Bernoulli's Equation

Why it matters

Bernoulli's Equation is fundamental in fluid mechanics, providing insights into the behavior of fluid flow in various engineering applications such as pipe systems, open channels, and hydraulic machines. Understanding this principle helps engineers design efficient systems for water supply, irrigation, and drainage.

Key ideas

• Conservation of Energy: Bernoulli's Equation is based on the principle of conservation of energy for a fluid particle moving along a streamline.
• Assumptions: The fluid is incompressible and non-viscous, and the flow is steady; compare points along a streamline with no pump/turbine work between them.
• Components: The equation relates pressure energy, kinetic energy, and potential energy per unit weight of the fluid.
• Applications: Used in calculating pressure drops, flow rates, and velocities in various fluid systems.

Formulas

• Bernoulli's Equation: P/ρg + v²/2g + z = constant
  • P = Pressure energy per unit volume (Pa)
  • ρ = Density of fluid (kg/m³)
  • g = Acceleration due to gravity (9.81 m/s²)
  • v = Velocity of fluid (m/s)
  • z = Elevation head (m)

Worked example

Given: A pipe with water flowing at a velocity of 3 m/s at a height of 5 m. The pressure at this point is 200 kPa. Calculate the pressure at a second point where the velocity is 5 m/s and the height is 2 m.

1. Apply Bernoulli's Equation between the two points: P₁/ρg + v₁²/2g + z₁ = P₂/ρg + v₂²/2g + z₂
2. Substitute known values: 200,000/(1000*9.81) + 3²/(2*9.81) + 5 = P₂/(1000*9.81) + 5²/(2*9.81) + 2
3. Solve without intermediate rounding, using water density 1000 kg/m³: P₂ = P₁ + ρg(z₁ − z₂) + ρ(v₁² − v₂²)/2 P₂ = 200000 + 1000 × 9.81 × 3 + 1000 × (9 − 25)/2 P₂ = 221430 Pa.

Final Answer: 221.43 kPa, using the same gauge or absolute pressure reference as P₁. Neglect losses between the stated sections.

Common mistakes

• Ignoring the assumptions of incompressibility and non-viscosity.
• Misplacing terms in the equation, especially when rearranging.
• Forgetting to convert units, particularly pressure units.

For GATE CE

• Questions often involve calculating unknown pressures, velocities, or heights using Bernoulli's Equation.
• Practice problems with varying knowns and unknowns to strengthen understanding.
• Be prepared to apply the equation in conjunction with other fluid mechanics principles like continuity equation.

Quick check

1. What is the primary assumption behind Bernoulli's Equation?
2. How does Bernoulli's Equation relate pressure and velocity?
3. What is the unit of the pressure-head term P/(ρg)?

Answers: 1. Incompressible, non-viscous fluid flow; 2. At equal elevation and under the stated assumptions, greater speed corresponds to lower static pressure; elevation changes can alter that comparison. 3. Metres (m).