# Mechanical Energy and Bernoulli Equation

## The mechanical energy equation related to energy per unit mass, energy per unit volume and energy per unit weight involving head

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:

pin / ρ + vin2 / 2 + g hin + wshaft = pout / ρ + vout2 / 2 + g hout + wloss                                      (1)

where

p = static pressure (Pa)

ρ = density (kg/m3)

v = flow velocity (m/s)

g = acceleration of gravity (9.81 m/s2)

h = elevation height  (m)

wshaft = net shaft energy per unit mass for a pump, fan or similar (J/kg)

wloss = loss due to friction (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 can be written as:

pin / ρ + vin2 / 2 + g hin = pout / ρ + vout2 / 2 + g hout + wshaft + wloss                                           (2)

where

wshaft = net shaft energy out per unit mass for a turbine or similar (J/kg)

Equation (1) and (2) dimensions are

• energy per unit mass (ft2/s2 = ft lb/slug or m2/s2 = N m/kg)

### Efficiency

According to (1) a larger amount of loss - wloss - result in more shaft work required for the same rise of output energy. The efficiency of a pump or fan process can be expressed as:

η = (wshaft - wloss) / wshaft                                    (3)

The efficiency of a turbine process can be expressed as:

η = wshaft / (wshaft + wloss)                                     (4)

### The Mechanical Energy Equation in Terms of Energy per Unit Volume

The mechanical energy equation for a pump or a fan (1) can also be written in terms of energy per unit volume by multiplying (1) with the fluid density - ρ:

pin + ρ vin2 / 2 + γ hin + ρ wshaft = pout + ρ vout2 / 2 + γ hout + ρ wloss                                       (5)

where

γ = ρ g = specific weight   (N/m3)

The dimensions of equation (5) are

• energy per unit volume (ft lb/ft3 = lb/ft2 or Nm/m3 = N/m2)

### 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:

pin / γ + vin2 / 2 g + hin + hshaft = pout / γ + vout2 / 2 g + hout + hloss                         (6)

hshaft = wshaft / g = net shaft energy head per unit mass for a pump, fan or similar  (m)

hloss = wloss / 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.

hshaft can also be expressed as:

hshaft = wshaft / g = Wshaft / m g = Wshaft / γ Q                                 (7)

where

Wshaft = shaft power (W)

m = mass flow rate  (kg/s)

Q = volume flow rate  (m3/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 ft3/s.

Since vin = vout = 0, pin = pout = 0 and hin = 0 - equation (6) can be modified to:

hshaft = hout + hloss

or

hloss = hshaft - hout                                           (8)

Equation (7) gives:

hshaft = Wshaft / γ Q

= (4 hp)(550 ft lb/s/hp) / (62.4 lb/ft3)(2 ft3/s)

= 17.6  ft

Combined with (8):

hloss = (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

## Related Topics

• Fluid Mechanics - The study of fluids - liquids and gases. Involves velocity, pressure, density and temperature as functions of space and time
• Fluid Flow and Pressure Drop - Pipe lines - fluid flow and pressure loss - water, sewer, steel pipes, pvc pipes, copper tubes and more
• Pumps - Piping systems and pumps - centrifugal pumps, displacement pumps - cavitation, viscosity, head and pressure, power consumption and more
• Ventilation - Systems for ventilation and air handling - air change rates, ducts and pressure drops, charts and diagrams and more

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