The background to Bernoulli's Theorem.
Daniel Bernoulli (1700 - 1782) investigated not only mathematics but also such fields as medicine, biology, physiology, mechanics, physics, astronomy, and oceanography. Key Facts:
- v is velocity [m/s]
- P is pressure [N/m2]
- g is accretion due to gracity [m/s2]
- z is vertical elevation
- ρ is the density of the liquid
Bernoulli's principle states,
" For a perfect incompressible liquid, flowing in a continuous stream, the total energy of a particle remains the same, while the particle moves from one point to another."This statement is based on the assumption that there are no losses due to friction in the pipe. Mathematically,
- z = Potential Energy
- = Kinetic Energy
- = Pressure Energy
Note: The Bernoulli's theorem is also the law of conservation of energy, i.e. the sum of all energy in a steady, streamlined, incompressible flow of fluid is always a constant.
Consider a perfect incompressible liquid, flowing through a non-uniform pipe as shown in fig. Let us consider two sections AA and BB of the pipe and assume that the pipe is running full and there is a continuity of flow between the two sections. Let,
- = Height of AA above the datum
- = Pressure at AA
- = Velocity of liquid at AA
- = Cross sectional area of the pipe at AA, and
- = Corresponding values at BB
We know that the work done by pressure at AA, in moving the liquid to A'A'
Similarly, the work done by pressure at BB, in moving the liquid to B'B'
Note: Minus (-) sign is taken as the direction of is opposite to that of .Total work done by the pressure
Loss of potential energy = and again in Kinetic Energy = = We know that, Loss of potential energy + Work done by pressure = Gain in kinetic energy
which proves Bernoulli's Equation.
Example - Bernoulli's Theorem
The diameter of a pipe changes from 200mm at a section 5m above datum to 50mm at a section 3m above datum. The pressure of water at first section is 500kPa. If the velocity of the flow at the first section is 1m/s, determine the intensity of pressure at the second section.
- = Velocity of flow at section (2), and
- = Pressure at section (2).
and area of pipe at section (2),
Since the discharge through the pipe is continuous, therefore
Applying Bernoulli's equation for both the ends of the pipe,
The intensity of pressure at the second section = 392.4 kPa