Pump Affinity Laws are equations of proportionality that indicate how a change

in impeller diameter [D], or rotative speed [N], affects the capacity [Q], the

differential head [H], and the required brake horsepower [HP] of a centrifugal

pump. Affinity Laws are also known as Affinity Equations.

**According to Affinity Laws:**

Flow is directly proportional to change in impeller diameter, or speed:

Q2 = Q1 x [ D2 / D1 ]

Q2 = Q1 x [ N2 / N1 ]

**Head is directly proportional to the square of impeller diameter, or **

speed:

H2 = H1 x [ D2 / D1 ] ^2

H2 = H1 x [ N2 / N1 ] ^2

**Horsepower is directly proportional to the cube of impeller diameter, or **

speed:

HP2 = HP1 x [ D2 / D1 ] ^3

HP2 = HP1 x [ N2 / N1 ] ^3

**If both impeller diameter and speed are changed, the equations are **

combined to become:

Q2 = Q1 x [ (D2 x N2) / (D1 x N1) ]

H2 = H1 x [ (D2 x N2) / (D1 x N1) ] ^2

HP2 = HP1 x [ (D2 x N2) / (D1 x N1) ] ^3

The subscript 1 is for initial condition, subscript 2 is for new condition, and ^ is

an exponential symbol.

The equations can be rearranged to calculate either the diameter, or speed, if

the other parameters were known.

The term*head* usually refers to *differential head*, not to *discharge head*; the

discharge head may be used only if*suction head* is zero. The term head may

also refer to*total dynamic head* (TDH) that includes velocity head except in

vertical pumps where static or elevation head is a component of TDH because

static or elevation head remains the same regardless of pump speed. [See

discussion on different head parameters.]

The equations are valid for both U.S. customary and metric units provided

consistent units are used in the calculations. (For simplicity the data shown in

this article are based on U.S. customary units.)

The affinity equations do not imply that an impeller diameter, or speed, can be

changed arbitrarily. The change in impeller diameter shall be within the

allowable maximum and minimum diameters for the specific pump casing.

Likewise, the change in speed shall be within the allowable maximum and

minimum speeds for the pump. The factors affecting those maximum and

minimum values are discussed in a separate article.

**Effect of speed change on NPSHR**

Conventional understanding of Affinity Laws does not address the effect of

speed change on pump NPSHR but common sense practice indicates that as

pump*specific speed* (NS) does not change with speed the same holds true with

*suction specific speed* (NSS), i.e. NSS remains constant with speed change.

This common sense analysis is confirmed by numerous historical test results,

hence CENTRIFUGALPUMP.COM and The EMA Project conclude that the

NPSHR of centrifugal pumps varies directly with the square of the speed

change. However, this conclusion is limited to impellers whose eye diameter

peripheral speeds do not exceed 130 feet per second. (The NPSHR of those

exceeding 130 feet per second is discussed in a separate article.)

For example, consider a single suction pump with BEP at 3000 GPM and NPSHR

of 20 feet, running at 1780 RPM motor speed. Its suction specific speed (NSS) is

10,309. If it runs at twice the speed of 3560 RPM, its capacity will double to 6000

GPM and its NPSHR at 6000 GPM will quadruple to 80 feet. Its NSS remains at

10,309.

The Affinity Laws are based on same principles as Similarity Laws (or Laws of

Similitude, or Modeling Laws.) The main difference is in their use and

application. Affinity Laws are used mainly to predict the change in performance

within the same pump. Similarity Laws are used mainly to extrapolate the

performance of one pump into a predicted performance of another pump of

similar hydraulic or kinematic model, or in size-factoring the performance of a

new pump based on a model pump with similar specific speed.

In Chapter 5 of his book*Centrifugal and Axial Flow Pumps*, pump author A. J.

Stepanoff discusses the mathematical development of affinity laws. But there is

no better proof for affinity laws than the consistent pump test results conducted

under carefully controlled and monitored test laboratories confirming their

validity.

**Affinity Laws are not applicable to viscous performance**

The Affinity Laws cannot be applied directly to a viscous performance to predict

a new viscous performance at a different speed because the viscous correction

factors for flow, head, and efficiency change with their actual values. In such

instance, the viscous performance should first be converted to its equivalent

water performance before applying the Affinity Laws to obtain a new water

performance. The viscous correction factors should then be obtained for the

new water performance to generate a new viscous performance at the new

pump speed.

**DASHBOARD:**

The answers to these questions are discussed in the full version of this article.

**Use and application of Affinity Laws**

Q - Other than for predicting the performance of a pump due to a change in

impeller diameter, or speed, is there another practical use for affinity laws?

A - Affinity laws can be used in modeling, or size factoring, an existing pump to

design a new size pump from scratch. But they should be applied only to pumps

of similar design. Example, using affinity laws to model, or size factor, a single

suction pump to design a new double suction pump will have inaccurate result -

they should be both of the same type.

Q - The affinity laws show the changes to both capacity and head for a given

change in impeller diameter. How are the affinity laws used to estimate the new

impeller diameter if I want to reduce the head only, say by 10%, if the capacity

and speed are to remain the same?

A -See answer in full version of this article.

Q - We have a centrifugal pump driven by a diesel engine through a clutch and

a gearbox. During the summer months we want to increase its flow rate 10%,

from 1000 GPM to 1100 GPM. Based on Affinity Laws we can increase the

engine speed 10%, from 800 RPM to 880 RPM, to get the 10% increase in flow

rate. But this will also increase the head 21%. The pump is used for irrigation so

we only want to increase its flow rate, and not its head. Assuming the head is 30

feet, at what speed should we run the engine?

A - See answer in full version of this article.

[This is a raw article that is undergoing revision. Please bookmark this page and come back

later to read the latest update.]

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The copyright of the materials in this website is retained by its author. The

article/s shall not be reprinted or republished, in whole or in part, in any manner

or form, without the written permission of the author. To obtain permission

please contact: admin@centrifugalpump.com .

in impeller diameter [D], or rotative speed [N], affects the capacity [Q], the

differential head [H], and the required brake horsepower [HP] of a centrifugal

pump. Affinity Laws are also known as Affinity Equations.

Flow is directly proportional to change in impeller diameter, or speed:

Q2 = Q1 x [ N2 / N1 ]

speed:

H2 = H1 x [ N2 / N1 ] ^2

speed:

HP2 = HP1 x [ N2 / N1 ] ^3

combined to become:

H2 = H1 x [ (D2 x N2) / (D1 x N1) ] ^2

HP2 = HP1 x [ (D2 x N2) / (D1 x N1) ] ^3

The subscript 1 is for initial condition, subscript 2 is for new condition, and ^ is

an exponential symbol.

The equations can be rearranged to calculate either the diameter, or speed, if

the other parameters were known.

The term

discharge head may be used only if

also refer to

vertical pumps where static or elevation head is a component of TDH because

static or elevation head remains the same regardless of pump speed. [See

discussion on different head parameters.]

The equations are valid for both U.S. customary and metric units provided

consistent units are used in the calculations. (For simplicity the data shown in

this article are based on U.S. customary units.)

The affinity equations do not imply that an impeller diameter, or speed, can be

changed arbitrarily. The change in impeller diameter shall be within the

allowable maximum and minimum diameters for the specific pump casing.

Likewise, the change in speed shall be within the allowable maximum and

minimum speeds for the pump. The factors affecting those maximum and

minimum values are discussed in a separate article.

speed change on pump NPSHR but common sense practice indicates that as

pump

This common sense analysis is confirmed by numerous historical test results,

hence CENTRIFUGALPUMP.COM and The EMA Project conclude that the

NPSHR of centrifugal pumps varies directly with the square of the speed

change. However, this conclusion is limited to impellers whose eye diameter

peripheral speeds do not exceed 130 feet per second. (The NPSHR of those

exceeding 130 feet per second is discussed in a separate article.)

For example, consider a single suction pump with BEP at 3000 GPM and NPSHR

of 20 feet, running at 1780 RPM motor speed. Its suction specific speed (NSS) is

10,309. If it runs at twice the speed of 3560 RPM, its capacity will double to 6000

GPM and its NPSHR at 6000 GPM will quadruple to 80 feet. Its NSS remains at

10,309.

The Affinity Laws are based on same principles as Similarity Laws (or Laws of

Similitude, or Modeling Laws.) The main difference is in their use and

application. Affinity Laws are used mainly to predict the change in performance

within the same pump. Similarity Laws are used mainly to extrapolate the

performance of one pump into a predicted performance of another pump of

similar hydraulic or kinematic model, or in size-factoring the performance of a

new pump based on a model pump with similar specific speed.

In Chapter 5 of his book

Stepanoff discusses the mathematical development of affinity laws. But there is

no better proof for affinity laws than the consistent pump test results conducted

under carefully controlled and monitored test laboratories confirming their

validity.

a new viscous performance at a different speed because the viscous correction

factors for flow, head, and efficiency change with their actual values. In such

instance, the viscous performance should first be converted to its equivalent

water performance before applying the Affinity Laws to obtain a new water

performance. The viscous correction factors should then be obtained for the

new water performance to generate a new viscous performance at the new

pump speed.

- On what types of pumps are the affinity laws applicable?

- The affinity laws are accurate only under what specific condition? What

should be done if this condition were not met?

- Are the equations applicable to reverse-running centrifugal pump acting

as hydraulic power recovery turbine (HPRT)?

The answers to these questions are discussed in the full version of this article.

impeller diameter, or speed, is there another practical use for affinity laws?

A - Affinity laws can be used in modeling, or size factoring, an existing pump to

design a new size pump from scratch. But they should be applied only to pumps

of similar design. Example, using affinity laws to model, or size factor, a single

suction pump to design a new double suction pump will have inaccurate result -

they should be both of the same type.

Q - The affinity laws show the changes to both capacity and head for a given

change in impeller diameter. How are the affinity laws used to estimate the new

impeller diameter if I want to reduce the head only, say by 10%, if the capacity

and speed are to remain the same?

A -See answer in full version of this article.

Q - We have a centrifugal pump driven by a diesel engine through a clutch and

a gearbox. During the summer months we want to increase its flow rate 10%,

from 1000 GPM to 1100 GPM. Based on Affinity Laws we can increase the

engine speed 10%, from 800 RPM to 880 RPM, to get the 10% increase in flow

rate. But this will also increase the head 21%. The pump is used for irrigation so

we only want to increase its flow rate, and not its head. Assuming the head is 30

feet, at what speed should we run the engine?

A - See answer in full version of this article.

[This is a raw article that is undergoing revision. Please bookmark this page and come back

later to read the latest update.]

article/s shall not be reprinted or republished, in whole or in part, in any manner

or form, without the written permission of the author. To obtain permission

please contact: admin@centrifugalpump.com .