Topology optimization for the design of flow fields in a redox flow battery

Abstract

This paper presents topology optimization for the design of flow fields in vanadium redox flow batteries (VRFBs), which are large-scale storage systems for renewable energy resources such as solar and wind power. It is widely known that, in recent VRFB systems, one of the key factors in boosting charging or discharging efficiency is the design of the flow field around carbon fiber electrodes and in flow channels. In this study, topology optimization is applied in order to achieve optimized flow field designs. The optimization problem is formulated as a maximization problem for the generation rate of the vanadium species governed by a simplified electrochemical reaction model. A typical porous model is incorporated into the optimization problem for expressing the carbon fiber electrode; furthermore, a mass transfer coefficient that depends on local velocity is introduced. We investigate the dependencies of the optimized configuration with respect to the porosity of the porous electrode and the pressure loss. Results indicate that patterns of interdigitated flow fields are valid designs for VRFBs.

Appendix: Dimensionless form

To simplify the numerical implementation, the governing equations (3), (7), and (8) are replaced with dimensionless equations as follows:

$$\begin{array}{@{}rcl@{}} &&{}\nabla^{*}\cdot\mathbf{u}^{*}=0, \end{array} $$

(A.1)

$$\begin{array}{@{}rcl@{}} &&{}(\mathbf{u}^{*}\cdot\nabla^{*})\mathbf{u}^{*}=-\nabla^{*} p^{*}+\frac{1}{Re}\nabla^{*2}\mathbf{u}^{*}-\alpha^{*}\mathbf{u}^{*}, \end{array} $$

(A.2)

$$\begin{array}{@{}rcl@{}} &&{}\mathbf{u}^{*}\cdot\nabla^{*} c^{*}=\frac{1}{Pe}\nabla^{*2}c^{*} + K_{\text{m}}^{*}(1-c^{*}), \end{array} $$

(A.3)

where the asterisk represents dimensionless variables defined using a characteristic length L, a characteristic speed U, and the inlet concentration of vanadium c _in as follows:

$$\begin{array}{@{}rcl@{}} &&{}\nabla^{*}= L\nabla, \ \ \mathbf{u}^{*}=\frac{\mathbf{u}}{U}, \ \ p^{*}=\frac{p-p_{0}}{\rho U^{2}}, \ \ \alpha^{*}=\frac{\alpha L}{\rho U}, \\ &&{}c^{*}=\frac{c-c_{\text{in}}}{c_{\max}-c_{\text{in}}}, \ \ K_{\text{m}}^{*}=\frac{k_{\text{m}}A_{\text{V}}L}{U}. \end{array} $$

The Reynolds number, R e, and the Pélet number, P e, are respectively given by

$$\begin{array}{@{}rcl@{}} Re&=&\frac{\rho UL}{\mu}, \end{array} $$

(A.4)

$$\begin{array}{@{}rcl@{}} Pe&=&\frac{UL}{D}. \end{array} $$

(A.5)

In the numerical examples, the characteristic length L is defined as the inlet width. It is noted that the characteristic speed U cannot be defined using the magnitude of the inlet velocity since the pressure loss is fixed in this paper. According to previous research (Yaji et al. 2015), U can be defined using the pressure loss as follows:

$$ U=\sqrt{\frac{\Delta p}{\rho}}, $$

(A.6)

where Δp is the dimensional pressure loss, and thus U has the same units as velocity.