Coagulation in Turbulent Flow

The relative magnitudes of the turbulent shear and Brownian coagulation coefficients for equal-sized particles in the continuum regime is given by the ratio 

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The second method for aerosol coagulation in turbulent flows arises because of inertial differences between particles of different sizes. The particles accelerate to different velocities by the turbulence depending on their size, and they may then collide with each other. This mechanism is unimportant for a monodisperse aerosol. For a polydisperse aerosol of unspecified size distribution, Levich (1962) has shown that the agglomeration rate is proportional to the basic velocity of the turbulent flow raised to the 9/4 power, indicating that the agglomeration rate increases very rapidly with the turbulent velocity. Since very small particles are rapidly accelerated, this mechanism also decreases in importance as the particle size becomes very small, being most important for particles whose sizes exceed 10-6 to 10"4 cm in diameter. In all cases brownian diffusion predominates when particles are less than 10-6 cm in diameter. 

In practice, coagulation almost always takes place in turbulent flows. Examples are industrial aerosol reactors, combustion systems, process gas flows, and the atmosphere. Turbulent... 

The third mechanism, named turbulent coagulation, is characteristic of coagulation of particles suspended in turbulent flow, for example in a pipe or in some special mixing devices - mixers, agitators, etc. In some aspects, the turbulent coagulation is similar to the Brownian one, since in the first case the particles approach is due to random turbulent pulsations, and in the second case it is due to random thermal motion of particles. 

Aerosol field dynamical studies comprise those investigations of the factors that govern the (spacially) large-scale behavior of the particles. Turbulent transport of particles is the motion of particles in turbulent flow fields. Since the response of a particle to this flow is, by definition, dependent upon its aerodynamic properties, differential motion between particles of differing sizes can greatly influence coagulation and, therefore, the aerosol size distribution. 

When the stirring speed increases, there is a transition from laminar to turbulent flow. Treatment of this kind of orthokinetic coagulation is much more difficult than the case of laminar flow. An approximate way to treat this situation is to use the Smoluchowski equation for orthokinetic coagulation in laminar flow and to employ an average shear gradient in the turbulent flow. Various estimates of this average have been derived. One of these... 

Conceptually similar results were demonstrated by Krutzer et al. [14], who measured the orthokinetic coagulation rate under laminar Couette flow and isotropic turbulent flow (as well as other flow conditions). Despite equal particle collision rates, significance differences were observed in the overall rates indicating different collision efficiencies (higher collision efficiencies were found under a turbulent flow regime). Thus, identical chemical properties of a dispersion do not determine a single collision efficiency the collision efficiency is indeed dependent upon the physical transport occurring in the system. 

Turbulent coagulation occurs when particles in multiple flow line intersections collide, a process stimulated by turbulent flow or eddy conditions. 

The effects of turbulence on coagulation are only partially understood. There are uncertainties in the fundamental theory of turbulence and in the motion of small particles in close proximity in the turbulent flow field. As a result, theoretical predictions of turbulent coagulation rates must be considered approximate perhaps to within a factor of 10 and are likely to be on the high side. 

There have been few experimental tests of the theoretical predictions of turbulent coagulation under controlled conditions. Delichaisios and Probstein (1975) measured rates of coagulation of 0.6-mm latex particles suspended in an aqueous solution in turbulent pipe flow. The Reynolds numbers ranged from 17,000 to 51,000 for flow through a 1-in. (l.D.) smooth-walled pipe. For the core of the pipe flow, the turbulence was approximately isotropic. The energy dissipation per unit mass was calculated from the relation... 

In many, if not most, cases of practical interest, the fluid in which the particles are suspended is in turbulent motion. In Chapter 7. the effects of turbulence on the collision frequency function for coagulation were di.scussed. In the last chapter, nucleation in turbulent How was analy ,ed through certain scaling relations based on the form of the concentration and velocity fluctuations in the shear layer of a turbulent jet. In this section the GDE for turbulent flow is derived by making the Reynolds assumption that the fluid velocity and size distribution function cun be written as the sum of mean and fluctuating components ... 

If the rate of coagulation is rapid compared with the rate of loss to the walls of the pipe, the two processes—coagulation and surface deposition—can be treated. separately. For the purposes of the calculation, the flow can be broken into two parts In the turbulent core, coagulation controls the shape of the size distribution, which is then detemiined by the solution to the equation for coagulation under steady flow conditions. 

Miihle K. (1993), Floe stability in laminar and turbulent flow, In Coagulation and flocculation theory and... 

The mechanism of coagulation in a laminar flow has limited practical application since in the majority of applications the motion of liquid has a turbulent character. At turbulent motion, the collision frequency of particles increases very considerably in comparison with a quiescent environment or with laminar motion. Now consider the mechanism of particle coagulation in a turbulent flow, following the work [52]. 

Thus, in the case of coagulation in a turbulent flow, coagulation frequency corresponds to a reaction of the second order and is proportional to a , as in the laminar flow. 

The use of turbulent emulsion flow regime to facilitate integration of drops is justifled by the substantial increase of collision frequency that is achieved in a turbulent flow as compared to the collision frequency during the sedimentation of drops in a quiescent liquid or in a laminar flow. Particles suspended in the liquid are entrained by turbulent pulsations and move chaotically inside the volume in a pattern similar to Brownian motion. Therefore this pulsation motion of particles can be characterized by the effective factor of turbulent diffusion Dj, and the problem reduces to the determination of collision frequency of particles in the framework of the diffusion problem, as it was first done by Smoluchowsld for Brownian motion [18]. A similar approach was first proposed and realized in [19] for the problem of coagulation of non-interacting particles. The result was that the obtained frequency of collisions turned out to be much greater than the frequency found in experiments on turbulent flow of emulsion in pipes and agitators [20, 21]. 

Here t = (ve/so) is the characteristic time scale in a turbulent flow with specific energy dissipation so, W — AnfiRlrio/ i is the volume concentration of drops of type 2 in the flow, C is the parameter whose value depends on the chosen model of drop coagulation in a turbulent flow. 

The principal shortcoming of the turbulent coagulation model offered by Levich [19] and rejected by many researchers is that it seriously overestimates the collision frequency of drops. Therefore the shear coagulation model [110] of particle coagulation in a turbulent flow has emerged as by far the most popular one. Since Levidfs model does not take into account the hydrodynamic interaction of particles, let us estimate the effect of this interaction on the collision frequency. 

For Browrtian diffusion of small particles, the influence of hydrodynamic interaction on the collision frequency was studied in works [28, 29], which also mention the decrease in the collision frequency by a factor of 1.5-2. This decrease is not as large as in the case of turbulent coagulation. There are two reasons why the effect of hydrodynamic interaction on the collision frequency of particles differs so substantially in the cases of turbulent flow and Brownian motion. First, the particle size is different in these two cases (the characteristic size of particles participating in Brownian motion is smaller than that of particles in a turbulent emulsion flow). Second, the hydrodynamic force behaves differently (the factor of Browrtian diffusion is inversely proportional to the first power of the hydrodynamic resistance factor h, and the factor of turbulent diffusion - to the second power of h). 

The diffusion model of turbulent coagulation is applicable to homogeneous and isotropic turbulent flow. A developed turbulent flow of emulsion in a pipe, in the vicinity of the flow core, can be considered as isotropic. However, the turbulent motion of liquid in a stirrer is neither homogeneous nor isotropic. Therefore the applicability of diffusion model to process of coagulation in agitator is not established with certainty. 

Sinaiski E. G., Coagulation of Droplets in a Turbulent Flow of viscous Liquid, Colloid J., 1993,... 

In the considered case, the basic mechanisms of formation of droplets in the turbulent gas flow are processes of coagulation and breakage of drops. These two processes proceed simultaneously. As a result, the size distribution of the drops is established. Assuming homogeneity and isotropy of the turbulent flow, this distribution looks like a logarithmic normal distribution [1] ... 

Droplets in motion in a turbulent flow of gas in a pipe are subject to breakage and coagulation (coalescence). These processes occur simultaneously and as a result a dynamic balance is established between them, determined by some distribution of drops with average radius i av-... 

In a turbulent flow there are two chief mechanisms of drop coagulation [2], that of turbulent diffusion and that of inertia. The inertial mechanism is based on the assumption that turbulent pulsations do not completely entrain the drop. As a result, relative velocities attained by drops due to turbulent pulsations depend on their masses. The difference in the pulsation velocities of drops of various radii causes their approach and leads to an increase of collision probability. The mechanism of turbulent diffusion is based on an assumption of full entrainment of drops by turbulent pulsations with scales, playing the chief role in the mechanism of approach of drops. Since drops move chaotically under the action of turbulent pulsations, their motion is similar to the phenomenon of diffusion and can be characterized by a coefficient of turbulent diffusion. 

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