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Where ΞΌ Μ is the effective viscosity. Brinkman assumed the inertial terms are negligible, therefore, he only considered the viscous terms in N-S equations. He chose to use the fluid viscosity instead of the effective viscosity after refuting other available options. However, later studies have stated that the effective viscosity depends both on the fluid viscosity and the system geometry. [90,95] Uncertainties and rarity of knowledge on the relationship between geometry and effective viscosity obligates studies to use the fluid viscosity as the effective viscosity in Brinkmann correction term. [38,43,79] Here, it is assumed to be the same. Therefore, the corrected Darcyβs law is represented as: π»π =βππ£ β0.55π|π£ |π£ +πβ2π£ (3.8) ππΎπβπΎππ π The necessity of using the modified version for this study is investigated in Section 3.4. 3.1.3- Dimensional analysis Buckingham pi theorem is an analytical tool developed by E. Buckingham, which describes important parameters of the studied system in terms of dimensionless numbers (DN). These DNs provide a basis for experiment design. Empirical equations may be developed by using these numbers. Dimensional analysis is often used for scaling up or data analysis. [87,96β98] Based on the Buckingham pi theorem, the number of DNs, which describe the system, is the number of the variables subtracted by the number of independent units. For example, Mukherjee, B. et. al. [97] found that nine parameters affect oil droplet size generated during chemical dispersion of crude oil, and the involved independent units are mass, length, and time. Therefore, six DNs are sufficient to describe the system and control the experiments with. 3.1.4- Scaling analysis 22PDF Image | Analysis of Fluid Flow in Redox Flow Batteries
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