## The Flow through Pipes

**Introduction**: In general the flow of liquid along a pipe can be determined by the use of The Bernoulli Equation and the Continuity Equation. The former represents the conservation of energy, which in Newtonian fluids is either potential or kinetic energy, and the latter ensures that what goes into one end of a pipe must come out at the other end. It must be stressed that only incompressible liquids are being considered.

**Bernoulli's Equation: **

This is usually expressed as:

Where is the head lost due to friction?

**Proof:**The work done by the pressure on AB in time

All the work is expended in:

a) Doing work against the pressure at CD i.e.

b) Raising the weight of A, B, B’, A’ to C, D, C’D’ i.e. w a_{1}, v_{1}, dt.

c) Increasing the kinetic energy of A, B, B’A’ to that C, D, C’, D’ i.e.

Equating (1) to the work expended

Where C = Constant

This equation can be expressed in three ways:

Where C = Constant

**Notice that the above equations make no allowance for a head lost due to Friction and it is normal to write Bernoulli's Equation as:**

Where h_{L}is the head lost due to friction.