Pumps and Fans - Energy Equation and Head Rise

Actual Head Rise of a Pump or Fan

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

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

where

h_a = 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)

• Pressure vs. head

The actual head rise can be expressed as:

h_a = h_shaft - h_loss                                    (2)

where

h_shaft = shaft work in the pump or fan

h_loss = head loss through the pump or fan

The head loss - h_loss - 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:

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

Specific Work

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

w = h_a 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):

h_water = (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):

h_air = (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:

h_water = (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.