A Computational Model for Natural Convection and Forced Convection in Redox MHD Systems Based on Electroneutrality and Migration
Authors: Isaac KM , Yuan Fangping 
Published: February 2020

We present a new non-dimensional parameter for buoyancy effects in natural convection and mixed convection problems. The TN number [1] for mixed convection in redox MHD systems compares MHD forces to buoyancy forces to determine if buoyancy will have a significant effect on system behavior. When convection is caused solely by buoyancy, the Rayleigh number is a relevant parameter to consider the effect of natural convection. However, when the system is influenced by other competing forces such as the Lorentz force, the relative magnitudes of these additional phenomena would determine if buoyancy needs to be considered. If the velocity magnitudes due to forced convection are an order of magnitude higher than those due to natural convection, it would be reasonable to assume that natural convection can then be neglected. A factor that accounts for their respective regions of influence would provide a better representation of the overall influence of natural convection. Based on these arguments, we proposed the non-dimensional group called the TN number. Density change in the present context is a consequence of the charge balance required for electroneutrality which is attained by a complex process of species transport in the vicinity of the electrode including supporting electrolyte ions. Redox electrochemical systems do not involve deposition, but electrochemical reactions occur by electron transfer at the electrode-solution interface. The effect of natural convection in redox systems are less pronounced than in electroplating. It is generally accepted that, in redox electrochemistry, the main reason for natural convection is the migration of ionic species to maintain electroneutrality, a term used to describe the tendency of a fluid volume element to be electrically neutral. To achieve electroneutrality, cations and anions in the solution rearrange themselves, and cause density gradients and natural convection. The L3
-dependence in the TN number indicates the strong dependence on geometric scaling. Micro scale and nanoscale systems might show behavior not observed in larger-scale systems.
Bard and Faulkner [2] provide an insightful discussion of this topic by considering balance sheets for mass transfer. The redox reactions lead to concentration gradients of the electroactive species in the diffusion layer which result in a charge imbalance. Cations and anions of the supporting electrolyte migrate into or out of the diffusion layer dictated by their transference numbers in order to neutralize the charge imbalance caused by electron transfer. Our natural convection model is based on the charge balance sheet approach. For the redox reaction at the electrode, we can write an expression for the charge in the diffusion layer relative to the bulk solution. The supporting electrolyte ions moving in and out of the diffusion layer to compensate for the resulting charge imbalance and satisfy electroneutrality is quantitatively modeled and used to calculate the corresponding density change in the diffusion layer. The degree to which the density changes in the diffusion layer will depend on the difference in the molecular weights and the transference numbers of the anions and the cations of the supporting electrolyte. In our model, the changes in the concentrations of the anions and cations of the supporting electrolyte in the diffusion layer with reference to the bulk solution are calculated using the molecular weights and the transference numbers of the ions of the supporting electrolyte. This then enables the calculation of the solution density in the diffusion layer. Thus, we have a quantitative model for density change caused by redox electrochemical reaction. The change in density is then used to calculate the body force due to gravity (buoyancy force). This buoyancy force will be an additional term in the momentum equation similar to the Lorentz force term. This modified momentum equation is solved along with the continuity and species transport equations in a coupled manner using user defined functions (UDF) in the simulations.
Results show that three regimes can be delineated by calculating the TN number: pure natural convection, mixed convection and forced convection. The above model should be used in the first two regimes to account for density gradients. Results also show that the magnitude of natural convection depends on the orientation of the gravity vector relative to the electrode surface normal. The simulations also support the L3
-dependence present in the TN number.
References
[1] K. M. Isaac, C. Gonzales and D. Sen, Modeling of redox electrochemical MHD and three-dimensional CFD simulations of transient phenomena in microfluidic cells, Microfluidics and Nanofluidics, 17,5, 2014, 943-958. [2] A. J. Bard and L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications, 2nd Ed., 2001.

Figure 1