Mass transfer in hydrodynamically developed flow
In a perfectly laminar flow, with a perfectly hydrodynamically developed
flow, the mass transfer coefficient between parallel plates in a
rectangular channel can be described by a theoretical Leveque-type
equation, eq. 3a [12]. Here, \(\gamma\) is the aspect ratio of the
electrode: \(B/L\). In the work of Ong and Picket no significant effect
has been found of the developing diffusive layer on the rate of mass
transfer in laminar flow. As a consequence, eq.3a is valid for both
short and long electrodes. [12-13]
\(Sh\ =\ 1.467\left(\frac{2}{1+\gamma}\right)^{0.33}\left(\text{Re\ Sc}\frac{d_{H}}{L}\right)^{0.33}\ \)eq.3a
In some publications an alternative correlation is reported in the form
of eq.3b. [1, 13-14]. This correlation is based on a best fit of the
experimental data obtained by Ong and Pickett [13]. The difference
with eq. 3a has been explained as being the result of electro-organic
absorption of m-nitrobenzene. [12] These electro-organic impurities
poison the electrode surface resulting in lower than expected mass
transfer, especially at higher flowrates. Similar poisoning behavior is
seen for the hexacyanoferrate redox couple. [8-10]
\(Sh\ =\ 2.54\ \left(\text{Re\ Sc}\frac{d_{H}}{L}\right)^{0.296}\)eq.3b
The correlations for a perfectly developed turbulent regime are based on
analogies between momentum and mass transfer. Many different analogies
exist, both theoretical (Prandtl, Von Karman) and empirical
(Chilton-Colburn, Lin-Koulton-Putnam, Deissler, Vieth-Porter-Sherwood,
Wasan-Wilke, Dittus-Boelter). Most of these analogies can be rewritten
in the form of \(Sh\ \sim\ Re^{b}\text{\ S}c^{c}\), with varying powers
of Re and Sc. In the work of Ong, the Chilton-Colburn analogy is found
to be most representative of their dataset [12] :
\(Sh\ =\frac{f}{2}\text{Re\ S}c^{0.33}\) eq. 4a
This analogy sets c at a value of 0.33 and uses the friction factor f.
Typically, f is determined by a further correlation to the Reynolds
number, i.e.: \(f\ \sim\ a^{*}Re^{b^{*}}\). In some references, eq.4a is
written using the J-factor instead of the friction factor. [12-13,
15] The M subscript denotes mass transfer. The J-factor is equal to
f/2:
\(Sh=J_{M}\text{Re\ S}c^{0.33}\ \) eq. 4b
For fully developed turbulent flow, Pickett and Ong determined two
separate correlations for the J-factor: one for short electrodes with\(Le=L/d_{H}\ <\ 10\), and another for long electrodes with\(Le=L/d_{H}>12\) [13]. Filled into eq.4b, these result in eq.5a
and eq.5b. For short electrodes (eq.5a) the Le number was introduced as
a variable, because mass transfer depended significantly on their
relative length. Hence eq. 5a corresponds to a situation that is
hydrodynamically developed, diffusively developing, whereas eq. 5b
corresponds to a hydrodynamically and diffusively developed situation.
\(Le<10:\ \ \ \ Sh\ =0.125\ Re^{0.66}Sc^{0.33}Le^{-0.25}\) eq. 5a
\(Le>12:\ \ \ \ Sh\ =0.023\ Re^{0.8}Sc^{0.33}\) eq. 5b