Pump Affinity Laws for Centrifugal and Positive displacement pumps

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Pump Affinity Laws: The pump speed, head, and flow relationships are expressed by the affinity laws. These laws are mathematical expressions that define changes in pump Brake horsepower (BHP), head and capacity when a change is made to pump speed, impeller diameter, or both. Affinity laws also called as laws of similitude

Affinity laws are a set of equations or laws governing the operation (discharge-Q , head-H, power-P) of a single pump operated at different speeds or, by close approximation, governing the relationship of operation of pumps or impellers of different sizes operated under dynamically similar conditions.

A. Pump Affinity laws for centrifugal pumps or fans

1. Pump affinity laws constant impeller Dia

With pump impeller diameter ” D ” held constant and speed (N) change

Q = Capacity of pump (Flow rate) in Liters/Sec

H = Total Head in meters

N = Pump speed (RPM)

BHP = Brake Horse Power in kW ( Pump input power)

1 . 1 Flow rate (Q )changes directly proportional to the change in Speed of the pump(N)

Q α N $\frac{Q_{{1}}}{Q_{{2}}} = \frac{N_{{1}}}{N_{{2}}}$

1 . 2 Pump Head (H) Changes directly proportional to the Square of the change in Speed of the pump(N)

H α (N)² i.e $\frac{H_{{1}}}{H_{{2}}} = \frac{(N_{{1}})^2}{(N_{{2}})^2}$

1 . 3 Brake Horse Power (BHP) Changes directly proportional to the Cube of the change in Speed of the pump (N)

BHP α (N)³ i.e $\frac{BHP_{{1}}}{BHP_{{2}}} = \frac{(N_{{1}})^3}{(N_{{2}})^3}$

2. Pump affinity laws constant speed

With pump speed (N) held constant and impeller diameter ( D) change

2 . 1 Flow rate (Q )changes directly proportional to the change in impeller diameter ( D) of the pump

Q α D i.e $\frac{Q_{{1}}}{Q_{{2}}} = \frac{D_{{1}}}{D_{{2}}}$

2 . 2 Pump Head (H) Changes directly proportional to the Square of the change in impeller diameter ( D )

H α (D)² i.e $\frac{H_{{1}}}{H_{{2}}} = \frac{(D_{{1}})^2}{(D_{{2}})^2}$

2 . 3 Brake Horse Power (BHP) Changes directly proportional to the Cube of the change in impeller diameter ( D )

BHP α (D)³i.e $\frac{BHP_{{1}}}{BHP_{{2}}} = \frac{(D_{{1}})^3}{(D_{{2}})^3}$

The Affinity Laws are valid only under conditions of constant efficiency.

3. Affinity laws while changing both pump rotational speed & pump impeller diameter

Pump Affinity Laws for a Family of Geometrically Similar Pumps as follow as

3.1 – $\frac{Q_{{1}}}{Q_{{2}}} = \frac{N_{{1}}D_{{1}}}{N_{{2}}D_{{2}}}$

3.2 – $\frac{H_{{1}}}{H_{{2}}} = \frac{(N_{{1}}D_{{1}})^2}{(N_{{2}}D_{{2}})^2}$

3.3 – $\frac{P_{{1}}}{P_{{2}}} = \frac{(N_{{1}}D_{{1}})^3}{(N_{{2}}D_{{2}})^3}$

4. These affinity law relationships can be expressed by

4.1 – $\frac{Q_{{1}}}{Q_{{2}}} = \frac{N_{{1}}}{N_{{2}}} = \frac{(H_{{1}})^{1/2}}{(H_{{2}})^{1/2}}$

4.2 – $D_{{1}} = D_{{2}} \sqrt{\frac{H_{1}}{H_{2}}}$

4.3 – $Q_{{1}} = Q_{{2}} \sqrt{\frac{H_{1}}{H_{2}}}$

4.4 – $P_{1} \ = \ P_{2} \ \left [ \frac{H_{1}}{H_{2}} \right ]^{3/2}$

4.5 – $D_{1} \ = \ D_{2} \ \left [ \frac{Q_{1}}{Q_{2}} \right ]$

4.6 – $H_{1} \ = \ H_{2} \ \left [ \frac{Q_{1}}{Q_{2}} \right ] ^{2}$

4.7 – $P_{1} \ = \ P_{2} \ \left [ \frac{Q_{1}}{Q_{2}} \right ] ^{3}$

4.8 – $M_{1} \ = \ M_{2} \ \left [ \frac{N_{1}}{N_{2}} \right ] ^{2}$

where Q = flow rate
N = speed
H = head
P = power
M = torque

5. NPSH vs pump speed

To adjust pump Net positive suction head ( NPSH ) from the values recorded during test to another speed ( N ), the following formulas should be used

$NPSH_{1} \ = \ NPSH_{2} \ \left [ \frac{N_{1}}{N_{2}} \right ] ^{2}$

The affinity laws helpful to trim the impellers at site to suit the actual site conditions because the motor draws more current than that written by pump manufacturer ( on the name plate), the reason being the actual operating head is much lower than the estimated. Thus the pump operates at excessive flow.

Some times at actual site, flooded suction is available instead of suction lift specified. The pump discharge increases alerting under such operating conditions and the motor is likely to get overloaded under the situation. One of the ways to resolve such problems is to trim the impeller at site to suit actual operating conditions.

Pump affinity laws example for centrifugal pumps

As per the pump manufacture the specifications for that pump given as flow rate Q = 100 M³/hr, head = 70 metres, RPM = 1440, pump input power (BHP) = 21 kW and its impeller dia 240mm. The pump installed at site and its actual head requirement only 50 metres according to site condition.

Now to reduce the power consumption there is two options

Option – 1 : Reduces the speed of the pump by using the VFD

$\frac{H_{{1}}}{H_{{2}}} = \frac{(N_{{1}})^2}{(N_{{2}})^2}$

Here H1 = 70 , H2 = 50 , N1 = 1440 and N2 = ?

(N2)² = (50/ 70 ) x (1440)²

N2 = 1217 RPM

At the same time the power reduced

$\frac{P_{{1}}}{P_{{2}}} = \frac{(N_{{1}})^3}{(N_{{2}})^3}$

Here P1 = 21 kW , N1 = 1440 , N2 = 1217 and P2 = ?

P2 = 21 / 1.6566 = 12.67 kW

Option – 2 : Reduces the impeller Dia of the pump by trimming

$\frac{H_{{1}}}{H_{{2}}} = \frac{(D_{{1}})^2}{(D_{{2}})^2}$

Here H1 = 70 , H2 = 50 , D1 = 240 and D2 = ?

(D2)2 = (50/ 70 ) x (240)²

D2 = 203 mm

At the same time the power reduced

$\frac{P_{{1}}}{P_{{2}}} = \frac{(D_{{1}})^3}{(D_{{2}})^3}$

Here P1 = 21 kW , D1 = 240 , D2 = 203 and P2 = ?

P2 = 21 / 1.6525 = 12.70kW

The above estimated power is approximate value. Because the affinity laws are valid only under conditions of constant efficiency.The efficiency of the pump varies according to its head and flow.

B. Positive displacement pumps Affinity laws

6. Affinity laws for positive displacement pumps

Positive displacement pumps includes Reciprocating pumps, Rotary pumps, piston type pumps, Diaphragm pumps, Rotary Lobe Pumps, Progressive Cavity Pumps, Screw Pumps. All these pumps act different than centrifugal pumps because

Positive displacement pumps do not have a single best efficiency point (BEP)

There is no impeller shape (specific speed) to consider

There is no parallels for radial thrust loads exist as a function of the position on the curve relative to BEP.

There is no system curve to match

Positive displacement pumps capacity is a constant even if the head changes

Positive displacement pumps simply use the flow per revolution to make general comparisons. So the affinity laws for displacement pumps for speed change as follow as

6.1 Flow rate (Q )changes directly proportional to the change in Speed of the pump(N)

Q α N $\frac{Q_{{1}}}{Q_{{2}}} = \frac{N_{{1}}}{N_{{2}}}$

6.2 Pump Speed(N) does not direct effect on its differential pressure (Pump head – H)

6.3 Brake Horse Power (BHP) Changes directly proportional to the change in Speed of the pump (N)

BHP α N $\frac{BHP_{{1}}}{BHP_{{2}}} = \frac{N_{{1}}}{N_{{2}}}$

6.4 Pump Net positive suction head ( NPSH ) varies as per the speed of the pump is

$\frac{N_{1}}{N_{1}} = \left [ \frac{NPSH_{1}}{NPSH_{2}} \right ]^{X}$ Here X varies from 1.5 to 2.5

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2 thoughts on “Pump Affinity Laws for Centrifugal and Positive displacement pumps”

Ekeriode obaro peter

(January 25, 2020 - 9:40 pm)

A centrifugal pump having this data impeller dia192mm, rotational speed 2900rpm,flow100m cube perhour,head38.61mc…the pump is experiencing overload which result to pump tripping off when the discharge Valve is fully open.what is the problem with this pump

siva alluri

(January 26, 2020 - 4:31 pm)

What is the motor rated kW
Go through the below link and calculate required motor capacity
https://www.sugarprocesstech.com/pump-power-calculation/
http://siva.sugarprocesstech.com/PumpPowerCalculation/PumpPowerCalculation.htm

Pump Affinity Laws for Centrifugal and Positive displacement pumps

Affinity Laws Energy Savings | Pump Affinity Law Online Calculator