Changes

Péclet number (Pe) - Nishanth Saldanha

, 07:23, 16 February 2017
Derivation
== Derivation ==
For a one dimensional 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 $V$ . From this system, the Péclet number can be derived.
[[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 ]]
$V\frac{dC}{dt} = A\cdot J_{in} - A\cdot J_{out}$

This mass balance can be used to describe the flux into and out of the system. Flux is mass flow rate per unit area. Assuming inlet and outlet areas are constant, the mass balance can be simplified.
$\frac{dC}{dt} = \frac{A}{V}\cdot (J_{in} - J_{out})$

If there are gradients in the system, the flux out of the system can be described as follows.
$J_{out} = \frac{\partial J}{\partial x}\cdot (\Delta x)+J_{in}$

Thus, the mass balance can be simplified as shown below. The balance can also be described in three dimensions.
$\frac{dC}{dt} = \frac{A}{V}\cdot \frac{\partial J}{\partial x}\cdot (\Delta x)$
Due to how the system is defined, $\frac{A}{V} = \frac{1}{\Delta x}$. Thus,
$\frac{dC}{dt} = \frac{\partial J}{\partial x}\rightarrow \frac{\partial C}{\partial t} =\triangledown J$
The assumption of negligible sources and sinks are made, so as to focus the system on diffusion and convection. Thus, $J = J_{convection}+J_{diffusion}$
Mass flow rate, $Convection Q$, can be defined as $Q = C \Delta x\cdot A$. Since the convection is the prime source of this mass flow rate in this system, convective flux, $J_{convection}$ is defined as such.
$J_{convection} = \frac{Q}{\Delta t\cdot A} = \frac{C \Delta x }{\Delta t} \rightarrow \frac{\partial x}{\partial t}C$
Diffusion equation can be derived from first Fick's law as shown below. $Diffusion D$is the Diffusivity constant.
$J_{diffusion} = -D \frac{\partial C }{\partial x}$