Bernoulli's Equation

Bernoulli’s Equation is a fundamental principle in fluid mechanics that describes the conservation of energy along a streamline for an ideal fluid.

It states that the total mechanical energy per unit volume remains constant in steady, incompressible, inviscid flow.

Equations

Basic Form

$P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$

Where:

• P = Static pressure
• ρ = Fluid density
• v = Flow velocity
• g = Gravitational acceleration
• h = Elevation head

Per unit mass (specific energy)

$\frac{P}{\rho }+\frac{v^2}{2}+gh=\text{constant}$

Head form (per unit weight)

$\frac{P}{\rho g}+\frac{v^2}{2g}+h=H_{\text{total}}$

Where:

• • H = Total head

With Energy Losses

$\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+h_1=\frac{P_2}{\rho g}+\frac{v_2^2}{2g}+h_2+h_L$

Where $h_L$ = head loss due to Friction and minor losses

With Energy Addition

$\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+h_1+h_{\text{pump}}=\frac{P_2}{\rho g}+\frac{v_2^2}{2g}+h_2+h_L$

Where $h_{pump}$ = head added by Pump

Energy Components

Static Pressure Energy

• P/ρ: Pressure energy per unit mass
• Energy due to fluid pressure at a point
• Can do work by pushing against surfaces

Kinetic Energy

• v²/2: Kinetic energy per unit mass
• Energy due to fluid motion
• Increases with velocity squared

Potential Energy

• gh: Gravitational potential energy per unit mass
• Energy due to elevation above reference level
• Constant for horizontal flow

Assumptions & Limitations

Valid When:

• Steady flow: No time-dependent changes
• Incompressible fluid: Constant density (ρ)
• Inviscid flow: No viscous losses
• No energy addition/removal: No pumps, turbines, or heat transfer
• Along a streamline: Energy balance applies to same fluid particle

Invalid When:

• High-speed compressible flow (Mach > 0.3)
• Significant viscous effects (Viscosity)
• Unsteady flow conditions
• Energy input/output devices present
• Across streamlines in rotational flow

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Related

• Navier-Stokes Equations