## P

is a constant for a given pump geometry. Mockridge, in a discussion attached to an ASME paper by Stepanoff, reasoned that a wider impeller (larger b2 at the same diameter D2) would recirculate more fluid at shut-off and therefore have a higher value of this coefficient. His correlation is shown in Figure 23 and is probably the most significant quantitative result available for predicting the performance of centrifugal pumps at shut-off conditions36.

or k c) Complete prediction via CFD. The uncertainties that have characterized the prediction of pump performance are now being overcome through advances in computational fluid dynamics. CFD entails three-dimensional solution of the flow fields within pumps via the Reynolds-averaged Navier-Stokes equations. Graf demonstrated the ability of a CFD computer code to calculate recirculation, the consequent prediction of the head curve for the impeller comparing favorably with experimental data46. The resulting distorted flows entering and leaving adjacent systems of impeller blades and stator vanes produce time-varying boundary conditions on each, the associated computational grids also moving relative to each other. This involves extensive, time-dependent computation. To provide solutions quickly on conventional, storage- and speed-limited workstations, some steady-flow codes treat these interfaces by circumferentially averaging the conditions at each point of the blade and vane leading and trailing edges as they appear in the meridional plane. Even with this simplification, pump analysts can now predict the entire performance curve of head within about two percent and the power curve with slightly less accuracy47.

FIGURE 23 Shut-off power coefficient

The design task therefore resolves itself into an iteration between an efficient geometry-generating scheme and a rapid CFD flow and performance analysis of the geometry resulting from each iteration48. This is especially useful if a non-traditional geometry is involved, or if an efficient design is sought that will produce a desired performance curve shape. Nevertheless, many turbomachinery designers can make more rapid and valid judgments about their respective classes of machines through the time-honored iteration between a proprietary direct or inverse design and performance-prediction scheme and inviscid quasi-3D analysis 41,43. They have developed reliable diffusion criteria (computed, for example, from Eqs. 59a and 59b) for interpreting the acceptability of the free-stream relative velocity distributions Ws and Wp on the blade surfaces (Figure 17) produced by the Q3D blade-to-blade solutions43. Because CFD codes solve the actual viscous flow field, the boundary condition on the blade surface is zero relative velocity. This can be at least partly overcome by displaying the CFD-distributions of pressures on the blade surfaces, the interpretation of which would require knowledge of the corresponding criteria for these pressures46. Also, the velocities at the edge of the boundary layer could be extracted from the CFD solution and displayed in familiar terms. A useful design approach for the present may therefore be to a) produce the final design by the more traditional methods and b) predict the performance curves via CFD49.

Predicting Axial Thrust The prediction of pump performance is not truly complete without the corresponding prediction of the hydrodynamic axial and radial thrust that the impeller(s) can be expected to encounter. A comprehensive treatment of radial thrust appears in Section 2.3.1, and a review of axial thrust and thrust balancing devices is cov ered in Section 2.2.1. However, obscure flow phenomena can profoundly affect the radial distributions of pressure on the outside surfaces of a shrouded impeller that give rise to the net axial thrust. These phenomena become even more complex when discharge recirculation occurs and can cause adverse mechanical response in high-energy pumps, as will be explained further on. As a basis for tackling such problems, the fundamentals of axial thrust are presented in Table 4 for shrouded centrifugal impellers that have leaking fluid flowing in the gaps between the impeller shrouds and the adjacent casing walls. The positive direction of the thrust T is taken toward the suction or eye of the single-suction impeller shown. The incoming axial momentum pQVz1 is generally quite small for radial impellers and has been omitted from the Table. It serves, however, to reduce T.

The centrifugal effect of the fluid spinning in the sidewall gaps causes a reduction in static pressure from the outer periphery (OD) of the impeller to the sealing ring, and this

TABLE 4 Leakage effects on axial thrusts

TABLE 4 Leakage effects on axial thrusts

TABLE 4 Continued.

Specifically,

or x /outflow (®lkg) x /width where x /outflow (®lkg) x /width where where

/inflow

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