Ind. Eng. Chem. Res. 1996, 35, 3155-3162
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Coalescence in Creaming Emulsions. Existence of a Pure Coalescence Zone Sanjeev Kumar,†,‡ G. Narsimhan,§ and D. Ramkrishna*,‡ School of Chemical Engineering and Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, Indiana 47907
Stored emulsions undergo various destabilizing processes including creaming and coalescence of drops. Many experimental studies have been carried out in the past to quantify drop coalescence; however, the presence of two distinct types of coalescence mechanismssone due to the relative motion between the drops and the other due to their permanent proximity to each other in the creamshas not been recognized. Furthermore, the effects of creaming in these investigations could not be eliminated in ways that do not introduce new modes of coalescence or alter the existing ones. The population balance equation for an emulsion in a column in which creaming, Brownian diffusion, and coalescence of drops occur simultaneously is analyzed. The analysis reveals the existence of a dynamic zone at the top of the column in which the net effect of creaming and Brownian diffusion of drops is eliminated. Thus, while the drops cream, diffuse, and coalesce as in an actual emulsion, the measurements from this zone allow the effects of drop coalescence to be isolated from other destabilizing processes. Based on this finding, a new methodology to investigate coalescence is proposed. Experimental support for the proposed theory is also provided. If the size distribution in the uniform zone evolves in a self-similar manner, it is shown that the techniques already available in the literature can be used directly to estimate the coalescence frequency kernel. 1. Introduction A quantitative prediction of the shelf-life of emulsionbased products is important for their successful commercial use. In general, the overall appearance and the quality of a stored emulsion deteriorate with time and after a while reach unacceptable levels. The finite time for which an emulsion retains acceptable characteristics while sitting undisturbed on a shelf is taken as its shelflife. (Clearly, better definitions of shelf-life while do not rely on the interpretation of “what is acceptable?” are needed. We will, however, not dwell on this issue in this paper.) The appearance and quality of a stored emulsion product are affected by three processes: creaming of dispersed-phase drops due to buoyancy, their flocculation, and their coalescence. The first two processes are reversible and harmless by themselves, but they can aid coalescence of drops which is irreversible and eventually leads to complete destruction of the emulsion. A creamed or flocculated emulsion can be shaken to easily restore the original homogeneous emulsion. In comparison, if drops become bigger or oil appears on the top (known as “oiling off”) due to the coalescence process, the emulsion cannot be restored to its original state unless it is homogenized again. Let us consider various stages of the stabilization process. Figure 1 shows the state of a stored emulsion at various times in which creaming and coalescence are the only rate processes present. Figure 1a shows a uniform emulsion at an initial time. Figure 1b shows the same emulsion at a later time when some drops have creamed to the top and have arranged themselves in a tight packing while the others have moved away * Author to whom correspondence should be addressed. Phone: (317)-494-4066. Fax: (317)-494-0805. e-mail:
[email protected]. † Present address: Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India. ‡ School of Chemical Engineering. § Department of Agricultural and Biological Engineering.
S0888-5885(96)00014-0 CCC: $12.00
Figure 1. Various stages of evolution in a stored emulsion.
from the bottom. As the drops move, they collide with each other, coalesce, and form bigger drops. The drops in the tight packing at the top also continue to coalesce with others, and much later, as shown in Figure 1c, free oil appears on the top of the emulsion. It is important to note that the two types of coalescence mechanisms are different in nature and they have different implications for shelf-life. We shall term the former, which is caused by relative motion due to buoyancy or Brownian motion, as “coalescence in bulk” and the latter, which is caused by the permanent proximity of drops to each other in cream as “coalescence in cream”. Coalescence in bulk accelerates the creaming process by forming large drops which accumulate a large amount of dispersed phase in the cream layer in much shorter times. Furthermore, the cream undergoes accelerated coalescence because it now consists of large drops to begin with. Thus, increased coalescence in bulk has far more serious implications than the corresponding increase in coalescence in cream. If the two types of coalescence rates are combined into one overall rate of coalescence, the interesting and important implications of the geometry and the height of the container on the shelf-life of emulsion products will not manifest. As pointed out before, coalescence is an irreversible and the most deleterious process for the shelf-life of an emulsion. It is for this reason that coalescence in emulsions has been studied for a long time (King, 1941; © 1996 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Lawrence and Mills, 1954; Mita et al., 1973; Reeve and Sherman, 1988; Das and Kinsella, 1989; Dickinson and Semenova, 1992). Unfortunately, the two types of coalescence mechanisms, as outlined before, have not been recognized for their different roles. In fact, most of the investigators have not specified from where they drew the samples or which type of coalescence they were studying. It is interesting to note that even the recent review papers (Jaynes, 1983; Dickinson and Stainsby, 1988) have failed to recognize the two types of coalescence mechanisms and their different implications. Although not stated, some investigators appear to have been interested in studying coalescence in bulk alone. They, however, had to employ various means to eliminate the effects due to creaming. In this regard, it is interesting to recall Tadros and Vincent’s (1983) comment “Even in ‘fundamental studies’ the isolation of four processes such as creaming, flocculation, coalescence, and Ostwald ripening is experimentally difficult. For example, in order to eliminate sedimentation [or creaming] effects one should strictly work with two liquids having zero density difference, so that gravitational forces are absent, or to work under conditions of zero gravity”. Clearly if creaming is eliminated in this manner, gravity-induced coalescence in the bulk as well as in the cream layer at the top, both of which are responsible for degradation of emulsion, will also be eliminated. Aware of difficulties with creaming, Halling (1981) in his review paper writes, “...and the emulsion produced were allowed to stand (presumably with occasional mixing to reverse creaming).” Gentle stirring of emulsions at regular intervals to redisperse the cream collected at the top seems to have been a regular feature of coalescence studies. This obviously introduces additional effects. Das and Kinsella (1989, 1993) and Klemaszewski et al. (1992) avoided cream formation by storing emulsions in a gently rotating shaker. They point out that it also accelerates the coalescence process. Britten and Giroux (1991) indicate that an emulsion that did not show any coalescence under quiescent conditions coalesced when stirred. This is also supported by our own experiments in which an emulsion did not show any coalescence under quiescent conditions for as long as 20 days but coalesced rapidly when stirred. Clearly, stirring of emulsions to avoid creaming not only accelerates the coalescence but also can actually introduce new mechanisms of coalescence which may not have any bearing on coalescence in an actual emulsion sitting on a shelf. Lawrence and Mills (1954) in their widely quoted study on coalescence in bulk introduced a novel method to eliminate coalescence in cream by storing emulsion in a test tube which was slowly rotated in a vertical plane. Thus, the drops did not experience a net gravitational field and did not cream altogether. It is to be noted here, however, that, if drops do not cream, they will also not undergo gravity-induced coalescence in the bulk. Thus, although the coalescence in cream has been eliminated, the coalescence in bulk has not remained the same as that found in an emulsion kept on a shelf. Dickinson and Semenova (1992) have studied simultaneous creaming and coalescence in 10 vol % n-tetradecane-water emulsions stabilized by proteins and dextran. Creaming was studied by monitoring the serum layer thickness at the bottom, and coalescence was studied by measuring changes in the drop size distribution. It is not clear which coalescence was studied. In fact, in such systems, it is possible to observe changes in the drop size distribution to large drops even when there is no coalescence (Pirog, 1994).
Large drops reach the top layer faster than the smaller ones, thus increasing the concentration of larger drops there which can be mistakenly interpreted to be due to the coalescence process. It is clear from the foregoing that the presence of two different types of coalescence mechanisms has not been recognized in the past literature. This perhaps is also the reason why some investigators found the coalescence process to be second order whereas others found it as first order. In the present paper, we propose a new procedure for studying coalescence in bulk caused by Brownian or buoyancy-induced relative motion of drops while they continue to cream as in a stored emulsion. The focus here is on studying coalescence without introducing new mechanisms or altering any of the existing ones. In an infinitely long column, initially filled with a uniform emulsion, the size distribution at all locations evolves only due to the coalescence of drops and not due to their creaming. The effects due to creaming and Brownian motion are thus completely eliminated without altering the coalescence characteristics of the emulsion. The methodology employed here therefore aims at investigating what portions of a semiinfinite or a finite column, if any, possess the characteristics of an infinitely long columnschanges in size distribution occur due to coalescence alone. In order to carry out the mathematical analysis of the rate processes, we need the framework of population balances (Hulburt and Katz, 1964; Ramkrishna, 1985). The rest of the paper is therefore organized as given below. In section 2, we derive the population balance equation for simultaneous creaming and coalescence of drops. These equations are analyzed in section 3 to obtain the necessary condition for eliminating the effect of creaming although the particles continue to cream. Section 4 considers the effect of Brownian diffusion on the analysis presented in section 3. Based on the results of sections 3 and 4, section 5 presents a new experimental procedure for studying bulk coalescence while drops cream and diffuse as in an actual emulsion. Experimental support for the proposed theory is provided in section 6. The methodology for estimating coalescence frequency from the experimental data is provided in section 7. The final conclusions are presented in section 8. 2. Derivation of a Population Balance Equation We consider here a system in which drops undergo simultaneous coalescence and creaming. Population balance for such a system over a differential height dz, shown in Figure 2, and for drops in the size range {v, v + dv} is expressed as
{
change in population of drops in size range ) {v, v + dv} in time dt drops entering through surface at z in time dt birth of drops in size range {v, v + dv} due to coalescence of smaller drops in time dt
-
-
drop leaving through surface at z + dz in time dt loss of drops in size range {v, v + dv} due to their coalescence with any other drop in time dt
The number of drops in size range {v, v + dv} at time t in a differential height dz at location z in a column of
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3157
have not included the additional dependence on φ. However, it will eventually become clear that the inclusion of φ dependence cannot change the conclusions of the analysis contained in the next two sections. 3. Analysis Let continuous population density n(v,z,t) be approximated to population at discrete sizes, vi, i ) 1, 2, ..., such that (Kumar and Ramkrishna, 1996) M
n(v,z,t) )
Figure 2. Differential element for population balance.
cross-sectional area A is given by n(v,z,t) dv A dz, where n(v,z,t) is the number density. If Q(v,v′) is the coalescence frequency for drops of sizes v and v′ and Z˙ (v) and D(v) are respectively the creaming velocity and diffusion coefficient of drops of volume v, the above balance can be cast in the following mathematical form:
n(v,z,t+dt) dv A dz - n(v,z,t) dv A dz ) ∂ Z˙ (v) - D(v) n(v,z,t) dv A dt ∂z ∂ Z˙ (v) - D(v) n(v,z+dz,t) dv A dt + ∂z v 1 (A dz) Q (v-v′,v′) dt × 0 2 n(v-v′,z,t) d(v-v′) A dz n(v′,z,t) dv′ A dz A dz A dz n(v,z,t) d(v) A dz ∞ (A dz)[Q (v,v′) dt] × 0 A dz n(v′,z,t) dv′ A dz (1) A dz
[
[
] [
∫
[
∫
]
][ [
]
]
[
]
]
Factor 1/2 is introduced in the third term on the righthand side of the above equation to avoid double counting. Dividing the above equation by dv A dz dt, and taking limits dv f 0, dz f 0, and dt f 0, the following general population balance equation is obtained.
{[
]
}
∂ ∂ ∂ n(v,z,t) + Z˙ (v) - D(v) n(v,z,t) ) ∂t ∂z ∂z 1 v n(v-v′,z,t) n(v′,z,t) Q(v - v′,v′) dv′ 2 0
∫
∫0∞n(v,z,t) n(v′,z,t) Q(v,v′) dv′
(2)
∂2 ∂ n(v,z,t) - D(v) 2 n(v,z,t) ) ∂t ∂z 1 v n(v-v′,z,t) n(v′,z,t) Q(v-v′,v′) dv′ 2 0
∫
∫0
for i > j, vi > vj (4)
Although the upper value of index i in the above expression can be taken as infinity, in practical applications there always exists a finite size such that the drops larger than this size do not exist. In a coalescing system, this size will increase with time and therefore i ) M can be replaced by i ) M(t); i.e., index M keeps increasing with time. Thus, M(t)
n(v,z,t) )
Ni(z,t) δ(v-vi) ∑ i)1
n(v,z,t) n(v′,z,t) Q(v,v′) dv′ (3)
where change in location z for drops of size v is given by dz/dt ) Z˙ (v) and differentiation with respect to time holds only v constant. Kernel Q(v,v′) represents coalescence due to the combined effect of Brownian and gravity-induced motion. Strictly, Z˙ (v) and D(v) depend not only on drop size but also on local dispersed phase holdup φ which have been generally the way to account for hindering effects. To keep the analysis simple, we
(5)
Substituting for n(v,z,t) from the above equation in eq 2 and discretizing the right-hand side following the procedure given by Kumar and Ramkrishna (1996), we obtain
∂ ∂t
Ni(z,t) + Z˙ i i
∂ ∂z
Ni(z,t) ) Di
∂2 ∂z2
Ni(z,t) +
j
∑ ∑ Ri,j,kNj(z,t) Nk(z,t) Qj,k j)1 k)1 M(t)
Ni(z,t)
∑ Nk(z,t) Qi,k
i ) 1, 2, 3, ..., M(t) (6)
k)1
Here, Z˙ i ) Z˙ (vi) and Qj,k ) Q(vj,vk). Coefficients Ri,j,k depend on the grid used for discretization and the moments preserved in discrete equations. Since we shall be interested here only in the form of eq 6, the details of how Ri,j,k’s are evaluated are not relevant here but may be found elsewhere (Kumar and Ramkrishna, 1996). The set of equations in eq 6 can be more concisely presented as
∂n ∂2n ∂n +A - D 2 + b(n) ) 0 ∂t ∂z ∂z
In terms of moving coordinates, the above equation can be expressed as
∞
Ni(z,t) δ(v-vi), ∑ i)1
(7)
Vector n(z,t) consists of elements N1(z,t), N2(z,t), ..., NM(t)(z,t), and vector b consists of summation terms on the right-hand side of eq 6. Matrices A and D are diagonal matrices whose elements do not depend on n, z, or t. The set of equations in eq 7 is a system of first-order parabolic partial differential equations. If the term drop D ∂2n/∂z2 is dropped, eq 7 reduces to the following simpler form:
∂n ∂n +A + b(n) ) 0 ∂t ∂z
(8)
The eigenvalues of the diagonal matrix A in the above equation are distinct and real (equal to Z˙ 1, Z˙ 2, ..., Z˙ M(t); Z˙ i+1 > Z˙ i) for all z and t, which ensures that eq 8 is a system of first-order semilinear hyperbolic partial differential equations. These systems have been studied in detail in the mathematics literature (Courant and Hilbert, 1962; Garabedian, 1964; Lax, 1973; Zachmanoglou and Thoe, 1976). Our approach therefore will be to first analyze eq 8 and then incorporate the effect of
3158 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Figure 3. Phase portrait for an emulsion in which drops cream and coalescence.
the dropped term (which represents Brownian diffusion of drops) on the results of this analysis. A system of semilinear first-order hyperbolic partial differential equations possesses the property that, if initial condition n(z,0) and boundary condition n(0,t) are specified on noncharacteristic curves, it will lead to a unique determination of vector n(z,t) for all z > 0 and t > 0. Furthermore, the state of the column at a fixed height z and time t, i.e., vector n(z,t), is determined only by some portions of the initial condition or boundary condition or both. This property, characterized by the existence of a domain of dependence, is well documented in the literature. Therefore, we defer the details concerning the origin of this property to the quoted references and provide here only its graphic interpretation. Figure 3 shows the phase portrait {z, t} for the twodimensional problem at hand. The z-axis (or t ) 0 line) represents the column at time t ) 0. The state of the column at a later time t1 is represented by the vertical line t ) t1. The boundary condition at z ) 0 and t > 0 is represented by the t-axis (or z ) 0 line). The evolution of n(z1,t1) is influenced by the initial distribution contained in the segment AB where A is the intersection of the initial distribution (on line t ) 0) with the line passing through {t1, z1} with a slope equal to the smallest eigenvalue (min{Z˙ 1, Z˙ 2, ..., Z˙ M(t1)}) and B is the intersection of the initial distribution (on line t ) 0) with the line passing through {t1, z1} with a slope equal to the largest eigenvalue (max{Z˙ 1, Z˙ 2, ..., Z˙ M(t)}). Since Z˙ 1, Z˙ 2, ..., Z˙ M(t1) increase monotonically, the smallest and the largest eigenvalues are Z˙ 1 and Z˙ M(t1). Before we make use of this property to analyze eq 8, we attempt a physical visualization of how this property manifests itself. Consider drops of size v moving up with their creaming velocity Z˙ (v) from some starting location zv in the column. A change in the population of these drops until time t when they move to location zv + Z˙ (v) t can occur only if they coalesce with drops of other sizes in the immediate neighborhood as they move up or when smaller drops coalesce to form drops of size v at such times that, after they are born, they reach location zv + Z˙ (v) t exactly at time t. Since all the drops are moving with their creaming velocities, it is clear that the drops contained in a certain height range alone can come in the vicinity of the drops of size v to influence their population by coalescing with them. Similarly, the drops contained in a certain height range alone can produce drops of size v such that they will be located at height zv + Z˙ (v) t at time t. The two extreme events that define the height range that can influence the
population of drops of size v at location zv + Z˙ (v) t at time t are as follows: (i) drops of the largest size, say vmax, start from location zvmax, reach location zv + Z˙ (v) t after time t, and coalesce with drops of size v there to change their population; (ii) the smallest drops, say of size vmin, start from location zvmin, reach the location zv + Z˙ (v) t after time t, and right at that instant coalesce with drops of size v - vmin to produce drops of size v to change their population. Clearly, locations zvmin and zvmax are obtained as the intersection points of line t ) 0 with the lines passing through the phase point {zv + Z˙ (v) t, t} with slopes Z˙ 1 and Z˙ M, respectively, but this is exactly how locations A and B were also obtained. It is also easy to see that all other single or a sequence of events that can change the population of interest at the desired point will involve only those drops whose origins can be traced to drops initially contained in the height range zvmax to zvmin. Interestingly, populations of other sizes at the same phase point, i.e., location zv + Z˙ (v) t and time t, will also be determined by the particles contained in the same height range only. Thus, we have qualitatively shown that the whole size distribution [or vector n(z,t)] at any given height z (above a characteristic height which is defined later) and time t is uniquely determined by the drops initially contained in height range z - Z˙ Mt to z - Z˙ 1t only. Let us now consider another point {z′1, t1} in the phase space. The evolution of n(z′1,t1) will be correspondingly influenced by the portion of the initial distribution contained in segment A′B′ as shown in the figure. If the initial distributions [n(z,0)] contained in the segments AB and A′B′ are identical (the two segments can overlap to any extent), it is clear from the figure that not only will the size distributions n(z1,t1) and n(z′1,t1) to be identical but their further subsequent evolution will be also. Extending this reasoning to all the points on line t ) t1, we see that, if the emulsion poured into the column at initial time is uniform (identical at all heights), at time t1 it will be uniform from height H(t1) to the top of the column. The variation of height H(t) with t, which is also known as the characteristic curve, is given by
dH(t)/dt ) Z˙ M(t)
(9)
Curve H(t) should be a straight line if the size of the largest particle or equivalently index M(t) does not increase with time. In a coalescing system the size of the largest particle keeps increasing with time and therefore the slope of the characteristic curve which is equal to Z˙ M(t) also keeps increasing. This makes H(t) vs t curve concave upward. For phase points located below the above characteristic curve, the domain of influence for any two points at a fixed time will not have identical startup distributions and therefore size distributions in this zone will manifest creaming effectssthey will change with height and time both. Phase points lying in region T below curve G(t) (Figure 3), which is defined as
dG(t)/dt ) Z˙ 1
(10)
once again have identical startup distributions [n(0,t) ) 0] and hence the size distributions do not show any height dependence. Since n(0,t) ) 0, zone T is a uniform zone with no drops in it. To rephrase the above discussion in mathematical terms, if ∂n(z,0)/∂z ) 0 in the range z > 0, ∂n(z,t1)/∂z remains identically equal to zero in the height range z > H(t1) (zone R). Looking back at eq 8, this result
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3159
allows us to drop A ∂n/∂z from it for all the points lying above the characteristic curve. thus, in the zone above the characteristic curve, the drop size distribution changes only due to coalescence and the effect of creaming has been eliminated completely while the drops continue to move as in a real system! Of course, creaming of drops does continue to occur, except that its net effect does not destroy the prevailing spatial independence of the number distribution. This result forms the basis of the procedure that we suggest here for studying bulk coalescence in the absence of creaming effects while the drops continue to cream as in an actual system. We have to, however, consider the influence of the diffusion term that had to be dropped to arrive at this result.
It is clear from the previous section that the reason why creaming does not affect the evolution of size distribution in the zone above the characteristic curve is (i) the emulsion is taken to be uniformly distributed at the initial time and (ii) the effect of boundary condition at the bottom of the column that there are no drops coming from the z ) 0 plane is felt at a given point in the column only after some finite time. However, once the neglected diffusion terms are incorporated, the nature of the governing equation changes from hyperbolic to parabolic. Unlike hyperbolic equations, parabolic equations transmit the effect of conditions at the boundary in the entire domain (z > 0, t > 0) with infinite speed. From a strictly mathematical viewpoint, the elegant results of the preceding section can no longer hold in the presence of particle diffusion. However, we shall investigate this from an approximate standpoint using arguments such as those arising in boundary layer theory. The argument is built on viewing the effect of diffusion (which strictly extends to an infinite region at all times) as being confined to a boundary of thickness 2γxDt where D is the diffusion coefficient and γ is a constant which increases with the fineness of the approximation. Although for the exact region of influence γ is infinitely large, for reasonably small values of it, a suitably chosen approximate region (boundary layer) is obtained. To this end, we review eq 6 for the largest and the smallest sizes which decide the domain of dependence as shown in Figure 3. These equations are given by
+ Z˙ M
∂t DM
∂2
∂NM(z,t) ) ∂z M
∂z2
NM(z,t) +
j
∑ ∑ RM,j,kNj(z,t) Nk(z,t) Qj,k j)1 k)1 M
NM(z,t)
∑ Nk(z,t) QM,k
(11)
k)1
∂N1(z,t) ∂t D1
+ Z˙ 1
∂2 ∂z2
∂N1(z,t) ) ∂z 1
N1(z,t) +
∂N1(z,t) ∂t
4. Effect of Diffusion of Drops
∂NM(z,t)
Creaming occurs most rapidly for the largest drops. For the creaming emulsions of main interest to this paper, the Brownian motion of the largest size drops may be considered negligible in comparison with their creaming. Thus, the diffusion term in eq 11 may be dropped since its contribution to the domain of dependence will eventually be found to be negligible. For the population of the smallest size drops, their formation through coalescence (of small drops) is identically equal to zero. For other drops in the small size range, the “sink” term dominates the “source” term in the population balance equation (12). Thus, eq 12 can be reduced to
j
∑ ∑ R1,j,kNj(z,t) Nk(z,t) Qj,k j)1 k)1
+ Z˙ 1 D1
∂N1(z,t) ) ∂z ∂2 ∂z2
M(t)
N1(z,t) - N1(z,t)
∑ Nk(z,t) Q1,k
(13)
k)1
In terms of new variables, v1(ζ,t) ) N1(z,t) and ζ ) z Z˙ 1t, the above can be further simplified to the following form:
∂ ∂2 v1(ζ,t) ) D1 2 v1(z,t) - Kv1(z,t) ∂t ∂ζ
(14)
When K ) 0, the structure of the above equations is identical to those used for defining penetration thicknesses for momentum and temperature fields. When v1 is replaced by the concentration of a solute and K ) 0, the same equation forms the basis of the well-known “penetration theory in mass transfer”. As stated earlier, the effect of the boundary condition is felt all through the domain, but engineers have found it useful to dispense with its negligible influence (less than 1%) beyond the penetration thickness defined as 2xDt which corresponds to γ ) 1. For the present situation, when K ) 0, the penetration thickness is analogously given by 2xD1t, but it is measured with respect to a moving surface (constant ζ or dz/dt ) Z˙ 1). Interestingly, when K is a positive number, which is always the case in the present situation, the penetration thickness becomes smaller than 2xDt [the corresponding problem in mass transfer; i.e., diffusion with first-order reaction is solved in Bird et al. (1960)]. This is because a diffusing species that is being destroyed continuously will be able to penetrate only smaller distances as compared to when it is not being destroyed. Quantity 2xD1t can therefore be taken as an upper estimate of the penetration thickness at time t. The implication of completely ignoring the negligibly small influence of diffusion of drops in regions beyond the penetration thickness is that the domain of influence for evolution at phase point {z1, t1} now includes a portion of a strip of width 2xD1t1 running along the convected z coordinate which coincides with the characteristic direction corresponding to the smallest eigenvalue Z˙ 1. This strip is shown by the lightly shaded region in Figure 4. The exact portion of this strip at time t1 that adds to the domain of influence already obtained from eq 8 is contained between the curves
z ) (z1 - Z˙ 1t1) + Z˙ 1t
M
N1(z,t)
∑ Nk(z,t) Q1,k
k)1
(12)
(equation for the characteristic corresponding to the eigenvalue Z˙ 1) and
3160 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
Figure 4. Phase portrait for an emulsion in which drops cream, execute Brownian motion, and coalesce.
z ) (z1 - Z˙ 1t1) + 2xD1t1 + Z˙ 1t - 2xD1t These curves are presented in Figure 4 as curves 1 and 2, respectively. Also shown in this figure is the detailed representation of various terms contained in the equations for these two curves. Since the diffusion coefficient of particles larger than v1 is smaller than D1 and their creaming velocity is larger than Z˙ 1, it can be shown exactly that the domain of dependence of every phase point lying in the domain of dependence for phase point {z1, t1} is completely contained in the domain of dependence for {z1, t1}. Thus, parabolic equations combined with the concept of penetration thickness retain the much needed property of the hyperbolic equationssthat the domain of dependence is finite at least from the viewpoint of engineering applications. Clearly, the effect of including the diffusion term is only to slighly broaden the domain of dependence, and therefore all the results of the analysis presented in section 3 still hold true. If the diffusion term is not dropped from eq 11 for the largest particle, the domain of dependence will be increased on the lower side also exactly in the same way as discussed before. This increase will, however, be extremely small as the diffusion coefficient for large drops is very small. In extreme situations, i.e., when v1 f 0 or equivalently D1 f ∞, the domain of dependence will expand to include the whole of the initial condition and also the boundary condition. Emulsions, in general, do not contain such small drops. Also when creaming begins to manifest, the amount of dispersed phase contained in very small drops is negligibly small to have any appreciable effect on the coalescence instability caused mostly by large drops. The populations of drops of such small sizes can therefore be safely ignored. However, in a pathological situation when a significant amount of dispersed-phase volume is contained in very small drops that have very high diffusion coefficients and also in large drops that can cream and coalesce, the conclusions of this theory will not hold. We now address the assumption that Z˙ (v) and D(v) do not depend on the local dispersed-phase holdup. It can be seen from the analysis that in the uniform zone φ remains constant (pure coalescence conserves mass), and therefore even if the assumption that quantities Z˙ (v) and D(v) do not depend on φ is relaxed, these quantities will not change their values in the uniform zone and therefore the conclusions based on constant values of Z˙ (v) and D(v) will continue to hold. The analysis presented here is based on the assumption that the conditions at the top boundary do not affect dynamic processes in the bulk, which is akin to the assumption of a semiinfinite column. Such an assump-
tion will hold for all finite size columns that display a shock in the top region of the column. Under these conditions, the dynamics of the column above the shock surface does not influence the dynamic processes below the shock surface (supersonic flow is a well-known example). The processes below the shock layer do not see the end effects and thus a finite column continues to behave like a semiinfinite column. Physically, shock manifests itself through a clear interface (across which the local holdup changes abruptly to very high values) between the cream layer and the bulk of the emulsion. When such a shock does not exist, effects from the top boundary travel downward (Kynch, 1952) while the effects from the bottom boundary travel upward and in between the two fronts exists the uniform zone. Development of a general theory for predicting the extent of the uniform zone in such conditions for polydispersed emulsions is currently in progress. One final comment concerns the assumption of negligible flocculation. Weak flocculation results in the joining together of several drops to form a floc which creams at a much faster rate than the creaming rates of the constituents drops. The net effect of flocculation can therefore be taken as increasing the creaming velocity of a drop of size v from Z˙ (v) to a much larger value corresponding to the creaming velocity of the floc. Since creaming velocity does not affect evolution in the uniform zone, any changes in the size distribution will still be due to coalescence alone provided flocs are broken when the size distribution is measured (which is easily done for emulsions). Flocculation will, however, increase the rate at which the uniform zone shrinks, and thus the time window of observation for effects of coalescence alone will be reduced. 5. A New Procedure for Studying Bulk Coalescence The analysis presented in previous sections suggests that coalescence in the bulk should be studied in a tall column. Samples for measuring changes in drop size distribution should be taken from locations that lie above the characteristic line. This simple strategy ensures that, if the emulsion is initially uniform in the whole column, any change in the measured size distribution can only be due to the coalescence of drops. Drops can continue to cream and execute Brownian motion as in an actual emulsion, and still a change in the size distribution will be solely due to the rate process under investigation. It is to be realized that the complications due to creaming which required earlier investigators to either rotate the tube containing emulsion or shake it gently or even mix the content at regular intervals (all of which introduce additional mechanisms) are avoided only if samples are taken from the zone above the characteristic line. This requires a tall columnsin a short column, characteristic curve hit the top of the column in a short time thus providing only a short time window for monitoring coalescence without interference from the creaming effects. It is not always possible to determine the characteristic curve as it requires an estimate of the creaming velocity of the largest drop present in the emulsion, which is not known a priori. The suggested procedure therefore is to take samples from two locations near the top of the column. If the locations are above the characteristic curve, the drop size distributions at both locations will evolve identically. At some stage, however, the sample from the lower location will show smaller drops as compared with those from the upper
Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3161
which, because of a very large number of data points, give the appearance of smooth curves. The figure shows that, as time proceeds, the drop size distribution shifts to the right. This indicates coalescence of drops. The figure also shows that, after 45 min, the distributions measured at the three ports are nearly identical, as predicted by the theory proposed in the previous sections. The size distributions, measured at long times for other experiments, indicate that eventually size distributions at all ports begin to shift to th left (indicating creaming of large size drops), first at the bottom most port followed by the ports above it in succession as time proceeds. Also, there exists a nodrop zone at the very bottom of the column which keeps expanding with time. All these findings support the theoretical predictions that the characteristic curves divide the column into three zonessthe uniform zone at the top where creaming does not manifest, a nonuniform zone below it that keeps expanding with time due to the creaming of larger drops, and a no-drop zone at the very bottom that also expands but at a much slower rate. Figure 5. Evolution of the drop size distribution in a uniform zone (20% hexadecane-water emulsion at pH 9.0).
location because the lower location will come under the characteristic curve first and begin to manifest creaming effects. The data from the upper location will show the effect of pure coalescence for only some more time, after which this data will also display simultaneous creaming and coalescence effects. 6. Experimental Validation We have used the method outlined above not only to show that it can be used successfully but also to provide experimental support for the theory that predicts the existence of a uniform zone in a tall column in which the size distribution changes with time identically at all locations. A 20% holdup hexadecane-water (pH 9.0 ( 0.1) emulsion was made using an Omni GLH homogenizer. The emulsion was then poured into a 2 m long column equipped with 20 uniformly distributed sampling ports covered with septum. The samples were taken using a 0.5 mL syringe with needles whose i.d. was at least 10 times larger than the largest drop size. These samples were analyzed using a Coulter counter interfaced with a personal computer. Whenever the samples could not be analyzed immediately, they were stored in a 1% sodium lauryl sulfate (SLS) surfactant solution to prevent any further coalescence. Hexadecane was obtained from Sigma, and water used of Millipore grade. Instead of using SLS to slow down the coalescence rate, a pH 9.0 solution was used to stabilize the emulsion. Figure 5 shows the evolution of the drop size distribution at ports 15, 13, and 11, which correspond to 150, 130, and 110 cm height from the bottom. The results are presented in terms of cumulative volume fraction, defined as
cumulative volume fraction )
1 φ
∫0vvn(v,t) dv
versus drop volume v plots. Since the Coulter counter produces a large number of very closely spaced data points, plotting them by using symbols only generates very thick curves and the fine details are lost. We have therefore connected the data points by straight lines,
7. Estimation of Coalescence Frequency Once the experimental data showing changes in drop size distribution due to coalescence alone become available, the next step is to recover the coalescence frequency from it. The objective of this paper is only to present an experimental method for isolating coalescence effects, and therefore we will not attempt to obtain the coalescence kernel here; however, for completeness, we very briefly discuss how this is to be done. In light of the conclusions reached in the previous sections, if the size distributions are measured in the uniform zone of the column where they do not display any spatial dependence, they can be adequately described by eq 2 even after the whole of the second term on its left-hand side is dropped. The new simpler equation for n(v,z,t) which does not depend on z is given as
∫0vn(v-v′,t) n(v′,t) Q(v-v′,v′) dv′ ∫0∞n(v,t) n(v′,t) Q(v,v′) dv′
∂ 1 n(v,t) ) ∂t 2
(15)
The above equation is the same as that used to describe pure coalescence in a perfectly mixed stirred dispersion. For self-similar evolution of drop size distribution, Wright and Ramkrishna (1992) have proposed an inverse problem technique to determine coalescence frequencies in systems described by the above equation. Clearly, the coalescence frequency (due to the combined effect of Brownian and gravity-induced motion) in stored emulsions can also be obtained by following their procedure exactly. We refer the interested reader to the above paper for relevant details. When the evolution of the size distribution is not selfsimilar, the inverse problem technique cannot be used. Efforts to develop an alternative procedure for quantifying coalescence are in progress. 8. Conclusions Coalescence of drops, which introduces irreversible changes in an emulsion, has been studied for a long time; however, the two types of coalescence mechanisms, (i) coalescence when drops are moving relative to each other and (ii) coalescence when they are tightly packed in top layers, have not been recognized for their separate
3162 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996
roles. We have termed i as coalescence in bulk and ii as coalescence in cream. In the past, several investigators who tried to study coalescence in bulk (though without mentioning it explicitly) either introduced new mechanisms or altered the existing ones to eliminate the effects due to the creaming of drops. In the present work, a new experimental procedure is suggested for studying pure coalescence in bulk while the drops continue to move due to buoyancy and Brownian motion. We have shown that, in an initially uniform column, there exists a “uniform zone” in which the creaming and diffusion effects do not manifest. This uniform zone shrinks with time, but changes in drop size distribution occur only due to coalescence of drops. Thus, the measurement of size distributions from the uniform zone provides a new experimental method for isolating the effect of drop coalescence in bulk on the overall destabilization dynamics. Experimental results supporting the predictions of the theorysexistence of a uniform zonesare also provided. It is pointed out that, if experimental data show selfsimilar evolution, the coalescence kernel (corresponding to Brownian and gravity-induced relative motion among various drops) can be estimated using the inverse problem technique. From the mathematical viewpoint, the set of parabolic partial differential equations which do not show a finite domain of dependence is analyzed using familiar concepts from boundary layer theory. It is shown that, if the effect of particle diffusion due to Brownian motion is assumed to be confined only to penetration thicknesses, the system of parabolic equations shows hyperbolicity; i.e., the domain of influence is finite.
Acknowledgment The authors thank Ms. Negar Boushehry for carrying out the experimental work reported here as part of her CHE 411 project at Purdue University. Financial assistance through USDA Grant No. 92-37500-8020 is also acknowledged. Nomenclature A ) cross-sectional area of the column A ) diagonal matrix with elements Z˙ i b ) vector whose ith element is given by the second and third terms on the right-hand side of eq 6 D(v) ) diffusion coefficient of a drop size of v Di ) diffusion coefficient of a drop size of vi D ) diagonal matrix with elements Di H(t) ) upper characteristic curve, defined by eq 9 G(t) ) lower characteristic curve, defined by eq 10 M NjQ1,j} K ) a function of z and t {)∑j)1 M(t) ) index for the largest size drop in emulsion at time t n(v,z,t) dv A dz ) number of drops in size range {v, v + dv} in height band {z, z + dz} of a column of crosssectional area A at time t Ni(z,t) A dz ) population of particle vi in height band {z, z + dz} of a column of cross-sectional area A at time t n ) vector consisting of elements Ni(z,t) Q(v,v′) ) coalescence frequency for drops of sizes v and v′ Qi,j ) coalescence frequency for drops of sizes vi and vj t ) time z ) height from the bottom of the column Z˙ (v) ) creaming velocity for a drop of size v Z˙ i ) creaming velocity for a drop of size vi
Greek Letters Ri,j,k ) fraction of drops of size vi formed when two drops of sizes vj and vk coalesce γ ) used in defining penetration thickness as 2γxDt, equal to 1 for the conventional definition of penetration thickness and ∞ for a strictly mathematical definition φ ) local dispersed-phase holdup
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Received for review January 16, 1996 Revised manuscript received January 23, 1996 Accepted April 23, 1996X IE9600147 X Abstract published in Advance ACS Abstracts, August 15, 1996.
