SUBJECT : Calculating the centrifugal pump first critical speed 5-4
In an other paper I addressed the radial deflection of a centrifugal pump shaft when the pump operated off of its best efficiency point. We calculated the magnitude of the deflection from the basic formula:
I reviewed this formula with you because we are going to use the same formula to learn the first critical speed of a centrifugal pump.
At this point it is important to note that any object made from an elastic material (and metal is an elastic material) has a natural period of vibration. This happens because the pump rotating assembly is not absolutely uniform around the center line of the shaft. We get variations in the density of the materials as well as manufacturing tolerances and casting irregularities contributing to the problem.
This eccentricity produces deflection when the rotating assembly rotates at the speed the centrifugal force exceeds the elastic restoring forces. At this speed the assembly will vibrate as if it were unbalanced, and could fail the seal, bearings or fatigue the shaft itself. The lowest speed at which this happens is called the first critical speed.
The first critical speed is linked to the pump's static deflection. We can calculate this deflection by going back to the original formula and substituting the weight of the rotating assembly for the "W" in the formula. You can use either pounds or Newtons.
It should also be noted that this critical speed can be very destructive in mixer and agitator applications because of their very high L3/D4 numbers.
Now that you have calculated the static deflection (sag) of the shaft as measured at the impeller, we will use this number to calculate the first critical speed of the pump. For all practical purposes you can calculate the first critical speed by using one of the following formulas:
As you can see, these numbers are well in excess of the 1750 or 1450 rpm. that we normally use for centrifugal pump speed. They are, however, lower than the higher speed pumps that run at 3500 rpm. or 3000 r.p.m. This means that higher speed pumps and variable speed pumps will experience shaft deflection as they pass through or run at these critical speeds.
Since operation off of the B.E.P. is common for centrifugal pumps, you will be experiencing shaft loads well in excess of those noted in the above examples&emdash; meaning that your critical speed will actually be experienced at a much lower rpm. than noted. The numbers we calculated reference a shaft running in air. In actual practice the impeller and a major portion of the rotating assembly is immersed in liquid that provides a hydrodynamic support to help stabilize the assembly. This hydrodynamic stabilizing force is referred to, by pump people, as the "Lomakin Effect".
Shaft packing provided an additional stabilization affect, but it was lost when the modern pumps were converted to mechanical face seals. Closed impeller pumps continue to retain some of the effect in their wear rings (this is, in fact, the major cause of wear ring wear).
In addition to the radial force created by passing through a critical speed, the rotating assembly is subjected to additional radial loads:
All of these radial forces will have a major affect on the life of the seal and bearings, as well as the shaft itself. Since it is almost impossible to calculate all of these changing forces in advance, it is important for you to stabilize the shaft as best you can to hold the deflection to an absolute minimum. Your options include:
Changing the shaft material will not help. All the common shaft materials have just about the same modulus of elasticity (stiffness):
If you are purchasing a new pump try to purchase larger diameter or shorter shafts when ever possible. The L3/D4 number referred to in other papers of this technical series is as good a guide as any thing else you can use.
Converting packed pumps to a mechanical seal presents a major shaft stabilization problem to the pump manufacturer. Some day the ANSI. (American) and ISO. (European) standards will be modified to compensate for this change. Between now and then you'll have to provide your own stabilization if you want to achieve satisfactory seal and bearing life.
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