Darcy Weisbach Equation Derivation

In the dynamic fluid, we know that the Darcy Weisbach equation is empirical as it nearly relates the loss head or loss pressure due to friction along a length which is given by pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after the great Sir Henry Darcy and Julius Weisbach.

The equation contains a dimensionless factor of friction which is known as the Darcy factor of friction. This is variously called the Darcy Weisbach factor of friction or the resistance coefficient or the flow coefficient.

The friction factor fD is not a constant, it depends on things such as the characteristics of the pipe - the diameter D and roughness height denoted as ε, the characteristics of the fluid that is its kinematic viscosity ν [nu] and the velocity of the fluid flow denoted as v. We can notice the value of fD that is measured by the experimenters is for many fluids which are different over a wide range of numbers and for the pipes of varying roughness and heights. There are three regime broadsides of fluid flow encountered in these data that are critical, laminar, and turbulent.

Friction Factor in Darcy Weisbach Equation

We know that friction is proportional to the square of the flow of velocity. The domain over many orders of magnitude of Re that is 4000 < Re < 108. The factor friction generally varies less than one order of magnitude that is 0.006 < fD < 0.06. The turbulence within the flow regime is  the nature of the flow that can be further divided into a regime where the pipe wall is smooth effectively and one where its height roughness is salient.

Darcy Weisbach Formula

We see that historically this equation arose as a variant on the Prony equation. This variant was said to be earlier developed by France Henry Darcy. In 1845, it was refined further into the form which is used today by Julius Weisbach of Saxony. The data which is on the variation of fD initially with velocity was lacking. Thus, the Darcy Weisbach equation has outperformed the Prony empirical equation in many cases. In years later it was eschewed that in many cases which were situations in favour of a variety of empirical equations is valid only for flow certain regimes. 

Darcy Weisbach Coefficient

The characteristics such as the flow are independent in nature of the position along the pipe. The key quantities are then the drop of pressure which is along the pipe per unit length that is:  denoted as Δp/L.

And we know the flow rate of volumetric. The rate of flow can be converted to a mean velocity flow denoted by letter V by dividing the area of wetted area of the flow by the equal area of cross-sectional of the pipe if the pipe is full of fluid.

We should recall that the pressure must be proportional to the pipe of  length which is between the two points denoted by L, as the pressure drop per unit length is a constant. The turning of the relationship into a coefficient of proportionality of dimensionless quantity. We can easily divide by the diameter of the hydraulic pipe denoted by letter D which is also constant along with the pipe. The coefficient of proportionality is the dimensionless "Darcy friction factor" or we can say that the "flow coefficient".