Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

# 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

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

## Related Topics

• ### Fluid Flow and Pressure Loss

Pipe lines - fluid flow and pressure loss - water, sewer, steel pipes, pvc pipes, copper tubes and more.
• ### Fluid Mechanics

The study of fluids - liquids and gases. Involving velocity, pressure, density and temperature as functions of space and time.
• ### 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.

## Related Documents

• ### 1st Law of Thermodynamics

The First Law of Thermodynamics simply states that energy can be neither created nor destroyed (conservation of energy). Thus power generation processes and energy sources actually involve conversion of energy from one form to another, rather than creation of energy from nothing.
• ### Bernoulli Equation

Conservation of energy in a non-viscous, incompressible fluid at steady flow.
• ### Energy

Energy is the capacity to do work.
• ### Energy Equation - Pressure Loss vs. Head Loss

Calculate pressure loss - or head loss - in ducts, pipes or tubes.
• ### Fluid Flow - Equation of Continuity

The Equation of Continuity is a statement of mass conservation.
• ### Hydropower

Power potential vs. head and flow rate.
• ### Liquid Flow from Containers - Emptying Time

Calculate liquid velocity, volume flow and draining time when emptying a container.
• ### Potential Energy - Hydropower

Elevation and potential energy in hydropower.
• ### Pumps - NPSH (Net Positive Suction Head)

An introduction to pumps and the Net Positive Suction Head (NPSH).

## Search

Search is the most efficient way to navigate the Engineering ToolBox.

## Engineering ToolBox - SketchUp Extension - Online 3D modeling!

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with older versions of the amazing SketchUp Make and the newer "up to date" SketchUp Pro . Add the Engineering ToolBox extension to your SketchUp Make/Pro from the Extension Warehouse !

We don't collect information from our users. More about

## Citation

• The Engineering ToolBox (2003). Mechanical Energy Equation vs. Bernoulli Equation. [online] Available at: https://www.engineeringtoolbox.com/mechanical-energy-equation-d_614.html [Accessed Day Month Year].

Modify the access date according your visit.

12.8.9