Section 2.1.1 - Centrifugal Pump Theory - Designing The Impeller

Designing the Impeller  Determination of the geometrical features of the impeller is
generally accomplished in the following order: a) the “eye” radius r_e, b) the exit radius r₂
or r_t,2, and c) the exit width b₂ or, in the case of mixed- and axial-flow impellers, the hub
exit radius r_h,2—all of which form the starting point for d) shaping the hub and shroud
profiles (Figure 13); and, finally, e) construction of the blades.

a) The eye. The inlet radius of the impeller eye r_e (Figure 13) is nearly the same as r_t,1,
which is the diameter of the tips of the impeller blades at the inlet. This emerges after
the eye flow coefficient φ_e = V_e/U_e [the ratio of the one-dimensional axial velocity entering
the eye (Figure 8) to the tangential speed of the impeller eye U_e = Ωr_e] is known:

(47)

(48)

Thus, r_e can be found from the following combination of Eqs. 47 and 48:

(49)

where the shaft-to-eye ratio r_s/r_e can be estimated at first. Typical values for φ_e vary from
0.2 to 0.3 for impellers and down to 0.1 or less for inducers, depending on suction conditions,
as can be seen from its relationship to suction-specific speed:

where the cavitation coefficient t (cf. τ₂ in Eq. 36) is defined in terms of the eye speed
U_e = Ωr_e (Table 1). t is related to φ_e through empirical correlations, such as those given in
Table 1. (Gongwer’s¹⁴ values for the correlation factors k₁ and k₂ apply to large pumps. The
larger “typical” values shown for 3% breakdown apply to the more common smaller sizes.
The inducer correlation is a curve fit to the data of Stripling and Acosta¹⁵ for the breakdown
value of τ.) Thus one can solve for φ_e from a given suction-specific speed and, through
Eq. 49, obtain the eye size. However, the value of φ_e at the BEP or design point rarely
exceeds 0.3, regardless of how much NPSH is available. This φ_e-limit therefore applies to
impellers that follow and are in series with the first stage in a multistage pump. These are
variously referred to as “series” or “intermediate” stages. (Slurry pump impellers are an
exception to this guideline, for then the relative velocity is minimized to avoid excessive
wear. In this case φ_e can be as high as 0.4 and NPSHA is generally more than adequate for
these slow-running machines.)

b) The exit radius r₂ (or diameter D₂). This is found from head-coefficient ψ by means
of the equation for r₂ in Figure 12. The curve for ψ can be used unless detailed performance
analysis or a desired performance characteristic curve indicates otherwise. Eq. 46
can be used for specific speeds Ω_s greater than 1.6 (N_s > 4373), where the maximum
radius r_t,2 is computed per the previous discussion accompanying that equation. At this
point, it can be seen that the equation in Figure 12 for exit radius r₂ and Eq. 49 for the

eye radius r_e can be combined to yield the ratio of eye size to that of the OD for centrifugal
impellers; viz.:

(51)

Except for inducers, which have very low values of φ_e, this ratio is affected mainly by specific
speed as illustrated in Figure 9.

c) The exit width b₂. The equation for exit passage width b₂ in Figure 12 can be used for
radial-outflow and mixed-flow impellers, r₂ being located halfway across the passage. This
involves the exit flow coefficient or meridional velocity ratio φ_i = V_m,2/U₂, the lower curve
of Figure 12 being for typical values of this quantity. The “openness” factor ε allows for
blockage due to blade thickness and to the buildup of boundary layers on the surfaces of
the passageways (blades and hub and shroud). The value of ε is generally between 0.8 and
0.9, the higher figure applying to larger machines. For axial-flow impellers or propellers
and inducers, a choice of the hub-to-tip radius ratio at the exit defines the passage width
instead. This ratio decreases with specific speed from about 2/3 at the right end of Figure 12
to 1/3 or less at the highest specific speeds.

d) Hub and shroud profiles. With the eye and the outlet sizing established, the two are
connected by specifying the hub and shroud profiles. Some texts illustrate the variation
of hub and shroud profiles with specific speed¹⁶. Although these are excellent guidelines
coming from experience, what follows is the approach one would take to synthesize these
shapes on the basis of fundamental fluid dynamical considerations, at the same time taking
experience into account. Referring to Figure 13, an acceptable geometry can be
achieved by following these guidelines:

1. Maintaining the meridional flow area 2πr_b,1b₁ at the blade leading edge at about the
   same as it is at the eye, namely π(r_e²- r_s²), or slightly larger to compensate for blade
   blockage; but then gradually increasing it versus meridional distance to the generally
   larger value already established at the exit, namely 2πr₂b₂.
   
2. Choosing the minimum radius of curvature R_sh of the shroud to be about half the
   radial opening at the eye. This avoids excessive local velocity V_1,sh at the blade leading
   edge, (Figure 14 in Table 1.) This has two consequences. Shaping the impeller blades
   to match a widely varying meridional approach velocity can complicate the construction
   of these blades. Also, on first-stage impellers, if V_1,sh is too great, the local pressure
   at that location will be closer to the vapor pressure, increasing the required NPSH (or
   NPSH_3%). This is due to a larger resulting value of the empirical factor k₁, presented
   in Table 1 as the nth power of the velocity ratio V_1,sh/V_e.n = 2 is expected from Bernoulli
   considerations, and if V_1,sh/V_e = 1.3, the value of 1.69 emerges for k₁ as shown in Table 1.
   V_1,sh/V_e = 1.25 is a typical choice of designers for setting the impeller blades at inlet.
   A CFD solution of the combined inlet passage and impeller blading would show the
   actual resulting value of this velocity ratio, which might then lead to a change in the
   impeller blading or radius of curvature R_sh. The powerful effect on this velocity ratio
   of the sharpness of the turn that the flow negotiates in coming into the impeller eye
   and turning toward the radial direction can be seen from the following finite-difference
   form of the differential equation governing axisymmetric inviscid flow in
   the impeller eye region:
   
   
   where R is the average radius of curvature of the flow in the meridional plane of Figure
   13, ΔV_m is the difference in V_m from hub to shroud over a distance be perpendicular
   to the direction of flow that moves at an average meridional velocity V_e. Thus a
   sharp turn with its small value of R produces a large value of ΔV_m/V_e, which in turn
   corresponds to a large value of the velocity ratio V_1,sh/V_e.
   
3. Shaping the hub profile compatibly with the guidelines as stated earlier. This is best
   done after making an initial estimate of the shroud profile as outlined previously. The
   distribution of meridional flow area from the eye to the exit should then be specified.
   From this, a hub profile will emerge. As seen in Figure 13, the hub of a radial-outflow
   impeller becomes essentially radial over the outer portion of its extent. If this does not
   result from this procedure, appropriate adjustments can be made to the shroud profile
   and the process repeated.
   
4. For a high-specific-speed, axial-flow impeller, or inducer, the hub profile is often a cone
   or a reverse curve between a smaller radial location at inlet to a larger one at outlet,
   the latter radius decreasing with increasing specific speed as mentioned earlier. A
   cone or cylinder for the shroud profile is often found in such machines.

e) Construction of the blades. The blades are designed by i) selecting the locus of the leading
and trailing edges in the meridional plane, ii) establishing the surfaces of revolution
(streamwise lines in the meridional plane) from inlet to outlet along which the construction
proceeds, iii) selecting the inlet angles, iv) selecting the outlet angles, v) establishing
the number of blades, and vi) obtaining the blade coordinates from inlet to outlet:

1. Leading and trailing edge loci. If every point along the leading and trailing edges is
   revolved about the axis of rotation so as to lie in one meridional plane, the loci of these
   edges appear as shown in Figure 13. The outer or shroud end of the blade leading edge
   is positioned at or near the minimum radial location; that is, at or near the eye plane,
   whereas the inner or hub end is typically well back and largely around the corner
   along the hub profile. These locations are desirable; first, at the shroud, because the
   absolute velocity V (typically = V_m) approaching the blade begins to decelerate beyond
   the eye plane, so starting the blade ahead of this decelerating region tends to prevent
   separation of the fluid from the shroud surface due to the pumping action in the blade
   channels¹⁷; and secondly, being far enough along the hub in the streamwise direction
   to avoid impractical blade shapes (excessive twist, rake, and so on) that would make
   both the construction and the flow inefficient. The locus of the blade trailing edges is
   normally straight in the meridional plane and is axial in orientation for most centrifugal
   pumps. At the higher specific speeds, this locus becomes more and more slanted
   until it takes on the nearly radial orientation it has for a propeller (Figure 9).
   
2. Surfaces of revolution for blade construction. Developing the coordinates of the
   blades along three streamwise surfaces of revolution—the hub, mean, and shroud,
   whose intersections with the meridional plane appear as streamwise lines in that
   plane—usually provides a sufficient framework for shaping the blades of an impeller.
   However, for high specific-speed impellers, where the passage width in the meridional
   plane b (Figure 13) is large (about equal to or greater that the meridional distance
   from leading to trailing edge), definition along two intermediate surfaces of revolution
   is also needed to achieve a satisfactory design.
   
   The “mean” line is one that is representative of the flow from a one-dimensional
   standpoint as well as for the construction of the blades. Precisely, this is the mass-
   averaged or “50%” streamline (that is, the streamline for ψ = 0.5 in Figure 14)—which
   evenly divides the mass flow¹⁷. This line is reasonably and conveniently approximated
   by the “rms streamline;” that is, the line that would result in a uniform meridional
   velocity distribution from hub to shroud and therefore equal areas 2πrΔn normal to
   the meridional velocity component V_m. In this case, Δn (= Δb) is the spacing between
   the rms streamline and the hub or shroud line. This would put each point on the mean
   line at the root mean square radial position along a true normal to the meridional
   streamlines; hence, the “rms” terminology.
   
3. Inlet blade angles. The blade angles are set to match the inlet flow field. This is done
   where each of the previously chosen surfaces of revolution (that intersect the meridional
   plane in the streamwise lines just described) crosses the chosen locus of the
   blade leading edges in the meridional plane. At each such crossing point, an inlet
   velocity diagram of the type shown in Figure 3 is plotted in a plane tangent to the surface
   of revolution at that point. (Figure 3, representing a purely radial-flow configuration,
   is a view of such a plane, as the surfaces of revolution are then simply disks.)
   Each such velocity diagram or triangle contains a specific value of the angle β_f,1
   between the relative velocity vector W₁ and the local blade speed vector U₁ = Ωr₁.
   
   The corresponding blade angle β_b,1 between the mean camber line of the blade and
   the circumferential direction is set equal to β_f,1 or slightly higher than this to allow for
   the higher V_m,1 caused by non-zero blade thickness at the leading edge and to allow for
   higher flow rates that may be called for at off-design conditions. To construct the triangle,
   one first plots U₁ and then V_m,1, which is taken from a CFD or other analysis of
   the meridional velocity variation across the inlet eye as discussed above under d) Hub
   and shroud profiles or is chosen as the mean value Q/2πr_b,1b₁ (Figure 13) at the rms
   streamline. It is adjusted from experience at the shroud and hub. Likewise, if any
   prewhirl V_θ,1 is delivered to the impeller, it must be taken into account as illustrated
   in Figure 3.
   
4. Outlet blade angles. Whereas the inlet velocity diagrams enable the designer to correctly
   set the blades to receive the incoming fluid with minimum loss, the outlet velocity
   diagram displays the evidence—through the magnitude of the circumferential
   velocity component V_θ,2 that the intended head will be delivered by the pump in accordance
   with Eq. 15c. As shown in Figure 3, V_θ,2 is determined—for the given impeller
   tip speed U₂—by the exit relative flow angle β_f,2 in conjunction with the exit meridional
   velocity component V_m,2. This value of V_m is somewhat larger than that given by
   Eq. 16 because of a) blockage due to blade thickness and boundary layer displacement
   thickness and b) the presence of any leakage flow Q_L (Figure 2 and Eq. 11) that may
   also be flowing through the impeller exit plane or Station 2.
   
   Well inward of the exit plane, the direction of the one-dimensional relative velocity
   vector W can be assumed to be parallel to the blade surface; however, in the last third
   of the passage, the blade-to-blade distribution of the local relative velocity changes
   due to the unloading of the blades at the exit. This produces a deviation of the direction
   of W₂ from that of the blade. This deviation, called “slip” in centrifugal pumps,
   results in less energy being delivered to the fluid by the impeller than would be the
   case if there were “perfect guidance” such as would occur with an infinite number of
   blades. Accordingly, in the outlet velocity diagram of Figure 15, the relative flow angle
   β_f, is less than the blade angle β_b. This deviation is quantified by the “slip velocity” V_s.
   The magnitude of V_s depends on the distribution of loading along the blades from inlet
   to exit and therefore on the geometry of the flow passages and the number of blades.
   (Without slip,W₂ is the same as the “geometric” relative velocity W_g,2 shown in the figure.)
   The slip factor μ = V_s/U₂—typically between 0.1 and 0.2—was determined theoretically
   by Busemann for frictionless flow through impellers with zero-thickness,
   logarithmic-spiral blades (constant-β from inlet to exit) and a two-dimensional, radial-
   flow geometry with parallel hub and shroud¹⁸. Applicability of this theory to typical
   impellers, despite the differences in geometry and the real fluid effects, was found to
   be good by Wiesner, who represented Busemann’s results by the following convenient
   approximation¹⁹:

A broader, empirical slip correlation for pumps was developed by Pfleiderer, taking
into account impeller geometry and blade loading, as well as the influence of the downstream
collecting system (volute or diffuser)²⁰. Pfleiderer computes the slip velocity as
the product of a slip factor p and the impeller exit tangential velocity V_θ,2, where p is
computed as shown in Table 2. This table also contains a simple example; namely, a

radial flow impeller of a volute pump, for which the resulting value of μ is 0.1826—
versus 0.1498 via Eq. 52; however, in this case the latter result is low by about 15%.
A study of the Busemann plots in Wiesner’s paper yields μ = 0.18.Yet, if this had been
a vaned-diffuser pump, Pfleiderer would have predicted μ = 0.1468 for the same
impeller, as it would have delivered more V_θ,2 for the same β_b,2,W_g,2, and, therefore,V_θ,2,∞,
(Figure 15). This stems from the factor “a” in Table 2 having the value 0.6 (for a vaned
diffuser) instead of 0.8 (for a volute). So by this combination of circumstances—and in
this example—Eq. 52 describes the slip of a diffuser pump impeller. But, despite the
simplicity of Eq. 52, Pfleiderer’s method (Table 2) would appear to be a more rational,
comprehensive, and satisfying method for estimating slip in real pumps.

So, to find the outlet blade angle, the designer begins by deciding upon the required
value of V_θ,2; finds the exit flow angle and other elements of the diagram assuming the
existence of slip. Next, the designer computes the slip and then obtains the value of
the outlet blade angle β_b,2. The process is iterative because the forementioned blockage
depends on the blade angle as well as the thickness.

5. Number of blades. The choice of the number of impeller blades is influenced by a)
   interaction of the flow and pressure fields of the impeller and adjacent vaned structures
   such as the volute tongues or diffuser vanes and b) the need to maintain smooth,
   attached—and therefore efficient—fluid flow within the impeller passages. The effect
   of the number of blades on the interaction phenomenon is addressed in the latter part
   of this section under the topic of high-energy pumps, where this issue becomes critical.
   Smooth, attached flow is assured if the product of the number of blades and their
   total arc length l along a given meridional streamline, as illustrated in Figure 16, is
   of sufficient magnitude. Divided by a representative circumference on that streamline,
   usually that of the impeller outer diameter (OD), this product is called the solidity σ:

(53)

If splitter blades are used, the value of n_b changes from impeller inlet to outlet (1 to 2).
In that case, the numerator of Eq. 53 is replaced by the implied sum of the arc lengths
of all the blades in the impeller.

In practice, solidity varies from about 1.8 at low specific speed (Ω_s > 0.4 or N_s <
1093) to slightly less than unity at Ω_s = 3 (N_s = 8199). For example, Dicmas’ curve²¹ is
useful for Ω_s > 1 (N_s > 2733). This limits the relative velocity reduction that occurs on
the blade surfaces. Illustrated in Figure 17, this reduction or diffusion arises from the
loading on the blades expressed in terms of the blade-to-blade relative velocity difference
ΔW_b:

(54)

V_m,o is the local meridional velocity component neglecting blockage. (One-dimensionally,
V_m,o is the value of V_m found from Eq. 16, where the radius r is that from the axis of
rotation to the center of the circle of diameter b in Figure 8, which in turn lies on an

imaginary line in Figure 8 that is normal to the hub, shroud, and intermediate stream
surfaces.) Here, ΔW_b emerges by applying Bernoulli equation [Eq. 21 with no change
in radius (that is, no change in U) or loss as one traverses from pressure side to
suction side of the passage] to the static pressure difference p_p - p_s arising from the
delivery of angular momentum to the fluid (Eq. 26). This in turn results from the
application of the shaft torque to the blades. It is also assumed in the derivation of
Eq. 54 that the blade-to-blade average relative velocity W lies halfway between the surface
velocities W_s and W_p, (which would exist just outside the boundary layers on the
blades,) as illustrated in Figure 17. This is a good assumption for efficient flow well
within a bladed channel²². ΔW_b is inversely proportional to the solidity because, on the
average, from inlet to outlet, Eq. 54 becomes

(55)

where it can be seen from Figure 16 and Eq. 53 that the fraction involving the number
of blades n_b is the reciprocal of the solidity σ because

(56)

If splitter blades are used, an appropriate average number of blades is substituted
for n_b in Eq. 53.

For unconventional impeller geometries, the foregoing solidity guidelines may be
inadequate to assure efficient flow. For any geometry, though, the concept of a diffusion
factor D, utilized by NACA researchers²³ to assess stationary cascades of airfoils
can be employed. In view of Eqs. 53—56, their equation for D takes the following form
for both axial- and non-axial-flow geometries, rotating or not:

(57)

This can be deduced from Figure 17 as follows:

(58)

Then, Eq. 57 is obtained through the definitions of the average value of ΔW_b (Eq. 55
with Eq. 56) and σ (Eq. 53). NACA researchers found that losses increase rapidly if D
> 0.6. However, many centrifugal pump impellers have virtually the same value of relative
velocity W at inlet and at outlet—along the rms streamline (Figure 17), so D
from Eq. 57 is less than 0.6 on the rms streamline and even negative along the hub
streamline. This situation was encountered in accelerating (turbine) cascades and led
to the use of local diffusion factors, one for each side of the blade, namely D_p and D_s.
Here, inspection of Figure 17 and Eq. 58 leads to

(59a and b))

where the 0.6 limit applies individually to D_p and D_s. 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 cavitation⁹ or impellers for pumps that must
produce lower levels of pressure pulsations.

6. 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 β (or more precisely, β_b) 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, θ, 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 β = 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 = rdθ,
   so the “wrap” angle θ is found from

(60)

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 θ 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 W_g—often linear as in Figure 17—after
determining the hub and shroud profiles and the corresponding distribution of V_m¹⁶.
In effect, one obtains the distribution of the blade angle β_b 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 nondeviated velocities, in order to get a
smooth variation of the blade angle β_b 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 θ 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 θ 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 θ 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 W_g or other velocity component in Figure 15; for example, V_θ,^.. 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 θ₂- θ₁ 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 β_b 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 surfaces⁵. In such a representation,
the blade angles are the same as they are on the physical surface of revolution
because tan β = dm/dy and dy = rdθ, also yielding Eq. 60.

If the associated distributions of W_g and V_m 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 β at inlet and outlet by the previous procedures as the starting point for
drawing each grid-line. This conformal blade shape is then transformed onto the physical
surface, the differential tangential distance dy becoming rdθ 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 β_s.

Designing the Impeller  Determination of the geometrical features of the impeller is
generally accomplished in the following order: a) the “eye” radius r_e, b) the exit radius r₂
or r_t,2, and c) the exit width b₂ or, in the case of mixed- and axial-flow impellers, the hub
exit radius r_h,2—all of which form the starting point for d) shaping the hub and shroud
profiles (Figure 13); and, finally, e) construction of the blades.

a) The eye. The inlet radius of the impeller eye r_e (Figure 13) is nearly the same as r_t,1,
which is the diameter of the tips of the impeller blades at the inlet. This emerges after
the eye flow coefficient φ_e = V_e/U_e [the ratio of the one-dimensional axial velocity entering
the eye (Figure 8) to the tangential speed of the impeller eye U_e = Ωr_e] is known:

(47)

(48)

Thus, r_e can be found from the following combination...