Bernoulli Equation: Derivation, Assumptions, Applications, and Limitations

Bernoulli’s Equation

Bernoulli’s equation is one of the most important equations in fluid mechanics. It expresses the conservation of energy along a streamline for steady, incompressible, inviscid flow:

P/ρg + V²/2g + z = constant

Or equivalently:

P₁/ρg + V₁²/2g + z₁ = P₂/ρg + V₂²/2g + z₂

Where:

• P = static pressure (Pa)
• ρ = fluid density (kg/m³)
• g = gravitational acceleration (9.81 m/s²)
• V = fluid velocity (m/s)
• z = elevation above datum (m)

Each term represents a head (pressure head + velocity head + elevation head = total head = constant).

Derivation from Work-Energy Theorem

Bernoulli’s equation can be derived by applying Newton’s second law (F = ma) along a streamline to a fluid element, or by applying the first law of thermodynamics to a control volume of inviscid, steady, incompressible flow. Key steps:

1. Consider a fluid element of mass dm moving along a streamline
2. Forces acting: pressure forces on both faces + component of gravity
3. Apply F = ma along the streamline (Euler’s equation)
4. Integrate Euler’s equation along the streamline
5. Result: P + ½ρV² + ρgz = constant

Assumptions of Bernoulli’s Equation

• Steady flow, fluid properties at any point do not change with time
• Incompressible flow, constant fluid density (valid for liquids; gases below Mach 0.3)
• Inviscid flow, no viscous (frictional) effects (no energy loss)
• Along a streamline, the equation applies between two points on the same streamline
• No work or heat transfer between the two points considered

Applications of Bernoulli’s Equation

Limitations of Bernoulli’s Equation

• Not valid for viscous flow, ignores friction losses (pipe flow needs the Darcy-Weisbach equation)
• Not valid for compressible flow at high Mach numbers
• Not valid across turbomachinery (pumps, fans, turbines), work is added or extracted
• Not valid across shock waves
• Only along a streamline, different streamlines can have different Bernoulli constants unless flow is irrotational

Modified Bernoulli’s Equation (with Head Loss)

For real fluid flow with friction losses and/or pump work:

P₁/ρg + V₁²/2g + z₁ + h_pump = P₂/ρg + V₂²/2g + z₂ + h_L

Where h_pump = energy added by pump per unit weight, and h_L = head loss due to friction and minor losses.

Lab Experiment

The Bernoulli’s Theorem Apparatus consists of a convergent-divergent section with pressure tappings at multiple points. Students:

1. Set a steady flow rate through the apparatus
2. Read the piezometric (static) head at each tapping using a manometer bank
3. Measure the flow velocity using a rotameter or by timing collection
4. Calculate the total head at each tapping and verify it is constant (Bernoulli’s theorem)
5. Observe how static pressure decreases and velocity increases at the throat

Contact Scientico India for CE-certified Bernoulli’s Theorem Apparatus for your fluid mechanics lab.