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Basic Hydraulics - Affinity Laws
Many pumping applications require a pump to be operated at speeds that are different from the published performance curves. The affinity laws are mathematical relationships that allow the performance of a pump to be determined at these different speeds.
The affinity laws are applied to a specific point on a performance curve. The speed in rpm, head, capacity, and power draw must be known for that point. The new speed is estimated and a calculation is made to determine if the new speed will meet the desired condition point. There is a certain amount of trial and error that must be done to find the new speed that satisfies the requirement. (It is assumed that there is no change in the pump's efficiency.)
Constant Impeller Diameter
For changes in pump speed with a constant impeller size, the affinity laws
are:
1. Capacity varies directly with the ratio of the change in rpm.
2. Head varies as the square of the ratio of the change in rpm.
3. Absorbed Power varies as the cube of the ratio of the change in rpm.
Expressed in the form of an equation:
Where:
| n | = | new speed in rpm | n1 | = | speed in rpm of the curve | Q | = | capacity at n | Q1 | = | capacity at n1 | H | = | head at n and Q | H1 | = | head at n1 and Q1 | P | = | absorbed power at n, Q and H | P1 | = | absorbed power at n1, Q1 and H1 |
Impeller Diameter Trim Normally,
centrifugal pump impellers are trimmed to the specific diameter required
to meet the operating condition points, and are operated at synchronous
motor speeds. However, elastomer lined and hard metal impellers in
slurry pumps are often impossible, or at the very least difficult
to trim, thus slurry pump operating speeds must be adjusted using
belt and sheave or variable frequency drive controls.
Slurry pump impellers are rarely trimmed to meet operating condition points.
The elastomer covered or hard metal impellers required by these pumps are difficult
to reduce in diameter and every effort should be made to meet the condition
by changing the pump speed. However, certain installations will require a pump
to operate at a fixed speed and the impeller must be trimmed.
Various published handbooks and data sheets use a modified form of the affinity laws to estimate the performance of a pump when the impeller diameter is reduced. In this scheme, the ratio of the impeller diameter change is substituted for the speed change ratio. This always results in impeller diameters that are too small to meet the new condition point. When the impeller diameter is changed, the geometric relationship of the pump parts is changed significantly and the theory behind the affinity laws no longer applies. This approach should not be used.
Better results can be obtained with the formula given below. This method allows for the effects of the geometric changes and results in a curve that is steeper than the original. For changes in impeller diameter at a constant rpm, the following approximation is used:
Where:
| D2 | = | original outlet diameter | Q1 | = | capacity at D2 | H1 | = | head at D2 | D2TR | = | trimmed outlet diameter | Q | = | capacity at D2TR | H | = | head at D2TR |
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