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Where r, v(r), D, ΞΌ, and l are the distance of an arbitrary point from the centerline of the pipe, fluid velocity at radius r, pipe diameter, fluid dynamic viscosity, and pipe length, respectively. Poiseuilleβs Law is achieved by integrating this velocity profile over the cross-sectional area: π = ππ·4βπ (3.28) 128ππ Where Q is the fluid flow rate. A similar relationship between the fluid flow rate and the pressure drop is obtained by applying a dimensional analysis on the fluid dynamics in the system: π=π(π·,π,π,βπ )βππ·3βπ=πΆπ βπ=ππ·4 βπ (3.29) π 4ππ π· 4πΆππ π Where C is a constant based on the geometry of the cross-sectional area of the pipe. By comparing Equations 3.28 and 3.29, it is concluded that C-value is 32 for round pipes. Some C-values for square pipes are mentioned in Equation 3.30: [101] πΆ = 31.1, ππ π = 0.5 π πΆ = 29,ππ π = 0.75 π { πΆ = 28.5, ππ π = 1 π (3.30) Where a and b are the height and width of the pipe, respectively. The pipe hydraulic diameter (Dh=2ππ) replaces the pipe diameter in Equation 3.29 for a square pipe. However, the actual π+π velocity profile differs from what is described here, and portions of the flow diffuse in the electrode. Therefore, the actual pressure drop is lower than the one predicted by Poiseuilleβs law. To accommodate for the deviation from the pipe assumption, the C-value in Equation 3.29 is adjusted to a value between zero and C-values for pipes, mentioned in Equation 3.30. This value is fitted to the data for each FF and electrode design to enhance the accuracy of the HEAM. The 34PDF Image | Analysis of Fluid Flow in Redox Flow Batteries
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