# 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

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### 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}= (p_{2}- p_{1}) /γ + h_{2}- h_{1}+ (v_{2}^{2}- v_{1}^{2}) / 2 g(1)

where

h_{a}= actual head rise

p= pressure

h= elevation height

γ=ρ g =specific weight

v= velocity

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 -
*q*^{2}. - 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 (*v _{2} = v_{1}*), and the inlet and outlet elevation are the same (

*h*), the generic equation (1) can be modified to:

_{2}= h_{1}

h_{a}= (p_{2}- p_{1}) /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 10 ^{5} N/m^{2})* and

*10 bar (10 10*. Density of water is

^{5}N/m^{2})*1000 kg/m*. The actual water head (water column) can be calculated using (3):

^{3}

h_{water}= (p_{2}- p_{1}) /γ

= (p_{2}- p_{1}) /ρ g

=((10 10^{5}N/m^{2}) - (1 10^{5}N/m^{2})) / (1000 kg/m^{3}) (9.81 m/s^{2})

= 91.7 m - water column

### Example - Head Rise of an Fan

An inline fan working with hot air with density *ρ** = 1,06 kg/m ^{3}* add a pressure of

*400 Pa (N/m*to the flow.

^{2})The **air head** (air column) can be calculated with (3):

h_{air}= (p_{2}- p_{1}) / ρ g

= (400 N/m^{2}) / (1.06 kg/m^{3}) (9.81 m/s^{2})

= 38.5 m - air column

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

h_{water}= (p_{2}- p_{1}) / ρ g

= (400 N/m^{2}) / (1000 kg/m^{3}) (9.81 m/s^{2})

= 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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