Analytical mass transfer coefficients for natural convection from vertical gas-evolving electrodes

The high mass transfer to or from gas-evolving electrodes is an attractive feature of electrochemical reactors, which can be partly attributed to the large convective flows that arise due to the buoyancy of bubbles. We derive exact analytical expressions for mass transfer coefficients for the case of constant gas flux boundary conditions. For the mass transport both Dirichlet and Neumann boundary conditions are considered. We deploy a recently derived self-similar solution of laminar two-phase flows, with density, hydrodynamic diffusivity, and viscosity dependent on the local gas fraction. Combining this with the Lévêque approximation, new mass transfer coefficients are obtained analytically. These new results are relevant for various electrochemical processes with gas evolution as well as boiling. The new formulation shows the mass transfer coefficient to scale with the vertical coordinate z proportional to z^−1/5 for short electrodes and low current densities and z^−4/15 for long ones and high current densities. The former limit also applies when buoyancy is due to temperature or concentration differences in the case that density differences are small. We provide a general overview considering all possible gas and mass boundary conditions combinations and a comparison with the Boussinesq approximation of small density differences.