Bernoullis Equation

What is Bernoullis Equation ?

Bernoullis equation is all about conservation of energy.
Pressure as we have already seen is a force spread over an area.
Gas Pressure is 1/3 of the kinetic energy of the gas (one of 3 dimensions = 1/3 ?) as pressure is perpendicular to a surface.
It is the momentum rate of change per unit area per unit time
m dv / a / t
md2v/a/dt2

pressure is perpendicular, when flow is directed average molecular motion is more directed so more acute with sides of pipe.
perpendicular portion is reduced, which reduces the pressure ! (by definition)
static pressure = ρgh , must relate to gpe if released, gpe = mgh
top of tank has different mgh to bottom of tank, if the water flows then pe is converted to ke
ke = ½ m v2
energy must be conserved so gpe(top) = pe(bottom) + ke(flow)
mgh1 = mgh2 + ½ m v2, to find v
mgh1 - mgh2 = ½ m v2
2mg(h1-h2) = m v2 (divide both sides by m)
2g(h1-h2) = v2
v is dependant only on the height ! flow rate is related to pipe size, velocity is not.
a hole in in tank allows a flow which is ke,
ρ * area * flow rate = volume

Mathematically

Pressure And Energy

 

Pressure = Force / Area
Force * Distance = energy (mgh) [kg m s-2 m] = kgm2s-2
Area * Distance = volume (m2 * m) = m3
Pressure = energy / volume, which is dimensionally equivalent to Force / Area
P * d= (F/A) * d
Dimensionally
Nm-2 = Force /Area = kg m-1 s-2
Force = kg m s - 2
Area = m2

Example Content

What this means

The height of fluid determines the velocity, the diameter of the outflow pipe controls the volume.

Web References

http://zebu.uoregon.edu/1998/ph101/l19.html
http://ccrma.stanford.edu/~jos/pasp/Bernoulli_Equation.html