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Is there any analytical solution for the most common expression of column mass balance:

$\dfrac{\partial C}{\partial t} + \left(\dfrac{1+\varepsilon}{\varepsilon}\right)\dfrac{\partial q} {\partial t} + u \dfrac{\partial C}{\partial t} \:\:=\:\: D_{\mathrm{axial}}\:\dfrac{\partial^2 C}{\partial t^2} $

This is the most common isothermal form for just one gaseous solution (two expressions for concentration change between the phases, convective flow and flow dispersion). If you consider adsorption from the "side" of adsorbent (solid phase), the density of fixed bed $\rho_{{\mathrm{bed}}}$ should be considered. $\partial q\big/\partial t$ is expressed by LDF (linear driving force) model which simplifies PDE to ODE.

What I would like to know:

  • I know that these equation can be always solved numerically, but is it possible to solve this balance analytically for certain boudary / initial conditions? If so, how would you express $\partial C\big/\partial t$ ? (I cannot find a direct question anywhere, some suggest it may be solved analytically with certain simplifications without giving any details)
  • There are several mathematical models for numerical solution. Which one would be suitable for adsorption? Unfortunately, literatures I found does not go into details.
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  • $\begingroup$ JoshE, You should consult the book of Guiochon, Golshan Shirazi and Atilla Felinger (postdocs of Guiochon). The book is titled "Fundamentals of Preparative and Nonlinear Chromatography". He spent his in solving these type of problems. $\endgroup$
    – ACR
    Commented Jun 26, 2020 at 3:13
  • $\begingroup$ @M.Farooq Thank you for your suggestion. I think I have seen this book in the past. I have gone through the chapters concerning mass balance and description of these equations and each expressions is very detailed. In Chapter 10, there are different solution methods (mostly mathematically based). However, despite defining initial and boundary conditions, I still miss the middle-step where someone would show step-by-step solution (even if it is dimensionless form - many other literatures suggest to rewrite mass balance) leading to obtain component gas conc. as a function of time, $C(t)$. $\endgroup$
    – Josh E.
    Commented Jun 27, 2020 at 16:22
  • $\begingroup$ If I understand your sentence about LDF correctly, it considers dq/dt a constant? If so, the equation becomes a homogeneous second-order differential equation in C, or a first-order homogeneous differential equation w.r.t. dC/dt, and would thus have analytical solutions as long as epsilon, u, and D_axial are fixed. Or does LDF assume that dq/dt can be approximated as a function of C? $\endgroup$
    – Curt F.
    Commented Dec 21, 2020 at 18:05
  • $\begingroup$ @CurtF. The term $\dfrac{\partial q_i}{\partial t}=k_i\left(q_i^* - q_i\right)$ is not constant. What I learned, the transport coefficient $k_i$ can be expressed as $k_i=f\left(D_{\rm eff}\right)=f\left(\rm Re, Sc\right)$ such that the column geometry and velocity of gas matters. The maximal adsorbed conc. is expressed by the adsorption isotherm, i.e. $q_i^*=f\left(P,T\right)=f\left(p_i,y_i,T\right)$. According to, e.g. Ruthven(1984), the average concentration through particle is defined as integral over microparticle radius $q_i = \dfrac{3}{r_c^3}\,\displaystyle\int_0^{r_c}qr^2\,\mathrm{d}r$. $\endgroup$
    – Josh E.
    Commented Dec 24, 2020 at 15:36

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