# Static Pressure vs. Head in Fluids

## Static pressure and pressure head in fluids

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 = - γΔh(1)

where

Δp = change in pressure (Pa, psi)

Δh = change in height (m, in)

γ = specific weight of fluid (N/m^{3}, lb/ft^{3})

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/m^{3}, slugs/ft^{3})

g= acceleration of gravity(9.81 m/s^{2},32.174 ft/s^{2})

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:

pΔp =_{2}- p_{1}

= - γ (h_{2}- h_{1})(3)

where

p_{2}= pressure at level 2 (Pa, psi)

p_{1}= pressure at level 1(Pa, psi)

h_{2}= level 2 (m, ft)

h_{1}= level 1(m, ft)

(3) can be transformed to:

pΔp =_{1}- p_{2}

= γ (h_{2}- h_{1})(4)

or

p_{1}- p_{2}= γhΔ(5)

where

hΔ= h_{2}- h_{1}= difference in elevation - the dept down from location h_{2}to h_{1}(m, ft)

or

p_{1}= γ+ phΔ_{2}(6)

#### Example - Pressure in a Fluid

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

*p _{1} = γ Δh + p_{2}*

* = (1000 kg/m ^{3}) (9.81 m/s^{2}) (10 m) + (101.3 kPa)*

* = (98100 kg/ms ^{2} or Pa) + (101300 Pa)*

* = 199400 Pa*

* = 199.4 kPa*

*where *

*ρ = 1000 kg/m ^{3}*

*g = 9.81 m/s ^{2}*

*p _{2} = pressure at surface level = atmospheric pressure = *

*101.3 kPa*

The gauge pressure can be calculated by setting *p _{2} = 0*

*p _{1} = γ Δh + p_{2}*

* = (1000 kg/m ^{3}) (9.81 m/s^{2}) (10 m) *

* = 98100 Pa*

* = 98.1 kPa*

### Pressure vs. Head

(6) can be transformed to:

= (phΔ_{2}- p_{1}) / γ(7)

*Δh* express

**head**- the height difference of a column of fluid of specific weight -

*γ*- required to give a pressure difference

*Δp = p*

_{2}- p_{1}.#### Example - Pressure vs. Head

A pressure difference of *5 psi (lb _{f}/in^{2})* is equivalent to head in water

*(5 lb _{f}/in^{2}) (12 in/ft) (12 in/ft) / (62.4 lb/ft^{3}) *

* = 11.6 ft of water*

or head in Mercury

*(5 lb _{f}/in^{2}) (12 in/ft) (12 in/ft) / (847 lb/ft^{3}) *

* = 0.85 ft of mercury*

Specific weight of water is* 62.4 (lb/ft ^{3})* and specific weight of mercury is

*847 (lb/ft*.

^{3})