Mechanical Energy Equation vs. Bernoulli Equation
The Mechanical Energy Equation compared to the Extended Bernoulli Equation.
The Energy Equation is a statement based on the First Law of Thermodynamics involving energy, heat transfer and work. With certain limitations the mechanical energy equation can be compared to the Bernoulli Equation .
The Mechanical Energy Equation in Terms of Energy per Unit Mass
The mechanical energy equation for a pump or a fan can be written in terms of energy per unit mass where the energy into the system equals the energy out of the system.
E _{ pressure,in } + E _{ velocity,in } + E _{ elevation,in } + E _{ shaft }
= E _{ pressure,out } + E _{ velocity,out } + E _{ elevation,out } + E _{ loss } (1)
or
p _{ in } / ρ + v _{ in } ^{ 2 } / 2 + g h _{ in } + E _{ shaft }
= p _{ out } / ρ + v _{ out } ^{ 2 } / 2 + g h _{ out } + E _{ loss } (1b)
where
p = static pressure (Pa, (N/m ^{ 2 } ))
ρ = density (kg/m ^{ 3 } )
v = flow velocity (m/s)
g = acceleration of gravity (9.81 m/s ^{ 2 } )
h = elevation height (m)
E _{ shaft } = net shaft energy per unit mass for a pump, fan or similar (J/kg)
E _{ loss } = hydraulic loss through the pump or fan (J/kg)
The energy equation is often used for incompressible flow problems and is called the Mechanical Energy Equation or the Extended Bernoulli Equation .
The mechanical energy equation for a turbine  where power is produced  can be written as:
p _{ in } / ρ + v _{ in } ^{ 2 } / 2 + g h _{ in }
= p _{ out } / ρ + v _{ out } ^{ 2 } / 2 + g h _{ out } + E _{ shaft } + E _{ loss } (2)
where
E _{ shaft } = net shaft energy out per unit mass for the turbine (J/kg)
Equation (1) and (2) dimensions are
 energy per unit mass (ft ^{ 2 } /s ^{ 2 } = ft lb/slug or m ^{ 2 } /s ^{ 2 } = N m/kg)
Efficiency
According to (1) more loss requires more shaft work to be done for the same rise of output energy. The efficiency of a pump or fan process can be expressed as:
η = (E _{ shaft }  E _{ loss } ) / E _{ shaft } (3)
The efficiency of a turbine process can be expressed as:
η = E _{ shaft } / (E _{ shaft } + E _{ loss } ) (4)
The Mechanical Energy Equation in Terms of Energy per Unit Volume
The mechanical energy equation for a pump or fan (1) can also be written in terms of energy per unit volume by multiplying (1) with the fluid density  ρ :
p _{ in } + ρ v _{ in } ^{ 2 } / 2 + γ h _{ in } + ρ E _{ shaft }
= p _{ out } + ρ v _{ out } ^{ 2 } / 2 + γ h _{ out } + ρ E _{ loss } (5)
where
γ = ρ g = specific weight (N/m ^{ 3 } )
The dimensions of equation (5) are
 energy per unit volume (ft lb/ft ^{ 3 } = lb/ft ^{ 2 } or Nm/m ^{ 3 } = N/m ^{ 2 } )
The Mechanical Energy Equation in Terms of Energy per Unit Weight involving Heads
The mechanical energy equation for a pump or a fan (1) can also be written in terms of energy per unit weight by dividing with gravity  g :
p _{ in } / γ + v _{ in } ^{ 2 } / 2 g + h _{ in } + h _{ shaft }
= p _{ out } / γ + v _{ out } ^{ 2 } / 2 g + h _{ out } + h _{ loss } (6)
h _{ shaft } = E _{ shaft } / g = net shaft energy head per unit mass for a pump, fan or similar (m)
h _{ loss } = E _{ loss } / g = loss head due to friction (m)
The dimensions of equation (6) are
 energy per unit weight (ft lb/lb = ft or Nm/N = m)
Head is the energy per unit weight .
h _{ shaft } can also be expressed as:
h _{ shaft } = E _{ shaft } / g
= E _{ shaft } / m g = E _{ shaft } / γ Q (7)
where
E _{ shaft } = shaft power (W)
m = mass flow rate (kg/s)
Q = volume flow rate (m ^{ 3 } /s)
Example  Pumping Water
Water is pumped from an open tank at level zero to an open tank at level 10 ft . The pump adds four horse powers to the water when pumping 2 ft ^{ 3 } /s .
Since v _{ in } = v _{ out } = 0, p _{ in } = p _{ out } = 0 and h _{ in } = 0  equation (6) can be modified to:
h _{ shaft } = h _{ out } + h _{ loss }
or
h _{ loss } = h _{ shaft }  h _{ out } (8)
Equation (7) gives:
h _{ shaft } = E _{ shaft } / γ Q
= (4 hp)(550 ft lb/s/hp) / (62.4 lb/ft ^{ 3 } )(2 ft ^{ 3 } /s)
= 17.6 ft
 specific weight of water  62.4 lb/ft ^{ 3 }
 1 hp ( English horse power ) = 550 ft lb/s
Combined with (8) :
h _{ loss } = (17.6 ft )  (10 ft)
= 7.6 ft
The pump efficiency can be calculated from (3) modified for head:
η = (( 17.6 ft)  ( 7.6 ft) ) / (17.6 ft)
= 0.58
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