Power-to-weight ratio

The inverse of power-to-weight, the weight-to-power ratio (power loading), is a calculation commonly applied to aircraft, cars, and other vehicles to enable 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" is 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_{Δ t \to 0} \frac{Δ W (t)}{Δ t} = lim_{Δ t \to 0} P_{a v g} = \frac{d}{d t} W (t)

The typically used metric unit of the power-to-weight ratio is $\frac{W}{kg}$ which equals $\frac{m^{2}}{s^{3}}$ . 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 $| v (t) |$ and angle $ϕ$ with respect to the centre and radial of a gravitational field by an onboard powerplant, then the associated kinetic energy is

E_{K} = \frac{1}{2} m | v (t) |^{2}

where:

m

is mass of the body

| 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 $\int_{C} F \cdot d x = \int_{t}^{t + Δ t} F \cdot v (t) d t$ , so the fundamental theorem of calculus has that power is given by $F (t) \cdot v (t) = m a (t) \cdot v (t) = τ (t) \cdot ω (t)$ .

where:

a (t) = \frac{d}{d t} v (t)

is acceleration of the center of mass of the body, changing with time.

F (t)

is linear force – or thrust – applied upon the center of mass of the body, changing with time.

v (t)

is velocity of the center of mass of the body, changing with time.

τ (t)

is torque applied upon the center of mass of the body, changing with time.

ω (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 enables the power plant to operate at peak power output. 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

P-to-W = | a (t) | | v (t) |

where:

| 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.

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Examples

Engines

Heat engines and heat pumps

Thermal energy is made up from molecular kinetic energy and latent phase energy. Heat engines can convert thermal energy in the form of a temperature gradient between a hot source and a cold sink into other desirable mechanical work. Heat pumps take mechanical work to regenerate thermal energy in a temperature gradient. Standard definitions should be used when interpreting the transfer of a jet or rocket engine's propulsive power to its vehicle.

Electric motors and electromotive generators

An electric motor uses electrical energy to provide mechanical work, usually through the interaction of a magnetic field and current-carrying conductors. By the interaction of mechanical work on an electrical conductor in a magnetic field, electrical energy can be generated.

Fluid engines and fluid pumps

Fluids (liquid and gas) can be used to transmit and/or store energy using pressure and other fluid properties. Hydraulic (liquid) and pneumatic (gas) engines convert fluid pressure into other desirable mechanical or electrical work. Fluid pumps convert mechanical or electrical work into movement or pressure changes of a fluid, or storage in a pressure vessel.

Thermoelectric generators and electrothermal actuators

A variety of effects can be harnessed to produce thermoelectricity, thermionic emission, pyroelectricity and piezoelectricity. The Electrical resistance and ferromagnetism of materials can be harnessed to convert electrical current into thermoacoustic energy.

Electrochemical (galvanic) and electrostatic cell systems

(Closed cell) batteries

All electrochemical cells deliver a changing voltage as their chemistry shifts from "charged" to "discharged". A battery's manufacturer typically specifies a nominal output voltage and a cutoff voltage. The output voltage falls to the cutoff voltage when the battery becomes "discharged". The nominal output voltage is always less than the open-circuit voltage produced when the battery is "charged". A battery's temperature can affect its power output, with lower temperatures reducing it. Total energy delivered over a single charge cycle is affected by both the battery temperature and the power delivered. If the temperature lowers or the power demand increases, the total energy delivered at the point of "discharge" is also reduced.

Battery discharge profiles are often described in terms of a factor of battery capacity. For example, a battery with a nominal capacity quoted in ampere-hours (Ah) at a C/10 rated discharge current (derived in amperes) may safely provide a higher discharge current – and therefore higher power-to-weight ratio – but only with a lower energy capacity. Power-to-weight ratio for batteries is therefore less meaningful without reference to the corresponding energy-to-weight ratio and cell temperature. This relationship is known as Peukert's law.

Electrostatic, electrolytic, and electrochemical capacitors

Capacitors store electric charge on two electrodes separated by an electric field in a semi-insulating (dielectric) medium. Electrostatic capacitors feature planar electrodes onto which electric charge accumulates. Electrolytic capacitors use a liquid electrolyte as one of the electrodes, and the electric double-layer effect at the dielectric-electrolyte boundary to increase the amount of charge stored per unit volume. Electric double-layer capacitors extend both electrodes with a nanoporous material such as activated carbon to significantly increase the surface area upon which electric charge can accumulate, reducing the dielectric medium to nanopores and a very thin high permittivity separator.

While capacitors tend not to be as temperature-sensitive as batteries, they are significantly capacity-constrained and, lacking the strength of chemical bonds, suffer from self-discharge. The power-to-weight ratio of capacitors is usually higher than that of batteries because the charge carriers in capacitors are smaller (electrons rather than ions); however, the energy-to-weight ratio is usually lower.

Fuel cell stacks and flow cell batteries

Fuel cells and flow cells, although perhaps using similar chemistry to batteries, do not contain the energy storage medium or fuel. With a continuous flow of fuel and oxidant, available fuel cells and flow cells continue to convert the energy storage medium into electric energy and waste products. Fuel cells contain a fixed electrolyte, whereas flow cells require a continuous flow of electrolyte. Flow cells typically have the fuel dissolved in the electrolyte.

Photovoltaics

Vehicles

Power-to-weight ratios for vehicles are usually calculated using curb weight (for cars) or wet weight (for motorcycles), i.e., excluding the driver and any cargo. This could be slightly misleading, especially for motorcycles, where the driver might weigh 1/3 to 1/2 of the vehicle's weight. In the sport of competitive cycling, an athlete's performance is increasingly being expressed in VAMs and thus as a power-to-weight ratio in W/kg. This can be measured using a bicycle power meter or calculated from the road's incline and the rider's time to ascend it.

Locomotives

A locomotive generally must be heavy to develop enough adhesion on the rails to start a train. As the coefficient of friction between steel wheels and rails seldom exceeds 0.25, improving a locomotive's power-to-weight ratio is often counterproductive. However, the choice of power transmission system, such as a variable-frequency drive versus a direct-current drive, may enable a higher power-to-weight ratio by better managing propulsion power.

Utility and practical vehicles

Most vehicles are designed to meet passenger comfort and cargo-carrying requirements. Vehicle designs trade off power-to-weight ratio to increase comfort, cargo space, fuel economy, emissions control, energy security, and endurance. Reduced drag and lower rolling resistance in a vehicle design can enable increased cargo space without increasing the (zero-cargo) power-to-weight ratio. This increases the vehicle's role flexibility. Energy security considerations can trade off power (typically decreased) and weight (typically increased), and therefore power-to-weight ratio, for fuel flexibility or drive-train hybridisation. Some utility and practical vehicle variants, such as hot hatches and sports-utility vehicles reconfigure power (typically increased) and weight to provide the perception of sports car like performance or for other psychological benefit.

Notable low ratio

Common power

Performance luxury, roadsters and mild sports

Increased engine performance is a consideration, as are other features associated with luxury vehicles. Longitudinal engines are common. Bodies vary from hot hatches, sedans (saloons), coupés, convertibles and roadsters. Mid-range dual-sport and cruiser motorcycles tend to have similar power-to-weight ratios.

Sports vehicles

Power-to-weight ratio is an important vehicle characteristic that affects the acceleration of sports vehicles.

Vehicles in order of power to weight ratio

Aircraft

Propeller aircraft depend on high power-to-weight ratios to generate sufficient thrust to achieve sustained flight, and then for speed.

Thrust-to-weight ratio

Jet aircraft produce thrust directly.

Human

Power-to-weight ratio is important in cycling, since it determines acceleration and speed during hill climbs. Since a cyclist's power-to-weight output decreases with fatigue, it is normally discussed in relation to the length of time that they maintain that power. A professional cyclist can produce over 20 W/kg (0.012 hp/lb) as a five-second maximum.

Power-to-weight (specific power)

Physical interpretation

Propulsive power

Engine power

Examples

Engines

Heat engines and heat pumps

Electric motors and electromotive generators

Fluid engines and fluid pumps

Thermoelectric generators and electrothermal actuators

Electrochemical (galvanic) and electrostatic cell systems

(Closed cell) batteries

Electrostatic, electrolytic, and electrochemical capacitors

Fuel cell stacks and flow cell batteries

Photovoltaics

Vehicles

Locomotives

Utility and practical vehicles

Notable low ratio

Common power

Performance luxury, roadsters and mild sports

Sports vehicles

Vehicles in order of power to weight ratio

Aircraft

Thrust-to-weight ratio

Human

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