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 static head pressure). Mathematically this is expressed as:
ψ = p γ = p ρ g {\displaystyle \psi ={\frac {p}{\gamma }}={\frac {p}{\rho \,g}}}
where
ψ {\displaystyle \psi } Psi (Greek) is pressure head (which is actually a length, typically in units of meters or centimetres of water)
p {\displaystyle p} is fluid pressure (i.e. force per unit area, typically expressed in pascals)
γ {\displaystyle \gamma } is the specific weight (i.e. force per unit volume, typically expressed in N/m3 units)
ρ {\displaystyle \rho } is the density of the fluid (i.e. mass per unit volume, typically expressed in kg/m3)
g {\displaystyle g} is acceleration due to gravity (i.e. rate of change of velocity, expressed in m/s2). Note that in this equation, the pressure term may be gauge pressure or absolute pressure, depending on the design of the container and whether it is open to the ambient air or sealed without air.
Head equation Pressure head is a component of hydraulic head, in which it is combined with elevation head. When considering dynamic (flowing) systems, there is a third term needed: velocity head. Thus, the three terms of velocity head, elevation head, and pressure head appear in the head equation derived from the Bernoulli equation for incompressible fluids:
h v + z elevation + ψ = C {\displaystyle h_{v}+z_{\text{elevation}}+\psi =C\,}
where
h v {\displaystyle h_{v}} is velocity head,
z elevation {\displaystyle z_{\text{elevation}}} is elevation head,
ψ {\displaystyle \psi } is pressure head, and
C {\displaystyle C} is a constant for the system
Practical uses for pressure head
Fluid flow is measured with a wide variety of instruments. The venturi meter in the diagram on the left shows two columns of a measurement fluid at different heights. The height of each column of fluid is proportional to the pressure of the fluid. To demonstrate a classical measurement of pressure head, we could hypothetically replace the working fluid with another fluid having different physical properties. For example, if the original fluid was water and we replaced it with mercury at the same pressure, we would expect to see a rather different value for pressure head. In fact the specific weight of water is 9.8 kN/m3 and the specific weight of mercury is 133 kN/m3. So, for any particular measurement of pressure head, the height of a column of water will be about [133/9.8 = 13.6] 13.6 times taller than a column of mercury would be. So if a water column meter reads "13.6 cm H2O", then an equivalent measurement is "1.00 cm Hg". This example demonstrates why there is some confusion surrounding pressure head and its relationship to pressure. Scientists frequently use columns of water (or mercury) to measure pressure (manometric pressure measurement), since for a given fluid, pressure head is proportional to pressure. Measuring pressure in units of "mm of mercury" or "inches of water" makes sense for instrumentation, but these raw measurements of head must frequently be converted to more convenient pressure units using the equations above to solve for pressure. In summary pressure head is a measurement of length, which can be converted to the units of pressure (force per unit area), as long as strict attention is paid to the density of the measurement fluid and the local value of g.
Implications for gravitational anomalies on ψ We would normally use pressure head calculations in areas in which g {\displaystyle g} is constant. However, if the gravitational field fluctuates, we can prove that pressure head fluctuates with it.
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