Perhaps the most common way to write the Bernoulli equation is:.

$\displaystyle{\left (\frac{1}{2}\rho \Delta v^2 + \rho g\Delta z + \Delta P \right ) = 0}$

If you allow for frictional losses in the Bernoulli equation, then this equation becomes:

$\displaystyle{\left (\frac{1}{2}\rho \Delta (v^2)_{avg} + \rho g\Delta z + \Delta P \right ) = \rho g h_L}$

When you include frictional losses it is no longer correct to assume that you have a plug flow, so you need to use the average value of the $v^2$ term at each point.

where $\rho g h_L$ represents the *frictional losses* in
the system.

Recall that we learned to calculate the $h_L$ calues earlier in the term.

One last tweak to the Mechanical Energy Balance equation is to note the following:

$\displaystyle{(v_{avg})^2 \neq (v^2)_{avg}}$

To allow the simplified statement above to be most conveniently used, one will often define a value $\alpha$ as $\alpha = \frac{(v^2)_{avg}}{v_{avg}^2}$ so that you may write:

$\displaystyle{\left (\frac{1}{2}\rho \Delta \alpha(v_{avg})^2 + \rho g\Delta z + \Delta P \right ) = \rho g h_L}$

Extend the Bernoulli's Equation for more general cases (i.e., write and use the mechanical form of the macroscopic energy balance equation)