Extended Bernoulli Concepts and Equations

Fluid Flow Table of Contents
Hydraulic and Pneumatic Knowledge

Extended Bernoulli Concepts and Equations

The Bernoulli equation can be modified to take into account gains and losses of head. The resulting equation, referred to as the Extended Bernoulli equation, is very useful in solving most fluid flow problems. In fact, the Extended Bernoulli equation is probably used more than any other fluid flow equation. Equation below is one form of the Extended Bernoulli equation.

z₁ = V ₁² / ( 2 g ) + P₁ ν₁ g_c / g + H_p - z₂ +V₂² / ( 2 g ) + P₂ v₂ g_c / g + H_f

where

z = height above reference level (ft)
V = average velocity of fluid (ft/sec)
P = pressure of fluid (lbf/ft²)
v = specific volume of fluid (ft³/lbm)
H_f = head loss due to fluid friction (ft)
H_p = head added by pump (ft)
g = acceleration due to gravity (ft/sec²)

The head loss due to fluid friction (H_f ) represents the energy used in overcoming friction caused by the walls of the pipe. Although it represents a loss of energy from the standpoint of fluid flow, it does not normally represent a significant loss of total energy of the fluid. It also does not violate the law of conservation of energy since the head loss due to friction results in an equivalent increase in the internal energy (u) of the fluid. These losses are greatest as the fluid flows through entrances, exits, pumps, valves, fittings, and any other piping with rough inner surfaces.

Most techniques for evaluating head loss due to friction are empirical (based almost exclusively on experimental evidence) and are based on a proportionality constant called the friction factor (f).