Laminar flow in the entrance region of circular sector ducts

A linearization method developed by Sparrow et al. (1964) is used to analyze developing flows in the entrance region of circular ducts with angles of pi/16, pi/8, pi/4, and pi/2. A nondimensional momentum equation in the axial direction is obtained and solved using a finite-difference approach. The solution is progressed from the inlet section to the fully developed section until all axial velocities are within 1% of the corresponding fully developed value, and the value of the dimensionless distance X there is taken as the entrance length Le. The resulting values of Le are presented for the four geometries considered along with the values of the Reynolds number and the pressure defect.