for Countercurrent Vapor/Liquid Packed Columns. 1. Pressure Drop Brian Hanley,’ Brian Dunbobbin, and Douglas Bennett Air Products and Chemicak, Inc., 7201 Hamilton Boulevard, Allentown, Pennsylvania 18195-1501
A model for the two-phase pressure drop in a packed column is developed from percolation concepts. Specifically, the macroscopic “flood” point for countercurrent liquid/vapor flow in a packed column is assumed to play the identical role that the incipient percolation threshold plays for the conductor/ insulator transition on an electrical lattice. The flow passages making up the void space of the packing are taken to be “conducting” when they are open to vapor flow and “insulating” when they are choked off by liquid (Le., localized flooding). The Wallis equation, developed to describe flooding for countercurrent flow in vertical tubes, is used to describe the flood point in the column. By an appropriate renormalization to the “flood” point, a simple equation for the pressure drop is derived.
Introduction Packed columns contain networks of contacting packing elements which are often distributed at random. The particular way in which these elements pack within a column gives rise to a spatial, as well as size, distribution of voidages oftentimes unique to that column. At any given set of flow conditions, some of these void spaces will contain very little or no liquid while others will be virtually filled with liquid because of interfacial forces, local packing conditions, or local gas and liquid flow conditions (see, for example, Tour and Lerman (1939,1944)). The open voidages allow for the passage of the majority of the vapor and thus appear to be analogous to electrical conductors while the filled voids allow very little vapor to pass and therefore behave in a fashion similar to electrical insulators. One might expect that the liquid-filled and open passages would be distributed in a fairly random manner, but this is not necessarily true due to poor initial liquid distribution and/or liquid channeling. As the operating conditions are changed so that the column is closer to “macroscopic flooding”,the proportion of filled voids would be expected to increase at the expense of open flow passages. In many respects, this is related to the replacement of conductors in a network with insulators. At certain particular combinations of vapor and liquid flow, the network of voidages changes in character from highly conducting to almost nonconducting. Others have recognized this fundamental change in the character of the column as the flood point is approached. According to Eckert (1963) The flooding point in the bed ...could be defined as that point in gas-liquid loading where the liquid phase becomes continuous in the voids, and the gas phase becomes discontinuous in the same voids of the bed. while Fair et al. (1973)state As the liquid hold-up increases, one of two changes may occur. If the packing is composed essentially of extended surfaces, the effective orifice diameter ~~
* Author to whom correspondence should be addressed.
becomes so small that the liquid surface becomes continuous across the cross section of the columngenerally at the top of the packing. ... The change in pressure drop is quite great with only a slight change in gas rate. This phenomenon is called flooding... . If the packing surface is discontinuous in nature, a phase inversion occurs, and gas bubbles through the liquid. ... Analogous to the flooding condition, the pressure drop rises rapidly as phase inversion occurs. These are undoubtedly statements alluding to percolation-the abrupt change in the “conductivity“ of the packed column (see Melli et al. (1990)for some interesting visualizations of two-phase cocurrent flow in a packed bed). Perhaps the most convincing evidence that there is indeed a gadliquid phase transition at the flood pointaside from visual observations-is the fact that there are large fluctuations in the pressure drop near the macroscopic flood point. Indeed, some researchers have abandoned the notion of a well-defined “flood” point because of the experimental difficulties in determining a representative “average” pressure drop as the flood point is approached. The gas/liquid phase transition here is an example of a “continuous” transition, i.e., a phase change which involves no latent heat effect. It is well-known that systems undergoing “continuous”phase transitions exhibit large fluctuations (Binney et al., 1992; Stauffer and Aharony, 1992). The large pressure drop fluctuations near flooding are clear evidence that a description of packed column operation in percolation terms is apropos. In a future paper we plan to investigate the nature and magnitude of packed column pressure drop fluctuations in light of the theory of continuous phase transitions. Our primary intent in this paper is to stress the analogy between the phase behavior of the macroscopic conductivity of a conductor insulator mixture and the pressure droplflow rate behavior for a packed column. Although the correspondencebetween our current model formulation and the realities of packed column operation is not exact, it does provide insight into the mechanism(s) governing the operation of packed towers. It also provides
OSSS-5SS5/94/2633-12QS~Q4.50/00 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 1209 a starting point for future percolation-based models which improve upon the simplifications and arguments we have made here. In particular, we will assume that the flood point in a packed column corresponds to the insulator/conductor incipient percolation threshold for an electrical network. By an appropriate renormalization of flow rates to the flood point, we arrive at a simple model for the pressure drop in a packed tower.
Description of the Model Electromagnetic analogies have been used with great success in the study of fluid mechanics (see, for example, Moore (1949)). Generally speaking, potential drop in the electrical problem is akin to pressure drop in the fluid flow problem while current in the electrical problem is comparable to flow rate in the hydrodynamic problem. potential drop * pressure drop current ==+ flow rate fc
This correspondenceplays a crucial role in the remainder of this paper. It is well-known that the macroscopic Conductivity of an electrical network, 2, near the conductor/insulator percolation threshold scales as
-
(1-f/fc)*
(1)
where f is the fraction of sites (bonds) which are insulators, fc is the fraction of sites (bonds) which are insulators at the incipient percolation threshold, and a is a uniuersal
scaling exponent, independent of the detailed structure of the underlying network (Stephen, 1983; Gingold and Lobb, 1990; Stauffer and Aharony, 1992). Scaling laws such as this are strictly valid only very close to the percolation threshold (away from fc there are correction terms), but in the case of the macroscopic conductivity the above scaling relation holds quite well even away from the percolation transition. If we define VO to be the potential drop required to maintain a constant current when f = 0, then the potential drop required to maintain this same current near the percolation threshold is given by
Much work has been done on the prediction of the percolation threshold, fc, and the scaling exponent, a,for linear resistors removed at random from well-defined trees and lattices. For bond problems (versus site problems), it has been found that fc can be calculated approximately for any lattice from the expression (Stephen, 1983) fc = %-I(-) d
d-1 where z is the “average” bond coordination number (the number of bonds which connect to a given site) and d is the dimension of the space. For the Bethe lattice (Cayley tree) fc can be calculated exactly for both the bond and site problem from the equation (Stauffer and Aharony, 1992) fc
1 =2-1
For a simple cubic network, z = 6 and the bond percolation threshold is predicted to be fc = 0.25. Equation
fraction of sites (bonds) occupied
Figure 1. Schematic of the potential drop on approach to the percolation threshold for lattices composed of conductors/semiconductors and conductors/insulators.
3b for the Bethe lattice predicts that fc = 0.2 for a lattice with a coordination number of 6. The above equations point out two important facts to keep in mind throughout the rest of this paper. First, the transition from conducting to nonconducting can take place even when a relatively small fraction of the conducting elements have been replaced by insulators for values of the average coordination number “typical” of most random packings (we estimate 4 5 z 5 9 by visual observation of a variety of random packing materials). Second, the exact value of the percolation threshold depends upon the underlying structure of the network. Monte Carlo simulations on an assortment of lattices of linear resistors and even on “random” networks of such resistors have yielded a value for the universal scaling exponent, a,of 2 (Stauffer and Aharony, 1992). By comparison to the electrical network, it seems reasonable to assume that, at some conditions of vapor and liquid flow, the isolated clusters of liquid-filled voidages aggregate to form a structure which spans the diameter of the column. This structure should form an effective barrier to the passage of vapor, so that we should expect the pressure drop under these conditions to be large relative to normal pressure drops encountered away from the flooding region. Indeed, if the liquid-filled passages allowed for the passage of no vapor, the pressure drop would become infinite at the flood point. In fact, the liquidfilled voids in a packed column do not completely prevent vapor from flowing, even at flooding. In this sense, the liquid might be considered a “semiconductor”rather than an insulator. When semiconductors are used to replace conductors on a lattice, the macroscopic conductivity near the percolation threshold rises abruptly, but it remains finite at fc; above fc the conductivity is more characteristic of that of the semiconducting phase. Figure 1illustrates the behavior of the potential drop near the incipient percolation threshold for the conductor/insulator transition and for the conductor/semiconductor transition. A transition similar to the conductor/semiconductortransition has been observed for the pressure drop in packed columns. Above the flood point, several researchers have observed a pressure drop “plateau” with pressure drops
1210 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994
occupied at any given set of conditions or the percolation threshold under those same conditions, we assume that the two-phase multiplier, 4, scales in the following way:
which is equivalent to postulating that the followingpower law relationship holds
Gas Rate (Ibsihr/ft2)
’F -
1
I
I
I
I
I
0
2
4
6
8
IO
I
;2
14x10’
Liquid Raw (Ibs/hr/ft2)
Figure 2. Two-phase pressure drop data of Lerner and Grove for 1-in.Raachig rings. Note the similarity between these data and the conductor/semiconductor curve of Figure 1.
characteristic of “bubbling” flow (Lerner and Grove, 1951; Teller, 1954). Figure 2 presentssome examples which show the behavior of the pressure drop above the flood point. It is interesting to note that the values of fc calculated for various three-dimensional lattices are very close to typical values for the fractional holdup observed for many types of packings near the flood point (Stauffer and Aharony, 1992). Experimentally it has been found that the fractional liquid holdup a t the flood point is often in the neighborhood of 0.15-0.45 (Shulmanet al., 1955;Bemer and Kalis, 1978; McNulty and Hsieh, 1982; Billet, 1988; Billet and Schultes, 1988). The holdup is clearly a small fraction of the total open volume of the packing. In addition, it has been found that for the majority of dumped packings the “fractional wetted surface area” away from the flood point is often less than the total specific surface area of the column packing (Shulman et al., 195513; Norman, 1960; Onda et al., 1967; Fair and Bravo, 1988, 1990). One would expect that the vapor would preferentially seek the “dry”paths since they offer the best vapor “conductance”. Thus much of the vapor flow may not be directly affected by interaction with the liquid phase. Coupled with the pressure drop evidence presented above and the fact that the incipient percolation threshold, fc, is very often in this same vicinity as the fractional liquid holdup observed at the flood point, it seems extremely plausible that a percolation mechanism underlies packed column operating behavior. In light of the discussion and evidence presented above, the increase in pressure drop in an irrigated packing relative to that in a wetted and drained packing (i.e., the two-phase pressure drop multiplier, 4) might be expected to scale with a power of (1- f / f c ) . Since, for all practical purposes, it is impossible to determine the number of voids
where v = v/a. In words, this power law relation states that the deviation in the fractional occupation of the voids in the packing from the critical fractional occupation is a simple power of the deviation of the liquid load from that at the flood point. In our development, we treat 71 as a fitting parameter specific to each experiment. CL*is the liquid volumetric flux (based on the total cross-sectional area of the column) necessary to flood the column for the density-corrected vapor volumetric flux, Kv, of interest. When KV is fixed, so is CL*for that packing. As the superficial liquid volumetric flux, CL, is increased from the pressure drop rises dramatically from zero toward CL*, the value for the wetted and drained packing. At CL = CL*,the column is flooded, and the above scaling form predicts that the pressure drop as well as the instantaneous power law slope of the pressure drop with respect to KV or CLdiverges to infinity. The fact that the pressure drop diverges at the flood point is not correct in light of the fact that the liquid phase is still able to %onduct” vapor (althoughpoorly relative to an open channel), but it should not be a serious hindrance until one is very close to the flood point. The problem could be alleviated by considering a scaling form which takes into account the “conductance” of the liquid (Hong et al., 1986),but the added complication hardly seems worthwhile at this point in our discussion because it does not add significantly to the underlying ideas we are trying to present and because of the fact that there is not enough data taken near or above the flood point to support the increase in complexity resulting from such a revised scaling hypothesis. We plan to investigate the effects of the “semiconducting” nature of the liquid in a future paper. It should be pointed out that there is no reason to believe that the “flooding”liquid rate determined from this scaling form and experimental flood point data should agree. For example, the appearance of liquid on the top of the packing does not necessarily correspond to the formation of a percolation cluster which spans the column diameter. We will discuss this point further in the section Relations among Flood Point Definitions. Note that the pressure drop of interest for the no liquid flow condition in eq 4a is that from a packing which has been wetted and then drained, not from a dry packing. This distinction is made because we are assuming that liquid flow is related to occupation of void spaces. When a dry packing is initially wetted, a certain amount of liquid will be held up, even with no gas flow. This small amount of static holdup can increase the pressure drop relative to the dry pressure drop a t the same vapor flow conditions by as much as 50% for packings in smaller size ranges (see, for example, White (1935)). In order that eq 4a be useful for the prediction of twophase pressure drops, a value, or at a minimum a range of potential values, for the scaling exponent 7 is necessary. Far from macroscopic flooding (Le., 1- CL/CL* =;) 1)we would expect that very few of the voids in the packing
-
would themselves be “flooded”. However, as the macroscopic flood point is approached (Le., 1- CL/CL* 0), we would anticipate an acceleration in the choking off of voids. In simple terms, this means that the slope of a plot of (1 - f/fc) (=y) versus (1- CL/CL*)( = x ) should be steeper as one approaches x = 0. The above thinking leads one to the conclusion that 0 Itl 5 CY. In the discussions that follow we will take CY = 2 for the sake of simplicity. Equation 4a deserves further comment. In the electrical lattice problem, it is common to think of the incipient percolation threshold, fc, as being fixed (Le., it is only a function of how the electrical conductors are removed and the topology of the lattice). The percolation threshold is approached by replacing conductors with insulators (or semiconductors). For the pressure drop model suggested here, the flood point can be approached in either of two distinct ways: CLcan be varied, with KVand hence CL* fixed, or CL* can be varied (through Kv) with CL fixed. Even though fc is fixed for a particular void distribution when the voids are removed randomly,CL*can vary widely so long as CLis adjusted at the same time to keep eq 4b true. Scaling equations with a constant scaling exponent in which the critical point can change with external conditions are not unknown in other areas of physics concerned with critical behavior (Khurana,1989;Greywall and Ahlers, 1973).
Flooding in Packed Columns Flood points for packings have been defined and determined in a number of ways. Some investigators have associated the flooded condition with the locus of vapor and liquid flow rates which produces a backup of liquid on the top (or sometimes on the bottom) of a bed of packing. This is, by its nature, a ”visualflood”. Others have defined flooding as the locus of vapor/liquid flow rates which give apressure drop of 2.5 in. of HzO/ft. Yet others have defined the flood point in terms of the instantaneous power law slope of the pressure drop/vapor flow rate curve (liquid flow rate as a parameter); flooding of the packing is defined as having occurred when this instantaneous power law slope is “infinite”. Of course, this condition is never actually achieved and some type of extrapolation, or local curve fit, of the data would be necessary to fulfill this definition of the flood point. In actual practice, an “infinite” slope is often taken to mean a power law slope on a (Ap/H)2@versus KV plot (with CL as a parameter) which has exceeded some critical value (often taken to be about 4-5). In contrast to the “visual flood” defined previously, the power law slope method can be said to yield a “graphicalflood” point. Many alternate definitions have also been developed and applied (Silvey and Keller, 1966; Bain, 1943). In general, the approaches discussed here yield flood point data which are in approximate agreement. However, there are cases where these definitions for flooding do not agree well. This point will be discussed in greater detail in the section Definition of the Flood Point for Random Packings. In order to use the percolation expressionfor the pressure drop (eq 4a), an equation for the superficial liquid volumetric flux at the “percolation flood point”, CL*, is required in terms of the vapor volumetric flux, the topology of the packing, and the physical properties of the liquid and vapor phases. Instead of using any of the common correlations for the flood point found in the literature, we choose to describe the percolation flood point of a packed column with what shall be termed here the Wallis equation (Tien et al., 1980,1981). The Wallis equation, developed to describe flooding in vertical tubes, can be cast in the
Ind. Eng. Chem. Res., Vol. 33, No. 5,1994 1211 following form for packed columns: K;12
+ mCL*’I2 = 3(c2gd,)’/4
(5)
where m and 3 are adjustable parameters. This equation is simpler than others found in the literature for the calculation of the flood point (for example, Prahl(l969) and Ward and Sommerfeld (1982)), but it retains the features essential for description of flooding in packed columns and it has been found to work well in this context (Wallis, 1971; Liu et al., 1982). When the Wallis expression is rearranged to match the groupings of the Sherwood correlation, it yields:
where we have assumed that (7) For turbulent flows in vertical tubes, m is expected to have a value close to 1. Note, however, that interfacial and wetting effects as well as the effect of liquid viscosity are not explicitly included. The fact that liquid viscosity is not accounted for should not be too serious since the effect is apparently rather weak, at least for liquids with viscosities like that of water (recall that the best estimate of the effect of liquid viscosity on the flood point in the currently accepted embodiment of Sherwood‘s correlation in the ordinate with is the inclusion of a factor of (c(L/c(w)@ 0 Io 50.25). In addition, implicit account of these effects will be available for any given experimental system through the parameters 3 and m. For structured packings with a given surface area to volume ratio and inclination angle from the horizontal, 9, it is necessary to modify some of the equations used above for random packings. Figure 3 is a depiction of the geometry of a typical structured packing. For angles greater than about 20°, the vapor tends to flow along the furrows of the packing (Focke et al., 1985; Gaiser and Kottke, 1989). For an inclination angle of 9,the volumetric flux of vapor in a furrow is approximately equal to
while the effective gravitational acceleration is geff = g sin(9). When these modifications are made, the Wallis equation becomes K;I2
+ mCL*’/2 = 3(e2g(sin3(9))d,)’/4
(9)
with de = 2e/a. Equation 9 can be rearranged to a form suggestive of the Sherwood correlation. When this is done, we find that
The major distinction between this equation and the earlier one for random packings is the appearance on the left-hand side of the divisor sinY9). If the parameters fl
1212 Ind. Eng. Chem. Res., Vol. 33, No. 5, 1994 Flow
Flow
Flow
Flow
Lknsity Corrected Suprficial Vapor Veloeity
Figure 4. Schematic showing the relationships among the various vapor and liquid flow rates wed in the percolation model.
multiplier, 4. Figure 5 is a plot of the error conversion factor as a function of E, the fraction of the vapor flooding uvelocity" (actually the superficial volumetric flux), for a fairly typical set of parameters (asspecified in the figure). Clearly,the flood point must be accurate to minimize errors in the calculation of the pressure drop when one is operating above E = 0.5. Figure 3. Schematic depicting the arrangement of a typical structured packing. Impingement of the vapor streams leads to a spiraling flow up the furrows of the packing.
and m can be treated as constanta, then the above equation predicts that packings with steeper inclination angles will flood later than packings with shallower inclination angles. Proximity to the flood point has a profound effect on the the magnitude of the pressure drop and ita rate of increase with liquid or vapor flow. This can be seen by estimating the error in the two-phase pressure drop or the two-phase multiplier, 4, which results for errors in the estimate of the flooding vapor rate. Very often, one works at fixed LIV and operates at a certain "percent of flood". This situation is depicted graphically in Figure 4. In this case, Kv* is the density-corrected vapor volumetric flux at flooding for the given LIV. It is straightforward to show that
Relation to Other Available Pressure Drop Models The percolation expression for the pressure drop can be reexpressed in a number of ways when the column is far from the flood point. For example, eq 4a can be written as follows:
For CL/CL*
