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$4\mathrm{.}1$ Coagulation

In this problem, we will explore the role that diffusive Brownian coagulation plays in removing

smaller particles from the particle size distribution. To do so$,$ we will use the Fuchs form of the

Brownian coagulation coefficient, as presented in Seinfeld and Pandis Table $13\mathrm{.}1$:

TABLE $13\mathrm{.}1$ Fuchs Form of the Brownian Coagutation Coefticient $K_{12}$

$K_{12}=2x(D_1+D_2)(D_{p1}+D_{p2})(\frac{D_{p1}+D_{p2}}{D_{p2}+D_{p2}+2(s_1^2+g_2^2{)}^{\frac{1}{2}}}+\frac{8(D_1+D_2)}{(vec(c{)}_1^2+\text{v}\text{e}\text{c}(c{)}_2^2{)}^{\frac{1}{2}}(D_{p1}+D_{p2})}{)}^{-1}$

where

$\{\begin{array}{c}\text{:}{D}_i=\frac{T{C}_e}{3D_{}}\end{array}$

The mean frese parh $t_{,}$ is defined shightly differenriy than in $(9.$hat$(K)).$

$($a$)$ To confirm that you can correctly implement the coagulation equations, compute $K_{12}$

$($units: $c{m}^3{s}^{-1})$ for $($i$)$ a $10nm$ particle to coagulate onto a $100nm$ particle, and $($ii$)$ for a

$100nm$ particle to coagulate onto another $100nm$ particle. Check your results against

Seinfeld and Pandis Table $13\mathrm{.}3,$ where they report these coagulation coefficients to be

respectively $2\mathrm{.}5{10}^{-8}$ and $15{10}^{-10}c{m}^3{s}^{-1}.$ Assume conditions of $25C$ in air.

$($b$)$ Consider a monodisperse aerosol with $100nm$ particles. For an initial number

concentration $N_0$ of $($i$){10}^4$ and $($ii$){10}^6$ particles $c{m}^{-3},$ compute the characteristic

coagulation timescale ${}_t$ for the total number concentration to be halved, where:

${}_c=\frac{2}{K{N}_0}$

Now, let's consider the archetypal polluted "urban" particle size distribution from Seinfeld and

Pandis which we have been using in previous assignments. As we discussed in lecture,

coagulation is most rapid for particles of highly dissimilar sizes. Let's compute the

characteristic timescale for a particle of diameter $i$ to be lost to coagulation onto any larger

particle of diameter $j>i.$ We can express this timescale as: