Power-to-weight ratio

Power-to-weight ratio (PWR, also called specific power, or power-to-mass ratio) is a calculation commonly applied to engines and mobile power sources to enable the comparison of one unit or design to another. Power-to-weight ratio is a measurement of actual performance of any engine or power source. It is also used as a measurement of performance of a vehicle as a whole, with the engine's power output being divided by the weight (or mass) of the vehicle, to give a metric that is independent of the vehicle's size. Power-to-weight is often quoted by manufacturers at the peak value, but the actual value may vary in use and variations will affect performance.

The inverse of power-to-weight, weight-to-power ratio (power loading) is a calculation commonly applied to aircraft, cars, and vehicles in general, to enable the comparison of one vehicle's performance to another. Power-to-weight ratio is equal to thrust per unit mass multiplied by the velocity of any vehicle.

Power-to-weight (specific power)

The power-to-weight ratio (specific power) is defined as the power generated by the engine(s) divided by the mass. In this context, the term "weight" can be considered a misnomer, as it colloquially refers to mass. In a zero-gravity (weightless) environment, the power-to-weight ratio would not be considered infinite.

A typical turbocharged V8 diesel engine might have an engine power of 250 kW (340 hp) and a mass of 380 kg (840 lb), giving it a power-to-weight ratio of 0.65 kW/kg (0.40 hp/lb).

Examples of high power-to-weight ratios can often be found in turbines. This is because of their ability to operate at very high speeds. For example, the Space Shuttle's main engines used turbopumps (machines consisting of a pump driven by a turbine engine) to feed the propellants (liquid oxygen and liquid hydrogen) into the engine's combustion chamber. The original liquid hydrogen turbopump is similar in size to an automobile engine (weighing approximately 352 kilograms (775 lb)) and produces 72,000 hp (54 MW) for a power-to-weight ratio of 153 kW/kg (93 hp/lb).

Physical interpretation

In classical mechanics, instantaneous power is the limiting value of the average work done per unit time as the time interval Δt approaches zero (i.e. the derivative with respect to time of the work done).

$P=\lim _{\Delta t\rightarrow 0}{\tfrac {\Delta W(t)}{\Delta t}}=\lim _{\Delta t\rightarrow 0}P_{\mathrm {avg} }={\frac {d}{dt}}W(t)\,$

The typically used metric unit of the power-to-weight ratio is ${\tfrac {\text{W}}{\text{kg}}}\;$ which equals ${\tfrac {{\text{m}}^{2}}{{\text{s}}^{3}}}\;$. This fact allows one to express the power-to-weight ratio purely by SI base units. A vehicle's power-to-weight ratio equals its acceleration times its velocity; so at twice the velocity, it experiences half the acceleration, all else being equal.

Propulsive power

If the work to be done is rectilinear motion of a body with constant mass $m\;$, whose center of mass is to be accelerated along a (possibly non-straight) line to a speed $|\mathbf {v} (t)|\;$ and angle $\phi \;$ with respect to the centre and radial of a gravitational field by an onboard powerplant, then the associated kinetic energy is

$E_{K}={\tfrac {1}{2}}m|\mathbf {v} (t)|^{2}$

where:

$m\;$ is mass of the body

$|\mathbf {v} (t)|\;$ is speed of the center of mass of the body, changing with time.

The work–energy principle states that the work done to the object over a period of time is equal to the difference in its total energy over that period of time, so the rate at which work is done is equal to the rate of change of the kinetic energy (in the absence of potential energy changes).

The work done from time t to time t + Δt along the path C is defined as the line integral $\int _{C}\mathbf {F} \cdot d\mathbf {x} =\int _{t}^{t+\Delta t}\mathbf {F} \cdot \mathbf {v} (t)dt$, so the fundamental theorem of calculus has that power is given by $\mathbf {F} (t)\cdot \mathbf {v} (t)=m\mathbf {a} (t)\cdot \mathbf {v} (t)=\mathbf {\tau } (t)\cdot \mathbf {\omega } (t)$.

where:

$\mathbf {a} (t)={\frac {d}{dt}}\mathbf {v} (t)\;$ is acceleration of the center of mass of the body, changing with time.

$\mathbf {F} (t)\;$ is linear force – or thrust – applied upon the center of mass of the body, changing with time.

$\mathbf {v} (t)\;$ is velocity of the center of mass of the body, changing with time.

$\mathbf {\tau } (t)\;$ is torque applied upon the center of mass of the body, changing with time.

$\mathbf {\omega } (t)\;$ is angular velocity of the center of mass of the body, changing with time.

In propulsion, power is only delivered if the powerplant is in motion, and is transmitted to cause the body to be in motion. It is typically assumed here that mechanical transmission allows the powerplant to operate at peak output power. This assumption allows engine tuning to trade power band width and engine mass for transmission complexity and mass. Electric motors do not suffer from this tradeoff, instead trading their high torque for traction at low speed. The power advantage or power-to-weight ratio is then

${\mbox{P-to-W}}=|\mathbf {a} (t)||\mathbf {v} (t)|\;$

where:

$|\mathbf {v} (t)|\;$ is linear speed of the center of mass of the body.

Engine power

The useful power of an engine with shaft power output can be calculated using a dynamometer to measure torque and rotational speed, with maximum power reached when torque multiplied by rotational speed is a maximum. For jet engines the useful power is equal to the flight speed of the aircraft multiplied by the force, known as net thrust, required to make it go at that speed. It is used when calculating propulsive efficiency.

See also

• Energy density – Energy per volume
• Engine power – Power output of an engine
• Propulsive efficiency – Measure of efficiency in aerospace engineering
• Specific output – Measure of internal combustion engine performance
• Thrust-to-weight ratio – Dimensionless ratio of thrust to weight of a jet or propeller engine
• Vehicular metrics – Metrics that denote the relative capabilities of various vehicles
• von Kármán–Gabrielli diagram – Diagram which compares the efficiency of transportation methods