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Péclet number (Pe) - Nishanth Saldanha

2,107 bytes added, 11:13, 15 February 2017
Derivation
== Derivation ==
 
For system as described in '''Figure 1''', with a specie concentration, <math>C</math> following relations can be made to relate flux in, <math>J_{in}</math> , flux out,<math>J_{out}</math> and control volume <math>V</math>
 
 
[[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 ]]
 
<math> V\frac{dC}{dt} = A\cdot J_{in} - A\cdot J_{out} </math>
 
<math> \frac{dC}{dt} = \frac{A}{V}\cdot (J_{in} - J_{out}) </math>
 
<math> J_{out} = \frac{\partial J}{\partial x}\cdot (\Delta x)+J_{in} </math>
 
<math>\frac{dC}{dt} = \frac{A}{V}\cdot \frac{\partial J}{\partial x}\cdot (\Delta x) </math>
 
<math>\frac{A}{V} = \frac{1}{\Delta x} </math>
<math>\frac{dC}{dt} = \frac{\partial J}{\partial x}\rightarrow \frac{\partial C}{\partial t} =\triangledown J </math>
 
<math>J = J_{convection}+J_{diffusion} </math>
 
<math> Convection </math>
<math> Q = C \Delta x\cdot A </math>
 
<math> J_{convection} = \frac{Q}{\Delta t\cdot A} = \frac{C \Delta x }{\Delta t} \rightarrow \frac{\partial x}{\partial t}C </math>
 
<math> Diffusion </math>
<math> J_{diffusion} = -D \frac{\partial C }{\partial x} </math>
 
<math>\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} </math>
 
<math> \frac{\partial x}{\partial t} = u </math>
 
<math>\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 </math>
 
<math> C^{*} = \frac {C}{C_{max}}; U^{*} = \frac{u}{U}; t^{*} = \frac {t}{t_{0}} </math>
 
<math> \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}} </math>
 
<math> \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}} </math>
 
<math> Pe = \frac {UL}{D} </math>
== References ==

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