Reciprocating pumps are often used in the chemical process industries (CPI) because of their ability to generate high pressures at low velocities. A subcategory of positive-displacement pumps, reciprocating pumps act through the recipricating motion of a piston, plunger or diaphragm. Such pumps work by way of a connecting-rod-and-crank mechanism with a piston.

By nature, reciprocating pumps generate pulsing flow, which, when plotted as a function of time, or of crank angle, produces a curve that resembles a sine wave to a first approximation. For example, manufacturers of pulsation dampeners and surge suppressors often use sinusoidal curves for piston pumps and compressors in their product literature and sizing formulas. However, a closer examination of the flow profile for a piston-and-crank pump or compressor reveals the curve to be a significantly distorted sine wave because of the interaction between the crank and the connecting rod.

**Calculating flowrate**

In graphical form, the crank and crankshaft of a reciprocating pump can be visualized by placing the crankshaft center at the 90-deg mark of a 180-deg x-axis, and placing the crank bearing at the origin (see figure). A connecting rod links the crank to the piston.

Determining the position of the piston at any crank angle can be accomplished by measuring on a piston pump, compressor, or piston engine, or it can be calculated using trigonometric relationships.

The degree to which the actual flow profile curve deviates from the sinusoidal curve is determined by the ratio of the connecting rod length to the crankshaft length. Smaller values of the ratio translate into greater levels of distortion. As the connecting rod becomes very long, the flow profile would approach the sine curve

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To calculate the flowrate at a given crank angle, use the following procedure and definitions:

Crank length =

*OC*Piston rod length =

*CP*For any angle a, Line

*AC*=*OC*sin (*a*)Line SA =

*OC*–*OC*cos (*a*)Line AP = (CP

^{2}– AC^{2})^{0.5}Line SP = AP + SA

1. Calculate the piston position for two crank angles, perhaps 2 deg apart.

2. The difference in piston positions equals piston displacement over the time interval between the two crank angles. The value is an average over the span of the two readings, not an instantaneous reading. As the step size approaches zero, displacement nears the true velocity.

3. This value can be converted into flowrates (gal/min or other units) if the piston diameter and speed (revolutions per minute, rpm) are known.

Observations of the plot

In an illustrative example, plots of piston velocity versus crank angle are shown (see graph). The ratios of the connecting rod length to crank shaft length are 1.05 to 1 (blue line), 2 to 1 (red line) and 5 to 1 (green line). The following observations can be made:

1. At the beginning of the discharge stroke, flowrate approaches zero asympotically, rather than as a sinusoidal curve

2. Peak flowrates do not occur at the 90-deg point, but rather at 95–120 deg, depending on the ratio of rod length to crank length

3. Peak flowrates are higher than would be predicted with a pure sine curve

4. From 180 to 360 deg (the suction portion of the pump cycle), the curve mirrors the 0-to-180-deg portion

5. Flowrates during the suction portion of the curve are also higher and occur earlier than the 270-deg point

**Effects of distorted sine curve**

Within the areas of fluid flow and mechanical pump design, there are a number of aspects that are affected by the deviation of flow profile from a perfect sine curve for pumps and compressors. The effects include the following:

• Check valves and passages will have higher-than-predicted peak flowrates and pressure drop will be higher, by the square of flowrate

• The higher flowrates and pressure drops will affect net positive suction head (NPSH) and possibly induce vaporization

• Maximum crank revolutions per minute will be lower than what would be allowed by the pure (non-distorted) sinusoidal curve

• Loads experienced by bearings will increase somewhat, especially in high-speed compressors

• Stress analysis of the connecting rods will be affected

• Surge dampeners must handle the sharper peak of a bell curve, rather than a smoother sine curve

• Multi-piston pumps and compressors would have less “smoothing” effect than would be predicted because the bell-shaped curve has a sharper peak

**References**

1. McGuire, J.T., “Pumps for Chemical Processing,” Marcel Dekkar, New York, 1990.

2. Henshaw, T.E., “Reciprocating Pumps,” Van Nostrand Reinhold Co., New York, 1987.

3. Krugler, A., Piston Pumps and Compressors: Exploring the Flow Profile, Self-published, 2010.