Head Rise and the Energy Equation for a Pump or a Fan

The energy equation can be used to calculate the actual head rise in a pump or fan

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

p = pressure

h = elevation height

γ = ρ g = specific weight

v = velocity

g = acceleration of gravity

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 with 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 - g -, specific work - w - of the pump or fan may be calculated:

w = h_a g (4)

where

w = specific work

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

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 applications as ventilation and air condition systems.

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