Measuring mass transfer
The limiting current density method is often used to determine the mass
transfer performance of electrolyzers. [6-7] Typically, this
involves reversible redox couples such as hexacyanoferrate or
hexachloroiridate , for which the rate of reaction is limited by mass
transfer at sufficient overpotential [8-10]. The limiting current is
related to the Sherwood number as shown in eq. 1.
\(Sh\ =\frac{d_{H}}{D}\text{\ .\ }k_{\text{LS}}=\ \frac{d_{H}}{D}.\frac{i_{A,lim}}{\text{n\ F\ }C_{\text{bulk}}}\)eq.1
To understand how mass transfer occurs in electrolyzers it is important
to realize that there are two different boundary layers (see figure 1).
The first, is the hydrodynamic boundary layer which is the region where
the velocity of the flow is lower due to friction with the wall. The
second is the diffusive boundary layer, which is where the concentration
of reactant species is lower due to the reaction at the electrode
surface. In liquids the diffusive boundary layer is much thinner than
the hydrodynamic boundary layer, since liquid diffusion coefficients are
typically at least a factor 1000 smaller than liquid viscosities.
Moreover, the hydrodynamic and diffusive boundaries do not necessarily
develop simultaneously. For instance, if an inert section of wall
precedes the electrode, then the flow can be hydrodynamically developed
before the diffusive boundary layer begins to form. Essentially, there
are three distinct situations:
- Hydrodynamically and diffusively developing flow
- Hydrodynamically developed, but diffusively developing flow
- Hydrodynamically and diffusively developed flow
[FIGURE 1]
Situation A is most common in electrolyzers, where the electrode
typically starts directly after the inlet. Situation B occurs when a
certain entrance length without electrodes is used in which the flow is
allowed to develop before reaching the electrodes. Situation C only
occurs after a certain length of electrode and can therefore only be
seen in the downstream segments of a segmented electrode.
The distance required for a flow to reach fully developed conditions is
known as the hydrodynamic entrance length. More specifically it is
defined as the length needed for the centerline velocity to reach 99%
of its fully developed value. In laminar flow, the entrance length
depends on the Reynolds number as shown in eq. 2.
\(L_{\text{entry}}=\Phi\text{\ d}_{h}\text{\ Re}\) eq. 2
Where \(\Phi\) is a parameter that depends on the geometry. For a
rectangular channel it is nonlinearly dependent on the aspect ratio of
the cross-section \(\gamma^{\prime}\). [11] Table 1 lists the six values Han
established. [11]
[TABLE 1]
To obtain fully developed turbulent flow, a hydrodynamic entrance length
is required that is 50 hydraulic diameters \(d_{H}\) long, regardless of
the flowrate [12].
In figure 2 the current in a typical mass transfer experiment is shown.
In such an experiment the flow is turned on prior to the potential,
giving the hydrodynamic boundary layer time to develop. When the
potential is applied (at t=0), a current appears and quickly trends
toward a stable value that is determined by the thickness of the
diffusive boundary layer. The high initial current is due to the
development of this layer. At most flowrates, the reaction is fully
limited by mass transfer from the bulk after at most 7 seconds. At low
flow rates (20 L/h and 30 L/h), it takes several seconds longer for the
diffusive boundary layer to develop. This is because at lower flow rates
the diffusive boundary layer is thicker and therefore there is more
reactant present in this layer.
[FIGURE 2]