L

, t + f — — 2 2 Dh where D|, = passage hydraulic diameter,

( = passage length f = pipe-type friction factor k = incidence loss coefficient

W2 = the average square of the relative velocity within the passage along the rms or mean meridional streamline

W) = relative velocity just upstream of the impeller blades on the rms streamline

TABLE 3 Continued.

inviscid solutions42, each on a surface of revolution generated by one of the meridional streamlines of the hub-to-shroud 2D solution and producing results like that of Figure 17. From this, one computes the diffusion factors (Eqs. 57—59) and decides whether the diffusion losses are significant—in which case a redesign is in order, followed by a further Q3D evaluation. This type of iterative design approach for impeller blading has led some designers to combine Q3D analysis with an "inverse" design approach and a performance prediction scheme as discussed in this subsection. Here, in distinction to the more common "direct" choice of the conformal blade shape (Figure 19) between inlet and outlet as described in paragraph (e) (vi) under "Designing the Impeller," one specifies the distribution of fluid dynamical quantities from inlet to outlet—such as UVe or W—and finally produces the corresponding blading17,43. In this sense, specifying Wg as described in the same paragraph (e) (vi) is an inverse design procedure.

Mechanical efficiency ■qm) as stated earlier, is largely the result of impeller disk friction. If the drag of bearings and seals is added, as in Eq. (d) of Table 3, the moment coefficient Cm in the disk friction formula (e) can be increased over known disk friction values4445 to include these effects. (On the other hand, the drag power loss of shaft seals, though usually quite small, is generally directly proportional to speed. Such losses can therefore be significant in small pumps running at lower-than-normal speeds.) The Cm-expression given in Formula (f) reflects this adjustment and includes the drag on both sides of a smooth impeller for a typical clearance ratio s/a = 0.05, where a is the disc radius. This works well for most impellers: The drag at the ring fits roughly compensates for the fact that the impeller eye has been cut out of the disk, and so on. (There is very little influence on Cm of the gap width s between impeller shroud and casing wall, Cm being proportional to (s/a)0.1 in general44. For very small s/a, Cm instead grows as s/a decreases; see Refs. 44 and 45 for formulas.)

The value of Cm can be even larger for semi- or fully-open impellers, if the neighboring fluid is rotating faster relative to the wall—as is the case with radial-bladed open impellers. The fluid between a shrouded impeller and adjacent wall, on the other hand, rotates at half speed44. (In cases where the impeller surface and adjacent wall are both rough, Cm is larger than just discussed45.) Finally, notice in Eq. (h) that very low specific speed Q,s produces a dramatically low value of hm. This drives C to the larger values of Figure 12 at low Q,s—also dictated by the W-deceleration considerations per Figure 22. Overall there is a benefit, despite possibly lower ■HY [Eq. (a)] due to the consequently greater C and collector loss.

Volumetric efficiency hv applies to leakage across impeller shroud rings or "neck rings" and balancing drums. Eq. (j) in Table 3 is an approximation for the leakage across a typical ring of a closed-impeller pump, assuming orifice-type flow at a discharge coefficient of 2, as reported by Stepanoff4. Referring to Figure 2, leakage QL occurs at r = rR, (rR being approximately 1.2 times re) under a pressure difference across the ring of about § that of the pump stage. If the shroud is removed and the open blades are fitted closely to the adjacent wall, as with open impellers, the consequent leakage from one impeller passage to the next across the blade tips does not affect hv, and Eq. (j) should be modified accordingly. Rather, the tip leakage causes a hydraulic efficiency loss as previously discussed. Finally, as with hm, Eq. (j) indicates that low-Hs pumps have low hv.

At flow rates Q other than QBEP, the analytical methods described previously for computing the hydraulic efficiency are utilized, together with computation of the inlet and outlet velocity diagrams, which yield the ideal head and power curves as illustrated in Figure 6. In this procedure, the slip velocity Vs (Figure 15) applies to the BEP and, at other flow rates, the exit relative flow angle b/,2 can be assumed constant. This accords with the fact that Vs for the narrower active jet at low flow rates must be smaller. A blockage model for the thickening wakes and narrower active jets that develop as Q is decreased can be introduced to compute the one-dimensional velocity diagrams, but ignoring this at non-recirculating flow rates appears not to be serious in determining the shapes of the head and power curves.

b) Shut-o// and low /low. The foregoing analyses apply over that portion of the flow rate range that does not involve recirculation, as illustrated in Figure 6. The complexity of recirculation has not been readily handled analytically, and this has forced pump designers to estimate the low-flow end of the H-Q curve with the help of empirical correlations. Nevertheless, insightful fluid dynamical reasoning about the physics of the flow have led to useful expressions for the head developed and the power consumed at shut-off. Shut-off, then, in addition to the BEP, becomes the other anchor point of the head and power curves; and this—together with the shapes established for these curves at the higher flow rates—gives the analyst an idea of the intervening shapes.

Shut-off head Hs/o can be viewed as the sum of two effects occurring at Q = 0, each being represented by a term in this equation:

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