Darcy's Equation

DARCY'S EQUATION

• The major energy losses i.e., the energy losses due to friction in the pipe can be calculated by using Darcy's equation.

• The Darcy's equation for the loss of head due to friction in pipes is as follows:

where HL = Loss of head due to friction in pipe in m,

f = Friction factor,

L = Length of pipe in m,

D = Inside diameter of the pipe in m,

V = Average velocity of liquid in m/s, and

G = Acceleration due to gravity in m/s2.

• The Darcy's equation holds good for both laminar and turbulent flow provided a proper value of friction factor 'f' is evaluated.

1. Frictional Losses in Laminar Flow

For laminar flow, the friction factor 'f' is function of Reynolds number only and is given by,

Substituting equation (3.24) into equation (3.23), we get

The above equation is known as the Hagen-Poiseuille equation, which is valid only for laminar flow.

2. Frictional Losses in Turbulent Flow

• Unlike for laminar flow, the friction factor 'f' for turbulent flow is a function of Reynolds number as well as the relative roughness of the pipe.

• The relative roughness is defined as the pipe inside surface roughness (ɛ) divided by the inside diameter of the pipe (D).

• In equation (3.26), the ‘ɛ' is called the absolute roughness. Table 3.1 gives typical values of absolute roughness for various types of pipes. For a given pipe, the appropriate 'ε' value may be selected using the Table 3.1.

Table 3.1. Typical values of absolute roughness

• Once we calculate the relative roughness (ɛ/D) and Reynolds number, we can determine the friction factor for use in Darcy's equation by using the Moody diagram. The Moody diagram is presented in Fig.3.16. The Moody diagram contains many curves. The curves indicate the value of friction factor as a function of Reynolds number and relative roughness.