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Numerical Solutions By Method of Lines Approach for Fluid Flow in a Modified Rotating Disk Electrode Apparatus

_{2}

^{+}/VO

^{2+}) and negative (V

^{3+}/V

^{2+}) redox couple reactions are not well understood, as there is no clear agreement of redox kinetics information available in literature

^{1-2}for these electrolytes. Porous rotating disk electrode (PRDE) is a commonly used technique to study the kinetics of graphite felt electrodes.

A semi-infinite condition is typically used to arrive at well-known Levich profiles for velocity and limiting current analysis for rotating disk electrodes^{3}. However, this type of analysis didn’t account for the data observed for felt electrodes in a RDE set up. To obtain a numerical solution for fluid flow around a rotating disk electrode in a modified RDE setup is proposed in this work as shown in Figure 1. In this setup, the boundary conditions at the stationary walls, rotating walls and axial symmetry are well-defined. This method can be used for obtaining primary and secondary distributions effectively in electrochemical systems efficiently and also for simulating flow problems^{4}. Numerical solutions with well-defined boundary conditions will help in enhancing the reliability of the simulated results.

In this 2D fluid flow model, an axially symmetric cylindrical geometry is used and fluid flow was assumed to vary in radial (*r*) and axial (*z*) directions, whereas azimuthal (*θ*) dependence of fluid flow is ignored. Three components of Navier-Stokes equations (*r,** θ *and* z*) and a continuity equation represent the fluid flow outside the rotating disk electrode. The velocities (*u _{r}, u*

*and*

_{θ}*u*), pressure and radial and axial distances were non-dimensionalized through a method, using fluid properties and angular velocity, as reported in literature

_{z}^{5}. The 2D model equations were simulated using a new technique in which numerical method is applied in the z-direction and the resulting equations are integrated using efficient Boundary value problem (BVP) solvers in

*r*. Efficiency and accuracy of the proposed method compared with different finite difference and discretization schemes in

*r*will be presented.

**Acknowledgement: **This work was supported by a contract from The American Chemical Society- Petroleum Research Fund (ACS-PRF) Grant PRF# 52588-ND10.

**References:**

1. Gattrell, M., Park, J., MacDougall, B., Apte, J., McCarthy, S., and Wu, C. W., *J. Electrochem. Soc.*, **151**, A123-A130 (2004).

2. Aaron, D., Sun, C., Bright, M., Papandrew, A. B., Mench, M., and Zawodzinski, T. A., *Electrochem. Lett.*, **2**, A1-A3 (2013).

3. Dong, Q., Santhanagopalan, S., and White, R. E., *J. Electrochem. Soc.*, **155**, B963-B968 (2008).

4. Rife, D., Lawder, M., and Subramanian, V. R., "Numerical method of lines approach with single coordinate discretization for efficient simulation of 2D cavity flow", Manuscript under preparation.

5. B. Nam, and R. T. Bonnecaze, *J. Electrochem. Soc.*, **154**, F191-F197 (2007).

Figure 1. Schematics of the modified electrochemical cell to study the 2D fluid flow generated by RDE.