Static Pressure and Pressure Head in Fluids
The 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 can be expressed as
[INDENT] [I]dp = - γ dz[/I][I] (1)[/I]
[I]where[/I]
[I]dp[/I][I] = change in pressure[/I]
[I]dz[/I][I] = change in height[/I]
[I]γ[/I][I] = specific weight[/I]
[/INDENT]The pressure gradient in vertical direction is negative - the pressure decrease upwards.
[h=3]Specific Weight[/h] Specific Weight can be expressed as:
[INDENT] [I]γ = ρ g[/I][I] (2)[/I]
[I]where[/I]
[I]γ[/I][I] = specific weight[/I]
[I]g[/I][I] = acceleration of gravity[/I][/INDENT]In general the specific weight - [I]γ[/I] - is constant for fluids. For gases the specific weight - [I]γ[/I] - varies with the elevation.
[h=3]Static Pressure in a Fluid[/h] For a incompressible fluid - as a liquid - the pressure difference between two elevations can be expressed as:
[INDENT] [I]p[SUB]2[/SUB] - p[SUB]1[/SUB] = - γ (z[SUB]2[/SUB] - z[SUB]1[/SUB])[/I][I] (3)[/I]
[I]where[/I]
[I]p[SUB]2[/SUB][/I][I] = pressure at level 2[/I]
[I]p[SUB]1[/SUB][/I][I] = pressure at level 1[/I]
[I]z[SUB]2[/SUB][/I][I] = level 2[/I]
[I]z[SUB]1[/SUB][/I][I] = level 1[/I][/INDENT](3) can be transformed to:
[INDENT] [I]p[SUB]1[/SUB] - p[SUB]2[/SUB] = γ (z[SUB]2[/SUB] - z[SUB]1[/SUB])[/I][I] (4)[/I]
[I]or[/I]
[I]p[SUB]1[/SUB] - p[SUB]2[/SUB] = γ h[/I][I] (5)[/I]
[I]where[/I]
[I]h[/I][I] = [/I][I]z[SUB]2[/SUB] - z[SUB]1[/SUB][/I][I] difference in elevation - the dept down from location [/I][I]z[SUB]2[/SUB].[/I]
[I]or[/I]
[I]p[SUB]1[/SUB] = γ h + p[SUB]2[/SUB][/I][I] (6)[/I][/INDENT][h=4]Example - Pressure in a Fluid[/h] The absoute pressure at water depth of 10 m can be calulated as:
[I]p[SUB]1[/SUB] = γ h + p[SUB]2[/SUB][/I]
[I] = (1000 kg/m[SUP]3[/SUP]) (9.81 m/s[SUP]2[/SUP]) (10 m) + (101.3 kPa)[/I]
[I] = (98100 kg/ms[SUP]2[/SUP] or Pa) + (101300 Pa)[/I]
[I] = [U]199.4[/U] kPa
[/I]
[I]where [/I]
[I]ρ = 1000 kg/m[SUP]3[/SUP][/I]
[I]g = 9.81 m/s[SUP]2[/SUP][/I]
[I]p[SUB]2[/SUB] = pressure at surface level = atmospheric pressure = [/I][I]101.3 kPa[/I]
The gauge pressure can be calulated setting [I]p[SUB]2[/SUB] = 0[/I]
[I]p[SUB]1[/SUB] = γ h + p[SUB]2[/SUB][/I]
[I] = (1000 kg/m[SUP]3[/SUP]) (9.81 m/s[SUP]2[/SUP]) (10 m)
[/I]
[I] = [U]98.1[/U] kPa[/I]
[h=3]The Pressure Head[/h] (6) can be transformed to:
[INDENT] [I]h = (p[SUB]2[/SUB] - p[SUB]1[/SUB]) / γ[/I][I] (6)[/I][/INDENT][I]h[/I] express [B]the pressure head[/B] - the height of a column of fluid of specific weight - [I]γ[/I] - required to give a pressure difference of ([I]p[SUB]2[/SUB] - p[SUB]1[/SUB]).[/I]
[h=4]Example - Pressure Head[/h] A pressure difference of [I]5 psi (lbf/in[SUP]2[/SUP])[/I] is equivalent to
[INDENT] [I](5 lb[SUB]f[/SUB]/in[SUP]2[/SUP]) (12 in/ft) (12 in/ft) / (62.4 lb/ft[SUP]3[/SUP]) [/I]
[I] = [U]11.6[/U] ft of water[/I]
[I](5 lb[SUB]f[/SUB]/in[SUP]2[/SUP]) (12 in/ft) (12 in/ft) / (847 lb/ft[SUP]3[/SUP]) [/I]
[I] = [U]0.85[/U] ft of mercury[/I][/INDENT]when specific weight of water is[I] 62.4 (lb/ft[SUP]3[/SUP])[/I] and specific weight of mercury is[I] 847 (lb/ft[SUP]3[/SUP])[/I].
Heads at different velocities are indicated in the table below:
[TABLE="class: small"]
[TR]
[TD="bgcolor: #ccffff"]Velocity
[I](ft/sec)[/I][/TD]
[TD="width: 50%, bgcolor: #ccffff"][CENTER]Head Water
[/CENTER]
[I](ft)[/I]
[/TD]
[/TR]
[TR]
[TD]0.5[/TD]
[TD]0.004[/TD]
[/TR]
[TR]
[TD]1.0[/TD]
[TD]0.016[/TD]
[/TR]
[TR]
[TD]1.5[/TD]
[TD]0035[/TD]
[/TR]
[TR]
[TD]2.0[/TD]
[TD]0.062[/TD]
[/TR]
[TR]
[TD]2.5[/TD]
[TD]0.097[/TD]
[/TR]
[TR]
[TD]3.0[/TD]
[TD]0.140[/TD]
[/TR]
[TR]
[TD]3.5[/TD]
[TD]0.190[/TD]
[/TR]
[TR]
[TD]4.0[/TD]
[TD]0.248[/TD]
[/TR]
[TR]
[TD]4.5[/TD]
[TD]0.314[/TD]
[/TR]
[TR]
[TD]5.0[/TD]
[TD]0.389[/TD]
[/TR]
[TR]
[TD]5.5[/TD]
[TD]0.470[/TD]
[/TR]
[TR]
[TD]6.0[/TD]
[TD]0.560[/TD]
[/TR]
[TR]
[TD]6.5[/TD]
[TD]0.657[/TD]
[/TR]
[TR]
[TD]7.0[/TD]
[TD]0.762[/TD]
[/TR]
[TR]
[TD]7.5[/TD]
[TD]0.875[/TD]
[/TR]
[TR]
[TD]8.0[/TD]
[TD]0.995[/TD]
[/TR]
[TR]
[TD]8.5[/TD]
[TD]1.123[/TD]
[/TR]
[TR]
[TD]9.0[/TD]
[TD]1.259[/TD]
[/TR]
[TR]
[TD]9.5[/TD]
[TD]1.403[/TD]
[/TR]
[TR]
[TD]10.0[/TD]
[TD]1.555[/TD]
[/TR]
[TR]
[TD]11.0[/TD]
[TD]1.881[/TD]
[/TR]
[TR]
[TD]12.0[/TD]
[TD]2.239[/TD]
[/TR]
[TR]
[TD]13.0[/TD]
[TD]2.627[/TD]
[/TR]
[TR]
[TD]14.0[/TD]
[TD]3.047[/TD]
[/TR]
[TR]
[TD]15.0[/TD]
[TD]3.498[/TD]
[/TR]
[TR]
[TD]16.0[/TD]
[TD]3.980[/TD]
[/TR]
[TR]
[TD]17.0[/TD]
[TD]4.493[/TD]
[/TR]
[TR]
[TD]18.0[/TD]
[TD]5.037[/TD]
[/TR]
[TR]
[TD]19.0[/TD]
[TD]5.613[/TD]
[/TR]
[TR]
[TD]20.0[/TD]
[TD]6.219[/TD]
[/TR]
[TR]
[TD]21.0[/TD]
[TD]6.856[/TD]
[/TR]
[TR]
[TD]22.0[/TD]
[TD]7.525[/TD]
[/TR]
[/TABLE]
[LIST][*][I]1 ft (foot) = 0.3048 m = 12 in = 0.3333 yd[/I][/LIST]
