Mass transfer resistance and intrinsic adsorption rate
A useful approach to investigate and to compare the mass transfer resistance with the kinetic resistance is by comparing\(\frac{1}{k_{c}\text{.a}}\) with \(\frac{1}{k_{\text{ads}}}\) , i.e. the time constant for mass transfer and adsorption, respectively. Since the diameter of adsorbent particle is small, 60 microns, and the AgNO3 layer thickness is much smaller than the silica particle diameter, the internal mass transport through the pores of the adsorbent has been neglected (For more information see supplementary material, S2). Therefore, the mass transfer resistance calculation is only based on the external mass transfer from the bulk fluid to the adsorbent surface.
In order to determine \(k_{c}\ \)first the Reynolds number based on the particle diameter, Re, should be investigated. The calculated Reynolds number, Re, in our system is smaller than 2. From the literature [43] for low Re number, the Sherwood number can be defined as
\begin{equation} Sh=\frac{k_{c}d_{p}}{D}=\frac{\phi_{s}}{6\left(1-\varepsilon_{b}\right)\xi}\frac{ud_{p}}{D}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3.6)\ \nonumber \\ \end{equation}
where \(\phi_{s}\), \(u{,\ d}_{p}\),\(\xi,\ \text{and}\ \text{D\ }\)represent particle sphericity factor, the superficial liquid velocity, the particle diameter, the channeling factor and the diffusion coefficient of TCO in the solvent. In the Eq. 3.6, it is assumed that particles are spherical; so\(\ \phi_{s}=1\). Unfortunately, the \(\xi\) value cannot be estimated exactly. However, in literature [43] the value of \(\xi\) has been predicted to be in the range of 1-10. Here the worst case value, \(\xi\)=10 (the case that gives highest mass transfer resistance) has been considered.
\(D\) is estimated on the basis of values in the similar systems[44]. As the fluid phase is very diluted, parameters such as density and viscosity are approximated based on solvent characteristics (n-hexane) at 25˚C.
On the other hand, in order to determine \(\frac{1}{k_{\text{ads}}}\)which has the dimension of time, an adsorption experiment can be performed in batch in a way that mass transfer resistance is eliminated as much as possible. By this experiment, the time to reach 50% of the equilibrium adsorption (\(t_{\frac{1}{2,\ ads}}\)) is calculated which can be considered to be \(\frac{1}{k_{\text{ads}}}\). Lastly,\(k_{\text{ov}}\) is calculated according to:
\begin{equation} \frac{1}{k_{\text{ov}}}=\frac{1}{k_{\text{ads}}}+\frac{1}{k_{c}\text{.a}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ (3.7)\nonumber \\ \end{equation}
In the mentioned experiment (adsorption of TCO on AgNO3) the batch vessel was well stirred so it is assumed that there is no external mass transfer limitation (more details on the experiment can be found in experimental section). Adsorption is an equilibrium process. According to the results, by plotting TCO concentration versus time it is possible to record \(t_{\frac{1}{2,\ ads}}\).