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The way the particles diffuse depends on the dimensions of the system. In a one-dimensional system there is only one direction and with it less dilution which facilitates diffusion. In a two-dimensional system and more in three-dimensional system there is the possibility of lateral displacements and therefore slower diffusion.
ID:(1022, 0)
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$c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$
c(r)=(c_1ln(r_2/r)+c_2ln(r/r_1))/ln(r_2/r_1)
$c(r)=\displaystyle\frac{r_1c_1(r_2-r)+r_2c_2(r-r_1)}{r(r_2-r_1)}$
c(r)=(r_1c_1(r_2-r)+r_2c_2(r-r_1))/(r(r_2-r_1))
$c(x)=c_1+\displaystyle\frac{c_2-c_1}{x_2-x_1}(x-x_1)$
c(x)=c_1+(c_2-c_1)(x-x_1)/(x_2-x_1)
$c(x,t)=\displaystyle\frac{1}{2}c_0\left(\textrm{erfc}\displaystyle\frac{h-x}{2\sqrt{Dt}}+\textrm{erfc}\displaystyle\frac{h+x}{2\sqrt{Dt}}\right)$
c(x,t)=c_0/2 (erfc(h-x/2 sqrt(Dt))+erfc(h+x/2 sqrt(Dt)))
$c(x,t)=\displaystyle\frac{1}{2}c_0\textrm{erfc}\displaystyle\frac{x}{2\sqrt{Dt}}$
c(x,t)=c_0/2 erfc(x/2 sqrt(Dt))
$c(x,t)=\displaystyle\frac{M}{\sqrt{\pi Dt}}e^{-x^2/4Dt}$
c(x,t)=M/sqrt(pi Dt) e^(-x^2/4Dt)
$\displaystyle\frac{\partial c}{\partial t}=\displaystyle\frac{D}{r}\displaystyle\frac{\partial}{\partial r}\left( r\displaystyle\frac{\partial c}{\partial r} \right)$
dc/dt=(D/r)d(rdc/dr)/dr
$\displaystyle\frac{\partial c}{\partial t}=D\left(\displaystyle\frac{\partial^2c}{\partial r^2}+\displaystyle\frac{2}{r}\displaystyle\frac{\partial c}{\partial r} \right)$
dc/dt=D(d^2c/dr^2+(2/r)dc/dr)
$\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$
dc/dt=Dd^2c/dx^2
$j=-D\displaystyle\frac{c_2-c_1}{x_2-x_1}$
j=-D(c_2-c_1)/(x_2-x_1)
$J=\displaystyle\frac{2\pi D(c_2-c_1)}{\ln(r_2/r_1)}$
J=2pi D(c_2-c_1)/ln(r_2/r_1)
$J=4\pi D\displaystyle\frac{r_1r_2}{r_2-r_1}(c_2-c_1)$
J=4pi Dr_1r_2(c_2-c_1)/(r_2-r_1)
ID:(15360, 0)
Diffusion Equation, 1D
Equation
The temporal and spatial evolution of concentration $c$ in one dimension is governed by the equation:
| $\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$ |
where $D$ is the diffusion constant.
ID:(8381, 0)
Solution, 1D, stationary
Equation
The solution to the equation
| $\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$ |
for the steady-state case with concentration $c_1$ at position $x_1$ and concentration $c_2$ at position $x_2$ yields the following distribution:
| $c(x)=c_1+\displaystyle\frac{c_2-c_1}{x_2-x_1}(x-x_1)$ |
ID:(8388, 0)
Solution, 1D, stationary, flow
Equation
With the solution
| $c(x)=c_1+\displaystyle\frac{c_2-c_1}{x_2-x_1}(x-x_1)$ |
and the Fick's law equation
| $ j =- D \displaystyle\frac{ dc_n }{ dz }$ |
the flux is calculated as follows:
| $j=-D\displaystyle\frac{c_2-c_1}{x_2-x_1}$ |
ID:(8389, 0)
Solution, 1D, point
Equation
The solution to the equation
| $\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$ |
for the case of a point concentration $c$ (Dirac delta) with a total volume of $M$ is as follows:
| $c(x,t)=\displaystyle\frac{M}{\sqrt{\pi Dt}}e^{-x^2/4Dt}$ |
ID:(8383, 0)
Solution, 1D, non-point zone
Equation
The solution to the equation
| $\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$ |
for the case of a concentration $c$ in a semi-infinite system with a fixed concentration at the origin $c_0$ is as follows:
| $c(x,t)=\displaystyle\frac{1}{2}c_0\left(\textrm{erfc}\displaystyle\frac{h-x}{2\sqrt{Dt}}+\textrm{erfc}\displaystyle\frac{h+x}{2\sqrt{Dt}}\right)$ |
ID:(8385, 0)
Solution, 1D, semi-infinite
Equation
The solution to the equation
| $\displaystyle\frac{\partial c}{\partial t}=D\displaystyle\frac{\partial^2 c}{\partial x^2}$ |
for the case of a concentration $c$ in a semi-infinite system with a fixed concentration at the origin $c_0$ is as follows:
| $c(x,t)=\displaystyle\frac{1}{2}c_0\textrm{erfc}\displaystyle\frac{x}{2\sqrt{Dt}}$ |
ID:(8384, 0)
Diffusion Equation, 2D
Equation
The temporal and spatial evolution of concentration $c$ in two dimensions with rotational symmetry is governed by the equation:
| $\displaystyle\frac{\partial c}{\partial t}=\displaystyle\frac{D}{r}\displaystyle\frac{\partial}{\partial r}\left( r\displaystyle\frac{\partial c}{\partial r} \right)$ |
where $D$ is the diffusion constant.
ID:(8382, 0)
Solution, 2D, stationary
Equation
The solution to the equation
| $\displaystyle\frac{\partial c}{\partial t}=\displaystyle\frac{D}{r}\displaystyle\frac{\partial}{\partial r}\left( r\displaystyle\frac{\partial c}{\partial r} \right)$ |
for the steady-state case with concentration $c_1$ at radius $r_1$ and concentration $c_2$ at radius $r_2$ yields a flux as follows:
| $c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$ |
ID:(8386, 0)
Solution, 2D, stationary, flow
Equation
For the solution
| $c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$ |
the flux is calculated as follows:
| $J=\displaystyle\frac{2\pi D(c_2-c_1)}{\ln(r_2/r_1)}$ |
ID:(8387, 0)
Diffusion Equation, 3D
Equation
The temporal and spatial evolution of concentration $c$ in two dimensions with rotational symmetry is governed by the equation:
| $\displaystyle\frac{\partial c}{\partial t}=D\left(\displaystyle\frac{\partial^2c}{\partial r^2}+\displaystyle\frac{2}{r}\displaystyle\frac{\partial c}{\partial r} \right)$ |
where $D$ is the diffusion constant.
ID:(8390, 0)
Solution, 3D, stationary
Equation
The solution to the equation
| $\displaystyle\frac{\partial c}{\partial t}=D\left(\displaystyle\frac{\partial^2c}{\partial r^2}+\displaystyle\frac{2}{r}\displaystyle\frac{\partial c}{\partial r} \right)$ |
for the steady-state case with concentration $c_1$ at radius $r_1$ and concentration $c_2$ at radius $r_2$ yields a flux as follows:
| $c(r)=\displaystyle\frac{r_1c_1(r_2-r)+r_2c_2(r-r_1)}{r(r_2-r_1)}$ |
ID:(8391, 0)
Solution, 3D, stationary, flow
Equation
For the solution
| $c(r)=\displaystyle\frac{c_1\ln(r_2/r)+c_2\ln(r/r_1)}{\ln(r_2/r_1)}$ |
the flux is calculated as follows:
| $J=4\pi D\displaystyle\frac{r_1r_2}{r_2-r_1}(c_2-c_1)$ |
ID:(8392, 0)