The Bernoulli Equation for Process Engineers: Energy Conservation in Fluids

Introduction: Conservation of Energy

The Bernoulli equation is the fundamental statement of the conservation of energy for flowing fluids. Named after Daniel Bernoulli (1738), it states that for an incompressible, frictionless fluid, the total mechanical energy along a streamline remains constant.

A fluid in a pipe possesses three types of mechanical energy:

1. Pressure Energy: Energy due to the static pressure of the fluid.
2. Kinetic Energy: Energy due to the fluid's velocity.
3. Potential Energy: Energy due to the fluid's elevation in a gravitational field.

The Classic Bernoulli Equation

If we compare two points (Point 1 and Point 2) in a flowing pipeline, the ideal Bernoulli equation is written as:

$$ \frac{P_1}{\rho} + \frac{v_1^2}{2} + g z_1 = \frac{P_2}{\rho} + \frac{v_2^2}{2} + g z_2 = \text{Constant} $$

Where:

• $P$ = Static pressure (Pa)
• $\rho$ = Fluid density ($kg/m^3$)
• $v$ = Fluid velocity (m/s)
• $z$ = Elevation (m)
• $g$ = Gravity (9.81 $m/s^2$)

Note: The terms in the equation above represent energy per unit mass (Joules/kg).

The "Head" Form of Bernoulli

Process engineers and pump designers usually prefer to express energy in terms of "Head" (meters or feet of fluid). By dividing the entire equation by gravity ($g$), we get the Head form:

$$ \frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 $$

• $\frac{P}{\rho g}$ is the Pressure Head.
• $\frac{v^2}{2g}$ is the Velocity Head.
• $z$ is the Elevation Head.

The Mechanical Energy Balance (Real-World Bernoulli)

The classic Bernoulli equation assumes a perfect, frictionless fluid with no pumps or turbines. In the real world of chemical engineering, pipes have friction, and we use pumps to add energy. We must expand Bernoulli into the Mechanical Energy Balance:

$$ \frac{P_1}{\rho g} + \frac{v_1^2}{2g} + z_1 + H_{pump} = \frac{P_2}{\rho g} + \frac{v_2^2}{2g} + z_2 + h_f $$

Where:

• $H_{pump}$ = Total head added to the fluid by a pump (m)
• $h_f$ = Total frictional head loss through pipes, valves, and fittings (m) (calculated via the Darcy-Weisbach equation)

Practical Applications

The Mechanical Energy Balance is the basis for almost all hydraulic design in a process plant:

• Sizing Pumps: If you know your source tank (Point 1) and destination tank (Point 2), you can rearrange the equation to solve for $H_{pump}$, which gives you the required Total Dynamic Head (TDH) to buy a pump.
• Siphons and Gravity Drains: If there is no pump ($H_{pump}=0$), you can calculate if the elevation difference ($z_1 - z_2$) is sufficient to overcome the friction ($h_f$).
• Venturi Meters / Orifice Plates: In a horizontal pipe ($z_1 = z_2$), as the fluid passes through a restriction, velocity increases ($v_2 > v_1$). To conserve energy, the pressure must drop ($P_2 < P_1$). Flow meters measure this exact pressure drop to calculate the flow rate.

Try the Calculator

Explore the kinetic energy component of the Bernoulli equation using the Pipe Hydraulics Calculator on this page. It instantly computes the velocity and dynamic velocity head ($\frac{1}{2}\rho v^2$) for your specific flow rate and pipe size!