Diffusion Flashcards - Math Modeling

Identify what is known and the assumptions.

This is a diffusion problem and thus a diffusion equation will be used with a term of relative concentration.

The relative concentration is denoted as

$\C(x,t) \x=\text{location of oil} \t=\text{time}$

This function has been normalized resulting in the following.

$\int C(x,t)dx=1$

The law of conservation of mass will also be useful is solving this problem.

$\frac{\partial C}{\partial t}=-\frac{\partial q}{\partial x}$

$\t=\text{Time since release of oil} \\mu=\text{Distance travelled by the patches center} \x=\text{Distance between patches center and town} \s=\text{Patches spread at time t} \\ \mu=2t \x=12-\mu \P=15, t=1 \s=1600, t=1$

The goal is to calculate the maximum pollution level in town.

The diffusion equation is,

$\frac{\partial C}{\partial t}=\frac{D}{2}\frac{\partial ^2C}{\partial x^2}$

Using the Fourier transforms to solve the diffusion equation is as follows.

$\hat C(k,t)=\int_{-\infty}^{\infty} e^{-ikx}C(x,t)dx$

$\frac{d \hat C}{dt}=\frac{D}{2}(ik)^2 \hat C=-\frac{D}{2} k^2\hat C$

$C(x,t)=\frac{1}{\sqrt{2\pi Dt}} e^{-\frac{x^2}{2Dt}}$

For this particular function

$\mu =2, \sigma=0.500$

so the interval will be

$\mu + 2\sigma\rightarrow s=4\sigma$

$D=\sigma^2=0.25$ $C(x,t)=\frac{1}{\sqrt{0.5 \pi t}}e^{-\frac{x^2}{0.5t}}$

$P=15=P_0\frac{1}{\sqrt{0.5\pi }}e^{-\frac{0}{0.5}}$

Calculate given

$\P=P_0C \ P_0=15\sqrt{0.5\pi} \\P=\frac{15}{\sqrt{t}}e^{-\frac{(12-2t)^2}{(0.5t)}}$

Now find where the maximum occurs in time.

Diffusion - Math Modeling