View source for Hydraulic head

{{short description|Specific measurement of liquid pressure above a vertical datum}}
{{distinguish|Head (hydrology)}}
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{{More footnotes|date=April 2020}}
{{Lead too short|date=October 2024}}
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[[File:Hydraulic head.PNG|thumb|upright=1.3|Available difference in hydraulic head across a [[hydroelectric dam]], before [[#Head loss|head losses]] due to turbines, wall friction and turbulence]]
[[File:Headpressure.GIF|thumb|Fluid flows from the tank at the top to the basin at the bottom under the pressure of the hydraulic head.]]
[[File:USGS-Armenia, March 2, 2016, Ararat Valley 05.jpg|thumb|Measuring hydraulic head in an [[artesian aquifer]], where the water level is above the ground surface]]

'''Hydraulic head''' or '''piezometric head''' is a measurement related to [[Fluid pressure#Hydrostatic pressure|liquid pressure]] (normalized by [[specific weight]]) and the liquid elevation above a [[vertical datum]].<ref name=Mulley_43_44>{{citation | title=Flow of Industrial Fluids: Theory and Equations | first=Raymond | last=Mulley | publisher=CRC Press | year=2004 | isbn=978-0849327674 }}, 410 pages. See pp. 43–44.</ref><ref name=Chanson_22>{{citation | title=Hydraulics of Open Channel Flow: An Introduction | first=Hubert | last=Chanson | author-link=Hubert Chanson | publisher=Butterworth–Heinemann | year=2004 | isbn=978-0750659789 }}, 650 pages. See p. 22, eq.3.2a.</ref>
It is usually measured as an equivalent liquid surface elevation, expressed in units of length, at the entrance (or bottom) of a [[piezometer]]. In an [[aquifer]], it can be calculated from the depth to water in a piezometric well (a specialized [[water well]]), 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.

==Definition==
In [[fluid dynamics]], ''head'' is a concept that relates the [[energy]] in an [[Incompressible flow|incompressible]] fluid to the height of an equivalent static column of that fluid. From [[Bernoulli's principle]], the total energy at a given point in a fluid is the kinetic energy associated with the speed of flow of the fluid, plus energy from [[static pressure]] in the fluid, plus energy from the height of the fluid relative to an arbitrary [[Datum (geodesy)|datum]].<ref>Streeter, Victor L. (1958) ''Fluid Mechanics'', Section 3.7 (Fourth edition) McGraw-Hill </ref> 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 ''&rho;g'' where ''&rho;'' is the density of the fluid, and ''g'' is the [[gravitational acceleration]]. 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).

Head is useful in specifying [[centrifugal pump]]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 ([[kinetic energy]]) of a fluid.<math display="block">h_v=\tfrac{1}{2}\rho v^2/\rho g = \frac{1}{2}\frac{v^2}{g} </math> Note that <math>\rho g h_v</math> is equal to the [[dynamic pressure]] for [[irrotational flow]].
#''Elevation head'' is due to the fluid's weight, the [[gravitational force]] acting on a column of fluid. The elevation head is simply the elevation (''h'') of the fluid above an arbitrarily designated zero point: <math display="block">h_e = \rho g h/\rho g</math>
#''[[Pressure head]]'' is due to the [[static pressure]], 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: <math display="block">h_p = p/\rho g</math>
#''Resistance head'' (or ''friction head'' or [[Hydraulic head#Head loss|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]] which relates volume flow rate ('''q''') to the gradient of the hydraulic head through the [[hydraulic conductivity]] ''K'': <math display="block">\mathbf{q} = -K\nabla h</math> while in a piped system head losses are described by the [[Hagen–Poiseuille equation]] and [[Bernoulli’s equation]].

==Components==
After [[free fall]]ing through a height <math>h</math> in a [[vacuum]] from an initial velocity of 0, a mass will have reached a [[speed]]
<math display="block">v=\sqrt{{2 g}{h}}</math>
where <math>g</math> is the acceleration due to gravity. Rearranged as a ''head'':
<math display="block">h = \frac{v^2}{2 g}.</math>

The term <math>\frac{v^{2}}{2 g}</math> 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''.<ref name=Mulley_43_44/><ref name=Chanson_22/> The pressure head is the equivalent [[Gauge (engineering)|gauge]] [[Pressure measurement|pressure]] 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's principle|Bernoulli principle]] for incompressible fluids, can be expressed as:
<math display="block">h = \psi + z </math>
where
*<math>h</math> is the hydraulic head ([[Length]] in m or ft), also known as the piezometric head.
*<math>\psi</math> is the [[pressure head]], in terms of the elevation difference of the water column relative to the piezometer bottom ([[Length]] in m or ft), and
*<math>z</math> 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:
<math display="block">\psi = \frac{P}{\gamma} = \frac{P}{\rho g}</math>
where
<math>P</math> is the gauge pressure (Force per unit area, often Pa or psi),
*<math>\gamma</math> is the [[unit weight]] of the liquid (Force per unit volume, typically N·m<sup>−3</sup> or [[Pound-force|lbf]]/ft<sup>3</sup>),
*<math>\rho</math> is the [[density]] of the liquid (Mass per unit volume, frequently kg·m<sup>−3</sup>), and
*<math>g</math> is the [[gravitational acceleration]] (velocity change per unit time, often m·s<sup>−2</sup>)

===Fresh water head===
The pressure head is dependent on the [[density]] 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:
:<math display="block">h_\mathrm{fw} = \psi \frac{\rho}{\rho_\mathrm{fw}} + z</math>
where
*<math>h_\mathrm{fw} </math> is the fresh water head (Length, measured in m or ft), and
*<math>\rho_\mathrm{fw} </math> is the [[density]] of fresh water (Mass per unit volume, typically in kg·m<sup>−3</sup>)

==Hydraulic gradient==
The '''hydraulic gradient''' is a [[gradient|vector gradient]] between two or more hydraulic head measurements over the length of the flow path. For [[groundwater]], 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]] where it is also known as ''[[stream gradient]]'' and can be used to determine whether a reach is gaining or losing energy. 

The hydraulic gradient norm <math>i</math>, a [[dimensionless]] quantity (of [[Kind of quantity|kind]] length per length), can be calculated between two points with known head values as a ratio:
<math display="block">i = \frac{dh}{dl}</math>
where
*<math>dh=h_2 - h_1</math> is the difference between two hydraulic heads (length, usually in m or ft), and
*<math>dl</math> is the flow path length between the two piezometers (length, usually in m or ft)

The hydraulic gradient vector <math display="inline">\nabla h</math> can be formulated using the [[del]] [[Differential operator|operator]] for a [[spatial gradient]]. This requires a hydraulic head [[scalar field|field]], which can be practically obtained only from numerical models, such as [[MODFLOW]] for groundwater or [[standard step method|standard step]] or [[HEC-RAS]] for open channels. In [[Cartesian coordinates]], this can be expressed as:
<math display="block">\nabla h = \left(
{\frac{\partial h}{\partial x}},
{\frac{\partial h}{\partial y}},
{\frac{\partial h}{\partial z}}
\right) = 
{\frac{\partial h}{\partial x}}\mathbf{i} + 
{\frac{\partial h}{\partial y}}\mathbf{j} + 
{\frac{\partial h}{\partial z}}\mathbf{k}</math>
This vector describes both the magnitude and 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]] and a [[tensor]] of [[hydraulic conductivity]] to determine the flux of water in three dimensions.

==In groundwater==
{| class="wikitable" align="right"
|+Relation between heads for a ''hydrostatic'' case and a ''downward flow'' case.
|-
|[[Image:relation between heads hydrostatic.svg|200px]]
|-
|[[Image:relation between heads flowing.svg|200px]]
|}

The distribution of hydraulic head through an [[aquifer]] determines where groundwater will flow. In a [[Fluid pressure|hydrostatic]] 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]] in the calculation of hydraulic head, it is more correct to use [[absolute pressure]] (gauge pressure + [[atmospheric pressure]]), since this is truly what drives groundwater flow. Often detailed observations of [[barometric pressure]] are not available at each [[Water well|well]] 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]] 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). [[Blaise Pascal|Pascal]] first qualitatively observed these effects in the 17th century, and they were more rigorously described by the [[soil physics|soil physicist]] [[Edgar Buckingham]] (working for the [[United States Department of Agriculture]] (USDA)) using air flow models in 1907.

==Head loss==
In any real moving fluid, energy is dissipated due to [[friction]]; [[turbulence]] 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]]. Older, more empirical approaches are the [[Hazen–Williams equation]] 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.<ref>{{Cite web|url=https://powderprocess.net/Pneumatic_Transport/Pipe_Equivalent_Length.html|title = Pipe equivalent length (Pneumatic conveying)}}</ref>

==See also==
* [[Borda–Carnot equation]]
* [[Dynamic pressure]]
* [[Minor losses in pipe flow]]
* [[Total dynamic head]]
* [[Stage (hydrology)]]
* [[Head (hydrology)]]
* [[Hydraulic accumulator]]

==Notes==
<references/>

==References==
* Bear, J. 1972. ''Dynamics of Fluids in Porous Media'', Dover. {{ISBN|0-486-65675-6}}.
* for other references which discuss hydraulic head in the context of hydrogeology, see that page's [[Hydrogeology#Further_reading|further reading section]]

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[[Category:Aquifers]]
[[Category:Water]]
[[Category:Hydrology]]
[[Category:Fluid dynamics]]
[[Category:Water wells]]

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[[Category:Pressure]]