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 \(t_{\frac{1}{2,\ ads}}\).