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Static Pressure vs. Head in Fluids

Static pressure and pressure head in fluids

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Pressure indicates the normal force per unit area at a given point acting on a given plane. Since there is no shearing stresses present in a fluid at rest - the pressure in a fluid is independent of direction.

For fluids - liquids or gases - at rest the pressure gradient in the vertical direction depends only on the specific weight of the fluid.

How pressure changes with elevation in a fluid can be expressed as

Δp = - γ Δh                                            (1)

where

Δp = change in pressure (Pa, psi)

Δh = change in height (m, in)

γ = specific weight of fluid (N/m3, lb/ft3)

The pressure gradient in vertical direction is negative - the pressure decrease upwards.

Specific Weight

Specific Weight of a fluid can be expressed as:

γ = ρ g                                             (2)

where

ρ = density of fluid (kg/m3, slugs/ft3)

g = acceleration of gravity (9.81 m/s2, 32.174 ft/s2)

In general the specific weight - γ - is constant for fluids. For gases the specific weight - γ - varies with elevation (and compression).

The pressure exerted by a static fluid depends only upon

  • the depth of the fluid
  • the density of the fluid
  • the acceleration of gravity

Static Pressure in a Fluid

For a incompressible fluid - as a liquid - the pressure difference between two elevations can be expressed as:

Δp = p2 - p1

     = - γ (h2 - h1)                                     (3)

where

p2 = pressure at level 2  (Pa, psi)

p1 = pressure at level 1   (Pa, psi)

h2 = level 2    (m, ft)

h1 = level 1   (m, ft)

(3) can be transformed to:

Δp = p1 - p2

     = γ (h2 - h1)                                      (4)

or

p1 - p2 = γ Δh                                     (5)

where

Δh = h2 - h1 = difference in elevation - the dept down from location h2 to h1  (m, ft)

or

p1 = γ Δh + p2                                           (6)

Example - Pressure in a Fluid

The absolute pressure at water depth of 10 m can be calculated as: 

p1 = γ Δh + p2

   = (1000 kg/m3) (9.81 m/s2) (10 m) + (101.3 kPa)

   = (98100 kg/ms2 or Pa) + (101300 Pa)

   = 199400 Pa

   = 199.4 kPa

where

ρ = 1000 kg/m3

g = 9.81 m/s2

p2 = pressure at surface level = atmospheric pressure = 101.3 kPa

The gauge pressure can be calculated by setting p2 = 0

p1 = γ Δh + p2

   = (1000 kg/m3) (9.81 m/s2) (10 m)

   = 98100 Pa

   = 98.1 kPa

Pressure vs. Head

(6) can be transformed to:

Δh = (p2 - p1) / γ                                                (7)

Δh express head - the height difference  of a column of fluid of specific weight - γ - required to give a pressure difference Δp = p2 - p1.

Example - Pressure vs. Head

A pressure difference of 5 psi (lbf/in2) is equivalent to head in water

(5 lbf/in2) (12 in/ft) (12 in/ft) / (62.4 lb/ft3)

    = 11.6  ft of water

or head in Mercury

(5 lbf/in2) (12 in/ft) (12 in/ft) / (847 lb/ft3)

    = 0.85  ft of mercury

Specific weight of water is 62.4 (lb/ft3) and specific weight of mercury is 847 (lb/ft3).

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