After learning how to incorporate LaTeX into html. I decided to run through a
sample Biomedical equation derivation on the transport of a solute through time
t and space z.
What we know
Concentration Profile:
$$\bar{c}_{z,t} = \frac{c_{z,t} - c_L}{c_o - c_L} = 1- erf(\frac{z}{\sqrt{4Dt}})\tag{1}$$
Fick's first law:
$$ J = -D \frac{dc}{dz} \tag{2}$$
Error Function (erf):
$$ \frac{2}{\sqrt{\pi}} \int^n_0 e^{-n^2} \tag{3}$$
Finding the Flux
We can start by simplifying the equation by setting:
$$ \frac{z}{\sqrt{4Dt}} = \eta \tag{4}$$
Then:
$$\frac{c_{z,t} - c_L}{c_o - c_L} = 1- erf(\frac{z}{\sqrt{4Dt}})$$
$$c_{z,t} - c_L = c_o - c_L -c_o erf(\eta) + c_L erf(\eta)$$
$$c_{z,t} = c_L + c_o - c_L -c_o erf(\eta) + c_L erf(\eta)$$
$$c_{z,t} = c_o + (-c_o +c_L) erf(\eta)\tag{5}$$
How to take the derivative of the erf function
$$\frac{d}{dn} \frac{2}{\sqrt{\pi}} \int^n_0 e^{-n^2}$$
$$ \frac{d}{dn}\frac{2}{\sqrt{\pi}}(erf(n) -erf(0)) $$
$$ \frac{d}{dn}\frac{2}{\sqrt{\pi}}(erf(n) -0) $$
$$ \frac{2e^{-n^2}}{\sqrt{\pi}}\tag{6}$$
Use equation (5) and (6) to take the derivative of the concentration
profile
$$\frac{d}{d{\eta}} = 0 + \frac{d}{d{\eta}}(-c_o +c_L) erf(\eta)$$
$$\frac{d}{d{\eta}} = d{\eta}(-c_o +c_L) \frac{2e^{-\eta^2}}{\sqrt{\pi}} $$
Using equation (4) to substitute for $\eta$
$$\frac{dc}{dz} = (-c_o +c_L)\frac{1}{2\sqrt{Dt}} \frac{2e^{\frac{-z^2}{4Dt}}}{\sqrt{\pi}}$$
$$\frac{dc}{dz} = (-c_o +c_L)\frac{e^{\frac{-z^2}{4Dt}}}{\sqrt{Dt\pi}}\tag{7} $$
Use equation (7) and (2) to find the flux
$$J = -D(-c_o +c_L)\frac{e^{\frac{-z^2}{4Dt}}}{\sqrt{Dt\pi}} $$
$$J = (c_o - c_L)\sqrt{\frac{D}{\pi t}}e^{\frac{-z^2}{4Dt}}\tag{8}$$
