The Mechanical Energy Balance Reference Page
Add Friction and Work to Bernoulli? Here’s What Really Happens!

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Intro to The Mechanical Energy Balance

The mechanical energy balance is a useful form of the First Law of Thermodynamics for flowing, approximately constant-density fluids. It relates pressure, velocity, elevation, shaft work, and frictional losses.

In many fluid-flow problems, the mechanical energy balance is more convenient than the full energy balance because it focuses on the forms of energy most important in pipe flow, pumps, turbines, and fittings.

This video shows how the Bernoulli equation fits into the larger mechanical energy balance and explains what changes when friction and shaft work are included.

Add Friction and Work to Bernoulli? Here’s What Really Happens!

This video introduces the mechanical energy balance, explains the role of friction losses and shaft work, and shows how the Bernoulli equation emerges as a special case.

Visuals

Examples and Definitions

Definitions

Friction Loss, \(\hat{F}\)
The mechanical energy lost because of friction within a flowing fluid and between the fluid and the walls of its container, such as a pipe.
Moody Chart
A chart showing the Darcy–Moody friction factor as a function of Reynolds number and relative roughness for fully developed pipe flow.
Moody Friction Factor, \(f\)
A friction factor used to calculate friction losses in pipe flow. It is often obtained from the Moody chart.
Fanning Friction Factor, \(f\)
A friction factor also used to calculate friction losses. The Fanning friction factor is one-fourth of the Darcy–Moody friction factor.
Mechanical Energy Balance
A form of the First Law of Thermodynamics for flowing, approximately constant-density fluids in which the relevant energy terms are pressure, kinetic energy, potential energy, shaft work, and friction loss. \[ \Delta \frac{P}{\rho} + \Delta \frac{u^2}{2} + g \Delta z + \hat{F} = \frac{\dot{W}_s}{\dot{m}} \]
Bernoulli Equation
The mechanical energy balance with shaft work and friction losses neglected. \[ \Delta \frac{P}{\rho} + \Delta \frac{u^2}{2} + g \Delta z = 0 \]
Reynolds Number
A dimensionless number comparing inertial effects to viscous effects in fluid flow. For flow in a pipe or tube,

\[ Re = \frac{\rho u D}{\mu} \]