6.3 DESIGN OF CENTRIFUGAL PUMPS

Because of its specific needs, the rocket industry has developed its own pump design approaches which may differ from those for conventional applications. In addition, designers may employ their individual methods of analysis and calculation. However, the broad underlying principles are quite similar. The range of speeds, proportions, design coefficients, and other mechanical detail for rocket engine pumps has been well established by earlier designs as well as through experiments.

General Design Procedures

As a rule, rated pump head-capacity ( $H-Q$ ) requirements and expected available NPSH at the pump inlets will be established by engine system design criteria. The first step then is to choose a suitable suction specific speed ( $N_{\text {SS }}$ ) and the type of inducer which will yield the highest pump speed ( $N$ ) at design conditions (eq. (6-10)). The pump specific speed $\left(N_{\mathrm{S}}\right)$ or type of impeller can now be established from the chosen pump speed and required head-capacity characteristics. Owing to its relatively light weight and simplicity of construction, a singlestage centrifugal pump may be given first consideration.

With suction specific speed and specific speed of the proposed pump design established, the designer can now look for a suitable "design model" among comparable existing pumps which approximate the desired performance. The latter includes satisfactory suction requirements, suitable head-capacity characteristics, and acceptable efficiency. If a suitable model is available, the design calculations of the new pump will include application of a scaling factor to the parameters of the existing model. The following correlations are valid for pumps with like specific speed, based on the pump affinity laws (eqs. (6-6a) and (6-6b)):

$$\begin{gather*} \frac{N_{1} Q_{1}^{0.5}}{\Delta H_{1}^{0.75}}=\frac{N_{2} Q_{2}^{0.5}}{\Delta H_{2}^{0.75}} \tag{6-24a}\\ Q_{2}=Q_{1} f^{3}\left(\frac{N_{2}}{N_{1}}\right) \tag{6-24b} \end{gather*}$$ $$\begin{equation*} \Delta H_{2}=\Delta H_{1} f^{2}\left(\frac{N_{2}}{N_{1}}\right) \tag{6-24c} \end{equation*}$$

where

$N_{1}, Q_{1}$, and	$\Delta H_{1}=$ rotating speed (rpm), flow rate (gpm), and developed head (ft) of the existing model at rated conditions
$N_{2}, Q_{2}$, and	$\Delta H_{2}=$ rotating speed (rpm), flow rate (gpm), and developed head ( ft ) of the new pump at rated conditions
$f=D_{2} / D_{1}$	= scaling factor
$D_{1}$	=impeller diameter of the existing model, ft
$D_{2}$	= impeller diameter of the new pump, ft

This approach assumes that other dimensions of the pump are in approximately linear proportion to the impeller diameter.

If a suitable model is not available for the design of a new pump, the designer can use "design factors" established experimentally by other successful designs. These may permit establishing relations between rated pump developed head and flow rate, and such parameters as velocity ratios. However, best results are obtained through experimental testing of proposed design itself. The test results then are used for design revisions and refinements.

In the discussions below, the following basic symbols are used:

c=flow velocities, absolute (relative to ducts
    and casing)
v=flow velocities, relative to inducer or im-
    peller
u=velocities of points on inducer or impeller

Subscript: 0 = inducer inlet 1 = inducer outlet = impeller inlet 2 = impeller outlet 3 = pump casing prime 1 = actual or design

Operating Principles of the Centrifugal Pump Impeller

In its simplest form, the impeller of a centrifugal pump can be regarded as a paddle wheel with radial vanes, rotating in an enclosure, with the fluid being admitted axially and ejected at the periphery. This is shown schematically in figure 6-33. The tangential velocity component of each fluid element increases as it moves out radially between the vanes. Therefore, the centrifugal force acting on these fluid elements increases as the fluid moves out radially. Assuming constant flow velocity in the radial direction and no energy losses, the ideal head rise due to centrifugal force between the central entrance (1) and the peripheral exit (2) is

$$\begin{equation*} \Delta H_{i c}=\frac{\omega^{2}}{2 g}\left(r_{2}^{2}-r_{1}^{2}\right) \tag{6-25} \end{equation*}$$

where

$$\begin{aligned} \Delta H_{i c}= & \text { ideal head rise due to centrifugal forces, } \\ & \mathrm{ft} \\ \omega & =\text { angular velocity of the wheel, } \mathrm{rad} / \mathrm{sec} \\ \mathrm{r}_{1}= & \text { vane radius at the entrance, } \mathrm{ft} \\ r_{2}= & \text { vane radius at the periphery, } \mathrm{ft} \\ g \quad & =\text { gravitational constant, } 32.2 \mathrm{ft} / \mathrm{sec}^{2} \end{aligned}$$

For optimum performance, most impellers in high-speed centrifugal rocket engine pumps have shrouded, backward curved vanes. The impeller width is tapered toward the periphery to keep the cross-sectional area of the radial flow path near constant. A typical impeller design of this type is shown in figure 6-34.

Velocity diagrams may be constructed to analyze the fluid flow vector correlations at various points of an impeller. Let us assume the following ideal conditions: (1) There are no losses, such as fluid-friction losses (2) The impeller passages are completely filled with actively flowing fluid at all times

Figure 6-33.-Paddle wheel (schematic).

Figure 6-34.-Typical shrouded centrifugal impeller with backward curved vanes.

(3) The flow is two dimensional (velocities at similar points on the now lines are uniform) (4) The fluid leaves the impeller passages tangentially to the vane surfaces (complete guidance of the fluid at the outlet)

The ideal inlet (point (1)) and outlet (point (2)) flow velocity diagrams of the impeller described in figure 6-34 are shown in figure 6-35. At this point with corresponding fluid velocities $u, v$, and $c$ (as identified above), $\alpha$ is the angle between $c$ and $u$, and $\beta$ is the angle enclosed by a

INLET VELOCITY DIAGRAM

OUTLET VELOCITY DIAGRAMS

Figure 6-35.-Flow velocity diagrams for the impeller shown in figure 6-34 (draw in a plane normal to the impeller axis). tangent to the impeller vane and a line in the direction of vane motion. The latter is equal to the angle between $v$ and $u$ (extended).

Based on these velocity diagrams, the following correlations have been established: ${ }^{1}$

$$\begin{equation*} \Delta H_{i p}=\frac{u_{2}^{2}-u_{1}^{2}+v_{1}^{2}-v_{2}^{2}}{2 g} \tag{6-26a} \end{equation*}$$ $$\begin{gather*} \Delta H_{i}=\frac{u_{2}^{2}-u_{1}^{2}+v_{1}^{2}-v_{2}^{2}+c_{2}^{2}-c_{1}^{2}}{2 g} \\ =\frac{1}{g}\left(u_{2} c_{u_{2}}-u_{1} c_{u_{1}}\right) \tag{6-26b}\\ Q_{1 \mathrm{mp}}=448.8 c_{m_{1}} A_{1}=448.8 c_{m_{2}} A_{2} \tag{6-27}\\ c_{u_{2}}=u_{2}-\frac{c_{m_{2}}}{\tan \beta_{2}} \tag{6-28} \end{gather*}$$

where

$$\begin{aligned} & \Delta H_{i p}=\text { ideal static pressure head rise of the } \\ & \text { fluid flowing through the impeller due } \\ & \text { to centrifugal forces and to a decrease } \\ & \text { of flow velocity relative to the impel- } \\ & \text { ler, } \mathrm{ft} \\ & \Delta H_{i}=\text { ideal total pressure head rise of the } \\ & \text { fluid flowing through the impeller }=\text { the } \\ & \text { ideal developed head of the pump } \\ & \text { impeller, } \mathrm{ft} \end{aligned} \quad \begin{gathered} Q_{\mathrm{imp}}=\begin{array}{c} \text { impeller flow rate at the design point } \\ \text { (rated conditions), gpm } \end{array} \\ A_{1}=\text { area normal to the radial flow at the } \\ \text { impeller inlet, } \mathrm{ft}{ }^{2} \end{gathered}$$ $$\begin{aligned} c_{m_{1}}= & \text { "meridional" or (by definition for radial } \\ & \text { flow impellers) radial component of the } \\ & \text { absolute inlet flow velocity, } \mathrm{ft} / \mathrm{sec} \end{aligned}$$

For pumping low-density propellants (such as liquid hydrogen), which is associated with very high developed heads, straight radial vanes are frequently used in centrifugal impellers, since they permit a higher obtainable head coefficient $\psi$. Figure 6-36 presents a typical radial vane impeller and its outlet velocity diagram. The vane discharge $\beta_{2}=90^{\circ}$ and $c_{y_{2}}=u_{2}$. The ideal developed head of a radial vane impeller becomes

$$\begin{equation*} \Delta H_{i}=\frac{u_{2}^{2}-u_{1} c_{u_{1}}}{g} \tag{6-29} \end{equation*}$$

For centrifugal pumps of the noninducer type (which are now rarely used in rocketry), proper selection of the impelier inlet vane angle $\beta_{1}$ or the provision of guide vanes at the inlet minimizes the absolute tangential component of fluid flow at the inlet, $c_{\mathrm{u}_{1}}$, which for best efficiency should be zero. This is defined as no prerotation, where $a_{1}=90^{\circ}$. Thus, equation (6-26) becomes

$$\begin{equation*} \Delta H_{i}=\frac{u_{2} c_{u_{2}}}{g} \tag{6-30} \end{equation*}$$

Figure 6-36.-Typical radial vane impeller and its outlet velocity diagram.

The above discussions assumed ideal conditions. For most rocket applications, centrifugal pumps are designed with an inducer upstream of and in series with the impeller. The flow conditions at the impeller inlet thus are affected by the inducer discharge flow pattern. In addition, two types of flow usually take place simultaneously in the flow channels; namely, the main flow through the passages, and local circulatory flows (eddy currents). The latter are relatively small but modify the former. The resultant effect at the impeller inlet is to make the flow enter at an angle $\beta_{1}{ }^{\prime}$, larger than the impeller inlet vane angle $\beta_{1}$. The fluid is also caused to leave the impeller at an angle $\beta_{2}{ }^{\prime}$, less than the impeller discharge vane angle $\beta_{2}$, and to increase the absolute angle $\alpha_{2}$ to $\alpha_{2}{ }^{\prime}$. This and the hydraulic losses in the impeller correspondingly change the relative flow velocities $v_{1}$ and $v_{2}$ to $v_{1}{ }^{\prime}$ and $v_{2}{ }^{\prime}$, the absolute flow velocities $c_{1}$ and $c_{2}$ to $c_{1}{ }^{\prime}$ and $c_{2}{ }^{\prime}$, and the absolute tangential components $c_{u_{1}}$ and $c_{u_{2}}$ to $c_{u_{1}}{ }^{\prime}$ and $c_{u_{2}}{ }^{\prime}$. Since the radial flow areas $A_{1}$ and $A_{2}$, and the impeller flow rate $Q_{\text {imp }}$ remain constant, the absolute radial or meridional components $c_{m_{1}}$ and $c_{m_{2}}$ also remain unchanged. The inlet and outlet flow velocity diagrams in figure 6-35 may now be redrawn as represented by the dotted lines.

The correlation established in equation (6-26b) may be rewritten as

$$\begin{equation*} \Delta H_{\mathrm{imp}}=\frac{u_{2} c_{u_{2}}{ }^{\prime}-u_{1} c_{u_{1}}{ }^{\prime}}{g} \tag{6-31} \end{equation*}$$

where

$$\begin{aligned} \Delta H_{\mathrm{imp}}= & \text { impeller actual developed head, } \mathrm{ft} \\ c_{u_{1}}{ }^{\prime}= & \text { tangential component of the design } \\ & \text { absolute inlet flow velocity, } \mathrm{ft} / \mathrm{sec} \\ c_{u_{2}}{ }^{\prime}= & \begin{array}{r} \text { tangential component of the design } \\ \text { absolute outlet flow velocity, } \mathrm{ft} / \mathrm{sec} \end{array} \end{aligned}$$

The ratio of the design now velocity $c_{u_{2}}{ }^{ }$to the ideal flow velocity $c_{u_{2}}$ can be expressed as

$$\begin{equation*} e_{v}=\frac{c_{u_{2}}{ }^{\prime}}{c_{u_{2}}} \tag{6-32} \end{equation*}$$

where $e_{v}=$ impeller vane coefficient. Typical design values range from 0.65 to 0.75.

Referring to figure 6-35, equation (6-28) may be rewritten as

$$\begin{equation*} c_{u_{2}}^{\prime}=u_{2}-\frac{c_{m_{2}}}{\tan \beta_{2}^{\prime}} \tag{6-33} \end{equation*}$$

By definition, the required impeller developed head can be determined as

$$\begin{equation*} \Delta H_{\mathrm{imp}}=\Delta H+H_{\mathrm{e}}-\Delta H_{\mathrm{ind}} \tag{6-34} \end{equation*}$$

where

$$\begin{aligned} \Delta H \quad= & \text { rated design pump developed head, } \mathrm{ft} \\ \Delta H_{\text {ind }}= & \text { required inducer head-rise at the rated } \\ & \quad \text { design point, } \mathrm{ft} \\ H_{e} \quad= & \quad \text { hydraulic head losses in the volute, } \mathrm{ft} . \\ & \quad \text { Typical design values of } H_{e} \text { vary from } \\ & 0.10 \text { to } 0.30 \Delta H . \end{aligned}$$

The required impeller flow rate can be estimated as

$$\begin{equation*} Q_{\mathrm{imp}}=Q+Q_{\mathrm{e}} \tag{6-35} \end{equation*}$$

where

$$\begin{aligned} Q_{\text {imp }}= & \text { required impeller flow rate at the rated } \\ & \text { design point, gpm } \\ Q & \text { rated delivered pump flow rate, gpm } \\ Q_{e}= & \text { impeller leakage losses, gpm. Most of } \\ & \text { these occur at the clearance between } \\ & \text { impeller wearing rings and casing. } \\ & \text { Typical design values of } Q_{e} \text { vary from } \\ & 1 \text { to } 5 \text { percent of } Q_{\text {imp }} . \end{aligned}$$

Centrifugal Impeller Design Elements

After general pump design parameters, such as developed head $\Delta H$, capacity $Q$, suction specific speed $N_{\text {SS }}$, rotating speed $N$, and specific speed $N_{S}$ have been set forth or chosen, the design of a centrifugal (radial) pump impeller may be accomplished in two basic steps. The first is the selection of those velocities and vane angles which are needed to obtain the desired characteristics with optimum efficiency. Usually this can be achieved with the help of available design or experimental data such as pump head coefficient $\psi$, impeller vane coefficient $e_{v}$, and leakage loss rate $Q_{e}$. The second step is the design layout of the impeller for the selected angles and areas. Considerable experience and skill are required from the designer to work out graphically the best-performing configuration based on the given design inputs.

The following are considered minimum basic design elements required for proper layout of a radial－flow impeller：

1．Radial velocity at the impeller entrance or eye，$c_{m_{1}}$ ．－This is a function of inlet conditions such as inducer discharge velocity and inlet duct size．For best performance，the value of $c_{m_{1}}$ should be kept reasonably low．Typical design values of $c_{m_{1}}$ range from 10 to $60 \mathrm{ft} / \mathrm{sec}$ ．

2．Radial velocity at the impeller discharge， $c_{m_{2}}$ ．－Its value is a function of the impeller peripheral velocity $u_{2}$ and the flow coefficient $\phi$ ． Typical design values for $c_{m_{2}}$ range from 0.01 to $0.15 u_{2}$ ．

3．Diameter of the impeller at the vane en－ trance，$d_{1}$ ．－Its value is determined by the in－ ducer design as well as by impeller shaft and hub size．

4．The impeller peripheral velocity at the discharge，$u_{2}$ ．－The value of $u_{2}$ can be calcu－ lated by equation（6－4）for a given pump devel－ oped head $H$ and a selected overall pump head coefficient，$\psi$ ．The maximum design value of $u_{2}$ is often limited by the material strength which thus determines the maximum developed head that can be obtained from a single－stage impel－ ler．Typical design values of $u_{2}$ range from 200 to $1500 \mathrm{ft} / \mathrm{sec}$ ．With $u_{2}$ and $N$ known，the impel－ ler discharge diameter $d_{2}$（in）can be calculated readily．

5．The inlet vane angle $\beta_{1}$ ．The value of $\beta_{1}$ is affected by the inlet flow conditions．Gener－ ally，$\beta_{1}$ should be made equal or close to the inlet flow angle $\beta_{1}{ }^{\prime}$ which can be approximated by

$$\begin{equation*} \tan \beta_{1}^{\prime}=c_{m_{1}} /\left(u_{1}-c_{u_{1}^{\prime}}\right) \tag{6-36} \end{equation*}$$

Typical design values for $\beta_{1}$ range from $8^{\circ}$ to $30^{\circ}$ ．

6．The discharge vane angle $\beta_{2}$ ．－In the special case of radial vane impeller designs $\beta_{2}=90^{\circ}$ ．For backward curved vane impellers， $\beta_{2}$ is the most important single design element． Usually the selection of $\beta_{2}$ is the first step in determining the other impeller design constants， since most of them depend on $\beta_{2}$ ．Pump effi－ ciency and head－capacity characteristics are important considerations for the selection．For a given $u_{2}$ ，head and capacity increase with $\beta_{2}$ ． Typical design values for $\beta_{2}$ range from 17 to $28^{\circ}$ ，with an average value of $22.5^{\circ}$ for most specific speeds．

Figure 6－37 presents the basic layout of a typical radial－flow impeller with backward curved vanes．The shaft diameter $d_{s}$ may be determined by the following correlations

$$\begin{align*} S_{s} & =\frac{16 T}{\pi d_{s}^{3}} \tag{6-37}\\ S_{t} & =\frac{32 M}{\pi d_{s}^{3}} \tag{6-38}\\ S_{s w} & =\frac{1}{2} \sqrt{4 S_{s}^{2}+S_{t}^{2}} \tag{6-39}\\ S_{t w} & =\frac{1}{2} S_{t}+\frac{1}{2} \sqrt{4 S_{s}^{2}+S_{t}^{2}} \tag{6-40} \end{align*}$$

where $d_{s}=$ impeller shaft diameter，in $T=$ shaft torque corresponding to yield or ultimate loads as defined by equations （2－9）and（2－10），lb－in $M=$ shaft bending moment corresponding to yield or ultimate loads as defined by equations（2－9）and（2－10），1b－in $S_{s}=$ shear stress due to torque， $\mathrm{lb} / \mathrm{in}^{2}$ $S_{t}=$ tensile stress due to bending moment， $\mathrm{lb} / \mathrm{in}^{2}$ $S_{\mathrm{SW}}=$ allowable working shear stress（yield or ultimate）of the shaft material， $\mathrm{lb} / \mathrm{in}^{2}$ $S_{t w}=$ allowable working tensile stress（yield or ultimate）of the shaft material， $\mathrm{lb} / \mathrm{in}^{2}$ Impeller hub diameter $d_{h}$ and eye diameter $d_{e}$ may be equal to hub diameter and tip diameter of the inducer．The maximum tensile stress in－ duced in an impeller by the centrifugal forces

Figure 6－37．－Basic layout of a typical radial－ flow impeller with backward curved vanes．

occurs as the tangential stress at the edge of the shaft hole. It may be checked by

$$S_{t \max }=\frac{\rho u_{2}^{2} \max (3+\mu)}{576 g}\left[1+\frac{1-\mu}{3+\mu}\left(\frac{d_{s}}{d_{2}}\right)^{2}\right] K_{s}(6-41)^{1}$$

where

$$\begin{aligned} S_{t \text { max }}= & \begin{aligned} & \text { maximum tensile stress, } \mathrm{lb} / \mathrm{in}^{2}(\text { should } \\ & \quad \text { be less than the allowable working } \\ & \quad \text { tensile stress of the impeller material }) \\ \rho & = \\ \mu & =\text { Poissity of the impeller material, } \mathrm{lb} / \mathrm{ft}^{3} \\ d_{\mathrm{s}}= & \text { impeller shaft hole diameter, in } \\ d_{2}= & \text { impeller outside diameter, in } \\ u_{2 \text { max }}= & \text { maximum allowable peripheral impeller } \\ & \quad \text { speed, } \mathrm{ft} / \mathrm{sec}=1.25 \times \text { design value of } \\ & u_{2}, \text { for most rocket engine applica- } \\ & \quad \text { tions } \end{aligned} \\ g & = \\ K_{\mathrm{s}} \quad= & \text { design factor, determined experimen- } \\ & \quad \text { tally. Typical values vary from } 0.4 \\ & \quad \text { to } 1.0, \text { depending on impeller shape. } \end{aligned}$$

The surface finish and contour of the impeller shaft hole should be free of stress concentrations. First-class splines are preferred rather than ordinary keyways.

The width of the impeller can be calculated by the following correlations:

$$\begin{align*} & b_{1}=\frac{Q_{\mathrm{imp}}}{3.12 \pi d_{1} c_{m_{1} \epsilon_{1}}} \tag{6-42}\\ & b_{2}=\frac{Q_{\mathrm{imp}}}{3.12 \pi d_{2} c_{m 2} \epsilon_{2}} \tag{6-43} \end{align*}$$

where

$$\begin{aligned} b_{1}= & \text { impeller width at the vane inlet, in } \\ b_{2}= & \text { impeller width at the discharge, in } \\ \epsilon_{1}= & \text { contraction factor at the entrance. It } \\ & \text { considers effective flow area reduction } \\ & \text { from vane thickness and other effects } \\ & \text { such as local circulatory flows. Typi- } \\ & \text { cal design values range from } 0.75 \text { to } \\ & 0.9 . \\ \epsilon_{2}= & \text { contraction factor at the discharge. } \\ & \text { Typical design values range from } 0.85 \\ & \text { to } 0.95 . \end{aligned}$$

[^6]

$$\begin{aligned} & Q_{\text {imp }}=\text { impeller flow rate at the rated design } \\ & \text { point, gpm } \end{aligned}$$

After the vane angles and other dimensions at inlet and discharge have been established, no set rule is available for designing the backward curved vanes. However, the number of vanes is usually between 5 and 12, and may be determined empirically by

$$\begin{equation*} z=\frac{\beta_{2}}{3} \tag{6-44} \end{equation*}$$

where

$$\begin{aligned} & \beta_{2}=\text { discharge vane angle } \\ & z=\text { number of vanes } \end{aligned}$$

If there is a space limitation at the impeller entrance, every other vane may be made a partial vane, starting at a larger radius. The contour of the vanes is designed to afford a gradual change of flow cross-sectional area (total divergence of $10^{\circ}$ to $14^{\circ}$ ), at reasonably short flow passage length. The flow passage shape should be as close to a square as possible. The vanes should be as thin as material strength and manufacturing processes will permit. They may be of constant thickness; i.e., a contour similar for both sides may be used. This allows a thinner edge (typical value: 0.12 inch) at the inlet and results in better efficiency if the angle $\beta_{1}$ has the correct value. The impeller is usually a highquality aluminum-alloy casting, the vanes being integral with the shrouds. In some high-speed applications, forged aluminum alloys or titanium alloys are used. A typical aluminum forging, the 7075 alloy with a T73 heat treat, has a yield strength of 63000 psi and an ultimate strength of 74000 psi . In this case a two-piece construction might be preferred to facilitate machining operations.

Mixed-flow-type vanes which extend into the impeller entrance or eye (shown in fig. 6-38a) are frequently used in radial-flow impellers or centrifugal pumps. This is done to match the impeller inlet flow path with the inducer discharge flow pattern and to provide more efficient turning of the flow.

The mixed-flow-type impeller as shown in figure 6-38b is also frequently used in a "centrifugal flow pump." The velocity correlations and design constants of a mixed-flow impeller are essentially the same as those of a radialflow impeller. Mean effective impeller diameters are used in the calculations for head rise, flow velocities, etc. They are presented in figure 6-38 a and b as:

$$\begin{align*} & d_{1}^{2}=\frac{\left(d_{10}^{2}+d_{1 i}^{2}\right)}{2} \tag{6-45}\\ & d_{2}^{2}=\frac{\left(d_{20}^{2}+d_{2 i}^{2}\right)}{2} \tag{6-46} \end{align*}$$

where

$$\begin{aligned} & d_{1}= \\ & \quad \text { mean effective impeller diameter at the } \\ & d_{10}=\text { outer vane diameter at the inlet, in } \\ & d_{1 i}=\text { inner vane diameter at the inlet, in } \\ & d_{2}=\text { mean effective impeller diameter at the } \\ & \quad \text { discharge, in } \\ & d_{20}=\text { outer vane diameter at the discharge, in } \end{aligned}$$

$d_{2 i}=$ inner vane diameter at the discharge, in Effective impeller widths at inlet and discharge, $b_{1}$ and $b_{2}$, are also presented in figure 6-38 $a$ and $b$. They are equal to the diameter of a circle which is tangent to the contours of both front and back shrouds. $\gamma$ is the angle between the meridional flow vectors ( $c_{m_{1}}$ and $c_{m_{2}}$ ) and the plane normal to the axis of rotation. It is also the angle between the plane of the velocity diagrams and the plane normal to the axis. The value of $\gamma$ varies along the flow passage. The layout of a mixed-flow impeller on the drawing board is a rather complicated drafting problem. This is due to the three-dimensional vane curvature and other complexities. The method of "error triangles" suggested by Kaplan may be used. Details of this method can be found in standard pump reference books.

Design of Cavitating Inducers

The cavitating inducer of a centrifugal propellant pump is a lightly loaded axial-flow impeller operating in series with the main pump impeller as shown in figure 6-5. The term "cavitating" refers to the fact that the inducer is capable of operating over a relatively broad range of incipient cavitation prior to a noticeable pump head dropoff. It produces from 5 to 20 percent of the total head rise of a pump. The conditions of pump critical NPSH at the 2-percent dropoff point may correspond to a 10 - to 30 percent inducer-developed-head reduction, depending upon its match to the main pump impeller. The required inducer head rise for a given design is expressed by the correlation

Figure 6-38.-(a) Radial-flow impeller with mixed-flow vanes at the impeller entrance; (b) Mixed-flow impeller.

$$\begin{equation*} \Delta H_{\mathrm{ind}}=(\mathrm{NPSH})_{\mathrm{imp}}-(\mathrm{NPSH})_{\mathrm{ind}} \tag{6-47} \end{equation*}$$

or

$$\begin{align*} \left[\frac{N(Q)^{0.5}}{\left(N_{\mathrm{S}}\right)_{\mathrm{ind}}}\right]^{1.333} = & \left[\frac{N(Q)^{0.5}}{\left(N_{\mathrm{SS}}\right)_{\mathrm{imp}}}\right]^{1.333} \\ & -\left[\frac{N(Q)^{0.5}}{\left(N_{\mathrm{SS}}\right)_{\mathrm{ind}}}\right]^{1.333} \tag{6-48} \end{align*}$$

where

$$\begin{aligned} \Delta H_{\text {ind }} \quad & =\text { required inducer head rise at the } \\ & \quad \text { design point, } \mathrm{ft} \\ (\mathrm{NPSH})_{\text {ind }}= & \quad \text { inducer critical } \mathrm{NPSH}=\text { pump } \\ & \quad \text { critical } \mathrm{NPSH} \text { or }(\mathrm{NPSH})_{c} \\ = & \quad \text { Thoma parameter } r \times \text { pump total } \\ & \quad \text { developed head } H \\ (\mathrm{NPSH})_{\text {imp }}= & \text { impeller critical } \mathrm{NPSH} \\ = & \text { pump shaft speed, rpm (same for } \\ & \quad \text { inducer and impeller } \\ = & \quad \text { rated pump flow rate, } \mathrm{gpm} \\ Q & \quad \text { inducer specific speed } \\ \left(N_{\mathrm{s}}\right)_{\text {ind }} \quad & \\ \left(N_{\mathrm{ss}}\right)_{\text {imp }}= & \text { impeller suction specific speed } \\ \left(N_{\mathrm{ss}}\right)_{\text {ind }}= & \text { inducer suction specific speed } \\ & \quad=\text { pump suction specific speed } \\ & \quad N_{\mathrm{ss}} \end{aligned}$$

Figure 6-39 presents the basic elements of a typical inducer design. The primary increase in static pressure occurs at the leading (upstream) edge of the vane through free stream diffusion: i.e., through a reduction of relative speed and by operating at a small angle of attack between relative inlet flow and inducer inlet vane. The cavitation performance of an inducer depends strongly on the inlet flow coefficient $\phi_{\text {ind }}$ (ratio of inlet axial flow velocity $c_{\text {mo }}$ to inlet tip speed $u_{o t}$ ). To obtain high suction specific speeds (for highest pump speed $N$ ), the inducer must have a low flow coefficient. This results in small angles ( $\theta_{t}, \theta_{h}$ ) between vanes and the plane normal to the axis of rotation. As a rule, the inducer vane angle $\theta$ varies radially according to a constant

$$\begin{equation*} c=d \tan \theta=d_{t} \tan \theta_{t}=d_{h} \tan \theta_{h} \tag{6-48a} \end{equation*}$$

It also often varies axially, according to the variation of the axial or meridional component $c_{m}$ of the absolute flow velocity. Inducer inlet flow coefficient $\phi_{\text {ind }}$, inducer diameter ratio $r_{d}$ (ratio of hub diameter $d_{h}$ to tip diameter $d_{t}$ ), and

Figure 6-39.-Elements of a typical inducer design (three-vanes; cylindrical hub and tip contour).

suction specific speed $\left(N_{\mathrm{ss}}\right)$ ind are related, based on theoretical one-dimensional fluid cavitation considerations, by the expression

$$\begin{equation*} \frac{\left(N_{\mathrm{ss}}\right)_{\mathrm{ind}}}{\left(1-\mathrm{r}_{d}{ }^{2}\right)^{0.5}}=\frac{8150\left(1-2 \phi_{\mathrm{ind}}{ }^{2}\right)^{0.75}}{\phi_{\mathrm{ind}}} \tag{6-49} \end{equation*}$$

Figure 6-40 shows this relationship graphically. The actual performance of a typical inducer is also shown for comparison.

The three vanes shown in figure 6-39 are equally spaced at a tip distance $P_{i}$. This is defined as "pitch" and can be expressed as

$$\begin{equation*} P_{\mathrm{i}}=\frac{\pi \mathrm{d}_{t}}{Z} \tag{6-50} \end{equation*}$$

where $P_{i}=$ pitch or vane spacing, in $d_{t}=$ inducer tip diameter, in $z=$ number of vanes The ratio of vane tip chord length $C_{i}$ to vane pitch $P_{i}$ is an important design element. It is defined as "vane solidity at the tip" of an inducer. Vane solidity $S_{V}$ is a descriptive term relating the vane area (actual or projected) to the area of the annuli normal to the axial flow. It can be expressed as

$$\begin{equation*} S_{v}=\frac{C_{i}}{P_{i}} \tag{6-51} \end{equation*}$$

The ratio of inducer length $L_{i}$ to inducer tip diameter $d_{t},\left(L_{i} / d_{t}\right)$ is another important design

Figure 6-40.-Relation between inducer inlet flow coefficient and inducer suction specific speed.

element used to describe the proportions of an inducer.

Frequently the design of high suction performance inducers dictates a relatively large inlet eve diameter, while the pump main impeller inlet eye diameter must remain small for best performance. This condition can be accommodated by tapering the inducer tip to lead over from one diameter to the other (as shown in fig. 6-41). To minimize the tip taper, tapering of the inducer inlet hub diameter may be added to maintain the desired inducer inlet flow area. In the calculation of tapered inducers, mean values may be used for $d_{t}$ and $d_{h}$. For structural reasons, the inducer vane elements sometimes are designed to cant forward instead of being normal to the axis of rotation. The angle between the canted vane and the plane normal to the axis is defined as the sweep angle.

Table 6-5 contains typical values for inducer design parameters and variables. Figure 6-42 presents inducer inlet and outlet velocity diagrams based on the mean effective diameters. For the design calculations of inducers, the following correlations may be used (figs. 6-39, 6-41 and 6-42). For inducers with cylindrical hub and tip contour:

$$\begin{equation*} d_{0}^{2}=d_{1}^{2}=\frac{d_{t}^{2}+d_{h}^{2}}{2} \tag{6-52} \end{equation*}$$

(assume $c_{u 0^{\prime}}=0$ )

$$\begin{gather*} c_{0}^{\prime}=c_{m 0}=c_{m_{1}}=\frac{Q_{\text {ind }}}{3.12 \times \frac{\pi}{4}\left(d_{t}{ }^{2}-d_{h}{ }^{2}\right)} \tag{6-53}\\ u_{0}=u_{1}=\frac{\pi N}{720} d_{0}=\frac{\pi N}{720} d_{1} \tag{6-54} \end{gather*}$$

Figure 6-41.-Taper contour inducer.

Table 6-5.-Cavitating Inducer Design Parameters and Variables

Parameter or variable	Typical design values	Design requirement
Specific speed, ( $N_{\mathrm{S}}$ ) ind	6000 to 12000	Head-capacity characteristics
Suction specific speed, ( $N_{\text {ss }}$ )ind	20000 to 50000	Suction characteristics
Head coefficient, $\psi_{\text {ind }}$	0.06 to 0.15	Head rise
Inlet flow coefficient, $\phi_{\text {ind }}$	0.06 to 0.20	Cavitation performance
Inlet vane angle, $\theta$	$8^{\circ}$ to $16^{\circ}$ (measured from plane normal to axis)	Flow coefficient, angle of attack
Angle of attack, i............................ .	$3^{\circ}$ to $8^{\circ}$	Performance, flow coefficient. vane loading
Diameter ratio, $r_{d} \ldots \ldots \ldots \ldots \ldots . . \ldots . . . . . . . . . . .$.	0.2 to 0.5	Performance, shaft critical speed
Vane solidity, $S_{V}$	1.5 to 3.0 at the tip	Desired flow area
Number of vanes, 2 .	3 to 5	Desired solidity
Hub contour	Cylindrical to $15^{\circ}$ taper	Compatibility with main impeller and shaft geometry
Tip contour	Cylindrical to $15^{\circ}$ taper	Compatibility with main impeller and shaft geometry
Vane loading	Leading edge loading, channel lead	Performance
Leading edge	Swept forward, radial, swept back (shown in fig. 6-39)	Vane stress, performance
Sweep angle	Normal to shaft to $15^{\circ}$ forward	Vane stress
Vane thickness	0.070 to 0.300 chord length $C_{i}$	Vane stress
Tip clearance (between inducer outside diameter and casing)	0.5 to 1 percent of inducer outside diameter	Shaft axial and radial detlections
Length to tip diameter ratio ( $L_{i} / d_{t}$ )	0.3 to 0.6	Head-capacity characteristics

INLET VELOCITY DIAGRAM

OUTLET VELOCITY DIAGRAM

Figure 6-42.-Typical flow velocity diagrams of an inducer based on the mean effective diameters.

For inducers with tapered hub and tip contour:

$$\begin{equation*} d_{0}^{2}=\frac{d_{0 t}{ }^{2}+d_{0 h}{ }^{2}}{2} \tag{$6\cdot55$} \end{equation*}$$ $$\begin{equation*} d_{1}{ }^{2}=\frac{d_{1 t}{ }^{2}+d_{1 h}{ }^{2}}{2} \tag{6-56} \end{equation*}$$ $$\begin{equation*} d_{t}=\frac{d_{0 t}+d_{1 t}}{2} \tag{6-57} \end{equation*}$$ $$\begin{equation*} d_{h}=\frac{d_{0 h}+d_{1 h}}{2} \tag{6-58} \end{equation*}$$ $$c_{0}^{\prime}=c_{m 0}=\frac{Q_{\mathrm{ind}}}{3.12 \times \frac{\pi}{4}\left(d_{0 t}{ }^{2}-d_{0} h^{2}\right)}$$

(no prerotation) (6-59)

$$\begin{equation*} c_{m 1}=\frac{Q_{\text {ind }}}{3.12 \times \frac{\pi}{4}\left(d_{1} t^{2}-d_{1} h^{2}\right)} \tag{6-60} \end{equation*}$$ $$\begin{equation*} u_{0}=\frac{\pi N}{720} d_{0} \tag{6-61} \end{equation*}$$ $$\begin{equation*} u_{1}=\frac{\pi N}{720} d_{1} \tag{6-62} \end{equation*}$$

For all inducers:

$$\begin{gather*} Q_{\mathrm{ind}}=Q+Q_{e e}+\frac{1}{2} Q_{e} \tag{6-63}\\ r_{d}=\frac{d_{h}}{d_{t}} \tag{6-64}\\ u_{t}=\frac{\pi N}{720} d_{t} \tag{6-65}\\ u_{0 t}=\frac{\pi N}{720} d_{0 t} \tag{6-65a}\\ \Delta H_{\mathrm{ind}}=\psi_{\mathrm{ind}} \frac{u_{t}^{2}}{g}=\frac{u_{1} c_{u_{1}}}{g} \tag{6-66}\\ \Delta H_{\mathrm{indt}}=\frac{H_{\mathrm{ind}}}{\eta_{\mathrm{ind}}}=\frac{u_{1} c_{u_{1}}}{g} \tag{6-67}\\ \phi_{\mathrm{ind}}=\frac{c_{m 0}}{u_{0 t}} \tag{6-68} \end{gather*}$$

where

Q	= rated pump flow rate, gpm
$Q_{\text {ind }}$	= required inducer flow rate at the rated design point, gpm
$Q_{e}$	= impeller leakage losses at the rated design point, gpm
$Q_{\text {ee }}$	= inducer leakage loss rate through the tip clearance, gpm. Typical design values vary from 2 to 6 percent of Q
$\Delta H_{\text {ind }}$	$=$ required inducer head rise at rated conditions, ft
$\Delta H_{\text {indt }}=$ ideal inducer head rise at rated conditions, ft = inducer efficiency
$d_{t}$	= inducer mean tip diameter, in
$d_{0 t}$	= inducer tip diameter at the inlet, in
$d_{1 t}$	= inducer tip diameter at the outlet, in
$d_{h}$	= inducer mean hub diameter, in
$d_{0 h}$	= inducer hub diameter at the inlet, in
$d_{1 h}$	= inducer hub diameter at the outlet, in
$d_{0}$	=inducer mean effective diameter at the inlet, in
$d_{1}$	= inducer mean effective diameter at the outlet, in
$u_{0}$	= inducer peripheral velocity at mean effective inlet diameter, ft/sec

$u_{1}$	= inducer peripheral velocity at mean effective outlet diameter, ft/sec
$u_{t}$	= mean tip speed of the inducer, ft/sec
$u_{0 t}$	$=$ inducer inlet tip speed, ft/sec
$u_{1 t}$	= inducer outlet tip speed, ft/sec
$v_{0}{ }^{1}$	$=$ inlet velocity of the flow relative to the inducer, ft/sec
$v_{1}{ }^{\prime}$	= outlet velocity of the flow relative to the inducer, $\mathrm{ft} / \mathrm{sec}$
$c_{0}{ }^{\prime}$	= absolute inlet velocity of the flow, $\mathrm{ft} / \mathrm{sec}$
$c_{u o^{\prime}}$	$=$ tangential component of the absolute flow velocity, ft/sec
$c_{m 0}$	=meridional or axial component of the absolute inlet flow velocity, ft/sec
$c_{1}$	= ideal absolute outlet flow velocity, $\mathrm{ft} / \mathrm{sec}$
$c_{1}{ }^{\prime}$	= absolute outlet flow velocity, ft/sec
$c_{u 1}{ }^{\prime}$	=tangential component of the absolute outlet flow velocity, $\mathrm{ft} / \mathrm{sec}$
$c_{m_{1}}$	=meridional component of the absolute outlet flow velocity, ft/sec
$r_{d}$	= hub to tip diameter ratio
$\psi_{\text {ind }}$	= inducer head coefficient (for range, see table 6-5)
$\phi_{\text {ind }}$	= inducer inlet flow coefficient (for range, see table 6-5)

The inducer is generally made from a highquality aluminum-alloy forging of single-piece construction. For manufacture, special machines and tooling are required for best results.

Experimental results have indicated that a high-pressure fluid-injection system can be designed to increase the suction performance of a pump with inducer by imparting an inlet "prewhirl" to the fluid entering the inducer. Fluid injection provides a tangential component $c_{u 0}$ in the proper direction to the absolute fluid inlet velocity $c_{0}{ }^{\prime}$ and thereby lowers the fluid inlet velocity $v_{0}$ relative to the inducer. Jet momentum and directed whirl in the direction of blade rotation combined should serve to reduce the tendency for the blade tips to cavitate as a result of high relative velocities and low static pressure. The suction specific speed of one typical inducer design was increased from 34000 to 44000 by applying "prewhirl." Fluid injection is introduced tangentially (at a small angle with the plane normal to the axis of rotation) several inches upstream of the inducer inlet. It is fed from the pump outlet fluid pressure.

Sample Calculation (6-7)

The following required design data and experimental model test results are given for the oxidizer pump of the A-1 stage engine:

Required pump developed head, $\Delta H=2930 \mathrm{ft}$ Required pump flow rate, $Q=12420 \mathrm{gpm}$ Pump shaft speed, $N=7000 \mathrm{rpm}$ Pump specific speed, $N_{S}=1980$ Pump critical NPSH, $(\text { NPSH })_{c}=58 \mathrm{ft}$ Pump actual suction specific speed, $N_{\text {SS }} =37230$ (from experimental tests) Pump overall head coefficient, $\psi=0.46$ Basic inducer configuration (tip and hub taper contours) (similar to fig. 6-41) Inducer head coefficient, $\psi_{\text {ind }}=0.06$ Inducer diameter ratio, $r_{d}=0.3$ Inducer ratio, $L_{i} / d_{t}=0.4$ Angle of attack at inducer inlet tip, $i=4^{\circ}$ maximum Inducer tip contour taper half angle $=7^{\circ}$ Inducer hub contour taper half angle $=14^{\circ}$ Inducer solidity based on mean tip diameter, $d_{t}=2.2$ Inducer leakage loss rate, $Q_{\mathrm{ee}}=0.032 Q$ Basic impeller configuration = radial-flow type with mixed flow vanes at the inlet (similar to fig. 6-38a) Impeller suction specific speed, $\left(N_{\mathrm{SS}}\right)_{\text {imp }} =11000$ Impeller discharge vane angle, $\beta_{2}=24^{\circ}$ Impeller contraction factor at the entrance, $\epsilon_{1}=0.82$ Impeller contraction factor at the discharge, $\epsilon_{2}=0.88$ Impeller coefficient, $e_{v}=0.74$ Impeller leakage loss rate, $Q_{e}=0.035 Q$ Pump volute head loss, $H_{e}=0.19 H$ Design and calculate the basic parameters and dimensions of: (a) pump inducer, (b) pump impeller.

Solution

(a) Oxidizer pump inducer

Modifying equation (6-10), the impeller critical NPSH can be calculated as

$$\begin{aligned} (\mathrm{NPSH})_{i \mathrm{mp}} & =\left[\frac{N Q^{0.5}}{\left(N_{\mathrm{ss}}\right)_{i \mathrm{mp}}}\right]^{1.333} \\ & \quad=\left[\frac{7000 \times(12420)^{0.5}}{11000}\right]^{1.333}=293 \mathrm{ft} \end{aligned}$$

Substitute (NPSH) ${ }_{\text {imp }}$ into equation (6-47) to obtain the required inducer head rise

$$\begin{aligned} \Delta H_{\mathrm{ind}}= & (\mathrm{NPSH})_{\mathrm{imp}}-(\mathrm{NPSH})_{\mathrm{ind}} \\ & =(\mathrm{NPSH})_{\mathrm{imp}}-(\mathrm{NPSH})_{c}=293-58=235 \mathrm{ft} \end{aligned}$$

From equation (6-66), inducer mean tip speed

$$u_{t}=\sqrt{\frac{235 \times 32.2}{0.06}}=355 \mathrm{ft} / \mathrm{sec}$$

Inducer mean tip diameter

$$d_{t}=\frac{720}{\pi N} u_{t}=\frac{720 \times 355}{\pi \times 7000}=11.62 \mathrm{in}$$

(from eq. (6-65)). For the given ( $L_{i} / d_{t}$ ) ratio of 0.4 , the axial length of the inducer becomes

$$L_{i}=11.62 \times 0.4=4.65 \mathrm{in}$$

For a given tip contour taper half angle of $7^{\circ}$. the tip diameter at the inducer inlet

$$\begin{aligned} d_{0 t}=d_{t}+2 \times \frac{L_{i}}{2} \times & \tan 7^{\circ} \\ & =11.62+4.65 \times 0.1228=12.19 \mathrm{in} \end{aligned}$$

Tip diameter at the inducer outlet

$$d_{1 t}=d_{t}-2 \times \frac{L_{i}}{2} \times \tan 7^{\circ}=11.62-0.57=11.05 \mathrm{in}$$

Mean hub diameter

$$d_{h}=d_{t} r_{d}=11.62 \times 0.3=3.49 \mathrm{in}$$

(from eq. (6-64)). For the given hub contour taper half angle of $14^{\circ}$, the hub diameter at the inducer inlet $d_{0 h}=d_{h}-2 \times \frac{L_{i}}{2} \tan 14^{\circ}=3.49-4.65 \times 0.2493=2.33 \mathrm{in}$ The hub diameter at the inducer outlet $d_{1 h}=d_{h}+2 \times \frac{L_{i}}{2} \tan 14^{\circ}=3.49+1.16=4.65$ in

Substitute the given leakage loss rates $Q_{\text {ee }} =0.032$ for the inducer and $\frac{1}{2} Q_{e}=0.0175$ for the impeller into equation (6-63) to obtain the required inducer flow rate

$$\begin{aligned} Q_{\text {ind }}=Q & +Q_{\mathrm{ee}}+\frac{1}{2} Q_{\mathrm{e}} \\ & =12420(1+0.032+0.0175)=13040 \mathrm{gpm} \end{aligned}$$

From equation (6-59), the actual inducer absolute inlet flow velocity $c_{0}{ }^{\prime}$ (=its meridional component $c_{m 0}$, assuming its tangential component $c_{\text {uo }}=0 ; a_{0}{ }^{\prime}=0$ )

$$\begin{aligned} c_{0}^{\prime}=c_{m 0}= & \frac{Q_{\text {ind }}}{3.12 \times \frac{\pi}{4}\left(d_{0} t^{2}-d_{0} h^{2}\right)} \\ & =\frac{13040}{3.12 \times \frac{\pi}{4} \times(148.6-5.43)}=37.2 \mathrm{ft} / \mathrm{sec} \end{aligned}$$

From equation (6-60), the meridional component of the inducer absolute outlet flow velocity

$$\begin{aligned} c_{m_{1}}= & \frac{Q_{\text {ind }}}{3.12 \times \frac{\pi}{4}\left(d_{1} t^{2}-d_{1} h^{2}\right)} \\ & \quad=\frac{13040}{3.12 \times \frac{\pi}{2} \times(122.1-21.6)}=53.1 \mathrm{ft} / \mathrm{sec} \end{aligned}$$

From equation (6-55), the inducer mean effective diameter at the inlet $d_{0}=\sqrt{\frac{d_{0 t^{2}}+d_{0 h^{2}}}{2}}=\sqrt{\frac{148.6+5.43}{2}}=\sqrt{77}=8.76 \mathrm{in}$ From equation (6-61), the inducer peripheral velocity at $d_{0}$

$$u_{0}=\frac{\pi \times 7000}{720} \times 8.76=268 \mathrm{ft} / \mathrm{sec}$$

From equation (6-56), the inducer mean effective diameter at the outlet

$$\begin{aligned} d_{1}=\sqrt{\frac{d_{1 t^{2}+d_{1 h}{ }^{2}}^{2}}{2}}=\sqrt{\frac{122.1+21.6}{2}} & \\ & =\sqrt{71.85}=8.45 \mathrm{in} \end{aligned}$$

From equation (6-62), the inducer peripheral velocity at $d_{1}$

$$u_{1}=\frac{\pi \times 7000}{720} \times 8.45=258.5 \mathrm{ft} / \mathrm{sec}$$

From equation (6-66), the tangential component of the inducer absolute outlet velocity

$$c_{\mathrm{u}_{1}}^{\prime}=\Delta H_{\mathrm{ind}} \frac{g}{\mathrm{u}_{1}}=\frac{235 \times 32.2}{258.5}=29.2 \mathrm{ft} / \mathrm{sec}$$

Refer to figure 6-42 for the flow velocity diagrams of the inducer, based on the mean effective diameters $d_{0}$ and $d_{1}$. Inducer design relative inlet flow velocity

$$v_{0}^{\prime}=\sqrt{c_{m 0}^{2}+u_{0}^{2}}=\sqrt{1384+71825}=270.6 \mathrm{ft} / \mathrm{sec}$$

Inducer design relative inlet flow angle

$$\sin \beta_{0}^{\prime}=\frac{c_{m_{0}}}{v_{0}^{\prime}}=\frac{37.2}{270.6}=0.135 ; \quad \beta_{0}^{\prime}=7^{\circ} 45^{\prime}$$

Inducer design absolute outlet flow velocity

$$c_{1}^{\prime}=\sqrt{c_{u 1}^{\prime 2}+c_{m 1}^{2}}=\sqrt{852.64+2819.61}=60.5 \mathrm{ft} / \mathrm{sec}$$

Inducer design absolute outlet flow angle

$$\tan a_{1}^{\prime}=\frac{c_{m_{1}}}{c_{u_{1}}}=\frac{53.1}{29.2}=1.82 ; \quad a_{1}^{\prime}=61^{\circ} 13^{\prime}$$

Inducer design relative outlet flow velocity

$$\begin{aligned} & v_{1}^{\prime}=\sqrt{\left(u_{1}-c_{u_{1}}^{\prime}\right)^{2}+c_{m_{1}}^{2}} \\ & \quad=\sqrt{52578+2820}=235 \mathrm{ft} / \mathrm{sec} \end{aligned}$$

Inducer design relative outlet flow angle

$$\tan \beta_{1}^{\prime}=\frac{c_{m_{1}}}{u_{1}-c_{u_{1}}}=\frac{53.1}{229.3}=0.232 ; \quad \beta_{1}^{\prime}=13^{\circ} 3^{\prime}$$

Since the cavitation performance of an inducer depends largely on the angle of attack of the vane leading edge at the inducer inlet tip, and on the inducer inlet flow coefficient $\phi_{\text {ind }}$, we now proceed to determine the vane angle $\theta_{0 t}$ at the inducer inlet tip, and to check $\phi_{\text {ind }}$ with the help of equation (6-49).

From equation (6-65a) the inducer inlet tip speed

$$u_{0 t}=\frac{\pi \times N}{720} d_{0 t}=\frac{\pi \times 7000}{720} \times 12.19=372.5 \mathrm{ft} / \mathrm{sec}$$

The relative flow angle at the inducer inlet tip

$$\tan \beta_{0 t}{ }^{\prime}=\frac{c_{m 0}}{u_{0 t}}=\frac{37.2}{372.5}=0.0998 ; \quad \beta_{0 t}{ }^{\prime}=5^{\circ} 42^{\prime}$$

If we use a vane angle $\theta_{0 t}=9^{\circ}$ at the inducer inlet tip, the angle of attack at the inlet tip $\theta_{0 t}-\beta_{0 t} t^{\prime}=9^{\circ}-\left(5^{\circ} 42^{\prime}\right)=3^{\circ} 18^{\prime}\left(<4^{\circ}\right.$, as desired)

The vane angle $\theta_{0}$ at the inducer inlet mean effective diameter $d_{0}$

$$\begin{array}{r} \tan \theta_{0}=\frac{d_{0 t}}{d_{0}} \tan \theta_{0 t}=\frac{12.19}{8.76} \times \tan 9^{\circ}=0.220, \\ \theta_{0}=12^{\circ} 25^{\prime} \end{array}$$

(see eq. (6-48a)). The vane angle $\theta_{o h}$ at the inducer inlet hub diameter $d_{0 h}$

$$\begin{aligned} \tan \theta_{0 h}=\frac{d_{0 t}}{d_{0 h}} \tan \theta_{0 t}=\frac{12.19}{2.33} \times \tan 9^{\circ}= & 0.829 ; \\ & \theta_{0 h}=39^{\circ} 40^{\prime} \end{aligned}$$

From equation (6-68), the inducer inlet flow coefficient

$$\phi_{\mathrm{ind}}=\frac{c_{m 0}}{u_{0 t}}=\frac{37.2}{372.5}=0.0998$$

Substitute this into equation (6-49), to obtain the theoretical inducer suction specific speed

$$\begin{aligned} \left(N_{\mathrm{SS}}\right)_{\text {ind }} & =\frac{8150\left(1-2 \phi_{\text {ind }}{ }^{2}\right)^{0.75}}{\phi_{\text {ind }}}\left(1-r_{d}{ }^{2}\right)^{0.5} \\ & =\frac{8150 \times(0.9601)^{0.75}}{0.0998} \times(0.91)^{0.5} \\ & =\frac{8150 \times 0.97 \times 0.954}{0.098}=75700 > N_{\mathrm{SS}}=37300 \end{aligned}$$

Our inducer exhibits characteristics similar to those shown in figure 6-40 for a typical inducer. If we use a vane angle $\theta_{1}$ of $14^{\circ} 30^{\prime}$ at the inducer outlet mean effective diameter $d_{1}$, the difference between $\theta_{1}$ and the relative outlet flow angle $\beta_{1}{ }^{\prime}$

$$\theta_{1}-\beta_{1}^{\prime}=\left(14^{\circ} 30^{\prime}\right)-\left(13^{\circ} 3^{\prime}\right)=1^{\circ} 27^{\prime}$$

This allows for the effect of local circulatory flow (boundary effects).

The vane angle $\theta_{1 t}$ at the inducer outlet tip diameter $d_{1 t}$ $\tan \theta_{1 t}=\frac{d_{1}}{d_{1 t}} \tan \theta_{1}=\frac{8.45}{11.05} \tan \left(14^{\circ} 30^{\prime}\right)=0.198 ;$

$$\theta_{1 t}=11^{\circ} 12^{\prime}$$

The vane angle $\theta_{1} h$ at the inducer outlet hub diameter $d_{1 h}$ $\tan \theta_{1 h}=\frac{d_{1}}{d_{1 h}} \tan \theta_{1}=\frac{8.45}{4.65} \tan 14^{\circ} 30^{\prime}=0.471 ;$

$$\theta_{1 h}=25^{\circ} 13^{\prime}$$

We will use three vanes ( $z=3$ ). The vane pitch at the mean tip diameter $d_{t}$ can be calculated from equation (6-50)

$$P_{i}=\frac{\pi d_{t}}{z}=\frac{\pi \times 11.62}{3}=12.18 \mathrm{in}$$

The chord length at vane tip can be calculated as

$$C_{i}=\frac{L_{i}}{\sin \left(\frac{\theta_{0} t+\theta_{1} t}{2}\right)}=\frac{4.65}{\sin 10^{\circ} 6^{\prime}}=\frac{4.65}{0.175}=26.57 \mathrm{in}$$

From equation (6-51) the inducer solidity based on the mean tip diameter $d_{t}$

$$S_{V}=\frac{C_{i}}{P_{i}}=\frac{26.57}{12.18}=2.18$$

A-1 Stage Engine Oxidizer Pump Inducer Design Summary

Following completion of calculations it is advisable to compile the results systematically in a summary, prior to start of layouts. This gives an opportunity for cross checks and reduces the probability of errors.

Required head rise and capacity, $\Delta H_{\text {ind }} =235 \mathrm{ft}, Q_{\text {ind }}=13040 \mathrm{gpm}$ Inlet velocity diagram (at inlet mean effective diameter $\mathbf{d}_{0}$ )

$$\begin{aligned} & a_{0}^{\prime}=90^{\circ}, \beta_{0}^{\prime}=7^{\circ} 45^{\prime} \\ & u_{0}=268 \mathrm{ft} / \mathrm{sec}, v_{0}^{\prime}=270.6 \\ & c_{0}^{\prime}=c_{m 0}=37.2 \mathrm{ft} / \mathrm{sec}, c_{\mathrm{u}_{0}}=0 \end{aligned}$$

Outlet velocity diagram (at outlet mean effective diameter $d_{1}$ )

$$\begin{aligned} & a_{1}^{\prime}=61^{\circ} 13^{\prime}, \beta_{1}^{\prime}=13^{\circ} 3^{\prime} \\ & u_{1}=258.5 \mathrm{ft} / \mathrm{sec}, v_{1}^{\prime}=235 \mathrm{ft} / \mathrm{sec} \\ & c_{1}^{\prime}=60.5 \mathrm{ft} / \mathrm{sec}, c_{u_{1}^{\prime}}=29.2 \mathrm{ft} / \mathrm{sec} \\ & c_{m_{1}}=53.1 \mathrm{ft} / \mathrm{sec} \end{aligned}$$

Axial length of inducer, $L_{i}=4.65$ in Taper half angle at tip: $7^{\circ}$; at hub: $14^{\circ}$ Inlet dimensions

$$\begin{aligned} & d_{o t}=12.19 \mathrm{in}, d_{o h}=2.33 \mathrm{in}, d_{0}=8.76 \mathrm{in} \\ & \text { vane angle at } d_{o t}, \theta_{o t}=9^{\circ} \\ & \text { vane angle at } d_{o h}, \theta_{o h}=39^{\circ} 40^{\prime} \\ & \text { vane angle at } d_{0}, \theta_{0}=12^{\circ} 25^{\prime} \end{aligned}$$

Outlet dimensions

$$\begin{aligned} & d_{1 t}=11.05 \text { in, } d_{1 h}=4.65 \mathrm{in}, d_{1}=8.45 \mathrm{in} \\ & \text { vane angle at } d_{1} t, \theta_{1 t}=11^{\circ} 12^{\prime} \\ & \text { vane angle at } d_{1 h}, \theta_{1 h}=25^{\circ} 13^{\prime} \\ & \text { vane angle at } d_{1}, \theta_{1}=14^{\circ} 30^{\prime} \end{aligned}$$

Number of vanes, $z=3$ Solidity at vane tip, $S_{V}=2.18$ Inlet flow coefficient, $\phi_{\text {ind }}=0.0998$ (b) Oxidizer pump impeller

We will use a radial-flow-type impeller with mixed-flow-type vanes extending into the impeller entrance eye, as shown in figure 6-38a. The flow path and velocity conditions at the impeller inlet can be assumed to be the same as those at the inducer outlet.

From equation (6-4), the tip or peripheral speed at the impeller discharge

$$u_{2}=\sqrt{\frac{g H}{\psi}}=\sqrt{\frac{32.2 \times 2930}{0.46}}=453 \mathrm{ft} / \mathrm{sec}$$

The impeller outlet diameter

$$d_{2}=\frac{720 \times u_{2}}{\pi \times N}=\frac{720 \times 453}{\pi \times 7000}=14.8 \mathrm{in}$$

Substitute the given hydraulic head losses $H_{e}=0.19 \Delta H$ into equation (6-34), to obtain the required impeller developed head

$$\begin{aligned} \Delta H_{\mathrm{imp}}=\Delta H+H_{\mathrm{e}}-\Delta H_{\mathrm{ind}} & \\ & =2930(1+0.19)-235=3252 \mathrm{ft} \end{aligned}$$

From equation (6-35), the required impeller flow rate

$$Q_{\mathrm{imp}}=12420(1+0.035)=12855 \mathrm{gpm}$$

From equation (6-31), the tangential component of the impeller design absolute outlet flow velocity

$$\begin{aligned} & c_{u_{2}^{\prime}}=\frac{g \Delta H_{\mathrm{imp}}+u_{1} c_{u_{1}}{ }^{\prime}}{u_{2}} \\ & \quad=\frac{32.2 \times 3252+258.5 \times 29.2}{453}=248 \mathrm{ft} / \mathrm{sec} \end{aligned}$$

From equation (6-32), the tangential component of the impeller ideal absolute outlet flow velocity

$$c_{u_{2}}=\frac{c_{u_{2}}{ }^{\prime}}{e_{v}}=\frac{248}{0.74}=335 \mathrm{ft} / \mathrm{sec}$$

Referring to figures 6-34 and 6-35, and to equation (6-28), the meridional component of the impeller design absolute outlet flow velocity $c_{m 2}=\left(u_{2}-c_{u_{2}}\right) \tan \beta_{2}$

$$=(453-335) \tan 24^{\circ}=52.5 \mathrm{ft} / \mathrm{sec}$$

The impeller design absolute outlet flow velocity

$$c_{2}^{\prime}=\sqrt{c_{u_{2}}{ }^{\prime 2}+c_{m_{2}}{ }^{2}}==\sqrt{248^{2}+52.5^{2}}=253.4 \mathrm{ft} / \mathrm{sec}$$

Impeller design absolute outlet flow angle:

$$\tan a_{2}^{\prime}=\frac{c_{m 2}}{c_{u 2}^{\prime}}=\frac{52.5}{248}=0.212 ; \quad a_{2}^{\prime}=11^{\circ} 58^{\prime}$$

Impeller design relative outlet flow velocity

$$\begin{aligned} & \begin{aligned} v_{2}^{\prime}=\sqrt{\left(u_{2}-c_{u_{2}}{ }^{\prime}\right)^{2}+c_{m 2}{ }^{2}} & \\ & =\sqrt{205^{2}+52.5^{2}}=211.6 \mathrm{ft} / \mathrm{sec} \end{aligned} \end{aligned}$$

Impeller design relative outlet flow angle:

$$\tan \beta_{2}^{\prime}=\frac{c_{m_{2}}}{\left(u_{2}-c_{u_{2}}{ }^{\prime}\right)}=\frac{52.5}{205}=0.256 ; \quad \beta_{2}^{\prime}=14^{\circ} 22^{\prime}$$

Referring to figure 6-38a, and to equation (6-42), the width of the impeller at the vane inlet

$$\begin{aligned} & b_{1}=\frac{Q_{\mathrm{imp}}}{3.12 \pi d_{1} c_{m 1} \epsilon_{1}} \\ & \quad=\frac{12855}{3.12 \times \pi \times 8.45 \times 53.1 \times 0.82}=3.56 \mathrm{in} \end{aligned}$$

From equation (6-43), the width of the impeller at the discharge

$$\begin{aligned} b_{2}=\frac{Q_{\mathrm{imp}}}{3.12 \pi d_{2} c_{m 2} \epsilon_{2}} & \\ & \quad=\frac{12855}{3.12 \times \pi \times 14.8 \times 52.5 \times 0.88}=1.91 \mathrm{in} \end{aligned}$$

From equation (6-5) the overall pump flow coefficient

$$\phi=\frac{c_{m_{2}}}{u_{2}}=\frac{52.5}{453}=0.116$$

A-1 Stage Engine Oxidizer Pump Impeller Design Summary
Required impeller developed head and $\Delta H_{\mathrm{imp}} =3252 \mathrm{ft}, Q_{\mathrm{imp}}=12855 \mathrm{gpm}$ flow capacity

Inlet velocity diagram (at mean inlet effective diameter $d_{1}$ )

$$\begin{aligned} & a_{1}^{\prime}=61^{\circ} 13^{\prime}, \beta_{1}^{\prime}=13^{\circ} 3^{\prime} \\ & u_{1}=258.5 \mathrm{ft} / \mathrm{sec}, v_{1}^{\prime}=235 \mathrm{ft} / \mathrm{sec} \\ & c_{1}^{\prime}=60.5 \mathrm{ft} / \mathrm{sec}, c_{u_{1}}^{\prime}=29.2 \mathrm{ft} / \mathrm{sec}, \\ & c_{m_{1}}=53.1 \mathrm{ft} / \mathrm{sec} \end{aligned}$$

Outlet velocity diagram (at outlet diameter $d_{2}$ )

$$\begin{aligned} & a_{2}^{\prime}=11^{\circ} 58^{\prime}, \beta_{2}^{\prime}=14^{\circ} 22^{\prime} \\ & u_{2}=453 \mathrm{ft} / \mathrm{sec}, v_{2}^{\prime}=211.6 \mathrm{ft} / \mathrm{sec} \\ & c_{2}^{\prime}=253.4 \mathrm{ft} / \mathrm{sec}, c_{\mathrm{u}_{2}^{\prime}}=248 \mathrm{ft} / \mathrm{sec}, \\ & c_{m_{2}}=52.5 \mathrm{ft} / \mathrm{sec} \end{aligned}$$

Inlet dimensions inlet eye diameter $=d_{1 t}=11.05 \mathrm{in}$ inlet hub diameter $=d_{1 h}=4.65$ in inlet mean effective diameter $=d_{1}=8.45 \mathrm{in}$ inlet vane angle at diameter $d_{1}=\beta_{1}=13^{\circ} 3^{\prime}$ inlet vane width $b_{1}=3.56 \mathrm{in}$

Outlet dimensions outside diameter $d_{2}=14.8 \mathrm{in}$ discharge vane angle $\beta_{2}=24^{\circ}$ impeller outlet width $b_{2}=1.91 \mathrm{in}$ Number of impeller vanes (eq. 6-44) $z_{i}=\beta_{2} / 3$

$$=24 / 3=8$$

Design of Casings

The main function of a pump casing is to convert the kinetic energy of high flow velocity at the impeller discharge into pressure. It does not contribute to the generation of head. The construction of a typical centrifugal pump casing is shown in figure 6-5. The front section of the casing, which provides the pump inlet and houses the inducer, is called the suction nozzle. The rear section of the casing, which collects the fluid from the impeller and converts the velocity head into pressure prior to discharge, is called the volute.

Since the flow path in a suction nozzle is short and the flow velocities are relatively low, the head loss in a suction nozzle due to friction is very small. The contour of the suction nozzle is designed to suit the inducer configuration. A tapered suction nozzle (as shown in fig. 6-14), also known as an end suction nozzle, together with a tapered inducer, yields best results in most respects. This nozzle, the area of which gradually decreases toward the impeller eye, greatly steadies the flow and assures uniform feed to the impeller. In liquid oxygen pumps, frequently a liner made of a material such as Kel-F is inserted between inducer and suction nozzle wall. This eliminates the possibility of metal-to-metal rubbing in the presence of narrow inducer tip clearances. Rubbing in liquid oxygen pumps may cause dangerous explosions. In turbopumps of the single-shaft type (fig. 6-18), the fuel is introduced to the fuel pump in a radial direction. Special guide devices are required in the inlet to minimize pressure drops because of the need of turning the flow axially into the inducer.

Two types of volute casing are used in rocket centrifugal pumps, the plain volute and the diffusing vane volute (see fig. 6-43). In the first, the impeller discharges into a single volute channel of gradually increasing area. Here, the

Figure 6-43.-Plain volute and diffusing vane volute centrifugal pump casings.

major part of the conversion of velocity to pressure takes place in the conical pump discharge nozzle. In the latter, the impeller first discharges into a diffuser provided with vanes. A major portion of the conversion takes place in the channels between the diffusing vanes before the fluid reaches the volute channel. The main advantage of the plain volute is its simplicity. However, the diffusing volute is more efficient. Head losses in pump volutes are relatively high. Approximately 70 to 90 percent of the flow kinetic energy is converted into pressure head in either volute type.

The hydraulic characteristics of a plain volute are determined by several design parameters which include: volute throat area $a_{v}$ and flow areas $\mathrm{a}_{\theta}$, included angle $\theta_{s}$ between volute side walls (fig. 6-44), volute tongue angle $\alpha_{V}$, radius $r_{t}$ at which the volute tongue starts, and volute width $b_{3}$. Their design values are somewhat influenced by the pump specific speed $N_{\mathrm{S}}$ and are established experimentally for best performance.

All of the pump flow $Q$ passes through the volute throat section $a_{v}$, but only part of it passes through any other section, the amount depending on the location away from the volute tongue. One design approach is to keep a constant average flow velocity $c_{3}{ }^{\prime}$ at all sections of the volute. Thus

$$\begin{equation*} c_{3}^{\prime}=\frac{Q}{3.12 \mathrm{a}_{v}} \frac{1}{3.12} \frac{\theta}{360} \frac{Q}{\mathrm{a} \theta} \tag{6-69} \end{equation*}$$

where $c_{3}{ }^{\prime}=$ average flow velocity in the volute, $\mathrm{ft} / \mathrm{sec}$ $Q=$ rated design pump flow rate, gpm $\mathrm{a}_{V}=$ area of the volute throat section, in ${ }^{2}$ $\mathrm{a}_{\theta}=$ area of a volute section ( $\mathrm{in}^{2}$ ), at an angular location $\theta$ (degrees) from the tongue

Figure 6-44.-Plain volute casing of a centrifugal pump.

The design value of the average volute flow velocity $c_{3}{ }^{\prime}$ may be determined experimentally from the correlation

$$\begin{equation*} c_{3}{ }^{\prime}=K_{V} \sqrt{2 g \Delta H} \tag{6-70} \end{equation*}$$

where

$$\begin{aligned} K_{v}= & \text { experimental design factor; typical values } \\ & \text { range from } 0.15 \text { to } 0.55 . K_{v} \text { is lower for } \\ & \text { higher specific speed pumps } \\ \Delta H= & \text { rated design pump developed head, } \mathrm{ft} \\ g= & \text { gravitational constant, } 32.2 \mathrm{ft} / \mathrm{sec}^{2} \end{aligned}$$

In order to avoid impact shocks and separation losses at the volute tongue, the volute angle $a_{V}$ is designed to correspond to the direction of the absolute velocity vector at the impeller discharge: $a_{v} \cong a_{2}{ }^{\prime}$. Higher specific speed pumps have higher values of $a_{2}{ }^{\prime}$ and thus require higher $a_{v}$. The radius $r_{t}$ at which the tongue starts should be 5 to 10 percent larger than the outside radius of the impeller to suppress turbulence and to provide an opportunity for the flow leaving the impeller to equalize before coming into contact with the tongue.

The dimension $b_{3}$ at the bottom of a trapezoidal volute cross section is chosen to minimize losses due to friction between impeller discharge flow and volute side walls. For small pumps of lower specific speeds, $b_{3}=2.0 b_{2}$, where $b_{2}$ is the impeller width at the discharge, in. For higher specific speed pumps, $b_{3}=1.6$ to $1.75 b_{2}$. The maximum included angle $\theta_{\mathrm{S}}$ between the volute side walls should be about $60^{\circ}$. For higher specific speed pumps, or for higher impeller discharge flow angles $a_{2}{ }^{\prime}$, the value of $\theta_{s}$ should be made smaller.

The pressure in the volute cannot always be kept uniform, especially under off-design operating conditions. This results in a radial thrust on the impeller shaft. To eliminate or reduce the radial thrust, double-volute casings have been frequently used (fig. 6-45). Here, the flow is divided into two equal streams by two tongues set $180^{\circ}$ apart. Although the volute pressure unbalances may be the same as in a singlevolute casing, the resultant of all radial forces may be reduced to a reasonably low value, owing to symmetry.

The diffusing vane volute has essentially the same shape as a plain volute, except that a number of passages are used rather than one. This permits the conversion of kinetic energy to pressure in a much smaller space. The radial clearance between impeller and diffuser inlet vane tips should be narrow for best efficiency. Typical values range from 0.03 to 0.12 inch, depending upon impeller size. The width of the diffuser at

Figure 6-45.-Typical single discharge, $180^{\circ}$ opposed double-volute casing of a centrifugal pump.

its inlet can be approximated in a manner similar to that used for the width of a plain volute (i.e., 1.6 to 2.0 impeller width $b_{2}$ ). A typical diffuser layout is shown in figure 6-46. The vane inlet angle $\alpha_{3}$ should be made equal or close to the absolute impeller discharge flow angle $a_{2}{ }^{\prime}$. The design value of the average flow velocity at the diffuser throat $c_{3}{ }^{\prime}$ may be approximated by

$$\begin{equation*} c_{3}^{\prime}=\frac{d_{2}}{d_{3}} c_{2}^{\prime} \tag{6-71} \end{equation*}$$

where $c_{3}{ }^{\prime}=$ average flow velocity at the diffuser throat, $\mathrm{ft} / \mathrm{sec}$ $d_{2}=$ impeller discharge diameter, in $d_{3}=$ pitch diameter of the diffuser throats, in $c_{2}{ }^{\prime}=$ absolute flow velocity at impeller discharge, ft/sec Since each vane passage is assumed to carry an equal fraction of the total flow $Q$, the following correlation may be established:

$$\begin{equation*} b_{3} h_{3} z=\frac{Q}{3.12 c_{3}^{\prime}} \tag{6-72} \end{equation*}$$

where $b_{3}=$ width of the diffuser at the throat, in $h_{3}=$ diffuser throat height, in $z=$ number of diffuser vanes $Q=$ rated design pump flow rate, gpm The number of diffuser vanes $z$ should be minimum, consistent with good performance, and should have no common factor with the number of impeller vanes to avoid resonances. If possible,

Figure 6-46.-Typical layout of the diffuser for a centrifugal pump volute casing.

the cross section of the passages in the diffuser are made nearly square; i.e., $b_{3}=h_{3}$. The shape of the passage below the throat should be diverging, with an angle between $10^{\circ}$ to $12^{\circ}$. The velocity of the flow leaving the diffuser is kept slightly higher than the velocity in the pump discharge line.

Rocket pump casings are frequently made of high-quality aluminum-alloy castings. In lowpressure pumps, the casing wall thickness is held as thin as is consistent with good foundry practice. Owing to the intricate shape of the castings, stress calculations are usually based upon prior experience and test data. For a rough check, the hoop stress at a casing section may be estimated as

$$\begin{equation*} S_{t}=p \frac{a}{a^{T}} \tag{6-73} \end{equation*}$$

where $S_{t}=$ hoop tensile stress, $\mathrm{lb} / \mathrm{in}^{2}$ $p=$ local casing internal pressure, psia (or pressure difference across the casing wall, psi) $a=$ projected area on which the pressure acts, in ${ }^{2}$ $a^{\prime}=$ area of casing material resisting the force pa, in ${ }^{2}$ The actual stress will be higher, because of bending stresses as a result of discontinuities and deformation of the walls, and thermal stresses from temperature gradients across the wall.

Sample Calculation (6-8)

The flow conditions at the outlet of the A-1 stage engine oxidizer pump impeller were derived in sample calculation (6-7). Calculate and design a double-volute (spaced $180^{\circ}$ ), single-discharge-type casing (as shown in fig. 6-45) for the same pump, assuming a design factor $K_{V}$ of 0.337.

Solution

From equation (6-70), the average volute flow velocity may be calculated as $c_{3}{ }^{\prime}=K_{V} \sqrt{2 g \Delta H}=0.337 \times \sqrt{2 \times 32.2 \times 2930}=146 \mathrm{ft} / \mathrm{sec}$ Referring to figures 6-44 and 6-45, and from equation (6-69), the required volute flow area at any section from $0^{\circ}$ to $180^{\circ}$ away from the volute tongue may be calculated for both volutes as

$$\mathrm{a}_{\theta}=\frac{\theta Q}{3.12 \times 360 \times \mathrm{c}_{3}}=\frac{12420}{3.12 \times 360 \times 146} \theta=0.076 \theta$$

At $\theta=45^{\circ}, \mathrm{a}_{45}=3.42 \mathrm{in}^{2}$; at $\theta=90^{\circ}, \mathrm{a}_{90}=6.84 \mathrm{in}^{2}$; at $\theta=135^{\circ}, \mathrm{a}_{135}=10.26 \mathrm{in}^{2}$; and at $\theta=180^{\circ}$. $\mathrm{a}_{180}=13.68 \mathrm{in}^{2}$.

Total volute throat area at the entrance to the discharge nozzle

$$\mathrm{a}_{v}=2 \times 13.68=27.36 \mathrm{in}^{2}$$

The volute angle $a_{v}$ can be approximated as

$$a_{V}=a_{2}^{\prime}=11^{\circ} 58^{\prime}, \text { say } 12^{\circ}$$

The radius $r_{t}$ at which the volute tongues start can be approximated as (assuming 5 percent clearance)

$$r_{t}=\frac{d_{2}}{2} \times 1.05=\frac{14.8}{2} \times 1.05=7.77 \mathrm{in}$$

The width at the bottom of the trapezoidal volute section shall be

$$b_{3}=1.75 b_{2}=1.75 \times 1.91 \mathrm{in}=3.34 \mathrm{in}$$

Allowing for a transition from the shape of the volute to round, we use a diameter of 6.25 inches, or an area of $30.68 \mathrm{in}^{2}$, for the entrance to the discharge nozzle. With a $10^{\circ}$ included taper angle and a nozzle length of 10 inches, the exit diameter of the discharge nozzle can be determined as

$$\begin{aligned} d_{e} & =6.25+2 \times 10 \times \tan 5^{\circ} \\ & =6.25+2 \times 10 \times 0.0875=6.25+1.75 \\ & =8 \mathrm{in}\left(\text { or an area of } 50.26 \mathrm{in}^{2}\right) \end{aligned}$$

Flow velocity at the nozzle inlet:

$$\frac{12420}{3.12 \times 30.68}=130 \mathrm{ft} / \mathrm{sec}$$

Flow velocity at the nozzle exit:

$$\frac{12420}{3.12 \times 50.26}=79.4 \mathrm{ft} / \mathrm{sec}$$

Balancing the Axial Thrust of Centrifugal Pumps

Unbalanced axial loads acting on the inducerimpeller assembly of centrifugal pumps are primarily the result of changes in axial momentum, and of variations in pressure distribution at the periphery of the assembly. These unbalanced forces can be reduced by mounting two propellant pumps back to back, as shown in figures 6-14 and 6-18. More subtle balancing of the axial loads can be accomplished by judicious design detail, which is especially important in highpressure and high-speed pump applications. Either one of the following two methods is frequently used.

With the first method (as shown in fig. 6-47), a balance chamber is provided at the back shroud of the impeller, between back wearing ring diameter $d_{b r}$ and shaft seal diameter $d_{s}$. Balancing of axial loads is effected by proper selection of the projected chamber area and of the admitted fluid pressure. The pressure level $p_{c}$ in a balance chamber can be controlled by careful adjustment of the clearances and leakages of the back wearing ring and the shaft seals. The required $p_{c}$ may be determined by the following correlation:

$$\begin{align*} & p_{c} \pi\left(d_{b r}{ }^{2}-d_{S}{ }^{2}\right)=p_{V} \pi\left(d_{b r}{ }^{2}-d_{f r}{ }^{2}\right) \\ & \quad+p_{1} \pi\left(d_{f r}{ }^{2}-d_{t}{ }^{2}\right)+p_{0} \pi d_{h}{ }^{2}+\frac{4 \dot{w}_{i} c_{m 0}}{g} \pm T_{\mathrm{e}} \tag{6-74} \end{align*}$$

where $p_{c}=$ balance chamber pressure, psia $p_{v}=$ average net pressure in the space between impeller shrouds and casing walls, psia $p_{1}=$ static pressure at the inducer outlet, psia $p_{0}=$ static pressure at the inducer inlet, psia $d_{s}=$ effective shaft seal diameter, in $d_{h}=$ hub diameter at the inducer inlet, in $d_{t}=$ inducer tip diameter=eye diameter at the impeller inlet, in $d_{t r}=$ front wearing ring diameter, in $d_{b r}=$ back wearing ring diameter, in $\dot{w}_{i}=$ inducer weight flow rate, $\mathrm{lb} / \mathrm{sec}$ $c_{m 0}=$ axial flow velocity at the inducer inlet, $\mathrm{ft} / \mathrm{sec}$ (converts to radial) $g$ = gravitational constant, $32.2 \mathrm{ft} / \mathrm{sec}^{2}$ $T_{e}=$ external axial thrust due to unbalanced axial loads of the other propellant and/or the turbine; lb. A positive sign indicates a force which tends to pull the impeller away from the suction side, a negative sign indicates the opposite. The static pressure at the inducer outlet, $p_{i}$, can be either measured in actual tests, or approximated by

$$\begin{equation*} p_{1}=k_{i} p_{0} \tag{6-75} \end{equation*}$$

where $k_{i}=$ design factor based on experimental data (ranging from 1.1 to 1.8 ) $p_{0}=$ static pressure at the inducer inlet, psia The average pressure in the space between impeller shrouds and casing side walls, $p_{v}$, may be approximated by

$$\begin{equation*} p_{v}=p_{1}+\frac{3}{576} \frac{u_{2}^{2}-u_{1}^{2}}{2 g} \rho \tag{6-76} \end{equation*}$$

where

$$\begin{gathered} u_{2}=\text { peripheral speed at the impeller outside } \\ \text { diameter } d_{2}, \mathrm{ft} / \mathrm{sec} \\ u_{1}=\text { peripheral speed at the impeller inlet mean } \\ \quad \text { effective diameter } d_{1}, \mathrm{ft} / \mathrm{sec} \\ \rho=\text { density of the pumped medium, } \mathrm{lb} / \mathrm{ft}^{3} \end{gathered}$$

The main advantage of the balance chamber method is flexibility. The final balancing of the turbopump bearing axial loads can be accomplished in component tests by changing the value of $p_{c}$ through adjustment of the clearances at the wearing ring and shaft seals. However, this tends to increase leakage losses.

In the second method (as shown in fig. 6-48), straight radial ribs are provided at the back shroud of the impeller to reduce the static pressure between the impeller back shroud and casing wall through partial conversion into kinetic energy. This reduction of axial forces acting on the back shroud of the impeller may be approximated by the following correlation:

$$\begin{equation*} F_{a}=\frac{3 \pi}{4608}\left(d_{r}^{2}-d_{s}^{2}\right) \frac{\left(u_{r}^{2}-u_{s}^{2}\right)}{2 g} \rho \frac{(s+t)}{2 s} \tag{6-77} \end{equation*}$$

where $F_{a}=$ reduction of the axial forces acting on the back shroud of impeller, lb $d_{I}=$ outside diameter of the radial ribs, in

Figure 6-47.-Balancing axial thrusts of a centrifugal pump by the balance chamber method.

Figure 6-48.-Balancing axial thrusts of a centrifugal pump by the radial rib method.

$d_{s}=$ effective shaft seal diameter $\cong$ inside diameter of the radial ribs, in $u_{I}=$ peripheral speed at diameter $d_{I}, \mathrm{ft} / \mathrm{sec}$ $u_{s}=$ peripheral speed at diameter $d_{s}, \mathrm{ft} / \mathrm{sec}$ $g=$ gravitational constant, $32.2 \mathrm{ft} / \mathrm{sec}^{2}$ $\rho=$ density of the pumped medium, $\mathrm{lb} / \mathrm{ft}^{3}$ $t=$ height or thickness of the radial ribs, in $s$ =average distance between casing wall and impeller back shroud, in

The required $F_{a}$ may be determined by the following correlation: $p_{V} \pi\left(d_{f r}{ }^{2}-d_{s}{ }^{2}\right)-4 F_{a}=p_{1} \pi\left(d_{f r}{ }^{2}-d_{t}{ }^{2}\right)$

$$\begin{equation*} +p_{0} \pi d_{h}{ }^{2}+\frac{4 \dot{w} c_{m 0}}{g} \pm T_{e} \tag{6-78} \end{equation*}$$

The pressures $p_{1}$ and $p_{V}$ may be approximated by equations (6-75) and (6-76). See equation (6-74) for other terms.

Sample Calculation (6-9)

Radial ribs (similar to those in fig. 6-48) are used on the back shroud of the A-1 stage engine oxidiser punp impeller, with the following dimensions:

Outside diameter of the radial ribs, $d_{r}=14.8$ in (equal to $d_{2}$ ) Inside diameter of the radial ribs, $d_{s}=4.8 \mathrm{in}$ Height of the radial ribs, $t=0.21 \mathrm{in}$ Width of the radial ribs, $w=0.25$ in (not critical) Average distance between the casing wall and impeller back shroud, $s=0.25$ in Estimate the reduction of the axial forces acting on the back shroud of the impeller, due to the radial ribs.

Solution

The peripheral speed at diameter $d_{r}$

$$u_{r}=\frac{\pi N}{720} d_{r}=\frac{\pi \times 7000}{720} \times 14.8=452 \mathrm{ft} / \mathrm{sec}$$

The peripheral speed at the diameter $d_{s}$

$$u_{s}=\frac{\pi N}{720} d_{s}=\frac{\pi \times 7000}{720} \times 4.8=147 \mathrm{ft} / \mathrm{sec}$$

From equation (6-77), the reduction of the axial forces

$$\begin{aligned} F_{a}=\frac{3 \pi}{4608}(219.04 & -23.04) \frac{(204304-21609)}{2 \times 32.2} \\ & \times 71.38 \times \frac{(0.25+0.21)}{2 \times 0.25}=74680 \mathrm{lb} \end{aligned}$$