We present a mathematical model to study the steady-state performance of a membrane-less reversible redox flow battery formed by two immiscible electrolytes that spontaneously form a liquid-liquid system separated by a well defined interface. The model assumes a two-dimensional battery with two coflowing electrolytes and flat electrodes at the channel walls. In this configuration, the analysis of the far downstream solution indicates that the interface remains stable in all the parameter range covered by this study. To simplify the description of the problem, we use the dilute solution theory to decouple the calculation of the velocity and species concentration fields. Once the velocity field is known, we obtain the distribution of the mobile ionic species along with the current and the electric potential field of the flowing electrolyte solution. The numerical integration of the problem provides the variation of the battery current density $I_{\mathrm{app}}$ with the State of Charge ($\mathrm{SoC}$) for different applied cell voltages $V_{\mathrm{cell}}$. A detailed analysis of the concentration density plots indicates that the normal operation of the battery is interrupted when reactant depletion is achieved near the negative electrode both during charge and discharge. The effect of the electrolyte flow on the performance of the system is studied by varying the Reynolds, $Re$, and Péclet, $Pe$, numbers. As expected, the flow velocity only affects the polarization curve in the concentration polarization region, when $V_{\mathrm{cell}}$ is well below the equilibrium potential, resulting in limiting current densities that grow with $Re$ as $j_{\lim }\sim R{e}^{0.3}$. In addition, both the single-pass conversion efficiency $\psi$ and the product $\psi j_{\lim }$ decrease with $Re$. Concerning the later, the decay rate with $Re$ exhibits a power law with an exponent that almost doubles previous theoretical predictions obtained by imposing a prescribed velocity profile for the electrolyte in a membrane-less laminar flow battery with a liquid oxidant and gaseous fuel. The present work constitutes the first modelling attempt that simultaneously solves the fluid dynamical system formed by the two immiscible electrolytes and the electrochemical problem that determines the response of the membrane-less battery. The proposed model could be used as a valuable tool to optimize future flow battery designs based on immiscible electrolytes.

我们提出了一个数学模型来研究无膜可逆氧化还原液流电池的稳态性能，该电池由两种不混溶的电解质自发形成由明确定义的界面分隔的液-液系统。该模型假设二维电池具有两个共同流动的电解质和通道壁处的扁平电极。在此配置中，对远下游解决方案的分析表明，界面在本研究涵盖的所有参数范围内保持稳定。为了简化问题的描述，我们使用稀溶液理论来解耦速度场和物质浓度场的计算。一旦知道速度场，我们就可以获得移动离子物质的分布以及流动电解质溶液的电流和电势场。${\mathrm{一世}}_{\mathrm{应用程序}}$ 与充电状态（$\mathrm{系统级芯片}$) 对于不同的施加电池电压 ${伏}_{\mathrm{细胞}}$. 对浓度密度图的详细分析表明，在充电和放电期间，当负极附近的反应物耗尽时，电池的正常运行会中断。通过改变雷诺数来研究电解液流量对系统性能的影响，$\mathrm{电阻}\mathrm{电子}$, 和佩克莱特, $磷\mathrm{电子}$，数字。正如预期的那样，流速只影响浓差极化区域的极化曲线，当${伏}_{\mathrm{细胞}}$ 远低于平衡电位，导致限制电流密度随 $\mathrm{电阻}\mathrm{电子}$ 作为 $j_{\mathrm{林}}～\mathrm{电阻}{\mathrm{电子}}^{0.3}$. 此外，无论是单程转换效率$\psi$ 和产品 $\psi j_{\mathrm{林}}$ 减少 $\mathrm{电阻}\mathrm{电子}$. 关于后者，衰减率与$\mathrm{电阻}\mathrm{电子}$展示了一个幂律，其指数几乎是之前通过对具有液体氧化剂和气体燃料的无膜层流电池中的电解质施加规定的速度分布而获得的理论预测的两倍。目前的工作构成了第一次建模尝试，同时解决了由两种不混溶电解质形成的流体动力学系统和决定无膜电池响应的电化​​学问题。所提出的模型可用作优化未来基于不混溶电解质的液流电池设计的宝贵工具。