## Predicting The Performance Curves

The choices made in the foregoing design procedures can and should be verified analytically, the objectives being first to generate the performance characteristic curves for head and power at constant speed and second to ensure stable behavior of the various systems in which the pump is to be applied. For the first objective, the solution involves analytical or empirical approaches: a) at non-recirculating flow conditions; that is, from flow rates Q somewhat below QBEP out to the maximum "runout" flow rate, b) at shut-off (Q = 0) and low flow, or c) the complete set of curves for a given pump predicted by means of computational fluid dynamics (CFD).

Generating Performance Curves The fluid dynamical limitation on the deceleration of the relative velocity W determines the shape of the head-versus-flow curves. This is inherent in the choice made for the head coefficient C in Figure 12, which sizes the impeller and is illustrated in Figure 22. The typical situation of zero (or nearly so) inlet whirl Vu,1 = 0 means that the ideal head coefficient C equals the most significant ratio of the outlet velocity diagram because from Eqs. 15 and 31 (with for Vu,1 = 0):

Figure 22 illustrates how specific speed Q,s affects the BEP value of C and therefore C. Overall, only a small reduction of W occurs in most impellers. So, at low Hs, the low value of W1 associated with the small eye relative to the maximum diameter (Figure 9) enables the outlet velocity diagram (Figure 22a) to have a high value of Ve 2/U2. On the other hand, this ratio drops as Hs increases and the eye grows to be as large as the maximum diameter of the wheel. Figure 22b is the result because the value of C at shut-off (about 2) is not based on the one-dimensional concept of velocity diagrams but primarily on the pressure generated by solid body rotation of stalled (though recirculating) fluid contained within the impeller. The BEP values of C in Figure 22b are consistent with Figure 12 and illustrate why a high-specific-speed impeller has such a substantial "rise to shut-off" of the head curve. This is dramatically illustrated in Figures 8-10 of Section 2.3.1 in which the head curves are normalized to that of the BEP36.

a) Non-recirculating flows. The BEP efficiency and head can be determined from correlations for typical pumps or from computation of the losses. Fluid dynamic procedures described in this section can be used to determine the shapes of the head and power curves at all flow rates to runout, using the BEP as an anchor point for such computations. For pumps designed conventionally, beginning with Figure 12, Anderson's overall (BEP) efficiency correlation (Eq. 44) as modified in Figure 10 is useful. Other similar charts, especially Figure 6 in Section 2.3.1, are in widespread use. The breakdown of the losses involved, as expressed by Eqs. 8-11, is quantified through the development of the three component efficiencies hHY, hm, and hv in Table 3. All three decrease with decreasing specific speed—as might be expected from the charts just mentioned.

This can be seen in the hiy-expression (a) of Table 3 because C is greater at low Hs as discussed relative to Figure 22. Jekat's hjy-expression (b) of the table works surprisingly well, largely because of the flow effect in Figure 9 (explained there as the "size effect" of larger relative roughness and clearances in smaller pumps) and because low Q,s tends to go hand-in-hand with low flow rate Q.

To compute hm at Q QBEP (and, if required, at Q = QBEP as well), it is necessary to go deeper into the prediction of hHY by developing expressions for the losses noted in Eq. 21, which are basically expansions of the expression for the collector loss coefficient Zc26 and for the impeller loss expression (c) of Table 337. In this expression, the incidence loss coefficient k can be obtained from cascade data or developed as a combination of a turning and a sudden expansion loss4,8,27. The "pipe-type friction factor" f can be increased to include secondary flow and diffusion losses due to blade loading (or turning38 of the absolute velocity vector V). The resulting f-value can thus be twice the usual pipe value associated with the skin friction losses in the passage. (The pipe value of f is found from the well-known FIGURE 22 Performance versus specific speed: a) velocity diagrams at BEP; b) head-versus flow curves.

FIGURE 22 Performance versus specific speed: a) velocity diagrams at BEP; b) head-versus flow curves.

pipe friction chart—Figure 31 in Section 8.1—by substituting a representative average passage hydraulic diameter Dh = 4Ap/p for the pipe diameter d.) A further increase in this /'-value occurs if the impeller is missing one or both rotating shrouds; that is, it is a semi- or fully-open impeller with blade tip leakage losses appearing in the main flow stream39. Multiphase flows in pumps often are accompanied by greater than normal hydraulic losses; for example, increasing the concentration of solids in the carrier liquid flowing through a slurry pump increases the /-value still further40 (see Section 9.16.2).

Quasi three-dimensional TQ3DJ analysis41 affords an assessment of the secondary flow and diffusion losses and gives results similar to inviscid three-dimensional (3D) flow analysis. Q3D analysis starts by solving the 2D meridional (hub-to-shroud) flow field—as in Figure 14, but with blades present. This is followed by a series of 2D blade-to-blade

TABLE S Component efficiency expressions developed

A) Hydraulic Efficiency

Breaking up the main flow losses into impeller (including inlet)

EV,2

Lj = Ljmp + Lc and defining Lc = Çc — , where Çc i is the collector loss coefficient or fraction of V22/2 not converted into static pressure rise in the collector, we have (Eq. 10)

a(uv9

CcVih-A(UVe)

¥¡.2 | <t>i.2 2 2\Vll or, for no inlet whirl (Ve i=0) =v|/¡,2 :1231 ## Survival Treasure

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