In reciprocating pumps, compressors, etc., the crankshaft and crank move a connecting rod and piston in a cylinder (Figure 1). The crankshaft center is placed at 90° on the 0°–180° x axis; the crank's rod bearing is shown at 45°. Clockwise rotation of the shaft will cause the crank bearing to generate a sinusoidal x-y curve between 0° and 360°. Figure 2 shows this sinusoidal velocity curve between 0° and 180°. It also shows the actual velocity curve for a piston when the rod length divided by crank length is 2.1. It is not sinusoidal.
Flow recorders could provide a pump flow profile but it's much easier to use simple trigonometry to calculate piston positions versus crank angles, then calculate piston travel per degree of crank rotation and plot the results. They are not sinusoidal. (Compressors and piston engines would follow similar curves.)
I've observed the following:
- As already mentioned, peak flow rates are up to 10% higher than a sinusoidal curve predicts.
- The curve shape depends on the ratio of rod length to crank length.
- Peak rate doesn't occur at the 90° point but rather at 95°–120° depending on the ratio.
- From 180° to 360° (the suction portion), the curve is a mirror image of the 0° to 180° discharge portion.
- During the suction portion of the curve, flow rates also are higher and peak earlier than the 270° point.
- The curve also will change if the centerline of the cylinder doesn't pass through the center of the crankshaft.
- Multi-piston pumps and compressors provide less "smoothing" effect than predicted because the bell-shaped curve has a sharper peak.
The difference between actual and sinusoidal curves affects many facets of operation and mechanical design:
- Check valves and passages will have higher-than-predicted peak flow rates, and pressure drop will be higher -- by the square of flow rate (20%).
- This will impact NPSHR and possibly induce vaporization as the column of liquid in the suction pipeline is accelerated through valves, etc.
- Even a small amount of vaporization or release of dissolved gas will reduce pump discharge rates substantially. Because flow meters in pulsation flow aren't practical, difficulties often first appear downstream as corrosion or chemical problems. Plant engineers with any indication of such problems should check the many NPSH factors such as air pockets in the pump, supply tank level error, plugs in valves or piping and check-valve backflow.
- Maximum rotation per minute (rpm) will be lower than indicated by the sinusoidal curve. If a pump is running near the manufacturer's maximum recommended speed, starved suction is a possibility.
- Loads on bearings will rise somewhat, especially in high-speed compressors.
- Stresses in connecting rods will increase but rod failure is rare.
- Manufacturers of pulsation dampeners and surge suppressors use the sinusoidal curves in their literature and possibly sizing formulas. Yet, surge dampeners must handle the sharper peak of a bell curve compared to a sinusoidal curve. While these devices can't provide constant or pulse-less flow, they do make a major contribution unless terribly undersized, improperly installed or the gas cushion is lost.
The above comparisons should help you understand the real flow environment and thus avoid NPSH and pipe-sizing errors. Calculating stresses due to velocity and acceleration, possibly with simple algebra (see sidebar), can aid in optimizing design of pumps, compressors and engines.
It's all too easy, even in engineering, to accept what appears to be logical. Unfortunately, incomplete or inaccurate analysis leads to less-than-optimum results in many different fields. Another common failing is inappropriately extrapolating data -- see, for instance, "Show Some Skepticism."
ARTHUR H. KRUGLER, P.E., is president of Krugler Engineering Group, Inc., Whittier, Calif. E-mail him at email@example.com.