Environ. Sci. Techno/. 1995, 29, 1346- 1352
Trajectory
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Introduction
PAUL LAFRANCEt A N D D 0MEN IC GRASS0 * m t , *
Environmental Engineering Program and Department of Civil & Environmental Engineering, The University of Connecticut, Storrs, Connecticut 06269-2037
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Pressure flotation of nitrocellulose (NC) fines was quantified by trajectory analysis. A model was formulated that incorporates recently developed extended DLVO (ExDLVO) theory and the short-range hydrodynamic interactions of a particle near a bubble surface. A repulsive Lifshitz-van der Waals (LW) force was found to exist between particle and bubble. Attractive long-range hydrophobic forces between NC particles and bubbles were responsible for particle adhesion. Model predictions of flotation efficiency compared well with experimental results.
Pressure flotation,or dissolved air flotation (DAF),is a solid/ liquid separation process first developed in the ore processing industry. DAF has since found many applications in the environmental engineering field including drinking water clarification,sludge thickening, and the clarification of wastewater from a variety of industrial and municipal processes. Malley (1) and Malley and Edzwald (2,3)have provided excellent summaries of DAF processes and have proposed a conceptual model for the dissolved air flotation process, derived from single-collectorcollision theory. Previous to their work, no conceptual models were available to describe DAF performance in water treatment. The design and operation of DAF facilities is based largely on experience and results from pilot plant studies. However, Okada etal. (4) have recently used a trajectory model to estimate flotation removal efficienciesof small particles. Their model incorporated short-range hydrodynamic interaction near the bubble surface as well as traditional DerjaguinLandau-Venvey-Overbeek (DLVO) forces. Classical DLVO forces are comprised of an electrodynamic or van der Waals component and an electrostatic component. Van Oss and colleagues have recently extended DLVO (ExDLVO)theory by accounting for Lewis acid-base interactions. The inclusion of acid-base forces is essential for describingpolar media interfacial interactions (5). Most work dealing with particle stability focuses on like-particle interaction where van der Waals forces between identical particles are attractive and electrostatic forces are repulsive. When examining flotation of fine or colloidal size particles, surface forces between the bubble and particle become important (6, 7). Heterocoagulation theory (8) describes interactions between dissimilar particles (e.g., particlebubble). In this case, both the van der Waals and the electrostatic force may be either attractive or repulsive. The work presented herein is part of a larger effort to explore techniques to recover and reuse nitrocellulosefines resulting from propellant manufacturing processes. A previous paper (9) investigated NC particle stability and interfacial thermodynamics. This paper builds on that work and presents the development andvalidation of a trajectory model to describe pressure flotation performance. In addition to describing short-range hydrodynamic forces, the model also incorporates an extended DLVO approach to describe interfacial forces. Model simulation results are subsequently compared to experimental data.
Model Development The limiting trajectory approach is a quantitative description of particle-deposition rates on collector surfaces, most notably on cylindrical or spherical collectors (10). Flotation efficiency can be predicted by calculating a particle’s trajectory around the surface of a bubble (Figure 1). Trajectory models have also been utilized to describe other environmental engineering processes, such as coagulation, filtration,and membrane processes (11-1 8). Surfaceforces on the particles and bubbles as well as short-range hydrodynamic interactions near the bubble surface must be quantified to accurately predict the particle’s trajectory. +
Environmental Engineering Program.
* Department of Civil & Environmental Engineering. 1346 ENVIRONMENTAL SCIENCE & TECHNOLOGY I VOL. 29, NO. 5, 1995
0013-936W95/0929-1346$09.00/0
D 1995 American Chemical Society
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affected by either the particle or neighboring bubbles (23); (iii) due to the small bubble size, rise velocity may be represented to Stokes’ law
where e is the density of the suspending medium (kg/m3); egis the density of the gas bubble (kg/m3);p is viscosity of the medium (N s/m2);and g is the gravitational constant (9.81 m/s2);(iv) particles experienced no gravitational or hydrodynamic couples and were modeled as cylinders in dynamic rotational equilibrium excluding a sphere described by Stokes’ diameter (-2.3 pm for NC particles) (24, 25); and (v) based on removal mechanisms expected for this particle size range, Brownian motion was not considered (1). The total force acting on the particle normal to the bubble surface, P,is the sum of interfacial and hydrodynamic forces
FT= PW + PL+ P + PR where Fwis the Lifshitz-van der Waals component (N); P L i sthe electrostaticcomponent (N);P i s the Lewis acidbase component (N); and EHR is the hydrodynamic component (N).Determination of LW, EL, and AB forces for NC particles have been described by Grasso et al. (9). Lifshitz-van der Waals (LW) Interactions. For small distances from the bubble surface, Fw(unretarded) can be represented for a sphere of radius rp and a semi-infinite flat slab as (26, 27)
where H i s a dimensionless parameter = h/rp (see Figure 1);A312 is the Hamaker constant for particle 3 and bubble 2 immersed in liquid 1 (24)
FIGURE 1. limiting trajectory of a particle around a bubble. The variable 4 represents the far-field maximum orthogonal distance from the center line where the particle will contact and adhere to the bubble.
Flotation can be regarded as the capture of particles from suspension flowing past a spherical collector. Rising bubbles may be considered an assembly of individual spheres (collectors) of radius rB that are immersed in a fluid of infinite extent. This type of approach is often referred to as an isolated-sphere collector model. Once system physics and chemistry are defined, particle capture can be quantified from the identification of the limiting trajectory. On the basis of these principles, a model was formulated to predict particle-bubble collision efficiency. The model can be used to find the limiting trajectory, lL, which is the largest far-field orthogonal off-centerdistance forwhich aparticle-bubble collision will be successful (see Figure 1) (8,11). Assumptions invoked in the model development include the following: (i) the bubble radius is much larger than the radius of the NC particle (9) and may be regarded as a flat plane when the particie is near its surface (19, 20); (ii) the bubble has a rigid surface (21, 22), and its motion is not
where h, (nm) is the minimum equilibrium distance which derives from Born repulsion, calculated as 0.16nm (28). In the above formation, y (mJ/m2)is the LW surface tension component for each phase. With the assumed uniformity of hovalues,the above equation shows that the flocculation of dissimilar particles may become energetically unfavorable, i.e., one obtains a negative Hamaker constant, when the LW component of the surface tension of the medium is intermediate between the surface tension of the two types of dispersed units (24). At larger distances (’10 nm), dispersion forces (eq 3) decrease due to retardation effects at a rate proportional to H-, rather than H-2 (28). Electrostatic (EL) Interactions. Healy and co-workers (29, 30) reported that double-layer interaction between particles is afunction ofthe product oftheir Stem potentials lyy and lyg. If the 5 potential is assumed to be the same as the Stern potential, the double-layer interaction between a bubble and a particle will be a function of the product of their 5 potentials (31). For the case of rB >> rp
pL= nEEorpKcpCB x 4 exp(-Krpm
(1
+ exp(-Kr,Hi))
]
- 2(cP - cB12e x p ( - 2 ~ r a (5)
CPCB(l - exp(-2~r,H)I
VOL. 29. NO. 5 , 1 9 9 5 I ENVIRONMENTAL SCIENCE &TECHNOLOGY
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Neglecting the second term on the right-hand side of eq 5 ( ( ( 5-~ CB)’/CPCB)
