The Correct Formula for Using The Affinity Laws When
There Is a Minimum Pressure Requirement©
Richard R. Vaillencourt, PE, LEED AP
Canterbury Energy, LLC
Abstract
When a pump or fan system, or its operation, is not simply overcoming friction, the
Affinity Laws appear to overstate the savings. However, by carefully using the Affinity
Laws, the correct adjustment to the speed ratio for a system with a minimum pressure
requirement can be mathematically defined and applied. With the correct speed ratio
formula, applying the various Affinity Law equations without modification will deliver
the correct results.
Introduction
The application of Variable Frequency Drives (VFD) has proven to save energy for
pumps and fans with variable flow requirements. However, there has been considerable
debate about how to apply the Affinity Laws to calculate the savings. Many individuals
have argued that the Affinity Laws are theoretical and do not apply to actual situations.
A common response has been to modify the Affinity Laws by arbitrarily “adjusting” the
exponent in the horsepower equation to some number between 2 and 3, based upon
personal experience.
Certainly, when there is a minimum system pressure requirement, the direct application
of the equations results in an overstated amount of savings. The problem, however, lies
in the choice of the formula for the speed ratio, not the exponent.
Discussion
The Affinity Laws define the relationship between RPM, flow, pressure, and horsepower
by equating the percent change in RPM with the percent change in these three
parameters. The three common formulas are:
Equation 1
 N2  F2


 N1  F1
Equation 2
2
H2
 N2 

 
H1
 N1 
Equation 3
3
HP 2
 N2 

 
HP1
 N1 
N = RPM; F = Flow (GPM, CFM); HP = Horsepower
© 2005 Richard R. Vaillencourt
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It will be useful to refer to the various ratios as the (%N), (%F), (%H), and (%HP).
(These are not true percentages unless multiplied by 100, but it is easier than continually
repeating something like “the ratio of (…) expressed as a decimal”, etc.)
These equations can be directly applied when the piping distribution system and end
loads represent only friction losses.
Eq. 1 is the equation of a straight line: y = x. The general “slope-intercept” equation for a
straight line is:
Equation 4
y = mx + b
Where m = the slope of the line and b = the y-intercept.
When there are only friction losses: m = 1 and b = 0. Under these conditions y = x, or
 N2  F2


, and the curve is a straight line with a 1:1 slope passing through the origin
 N1  F1
on a graph and the point (1,1).
Figure 1
100%
90%
80%
70%
%N
60%
50%
40%
30%
20%
10%
0%
0%
10%
20%
30%
40%
50%
%F
© 2005 Richard R. Vaillencourt
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60%
70%
80%
90%
100%
A system curve relates pressure to flow in the basic equation: y = ax2. A system curve
with only friction losses will take the shape in Fig. 2. When x = 0, y = 0 and the curve
starts at (0,0).
Figure 2
60.00
50.00
H
40.00
30.00
20.00
10.00
150
140
130
120
110
90
100
80
70
60
50
40
30
20
0
10
0.00
GPM
However, if there is a static pressure head to overcome, i.e., an open cooling tower, or
any other minimum pressure requirement, the system curve is moved up the Y-axis by
the minimum pressure. Fig. 3 represents the system curve for a system with a 35’
minimum pressure requirement (Hmin). This could be could be the actual height
difference between the suction level of a sump and the discharge level of the system. Or
it could be a control setpoint for a variable frequency drive to provide enough pressure to
meet the P requirements of an evaporator bundle and chilled water coils of a chilled
water distribution system, etc.
Figure 3
90.00
80.00
70.00
60.00
H
50.00
40.00
Hmin
30.00
20.00
10.00
GPM
© 2005 Richard R. Vaillencourt
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150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0.00
If the pump or fan is operating at a speed that will not produce sufficient pressure to
overcome the minimum pressure requirement, there will be zero flow in the system. The
correct application of the second Affinity Law will tell you what that minimum pump
speed is that will be necessary to produce that minimum pressure even at zero flow.
Under these conditions (a 35’ minimum pressure requirement), Eq. 1 is no longer simply
y = x. The graph of Eq. 1 will now look like Fig. 4. We now need to determine the
values of the slope (m) and the y-intercept (b) to develop the correct speed vs. flow
equation.
Figure 4
100%
90%
80%
70%
%N
60%
50%
40%
30%
20%
10%
0%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90% 100%
%F
To find (m) and (b) we must look at the operating curve of the pump (often called the
“pump curve”). The following is the typical shape of a pump curve:
© 2005 Richard R. Vaillencourt
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Figure 5
100.00
90.00
80.00
70.00
H
60.00
50.00
40.00
30.00
20.00
10.00
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0.00
GPM
At a specified design flow rate (FD), there is a corresponding design pressure (HD). This
information is readily available on the nameplate of most pumps. These are the
parameters when the pump is operating at the nameplate speed of the motor (N1). We
can now refer to N1 as ND.
Figure 6
100.00
90.00
80.00
70.00
HD
H
60.00
50.00
40.00
30.00
20.00
10.00
GPM
150
140
130
120
110
100
FD
90
80
70
60
50
40
30
20
10
0
0.00
Adding the system curve with minimum pressure requirement to the pump curve looks
like this:
© 2005 Richard R. Vaillencourt
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Figure 7
100.00
90.00
80.00
70.00
HD
H
60.00
50.00
40.00
Hmin
30.00
20.00
10.00
GPM
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0.00
FD
By applying the second Affinity Law we can determine the pump speed required to
develop Hmin.
Equation 5
2
H2
 N2 

 
H1
 N1 
2
Therefore:
H min
 N min 

 
HD
 ND 
Solving for the % N:
Equation 6
 N min 


 ND 
H min
HD
The ratio (Nmin / ND) can be called the “Minimum % Speed”. Operating the pump at an
RPM below Nmin will result in a pressure output below the minimum pressure
requirement for flow (Hmin).
The question is: what is the formula for the % N that will allow us to insert it into the
Affinity Law equations and get the right answer when there is a minimum pressure
requirement? In other words: what is the equation of the line for Fig. 4?
Eq. 7 is the equation for the line in Fig. 4 after substituting %N for y and %F for x.
Once we have this equation, we need to determine (m) and (b) in Eq. 7.
© 2005 Richard R. Vaillencourt
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Equation 7
 N2
 F2 

  m   b
 N1 
 F1 
Since %F (x) is zero when the %N is at the minimum speed which we established was
H min , then:
HD
equal to:
Equation 8
H min
 m0  b , therefore
HD
H min  b .
HD
Having determined the y-intercept, the slope (m) can be defined in terms of (b) by using
another standard equation for a straight line: the Point-Slope Form.
Equation 9
 y 2  y1 
m 

 x 2  x1 
We know the values of %N and %F at two points: the design point and the Hmin point. At
the design point, N2 = ND and F2 = FD. Therefore; equation 1 becomes:
Equation 10
 ND  FD
1


 ND  FD
Therefore, the first point we know is the point (1,1).
The other point that we know is the y-intercept: (0,b). That is: %F = 0 when the %N
drops to
H min . Therefore the Point-Slope formula becomes:
HD
Equation 11
 1 b 
m 

 1 0 
Substituting and simplifying, the slope can be defined in terms of the Minimum % Speed.
© 2005 Richard R. Vaillencourt
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Equation 12

 1 H min

HD
m 
1






 Therefore: m  1 H min

HD


Finally, the question after Eq. 6; “What is the formula for the % N that will allow us to
insert it into the Affinity Law equations and get the right answer when there is a
minimum pressure requirement?” can be answered.
Substituting our expressions for (m) and (b) into Eq. 7 gives us the general equation for
the % N that will provide the correct answer for all cases when applying the Affinity
Laws.
Equation 13
H min
 N 2  

  1 
HD
 N1  
  F2 
  
  F1 

H min
HD
 N1 
This entire formula is substituted for 
 when applying the Affinity Laws. Therefore
 N2
Eq. 2 becomes:
Equation 14

H min
1 
HD

  F2 
  
  FD 

2
H min 
 H2

 
HD 
 H1 
Eq. 3 becomes:
Equation 15

H min
1 
HD

  F2 
   
  FD 

3
H min
HD
© 2005 Richard R. Vaillencourt
  HP 2 

 
  HP1 
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Conclusion
The Affinity Laws do not require modifications to make the theory work for real-world
applications. The only requirement is to address the fact that there may be a minimum
speed requirement to meet a minimum pressure requirement. Once that is included in the
formula, no changes to the Affinity Law equations are required.
Note that as Hmin approaches 0, the equations become the original Affinity Laws. Also,
as Hmin approaches HD the % speed approaches 100% indicating that the flatter the
system curve, i.e., the closer the minimum pressure requirement is to the design pressure,
the lower the savings potential from a VFD. Savings are only available for pressures
between the minimum pressure and the design pressure.
Biography
Richard R. Vaillencourt, PE, is the sole proprietor of Canterbury Engineering Associates,
a consulting firm specializing in supporting other energy engineering companies with
energy audits and assisting customers in the intelligent use of energy.
Canterbury Engineering Associates
PO Box 459
Canterbury, CT 06331
(860) 546-1124
rrvaillencourt@aol.com
© 2005 Richard R. Vaillencourt
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