Darcy-Weisbach formula
The Darcy-Weisbach equation is an empirical approach for the calculation of pressure losses in straight open channels and piping.
The analytical approach with a system of differential equations provided by [Navier 1822] can only be integrated for a few particular
cases. Specialized software packages provide iterative solutions. In everyday life of engineers the Darcy-Weisbach equation has
established itself due to easy handling.
Table 1 shows equations of major authors from the first half of the 19th century regarding pressure loss in straight piping.
The empirical equations were derived from experiments with turbulent water flow in iron and cast iron pipes.
Table 1
[Weisbach 1850, page 529] noted equation (1) as we know it today. Darcy Friction Factors f in Table 1 become dimensionless by the dimensions
of their constants.
The approach of Eytelwein features a constant Darcy Friction Factor.
Weisbach, Prony and Aubuisson show that Dracy Friction Factor f falls in value with rising velocity. Darcy points out that
Friction Factor f falls in value with rising diameters. Both observations are true. The absolute Darcy Friction Factor values
provided by equations of Table 1 match the range of Frictions Factors shown in the Moody diagram. In his equations Colebrook used
dimensionless values for describing the Darcy Friction Factor f (roughness/diameter and Reynolds number).
 Moody diagram
The work of Navier, Stokes, Hagen, Poiseuille, Reynolds, Karman, Prandtl and Colebrook - to name the most famous - led to the
formulation of the Dracy Friction Factor as we know it today (diagram by Moody). It comprises flow regimes from laminar to completely
turbulent for all Newtonian fluids.
Literature
|
| [Eytelwein 1801]
J.A. Eytelwein, Handbuch der Mechanik fester Körper und der Hydraulik, Berlin bei F.L. Lagarde, 1801 |
| [Prony 1804]
R. Prony, Recherches Physico-Mathématiques
sur la théorie des eaux courantes, Paris, de L´imprimerie Impériale, 1804 |
| [Navier 1822]
M. Navier, Mémoire de l´ Académie des Sciences, Tome 6, 1823, 389 ff |
| [Aubuisson 1834]
J.F.Aubuisson de Voisins, Traité D´Hydraulique,
Paris, Chez F.G. Levrault, 1834 |
| [Weisbach 1850]
J. Weisbach, Lehrbuch der Ingenieur- und Maschinenmechanik,
Erster Teil Theoretische Mechanik, 2. Auflage, Braunschweig F. Vieweg und Sohn, 1850 |
| [Darcy 1858]
Darcy Mémoire a l´Académie des Sciences, Tome 15, 1858, 141 ff |
|