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
H2 = H1 x [ D2 / D1 ] ^2
H2 = H1 x [ N2 / N1 ] ^2
Horsepower is directly proportional to the cube of impeller diameter, or
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
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
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
- 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.
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.
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