Power, energy, work and flow rate: formulas

This power, energy, work and flow rate formula memo recalls that a power in watts is a work in joules divided by a time in seconds, and that work is a force times a distance. This page gathers, as a memo, the relations we use constantly for a propeller, a turbine or a pump: work, energies, power, flow rates and the discharge speed of a water head.

In this page:

1: Work in joules: formula

Work describes a force applied over a distance. It is expressed in joules: 1 joule corresponds to a force of 1 newton moving an object by 1 meter, the force being parallel to the displacement.

W (J) = F (N) × D (m)

2: Kinetic energy and potential energy: formulas

The energy of an object represents the work it can produce. It is therefore measured in the same unit as work, the joule.

Kinetic energy measures the work the object can do thanks to its motion:

Ek (J) = 0.5 × mass (kg) × speed² (m/s)²

Potential energy (also called gravitational) measures the work the object can do by changing altitude:

Ep (J) = mass (kg) × g × H (m)

Example: a 2 kg object that drops by 10 m can provide a work of 2 × 9.81 × 10 = 196.2 J. Its potential energy is 196.2 J.

To simplify some conversions and power calculations, we express energy per unit of mass: this is the specific energy, in joules per kilogram (J/kg). For the previous object, 196.2 / 2 = 98.1 J/kg. We simply remove the mass factor:

Specific Ep (J/kg) = g × H (m)
Specific Ek (J/kg) = 0.5 × speed² (m/s)²

3: Power in watts: formulas

Power is the ability to do work in one second. A motor that applies 1 newton over 1 meter in 1 second delivers 1 watt; another that does the same work in 0.5 second is twice as powerful.

P (W) = energy (J) / time (s)

Power can also be expressed by a force and a speed:

P (W) = V (m/s) × F (N)

Finally, it can be expressed by a torque and a rotational speed. A torque of 1 N·m corresponds to a force of 1 newton on a lever of 1 meter, and 1 rad/s is about 9.55 revolutions per minute (60 / 2π):

P (W) = T (N·m) × ω (rad/s)

4: Volume flow rate and mass flow rate: formulas

The flow rate expresses how much matter moves in one second. We describe it either by its volume or by its mass.

The volume flow rate Qv is the volume moved in one second, in m³/s. For a fluid crossing a section S (m²) at speed V (m/s):

Qv (m³/s) = S (m²) × V (m/s)

The mass flow rate Qm is the mass moved in one second, in kg/s. We deduce it from the volume flow rate with the fluid density ρ (read "rho"). For water, ρ is about 1,000 kg/m³:

Qm (kg/s) = Qv (m³/s) × ρ (kg/m³)   and   Qv (m³/s) = Qm (kg/s) / ρ (kg/m³)

In a water head, potential energy turns into kinetic energy. The discharge speed is given by Torricelli's formula:

V = √(2 × g × H)

Example: a tank is filled with 1 meter of water and plugged at its base. The kinetic energy at the outlet is zero, since nothing flows. If we remove the plug, the water leaves at V = √(2 × 9.81 × 1) = 4.43 m/s.

To go further, we can see how momentum links these quantities to thrust with the definition of momentum, then how useful power compares with kinetic power in the propulsive efficiency of a propeller.