The affinity laws (also known as the "Fan Laws" or "Pump Laws") for pumps/fans are used in
hydraulics
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counter ...
,
hydronics
Hydronics () is the use of liquid water or gaseous water ( steam) or a water solution (usually glycol with water) as heat-transfer medium in heating and cooling systems. The name differentiates such systems from oil and refrigerant systems.
H ...
and/or
HVAC
Heating, ventilation, and air conditioning (HVAC) is the use of various technologies to control the temperature, humidity, and purity of the air in an enclosed space. Its goal is to provide thermal comfort and acceptable indoor air quality. HV ...
to express the relationship between variables involved in pump or fan performance (such as
head
A head is the part of an organism which usually includes the ears, brain, forehead, cheeks, chin, eyes, nose, and mouth, each of which aid in various sensory functions such as sight, hearing, smell, and taste. Some very simple animals may ...
,
volumetric flow rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
, shaft speed) and
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
. They apply to
pump
A pump is a device that moves fluids (liquids or gases), or sometimes slurries, by mechanical action, typically converted from electrical energy into hydraulic energy. Pumps can be classified into three major groups according to the method they u ...
s,
fans
Fan commonly refers to:
* Fan (machine), a machine for producing airflow, often used for cooling
** Hand fan, an implement held and waved by hand to move air for cooling
* Fan (person), short for fanatic; an enthusiast or supporter, especially wit ...
, and
hydraulic turbine
A water turbine is a rotary machine that converts kinetic energy and potential energy of water into mechanical work.
Water turbines were developed in the 19th century and were widely used for industrial power prior to electrical grids. Now, th ...
s. In these rotary implements, the affinity laws apply both to centrifugal and axial flows.
The laws are derived using the
Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
. The affinity laws are useful as they allow prediction of the head discharge characteristic of a pump or fan from a known characteristic measured at a different speed or impeller diameter. The only requirement is that the two pumps or fans are dynamically similar, that is, the ratios of the fluid forced are the same. It is also required that the two impellers' speed or diameter are running at the same efficiency.
Law 1. With impeller diameter (D) held constant:
Law 1a. Flow is proportional to shaft speed:
:
Law 1b. Pressure or Head is proportional to the square of shaft speed:
:
Law 1c. Power is proportional to the cube of shaft speed:
:
Law 2. With shaft speed (N) held constant:
Law 2a. Flow is proportional to the impeller diameter:
:
Law 2b. Pressure or Head is proportional to the square of the impeller diameter:
:
Law 2c. Power is proportional to the cube of the impeller diameter (assuming constant shaft speed):
:
where
*
is the volumetric flow rate (e.g.
CFM, GPM or L/s)
*
is the impeller diameter (e.g. in or mm)
*
is the shaft rotational speed (e.g.
rpm
Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines.
Standards
ISO 80000-3:2019 defines a unit of rotation as the dimensionl ...
)
*
is the pressure or head developed by the fan/pump (e.g. psi or Pascal)
*
is the shaft power (e.g. W).
These laws assume that the pump/fan
efficiency
Efficiency is the often measurable ability to avoid wasting materials, energy, efforts, money, and time in doing something or in producing a desired result. In a more general sense, it is the ability to do things well, successfully, and without ...
remains constant i.e.
, which is rarely exactly true, but can be a good approximation when used over appropriate frequency or diameter ranges (i.e., a fan will not move anywhere near 1000 times as much air when spun at 1000 times its designed operating speed, but the air movement may be increased by 99% when the operating speed is only doubled). The exact relationship between speed, diameter, and efficiency depends on the particulars of the individual fan or pump
design
A design is a plan or specification for the construction of an object or system or for the implementation of an activity or process or the result of that plan or specification in the form of a prototype, product, or process. The verb ''to design'' ...
.
Product testing
File:Consumer Reports - product testing - electric light longevity and brightness testing.tif, Testing electric light longevity and brightness testing
File:Consumer Reports - product testing - television testing laboratory.tif, Television testin ...
or
computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
become necessary if the range of acceptability is unknown, or if a high level of accuracy is required in the calculation.
Interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In engineering and science, one often has a n ...
from accurate data is also more accurate than the affinity laws. When applied to pumps, the laws work well for constant diameter variable speed case (Law 1) but are less accurate for constant speed variable impeller diameter case (Law 2).
For radial flow
centrifugal pumps, it is common industry practice to reduce the impeller diameter by "trimming", whereby the outer diameter of a particular impeller is reduced by machining to alter the performance of the pump. In this particular industry it is also common to refer to the mathematical approximations that relate the volumetric flow rate, trimmed impeller diameter, shaft rotational speed, developed head, and power as the "affinity laws". Because trimming an impeller changes the fundamental shape of the impeller (increasing the
specific speed
Specific speed ''N's'', is used to characterize turbomachinery speed.
Common commercial and industrial practices use dimensioned versions which are of equal utility. Specific speed is most commonly used in pump applications to define the su ...
), the relationships shown in Law 2 cannot be utilized in this scenario. In this case, the industry looks to the following relationships, which is a better approximation of these variables when dealing with impeller trimming.
With shaft speed (N) held constant and for small variations in impeller diameter via trimming:
The volumetric flow rate varies directly with the trimmed impeller diameter:
[
:
The pump developed head (the ]total dynamic head
In fluid dynamics, total dynamic head (TDH) is the total equivalent height that a fluid is to be pumped, taking into account friction losses in the pipe.
: {\rm h_{total} = \frac{P_2-P_1}{\rho g} + \frac{{v_2}^2-{v_1}^2}{2g
: TDH = Static Heig ...
) varies to the square of the trimmed impeller diameter:[
:
The power varies to the cube of the trimmed impeller diameter:][
:
where
* is the volumetric flow rate (e.g. CFM, GPM or L/s)
* is the impeller diameter (e.g. in or mm)
* is the shaft rotational speed (e.g. ]rpm
Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines.
Standards
ISO 80000-3:2019 defines a unit of rotation as the dimensionl ...
)
* is the total dynamic head
In fluid dynamics, total dynamic head (TDH) is the total equivalent height that a fluid is to be pumped, taking into account friction losses in the pipe.
: {\rm h_{total} = \frac{P_2-P_1}{\rho g} + \frac{{v_2}^2-{v_1}^2}{2g
: TDH = Static Heig ...
developed by the pump (e.g. m or ft)
* is the shaft power (e.g. W or HP)
See also
* Centripetal force
A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
References
{{DEFAULTSORT:Affinity Laws
Hydraulics
Pumps
Ventilation fans
Turbines