 ### Nikhil Kumar

A stylish blog on my code

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# Derivation of Flux from Concentration Profile

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

$$J = -D(-c_o +c_L)\frac{e^{\frac{-z^2}{4Dt}}}{\sqrt{Dt\pi}}$$ $$J = (c_o - c_L)\sqrt{\frac{D}{\pi t}}e^{\frac{-z^2}{4Dt}}\tag{8}$$