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# Pumps and Fans - Energy Equation and Head Rise

## The energy equation can be used to calculate the head rise in pumps and fans.

### Actual Head Rise of a Pump or Fan

Using the Energy Equation the head rise through a pump or fan can be expressed as:

ha = (p2 - p1) / γ + (h2 - h1) + (v22 - v12) / 2 g                                    (1)

where

ha = actual head rise (m fluid column)

p = pressure (Pa, N/m2)

h = elevation (m)

γ = ρ g = specific weight of fluid (N/m3)

v = velocity (m/s)

ρ = density of fluid (kg/m3)

g = acceleration of gravity (9.81 m/s2)

The actual head rise can be expressed as:

ha = hshaft - hloss                                    (2)

where

hshaft = shaft work in the pump or fan

hloss = head loss through the pump or fan

The head loss - hloss - through a pump or fan is related to the

• skin friction in the blade passages and is proportional to volume flow - q2.
• flow separation
• impeller blade casing clearance flows
• other three dimensional flow effects

### Actual Head Rise for an Inline Pump

For a very common installation - the inline pump or fan - where the inlet velocity and the outlet velocity are the same (v2 = v1), and the inlet and outlet elevation are the same (h2 = h1), the generic equation (1) can be modified to:

ha = (p2 - p1) / γ                                  (3)

### Specific Work

By multiplying (3) with acceleration of gravity, specific work of the pump or fan may be calculated:

w = ha g                                      (4)

where

w = specific work (Nm/kg, J/kg)

### Example - Head Rise of an Inline Pump

An inline water pump works between the pressure 1 bar (1 105 N/m2) and 10 bar (10 105 N/m2). Density of water is 1000 kg/m3. The actual water head (water column) can be calculated using (3):

hwater = (p2 - p1) / γ

= (p2 - p1) / ρ g

= ((10 105 N/m2) - (1 105 N/m2)) / (1000 kg/m3) (9.81 m/s2)

= 91.7 m Water Column

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.

### Example - Head Rise of an Fan

An inline fan working with hot air with density ρ = 1.06 kg/m3 add a pressure of 400 Pa (N/m2) to the flow.

The air head (air column) can be calculated with (3):

hair = (p2 - p1) / ρ g

= (400 N/m2) / (1.06 kg/m3) (9.81 m/s2)

= 38.5 m Air Column

The water head (water column) can be calculated with (3) using the density of water:

hwater = (p2 - p1) / ρ g

= (400 N/m2) / (1000 kg/m3) (9.81 m/s2)

= 0.041 m

= 41 mm Water Column

Measuring pressure with water column in an U-tube manometer is common in air distribution systems for ventilation and air condition.

## Related Topics

• Pumps - Piping systems and pumps - centrifugal pumps, displacement pumps - cavitation, viscosity, head and pressure, power consumption and more.

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## Citation

• Engineering ToolBox, (2004). Pumps and Fans - Energy Equation and Head Rise. [online] Available at: https://www.engineeringtoolbox.com/pump-energy-equation-d_631.html [Accessed Day Mo. Year].

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