Exercise $5\mathrm{.}19$

$($b$)$ Which equation is correct? Hints: Remember that the

Poiseuille equation assumes laminar flow and completely

developed flow. According to Welty et al$.(1976),$ the

entrance length $L_e$ required for the fully developed

laminar velocity profile to develop is

$\frac{L_e}{D}=0\mathrm{.}0575Re$

Remember also that laminar flows start to become

turbulent in smooth pipes around $Re=2000.$

$($Comment: The Bernoulli equation should overpredict flow

rates as the tube diameter becomes smaller. Nevertheless,

the Bernoulli equation can be used in this situation if

sufficient "major" and "minor" energy losses are accounted

for. These corrections can be found in fluid mechanics and

engineering reference books.$)$