Statistics of Size distributions The “moments” will come in when you do area, volume distributions We also define “effective areal” diameter and “effective volume” diameter Mean Diameter Standard Deviation Geometric Mean n th moment Histogram Discrete distribution Continuous dist.

Effective Diameters Consider N t aerosol particles, each with Diameter D. This is a “monodisperse” distribution. The Total surface area and volume will be: Now consider the total surface area and volume of a polydisperse aerosol population Substituting our definition for moments, we have

Effective Diameters Substituting our definition for moments, we have We now define an effective areal diamater, D a, and an effective volumetric diameter, D v, which are the diameters that would produce the same surface area and volume if the distribution were monodisperse. So… Board Illustration: Consider a population of aerosols where 900 cm -3 are 0.1  m, and 100 cm -3 are 1.0  m. Compute Da, Dv.

Converting size distributions Concentration: Rule of thumb: Always use concentration, not number distribution, when converting from one type of size distribution to another NOT

Converting size distributions Example: Let n(D) = C = constant. What are the log-diameter and ln-diameter distributions? Problem: Let n(D) = C = constant. What is the volumetric number distribution, dN/dv? Even though n(D) is constant w/ diameter, the log distributions are functions of diameter.

The Power-Law (Junge) Size Distribution n(D)=1000 cm -3  m 2 D -  Linear-linear plot Log-Log plot slope = -3 What is the total concenration, N t ? What is volume distribution, dV/dD? What is total volume? What is log-number distribution? Major Points for Junge Distr. Need lower+upper bound Diameters to constrain integral properties Only accurate > 300 nm or so. Linear in log-log space…. n(D)=C D - 

The log-normal Size Distribution Note that power-law is simply linear in log-log space, and was unbounded Let’s make a distribution that is quadratic in log-log space (curvature down)

The Log-Normal Size Distribution Linear-linear plot Log-Log plot What is the total concenration, N t ? What is volume distribution, dV/dD? What is total volume? What is log-number distribution? Major Points for Log-normal 3 parameters: N t, D g and  g No need for upper/lower bound constraints  goes to zero both ways Usually need multiple modes. N t =1000 cm -3, D g = 1  m,  g = 1.0

More Statistics

Medians, modes, moments, and means from lognormal distributions Median – Divides population in half. i.e. median of # distribution is where half the particles are larger than that diameter. Median of area distribution means half of the area is above that size Mode – peak in the distribution. Depends on which distribution you’re finding mode of (e.g. dN/dlogD or dN/dD). Set dn(D)/dD = 0 Secret to S+P 8.7… Get distributions in form where all dependence on D is in the form exp(-(lnD – lnD x ) 2 ). Complete the squares to find D x. Then, the median and mode will be at D x due to the symmetry of the distribution The means and the moments are properties of the integral of the size distribution. In the form above, these will appear outside the exp() term. (i.e. what is leftover after completing the squares).

Standard 3-mode distributions

Typical measured/parameterized urban size distributions

Southern AZ size distributions

Vertical distributions Often aerosol comes in layers Averaged over time, they form an exponentially decaying profile w/ scale height of ~1 to 2 km.

Particle Aerodynamics S+P Chap 9. Need to consider two perspectives Brownian diffusion – thermal motion of particle, similar to gas motions Forces on the particle –Body forces: Gravity, electrostatic –Surface forces: Pressure, friction Relevant Scales Diameter of particle vs. mean free path in the gas – Knudsen # Inertial “forces” vs. viscous forces – Reynolds #

Knudsen # = mean free path of air molecule D p = particle diameter Quantifies how much an aerosol particle influences its immediate environment Kn Small – Particle is big, and “drags” the air nearby along with it Kn Large – Particle is small, and air near particle has properties about the same as the gas far from the particle Kn Gas molecule self- collision cross-section Gas # concentration Free Molecular Regime Continuum Regime Transition Regime