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Mechanical Energy Equation vs. Bernoulli Equation

The Mechanical Energy Equation compared to the Extended Bernoulli Equation.

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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

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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