Coagulation turbulent flow

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. 

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. 

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

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 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. 

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 ... 

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

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. 

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. 

The mechanism of drop coagulation depends on the conditions of mixture flow. In laminar flow, the coagulation is caused by the approach of drops due to different velocities of their motion or in the non-uniform field of velocities of an external medium, or on sedimentation in the gravity field. In a turbulent flow, the approach of drops occurs due to chaotic turbulent pulsations. In comparison with the laminar flow, the number of collisions of drops in unit time increases. Any, even insignificant, mixing of the flow increases the collision frequency. 

The flow of biphasic mixture in wells, pipes, and elements of the field equipment is characterized by a developed turbulence, which results in intensive agitation of the mixture and a faster heat and mass exchange of drops with the gas flow. In addition, turbulence causes breakup and coagulation of drops. For the turbulent flow of a biphasic mixture with a small content of the liquid phase, there emerges a distribution of drops with the average radius... 

The most efficient method of inputting hydrate inhibitor into the flow of natural gas in the pipeline is by spraying with the help of atomizers. This results in the formation of a spectrum of drops with distribution (21.3) inside the flow. The average size of the resultant drops is smaller than the stable size that is characteristic of the turbulent flow. During the motion, drops change their size as a result of mass exchange with the gas, and also coagulation and breakup. Besides, drops can be deposited at the pipe wall and then swept away from the surface of the liquid film that forms on the wall. 

In a laminar flow, the coagulation constant K is proportional to the capture cross section of bubbles of volume to by a bubble of volume V, while in a turbulent flow, it is proportional to the flux of bubbles of volume w towards the test bubble of volume V. 

Note that, as opposed to the coagulation of bubbles in a laminar flow, coagulation in a turbulent flow is impossible in the absence of molecular forces, since Dt- at<5=r a )—>0. Therefore the integral term entering the solution of the diffusion equation has a nonintegrable singularity that can be eliminated only by the introduction of a molecular attraction force growing as at > 0. 

The expressions for frequencies of bubble collision in laminar and turbulent flow which derived in the previous paragraphs make it possible to And the kernels of coagulation K co, V) and then proceed to solve the kinetic equation (25.1). Because the solution, generally speaking, presents significant mathematical difficulties, we shall only consider some simple special cases. 

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