Energy Equation

The energy equation, based on the first law of thermodynamics, forms the basis of many engineering analyses of flow in conduits. For steady, incompressible flow in a conduit, the equation reads

where "1" and "2" refer to two locations in the piping system, p is pressure, is the unit weight of the fluid, z is elevation, V is average velocity, is the kinetic energy correction factor, hp is the head supplied by a pump, ht is the head supplied to a turbine, and hL is the head loss in the pipe due to friction, turbulence, and minor losses.

Theoretically, hL can be calculated from thermodynamic considerations, which takes the form

where u is the internal energy, is the rate heat is lost from the system, and is the mass flow rate. Unfortunately, this head loss expression is difficult to implement in practice so empirical formulas are employed. Three dominate the literature: Darcy-Weisbach equation, Hazen-Williams equation, and Manning equation. The first two find more use in problems dealing with flow in pressure conduits, while the latter is used primarily for calculations in open channel hydraulics.

Darcy-Weisbach Equation

Head loss is computed from the formula

where f is the friction factor, L is the pipe length, D is the pipe diameter, and V is the average velocity. Use a consistent set of units; f is dimensionless and is obtained either from the Moody diagram or from the following formulas:

for Re < 2100

for smooth pipes and 4000 < Re < 105

which is attributed to Blasius;

for smooth pipes and Re > 105

which is referred to as the Prandtl equation;

for rough pipes with Re > 4000

which is referred to as the Colebrook equation. The parameter e is the absolute pipe roughness.

Hazen-Williams Formula

For English units (ft, sec) head loss is computed from

where R is the hydraulic radius (A/P) and CH is the roughness coefficient.

For SI units (m, sec) head loss is computed from

Manning Equation

For English units (ft, sec) head loss is computed from

where n is the roughness coefficient.

For SI units (m, sec) head loss is computed from

Minor Losses

Valves, bends, expansions, contractions, and other appurtenances create additional losses over and above the losses predicted from the above formulas. The term "minor" is somewhat of a misnomer, for on short lengths of pipes with a number of fittings, minor losses can exceed the other losses. Typically, minor losses are computed from the empirical formula

where Km is a minor loss coefficient obtained from experiments and reported in the literature.

In analyzing pipe distribution systems, it is often convenient to cast minor loss in terms of equivalent length of pipe, that is, the length of pipe that produces the same loss as the fitting. For example, if the Hazen-Williams formula is being used to compute the head losses in a straight length of pipe, then the equivalent length of pipe is found by equating the Hazen-Williams formula to the minor loss formula. Assume the units are English and the pipe is circular; then

Substituting Q/A for V and solving for L gives

which can be added to the actual length of pipe used in the analysis.

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