In this chapter, we focus on the processes involving a single aerosol particle in a suspending fluid and the interaction of the particle with the suspending fluid itself. We begin by considering how to characterize the size of the particle in an appropriate way in order to describe transport processes involving momentum, mass, and energy. We then treat the drag force exerted by the fluid on the particle, the motion of a particle through a fluid due to an imposed external force and resulting from the bombardment of the particle by the molecules of the fluid. Because of its importance in atmospheric aerosol processes and aqueousphase chemistry, mass transfer to single particles will be treated separately in Chapter 12.

9.1 CONTINUUM AND NONCONTINUUM DYNAMICS: THE MEAN FREE PATH

As we begin our study of the dynamics of aerosols in a fluid (e.g., air), we would like to determine, from the perspective of transport processes, how the fluid "views" the particle or equivalently how the particle "views" the fluid that surrounds it. On the microscopic scale fluid molecules move in a straight line until they collide with another molecule. After collision, the molecule changes direction, moves for a while until it collides with another molecule, and so on. The average distance traveled by a molecule between collisions with other molecules is defined as its mean free path. Depending on the relative size of a particle suspended in a gas and the mean free path of the gas molecules around it, we can distinguish two cases. If the particle size is much larger than the mean free path of the surrounding gas molecules, the gas behaves, as far as the particle is concerned, as a continuous fluid. The particle is so large and the characteristic lengthscale of the motion of the gas molecules so small that an observer of the system sees a particle in a continuous fluid. At the other extreme, if the particle is much smaller than the mean free path of the surrounding fluid, an outside observer of the system (Figure 9.1) sees a small particle and gas molecule moving discretely around it. The particle is small enough that it resembles another gas molecule.

As usual in transport phenomena, one seeks an appropriate dimensionless group that reflects the relative lengthscales outlined above. The key dimensionless group that defines the nature of the suspending fluid relative to the particle is the Knudsen number (Kn)

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Second Edition, by John H. Seinfeld and Spyros N. Pandis. Copyright © 2006 John Wiley & Sons, Inc.

Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Second Edition, by John H. Seinfeld and Spyros N. Pandis. Copyright © 2006 John Wiley & Sons, Inc.

FIGURE 9.1 Schematic of the three regimes of suspending fluid-particle interactions: (a) continuum regime (Kn —> 0), (b) free molecule (kinetic) regime (Rn —> oo), and (c) transition regime (Kn ~ 1).

FIGURE 9.1 Schematic of the three regimes of suspending fluid-particle interactions: (a) continuum regime (Kn —> 0), (b) free molecule (kinetic) regime (Rn —> oo), and (c) transition regime (Kn ~ 1).

where X is the mean free path of the fluid, Dp the particle diameter, and Rp its radius. Thus the Knudsen number is the ratio of two lengthscales, a length characterizing the "graininess" of the fluid with respect to the transport of momentum, mass, or heat, and a length scale characterizing the particle size, its radius.

Before we discuss the role of the Knudsen number, we need to consider the calculation of the mean free path for a vapor. It will soon be necessary to calculate the mean free path both for a pure gas and for gases composed of mixtures of several components. Note that even though air consists of molecules of N2 and 02, it is customary to talk about the mean free path of air, ^air* ¿is if air were a single chemical species.

Mean Free Path of a Pure Gas Let us start with the simplest case, a particle suspended in a pure gas B. If we are interested in characterizing the nature of the suspending gas relative to the particle, the mean free path that appears in the definition of the Knudsen number is Xbb- The subscript denotes that we are interested in collisions of molecules of B with other molecules of B. Ordinarily, air will be the predominant vapor species in such a situation. The mean free path /-bb has been defined as the average distance traveled by a B molecule between collisions with other B molecules. The mean speed of gas molecules of B, cB is (Moore 1962, p. 238)

where MB is the molecular weight of B. Note that larger molecules move more slowly, while the overall mean speed of a gas increases with temperature. The mean speed of N2 at 298 K is, according to (9.2), cN2 = 474 m s 1 and for oxygen co2 = 444 m s_1. Molecular velocities of other atmospheric gases at 298 K are shown in Table 9.1.

Let us estimate what happens to a B molecule during a unit of time, say, a second. During this second the molecule travels on average (cB x 1 s) m. If during the same

Gas |
Molecular Weight |
Mean Velocity, m s 1 |

NH3 |
17 |
609 |

Air |
28.8 |
468 |

HCl |
36.5 |
416 |

hno3 |
63 |
316 |

h2so4 |
98 |
254 |

(CH2)3(COOH)2 |
132 |
219 |

second it undergoes Zbb collisions, then its mean free path will be by definition

Thus to calculate Xbb we need to first calculate the collision rate of B molecules, ZBb- Let ctb be the diameter of a B molecule. In 1 s a molecule travels a distance cb and collides with all molecules whose centers are in the cylinder of radius Ctb and height cb - Note that two molecules of diameter ctb collide when the distance between their centers is ctb- If ZVB is the number of B molecules per unit volume, then the number of molecules in the cylinder is tictIcbNb- Above we have calculated the number of collisions assuming that one molecule of B is moving while the rest are immobile and in the process we have underestimated the frequency of collisions. In general, all particles are moving in random directions and we need to account for this motion by estimating their relative speed. If two particles move in opposite directions, their relative speed is 2 cb (Figure 9.2). If they move in the same direction, their relative speed is zero, while for a 90° angle their relative

(a) Relative (b) Relative (c) Relative speed 2 c speed 0 speed J2 c

FIGURE 9.2 Relative speeds (RSs) of molecules for (a) head-on collision (RS = 2c), (b) grazing collision (RS = 0), and (c) right-angle collision (RS = y/2 c). For molecules moving in the same direction with the same velocity, the relative velocity of approach is zero. If they approach head-on, the relative velocity of approach is 2c. If they approach at 90°, the relative velocity of approach is the sum of the velocity components along the line.

velocity of approach is \/2 Cb (Figure 9.2). One can prove that the Salter situation represents the average, so we can write

and the mean free path Xbb is given by

Note that the larger the molecule size, aB, and the higher the gas concentration, the smaller the mean free path.

Unfortunately, even though (9.5) provides valuable insights into the dependence of A,bb on the gas concentration and molecular size, it is not convenient for the estimation of the mean free path of a pure gas, because one needs to know the diameter of the molecule Ob, a rather ill-defined quantity as most molecules are not spherical. To make things even worse, the mean free path of a gas cannot be measured directly. However, the mean free path can be theoretically related to measurable gas microscopic properties, such as viscosity, thermal conductivity, or molecular diffusivity. One therefore can use measurements of the above gas properties to estimate theoretically the gas mean free path. For example, the mean free path of a pure gas can be related to the gas viscosity using the kinetic theory of gases where the §as viscosity (in kg m 's '), p is the gas pressure (in Pa), and Mb is the molecular weight of B.

Calculation of the Air Mean Free Path The air viscosity at T = 298 K and p — 1 atm is uajt — 1.8 x 10~5kgm_ls The air mean free path at T = 298 K and p = 1 atm is then found using (9.6) to be

Thus for standard atmospheric conditions, if the particle diameter exceeds 0.2 pm or so, Kn < 1, and with respect to atmospheric properties, the particle is in the continuum regime. In that case, the equations of continuum mechanics are applicable. When the particle diameter is smaller than 0.01 pm, the particle exists in more or less a rarified medium and its transport properties must be obtained from the kinetic theory of gases. This Kn 3> 1 limit is called the free molecule or kinetic regime. The particle size range intermediate between these two extremes (0.01-0.2 pm) is called the transition regime, and there the particle transport properties result from combination of the two other regimes.

The mean free path of air varies with height above the Earth's surface as a result of pressure and temperature changes (Chapter 1). This change for standard atmospheric

FIGURE 9.3 Mean free path of air as a function of altitude for the standard U.S. atmosphere (Hinds 1999).

Mean Free Path, |j.m

FIGURE 9.3 Mean free path of air as a function of altitude for the standard U.S. atmosphere (Hinds 1999).

conditions (see Table A.7) is shown in Figure 9.3. The net result is an increase of the air mean free path with altitude, to roughly 0.2 pm at 10 km.

Mean Free Path of a Gas in a Binary Mixture If we are interested in the diffusion of a vapor molecule A toward a particle, both of which are contained in a background gas B (e.g., air), then the description of the diffusion process depends on the value of the Knudsen number defined on the basis of the mean free path aAb • The mean free path AAb is defined as the average distance traveled by a molecule of A before it encounters another molecule of A or B. Note that because ordinarily the concentration of A molecules is several orders of magnitude lower than that of the background gas B (air), collisions between A molecules can be neglected, and the collisions between A and B are practically equal to the total number of collisions for an A molecule. The Knudsen number in the case of interest is given by and we need to estimate X,AB. Jeans showed that the effective mean free path of molecules of A, A,ab, in a binary mixture of A and B is (Davis 1983)

where Na and NK are the molecular number concentrations of A and B, a a and ctAr are the collision diameters for binary collisions between molecules of A and molecules of A and B, respectively, where

and z = mA/mii = Ma/Mb is the ratio of molecular masses (or molecular weights) of A and B. The first term in the denominator of (9.9) accounts for the collisions between A molecules, while the second for the collisions between A and B molecules. If the concentration of species A is very low (a good assumption for almost all atmospheric situations), NA <C /VB and (9.9) can be simplified by neglecting the collisions between A molecules as

Note that the molecular concentration A^ can be calculated from the ideal-gas law Nb = p/kT, where p is the pressure of the system. The mean free path of the trace gas A in the background gas does not depend on the concentration of A itself. This is not a surprise, as we have assumed that the concentration of A is so low that A molecules never get to interact with each other. However, the mean free path of A depends on the sizes of the A and B molecules, and on the temperature and pressure of the mixture.

The mean free path once more is usually calculated not by (9.11) because of the difficulty of directly measuring c?ab, but from the binary diffusivity of A in B, Dab-This diffusivity can be either measured directly or calculated theoretically from the Chapman-Enskog theory for binary diffusivity (Chapman and Cowling 1970) by

where iij^ is the collision integral, which has been tabulated by Hirschfelder et al. (1954) as a function of the reduced temperature T* = kT/zab, where sab is the Lennard-Jones molecular interaction parameter. For hard spheres ' = 1, and for this case the following relationship connects the mean free path X.Ab, and the binary diffusivity DAB

Note the appearance of the molecular mass ratio z -= MA/MB in (9.13). Many investigators have assumed z<l either explicitly or implicitly and this has been the source of some confusion. We can identify certain limiting cases for (9.13):

3.397 Dab

2 CA

Additional relationships have been proposed to determine the mean free path in terms of Z)ab- From zero-order kinetic theory, Fuchs and Sutugin (1971) showed that while Loyalka et at (1989) used

Vit CA Ca

An additional relationship between the mean free path and the binary diffusivity can be derived using the kinetic theory of gases. The derivation relies on a simple argument involving the flux of gas molecules across planes separated by a distance X. Consider the simplest case, only a single gas, some of the molecules of which are painted red. Assume that the number N' of red molecules is greater in one direction along the x axis, and consequently, if the total pressure is uniform throughout the gas, the number N" of unpainted molecules must also vary along the Jt direction. Let us define the "mean free path" for diffusion as X, so that X is the distance both left and right of the plane at x where the molecules (both painted and unpainted) experienced their last collisions. We are purposely not defining X precisely at this point. Figure 9.4 depicts planes at a-* -f X and x' — X.

For molecules in three-dimensional random motion, the number of molecules striking a unit area per unit time is^Nc. If is the average distance from the control surface at which the molecules crossing the a:* surface originated, then the left-to-right flux of painted molecules is ^cN'(x* — X), while the right-to-left is |cN'(.1* 4- X).

The net left-to-right flux of painted molecules through the plane of x* is (in molecules cm"2 s-1)

Expanding both N'(x* — X) and N'{x* + X) in Taylor series about x', we obtain

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