Theoretical Solution for Fully Developed Laminar Flow in a Noncircular Duct

In this study, the friction factor of a fully developed laminar flow in a noncircular duct, which is used in an automotive cooler system, is investigated. The Poisson equation of velocity with homogeneous boundary conditions is solved by using the method of separation of variables and analytical solutions of velocity, wall shear stress, and friction factor for arbitrary symmetric noncircular ducts are obtained. The analytical solutions are validated by comparing against analytical and experimental data available for rectangular and triangular ducts. Cross-sectional distribution of velocity, wall shear stress along the duct boundary, and friction factor are analyzed in depth. It is obtained that the friction factor for this specific duct is lambda = (95)/Re, with Re defined with the equivalent diameter as length scale. Moreover, the applicability of a general equation of friction factor proposed by Muzychka and Yovanovich is validated for this specific duct by comparing with the analytical solution. The result shows that the general equation is accurate enough and can be used for the evaluation of flow resistance in the design of the cooler system.