I came across this publication:
Do, H. D., & Do, D. D. (2001). A new diffusion and flow theory for activated carbon from low pressure to capillary condensation range. Chemical Engineering Journal, 84(3), 295–308. https://doi.org/10.1016/S1385-8947(00)00383-1
In there the condensation process of benzene is described in porous activated carbon and then a nice mass transport equation is derived. I want to adapt this set of equations for another system. I couldn't get my head around the derivation of the condensation pressure in the pores of the carbon. Of course they are using the Kelvin equation . But for the thickness of the adsorbed film they are using the BET equation and combining it with a adapted form of the kelvin equation.
Using the equation for the film thickness *t*:
\begin{align}
t=\sigma_{ff} \left[\frac{C_{\mathrm{BET}} P}{\left(P_{0}-P\right)\left[1+\left(C_{\mathrm{BET}}-1\right)\left(P / P_{0}\right)\right]}\right]
\end{align}
and combining it with the equation for the condensation pressure (the adapted Kelvin equation):
\begin{align}
\frac{r_{\mathrm{k}}(P)-t(P, r)-\sigma_{ss}}{2}=-\frac{\gamma v_{\mathrm{M}}}{R T \ln \left(P / P_{0}\right)}
\end{align}
As stated in the paper one should end up with a description of the threshold pressure, where the pores are completely filled with condensate. From this two radii, can be extracted which then can be used to calculate the mass transport through the pores.
I get stuck at the point where combining those two equations doesn't yield me with a equation of the condensation pressure as a function of pore radius:
\begin{align}
P^{*} = f(r_K)
\end{align}
My first thought was, that the threshold pressure $P^*$ should exactly be the pressure where $t = r_K$, because this is the point where the whole pore is filled with condensate... Or am I doing something wrong?
But then I end up with an equation without any description of the radius and I have nothing.
So anybody with a clearer mind in calculus?
Thank you very much, any help is highly appreciated!
cheers
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