Turbulent pulsation

The present investigation applies deterministic methods of continuous mechanics of multiphase flows to determine the mean values of parameters of the gaseous phase. It also applies stochastic methods to describe the evolution of polydispersed particles and fluctuations of parameters [4]. Thus the influence of chaotic pulsations on the rate of energy release and mean values of flow parameters can be estimated. The transport of kinetic energy of turbulent pulsations obeys the deterministic laws. 

Theoretical investigations of the problem were carried out on the base of the mathematical model, combining both deterministic and stochastic approaches to turbulent combustion of organic dust-air mixtures modeling. To simulate the gas-phase flow, the k-e model is used with account of mass, momentum, and energy fluxes from the particles phase. The equations of motion for particles take into account random turbulent pulsations in the gas flow. The mean characteristics of those pulsations and the probability distribution functions are determined with the help of solutions obtained within the frame of the k-e model. 

Another kind of wall-effect was proposed by El perin (1967). He suggested that an adsorbed layer of polymer molecules could exist at the pipe wall during flow and this could lower the viscosity, create a slip, dampen turbulence pulsations, and prevent any initiation of vortices at the wall. Later work (Little 1969), however, with a transparent pipe and dyed polymer, showed that the adsorption could in be fact an experimental artifact (a quantity of polymer solution, trapped in pressure gage piping, slowly diffused back into the solvent flow). Although polymer molecules do more or less adhere to clean surfaces in thin films, there is no interaction with the bulk of the solution which could alter the flow properties (Gyr, 1974). Thus, it is evident that adsorption of the additives on surfaces is not the reason for the drag reducing effect. 

Levich (L8, L9) has given an interesting treatment of fully turbulent film flow. In the absence of a flowing gas stream at the interface, Levich deduced that the scale of turbulence and the turbulent velocity normal to the interface must be proportional to the distance from the interface, so that all turbulent pulsations must disappear at the interface itself, leaving there a nonturbulent layer of thickness... 

We imagine a distribution of a which is characterized by an amplitude o0 and a length scale L which exceeds the maximum scale of the turbulent pulsation, l. We denote the pulsating velocity by u the turbulent coefficient of diffusion, the coefficient of thermal conductivity and the effective turbulent kinematic viscosity are all expressed by the formula k = ul. For an initial uniform distribution of a, obviously,... 

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. 

Let particles be suspended in turbulent flow, with the average particle concentration re. Turbulent pulsations are characterized by the velocity Vx, and by the length X over which the pulsation velocity undergoes a noticeable change. In a turbulent flow, there are large-scale pulsations (with the upper cap on them imposed by the characteristic linear size of the flow region I, for example, the diam-... 

Consider particles of radius a Ao and assume that in the course of their motion in the liquid, they are completely entrained by turbulent pulsations that play the basic role in the mechanism of mutual approach of suspended particles. Then it can be assumed that particle transport is performed via isotropic turbulence. Since particles move chaotically in the liquid volume, their motion is similar to Brownian one and can be considered as diffusion with some effective factor of turbulent diffusion Dr. In the same manner as in the case of Brownian coagulation, it is possible to consider the diffusion flux of particles of radius U2 toward the test particle of radius Uj. The distribution of particles U2 is characterized by the stationary diffusion equation... 

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]. 

In turbulent flow of liquid, random pulsation motions characterized by a set of pulsation velocities are imposed on average movement with velocity 1/ in a certain direction. Turbulent pulsations are characterized not only by velocities, but also by distances at which these velocities undergo noticeable change. These distances are referred to as pulsation scales and are denoted as 2. The set of values X represents a spectrum of turbulent pulsations varying from 0 up to a maximal value, having the order of linear scale of cross-sectional area of current flow. So, at motion in a pipe of diameter I the greatest value X is equal to I. Every pulsation movement is characterized by its Reynolds number Re = Xuxjv, where... 

Combining (11.43) and (11.45), one receives general expression for velocity of turbulent pulsations of scale 1... 

In case of turbulent diffusion, the situation is somewhat different. Motion of particles under action of turbulent pulsations is not connected to thermal fluctuations. Therefore B = const and the factor of turbulent diffusion is inversely proportional to the second power of factor of hydrodynamic resistance. 

We start with the case when the drop surface is completely retarded, in other words, drops can be considered as undeformed particles. We also assume that coalescence occurs only due to the joint action of turbulent pulsations and molecular attractive forces. The force of molecular attraction between two spherical par-tides is given by the formula (11.100), which implies that this force is determined by the distance between particle surfaces and does not depend on their mutual orientation, i.e. is spherically-symmetrical with respect to the center of the particle of radius. Since the force of molecular attraction manifests itself only at small clearances A between particles, we shall take its asymptotic expression at A->0... 

Breakage of drops in a turbulent gas flow occurs due to the inertial effect caused by a significant difference of density of liquid and gas, and also due to difference of pulsation velocities, i.e. velocities of turbulent pulsations flowing around a drop, at opposite ends of the drop. Breakage of a drop thus occurs due... 

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 collision frequency and the breakage probability are determined by consideration of the interactions between drops in a field of turbulent pulsations. Their definition represents an independent problem. The problem of drop interactions in an emulsion will be considered in detail in Section V. 

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 assessment of hydrodynamic and molecular interactions of drops can be made in the same manner as previously described for emulsions in Part V. Upon approach of the drops to each other under the action of turbulent pulsations up to distances smaller than Ao, they are subject to significant resistance from the environment, and the force of moleetilar attraction leads to collision and coalescence of the drops. If the basic mechanism of drop coagulation is that of turbulent diffusion, the coefficient of turbulent diffusion depends on the coefficient of hydrodynamic resistance [see Eqs. (11.70), (11.72), and (11.74)] and hence on the relative distance between the approaching drops ... 

The basic mechanism of component delivery to the interface in the gas phase is transfer by turbulent pulsations characterized by a scale of X. The amount of component flux, Ji, to a drop (or from it) depends on the ratio J av/ o and is equal to [2] ... 

The mass exchange for a drop suspended in a turbulent flow, occurs due to the delivery of substance to the drop surface by turbulent pulsations, and also via molecular diffusion. As it was shown in Section 16.2, the expression for the mass flux of the substance at the drop surface depends on the ratio between the drop radius and the internal scale of turbulence Xo = Djt/Re ", where Dk is the diameter of the working cross-section of the absorber, and Re is Reynolds number. For the characteristic parameter values = 50 kg/m U = 3 m/s Dk = 0.4 m Pq = 10 Pa s, we have Re = 6 10 and 2o = 5 lO" m. Since the drop size is R 2 10 m, the inequality Xo < R holds, and the delivery of substance to the drop surface is mostly performed by turbulent pulsations. Thus the mass flux on the drop surface is equal to (see (16.55))... 

The motion of formed ensemble of drops with gas flow is accompanied by continuous change of drops distribution over sizes this results from the concurrent processes of mass-exchange between the drops and the gas, coagulation and breakup of drops under action of intensive turbulent pulsations of various scales. 

In a turbulent flow, turbulent pulsations with different amplitudes are superimposed on the averaged motion. These pulsations are characterized by both the velocity un and the distance (known as the scale of pulsations), at which the velocity of pulsations undergoes an appreciable change. Each of them can be associated with Reynolds number Re i = UxX/vi. For large pulsations, 2 L, where L is the characteristic linear scale of the region where viscous forces do not have a noticeable influence, while for small pulsations, they can dominate. The scale 2o for which Re o = 1, is called the internal scale of turbulence. Viscous forces are important for pulsations with X < Xq. Under the action of pulsations, bubbles can move chaotically. In [10], it is shown that the form of Do depends on the type of pulsations that cause bubbles to perform random motion ... 

The important factor influencing on specific surface area of phase interface is deformation of drops (bubbles) surface that in general case is caused by dynamic head under the effect of turbulent pulsations of disperse medium rate and (or) phases movement rate because of the difference in their densities. In this case the minimal size of dispersion phase particles dcr undergoing to deformation may be calculated from the ratio characterizing stability of phase interface (1.23) and (1.24). 

Where Ptuji, is the dynamic factor of turbulent viscosity, which can be expressed in terms of turbulent pulsation energy K and the specific rate of its dissipation, e. In turn, the corresponding phenomenological defining equations are worked out for these parameters [36, 51]. 

An important factor for the specific interface area is the deformation of droplets (bubbles), which is generally determined by the dynamic influx caused by turbulent pulsations from the dispersion medium, and/or the ratio of phase rates, as a result of their different densities (gravitation component). The minimal size d of dispersed phase particles undergoing deformation can be, in this case, calculated using the equation which characterises the stability of the interphase boundary. 

The negative influence of viscosity on the interphase boundary can be avoided by the addition of surfactants, leading to a decrease of the interfacial tension. The selective distribution of alcohol molecules, near the separating surface level, results in differences in the composition of the nonmixing phases and reduces the work required to form a new surface. In this case, with the continuous phase turbulent pulsation parameters unchanged, reaction mixture dispersion is favoured in a tubular diffuser-confusor device. It should be considered, that interfacial chemical reactions assume the presence of at least three substances in a reaction zone, different in chemical nature, with some able to adsorb onto the interphase boundary. It can influence the size of dispersion inclusions in real conditions. 

Adhesion of Particles to Walls (Sides) of Air Duct. Adhesion to vertical walls takes place as a result of the action of the normal velocity component of the dust-containing air stream. This component arises from turbulent pulsations of the flow in a direction perpendicular to the wall surface of the air duct. The correctness of this view is confirmed by the studies of Ryzhenko and Shcherbina [246], who showed that the amount of dust sticking to 80 X 80 mm duralumin plates mounted around the perimeter of a vent drift in the Kochegarka mine was approximately the same on the sidewalls and the roof. 

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