# Changes

## Péclet number (Pe) - Nishanth Saldanha

, 07:33, 16 February 2017
Derivation
In diffusion dominated regimes, the Peclet number is less than zero. Such is the case with microfluidic systems, where turbulence is low. In convection dominated systems, this number is greater than one.
== 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.
$J_{out} = \frac{\partial J}{\partial x}\cdot (\Delta x)+J_{in}$
Thus, the mass balance can be simplified as shown below. The balance can also be described in three dimensions.
$\frac{dC}{dt} = \frac{A}{V}\cdot \frac{\partial J}{\partial x}\cdot (\Delta x)$
$\frac{dC}{dt} = \frac{\partial J}{\partial x}\rightarrow \frac{\partial C}{\partial t} =\triangledown J$
This equation is also described in three dimensions. 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, $Q$, 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.
$J_{diffusion} = -D \frac{\partial C }{\partial x}$

Diffusive flux and Convective flux can be combined into the overall mass balance.
$\frac{\partial C}{\partial t} = -\frac{\partial \left [ J_{convection}+J_{diffusion} \right ]}{\partial x} = -\frac{\partial \left [ \frac{\partial x}{\partial t}C + -D \frac{\partial C }{\partial x} \right ]}{\partial x}$

Because the relation below can be applied, where $u$ equals velocity of a particle, the mass balance can be simplified and described in multiple dimensions
$\frac{\partial x}{\partial t} = u$
$\frac{\partial C}{\partial t} = D\frac{\partial^{2} C}{\partial x^{2}} - u \frac{\partial C}{\partial x}\rightarrow \frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x}= D\frac{\partial^{2} C}{\partial x^{2}} \rightarrow \frac{\partial C}{\partial t} + u \triangledown C= D \triangledown^{2} C$

Dimensionless numbers, as shown below can be used to restate the mass balance. $U$ equals the convective linear flow rate.
$C^{*} = \frac {C}{C_{max}}; U^{*} = \frac{u}{U}; t^{*} = \frac {t}{t_{0}}$

When these numbers are applied, the balance is described as shown.
$\frac{C}{t_{0}}\frac{\partial C^{*}}{\partial t} + \frac{UC}{L}u^{*} \frac{\partial C}{\partial x}= \frac{DC}{L^{2}}\frac{\partial^{2} C}{\partial x^{2}}$
$\frac{L^{2}}{D\cdot t_{0}}\frac{\partial C^{*}}{\partial t} + \frac{UL}{D}u^{*} \frac{\partial C}{\partial x}= \frac{\partial^{2} C}{\partial x^{2}}$

The first term in the above balance is referred to as the unsteady term. In time invariant flows, this term equals zero. The ratio between the remaining two terms (i.e. the diffusive and convective terms), equals the Peclet number, as described below.
$Pe = \frac {UL}{D}$