# Changes

## Péclet number (Pe) - Nishanth Saldanha

, 11:13, 15 February 2017
Derivation
== Derivation ==

For system as described in '''Figure 1''', with a specie concentration, $C$ following relations can be made to relate flux in, $J_{in}$ , flux out,$J_{out}$ and control volume $V$

[[Image:For_Wiki_page.jpg|150px|right|thumbnail|'''Figure 1''' This is an illustration of a control volume with fluxes in and fluxes out. If sources and sinks can be neglected, these fluxes can be contributed to convection and diffusion ]]

$V\frac{dC}{dt} = A\cdot J_{in} - A\cdot J_{out}$

$\frac{dC}{dt} = \frac{A}{V}\cdot (J_{in} - J_{out})$

$J_{out} = \frac{\partial J}{\partial x}\cdot (\Delta x)+J_{in}$

$\frac{dC}{dt} = \frac{A}{V}\cdot \frac{\partial J}{\partial x}\cdot (\Delta x)$

$\frac{A}{V} = \frac{1}{\Delta x}$
$\frac{dC}{dt} = \frac{\partial J}{\partial x}\rightarrow \frac{\partial C}{\partial t} =\triangledown J$

$J = J_{convection}+J_{diffusion}$

$Convection$
$Q = C \Delta x\cdot A$

$J_{convection} = \frac{Q}{\Delta t\cdot A} = \frac{C \Delta x }{\Delta t} \rightarrow \frac{\partial x}{\partial t}C$

$Diffusion$
$J_{diffusion} = -D \frac{\partial C }{\partial x}$

$\frac{\partial C}{\partial t} = -\frac{\partial \left [ J_{convection}+J_{diffusion} \right ]}{\partial x} = -\frac{\partial \left [ \frac{\partial x}{\partial t}C + -D \frac{\partial C }{\partial x} \right ]}{\partial x}$

$\frac{\partial x}{\partial t} = u$

$\frac{\partial C}{\partial t} = D\frac{\partial^{2} C}{\partial x^{2}} - u \frac{\partial C}{\partial x}\rightarrow \frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x}= D\frac{\partial^{2} C}{\partial x^{2}} \rightarrow \frac{\partial C}{\partial t} + u \triangledown C= D \triangledown^{2} C$

$C^{*} = \frac {C}{C_{max}}; U^{*} = \frac{u}{U}; t^{*} = \frac {t}{t_{0}}$

$\frac{C}{t_{0}}\frac{\partial C^{*}}{\partial t} + \frac{UC}{L}u^{*} \frac{\partial C}{\partial x}= \frac{DC}{L^{2}}\frac{\partial^{2} C}{\partial x^{2}}$

$\frac{L^{2}}{D\cdot t_{0}}\frac{\partial C^{*}}{\partial t} + \frac{UL}{D}u^{*} \frac{\partial C}{\partial x}= \frac{\partial^{2} C}{\partial x^{2}}$

$Pe = \frac {UL}{D}$
== References ==