ATx = external film temperature difference, "C mole fraction benzene X H = mole fraction hydrogen X T = mole fraction thiophene V N D=~ observable rate parameter, XB =
(Robpa CO)
Cassiere, G., Carberry, J. J., Chem. Eng. €duc., 7, 22 (1973). Hutchings, J., Carberry, J. J., AiChEJ., 12, 20 (1966). Kehoe. J. P. G., Butt, J. E., AiChEJ., 18, 347 (1972). Lee, J. C.M., Luss, D., Ind. Eng. Chem., fundam., 8, 596 (1969). Lee. J. W., Ph.. Dissertation, Northwestern University, Evanston, Ill., June, 1976. (Available from University Microfilms.) Weng, H-S., Eigenberger, G., Butt, J. E., Chem. Eng. Sci., 30, 1341 (1975).
reaction mixture density, g-mol/cm3
pP =
= effective thermal conductivity, callcm-s-"C 4 = Thiele Modulus = R2a exp(-E/RTb)/D,ff
X,f
Literature Cited
Received for review July 6, 1976 Accepted December 15,1976
Carberry, J. J., lnd. Eng. Chem., Fundam., 14, 129 (1975).
Coalescence of Oil-in-Water Suspensions by Flow through Porous Media Lloyd A. Spielman" and Yeang-Po Su Division of Engineering and Applied Physics, Harvard University, Cambridge, Massachusetts 02 138
Dilute oil-in-water suspensions at high ionic strength have been coalesced at steady state by flow through beds of glass beads having uniform diameters (0.26, 0.36, 0.50 mm). The diameters of the oil droplets in the incoming suspensions were 1-5 pm with oil viscosities (50, 500 cP), total suspended volume fractions (2 X 8 X and superficial velocities (0.3, 0.6 cm/s). Suspended drop size distributions were determined along the depth of the beds by Coulter counting. The local held-up oil was measured as a function of bed depth by x-ray absorption. The data are interpreted using equations that assume three distinct regimes of coalescing phase: oil microdrops suspended in capillary-conducted water; capillary-conducted oil forming well connected channels; held-up oil as discrete coalescing globules that act as an intermediary between the oil microdrops and the capillary-conducted oil. Filter coefficients characterizing microdrop capture are reported and correlated assuming capture is governed by van der Waals forces. The rate of coalescence of the intermediary held-up oil is found to be approximately first order in the fraction saturation of the intermediary held-up oil with a rate coefficient of about 1 X 10-4s-1.
Introduction The separation of finely dispersed liquid-liquid suspensions occurs in various engineering applications, such as the dewatering of petroleum and aviation fuels, prevention of product contamination in liquid-liquid extraction, and the removal of dispersed oils and other organics from wastewaters prior to discharge into receiving waters, among others. Although gravity settling is often sufficient for the primary separation of suspended globules greater than about 10 pm, it is frequently ineffective for the residual smaller droplets, which settle much more slowly. Depending on the situation, droplets on the order of a few microns or less may constitute to volume fraction of the dispersion, which is sufficiently dilute that droplet encounters are infrequent and spontaneous coalescence is slow. Some dispersions contain surfactants which may further inhibit coalescence. It is well known that coalescence of such suspensions can often be induced by flow through a porous medium consisting of fibrous or granular packing, such as glass filaments or pebbles. Fibrous media, which can be made to have both higher porosities and higher specific surface than coarse granular media, are observed to give more complete phase separation than granular solids for the same bed depth and operating conditions. This makes fibrous media appear more Address correspondence t o t h i s author a t Departments of C i v i l and Chemical Engineering, U n i v e r s i t y of Delaware, Newark, Del.
19711. 272
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
attractive from the standpoint of space requirements. However, even small amounts of suspended solids, which are sometimes unavoidably present, can rapidly accummulate to clog fibrous media to such a degree that their use over extended periods is prevented. Also, their permanent structure hampers periodic cleaning so frequent replacement would likely be necessary, although Shah (1975) has made progress in regenerating fibrous media. Unconsolidated, coarse granular media have a much longer operating period and in the less frequent event of excessive solids buildup they can be periodically cleaned by fluidization. There are also tradeoffs in material costs since common granular materials such as sand or pebbles are less expensive than manufactured synthetic fibers; so while the complete economics are not yet clear, it appears that granular media are likely to play a dominant role in certain separation schemes, particularly those involving oily wastewater treatment (see, e.g., Madia et al., 1975). Spielman and Goren (1970a) reviewed coalescence by flow through porous media. Recent work not covered by that review includes studies by Spielman and Goren (1972),Cordes (1972),Hazlett (19691, Hazlett and Carhart (1972), Bitten (1970), Bitten and Fochtman (1971), Sherony and Kintner (1971), Langdon et al. (1972),Rosenfeld and Wasan (1974),Chieu et al. (1975)and Madia et al. (1975). Although numerous efforts have been directed toward understanding the underlying phenomena, considerable disagreement still exists concerning the mechanisms by which the suspended phase is coalesced and transported through the porous matrix. The present work attempts to resolve aspects
of major disagreement and to set down rational, comprehensive equations governing coalescence by both granular and fibrous media. Spielman and Goren (1972b) experimented with oil-inwater dispersions using thin glass fibrous mats, a few millimeters thick a t most. In that work the upstream and downstream drop size distributions, pressure drops across the mats, and total oil held up in the mats were reported for steady-state conditions. The change in drop size distribution and pressure drop with increasing medium thickness are principal indicators of process perform,ance, and these are coupled through the coalescing phase held up within the matrix, which affects both. The present work reports results for coalescence of oilin-water dispersions using beds of unconsolidated glass beads rather than fibrous mats. One reason for this is that comparatively deep granular beds facilitate measurement in situ of the drop size distribution, pressure drop and held-up oil as profiles over bed depth. This cannot readily be done with thin fibrous mats, and such data have proved to be very informative. Another reason for using glass beads is that reliable measurements of oil-water relative permeabilities for unconsolidated granular media, including glass beads, already exist (Leverett, 1939; Jones, 1949; Carpenter et al., 1962). Such data form an important part of the analysis and corresponding data are not available for fibrous media. Finally, as discussed above, granular media are likely to be important for certain applications such as oily wastewater treatment and quantitative information on them could be of value in development work. Experiments have been carried out with systematic variation of flow rate, bead diameter, oil viscosity, and system chemistry. Measurements include the suspended oil drop size distribution, quantity of oil held up in the bed pores, and aqueous phase pressure, all as simultaneous functions of position, progressing through the bed at steady state.
Theoretical Background Sherony and Kintner (1971s) and Rosenfeld and Wasan (1974), among others, have proposed that fibrous coalescers act by capturing suspended droplets that grow on the fibers by further capture and coalescence until they eventually become so large that hydrodynamic drag causes breakaway. After breakaway these large drops are pictured to travel through the pores of the medium, eventually being released a t the downstream face. Although the analyses presented in these studies differ in quantitative detail, the traveling globule idea is shared by them, and it is this aspect with which we take issue here, both for fibrous and granular media. We further remark that visual observations of coalescence on individual fibers or very sparsely packed media, which provide the basis for their approach, are not wholly representative of phenomena in more densely packed media. In the Iatter, globules that grow by coalescence eventually extend over many pore dimensions and engulf many fibers; likewise for granular media. To show this, let us consider an assemblage of fibers and assume, for simplicity, that fibers have equal diameters and are arranged with axes in square array. This gives interfiber spacings of 2 and 3 fiber diameters for typical fibrous media voids fractions of 0.90 and 0.95, respectively (random arrays should not be different enough to change the argument). Bitten (1970) described visual observations of coalescence of water drops in jet fuel on individual fibers of treated and untreated glass, Teflon, Dacron, and Nylon. With the exception of Teflon, individual fibers were reported to retain globules having diameters many times that of the fibers, without reentrainment by the flow a t velocities reaching 3 to 4 ft/min. For untreated and treated glass fibers of diameter 5-6 pm, adhering globules reached reported diameters of 400 to 500 pm, or roughly 9 fiber diameters without being released.
Even for the least retentive fiber tried, which was Teflon, adhering drops reached diameters of 65 to 75 pm before release. It should be noted that the Teflon fibers used by Bitten were ribbon-like which might have contributed to their lower retention. Even so, the ratio of held drop diameter to Teflon fiber width (16-17 pm) reached about 5 before release. One is therefore led to conclude that characteristic interfiber spacings of 2 or 3 fiber diameters for typical packed fibrous media will not permit progressive travel of globules whose diameters are 9 (or even 5 ) fiber diameters. Such globules, if unattached, should soon become wedged in place where continued growth by coalescence with impinging drops would further ensure their immobility. Also, interfacial tension resists deformation of globules that might permit them to be squeezed through the pores by drag forces. The traveling drop mechanism is thus improbable except possibly for media that are packed to densities far below those cited above. Another observation not well explained by the traveling drop model is the release from fixed points on the downstream face, of globules whose diameters exceed the dimensions of a typical pore by a factor of 10 or more (Spielman and Goren, 1972b). Spielman and Goren (1970a) assumed that two readily distinguishable regimes of the coalescing oil phase exist within the pores. One regime consists of microdroplets that are suspended in the noncoalescing water; the other is coalesced oil that is held up within the pores and assumed to form a network of channels which are sufficiently well connected to support viscous flow of oil by capillary-conduction. This latter regime is consistent with established ideas from petroleum reservoir science about immiscible fluid flow through porous media (see, e.g. Scheidegger, 1960). Further pursuing established multiphase flow concepts, this second regime was taken to be in local capillary-equilibrium with the solid and water phases (hydrostatic or Laplace equilibrium: Everett, 1975). Transport from the finely suspended regime was considered to occur by irreversible capture of the suspended microdrops upon encountering the solid/coalesced-oil matrix. The drop capture step was assumed to be rate-controlling, with subsequent coalescence and interfacial adjustment of the held-up oil thought to be so rapid as to prevent appreciable buildup of captured droplets which had not coalesced into the capillary-conducted regime. With this physical picture Spielman and Goren (1970a) formulated a set of equations and boundary conditions which they solved under appropriate simplications (Spielman and Goren, 1972a). Their scheme of equations relates the local drop size distribution, phase pressures and held-up oil through four fundamental coefficients: the relative permeabilities of the two phases, the capillary pressure and the filter coefficient, all of which are functions of S2, the coalesced oil held up in the pores and which have to be known by direct measurement or estimated in order for the equations to yield solutions. The investigations of Voyutski et al. (19531, Hazlett (1969a),and Hazlett and Carthart (1972) support the view that the coalesced phase gets transported by channelling through the matrix, rather than as traveling globules. These studies deal with water removal from hydrocarbons using fibrous media. Hazlett (1969a) describes observations for which ‘‘. . . threads of water snake through the coalescer.” An outcome of Spielman and Goren’s analysis is that Sl(n), the fraction saturation (pore volume fraction) of locally held-up oil must be a nondecreasing function of position in the direction of flow and, under usual conditions, S2 cannot attain values significantly larger than SzC, the minimum fraction saturation of coalesced oil necessary to sustain its flow by capillary conduction. In fact, for small incoming suspended phase volume fractions and moderate oil viscosities, the equations predict S2 = Szc = constant throughout the medium Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
273
at steady state, and these features are predicted for any values of the basic coefficients that underly the governing equations. However, inspection of Figure 4, which shows a typical profile of held-up oil (upper curve), reveals behavior near the inlet that contradicts Spielman and Goren’s prediction. Figure 4 shows data from the present study for oil-in-water dispersions with incoming volume fraction, f o = 2 X 10-4,’beingcoalesced during steady state flow through a bed of glass beads. There S2(x) is seen to be a decreasing function of x , having a value a t the inlet of Sz = 0.50, which is appreciably larger than the minimum value necessary to support its viscous flow, which turns out to be closely given by its downstream value, Sz = 0.104.20. Qualitatively similar observations were made earlier by Bitten and Fochtman (1971) who measured held-up coalescing phase profiles for coalescence of water-in-hydrocarbon dispersions by fibrous media and found the held-up coalescing phase to decline with increasing bed depth. This is therefore concluded to be quite general and should be properly accounted for because most of the drop capture takes place near the inlet in the region where S2 is greatest and coalescence performance can be affected. Figure 4 shows the total suspended oil, f , and held-up oil, Sz, to be closely related functions of position and this result is typical. Modified Theory The discrepancy between experiment and Spielman and Goren’s theory can be explained by the presence of captured, held-up oil that is coalescing in place, but not flowing by capillary-conduction within the pore spaces. Although Spielman and Goren (1970a) suggested the possibility of such a regime, it was excluded from their formulation. This third regime of coalescing phase must be largely discrete in structure and acts as an intermediary for transport between the microdrops in suspension and the regime of ultimately coalesced oil that flows by capillary-conduction. As proposed here, suspended microdrops are first captured by the two-phase matrix of solid and held-up oil where it becomes identified with the intermediary held-up regime of coalescing, but not capillary-conducted oil. The intermediary regime is not in capillary-equilibrium with the solid and water, but exists as discrete globules that grow by capturing microdrops, eventually crowd against neighboring globules and undergo a cascade-like sequence of coalescences by which held-up oil is transferred locally from smaller to larger globules, until finally coalescing with the capillary-conducted oil that is identified as a separate regime consisting of well connected channels. This latter regime, as in Spielman and Goren’s treatment, is assumed to be in capillary-equilibrium with the solid and water, and supports capillary-conduction by viscous flow which is the only mode by which captured oil is taken to be transported through the matrix. Governing Equations. For completeness, the following analysis includes transient effects, although measurements were carried out primarily at steady state. The subscript 1 refers to the noncoalescing (aqueous) phase; 2 refers to the coalescing (oil) phase. Setting the total fraction saturation of held-up oil equal to the sum of that constituting the capillary-conducted regime, S i , and the intermediary regime, Sa”, gives s2
= S2’
+ S2”
(1)
Letting S1 be the fraction saturation of aqueous suspension, we note that S I + S2 = S I + S2’ Sa’’= 1. Because only the capillary-conducted oil regime is considered to be in local capillary-equilibrium with the solid and water, we write
+
P2 - P1 = PC(S2’)
(2)
where the argument of the capillary pressure function pc(Sz’) 274
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
includes only Sz‘, the equilibrated portion. In eq 2, p 1 and p 2 are the local pressures in the capillary-conducted aqueous and oil phases, respectively. The capillary pressure function can be measured using conventional methods (e.g., Scheidegger, 1960). Following Spielman and Goren (1970a), it is suggested that the first drainage function be used because usually the medium is initially saturated with the phase that preferentially wets it and the nonwetting phase enters by displacement (although hysteresis is, in general, a questionable aspect of multiphase flow). Because capillary flow of aqueous suspension is locally resisted by both the intermediary and capillary-conducted held-up oil, the equation governing the flow of water suspension is written
(3) where q1 is the local volume flux (superficial velocity) of water suspension, kl(S2) is the conventional permeability (capillary conductivity) to water as a function of fraction saturation by oil (Appendix), MI is the water viscosity, and d p l l b x is the local pressure gradient. Here it is presumed that k 1 is a function of the total local deposit of held-up oil, independent of the interfacial structure of the latter (part of the held-up oil consists of discrete globules while part forms channels). Although this assumption is somewhat questionable, to justify it we argue that a similar assumption is commonly used in deep bed filtration by sand to remove suspended solids, where water permeability is likewise assumed to depend only on the total local solids deposit, without regard to the particle size distribution or detailed structure of the deposit (see, e.g. Ives, 1975). However, the best justification of eq 3 is that it is borne out experimentally. This is discussed below in connection with Figures 8 and 9. The local flux, q2, of oil in viscous flow by capillary-conduction is accordingly given by (4) where the argument of k z , the conventional permeability to oil (Appendix), is taken to be Sz’, the fraction pore saturation of the capillary-conducted held-up oil only. This is because the interfaces bounding the capillary-conducted oil are fixed by capillary equilibrium. Specifying S i therefore determines the oil permeability, k ~regardless , of the existence of discrete oil globules lying outside the capillary-conducted channels. The conservation equation governing aqueous suspended oil microdrops is
where the local drop size distribution n(ap)dapis the number of oil microdrops per unit volume suspension, with radii on the interval (ap,ap da,). c is the porosity of the solid. In eq 5 it has been assumed that the suspension is sufficiently dilute that its flux, q1, is virtually constant and so appears parametrically. If this is not the case, appropriate modifications can be made (Spielman and Goren, 1970; Su, 1974). X(S2) is the local filter coefficient (analogous to a mass transfer coefficient) which is conventionally defined and can account for capture by any of several possible mechanisms that bring about individual microdrop encounters with the static matrix of solid and held-up oil (Spielman and Goren, 1970a). Here h(S2)is taken to be a function of the total held-up oil, S2, since all held-up oil (both discrete and capillary-conducted) acts to capture suspended microdrops. As with kl(S2) in eq 3, the interfacial structure of the held-up oil is presumed to be of secondary importance compared with the total amount present, in this case with respect to its influence on droplet
+
capture. This is analogclus to assuming the filter coefficient depends on the specific deposit of solids in conventional deep bed filtration (e.g. Ives, 1975). The filter coefficient depends parametrically on up, 41, and grain diameter, d,, as well as other physical and chemical properties that can affect drop capture. Its dependence on Sz is shown explicitly, however, because Sz occurs in thia scheme as an unknown. One of the important objectives of experiment has been to acquire information about the quantitative dependence of X on Sz. The total local volume fraction, f , of oil in the form of suspended microdrops is straightforwardly expressed in terms of the microdrop size distribution, giving
Toto! P r e o
:
S,
i
Sh + S z
-0 u0
t ’ swrce of Captured
held-up globule volume
sink 10 capillary f l o w
droplets
Additional equations are obtained by material balances. A material balance on the intermediary regime of held-up oil over a differential thickness of bed gives
Figure 1. Qualitative distribution of the total volume of held-up oil over individual held-up globule volumes. Microdrops are captured at the left-hand end of the spectrum. Oil is transferred to successively larger globules by coalescence eventually leaving by capillary-conduction a t the right.
The first two terms on the right-hand side of eq 7 represent net addition of material to the intermediary regime by capture of locally suspended microdrops. The last term is a sink representing export of oil from the intermediary regime by coalescence into the capillary-conducted regime. By definition, R is the volume of heldl-up oil that coalesces from the intermediary regime into the capillary-conducted regime, per unit pore volume, per unit time (e.g., cm3 oil/cm3 void&). At steady state this just elquals the local rate of capture of suspended droplet volume, per unit volume of bed voids. An unsteady material balance on the capillary-conducted oil gives
cence of the held-up oil within the pores is viewed as a steady-state (or quasi-steady-state) rate process in which newly captured microdrops provide a continual source of oil at the lowest end of the size spectrum of held-up globules. The individual held-up globule volumes range from those of the newly captured microdrops to those participating in channel flow by capillary-conduction. Oil gets locally transferred from smaller to larger held-up globules by coalescence. While the smallest globules take the form of adhering spherical drops, the largest globules necessarily deviate from spherical because of contact with the solid and conformity to the boundaries of the pores. It also follows that material constituting the larger globules has resided in the bed longer than that in the smaller globules. We now assume that because held-up globules whose volumes are appreciably larger than those of the incoming source droplets have undergone many coalescences with other globules, they are statistically independent of the source droplets. That is, the region of the held-up globule spectrum constituted by all but the smallest globules attains a statistical form which, in conjunction with the detailed mechanisms governing the kinetics of the coalescence process, is determined primarily by the overall volume rate at which microdrops are fed, R , rather than by the details of the microdrop size spectrum. (This picture is closely analogous to Kolmogorov’s (1941) theory of locally isotropic turbulence, which assumes statistical independence of the large and small eddies; also Friedlander’s (1960) quasi-steady-state theory of coagulating atmospheric particle size spectra.) Our description of the coalescence cascade process presently makes no commitment to certain details of the coalescence mechanisms, which are expected to depend intricately on steric and chemical factors: internal solid geometry, preferential wetting, surfactants that affect interfacial film stability, etc., in a manner that is considerably more complicated than homogeneous emulsion coalescence or suspension coagulation by common binary mechanisms. I t is further assumed that only a negligible portion of the total held-up volume is comprised of newly captured microdrops; i.e., the total local held-up oil volume consists mainly of globules whose sizes are much bigger than the source droplets and whose size spectrum is therefore independent of the size distribution of the source droplets (Figure 1).Then that part of the globule spectrum containing most of the held-up volume should adjust to a definite form once both the volume input rate, R , and the capillary-conducted (equilibrium) volume, Sz’, are independently specified. (This differs a little from the analogous turbulence theory where the entire right-hand part of the turbulent energy spectrum gets de-
where the first term on the right represents net oil influx by capillary-conduction and the second term is the source to the capillary-conducted regime due to coalescence from the intermediary regime. The instantaneous volume rate of coalescence, R , requires further analysis. An oversimplified but useful first analysis of the coalescence kinetics can be done by using a crude analogy with chemical reaction kinetics. The process discrete held-up (globules)
-
coalescence
capillaryconducted (oil
)
can be thought of as a tiransformation involving the two species shown. Then the coalescence rate R would be a function of the local concentrations, Sp” and Sz’ of the reactant and product, so ,R = R(Sp”, Sz’) (9) Equation 9 states that the rate a t which oil coalesces from the intermediary regime into the capillary-conducted regime depends only on the total amounts of oil within those two regimes for a given oil-water-solid chemical system. Although the analogy with chemical reaction kinetics gives primitive understanding of the coalescence kinetics, it cannot be viewed as rigorous justification of the existence of a function of the type given by eq 9. In the following we therefore attempt to establish eq 9 more rigorously, by considering the actual processes that occur. Figure 1 qualitatively represents the volume of held-up oil (ordinate) statistically distributed over globules of different volumes (abscissa) a t a given depth in the bed. The coales-
Ind. Eng. Chem., Fundam., Vol. 16, No. 2, 1977
275
termined by fixing only the energy dissipation rate, which is analogous to the coalescence rate; here SZ’ can be specified independent of the coalescence rate, R , and S2’alone determines the configuration of oil/water and oil/solid interfaces in the capillary-conducted oil regime because the latter is taken to be in capillary-equilibrium with the solid and water phases.) Now, because the volume-containing part of the held-up globule spectrum is fixed by specifying R and S2‘, so must be the integral over the spectrum giving the intermediary volume, Sz”;that is, S2’’ = S2” (R,S2’). Inverting, we obtain eq 9. In addition to the functional dependences on the unknowns, S2’ and Sz”,which are shown explicitly, eq 9 contains implicit parameters reflecting steric and interfacial phenomena that affect the coalescence kinetics. This implies that the form of eq 9 could vary widely from system to system. So far, little has been said about the specific form eq 9 might have. An important part of the present study has been to investigate the form of the function experimentally for a specific system. However, certain of its properties can be anticipated from qualitative considerations. One such property is
R +Oa~S2’’+0
(10)
This just says that the rate of coalescence should tend to zero as the local amount of held-up oil in the intermediary regime goes to zero; then all the held-up oil will be capillary-conducted and in capillary-equilibrium with the solid and water, so no further coalescence can occur. Another expected property is that R should be an increasing function of Sa‘’ with other variables fixed; that is, the more uncoalesced held-up oil there is present, the more rapidly it should coalesce. R might also be expected to be an increasing function of S 2 ‘ since filling the pores with more capillary-conducted oil should increase the probability of its contact (and coalescence) with intermediary held-up oil. A plausible form to assume for R(S2”, S2’) would therefore be where k‘, 1, and m are parameters to be determined empirically. As indicated above, the values of these parameters could vary widely from one system to another, depending on internal solid geometry and interfacial chemistry. It is also evident that eq 11must be confined to values of S 2 ’ and S2” that are consistent with the physical restriction SI S2’ S 2 ” = 1. Once R(S2”, S2’) is specified along with the functions pc(S2’), k1(S2), k2(S2’), and X(S2), eq 1-8 provide eight equations in the eight unknown functions of 1: and t: S 2 , S2‘, SZ”,p1, p2, q2, n(ap),f . These may be reduced immediately to six by eliminating S 2 and f through direct substitution from eq 1and 6. In principle, the transient behavior of the system is obtained by solving these equations under appropriate initial and boundary conditions. Solution of the transient system is beyond the scope of this paper, which reports steady-state results. However, methods for treating systems of nonlinear first-order equations, including the transient terms, can be found in the literature (see, e.g. Aris and Amundson, 1973). The nature of these equations makes them capable of describing various fascinating unsteady phenomena including progressing frontal movement of coalesced material such as that observed by Douglas and Elliot (1962). Steady-State Equations. A t steady state, the above equations simplify. Equation 5 becomes
+ +
(5’) Equation 7 becomes (7’) 276
Ind. Eng. Chern., Fundarn., Vol. 16, No. 2, 1977
Equation 8 becomes
Equations 1-4 are unchanged except that b/dx gets replaced by d/dx . Boundary Conditions.Proposed boundary conditions for the above equations are the same as those given by Spielman and Goren (1970a),with the straightforward modification that S2 of the earlier treatment gets replaced by S i of the present treatment, since here only the latter is taken to be capillaryconducted and in capillary-equilibrium with the solid and aqueous phases. A t steady state, no initial conditions are needed and no boundary conditions on S 2 ” are required because no spacial derivatives of S2” appear in the governing equations, i.e., S2“ is determined only by satisfying the governing equations and boundary conditions on the other variables that appear. The reader is referred to Spielman and Goren (1970a) for a thorough discussion of boundary conditions, bearing in mind the above modification. Simplifications and Use of Equations to Analyze Data. Differentiating eq 2 with respect to x gives
Under most operating conditions, the suspension flow rate is sufficiently large that the capillary pressure gradient, given by the right-hand side of eq 12, is negligible compared with the aqueous phase pressure gradient dplldx, so eq 12 simplifies to (13)
Equation 13 says that the local pressure gradients in the capillary-conducted oil and water are approximately the same. This holds provided the slope of the capillary pressure function dpc(S2’)/dS2’ in eq 12 is sufficiently small. From detailed measurements of the capillary pressure function, this approximation has been shown to apply throughout the range of experiments covered here (Su, 1974). Equating the left-hand sides of eq 7’ and 8’ and integrating gives
Combining eq 13,14,3, and 4 and rearranging gives
Equation 15 is important because it permits calculation of the fraction saturation of capillary-conducted oil S i at any location where both the total held-up oil S 2 and total suspended oil f are known from direct measurements, because then S i is the only unknown appearing in eq 15. Having evaluated S2’ in this way, the fraction saturation of intermediary held-up oil S2”can be obtained from eq 1 simply by taking the difference Sa - S2‘ = 5’2’’. With f and S2 known as functions of position from direct measurements, and S2‘and S2”evaluated at a given position x using eq 15 and 1,R can be evaluated at the same position by graphical differentiation of f ( x ) , using eq 7’. The corresponding values of R, S2’, and S2” obtained in this way can then be associated to study the function R(S2”, Si). The local filter coefficient characterizing capture of microdrops of a given size can be evaluated from eq 5’ by plotting In n ( a p )vs. x and graphically differentiating. This gives values of X = -d In n/dx which can be paired with measured S 2 to
study the function
X(S2)
in eq 5'.
Experimental Methods Suspensions. In the iexperiments reported here, dispersions of micron-sized oil droplets in water were coalesced at steady state by passing them through columns of glass beads. Suspensions were prepared using a Manton-Gaulin laboratory homogenizer, This produces a nearly flat distribution expressed as suspended oil volume versus drop diameter, with almost all the suspended volume between about 1and 5 pm. An advantage of this method of drop dispersion is that it can be done continuously without introducing air bubbles into the system. The oils used were 50 and 500 CPDow Corning 200 silicone fluids, whose specific gravities are very close to unity, thus minimizing gravity settling effects. The interfacial tension of the oillwater system is estimated to be 45.0 f 5 dyn/cm in the absence of added surfactant. For most of the experiments the incoming suspended volume fraction was kept fixed at 2 X 10-4, although EI higher value, 8 X was used for a few experiments which were designed to test specific aspects of the theory. For most of the experiments the ionic strength of the water phase was kept at a high value (pH 2.0) by adding "03. This was to minimize double layer repulsion so the suspensions would coalesce readily. It also makes coalescence insensitive to trace impurities which adversely affect reproducibility. In some experiments electrolyte concentration was varied to study inhibited coalescence. Packing Media. Nonporous glass beads with mean diameters, 0.26,0.36, and 0.50 mm (uniform to approximately 10%) were used as the packing media. The bed voids fraction was kept at 0.38. Beads were reused after very thorough cleaning and a standardized surface treatment to maintain reproducibility (Su, 1974). Clean glass beads are preferentially waterwetted. Coalescence Apparatus. Figure 2 shows a diagram of the experimental setup. The effluent from the test column was circulated back to the homogenizer supply tank. Oil was periodically added to the homogenizer supply tank to maintain constant oil concentration in the test column influent during approach to steady state. It took several hours to reach steady state after startup. The temperature of the system remained nearly constant a t 21 "C throughout each run. The 1-2 in. i.d. Lucite test columns were fabricated with sample taps at closely spaced intervals over their lengths. This permitted obtaining samples of the flowing water suspension which were subsequently analyzed by Coulter counting to obtain the microdrop concentration size distribution, n(a,). This also permitted determination of the total suspended oil volume fraction f . The total held-up oil S 2 was determined by calibrated x-ray absorption. The depth of the column was scanned with the x-ray beam collimated to about 1mm. X-ray absorption has the advantages of giving good spacial resolution without disturbing the operation of the column. Aqueous phase pressure differences across the entire column were obtained routinely in each experiment from manometer taps at the inlet and outlet. Certain experiments were performed to study pressure profiles in detail. For these, columns were fitted with pressure taps placed a t closely spaced intervals over their lengths. For complete details concerning the experiimental methods, see Su (1974). Results a n d Discussion Figure 3 contains typical experimental data showing the change in suspended imicrodrop concentration with increasing depth into the bed a t steady state. These are plotted as the ratio of droplet number concentration at a given depth to that entering the bed, eaclh curve corresponding to a fixed droplet
Figure 2. Experimental setup with recirculation.
l
'
-
7
i
\
n/no\ lo-'
\
kpE:5;cp
,cz 0
q, fo
i
I
:0.3cm/sec I
2I
4 8 12 FILTER DEPTH, X k m )
1 6 Z U
Figure 3. Typical data showing the steady-state change in suspended microdrop concentrations with increasing depth into the bed. Each curve corresponds to the microdrop diameter shown.
diameter. The ionic strength has been kept high (pH 2.0) to suppress double layer repulsion and promote microdrop capture. The largest microdrops shown (4.5 pm diameter) are seen to be removed most efficiently. As the droplet diameter decreases, the droplets are seen to be removed progressively less efficiently, showing that capture of microdrops is highly selective with penetration of the smallest droplets being responsible for most of the effluent turbidity. Figure 4 contains typical results showing total oil as suspended microdrops (lower curve) and total held-up oil (upper curve), both as corresponding functions of bed depth on the same plot. The suspended oil on the left ordinate is expressed as the total volume fraction divided by the incoming volume fraction and is obtained by integrating the volume-weighted drop number concentrations, in this case those shown in Figure 3. The fraction pore saturation of total held-up oil determined by x-ray absorption is given on the right ordinate. Capillary-Conduction of Oil. The assertion that movement of coalesced oil occurs by capillary-conduction could be tested by changing the group (pzfolpl) which appears in eq 15. Well downstream of the inlet face nearly all the suspended oil should be captured so f l f o
