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.
