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Elusive Energy Savings:
Centrifugal Pumps and Variable Speed Drives - Part II

Copyright © 2002 Francis J. Martino

To calculate an operational point on the Head vs. Flow curve, the equation
of a specific system may be used if any two points on the curve are known.
The two points will be selected from the pump manufacturer's curve.
See Figure 4

The first point is at the maximum rating of the system of 154 feet of
head at 863 GPM. The second point is at the intended operational point on
Curve E of Figure 4 at 110 feet of total head at 680 GPM.

The general equation for a centrifugal pump is:

H = a + bQ + cQ2

where
H = total head, which is the sum of both static and dynamic head in feet.
a = static head in feet.
b, c = system constants that will be a different value for each system.
Q = flow in GPM.
bQ = head required to move the fluid.
cQ2 = head required to overcome frictional losses.

With two unknown quantities (b and c), two equations are needed which
must be solved simultaneously. The two above selected points give the
following equation:

154 = 30 + b(863) + c(863)2
110 = 30 + b(680) + c(680)2

Solving for b and c:

    154           30          b(863)       c(863)2
             =              +              +             
    863          863           863           863


.17845 = .03476 +    b     + c(863)
.14369 =      b    + c(863)


    110           30          b(680)       c(680)2
             =              +              +             
    680          680           680           680


.16176 = .04412 +    b     + c(680)
.11764 =      b    + c(680)


               .14369 =     b    + c(863)
minus:     .11764 =     b    + c(680)
                                                 
               .02605 =              c(183)

solving for c:

       .02605 
c =              = .00014
          183


solving for b:

.14369 =     b    + (.00014)(863)

b = .14369 - (.00014)(863) = .02287


Checking the results by placing b and c into the original equations:

154 = 30 + (.02287)(863) + (.00014)(863)2
         30 + 19.73681 + 104.26766 = 154.0047

110 = 30 + (.02287)(680) + (.00014)(680)2
         30 + 15.5516 + 64.736 = 110.2876

The system constants can now be used to determine the system head requirement for
any amount of GPM, and the GPM may be calculated for any system head requirement.

What will be the system head on curve E for a flow of 600 GPM?

H = a + bQ + cQ2

H = 30 + (.02287)(600) + (.00014)(600)2
H = 30 + 13.722 + 50.4
H = 94.122 Feet


What will be the GPM at a total head of 94 feet?

H = a + bQ + cQ2


The Quadratic Equation must be used to solve the above equation:

      -b +/- (b2 - 4ac)
Q =                         
            2c

In order to use the Quadratic Equation, the terms must be transposed
so that the sum of the terms equals zero:

94 = 30 + (.02287)Q + (.00014)Q2
0 = -64 + (.02287)Q + (.00014)Q2


      -.02287 +/- [(.02287)2 - 4(-64)(.00014)]
Q =                                                           
                              2(.00014)



      -.02287 +/- (.000523 + .03584)
Q =                                                
                           .00028


      -.02287 +/- .03636
Q =                                
                  .00028


      -.02287 +/- .19069
Q =                                
                  .00028


The equation has two solutions of which only one is correct for the pump system:

        -.02287 + .19069             .16782
Q =                                 =                   = 599.3571
                  .00028                  .00028


        -.02287 - .19069              .21356
Q =                                 =                   = -762.714
                  .00028                  .00028


The Quadratic Equation defines a parabolic curve that crosses the
vertical y-axis at y = -64, and crosses the x-axis at two points which are
+599.3571 and -762.714. Since the flow can not be negative, a GPM of
599.3571 is the solution for the system.

Transposing the terms to give the original system equation of

94 = 30 + (.02287)Q + (.00014)Q2

moves the parabolic curve upward on the y-axis so that the y-axis is
crossed at y = 30 and the x-axis is not crossed at all, however, the
coordinates of the curve with respect to the x-axis remain constant. Thus,
the GPM at 94 feet of head is 599 which agrees with Curve E of figure 4.

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