## Va2 a a

is quantified in the expressions given for the swirl velocity component Vs. These expressions are curve fits to experimental data for the leakage flowing radially inward on either or both sides of the impeller50 and for outflowing leakage as occurs on the back side (away from the "front" or suction side) of multistage pump impellers due to the higher pressure arising from the diffusing system downstream of each impeller51. In the absence of leakage, the fluid in the sidewall gap rotates at about half the local impeller wheel speed; that is, V = Hr/2, and this half speed is typical of the gap flow near the impeller OD, even in the presence of leakage. The greater the inflow leakage, the lower the pressure becomes at the entrance to the ring clearance. The major effect is that of swirl or the tangential component of velocity Ve, which varies inversely with radius unless casing wall drag interferes. More leakage flow is less influenced by this drag and so experiences a greater centrifugal effect. This in turn means more pressure drop from OD to ring. (The leakage rate, of course, is affected, the solutions for both leakage and pressure distribution being linked and usually requiring iteration.) The opposite effect happens for outflow on the back side of multistage pump impellers. Here the fluid enters the sidewall gap at a small radius (see Section 2.2.1) and so with negligible swirl. It flows outward without picking up much swirl, especially if there is substantial radial outflow leakage, which means the centrifugal effect is small, yielding a nearly constant pressure versus radius. The overall result is more net thrust than might be expected from a cursory look at the pressure-loaded surfaces.

If wear ring clearances increase during the life of the pump, the net thrust of multistage pump impellers increases. Likewise, unequal ring wear leads to uncertain changes in the thrust of a "balanced" single- or double-suction impeller with inflow to wear rings on both sides. Similarly, these theories can be applied to balancing drums and other such devices described in Section 2.2.1.

Integration of the pressure equation in Table 4 becomes a chore unless the whole theory is computerized. A quick estimate of the thrust is possible, however, if the distributions of pressure in the separable domains of the surfaces are assumed to be linear; in that case,

2.1 CENTRIFUGAL PUMP THEORY TABLE 5 Approximate axial thrust calculation

the integration is simple and yields the closed-form results of Table 5. This table also indicates how to account for each element of the thrust, including the axial momentum terms, which become significant for higher-specific-speed mixed-flow impellers. In all cases, in order to proceed with the calculation, the static pressure at the impeller OD must be known, as it is from the boundary condition imposed at the OD that the rest of the pressure distribution emerges. Even for substantial leakage, the pressure drop of the fluid entering the sidewall gaps from the impeller exit is small or negligible; therefore, the impeller pressure essentially applies in the gap (at the OD) as well. The foregoing methods of predicting pump head also yield the impeller OD pressure, which is usually between 75 and 80% of the stage pressure rise above inlet.

Thrust computations can therefore be coupled with the head-curve prediction scheme being employed for the pump, thereby yielding predicted thrust curves together with the predictions of hydraulic performance.

Ensuring Stable Performance The ability for a pump to run smoothly with minimal pressure-rise and flow-rate excursions is dependent on the shape of the pump head-flow performance curve and the characteristics of the system in which it operates. There are two types of pump-system instability; namely, a) static instability, which can be ascertained by studying the pump and system head curves, and b) dynamic instability, which requires more detailed knowledge of the system52. (In addition to these system-related instabilities, there is the unsteady behavior of the separated and recirculating flows that occur when a pump operates a flow rate substantially below the BEP. Called hydraulic instability, this becomes important in higher-energy applications and is therefore discussed later.)

a) Static stability and instability. Figure 24 illustrates two pumping systems; namely, a) a piping system in which the flow is turbulent and largely independent of Reynolds number; so, the head drop AH through it is proportional to Q2, and b) two reservoirs with a constant difference AH between the two liquid surfaces and comparatively negligible head loss in the pipes connecting them. In each case, the pump is designed to produce head AH, as required to deliver the desired flow rate Q. The influence of the pump head curve shape is immediately appreciated in Case (b): the curve "droops" as Q is reduced to shut-off, thereby producing two vastly different flow rates at the same head. In fact, however, the pump will not operate at the lower-Q intersection point of the two curves. The pump shut-off head is less than AH, so it will produce no positive flow rate. Instead, as discussed in Section 2.3.1, fluid will flow backwards through the pump. Further, if circumstances could allow operation at this lower-Q point, even a vanishingly small increase of Q would cause a further, divergent increase because the head of the pump exceeds the AH of the system. Likewise, a small decrease leads to even lesser Q because the system AH exceeds that of the pump. This is called "static instability." Conversely, the higher-Q point of Figure 24b is "statically stable," small departures in Q being suppressed by algebraically opposite signs of the difference between the system and pump heads. Both intersection points of Figure 24a are seen by this type of analysis to be statically stable. If the operator increases the frictional resistance by closing up a valve in the piping system, the operating point simply moves to the left on the curve and remains stable. Thus, it is concluded that if the slope of the pump H-versus-Q curve is less than that of the system, operation will be statically stable—and vice versa.

Most pumping systems are combinations of the "pure friction" type of Figure 24a and the "purely elevation" type of Figure 24b. In this case of static stability, the drooping pump head curve presents no problem. Theoretically, it is possible to have a pump head curve with a kink that could have a more positive slope than that of a system curve, which might intersect it at such a kink. The high-Hs curve of Figure 22b depicts a kink or dip, which is due to stalled flow within a mixed- or axial-flow impeller that is not sufficiently confined by the impeller to be maintained in solid body rotation. (It is the centrifugal effect of such rotation that maintains the pressure rise in radial-flow impellers despite the stalling.) That particular pump, if applied to a system that never intersects the pump head curve at the kink, would never experience static instability. On the other hand, the designer may want to take on the challenge of designing a machine without such a kink. The previously mentioned design procedure utilizing CFD in both the impeller and the diffuser to check whether the kink is gone would be a way of tackling this problem53.

In-depth discussion of the variety of systems that can be encountered, including multiple systems and parallel operation of multiple pumps, can be found in Sections 2.3.1, 8.1, and 8.2. Purely from a static stability standpoint, most of these situations demand a substantially negative slope of the head curve throughout the range of flow rate Q—or what is commonly specified as a pump with head "continuously rising to shut-off." (This "rise in head" versus a drop in Q should not be confused with the pump developed head H, which is also properly termed the "head rise" AH, produced by the pump at a given value of Q.)

b) Dynamic instability. If the system has appreciable capacitance, operation may not be stable if the slope of the pump head curve is positive or even zero54. This is true even though the slope of the pump head curve is less than that of the system head curve as required above for static stability—as with the lower-Q intersection point of Figure 24a. Dynamic instability can be manifested as pump surge, a phenomenon wherein the flow

rate oscillates and can even be alternately positive and negative through the pump54. This is characteristic of "soft" systems that contain vessels with free surfaces and, therefore, appreciable capacitance. Two-phase flow increases the capacitance of a system and can cause dynamic instability. For example, fluctuating vapor volume within the propellant pump inducers can contribute to the dynamic instability of a rocket propulsion system55.

On the other hand, a "hard" piping system with no capacitance is theoretically capable of accommodating a pump that has a flat or drooping head curve and that operates on that flat or drooping section of the curve. Low-specific-speed pumps can have drooping curves (Figure 22b), especially if designed with a high head coefficient C at the BEP. Figures 5 and 7 of Section 2.3.1 depict flat and drooping curves of low-Hs radial-bladed pumps, a type that is widely used in low-cost, small sizes. If properly applied, such machines will operate with stability. On the other hand, a conservative approach that guarantees both static and dynamic stability for the widest range of applications is to design all pumps without flat spots or droops in the head curves.

## Survival Treasure

This is a collection of 3 guides all about survival. Within this collection you find the following titles: Outdoor Survival Skills, Survival Basics and The Wilderness Survival Guide.

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