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Hydraulic head or piezometric head is a specific measurement of
liquid pressure above a
vertical datum
In geodesy, surveying, hydrography and navigation, vertical datum or altimetric datum, is a reference coordinate surface used for vertical positions, such as the elevations of Earth-bound features (terrain, bathymetry, water level, and built str ...
.
[, 410 pages. See pp. 43–44.][, 650 pages. See p. 22.]
It is usually measured as a liquid surface elevation, expressed in units of length, at the entrance (or bottom) of a
piezometer. In an
aquifer
An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials ( gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their characte ...
, it can be calculated from the depth to water in a piezometric well (a specialized
water well
A well is an excavation or structure created in the ground by digging, driving, or drilling to access liquid resources, usually water. The oldest and most common kind of well is a water well, to access groundwater in underground aquifers. T ...
), and given information of the piezometer's elevation and screen depth. Hydraulic head can similarly be measured in a column of water using a standpipe piezometer by measuring the height of the water surface in the tube relative to a common datum. The hydraulic head can be used to determine a ''hydraulic gradient'' between two or more points.
"Head" in fluid dynamics
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 ...
, ''head'' is a concept that relates the
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
in an
incompressible
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
fluid to the height of an equivalent static column of that fluid. From
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 ...
, the total energy at a given point in a fluid is the energy associated with the movement of the fluid, plus energy from
static pressure
In fluid mechanics the term static pressure has several uses:
* In the design and operation of aircraft, ''static pressure'' is the air pressure in the aircraft's static pressure system.
* In fluid dynamics, many authors use the term ''static pres ...
in the fluid, plus energy from the height of the fluid relative to an arbitrary
datum
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted. ...
. Head is expressed in units of distance such as meters or feet. The force per unit volume on a fluid in a gravitational field is equal to ''ρg'' where ''ρ'' is the density of the fluid, and ''g'' is the
gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodie ...
. On Earth, additional height of fresh water adds a static pressure of about 9.8 kPa per meter (0.098 bar/m) or 0.433 psi per foot of water column height.
The ''static head'' of a pump is the maximum height (pressure) it can deliver. The capability of the pump at a certain RPM can be read from its Q-H curve (flow vs. height).
A common misconception is that the head equals the fluid's energy per unit
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
, while, in fact, the term with pressure does not represent any type of energy (in the
Bernoulli equation
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 ...
for an incompressible fluid this term represents
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal t ...
of pressure forces). Head is useful in specifying
centrifugal pump
Centrifugal pumps are used to transport fluids by the conversion of rotational kinetic energy to the hydrodynamic energy of the fluid flow. The rotational energy typically comes from an engine or electric motor. They are a sub-class of dynamic ...
s because their pumping characteristics tend to be independent of the fluid's density.
There are generally four types of head:
#''
Velocity head'' is due to the bulk motion of a fluid (
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 ...
).
Note that
is equal to the
dynamic pressure
In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5
:q = \frac\rho\, u^2
where (in SI units):
* is the dynamic pressure in pascals ( ...
for
irrotational flow
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...
.
#''Elevation head'' is due to the fluid's weight, the
gravitational force
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
acting on a column of fluid. The elevation head is simply the elevation (''h'') of the fluid above an arbitrarily designated zero point:
#''
Pressure head
In fluid mechanics, pressure head is the height of a liquid column that corresponds to a particular pressure exerted by the liquid column on the base of its container. It may also be called static pressure head or simply static head (but not ''sta ...
'' is due to the
static pressure
In fluid mechanics the term static pressure has several uses:
* In the design and operation of aircraft, ''static pressure'' is the air pressure in the aircraft's static pressure system.
* In fluid dynamics, many authors use the term ''static pres ...
, the internal molecular motion of a fluid that exerts a force on its container. It is equal to the pressure divided by the force/volume of the fluid in a gravitational field:
#''Resistance head'' (or ''friction head'' or
Head Loss) is due to the frictional forces acting against a fluid's motion by the container. For a continuous medium, this is described by
Darcy's law
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...
which relates volume flow rate (q) to the gradient of the hydraulic head through the
hydraulic conductivity
Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...
''K'':
while in a piped system head losses are described by the
Hagen–Poiseuille equation
In nonideal fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar fl ...
and the
Bernoulli Equation
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 ...
.
Components of hydraulic head
After
free fall
In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on ...
ing through a height
in a
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
from an initial velocity of 0, a mass will have reached a
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (ma ...
where
is the acceleration due to gravity. Rearranged as a ''head'':
The term
is called the ''velocity head'', expressed as a length measurement. In a flowing fluid, it represents the energy of the fluid due to its bulk motion.
The total hydraulic head of a fluid is composed of ''pressure head'' and ''elevation head''.
[ The pressure head is the equivalent ]gauge
Gauge ( or ) may refer to:
Measurement
* Gauge (instrument), any of a variety of measuring instruments
* Gauge (firearms)
* Wire gauge, a measure of the size of a wire
** American wire gauge, a common measure of nonferrous wire diameter, ...
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 and e ...
of a column of water at the base of the piezometer, and the elevation head is the relative potential energy in terms of an elevation. The ''head equation'', a simplified form of the Bernoulli 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 ...
for incompressible fluids, can be expressed as:
where
* is the hydraulic head ( Length in m or ft), also known as the piezometric head.
* is the pressure head
In fluid mechanics, pressure head is the height of a liquid column that corresponds to a particular pressure exerted by the liquid column on the base of its container. It may also be called static pressure head or simply static head (but not ''sta ...
, in terms of the elevation difference of the water column relative to the piezometer bottom ( Length in m or ft), and
* is the elevation at the piezometer bottom ( Length in m or ft)
In an example with a 400 m deep piezometer, with an elevation of 1000 m, and a depth to water of 100 m: ''z'' = 600 m, ''ψ'' = 300 m, and ''h'' = 900 m.
The pressure head can be expressed as:
where
is the gauge pressure (Force per unit area, often Pa or psi),
* is the unit weight
The specific weight, also known as the unit weight, is the weight per unit volume of a material.
A commonly used value is the specific weight of water on Earth at , which is .National Council of Examiners for Engineering and Surveying (2005). ''Fu ...
of the liquid (Force per unit volume, typically N·m−3 or lbf
The pound of force or pound-force (symbol: lbf, sometimes lbf,) is a unit of force used in some systems of measurement, including English Engineering units and the foot–pound–second system.
Pound-force should not be confused with pound-m ...
/ft3),
* is the density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of the liquid (Mass per unit volume, frequently kg·m−3), and
* is the gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodie ...
(velocity change per unit time, often m·s−2)
Fresh water head
The pressure head is dependent on the density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of water, which can vary depending on both the temperature and chemical composition ( salinity, in particular). This means that the hydraulic head calculation is dependent on the density of the water within the piezometer. If one or more hydraulic head measurements are to be compared, they need to be standardized, usually to their ''fresh water head'', which can be calculated as:
:
where
* is the fresh water head (Length, measured in m or ft), and
* is the density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
of fresh water (Mass per unit volume, typically in kg·m−3)
Hydraulic gradient
The hydraulic gradient is a vector gradient between two or more hydraulic head measurements over the length of the flow path. For groundwater
Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidated ...
, it is also called the Darcy slope, since it determines the quantity of a Darcy flux or discharge. It also has applications in open-channel flow
In fluid mechanics and hydraulics, open-channel flow is a type of liquid flow within a conduit with a free surface, known as a channel. The other type of flow within a conduit is pipe flow. These two types of flow are similar in many ways but di ...
where it is also known as ''stream gradient Stream gradient (or stream slope) is the grade (slope), grade (or slope) of a stream measured by the ratio of drop in elevation per unit horizontal distance, usually expressed as metre, meters per kilometre, kilometer or Foot (length), feet per mile ...
'' and can be used to determine whether a reach is gaining or losing energy. A dimensionless
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
hydraulic gradient can be calculated between two points with known head values as:
where
* is the hydraulic gradient (dimensionless),
* is the difference between two hydraulic heads (length, usually in m or ft), and
* is the flow path length between the two piezometers (length, usually in m or ft)
The hydraulic gradient can be expressed in vector notation, using the del
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
operator. This requires a hydraulic head field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, which can be practically obtained only from numerical models, such as MODFLOW
MODFLOW is the U.S. Geological Survey modular finite-difference flow model, which is a computer code that solves the groundwater flow equation. The program is used by hydrogeologists to simulate the flow of groundwater through aquifers. The so ...
for groundwater or standard step or HEC-RAS
HEC-RAS is a computer program that models the hydraulics of water flow through natural rivers and other channels. Prior to the 2016 update to Version 5.0, the program was one-dimensional, meaning that there is no direct modeling of the hydraulic ...
for open channels. In Cartesian coordinates, this can be expressed as:
This vector describes the direction of the groundwater flow, where negative values indicate flow along the dimension, and zero indicates 'no flow'. As with any other example in physics, energy must flow from high to low, which is why the flow is in the negative gradient. This vector can be used in conjunction with Darcy's law
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...
and a tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
of hydraulic conductivity
Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...
to determine the flux of water in three dimensions.
Hydraulic head in groundwater
The distribution of hydraulic head through an aquifer
An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials ( gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their characte ...
determines where groundwater will flow. In a hydrostatic
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imme ...
example (first figure), where the hydraulic head is constant, there is no flow. However, if there is a difference in hydraulic head from the top to bottom due to draining from the bottom (second figure), the water will flow downward, due to the difference in head, also called the ''hydraulic gradient''.
Atmospheric pressure
Even though it is convention to use gauge pressure
Pressure measurement is the measurement of an applied force by a fluid ( liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pre ...
in the calculation of hydraulic head, it is more correct to use total pressure (gauge pressure + 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, ...
), since this is truly what drives groundwater flow. Often detailed observations of barometric 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 ...
are not available at each well
A well is an excavation or structure created in the ground by digging, driving, or drilling to access liquid resources, usually water. The oldest and most common kind of well is a water well, to access groundwater in underground aquifers. The ...
through time, so this is often disregarded (contributing to large errors at locations where hydraulic gradients are low or the angle between wells is acute.)
The effects of changes in 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, ...
upon water levels observed in wells has been known for many years. The effect is a direct one, an increase in atmospheric pressure is an increase in load on the water in the aquifer, which increases the depth to water (lowers the water level elevation). Pascal first qualitatively observed these effects in the 17th century, and they were more rigorously described by the soil physicist Edgar Buckingham
Edgar Buckingham (July 8, 1867 in Philadelphia, Pennsylvania – April 29, 1940 in Washington DC) was an American physicist.
He graduated from Harvard University with a bachelor's degree in physics in 1887. He did graduate work at Strasbourg ...
(working for the United States Department of Agriculture
The United States Department of Agriculture (USDA) is the federal executive department responsible for developing and executing federal laws related to farming, forestry, rural economic development, and food. It aims to meet the needs of com ...
(USDA)) using air flow models in 1907.
Head loss
In any real moving fluid, energy is dissipated due to 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 ...
; turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
dissipates even more energy for high Reynolds number flows. This dissipation, called ''head loss'', is divided into two main categories, "major losses" associated with energy loss per length of pipe, and "minor losses" associated with bends, fittings, valves, etc. The most common equation used to calculate major head losses is the Darcy–Weisbach equation
In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation ...
. Older, more empirical approaches are the Hazen–Williams equation The Hazen–Williams equation is an empirical relationship which relates the flow of water in a pipe with the physical properties of the pipe and the pressure drop caused by friction. It is used in the design of water pipe systems such as fire spr ...
and the Prony equation.
For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses. In design, minor losses are usually estimated from tables using coefficients or a simpler and less accurate reduction of minor losses to equivalent length of pipe, a method often used for shortcut calculations of pneumatic conveying lines pressure drop.
See also
* Borda–Carnot equation
* Dynamic pressure
In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5
:q = \frac\rho\, u^2
where (in SI units):
* is the dynamic pressure in pascals ( ...
* Minor losses in pipe flow
* Total dynamic head
In fluid dynamics, total dynamic head (TDH) is the total equivalent height that a fluid is to be pumped, taking into account friction losses in the pipe.
: {\rm h_{total} = \frac{P_2-P_1}{\rho g} + \frac{{v_2}^2-{v_1}^2}{2g
: TDH = Static Heig ...
* Stage (hydrology)
* Head (hydrology)
In hydrology, the head is the point on a watercourse up to which it has been artificially broadened and/or raised by an impoundment. Above the head of the reservoir natural conditions prevail; below it the water level above the riverbed has be ...
Notes
References
* Bear, J. 1972. ''Dynamics of Fluids in Porous Media'', Dover. .
* for other references which discuss hydraulic head in the context of hydrogeology, see that page's further reading section
Aquifers
Water
Hydrology
Fluid dynamics
Water wells
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