6.3. Gravitational deposition of aerosol particles
The work of gravitational dust-collecting devices is based on the laws of gravitational deposition, i.e. deposition of dust particles under the action of gravity. Deposition phenomena also occur in apparatuses whose action is based on the use of other forces.
Consider the rectilinear uniform motion of a particle that obeys Newton's law. Possible convective currents are not taken into account. When moving, the particle meets the resistance of the medium, which can be determined
where ζ h is the aerodynamic drag coefficient of the particle; S ч - the projection of the cross section of the particle on the direction of its motion (the area of the midsection), m; w h - particle velocity, m/s; p s is the density of the medium, kg/m.
The particle resistance coefficient ζ h depends on the Reynolds number Re h . For a spherical particle
here d h - particle diameter, m; μ s is the dynamic viscosity of air (gas), Pa · s.
The graph expressing the dependence of ζ h on Re h (Figure 6.3.), consists of three parts. At 5 x 10 & lt; Re h & lt; 5 · 105 resistance is characterized in the region of developed turbulence by Newton's law. In this section, the resistance coefficient ζ h of the automodels with respect to the Reynolds number (ζ h = 0.44). For Re h & lt; 1, the resistance force is determined by the Stokes law. The dependence of ζ h on Re h is expressed as a direct segment in logarithmic coordinates.
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Fig. 6.3. Dependence of the coefficient of drag of a spherical particle ζ, on the criterion Re ч :
l - the scope of the Stokes law; 2 - standard curve; 3 - range of Newton's formula
According to experimental data, the resistance coefficients for a spherical dust particle have the values given in Table. 6.1.
Table 6.1. Dependence of the drag coefficient on the driving mode
Re h & lt; 2
2 & lt; Re h & lt; 500
500 & lt; Re h & lt; 150 000
ζ h = 24/Re
ζ h = 18.5/Re
ζ h 0.44
Assuming ζ h = 24/Re h for the laminar motion in the Re h & lt; 2, we substitute its value in the Newton formula (6.17):
This formula expresses the Stokes law: the resistance force experienced by a solid spherical body with a slow motion in an unbounded viscous medium is directly proportional to the speed of translational movement, the diameter of the body and the viscosity of the medium.
If the motion of a non-spherical particle is considered, in the calculation formulas the value ζ h is multiplied by the dynamic coefficient of the form χ, instead of d h , the equivalent diameter is introduced:
where d e is the equivalent diameter of a particle equal to the diameter of a sphere whose volume is equal to the volume of a given particle, m
The values of χ for particles of different shapes: spherical - 1; rounded with an uneven surface - 2.4; oblong - 3; lamellar - 5; for mixed bodies - 2.9.
In the motion of a particle precipitated by gravity in a stationary medium, it is possible to distinguish three stages: the initial moment of the fall; movement with increasing speed until the moment when the forces of resistance and gravity are balanced; uniform motion with constant speed. The first two stages have a short duration.
In the region of the Stokes law, the velocity of deposition of a spherical particle is
where ρ ч is the particle density, kg/m; g = 9.81 m/s - acceleration of gravity; is the particle relaxation time, s.
The density of air (gas) is neglected.
If the air velocity is equal to the deposition rate and is directed against it, then the rate of precipitation of the dust particle in the air is zero.
The speed of air in the ascending stream, at which the particle is stationary (or oscillating), is called the speed of winding. Thus, the constant rate of precipitation of dust particles in still air is equal to the speed of its waning.
The concept of wobble rate It is important for systems and devices in which the gaseous medium moves with particles suspended in it (pneumatic transport, aspiration, dust catchers, working mainly on the principle of gravity).
Sedimentation is most fully manifested in a medium at rest and in laminar flows. A simple model for the deposition of particles in a precipitation chamber (Figure 6.4) is obtained on the basis of the assumption of the frontal character of the gas flow through the chamber and the uniform arrangement of the particles in the gas.
From the equation of motion of a dust particle at rest at time t = 0, and then slowly settling at a speed of w by gravity (0
The particle deposition rate is:
where w s is the constant speed of waning, equal to
Fig. 6.4. Scheme of deposition of particles in the chamber:
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1 - cleared area; 2 - limiting trajectory
Typically, the residence time t of the particle in the channel is much longer than its relaxation time τ. Consequently, the quantity e/τ can be neglected and the sedimentation velocity of the unequal wake velocity w s can be assumed. Precipitation of particles under the action of their own weight occurs very slowly (w = gτ).
In a laminar flow, the flow velocity components at any of its points can be expressed in terms of the stream function ψ:
If we assume that the velocity w L of particle motion in the direction of flow is equal to the flow velocity v L , then the velocity components of the particle can be expressed by the following equations:
As a result, we obtain the differential particle trajectory equation w s dL = - dψ. Integrating along the length of the channel L, we obtain
where ψ 0 and ψ L are the values of the current function at the points occupied by the particle, respectively, when entering and leaving the channel.
The current function expresses the volume of air flowing per unit time between the bottom of the channel of unit width and the given line (surface) of current. Since the trajectories of the precipitating particles cross the channel bottom, ψ L = 0 demarcates the trajectories of the precipitating and non-precipitating particles. For the precipitating particles, φ 0 = w s L. If we denote the total flow rate through v cp H, then the sedimentation efficiency of the particles? Will be characterized by the ratio
The length of the channel, which is necessary for the complete precipitation of all particles with a speed of w s , is:
As seen from the last expression, the deposition efficiency does not depend on the nature of the velocity distribution.
To separate small particles, the diffusion factor is more effective. The smaller the particles, the more their ability to molecular (Brownian) diffusion in all cases and to turbulent diffusion in turbulized aerosol flows is manifested.
During the diffusion process, the particles do not remain on the same current line. In an irregular motion, they also mix in directions that are transverse to the streamline, approaching the flow boundaries and the surface of streamlined obstacles, up to and including collisions with them.
For the deposition of particles of size d h & lt; 0.2 μm molecular diffusion is the determining one. The result of Brownian motion of particles of size d h & gt; 1 μm, when the diffusion coefficient D & lt; 10 cm/s, very small. Molecular diffusion is taken into account in the theory of high-efficiency filters designed to trap very fine dusts.
The effect of turbulent diffusion extends to particles of much larger size. Studies show that particles of size d h & lt; 30 μm are completely carried away by turbulent pulsations. The completeness of the capture of particles by turbulent pulsations depends on their mass or inertia. Thus, even in diffusion deposition, inertia forces play an important role, and in most cases the practice of dedusting ventilation air forces is the determining factor.
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