![Flight](https://upload.wikimedia.org/wikipedia/commons/0/03/Flight.rob.arp.750pix.jpg)
In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, disk loading or disc loading is the average
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
change across an
actuator disk, such as an airscrew. Airscrews with a relatively low disk loading are typically called rotors, including
helicopter
A helicopter is a type of rotorcraft in which lift and thrust are supplied by horizontally spinning rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forward, backward and laterally. These attribu ...
main rotor
A helicopter main rotor or rotor system is the combination of several rotary wings (rotor blades) with a control system, that generates the aerodynamic lift force that supports the weight of the helicopter, and the thrust that counteracts aerody ...
s and
tail rotor
The tail rotor is a smaller rotor mounted vertically or near-vertically at the tail of a traditional single-rotor helicopter, where it rotates to generate a propeller-like horizontal thrust in the same direction as the main rotor's rotation. Th ...
s;
propeller
A propeller (colloquially often called a screw if on a ship or an airscrew if on an aircraft) is a device with a rotating hub and radiating blades that are set at a pitch to form a helical spiral which, when rotated, exerts linear thrust upon ...
s typically have a higher disk loading.
The
V-22 Osprey
The Bell Boeing V-22 Osprey is an American multi-mission, tiltrotor military aircraft with both vertical takeoff and landing (VTOL) and short takeoff and landing (STOL) capabilities. It is designed to combine the functionality of a conventio ...
tiltrotor
A tiltrotor is an aircraft which generates lift and propulsion by way of one or more powered rotors (sometimes called ''proprotors'') mounted on rotating shafts or nacelles usually at the ends of a fixed wing. Almost all tiltrotors use a trans ...
aircraft has a high disk loading relative to a helicopter in the hover mode, but a relatively low disk loading in
fixed-wing
A fixed-wing aircraft is a heavier-than-air flying machine, such as an airplane, which is capable of flight using wings that generate lift caused by the aircraft's forward airspeed and the shape of the wings. Fixed-wing aircraft are distinct ...
mode compared to a
turboprop
A turboprop is a turbine engine that drives an aircraft propeller.
A turboprop consists of an intake, reduction gearbox, compressor, combustor, turbine, and a propelling nozzle. Air enters the intake and is compressed by the compressor. ...
aircraft.
Rotors
Disc loading of a
hovering helicopter
is the ratio of its weight to the
total main rotor disc area. It is determined by dividing
the total helicopter weight by the rotor disc area,
which is the area swept by the blades of a rotor. Disc
area can be found by using the span of one rotor blade
as the radius of a circle and then determining the area
the blades encompass during a complete rotation. When a helicopter is being maneuvered, its disc loading changes.
The higher the loading, the more power needed to
maintain rotor speed.
A low disc loading is a direct indicator of high lift thrust efficiency.
[Maisel, Martin D., Demo J. Giulianetti and Daniel C. Dugan]
NASA SP-2000-4517, "The History of the XV-15 Tilt Rotor Research Aircraft: From Concept to Flight" (PDF)
p2. ''NASA
The National Aeronautics and Space Administration (NASA ) is an independent agency of the US federal government responsible for the civil space program, aeronautics research, and space research.
NASA was established in 1958, succeedin ...
'', 2000. Accessed: 17 March 2012.
Increasing the weight of a helicopter increases disk loading. For a given weight, a helicopter with shorter rotors will have higher disk loading, and will require more engine power to hover. A low disk loading improves
autorotation
Autorotation is a state of flight in which the main rotor system of a helicopter or other rotary-wing aircraft turns by the action of air moving up through the rotor, as with an autogyro, rather than engine power driving the rotor. Bensen, Ig ...
performance in
rotorcraft
A rotorcraft or rotary-wing aircraft is a heavier-than-air aircraft with rotary wings or rotor blades, which generate lift by rotating around a vertical mast. Several rotor blades mounted on a single mast are referred to as a rotor. The Internat ...
.
[Leishman, J. Gordon.]
Development of the Autogiro : A Technical Perspective
" page 5. ''Hofstra University
Hofstra University is a private university in Hempstead, New York. It is Long Island's largest private university. Hofstra originated in 1935 as an extension of New York University (NYU) under the name Nassau College – Hofstra Memorial of New ...
'', New York, 2003.
Typically, an
autogyro
An autogyro (from Greek and , "self-turning"), also known as a ''gyroplane'', is a type of rotorcraft that uses an unpowered rotor in free autorotation to develop lift. Forward thrust is provided independently, by an engine-driven propeller. Whi ...
(or gyroplane) has a lower rotor disc loading than a helicopter, which provides a slower rate of descent in autorotation.
[
]
Propellers
In reciprocating and propeller engines, disk loading can be defined as the ratio between propeller-induced velocity and freestream velocity. Lower disk loading will increase efficiency, so it is generally desirable to have larger propellers from an efficiency standpoint. Maximum efficiency is reduced as disk loading is increased due to the rotating slipstream; using contra-rotating propellers
Aircraft equipped with contra-rotating propellers, also referred to as CRP, coaxial contra-rotating propellers, or high-speed propellers, apply the maximum power of usually a single piston or turboprop engine to drive a pair of coaxial propell ...
can alleviate this problem allowing high maximum efficiency even at relatively high disc loadings.
The Airbus A400M
The Airbus A400M AtlasNamed after the Greek mythological figure. is a European four-engine turboprop military transport aircraft. It was designed by Airbus Military (now Airbus Defence and Space) as a tactical airlifter with strategic capa ...
fixed-wing aircraft will have a very high disk loading on its propellers.
Theory
The ''momentum theory'' or ''disk actuator theory'' describes a mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
of an ideal actuator disk, developed by W.J.M. Rankine (1865), Alfred George Greenhill
Sir Alfred George Greenhill, FRS FRAeS (29 November 1847 in London – 10 February 1927 in London), was a British mathematician.
George Greenhill was educated at Christ's Hospital School and from there he went to St John's College, Cambridge ...
(1888) and R.E. Froude
Re or RE may refer to:
Geography
* Re, Norway, a former municipality in Vestfold county, Norway
* Re, Vestland, a village in Gloppen municipality, Vestland county, Norway
* Re, Piedmont, an Italian municipality
* Île de Ré, an island off the we ...
(1889). The helicopter
A helicopter is a type of rotorcraft in which lift and thrust are supplied by horizontally spinning rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forward, backward and laterally. These attribu ...
rotor
Rotor may refer to:
Science and technology
Engineering
* Rotor (electric), the non-stationary part of an alternator or electric motor, operating with a stationary element so called the stator
*Helicopter rotor, the rotary wing(s) of a rotorcraft ...
is modeled as an infinitely thin disc with an infinite number of blades that induce a constant pressure jump over the disk area and along the axis of rotation. For a helicopter that is hovering, the aerodynamic force is vertical and exactly balances the helicopter weight, with no lateral force.
The upward action on the helicopter results in a downward reaction on the air flowing through the rotor. The downward reaction produces a downward velocity on the air, increasing its kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
. This energy transfer from the rotor to the air is the induced power loss of the rotary wing, which is analogous to the lift-induced drag
In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings o ...
of a fixed-wing aircraft.
Conservation of linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
relates the induced velocity downstream in the far wake field to the rotor thrust per unit of mass flow
Mass flow, also known as mass transfer and bulk flow, is the movement of fluids down a pressure or temperature gradient,Moyes & Schulte (2008). Principles of Animal Physiology. Pearson Benjamin Cummings. San Francisco, California particularly in ...
. Conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
considers these parameters as well as the induced velocity at the rotor disk. Conservation of mass
In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...
relates the mass flow to the induced velocity. The momentum theory applied to a helicopter gives the relationship between induced power loss and rotor thrust, which can be used to analyze the performance of the aircraft. Viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
and compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
of the air, friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
al losses, and rotation of the slipstream in the wake are not considered.[
]
Momentum theory
For an actuator disk of area , with uniform induced velocity at the rotor disk, and with as the density of air
The density of air or atmospheric density, denoted '' ρ'', is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variation in atmospheric pressure, temperature a ...
, the mass flow rate
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot ('' ...
through the disk area is:
:
By conservation of mass, the mass flow rate is constant across the slipstream
A slipstream is a region behind a moving object in which a wake of fluid (typically air or mustard) is moving at velocities comparable to that of the moving fluid, relative to the ambient fluid through which the object is churning. The term sli ...
both upstream and downstream of the disk (regardless of velocity). Since the flow far upstream of a helicopter in a level hover is at rest, the starting velocity, momentum, and energy are zero. If the homogeneous slipstream
A slipstream is a region behind a moving object in which a wake of fluid (typically air or mustard) is moving at velocities comparable to that of the moving fluid, relative to the ambient fluid through which the object is churning. The term sli ...
far downstream of the disk has velocity , by conservation of momentum the total thrust developed over the disk is equal to the rate of change of momentum, which assuming zero starting velocity is:
:
By conservation of energy, the work done by the rotor must equal the energy change in the slipstream:
:
Substituting for and eliminating terms, we get:
:
So the velocity of the slipstream
A slipstream is a region behind a moving object in which a wake of fluid (typically air or mustard) is moving at velocities comparable to that of the moving fluid, relative to the ambient fluid through which the object is churning. The term sli ...
far downstream of the disk is twice the velocity at the disk, which is the same result for an elliptically loaded fixed wing predicted by lifting-line theory
The Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory.
The theory was expressed indepen ...
.
Bernoulli's principle
To compute the disk loading using Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematici ...
, we assume the pressure in the slipstream far downstream is equal to the starting pressure , which is equal to the atmospheric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, 7 ...
. From the starting point to the disk we have:
:
Between the disk and the distant wake, we have:
:
Combining equations, the disk loading is:
:
The total pressure in the distant wake is:
:
So the pressure change across the disk is equal to the disk loading. Above the disk the pressure change is:
:
Below the disk, the pressure change is:
:
The pressure along the slipstream is always falling downstream, except for the positive pressure jump across the disk.[
]
Power required
From the momentum theory, thrust is:
:
The induced velocity is:
:
Where is the disk loading as before, and the power required in hover (in the ideal case) is:
:
Therefore, the induced velocity can be expressed as:
:
So, the induced velocity is inversely proportional to the power loading
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 measuremen ...
.
Examples
See also
*Wing loading
In aerodynamics, wing loading is the total mass of an aircraft or flying animal divided by the area of its wing. The stalling speed of an aircraft in straight, level flight is partly determined by its wing loading. An aircraft or animal with a ...
References
{{DEFAULTSORT:Disk Loading
Aerospace engineering