Mtl

FIGURE 9.4 Control surfaces for molecular diffusion as envisioned in the elementary kinetic theory of gases.

Comparing (9.18) with the continuum expression J = —D(gN'/gx) results in D = 0.5c or, equivalently

Since the red molecules differ from the others only by a coat of paint, X and D apply to all molecules of the gas. Thus the diffusional mean free path X is defined as a function of the molecular diffusivity of the vapor and its mean speed by (9.19).

Expressions (9.13), (9.15), (9.16), and (9.19) have different numerical constants and their use leads to mean free paths A,ab that differ by as much as a factor of 2 for typical atmospheric gases. The consequences of using these different expressions are discussed in Chapter 12. In the remaining sections of this chapter we focus on the interactions of particles with a single gas, air, with a mean free path given by (9.6) and (9.7).

9.2 THE DRAG ON A SINGLE PARTICLE: STOKES' LAW

We start our discussion of the dynamical behavior of aerosol particles by considering the motion of a particle in a viscous fluid. As the particle is moving with a velocity urxj, there is a drag force exerted by the fluid on the particle equal to Fdng. This drag force will always be present as long as the particle is not moving in a vacuum.To calculate /*drag> one must solve the equations of fluid motion to determine the velocity and pressure fields around the particle.

The velocity and pressure in an incompressible Newtonian fluid are governed by the equation of continuity (a mass balance)

ox oy S z and the Navier-Stokes equations (a momentum balance) (Bird et al. 1960), the x component of which is

where u = (ux, uy, uz) is the velocity field, p(x, y, z) is the pressure field, p is the viscosity of the fluid, and gx is the component of the gravity force in the x direction. To simplify our discussion let us assume without loss of generality that gx = 0. The y and z components of the Navier-Stokes equations are similar to (9.21).

Let us nondimensionalize the Navier-Stokes equations by introducing a characteristic velocity uq and characteristic length L and defining the dimensionless variables

and the dimensionless time and pressure:

Then (9.20) and (9.21) can be rewritten using the definitions presented above dux Wy 9 U

where Re = woLp/p is the Reynolds number, representing the ratio of inertial to viscous forces in the flow. Note that all the parameters of the problem have been neatly combined into one dimensionless number, Re. The above nondimensionalization provides us with considerable insight, namely, that the nature of the flow will depend exclusively on the Reynolds number.

For flow around a particle submerged in a fluid, the characteristic lengthscale L is the diameter of the particle Dp, and mq can be chosen as the speed of the undisturbed fluid upstream of the body, Uoo. Therefore

Re puxDp

One could also use the radius Rp of the particle as L and then define Re as pu-^Rp / Clearly, these differ only by a factor of 2. We will use the Reynolds number Re defined on the basis of the particle diameter in our subsequent discussion.

When viscous forces dominate inertial forces, Re <C 1, and the type of flow that results is called a low-Reynolds-number flow or creeping flow. In this case the Navier-Stokes equations can be simplified as one can neglect the left-hand-side (LHS) terms of (9.24) (note that 1 /Re then is a large number) to obtain at steady state:

The solution of (9.23) and (9.25) to obtain the velocity and pressure distribution around a sphere was first obtained by Stokes. The assumptions invoked to obtain the solution are (1) an infinite medium, (2) a rigid sphere, and (3) no slip at the surface of the sphere. For the solution details, we refer the reader to Bird et al. (1960, p. 132).

Using the spherical coordinate system defined in Figure 9.5, the pressure field around the particle is given by (Bird et al. 1960)

3^cos0 2 rL

where Rp is the particle radius, po is the pressure in the plane z == 0 far from the sphere, «oo is the approach velocity far from the sphere, and gravity has been neglected.

Our objective is to calculate the net force exerted by the fluid on the sphere in the direction of the flow. This force consists of two contributions. At each point on the surface of the sphere there is a force acting perpendicularly to the surface. This is the normal force.

FIGURE 9.5 Coordinate system used in describing the flow of a fluid about a rigid sphere.

At each point there is also a tangential force exerted by the fluid due to the shear stress caused by the velocity gradients in the vicinity of the surface.

To obtain the normal force on the sphere, one integrates the component of the pressure acting perpendicularly to the surface. Then the normal force Fn is found to be

The calculation of the tangential force requires the calculation of the shear stress xrg and then its integration over the particle surface to find the tangential force F,

Both forces act in the z direction (Figure 9.5) and the total drag exerted by the fluid on the sphere is

which is known as Stokes' law. Note that the case of a stationary sphere in a fluid moving with velocity u00 is entirely equivalent to that of a sphere moving with a velocity u-x through a stagnant fluid. In both cases the force exerted by the fluid on the particle is given by (9.29).

9.2.1 Corrections to Stokes' Law: The Drag Coefficient

Stokes' law has been derived for Re <C 1, neglecting the inertial terms in the equation of motion. If Re= 1, the drag predicted by Stokes' law is 13% low, due to the errors introduced by the assumption that inertial terms are negligible. To account for these terms, the drag force is usually expressed in terms of an empirical drag coefficient CD as

where Ap is the projected area of the body normal to the flow. Thus for a spherical particle of diameter Dp

where the following correlations are available for the drag coefficient as a function of the Reynolds number:

Note for CD = 24/Re, the drag force calculated by (9.31) is Fdrag = 3n \iDpu^, equivalent to Stokes' law.

To gain a feeling for the order of magnitude of Re for typical aerosol particles, the Reynolds numbers of spherical particles falling at their terminal velocities in air at 298 K and 1 atm are shown in Table 9.2. Thus for particles smaller than 20 pm (virtually all atmospheric aerosols) Stokes' law is an accurate formula for the drag exerted by the air. For larger particles (rain and large cloud droplets) or for particles in rapid motion one needs to use the drag coefficient correlations presented above.

9.2.2 Stokes' Law and Noncontinuum Effects: Slip Correction Factor

Stokes' law is based on the solution of equations of continuum fluid mechanics and therefore is applicable to the limit Kn —> 0. The nonslip condition used as a boundary condition is not applicable for high Kn values. When the particle diameter Dp approaches the same magnitude as the mean free path X of the suspending fluid (e.g., air), the drag force exerted by the fluid is smaller than that predicted by Stokes' law. To account for

TABLE 9.2 Reynolds Number for Particles in Air Falling at Their Terminal Velocities at 298 K

Diameter, (im

Re

0.1

7 x 10~9

1

2.8 x 10~6

10

2.5 x 10"3

20

0.02

60

0.4

100

2

300

20

TABLE 93 Slip Correction Factor Cc for Spherical Particles in Air at 298 K and 1 atm

Dp, um

Cc

0.001

216

0.002

108

0.005

43.6

0.01

22.2

0.02

11.4

0.05

4.95

0.1

2.85

0.2

1.865

0.5

1.326

1.0

1.164

2.0

1.082

5.0

1.032

10.0

1.016

20.0

1.008

50.0

1.003

100.0

1.0016

noncontinuum effects that become important as Dp becomes smaller and smaller, the slip correction factor Cc is introduced into Stokes' law, written now in terms of particle diameter Dp as

''drag —

(9.33)

where

C^l+^i^ + O.iexpp'^)]

(9.34)

A number of investigators over the years have determined the values for the numerical coefficients used in the expression above. Allen and Raabe (1982) have reanalyzed Millikan's data (based on experiments performed between 1909 and 1923) to produce the updated set of parameters shown above.

Values of Cc as a function of the particle diameter Dp in air at 25°C are given in Table 9.3. The slip correction factor is generally neglected for particles exceeding 10 pm in diameter, as the correction is less than 2%. On the other hand, the drag force for a 0.1 pm in diameter particle is reduced by almost a factor of 3 as a result of this slip correction.

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