Deriving Fick's First Law in Combustion: A Step-by-Step Approach

[Saladsamurai]

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.

So, I am trying to follow the part where we derive our conservation expressions. Mass concentration is straightforward. The species conservation 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

$\dot{m}''$ and $\dot{m}'''$ 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,

$$m \equiv kg$$ (total mass)

$$m_i\equiv kg$$ (mass of i^th species)

$$\dot{m}''\equiv\frac{kg}{m^2\cdot s}$$

$$\dot{m}'''\equiv\frac{kg}{m^3\cdot s}$$

$$Y_i \equiv \frac{m_i}{m}$$

$$D \equiv \frac{m^2}{s}$$ (diffusivity)

$$\rho \equiv \frac{m}{vol}\equiv\frac{kg}{m^3}$$ (density of total mass)

$$\phi_i \equiv \frac{kg}{m^2}$$ (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

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

$$\dot{m}_i'' = -D\frac{\partial{\phi_i}}{\partial{x}}\qquad(2)$$

(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.

 

[Saladsamurai]

Presumably, the way to start this is to write down (2) and end up with the term in brackets in (1) and along the way, reason out what they have done. So,

Fick's 1st Law states that the 1-dimensional mass flux of species i is proportional to to the concentration gradient of species i along that dimension. That is,

$$\dot{m}_i'' = -D\frac{\partial{\phi_i}}{\partial{x}}$$

I assume that we would like to write the concentration of species i, $\phi_i$, in terms of the mass fraction of species i, Y_i, because it is a quantity that we can generally find for a given reaction mechanism. So, knowing that $\phi_i$ is the mass of the i^th species, per unit volume of total mass (V_T). We can say,

$$\phi_i = \frac{m_i}{V_T} = \frac{m_i}{m}\frac{m}{V_T} = \rho Y_i$$

So Ficks Law in terms if mass fractions becomes,

$$\dot{m}_i'' = -D\frac{d[\rho Y_i]}{dx}\qquad(3)$$

They must be assigning that the total mixture density remains constant which is why they were able to pull the density through the differential operator in (1). So I now have,

$$\dot{m}_i'' = -\rho D\frac{d Y_i}{dx}\qquad(4)$$

I am now almost there. They have an additional term preceding the right hand side of my (4) in their (1). I feel like this has something to do with the fact that there is another dependency here. That is, the instantaneous mass flux is dependent on what is left, in terms of concentration, from the instant that preceded it. I.e., the flux $\dot{m}_i''$ is not constant and is time dependent.

Does that sound reasonable?