# Pump Affinity Laws

## Turbo machines affinity laws are used to calculate volume capacity, head or power consumption in centrifugal pumps when changing speed - rpm - or wheel diameters

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**The Affinity Laws** of centrifugal pumps or fans indicates the influence on volume capacity, head (pressure) and/or power consumption of a pump or fan due to

- change in
speedof wheel - revolutions per minute (rpm)- geometrically similarity - change in
impeller diameter

Be aware that there are two sets of affinity laws

- affinity laws for a specific centrifugal pump - to approximate head, capacity and power curves for different motor speeds and /or different diameter of impellers
- affinity laws for a family of geometrically similar centrifugal pumps - to approximate head, capacity and power curves for different motor speeds and /or different diameter of impellers

### Pump Affinity Laws for a Specific Centrifugal Pump

#### Volume Capacity

*The volume capacity of a centrifugal pump can be expressed like*

q_{1}/ q_{2}= (n_{1}/ n_{2}) (d_{1}/ d_{2}) (1)

where

q= volume flow capacity (m^{3}/s, gpm, cfm, ..)

n= wheel velocity - revolution per minute - (rpm)

d = wheel diameter

#### Head or Pressure

The head or pressure of a centrifugal pump can be expressed like

dp_{1}/ dp_{2}= (n_{1}/ n_{2})^{2}(d_{1}/ d_{2})^{2}(2)

where

dp= head or pressure (m, ft, Pa, psi, ..)

#### Power

The power consumption of a centrifugal pump can be expressed as

P_{1}/ P_{2}= (n_{1}/ n_{2})^{3}(d_{1}/ d_{2})^{3}(3)

where

P= power (W, bhp, ..)

**Changing the Wheel Velocity**

If the **wheel diameter is constant** - change in pump wheel velocity can simplify the affinity laws to

##### Volume Capacity

q_{1}/ q_{2}= (n_{1}/ n_{2}) (1a)

##### Head or Pressure

dp_{1}/ dp_{2}= (n_{1}/ n_{2})^{2 }^{ }(2a)

##### Power

P_{1}/ P_{2}= (n_{1}/ n_{2})^{3}^{ }(3a)

**Note!** If the speed of a pump is increased with *10%*

- the volume flow increases with
*10%* - the head increases with
*21%* - the power increases with
*33 %*

If we want to increase the volume flow capacity of an existing system with *10%* we have to increase the power supply with *33%*.

#### Pump Affinity Laws Calculator - Changing Wheel Velocity

Replace the default values with the actual values. The calculator is generic and can be used with all common units as long as the use is consistent.

**Changing the Impeller Diameter**

If wheel velocity is constant **a change in impeller diameter** can simplify the affinity laws to

##### Volume Capacity

q_{1}/ q_{2}=d_{1}/ d_{2}(1b)

##### Head or Pressure

dp_{1}/ dp_{2}=(d_{1}/ d_{2})^{2}(2b)

##### Power

P_{1}/ P_{2}=(d_{1}/ d_{2})^{3}(3b)

#### Pump Affinity Laws Calculator - Changing Wheel Diameter

Replace the default values with the actual values. The calculator is generic and can be used with all common units as long as the use is consistent.

#### Example - Pump Affinity Laws - Changing Pump Speed

The pump speed is changed when the impeller size is constant. The initial flow is *100 gpm*, the initial head is *100 ft*, the initial power is *5 bhp*, the initial speed is *1750 rpm* and the final speed *3500 rpm*.

The final flow capacity can be calculated with (1a):

q_{2}= q_{1}n_{2}/ n_{1}

=(100 gpm) (3500 rpm) / (1750 rpm)

= 200 gpm

The final head can be calculated with (2a):

dp_{2}= dp_{1}(n_{2}/ n_{1})^{2}

= (100 ft) ((3500 rpm) / (1750 rpm))^{2}

= 400 ft

The final power consumption can be calculated with (3a):

P_{2}= P_{1}(n_{2}/ n_{1})^{3}

= (5 bhp) ((3500 rpm) / (1750 rpm))^{3}

= 40 bph

#### Example - Pump Affinity Laws - Changing Impeller Diameter

The diameter of the pump impeller is reduced when the pump speed is constant. The diameter is changed from *8 to 6 inches*.

The final flow capacity can be calculated with (1b):

q_{2}= q_{1}(d_{2}/ d_{1})

= (100 gpm) ((¨6 in) / (8 in))

= 75 gpm

The final head can be calculated with (2b):

dp_{2}= dp_{1}(d_{2}/ d_{1})^{2}

= (100 ft) ((6 in) / (8 in))^{2}

= 56.3 ft

The final power consumption can be calculated with (3b):

P_{2}= P_{1}(d_{2}/ d_{1})^{3}

= (5 bhp) ((6 in) / (8 in))^{3}

= 2.1 bph

### Pump Affinity Laws for a Family of Geometrically Similar Pumps

*The volume capacity can be expressed like*

q_{1}/ q_{2}= (n_{1}/ n_{2})(d_{1}/ d_{2})^{3}(4)

where

q= volume flow capacity (m^{3}/s, gpm, cfm, ..)

n= wheel velocity - revolution per minute - (rpm)

d = wheel diameter

#### Head or Pressure

The head or pressure of a centrifugal pump can be expressed like

dp_{1}/ dp_{2}= (n_{1}/ n_{2})^{2}(d_{1}/ d_{2})^{2}(5)

where

dp= head or pressure (m, ft, Pa, psi, ..)

#### Power

The power consumption of a centrifugal pump can be expressed as

P_{1}/ P_{2}= (n_{1}/ n_{2})^{3}(d_{1}/ d_{2})^{5}(6)

where

P= power (W, bhp, ..)

*Note that the affinity laws for fans are not identical with pumps.*

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