# Stoichiometry in Gas-Solid Flow System

I am a bit confused on how to account for the stoichiometry of a reaction as follows:

$$\ce{A}(s) + \delta \ce{B}(g) \ce{->} \ce{C}(s) + \delta \ce{D}(g)$$

The solid $\ce{A}$ is stationary in the reactor, and gas $\ce{B}$ flows through it with known inlet molar flow rate. Since 1 mol of $\ce{A}$ produces $\delta$ mol of $\ce{D}$, but the solids are static, how can I determine the amount of $\ce{D}$ formed (molar flow rate)? The other known piece of information is the normalized extent of reaction, $\alpha$, which can be defined as the amount of $\ce{D}$ formed at time $t$ divided by its total amount.

Please let me know if it's not clear enough. It's been difficult for me to find clear explanations about the modeling of such type of reaction in the papers and theses I've read. The closest I've found to a definition of $\alpha$ is as follows:

$$\alpha = \int \frac{\ce{D}(t)}{\ce{D}_\text{total}} \mathrm{d}t$$

(yes, written exactly like that)

My Approach So Far

I believe I have to somehow consider both the flow rate of $\ce{B}$ and the amount of $\ce{A}$ in the reactor. If the molar flow rate of $\ce{B}$ is "too large", then the stoichiometric coefficient $\delta$ will limit the flow rate of $\ce{D}$. I've thought of doing something like this:

$$\dot{n}_{\ce{D}} = \min\{\dot{n}_{\ce{B}},\delta n_{\ce{A}}\}$$

where $\dot{n}_i$ is the molar flow rate of species $i$ and $n_\ce{A}$ is the number of moles of solid $\ce{A}$, but as you can see one of the arguments of the $\min$ operator is a molar flow rate and the other is just moles, which is problematic. Any suggestions?

Edit

Clearly, I have to use $\alpha$ somewhere. But I'm still unsure what to do here. Generically, I'd write something like,

$$\dot{n}_i = \dot{n}_{i,0} + \dot{n}_0 \nu_i \alpha$$

but I'm not sure if this is correct in this case (again, how do I use the amount of solid in the reactor?).