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Chapter 3- The analysis of fluid dynamics in the RFB 3.1- Introduction 3.1.1- Navier-Stokes equations Navier-Stokes (N-S) equations are a set of differential equations describing the motion of viscous fluids. They are stated as: 𝜌(𝜕𝑣𝑥+(⃗𝑣⃗⃗.𝛻)𝑣)=−𝜌𝑔 −𝜕𝑃h+𝜇𝛻2𝑣 𝜕𝑇 𝑐 𝑥 𝑥 𝜕𝑥 𝑥 𝜌(𝜕𝑣𝑦 +(⃗𝑣⃗⃗.𝛻)𝑣 )=−𝜌𝑔 −𝜕𝑃h +𝜇𝛻2𝑣 𝜕𝑇 𝑐 𝑦 𝑦 𝜕𝑦 𝑦 {𝜌(𝜕𝑣𝑧+(⃗𝑣⃗⃗.𝛻)𝑣)=−𝜌𝑔 −𝜕𝑃h+𝜇𝛻2𝑣 𝜕𝑇 𝑐 𝑧 𝑧 𝜕𝑧 𝑧 (3.1) Where ρ, v, Ph, μ, T, and g are the fluid density, velocity vector, hydraulic pressure, viscosity, time, and gravitational constant, respectively. The subscriptions x, y, and z represent the magnitude of the assignee vector mirrored on these Cartesian axes. The first term on the left-hand side (LHS) of each N-S equation accounts for the transient pressure effect, as the second term represents the impact of inertial forces. The first term on the right-hand side (RHS) expresses gravitational forces effect, the second term embodies the hydraulic pressure effect, and the last one accounts for the impact of viscous forces. N-S equations may be represented in fewer terms than their depictions in Equation 3.1. gravitational forces and hydraulic pressure terms are combined to form a term called modified pressure (Pm). Modified pressure is defined as: [87] 𝑃 =𝑃 +𝜌𝑔h (3.2) 𝑚h gravitational forces are path-independent and only the flow elevation effects their magnitude. Therefore, the modified pressure gradient is obtained as: 19PDF Image | Analysis of Fluid Flow in Redox Flow Batteries
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