Bernoulli Equation
Conservation of energy in a nonviscous, incompressible fluid at steady flow.
The statement of conservation of energy is useful when solving problems involving fluids. For a nonviscous, incompressible fluid in a steady flow, the sum of pressure, potential and kinetic energies per unit volume is constant at any point.
Bernoulli's principle: At points along a horizontal streamline, higher pressure regions have lower fluid speed and lower pressure regions have higher fluid speed.
A special form of the Euler’s equation derived along a fluid flow streamline is often called the Bernoulli Equation:
Energy Form
For steady state incompressible flow the Euler equation becomes
E = p_{1} / ρ + v_{1}^{2}/ 2 + g h_{1}
= p_{2}/ ρ + v_{2}^{ 2 } / 2 + g h_{2} E _{ loss }
= constant (1)
where
E = energy per unit mass in flow (J/kg, Btu/slug)
p = pressure in the fluid (Pa, psf)
ρ = density of fluid (kg/m^{3}, slug/ft^{3} )
v = velocity of fluid (m/s, ft/s)
E _{ loss } = energy loss per unit mass in flow (J/kg, Btu/slug)
1 psi = 144 psf
Head Form
(1) can be modified by dividing with gravity like
h = p_{1} / γ + v_{1}^{2}/ (2 g) + h_{1} = p_{2}/ γ + v_{2}^{ 2 } / (2 g) + h_{2} E _{ loss } / g = constant (2)
where
h = head (m fluid column, ft fluid column)
γ = ρ g = specific weight of fluid (N/kg, lb_{f} /slug)
Equation (2) is often referred to as the " head " because all elements has the unit of length.
Note!  the head unit is with reference to the density of the flowing fluid. For other units  like mm Water Column  check Velocity Pressure Head .
Dynamic Pressure
(1) and (2) are two forms of the Bernoulli Equation for a steady state incompressible flow. If we assume that the gravitational body force is negligible  the elevation is small  then the Bernoulli equation can be modified to
p = p_{1} + ρ v_{1}^{2}/ 2
= p_{2}+ ρ v_{2}^{ 2 } / 2  p _{ loss }
= p_{1} + p _{ d }_{1} = p_{2}+ p _{ d2 }  p _{ loss } (3)
where
p = pressure (Pa, psi)
p _{ loss } = pressure loss (Pa, psi)
p _{ d } = 1/2 ρ v^{2}= dynamic pressure (Pa, psf)
It is common to refer to the flow velocity component as the dynamic pressure of the fluid flow.
Note!  increased flow velocity reduces pressure  decreased flow velocity increases pressure.
This phenomena can be observed in a venturi meter where the pressure is reduced in the constriction area and regained after. It can also be observed in a pitot tube where the stagnation pressure is measured. The stagnation pressure is where the velocity component is zero.
Bernoulli Equation and Flow from a Tank through a small Orifice
Liquid flows from a tank through a orifice close to the bottom. The Bernoulli equation can be adapted to a streamline from the surface (1) to the orifice (2) :
p_{1} / γ + v_{1}^{2}/ (2 g) + h_{1}
= p_{2}/ γ + v_{2}^{ 2 } / (2 g) + h_{2} E _{ loss } / g (4)
By multiplying with g and assuming that the energy loss is neglectable  (4) can be transformed to
p_{1} / ρ + v_{1}^{2}/ 2 + g h_{1}
= p_{2}/ ρ + v_{2}^{ 2 } / 2 + g h_{2}(4b)
Discharge Velocity
If
h = h_{1}  h_{2}(4c)
and (according the continuity equation)
v_{1} = (A_{2}/ A_{1} ) v_{2}(4d)
then the velocity out of the orifice can be expressed as
v_{2}= ([2 / (1  A_{2}^{ 2 } / A_{1}^{2})][g h + (p_{1}  p_{2}) / 2]) ^{ 1/2 } (5)
Vented Tank
For a vented tank where the inside pressure equals the outside pressure
p_{1} = p_{2}(5b)
and the surface area is much larger than the orifice area
A_{1} >> A_{2}(5c)
 then eq. 5 can be modified to
v_{2}= (2 g h) ^{ 1/2 } (6)
"The velocity out from the tank is equal to speed of a freely body falling the distance h ."  also known as Torricelli's Theorem.
Example  outlet velocity from a vented tank
The outlet velocity from a tank with level 10 m can be calculated as
v_{2}= (2 (9.81 m/s^{2}) (10 m)) ^{ 1/2 }
= 14 m/s
Orifice Discharge Coefficient
Eq. 6 is for ideal flow without pressure loss in the orifice. In the real worls  with pressure loss  eq. 6 can be expressed with a coefficient of discharge  friction coefficient  as
v_{2}= c (2 g h) ^{ 1/2 } (6b)
where
c = coefficient of discharge
The coefficient of discharge can be determined experimentally. For a sharp edged opening it may bee as low as 0.6 . For smooth orifices it may bee between 0.95 and 1 .
Pressurized Tank
If the tank is closed, pressurized and the level between the surface and the discharge outlet minimal (the influence from level difference is very small compared to pressure influence in eq. 5)  the discharge velocity can be expressed as
v_{2}= c (2 (p_{1}  p_{2}) / ρ) ^{ 1/2 } (7)
Example  Outlet Velocity from a Pressurized Tank
The outlet velocity of a pressurized tank where
p_{1} = 0.2 (MN/m^{2}, 10^{6} Pa)
p_{2}= 0.1 (MN/m^{2}, 10^{6} Pa)
A_{2}/ A_{1} = 0.01
h = 10 (m)
can be calculated as
V_{2}= ( (2 / (1  (0.01)^{2}) ((0.2 10^{6} N/m^{2})  (0.1 10^{6} N/m^{2})) / (1000 kg/m^{3} ) + (9.81 m/s^{2}) (10 m))) ^{ 1/2 }
= 19.9 m/s
Energy Loss through a Reduction Valve
When fluid flows through a reduction valve and the pressure is reduced  there is an energy loss. By neglecting the change in elevation (h_{1} = h_{2}) and the change in fluid velocity (v_{1} = v_{2}) the pressure energy before the valve and the pressure energy after the valve including the energy loss through the valve  is constant. The Bernouilli equation can be modified to
p_{1} / ρ = p_{2}/ ρ + E _{ loss } (8)
where
E _{ loss } = energy loss through valve (J)
(8) can be transformed to:
E _{ loss } = ( p_{1}  p_{2}) / ρ (8b)
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