Analysis of Fluid Flow in Redox Flow Batteries

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Analysis of Fluid Flow in Redox Flow Batteries ( analysis-fluid-flow-redox-flow-batteries )

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βˆ‡π‘ƒ =βˆ‡π‘ƒ +βˆ‡πœŒπ‘”h=βˆ‡π‘ƒ +πœŒπ‘”βˆ‡h=βˆ‡π‘ƒ +πœŒπ‘” π‘šhhh Now, N-S equations can be reiterated in fewer terms: 𝜌 ( πœ• 𝑣 π‘₯ + ( ⃗𝑣⃗ βƒ— . 𝛻 ) 𝑣 ) = βˆ’ πœ• 𝑃 π‘š + πœ‡ 𝛻 2 𝑣 πœ•π‘‡ 𝑐 π‘₯ πœ•π‘₯ π‘₯ 𝜌(πœ•π‘£π‘¦ +(⃗𝑣⃗⃗.𝛻)𝑣 )=βˆ’πœ•π‘ƒπ‘š +πœ‡π›»2𝑣 πœ•π‘‡ 𝑐 𝑦 πœ•π‘¦ 𝑦 { 𝜌 ( πœ• 𝑣 𝑧 + ( ⃗𝑣⃗ βƒ— . 𝛻 ) 𝑣 ) = βˆ’ πœ• 𝑃 π‘š + πœ‡ 𝛻 2 𝑣 πœ•π‘‡ 𝑐 𝑧 πœ•π‘§ 𝑧 (3.3) (3.4) Moving forward, the modified pressure is called pressure in the body of thesis. Other definitions of pressure will be mentioned if needed. 3.1.2- Darcy’s law for porous media The fluid dynamics in porous media is dominated by viscous forces at relatively low flow rates. Darcy’s experimental law is an established equation to predict the pressure gradient for laminar flow passing through porous media: 𝛻𝑃 =βˆ’πœ‡π‘£ (3.5) 𝐾 Where K is an experimental term called the permeability of porous media. Darcy’s law predicts the dominancy of viscous forces. Additionally, it shows the pressure gradient depends on the flow average velocity, viscosity, and permeability. However, when the fluid flow rate increases, microscopic inertial forces grow rapidly. Therefore, the pressure gradient deviates from what Darcy’s law predicts. Many experimental terms are suggested to account for this deviation. [88] The most well-known ones are Forchheimer and Brinkman correction terms in the literature. Forchheimer presented a general correction term to Darcy’s law for fluids at high flow rates in his article at 1901: [89] π‘š 20

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