I am not really sure how to start or what I am doing here. I am just trying to talk this out with myself and might need some guidance. My end goal is to derive a particular model used to measure laminar flame speeds. I am following along in An Introduction to Combustion by Stephen Turns and have hit a slight snag. Chemistry is not exactly my forte, but I need to get there.(adsbygoogle = window.adsbygoogle || []).push({});

So, I am trying to follow the part where we derive our conservation expressions. Mass concentration is straightforward. Thespeciesconservation of the reaction is where I am getting jammed. So let's get to it!

Notation: a *prime* symbol (') denotes "per each spatial dimension." So the symbols

[itex]\dot{m}''[/itex] and [itex]\dot{m}'''[/itex] mean mass flow rate per unit area (mass flux) and mass flow rate per unit volume (this is synonymous with production rate per unit volume), respectively.

So we have in all,

[tex]m \equiv kg[/tex] (total mass)

[tex]m_i\equiv kg[/tex] (mass of i^{th}species)

[tex]\dot{m}''\equiv\frac{kg}{m^2\cdot s}[/tex]

[tex]\dot{m}'''\equiv\frac{kg}{m^3\cdot s}[/tex]

[tex]Y_i \equiv \frac{m_i}{m}[/tex]

[tex]D \equiv \frac{m^2}{s}[/tex] (diffusivity)

[tex]\rho \equiv \frac{m}{vol}\equiv\frac{kg}{m^3}[/tex] (density of total mass)

[tex]\phi_i \equiv \frac{kg}{m^2}[/tex] (concentration of species i)

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I need to prove to myself that the text is not lying to me when they say

[tex]\frac{d[\dot{m}_i'']}{dx} = \dot{m}_i''' = \frac{d\left[\dot{m}_i''Y_i - \rho D\frac{dY_i}{dx}\right]}{dx}\qquad(1)[/tex]

It is the right hand side of (1) that is bothering me. I know that that the term inside the brackets is the mass flux by comparing it to the left hand side of (1). It looks like it makes sense, but I really would like to derive the expression using Fick's Law:

[tex]\dot{m}_i'' = -D\frac{\partial{\phi_i}}{\partial{x}}\qquad(2)[/tex]

(2) is the definition of Fick's Law in one dimension.

I need a coffee! Back in a moment to see what the next step is.

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# Combustion and Fick's 1st Law

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