Wp minW2

where the 0.6 limit applies individually to Dp and Ds—or to the sum of the two, in which case the limit is 1.2. Eqs. 59a and 59b, therefore, constitute a more useful form of the diffusion factor concept for assessing the blade loading and the choice of the number of blades in centrifugal pump impellers24.

Finally, the total blade length or number of blades, should not exceed that necessary to limit the diffusion as just described, as this adds unnecessary skin friction drag, which causes a reduction in efficiency. Thus the solidity values given in conjunction with Eq. 53 should not be appreciably exceeded, unless blade load needs to be reduced to lower levels, as with inducers to limit cavitation8 or impellers for pumps that must produce lower levels of pressure pulsations.

vi. Development of the blade shape. Blades are developed by defining the intersection of the mean blade surface (really an imaginary surface) or camber line on one or more nested surfaces of revolution. Two such surfaces are formed by the hub and shroud profiles. If the blade shape is two-dimensional (that is, the same shape at all axial positions z), the mean blade surface is completely defined by constructing it on only one such surface of revolution. Generally, however, the shape is three-dimensional and is a fit to the shapes constructed on two or more of these surfaces of revolution; namely, the hub and shroud and usually at least one surface between them. After this final shape is known, half of the blade thickness is added to each side. (Sometimes the full blade thickness is added to one side only, meaning that the constructed surface just mentioned ends up—usually—as the pressure side of the finished blade rather than the mean or "camber" surface. The effective blade angles are then slightly different from those of the pressure side used in the construction process.) The construction along a mean surface of revolution is illustrated in Figure 18. The distribution of the local blade angle b (or more precisely, bb) is found first by either the "point-by-point" method or the conformal transformation method—both of which yield the polar coordinates of the blade, r, d, and z. These coordinates also depend on the chosen shapes of the intersections of the surfaces of revolution with the meridional plane; that is, the hub, shroud, and mean meridional "streamline" or rms line, as in Figure 18c, and the fact that, on the surface of revolution Figure 18a, tan b = arc bc/arc ac = dm/dy. The elemental tangential length dy (= arc ac) is the same on both the surface of revolution (Figure 18a) and in the polar view (Figure 18b). From Figure 18b, it is seen that dy = rdd, so the "wrap" angle d is found from

FIGURE 18 Blade construction: a) view of construction surface of revolution; b) polar view; c) meridional view

and r and z are found from the fact that the coordinate m along each of the construction surfaces is a function of r and z (Figure 18c).

If the blade is two-dimensional, its mean surface consists of a series of straight-line axial elements, each having a unique r and i at all z. Such a blade is typical of low-specific-speed, radial-flow impellers, and can be easily constructed by the "point-by-point" method. Here, one specifies the distribution of Wg—often linear as in Figure 17—after determining the hub and shroud profiles and the corresponding distribution of Vm15. In effect, one obtains the distribution of the blade angle bb by constructing a velocity diagram like the one in Figure 15 at every m-location from inlet (1) to outlet (2) in Figure 18c, dealing only with the "geometric" or non-deviated velocities, in order to get a smooth variation of the blade angle bb vs m. Allowance is made for blockage due to the thickness of the blades and the displacement thickness of the boundary layers in the passage. The resulting wrap angle i for each m-point—as well as the corresponding r and z—is then found from Eq. 60. (For convenience in designing the blades, the construction angle i is often taken as positive as one advances from impeller inlet to exit. For most impellers, this turns out to be opposite to the direction of rotation; and i is taken in the direction of rotation for most other purposes of pump design and analysis.) As discussed previously in Paragraph iv and illustrated in Figure 17, the actual flow will deviate from the resulting blade via the "slip" phenomenon.

The point-by-point method allows the designer to exercise control over the relative velocity distributions on the blade surfaces (Eq. 54 and Figure 17) via specification of the distribution of Wg or other velocity component in Figure 15; for example, V8;. This becomes more important if an unconventional impeller geometry is being developed17.

The point-by-point method can also be used for three-dimensional blades. A simple approach in this respect would be to use this method to determine the blade shape along the rms- or 50%-streamline (that is, on the mean surface of revolution depicted in Figure 18). The shapes on the other streamlines, generally the hub and the shroud, can also be found by this method. The resulting overall blade shape, however, is subject to the condition that the resulting wrap i2 — i1 cannot greatly differ on all streamlines without the blade taking on a shape that is difficult to manufacture and which may turn out to be structurally unsound or create additional flow losses. This is because the final blade shape is the result of stacking the shapes that have been established on the nested stream surfaces defined by these meridional streamlines. Blade forces due to twists arising from this stacking could modify the expected flow and cause unexpected diffusion losses.

One way to generate blade shapes along the hub and shroud that have the same (or nearly the same) wrap as that obtained from point-by-point construction of the blade on the mean surface of revolution is to establish the desired inlet and outlet blade angles bb on each such surface and then mathematically fit a smooth shape y(m) to these end and wrap conditions, where y is the tangential coordinate seen in Figure 18 and defined in Figure 19. A conformal representation of the shapes of the blades resulting from such a procedure on each of the three surfaces is seen in Figure 19. These shapes are sometimes called "grid-lines" or simply "grids"—from the description of the graphical procedure that relates these shapes in the conformal representation to those on the actual, physical surfaces4. In such a representation, the blade angles are the same as they are on the physical surface of revolution because tan b = dm/dy and dy = rdi, also yielding Eq. 60.

If the associated distributions of Wg and Vm are smooth, one can expect to have a satisfactory result if these conformal representations are also smooth. Thus, many skilled designers bypass the computations just described for the point-by-point method and use the conformal transformation method of blade design. Here, one simply establishes the grid-line shapes by eye in the conformal plane of Figure 19, specifying the blade angles b at inlet and outlet by the previous procedures as the starting point for

Transformed Distante from Trailing Fdge, v = J rdO'. in FIGURE 19 Conformai transformation of blade shape: "grid-lines"

Transformed Distante from Trailing Fdge, v = J rdO'. in FIGURE 19 Conformai transformation of blade shape: "grid-lines"

drawing each grid-line. This conformal blade shape is then transformed onto the physical surface, the differential tangential distance dy becoming rdd on the physical surface (Figure 18) and the differential meridional distance dm being identical in both the conformal and physical representations. If the resulting blade shape appears to be unsatisfactory, the designer repeats this process, possibly first altering the hub and shroud profiles or the blade leading and trailing edge locations on these profiles and recomputing the b's.

Designing the Collector The fluid emerging from the impeller is conducted to the pump discharge port or entry to another stage by the collecting configuration, which can employ one or more of the following elements in combination: a) volutes, which can be used for designs of all specific speeds, b) diffuser or stator vanes, which are often more economical of space in high-specific-speed single-stage pumps and in multistage pumps, and for the latter, c) return or crossover passages, which bring the fluid from the volute or diffuser to the eye of the next-stage impeller. Generally, the most efficient impeller has a steady internal relative flow field as it rotates in proximity to these configurations. This is assured by all of these elements because they are designed to maintain uniform static pressure around the impeller periphery—at least at the design point or BEP. An exception to this rule is the concentric, "doughnut"-type, "circular-volute" collector, which is used on small pumps or in special instances where the uniform pressure condition is desired at zero flow rate.

The proximity of stationary vanes in these collecting configurations to the impeller must be considered in their design. Called "Gap B," the meridional clearance between the exit of the impeller blades and adjacent vanes ranges from 4 to 15 percent of the impeller radius, volutes having higher values in this range than diffusers, and pumps of higher energy level requiring the larger values. If these gaps are too small, the interactions of the pressure fields of the adjacent blade and vane rows passing each other can cause vibration and structural failure of impeller blades, diffuser vanes, and volute tongues.

a) Volutes. A volute is built by distributing its cross-sectional area on a "base circle" that touches the tongue or "cutwater" and is meridionally removed from the impeller exit by

FIGURE 20 Volute casing: a) polar view; b) meridional view including Section A-A of throat T

Gap B. (For radial-discharge impellers, as in Figure 20, this is a radial gap, and the base circle has radius r3.) Beginning at the tongue, the cross-sectional area Av of the volute passage is zero, but it increases with angle d in the direction of rotation, ending up at area At in the "throat" T, as depicted in Figure 20. Worster demonstrated that the desired peripheral uniformity of static pressure can be achieved if the product rVs is constant everywhere in the volute25. One-dimensionally, this means that rTVeT = r2Ve,2; and, if the velocity VT is essentially tangential (in the 0-direction), rTVT = r2Ve,2. The diffusion or reduction of the velocity V from the impeller periphery at r2 to the larger rT of the throat produces a static pressure increase above that at the impeller exit; however, friction losses in the volute would cause a reduction in static pressure around the impeller at r2 from tongue to throat unless the throat area AT = Q/VT is slightly enlarged, creating a little more diffusion to compensate for this loss. Thus, in practice, at the BEP, rTVT = (0.9 to 0.95) X r2Ve, 2 (61)

At off-BEP conditions, the volute will be either too large or too small and Eq. 61 will not be satisfied. When the flow coefficient (or Q/N) drops below the BEP value, there will be excessive diffusion and an increase of static pressure around the volute from the zero area point around to the maximum area point at the throat. Proceeding around further, past the throat, a sudden drop in pressure occurs across the tongue to bring the pressure back to what it was at the starting point25. The opposite situation occurs above BEP.

Each of these off-BEP circumferential static pressure distributions is properly viewed as the consequence of a mismatch between the head-versus-flow characteristics of the impeller and volute26. For the impeller, there is the falling, straight, Hrversus-Q line or "impeller line" of Figure 6, whereas the volute characteristic or "casing line" would be a straight line starting at the origin of Figure 6 and crossing the impeller line at the match point, which is generally at or close to the BEP flow rate. This casing line is straight because the throat velocity VT varies directly with flow rate Q and, through Eq. 61, directly with the ideal head H—because H X r2Ve,,2 = H (Eq. 15b for Ve,1 = 0). In other words, the same volute could be optimum at a different value of Q if it were paired with another impeller whose Hi-versus-Q line crossed this same casing line at that different Q.

To essentially eliminate the consequent radial thrust on the impellers of large pumps at off-BEP conditions, a double volute is used; that is, there are two throats, 180 degrees apart, there being either two discharge ports or a connecting "back channel" to carry the fluid from one of the throats around to join the flow emerging from the other—to form a single discharge port.

The value of the volute cross-sectional area Av at a given polar position 6v can be found for the portion of the total pump flow rate Q being carried in the volute at that exposition together with the condition rVe = constant versus radius. This will produce a distribution A„(0) that is slightly below a straight-line variation versus 6v from zero to AT Often, the practice is to use the latter straight-line design because this produces larger values of Av where the hydraulic radius of the volute is small, thus compensating for the greater friction loss in that region through lower velocity—particularly for the smaller pump sizes. The cross-sectional shape of the volute is dictated by the need to make a minimum-loss transition from a small area at the beginning of the volute where the height (as can be deduced from Figure 20b) is much smaller than the width b3 to the throat, for which the height (to the outer casing wall at rw) and the width bmax are more nearly equal. Too small an aspect ratio (heightIwidth) decreases the hydraulic diameter too much and increases the loss. There is another transition from the throat through an essentially conical diffuser (which may negotiate a turn) to a larger, circular exit port. This diffuser can be designed with the help of charts of flow elements and will normally have a 7-deg. angle of divergence and a discharge area up to twice that of the throat AT27. Thus, there is a substantial diffusion from the impeller periphery to the pump or stage exit port. This generally produces a static pressure rise in the collection system that is 20 to 25 percent of that of the whole stage.

b) Vaned diffusers. A vaned diffuser is rotationally symmetric and, if properly applied, produces minimal radial thrust over the whole flow rate range of a pump. Although diffusion can be accomplished in a radial outflow configuration without vanes due to the essential constancy of the angular momentum per unit mass rVe, one rarely finds a pump with a vaneless diffuser, partly because so much radial distance is needed to effect the reduction of tangential velocity required, as well as the still larger volute needed on single-stage

FIGURE 21 Vaned diffuser

pumps to collect the fluid at the exit of such a diffuser. Also, the absolute flow angle a2 (Figure 15) of the fluid leaving the impeller is usually too small to satisfy the conditions for stall-free flow in a vaneless diffuser28. A vaned diffuser, on the other hand, can accomplish the reduction of velocity in a shorter radial distance. Also it can diffuse axially and, to a degree, even with radially inward flow.

Vaned diffusers are similar to multiple volutes in concept, except they are subject to offdesign flow instabilities if not shaped correctly. Width b3 is usually slightly greater than b2 in order to accommodate discrepancies in the axial positions of impellers that feed them. With reference to Figure 21, "Gap B" (= r3 — r2) is in effect a short vaneless diffuser, and by the time the fluid has reached the throat (the dashed line at Station 4), it has gained a substantial portion of the static pressure recovery that takes place via diffusion from Station 2 to Station "ex." This "pre-diffusion" is enhanced by the fact that the throat area at Station 4 (= b3w: per passage for parallel-walled radial-flow diffusers) is larger than it is for volutes, the following relation applying to diffusers29:

Therefore, more diffusion than would result from applying Eq. 61 occurs in a vaned diffuser, the skin friction loss due to an otherwise higher velocity at the throat being offset by an efficient reduction of the velocity up to that point and a lower velocity onward.

The fully vaned portion from throat to exit (Figure 21), which performs most of the rest of the diffusion and associated static pressure recovery of the stage, is designed to perform efficiently and maintain stable flow. For typical radial-flow geometries with parallel walls, the vanes can be of constant thickness and comparatively thin or can thicken up to form "islands." The latter approach usually produces a channel that is two-dimensional with straight sides diverging at an included angle, length-to-entrance width / /w:, and area ratio Aex/AT in a combination that avoids appreciable stall30. A typical combination is an included angle of 11^ deg., //w: = 4, and Aex/AT = 1.8, which also applies for vanes of constant thickness, as illustrated in Figure 21. Constant thickness vanes have curvature. This modifies the performance somewhat31,32, but it allows a smaller overall radius ratio of the diffuser, rex/r3.

Also, this ratio rex/r3 will be smaller as the number of vanes nv increases. The best experience seems to be with diffusers that have only a few more vanes than the number of impeller blades nb (nb rarely exceeds 7 in traditional commercial pumps). For pumps of higher energy levels (or high head per stage, as discussed further on in connection with high-energy pumps), it is important that nv be chosen so as to avoid a difference of 0 or 1 between nb and nv or their multiples—up to at least the third multiple or "order" of each. A difference of 2 should also be avoided for at least the lower orders33.

At off-design flow-coefficients (or off-design flow rate at a constant speed), the angle a of the absolute velocity vector V (Figure 15) approaching the diffuser will vary; yet, for typ ical stages, a wide range of flow coefficient is possible without damaging instabilities, even at high energy levels. This is likely the case because a is rather small at the design point or BEP (except for designs having high specific speed), so variations of the angle that occur with flow changes are within the unstalled performance range of the diffuser vane system.

c) Return passages. Conducting the relatively low-velocity fluid from the diffuser to the eye of the next impeller in a multistage pump is accomplished with return vanes or passages that also deswirl the fluid wholly or partially. Except for development of stall in the diffuser, these passages will not see a changing angle of the approaching velocity vector because the diffuser feeding them is a stationary element. In radial-flow pumps, there is a sharp turn in the meridional plane in order to redirect the fluid inward. The fluid, still possessing a circumferential component of velocity that is greater than the meridional component, actually sees a much gentler turn. However, downstream of this point, a sharp turn of the blades is invariably a feature of a return passage; and this, together with the need to ensure undistorted flow into the following impeller, often dictates that the vane system accelerate the fluid as it approaches the eye. Although losses in the return passages —being related to the low velocity within them—have a minor effect on the overall stage efficiency, the design of such passages must ensure unstalled flow into the impeller in order to avoid the negative impact of a distorted inlet flow on the efficiency and to promote pulsation-free operation of the impeller.

A variety of return-passage geometries exist, some of which are presented in the literature 29,34. The continuous-vane type is integral with the upstream diffuser, thereby eliminating the entry losses into yet another vane system after the diffuser4,34. Improvements in manufacturing technology have made this potentially more efficient approach more viable for radial machinery. The continuous-vane concept is standard practice in the design of mixed-flow "bowl"-type pumps21. The diffusing stator vane row that receives the fluid from the impeller of an axial-flow pump—being an axial-flow element itself—possesses the return feature already. Diffusion in axial-flow stators is typically accomplished by a reduction in velocity of about 30 to 40 percent. The actual value is governed by an acceptable level of the diffusion factor, Eq. 57. (A similar reduction in relative velocity is needed for an axial-flow impeller to generate static pressure, as can be seen from Eq. 21. By comparison, centrifugal impellers, on the rms streamline, usually have W2 about equal to W1—as seen in Figure 17.)

Axial-Flow Pumps The preceding development, though general, is applicable mainly to centrifugal and mixed-flow pumps. In that procedure, the impellers have appreciable solidity, and original blade shapes are constructed from the viewpoint of one- or two-dimensional channel flow. The collectors are often volutes or non-axial-flow vane systems. Performance is not known a priori and so must be estimated, as outlined further on. On the other hand, the extensive two-dimensional, experimental, axial-flow cascade data amassed by NACA researchers23 and others enables the designer to adopt existing airfoil blade shapes and so predict the performance with greater confidence. The procedure for utilizing these shapes and the corresponding experimental results has long been the basis for designing axial-flow compressors for gas-turbine engines and is clearly described by Hill and Peterson35. This approach is widely used, especially for high-specific-speed, low-solidity axial-flow propeller pumps—in designing both rotating and stationary blade rows. Insights for propeller pump design and performance characteristics can be found in Stepanoff4.

An exception to this axial-flow pump design approach is the case of inducers. Although they are axial flow pumps, they have high solidity and are usually designed as channel-flow machines. The design philosophy outlined in the preceding paragraphs is applicable, except that the blades usually approximate constant- or variable-pitch helices. Performance prediction is generally accomplished via one-dimensional calculations and the correlations described in the following paragraphs.

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