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 3-12 is one form of the Extended Bernoulli
The head loss due to fluid friction (Hf)
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), which will be discussed
in the next section.
Example: Extended Bernoulli
To use the modified form of Bernoullis equation, reference
points are chosen at the surface of the
reservoir (point 1) and at the outlet of the pipe (point 2). The pressure at
the surface of the reservoir is the same as the
pressure at the exit of the pipe, i.e., atmospheric
pressure. The velocity at point 1 will be essentially zero.
Using the equation for the mass flow rate to determine the
velocity at point 2:
Now we can use the Extended Bernoulli equation to determine
the required pump head.
The student should note that the solution of this example
problem has a numerical value that "makes
sense" from the data given in the problem. The total head increase of 67
ft. is due primarily to the 65 ft.
evaluation increase and the 2 ft. of friction head.