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v. 3.3.1- The pressure gradient in lateral directions is negligible. The strength of this assumption is supported by implementing scaling analysis on N-S equations, as done in Section 3.3.1. Scaling analysis on N-S equations 𝜕𝑣 𝜕𝑃 𝜌( 𝑥+(𝑣𝑐.𝛻)𝑣𝑥)=− 𝑚+𝜇𝛻2𝑣𝑥 𝜕𝑡 𝜕𝑥 𝜕𝑣 𝜕𝑃 𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 𝜌( 𝑦+(𝑣𝑐.𝛻)𝑣𝑦)=− 𝑚+𝜇𝛻2𝑣𝑦→ 𝜕𝑡 𝜕𝑦 𝜕𝑣 𝜕𝑃 {𝜌( 𝑧+(𝑣𝑐.𝛻)𝑣𝑧)=− 𝑚+𝜇𝛻2𝑣𝑧 𝜕𝑡 𝜕𝑧 𝜕𝑣 𝜕𝑣 𝜕𝑣 𝜕𝑃 𝜕2𝑣 𝜕2𝑣 𝜕2𝑣 𝑥 +𝑣 𝑥 +𝑣 𝑥)= 𝑚 +𝜇( 𝑥 + 𝑥 + 𝑥) 𝜌(𝑣 𝑥 𝜕𝑥 𝑦 𝜕𝑦 𝑧 𝜕𝑧 𝜕𝑥 𝜕𝑥2 𝜕𝑦2 𝜕𝑧2 𝜕𝑣 𝜕𝑣 𝜕𝑣 𝜕𝑃 𝜕2𝑣 𝜕2𝑣 𝜕2𝑣 𝜌(𝑣 𝑦 +𝑣 𝑦 +𝑣 𝑦)=− 𝑚 +𝜇( 𝑦 + 𝑦 + 𝑦) 𝑥 𝜕𝑥 𝑦 𝜕𝑦 𝑧 𝜕𝑧 𝜕𝑦 𝜕𝑥2 𝜕𝑦2 𝜕𝑧2 𝜕𝑣 𝜕𝑣 𝜕𝑣 𝜕𝑃 𝜕2𝑣 𝜕2𝑣 𝜕2𝑣 𝜌(𝑣 𝑧 +𝑣 𝑧 +𝑣 𝑧)=− 𝑚 +𝜇( 𝑧 + 𝑧 + 𝑧) { 𝑥𝜕𝑥 𝑦𝜕𝑦 𝑧𝜕𝑧 𝜕𝑧 𝜕𝑥2 𝜕𝑦2 𝜕𝑧2 The second and third rows contain six terms containing negligible vy and vz and two terms of pressure gradients in y- and z-direction, respectively. Equating these negligible terms to their respective pressure gradients results in them being negligible too. Therefore, the first row suffices for modeling the system: 𝜕𝑣 𝑑𝑃 𝜌(𝑣 𝑥)=− 𝑚 +𝜇( 𝑑𝑥 𝜕2𝑣 𝑥 + 𝜕𝑥2 𝜕2𝑣 𝜕2𝑣 𝑥 + 𝑥) 𝑥 𝜕𝑥 𝜕𝑦2 𝜕𝑧2 𝑑𝑃𝑚 =𝜇𝑑2𝑣𝑐 +𝜇𝑑2𝑣𝑐 +𝜇𝑑2𝑣𝑐 −𝜌𝑣 𝑑𝑣𝑐 𝑑𝑥 𝑑𝑦2 𝑑𝑧2 𝑑𝑥2 𝑐 𝑑𝑥 (3.15) Macroscopic mass conservation equation for RFBs demonstrates that the velocity vector magnitude for the efflux and influx is equal: 𝜌=𝑐𝑡𝑒 𝐴𝑖𝑛=𝐴𝑜𝑢𝑡 𝑚̇𝑖𝑛 =𝑚̇𝑜𝑢𝑡 → (𝑣𝐴)𝑖𝑛 =(𝑣𝐴)𝑜𝑢𝑡 → 𝑣𝑖𝑛 =𝑣𝑜𝑢𝑡 27PDF Image | Analysis of Fluid Flow in Redox Flow Batteries
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