During the motion of viscous flow over a stationary body or particle, certain resistances arise. To overcome these resistances or drag and to provide more uniform fluid motion, a certain amount of energy must be expended. The developed drag force and, consequently, the energy required to overcome it, depend largely on the flow regime and the geometry of the solid body. Laminar flow conditions prevail when the fluid medium flows at low velocities over small bodies or when the fluid has a relatively high viscosity. Flow around a single body is illustrated in Figure 1. As shown in Figure 1(A), when the flow is laminar a well-defined boundary layer forms around the body and the fluid conforms to a streamline motion. The loss of energy in this situation is due primarily to fiction drag. If the fluid's average velocity is increased sufficiently, the influence of inertia forces becomes more pronounced and the flow becomes turbulent. Under the influence of inertia forces, the fluid adheres to the particle surface, forming only a very thin boundary layer and generating a turbulent wake, as shown in Figure 1(B). The pressure in the wake is significantly lower than that at the stagnation point on the leeward side of the particle. Hence, a net force, referred to as the pressure drag, acts in a direction opposite to that of the fluid's motion. Above a certain value of the Reynolds number, the role of pressure drag becomes
Figure 1. Flow around a single particle.
Figure 1. Flow around a single particle.
We shall begin discussions by analyzing a dilute system that can be described as a low concentration of noninteracting solid particles carried along by a water stream. In this system, the solid particles are far enough removed from one another to be treated as individual entities. That is, each particle individually contributes to the overall character of the flow. Let's consider the dynamics of motion of a solid spherical particle immersed in water independent of the nature of the forces responsible for its displacement. A moving particle immersed in water experiences forces caused by the action of the fluid. These forces are the same regardless whether the particle is moving through the fluid or whether the water is moving over the particle's surface. For our purposes, assume the water to be in motion with respect to a stationary sphere. The fluid shock acting against the sphere's surface produces an additional pressure, P. This pressure is responsible for a force, R (called the drag force) acting in the direction of fluid motion. Now consider an infinitesimal element of the sphere's surface, dF, having a slope, a, with respect to the normal of the direction of motion (Figure 2). The pressure resulting from the shock of the fluid against the element produces a force, dF, in the normal direction. This force is equal to the product of the surface area and the additional pressure, PdF0. The component acting in the direction of flow, dR, is equal to dxcosa. Hence, the force, R, acting over the entire surface of the sphere will be:
where dF is the projection of dF0 on the plane normal to the flow. The term F refers to a characteristic area of the particle, either the surface area or the maximum cross-sectional area perpendicular to the direction of flow. The pressure P represents the ratio of resistance force to unit surface area (R/F), and it depends on several factors, namely the diameter of the sphere (d), its velocity (u), the fluid density (p) and the fluid viscosity (p,).: i.e., P = f(d, u, p, ¡x). Applying
dimensional analysis, the following dimensionless groups are identified:
where Eu is the dimensionless Euler number, defined as P/u2 p, and Re is the Reynolds number (Re = du p//t). By substituting for density using the ratio of specific gravity to the gravitational acceleration, an expression similar to the well-known Darcy-Weisbach expression is obtained:
where CD is the drag coefficient, which is a dimensionless parameter that is related to the Reynolds number. The relationship between CD and Re for flow around a smooth sphere is given by the plot shown in Figure 3. As shown in this plot, there are three regions that can be approximated by expressions for straight lines. These three regions are the Stokes law region, Newton's law region, and the intermediate region. Refer to the sidebar discussion for these expressions. By substituting the expressions for the drag coefficient into equation (3), we obtain a convenient set of expressions that will enable us to calculate settling velocities. The details are left to you. There are plenty of empirical correlations in the literature for the drag
Figure 3. Drag coefficients for spheres.
coefficient for different geometry objects, but practice is to use a simple sphere and then account for geometry effects by means of a correction or shape factor. We will cover this further on. For now, let's examine the phenomenon of particle settling more closely.
If a particle at rest (with mass 'm' and weight 'mg') begins to fall under the influence of gravity, its velocity is increased initially over a period of time. The particle is subjected to the resistance of the surrounding water through which it descends. This resistance increases with particle velocity until the accelerating and resisting forces are equal. From this point, the solid particle continues to fall at a constant maximum velocity, referred to as the terminal velocity, ut. You should recall from the last chapter that we called the settling velocity. The force responsible for moving a spherical particle of diameter'd' can be expressed by the difference between its weight and the buoyant force acting on the particle.
Laminar Regime (Stokes Law Region)
Transitional Flow Regime (Intermediate Law Region) CD = 18.5/Re06; for 2 <Re < 500
Turbulent Flow Regime (Newton's Law Region)
The buoyant force is proportional to the mass of fluid displaced by the particle, that is, as the particle falls through the surrounding water, it displaces a volume of fluid equivalent to its own weight:
where pp = density of the solid particle p = density of the fluid
We now have all the information necessary to develop some working expressions for particle settling. Look back at equation 3 (the resistance force exerted by the water), and the expressions for the drag coefficient (sidebar discussion on page 261). The important factor for us to realize is that the settling velocity of a particle is that velocity when accelerating and resisting forces are equal:
THE THREE REGIMES OF SETTLING
The Laminar Regime (Best known as Stokes Law; for Re < 2)
The Intermediate Regime (Best known as the Transition Regime Law, for 2 <Re <500)
The Turbulent Regime (Best known as Newton's Law, for 500 < Re < 200,000)
From this point on, it's some very simple algebra. You can solve equation 5 for us, and then substitute an expression in for CD for each of the three flow regimes (laminar, turbulent and intermediate). You can work through the details, but the working expressions are summarized for you in the sidebar discussion on this page. Remember that to apply these equations you need to know the flow regime, and so you need to make some assumptions when applying any one of these expressions. We will get to this in a moment. One point we can make is that the expressions can be further developed to give us an idea of the maximum size particles that will settle out in the first two regimes of settling. If we take Stokes law for example, the maximum size particle whose velocity follows Stokes' law can be found by substituting /¿Re/c^p for the settling velocity into the first sidebar equation, and then setting Re = 2 (the limiting Reynolds number value for the flow regime). This then gives us the following useful expression:
The minimum size particles that do not follow Stokes' law occurs at Re ~ 10"4. The settling velocity in this lower bound regime is less than that computed by the Stokes' Law expression, and generally an empirical correction factor is applied to account for particle slippage. This correction factor, which is applied by dividing the value into the Stokes' law calculated us value is: K= 1 + A A./d, where X is defined as the mean free path of a fluid molecule, and constant A varies between 1.4 and 20 (as a point of reference, A = 1.5 for air). But this is a correction factor we will likely never have to consider in a conventional water treatment assignment. A more convenient set of expressions for settling velocity can be derived by expressing the three settling regime equations in terms of dimensionless groups. We won't get tangled in the derivations, although they are reasonably straightforward, but rather just list these expressions. The relationships are based on the dimensionless Archimedes number, defined as:
Note that the Archimedes number is a dimesnionless group that describes the physical properties of the heterogeneous system. It can be related to the Reynolds number (and hence the settling velocity, us) for each settling regime as follows: For the Stake's settling regime:
Note that the upper limiting or critical value of the Archimedes number for this range occurs at Re = 2, and hence Arcr [ = 18 X 2 = 36. This means that the laminar settling regime corresponds to Ar < 36.
For the intermediate settling regime, where 2 < Re < 500, we have the following expression:
For the critical value Re = 500, the limiting value of Ar for the intermediate settling regime is Arcr 2 = 83,000. In other words, the intermediate settling regime corresponds to 36 < Ar < 83,000.
For the Newton's law (turbulent settling regime) region, where Ar > 83,000, the expression of interest is:
The usefulness of these relationships lies in the recognition that by evaluating the Archimedes number, we can establish the theoretical settling range for the particles we are trying to separate out of a wastewater stream. This very often gives us a starting point for evaluating the settling characteristics of suspended solids for dilute systems. Note that from the definition of the Reynolds number, we can readily determine the settling velocity of the particles from the application of the above expressions (us = ¿tRe/dpP). The following is an interpolation formula that can be applied over all three settling regimes:
For low values of Ar, the second term in the denominator may be neglected, and equation 11 simplifies to equation 8; at high Ar values, we may neglect the first term in the denominator and the expression simplifies to equation 10, which corresponds to the Newton's law range.
The settling velocity of a nonspherical particle is less than that of a spherical one. A good approximation can be made by multiplying the settling velocity, us, of spherical particles by a correction factor, , called the sphericity factor. The sphericity, or shape factor is defined as the area of a sphere divided by the area of the nonspherical particle having the same volume:
The factor < 1 must be determined experimentally for particles of interest. Typical values are ij; = 0.77 for particles of rounded shape; ij; = 0.66 for particles of angular shape; = 0.43 for particles of a flaky geometry. The above analysis applies only to the free settling velocities of single particles and does not account for particle-particle interactions. Hence, the application of these formulas only applies to very dilute systems. At high particle concentrations, mutual interference in the motion of particles exists, and the rate of settling is considerably less than that computed by the given expressions. In the latter case, the particle is settling through a suspension of particles in a fluid, rather than through a simple fluid medium.
The above provides us with a theoretical staring basis for particle settling. Let's now take a closer look at some of the standard hardware.
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