Hydraulic head

In fluid dynamics, head is a concept that relates the energy in an 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 energy associated with the movement of the fluid, plus energy from static pressure in the fluid, plus energy from the height of the fluid relative to an arbitrary datum. Head is expressed in units of height such as meters or feet.

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, while, in fact, the term with pressure does not represent any type of energy (in the Bernoulli equation for an incompressible fluid this term represents work of pressure forces). Head is useful in specifying centrifugal pumps because their pumping characteristics tend to be independent of the fluid's density.

There are four types of head used to calculate the total head in and out of a pump:

1. Velocity head is due to the bulk motion of a fluid (kinetic energy). Its pressure head correspondent is the dynamic pressure.
2. Elevation head is due to the fluid's weight, the gravitational force acting on a column of fluid.
3. Pressure head is due to the static pressure, the internal molecular motion of a fluid that exerts a force on its container.
4. Resistance head (or friction head or Head Loss) is due to the frictional forces acting against a fluid's motion by the container.

After free falling through a height ${\displaystyle h}$ in a vacuum from an initial velocity of 0, a mass will have reached a speed

${\displaystyle v={\sqrt {{2g}{h}}}}$

where ${\displaystyle g}$ is the acceleration due to gravity. Rearranged as a head:

${\displaystyle h={\frac {v^{2}}{2g}}}$.

The term ${\displaystyle {\frac {v^{2}}{2g}}}$ 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.[1][2] The pressure head is the equivalent gauge 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 Principle for incompressible fluids, can be expressed as:

${\displaystyle h=\psi +z\,}$

where

${\displaystyle h}$ is the hydraulic head (Length in m or ft), also known as the piezometric head.

${\displaystyle \psi }$ 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

${\displaystyle z}$ 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:

${\displaystyle \psi ={\frac {P}{\gamma }}={\frac {P}{\rho g}}}$

where

${\displaystyle P}$ is the gauge pressure (Force per unit area, often Pa or psi),

${\displaystyle \gamma }$ is the unit weight of the liquid (Force per unit volume, typically N·m⁻³ or lbf/ft³),

${\displaystyle \rho }$ is the density of the liquid (Mass per unit volume, frequently kg·m⁻³), and

${\displaystyle g}$ is the gravitational acceleration (velocity change per unit time, often m·s⁻²)

The hydraulic gradient is a 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 can be used to determine whether a reach is gaining or losing energy. A dimensionless hydraulic gradient can be calculated between two points with known head values as:

${\displaystyle i={\frac {dh}{dl}}={\frac {h_{2}-h_{1}}{\mathrm {length} }}}$

where

${\displaystyle i}$ is the hydraulic gradient (dimensionless),

${\displaystyle dh}$ is the difference between two hydraulic heads (Length, usually in m or ft), and

${\displaystyle dl}$ 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 operator. This requires a hydraulic head field, which can be practically obtained only from numerical models, such as MODFLOW for groundwater or standard step or HEC-RAS for open channels. In Cartesian coordinates, this can be expressed as:

${\displaystyle \nabla h=\left({\frac {\partial h}{\partial x}},{\frac {\partial h}{\partial y}},{\frac {\partial h}{\partial z}}\right)={\frac {\partial h}{\partial x}}\mathbf {i} +{\frac {\partial h}{\partial y}}\mathbf {j} +{\frac {\partial h}{\partial z}}\mathbf {k} }$

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 and a tensor of hydraulic conductivity to determine the flux of water in three dimensions.

Relation between heads for a hydrostatic case and a downward flow case.

The distribution of hydraulic head through an aquifer determines where groundwater will flow. In a 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.

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.[3]

• Borda–Carnot equation
• Dynamic pressure
• Minor losses in pipe flow
• Total dynamic head
• Stage (hydrology)
• Head (hydrology)

• 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 further reading section