Survey

Survey

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Particle Aerodynamics S+P Chap 9. Need to consider two types of motion • Brownian diffusion – thermal motion of particle, similar to gas motions. – Direction is random, leads to diffusion – Directly involves the molecular nature of the gas • Forces on the particle – Body forces: Gravity, electrostatic • Direction follows force field – Surface forces: Pressure, friction • Direction opposes motion of particle with respect to the fluid – Treats the gas as a continuous fluid Relevant Scales • • Diameter of particle vs. mean free path in the gas – Knudsen # Inertial “forces” vs. viscous forces – Reynolds # Brownian Motion Review • We have looked at Brownian motion from both Einstein’s and Langevin’s points of view • Einstein came up with N D 2 N t Where D = BkT And B is particle mobility Multiply times x2 and do an average <>, and you get and then convert to 3-D space, noting x2 = x2 + y2 + z2 • Langevin looked at the problem stochastically and came up with N x Dx 2 2 N t 2 x2 t x2 t x2 t 2D 6D 6kTB Using Brownian Diffusion Problem: – Consider the DMA. If it takes 20 seconds for air to flow from the top of the column to the bottom, what is the rms lateral distance an uncharged particle of mobility B will travel during transit. Hint This is diffusion in 1 direction Gaussian Plumes • Random walk binomial distribution • Binomial distribution at large N Gaussian • We have seen for diffusion that <x> = 0 and <x2> = 2Dt in 1 dimension. • Thus our Gaussian plume looks like 2 N0 1 x x0 N ( x, t ) exp 4Dt 2 2 Dt • You can show that this satisfies the continuous diffusion equation N 2 D N t Drag and Mobility • 3 “drag force” results so far – vt = BFG – FD = -v / B – <v> = <v0>exp(-t/mB) – If no external force • t = mB “relaxation time” for a particle • 1st two are equilibrium results • Third is the approach towards equilibrium for an arbitrary initial velocity. • All three presume that the air is not moving, or rather that the particle velocity is expressed in the reference frame of the air. • The next step is to link the particle mobility to the properties of the particle and the air. Stokes Drag • This is the simplest case for drag. • Think of the air flowing around the particle at speed u • There are two drag stresses that must be integrated over surface: – Pressure drag – Friction drag stress GF = n(du/dr) • In the “Stokes” regime, these drag effects are equal in magnitude, producing a Stokes Drag Force – FS = 6nRu Mobility Vs. Diameter • Using Stokes drag with our definition of mobility, we have – FS = 6nRu – FD = -v / B = u/B – BS = 1/(6nR) • As Particle radius goes up, mobility goes down. • This is how the DMA works – it finds the mobility of a particle based on its drift velocity, and then infers the diameter from the assumed drag. • Stokes drag doesn’t work at all sizes and velocities, however. Other factors are at play Relaxation Time and Stop Distance • <v> = <v0>exp(-t/mB) –No external force • t = mB “relaxation time” for a particle • We can integrate this to see how far a particle gets that starts at x = 0 with v = v0 v v0 exp( t / t ) v0t t x x0 vdt x 0 x0 tv0 (1 exp( t / t ) “Stop Distance” ~ R2 t for stokes flow – increases w/ size Flow Around A Corner • Key to this problem is variable gas velocity and variable particle velocity • Drag force on particle takes more general form: – ] FD = -(v-u)/ B • Solve for particle trajectory in x and y directions. Knudsen # 2 Kn Dp = mean free path of air molecule Dp = particle diameter Gas molecule selfcollision cross-section 1 2 B N B Gas # concentration 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 Free Molecular Regime Transition Regime Continuum Regime DP Slip Correction for low Kn Relaxation Time and Start Distance • Now consider a particle starting at rest, but suddenly feeling an external force Fe • We can integrate this to see how far a particle gets that starts at x = 0 with v = v0 v v0 exp( t / t ) v0t t x x0 vdt x 0 x0 tv0 (1 exp( t / t ) “Stop Distance” ~ R2 t for stokes flow – increases w/ size