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$5\mathrm{.}19.$ At this point, we have two equations to compute the average velocity or flow in a pipe$$the Bernoulli and Poiseuille equations. However, the assumptions of these models are quite different: The Poiseuille equation includes viscous effects and the Bernoulli equation does not. This exercise is designed to compare the predictions of the two equations $($or models$).$

$($a$)$ Shown on the following page is a tank filled with water discharging through a smooth pipe to the atmosphere. What are the predicted flows $($m$3/$s$)$ according to the Bernoulli and Poiseuille equations for pipe diameters of $0\mathrm{.}1,1,$ and $10$ mm$?$ Make a figure comparing the two predicted discharges as a function of the pipe diameter. For the purpose of applying these equations, you can assume no entrance losses at the pipe entrance. Also recall that pressure is related to hydrostatic head $($H$)$ by Eq$.5\mathrm{.}11$:

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 Le required for the fully developed laminar velocity profile to develop is

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.$)$