5.3 Bernoulli's equation and applications

Bernoulli's equation connects pressure, velocity, and elevation in fluid flow. It's rooted in energy conservation — assuming steady, incompressible, frictionless flow along a streamline.

The equation has broad applications, from pipe flow to open channels, but it also has real limitations. Understanding both its uses and its constraints is key to solving practical fluid flow problems.

Bernoulli's Equation Derivation

Conservation of Energy Principle

The conservation of energy principle says that total energy in a closed system stays constant — nothing is added, nothing removed.

In fluid dynamics, total energy has three components: kinetic energy, pressure energy, and potential energy.

• Kinetic energy comes from the fluid's motion and is proportional to the square of velocity. Think of water flowing through a pipe — faster flow means more kinetic energy.
• Pressure energy relates to the force the fluid exerts due to its internal pressure. A compressed gas in a tank, for example, stores pressure energy.
• Potential energy depends on elevation relative to a reference level. Water sitting in an elevated reservoir has potential energy due to its height.

Applying Conservation of Energy to Fluid Flow

Bernoulli's equation is derived by applying energy conservation along a streamline, assuming steady, incompressible, frictionless flow.

The derivation equates total energy at two points along the streamline:

$P_1/\rho + v_1^2/2 + gz_1 = P_2/\rho + v_2^2/2 + gz_2$

where $P$ is pressure, $\rho$ is fluid density, $v$ is velocity, $g$ is gravitational acceleration, and $z$ is elevation.

Each term represents a form of specific energy (energy per unit mass). The equation states that the sum of these three energy forms remains constant along a streamline.

Applying Bernoulli's Equation

Fluid Flow in Pipes

Bernoulli's equation can be used to calculate pressure, velocity, or elevation at any point in a pipe, given conditions at another point.

When applying it to pipe flow, keep in mind:

• The continuity equation ($A_1 v_1 = A_2 v_2$) links velocity changes to pipe cross-sectional area. Where the pipe narrows, velocity increases.
• Friction losses, while not captured by the basic equation, can be accounted for using head loss terms (e.g., the Darcy-Weisbach equation or the Moody diagram).
• Pressure drops along a pipe can result from elevation changes, velocity increases, or energy dissipation due to friction.

Typical applications include:

• Determining pressure differences between two sections of a pipe with varying diameter
• Calculating required pump head to maintain a target flow rate

Open-Channel Flow

For open-channel flow — rivers, canals, spillways — Bernoulli's equation applies with some additional considerations:

• The fluid surface is open to the atmosphere, so surface pressure equals atmospheric pressure.
• Channel geometry (width, depth, slope) affects flow velocity and depth. A wider or deeper channel generally means slower surface velocity for a given flow rate.
• Energy losses from turbulence, bed friction, and channel bends are not included in the basic equation but can be significant.

Applications in open-channel flow include:

• Estimating discharge velocity from a reservoir or dam
• Analyzing flow over weirs and spillways (e.g., calculating flow rate over a rectangular weir)
• Evaluating velocity and depth changes as water moves through channels of varying cross-section

Limitations of Bernoulli's Equation

Simplifying Assumptions

Bernoulli's equation rests on several idealizations that don't always hold in practice:

• Incompressible flow: The fluid density is assumed constant. This works well for most liquids but breaks down for gases at high speeds or under large pressure changes.
• Steady flow: Conditions at any point are assumed not to change with time. Transient events — valve closures, pump startups, wave action — violate this assumption.
• Frictionless flow: The equation ignores viscous energy losses. In real systems, friction from pipe walls, fittings, and channel beds always dissipates some energy.

Energy Losses and Turbulence

Real fluid systems always involve some energy dissipation that the basic equation doesn't capture:

• Friction from pipe walls, valves, bends, and fittings causes pressure drops that must be modeled separately (using tools like the Darcy-Weisbach equation or empirical loss coefficients).
• Turbulence — chaotic, mixing flow — converts organized kinetic energy into heat. Turbulent losses are especially important in open-channel flow and in pipes with rough surfaces or complex geometry.
• Compressibility effects matter when fluid velocities are high or when density varies significantly along the flow path. For these cases, compressible flow equations (accounting for changes in $\rho$) are needed.

Despite these limitations, Bernoulli's equation remains one of the most widely used relationships in fluid mechanics. Recognizing where its assumptions apply — and where corrections are needed — is essential to using it effectively.