It is customary to express the energy contained in a fluid in terms of the potential energy contained in an equivalent height or “head” of a column of the fluid. Using this convention, Bernoulli’s theorem breaks down the total energy at a point in terms of:
1. The head due to its elevation above an arbitrary datum of zero potential energy.
2. A pressure head due to the potential energy contained in the pressure in the fluid at that point.
3. A velocity head due to the kinetic energy contained within the fluid
Assuming that no energy is added to the fluid by a pump or compressor, and that the fluid is not performing work as in a steam turbine, the law of conservation of energy requires that the energy at point “2” in the piping system downstream of point “1” must equal the energy at point “1” minus the energy loss to friction and change in elevation. Thus, Bernoulli’s theorem may be written:
Or
where:
Z = elevation head, ft
P = pressure, psi
p = density, lb/ft3
V = velocity, ft/sec
g = gravitation constant
HL = friction head loss, ft
Darcy’s Equation
This equation, which is also sometimes called the Weisbach equation or the Darcy-Weisbach equation, states that the friction head loss between two points in a completely filled, circular cross section pipe is proportional to the velocity head and the length of pipe and inversely proportional to the pipe diameter. This can be written:
where:
L= length of pipe, ft
D = pipe diameter, ft
f = factor of proportionality
Equations 1 and 2 can be used to calculate the pressure at any point in a piping system if the pressure, flow velocity, pipe diameter, and elevation are known at any other point. Conversely, if the pressure, pipe diameter, and elevations are known at two points, the flow velocity can be calculated.
In most production facility piping systems the head differences due to elevation and velocity changes between two points can be neglected. In this case Equation 2 can be reduced to:
where ∆P = loss in pressure between points 1 and 2, in psi