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:
  1. Hydrodynamically and diffusively developing flow
  2. Hydrodynamically developed, but diffusively developing flow
  3. 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]