The ultimate effects of the entrance region on startup flow are considered by one-dimensional steady-state analysis. By roughly accounting for the entrance loss, semiempirical formulas for the average velocity and the hydrodynamic entrance length in terms of a startup flow parameter and an entrance effect coefficient are provided. The startup flow parameter M = gHD4/2048v2L2 is identified as the fundamental parameter of the problem. While Szymanski’s solution is recovered in the long pipe limit, i.e., M → 0, the steady-state volume flux through short pipes (M > 0.05) is significantly reduced due to entrance region effects.

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