6.5 DESIGN OF TURBINES

For rocket engine applications, impulse turbines are preferred, for their simplicity and light weight. Our discussion will be confined to these turbines only. Figure 6-55 shows the general arrangement of a typical single-stage tworotor velocity-compounded impulse turbine.

General Design Procedure

The following steps are essential in the design of a rocket engine impulse turbine:

1. The first item of importance is the selection of the proper type. A single-stage singlerotor turbine (fig. 6-8) is used if the required turbine power is low, since in this case the efficiency of the turbine has less effect on overall engine systems performance. When the avail- able energy of the turbine working fluid and thus the gas spouting velocity $C_{0}$ is relatively low, a higher turbine velocity ratio $U / C_{0}$ may be achieved with a moderate turbine rotor blade speed $U$. As shown in figure 6-27, this suggests the use of a relatively simple single-stage single-rotor impulse turbine. We have selected this type for the A-2 stage oxidizer turbopump, at the same time taking advantage of its overall simplicity.

In most direct-drive turbopump configurations, such as the A-1 stage engine turbopump (fig. 6-63), where turbine rotating speed $N$ and consequently turbine velocity ratio $U / C_{0}$ tends to be lower than ideal, a single-stage two-rotor velocity-compounded impulse turbine (figs. 6-9 and 6-55) is selected for best results. Figure 6-27 indicates that the optimum efficiency of a velocity-compounded turbine can be achieved at a relatively low $U / C_{0}$ value.

On the other hand, if a reduction gear train is provided between pumps and turbine, such as in the turbopump shown in figure 6-14, the turbine can be operated at a much higher rotating speed (over 25000 rpm ). A higher value of $U / C_{0}$ can be achieved with reasonable turbine wheel size. Then a higher performance, two-stage, two-rotor, pressure-compounded impulse turbine (fig. 6-10) may be used. 2. After the type of impulse turbine has been selected, the next step is the determination of the turbine rotor size. Once the characteristics of the turbine working-fluid (i.e., inlet temperature $T_{0}$, specific heat ratio $\gamma$, etc.), the turbine pressure ratio $R_{t}$, and the pump or turbine rotative speed $N$ have been set forth, a larger diameter for the turbine rotor tends to result in a higher velocity ratio $U / C_{0}$, or higher efficiency. However, it also results in higher assembly weight, larger envelope, and higher working stresses. Thus, the final selection of the turbine rotor size, and consequently the $U / C_{0}$ ratio, is often a design compromise. 3. The required power output from the turbine shaft must be equal to the net input to the propellant pumps, plus the mechanical losses in the gear train (if any), plus the net power required for auxiliary drives. The required flow rate of the turbine working fluid can then be calculated by equation (6-19) after required turbine power, available energy of the working fluid (eq. 6-18), and overall turbine efficiency (estimated from

Figure 6-55.-Typical single-stage, two-rotor velocity compounded impulse turbine.

figure 6-27 for a given $U / C_{0}$ ratio and turbine type), have been established. 4. Now the dimensions of the stationary nozzles, as well as those of the rotor blades, can be calculated based on the characteristics and the flow rate of the turbine working fluid.

Design of Turbine Nozzles

The nozzles of most rocket engine turbines are basically similar to those of rocket thrust chambers. They are of the conventional converging-diverging De Laval type. The main function of the nozzles of an impulse-type turbine is to convert efficiently the major portion of available energy of the working fluid into kinetic energy or high gas spouting velocity. The gasflow processes in the thrust chamber nozzles are directly applicable to turbine nozzles. However, the gas flow in an actual nozzle deviates from ideal conditions because of fluid viscosity, friction, boundary layer effects, etc. In addition, the energy consumed by friction forces and flow turbulence will cause an increase in the temperature of the gases flowing through a nozzle, above that of an isentropic process. This effect is known as reheat. As a result of the above effects, the actual gas spouting velocity at the turbine nozzle exit tends to be less than the ideal velocity calculated for isentropic expansion (from stagnation state at the nozzle inlet to the static pressure at the rotor blade inlet). Furthermore, the effective flow area of a nozzle is usually less than the actual one, because of circulatory flow and boundary layer effects. The following correlations are established for the design calculations of turbine nozzles:

Nozzle velocity coefficient $k_{n}$

Actual gas spouting velocity at the nozzle exit, ft/sec $=$ Ideal gas velocity calculated for isentropic expansion from stagnation state at the nozzle inlet to static pressure at the rotor blade inlet, ft/sec

$$\begin{equation*} =\frac{C_{1}}{C_{0}} \tag{6-118} \end{equation*}$$

Nozzle efficiency $\eta_{n}$

$$=\frac{\text { Actual gas kinetic energy at the nozzle exit }}{\text { Ideal gas kinetic energy (isentropic }}$$

Nozzle throat area coefficient $\epsilon_{\mathrm{nt}}$

$$\begin{equation*} =\frac{\text { Effective area of the nozzle throat }}{\text { Actual area }} \tag{6-120} \end{equation*}$$

Actual gas spouting velocity at the nozzle exit, $\mathrm{ft} / \mathrm{sec}$ :

$$\begin{array}{r} C_{1}=k_{n} C_{0}=k_{n} \sqrt{2 g J C_{p} T_{0}\left[1-\left(\frac{p_{1}}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}\right]} \\ =k_{n} \sqrt{2 g J \Delta H_{0-1}} \tag{6-121} \end{array}$$

Amount of nozzle reheat:

$$\begin{equation*} q_{\mathrm{nr}}=\frac{\left(1-k_{n}^{2}\right) C_{1}^{2}}{k_{n}^{2} 2 g J}=\frac{\left(1-\eta_{n}\right) C_{1}^{2}}{\eta_{n} 2 g J} \tag{6-122} \end{equation*}$$

Required total nozzle throat area, in ${ }^{2}$ :

$$\begin{equation*} A_{\mathrm{nt}}=\frac{\dot{w}_{t}}{\epsilon_{\mathrm{nt}} p_{0} \sqrt{\frac{g \gamma\left[\frac{2}{\gamma+1}\right]^{\frac{\gamma+1}{\gamma-1}}}{R T_{0}}}} \tag{6-123} \end{equation*}$$

where

$$\begin{array}{ll} C_{p} & =\text { turbine gas (working fluid) specific } \\ & \text { heat at constant pressure, Btu/lb- } \\ & \text { deg } \mathrm{F} \end{array}$$

$\Delta H_{0-1}{ }^{\prime \prime}=$ isentropic enthalpy drop of the gases flowing through the nozzles, due to expansion, $\mathrm{Btu} / \mathrm{lb}$ The performance of a turbine nozzle, as expressed by its efficiency or velocity coefficient, is affected by a number of design factors, such as (1) Exit velocity of the gas flow (2) Properties of the turbine gases (3) Angles and curvatures at nozzle inlet and exit (4) Radial height and width at the throat (5) Pitch or spacing, and number of nozzles

Design values for the efficiency and velocity coefficients of a given turbine nozzle may be determined experimentally, or estimated from past designs. Design values of nozzle efficiency $\eta_{n}$ range from 0.80 to 0.96 . Design values of nozzle velocity coefficient $k_{n}$ vary from 0.89 to 0.98 . The nozzle throat area coefficient $\epsilon_{n t}$ generally will increase with nozzle radial height, with design values ranging from 0.95 to 0.99.

The cross-sectional shape (fig. 6-56) of rocket turbine nozzles is square, or, more frequently, rectangular. They are closely spaced on a circular arc extending over a part of (partial admission), or all (full admission), the circumference. Most high-power turbines use full admission for better performance.

While the gases are passing through a nozzle and expanding, the direction of flow is changing from an approximately axial direction to one forming the angle $\alpha_{1}$ (fig. 6-56) with the plane of rotation, at the nozzle exit. Thus the turning angle is $90^{\circ}-a_{1}$. The angle $\theta_{\Omega}$ of the nozzle centerline at the exit usually is the result of a design compromise. Theoretically, better efficiency is obtained through the use of a smaller nozzle exit angle, since the rotor blading work is larger and the absolute flow velocity at the rotor blade exit is smaller. However, a smaller nozzle exit angle means a larger angle of flow deflection within the nozzle, which causes higher friction losses. Design values of $\theta_{n}$ range from $15^{\circ}$ to $30^{\circ}$. The actual effective discharge angle $a_{1}$ of the gas jet leaving the nozzle tends to be greater than $\theta_{n}$, because of the unsymmetrical nozzle shape at the exit.

A sufficiently large nozzle passage aspect ratio, $h_{\mathrm{n} t} / b_{\mathrm{n} t}$, is desirable for better nozzle

Figure 6-56.-Nozzles, rotor blades, and velocity diagrams of a typical single-stage impulse turbine.

efficiency. For a given nozzle height, an increase in aspect ratio can be secured by decreasing the nozzle pitch, $P_{n}$. However, a small pitch, and consequently a large number of nozzles, $z_{n}$, with attendant increase in wall surface, tends to increase friction losses. The determination of nozzle pitch thus also requires a design compromise. The following correlations are established for the calculation of nozzle flow areas:

Total nozzle throat area, $\mathrm{in}^{2}$ :

$$\begin{equation*} A_{n t}=z_{n} b_{n t} h_{n t} \tag{6-124} \end{equation*}$$

Total nozzle exit area, in ${ }^{2}$ :

$$\begin{align*} A_{\mathrm{ne}}=\frac{144 \dot{w}_{t}}{\rho_{1} C_{1} \epsilon_{\mathrm{ne}}} & =z_{n} b_{\mathrm{ne}} h_{\mathrm{ne}} \\ & =z_{n} h_{\mathrm{ne}}\left(P_{n} \sin \theta_{n}-t_{n}\right) \tag{6-125} \end{align*}$$

Pitch or nozzle spacing:

$$\begin{equation*} P_{n}=\pi \frac{d_{m}}{z_{n}} \tag{6-126} \end{equation*}$$

where

$$\begin{aligned} \dot{w}_{t}= & \text { turbine gas mass flow rate, } \mathrm{lb} / \mathrm{sec} \\ \rho_{1}= & \text { density of the gases at nozzle exit, } \mathrm{lb} / \mathrm{ft}^{3} \\ C_{1}= & \text { gas spouting velocity at nozzle exit, } \\ & \quad \mathrm{ft} / \mathrm{sec} \\ \epsilon_{\mathrm{ne}}= & \text { nozzle exit area coefficient } \\ h_{\mathrm{nt}}= & \text { radial height at nozzle throat, in } \\ h_{\mathrm{ne}}= & \text { radial height at nozzle exit, in } \\ b_{\mathrm{nt}}= & \text { width normal to flow at nozzle throat, in } \\ b_{\mathrm{ne}}= & \text { width normal to flow at nozzle exit, in } \\ z_{n}= & \text { number of nozzles } \\ \theta_{n}= & \text { angle between nozzle exit centerline and } \\ & \quad \text { plane of rotation, deg } \\ t_{n}= & \text { thickness of nozzle partition at exit, in } \\ d_{m}= & \begin{array}{r} \text { mean diameter of nozzles and rotor blades, } \\ \text { in } \end{array} \end{aligned}$$

Turbine nozzle block and inlet gas manifold assembly can be made of, for instance, welded sections of forged Hastelloy C. However, the airfoil surfaces should be blended smoothly between the defined contour and the sections.

Design of Impulse Turbine Rotor Blades

The function of the rotor blades in an impulse turbine (figs. 6-55 and 6-56) is to transform a maximum of the kinetic energy of the gases ejected from the nozzles into useful work. Theoretically, there should be no change of gas pressure, temperature, or enthalpy in the rotor blades. In actual operation however, some gas expansion, i.e., reaction, usually occurs. Furthermore, the actual gas flow through the rotor blades deviates from ideal flow conditions because of friction, eddy currents, boundary layers, and reheating.

The velocity vector diagram shown in figure 6-56 describes graphically the flow conditions at the rotor blades of a single-stage, single-rotor turbine, based on the mean diameter $d_{m}$. The gases enter the rotor blades with an absolute velocity $C_{1}$, and at an angle $a_{1}$ with the plane of rotation. The tangential or peripheral speed of the rotor blades at the mean diameter is $U . V_{1}$ and $V_{2}$, the relative velocities at the blade inlet and outlet, differ, i.e., $V_{1}>V_{2}$, due to friction losses. Ideally, the gas should leave the blades at very low absolute velocity $C_{2}$ and in a direction close to axial for optimum energy conversion in the blades. The forces generated at the rotor blades are a function of the change of momentum of the flowing gases. The following correlations may be established for design calculations of the rotor blades of a single-stage, single-rotor turbine.

Tangential force acting on the blades ( $\mathrm{lb} / \mathrm{lb}$ of gas flow/sec):

$$\begin{align*} F_{t}=\frac{1}{g}\left(C_{1} \cos a_{1}+C_{2} \cos a_{2}\right) & \\ & =\frac{1}{g}\left(V_{1} \cos \beta_{1}+V_{2} \cos \beta_{2}\right) \tag{6-127} \end{align*}$$

Work transferred to the blades ( $\mathrm{ft}-\mathrm{lb} / \mathrm{lb}$ of gas flow/sec):

$$\begin{gather*} E_{b}=\frac{U}{g}\left(C_{1} \cos \alpha_{1}+C_{2} \cos \alpha_{2}\right) \\ =\frac{U}{g}\left(V_{1} \cos \beta_{1}+V_{2} \cos \beta_{2}\right) \tag{6-128}\\ U=\frac{\pi d_{m} N}{720} \tag{6-129} \end{gather*}$$

For subsequent calculations, the following relation will be useful:

$$\begin{equation*} \tan \beta_{1}=\frac{C_{1} \sin a_{1}}{C_{1} \cos a_{1}-U} \tag{6-130} \end{equation*}$$

Axial thrust at blades, $\mathrm{lb} / \mathrm{lb}$ of gas flow/sec

$$\begin{equation*} F_{a}=\frac{C_{1} \sin a_{1}-V_{2} \sin \beta_{2}}{g} \tag{6-131} \end{equation*}$$

Blade velocity coefficient:

$$\begin{equation*} k_{b}=\frac{V_{2}}{V_{1}} \tag{6-132} \end{equation*}$$

Blade efficiency:

$$\begin{equation*} \eta_{b}=\frac{\text { Work transferred to blades }}{\text { Kinetic energy input }}=\frac{E_{b}}{\frac{C_{1}{ }^{2}}{2 g}} \tag{6-133} \end{equation*}$$

Ideally, $\eta_{b}$ is a maximum for a single-rotor impulse turbine, when the turbine velocity ratio:

$$\frac{U}{C_{1}}=\frac{\cos a_{1}}{2}$$

i.e., when $U=\frac{1}{2} C_{1 t}$ where $C_{1 t}$ is the tangential component of $C_{1}$.

$$\text { Max. ideal } \eta_{b}=\frac{\cos ^{2} \alpha_{1}}{2}\left(1+k_{b} \frac{\cos \beta_{2}}{\cos \beta_{1}}\right)(6-134)$$

If there is some reaction or expansion of the gas flowing through the blades, the relative gas flow velocity at the rotor blade outlet can be calculated as

$$\begin{equation*} V_{2}=\sqrt{k_{b}^{2} V_{1}^{2}+2 g J \eta_{n} \Delta H_{1-2}{ }^{\prime}} \tag{6-135} \end{equation*}$$

Amount of reheat in the rotor blades, $\mathrm{Btu} / \mathrm{lb}$ of gas flow:

$$\begin{equation*} q_{\mathrm{br}}=\left(1-k_{b}^{2}\right) \frac{V_{1}^{2}}{2 g J}+\left(1-\eta_{n}\right) \Delta H_{1-2}{ }^{\prime} \tag{6-136} \end{equation*}$$

where

$$\begin{aligned} a_{1}, a_{2}= & \begin{array}{r} \\ \quad \text { absolute gas flow angles at the inlet } \\ \quad \text { and outlet of the rotor blades, deg } \end{array} \\ \beta_{1}, \beta_{2}= & \begin{array}{r} \text { relative gas flow angles at the inlet } \\ \quad \text { and outlet of the rotor blades, deg } \end{array} \\ C_{1}, C_{2}= & \begin{array}{r} \text { absolute gas flow velocities at the } \\ \quad \text { inlet and outlet of the rotor blades, } \\ \quad \mathrm{ft} / \mathrm{sec} \end{array} \\ V_{1}, V_{2}= & \begin{array}{r} \text { relative gas flow velocity at the inlet } \\ \quad \text { and outlet of the rotor blades, } \mathrm{ft} / \mathrm{sec} \end{array} \\ U & =\text { peripheral speed of the rotor, } \mathrm{ft} / \mathrm{sec} \\ d_{m}= & \text { mean diameter of the rotor, in } \\ \eta_{n}= & \begin{array}{r} \text { equivalent nozzle efficiency appli- } \\ \quad \text { cable to the expansion process in the } \\ \quad \text { blades } \end{array} \\ \Delta H_{1-2^{\prime}}= & \begin{array}{r} \text { isentropic enthalpy drop of the gases } \\ \quad \text { flowing through the rotor blades due } \\ \quad \text { to expansion or reaction, Btu/lb; } \\ \Delta H_{1-2^{\prime}}=0 \text { if only impulse is ex- } \\ \text { changed } \end{array} \end{aligned}$$

All parameters refer to the mean diameter $d_{m}$, unless specified otherwise. The turbine overall efficiency $\eta_{t}$ defined by equation (6-19) can be established for a single-stage, single-rotor impulse turbine as

$$\begin{equation*} \eta_{t}=\eta_{n} \eta_{b} \eta_{m} \tag{6-137} \end{equation*}$$

where $\eta_{n}=$ nozzle efficiency $\eta_{b}=$ rotor blade efficiency $\eta_{m}=$ machine efficiency indicating the mechanical, leakage, and disk-friction losses in the machine. Equation (6-134) shows that the blade efficiency $\eta_{b}$ improves when $\beta_{2}$ becomes much smaller than $\beta_{1}$. Reduction of $\beta_{2}$ without decreasing the flow area at the blade exit can be achieved through an unsymmetrical blade design (fig. 6-56), where the radial blade height increases toward the exit. In actual designs, the amount of decrease of $\beta_{2}$, or the increase of radial height, is limited considering incipient flow separation and centrifugal stresses. Generally, the $\beta_{2}$ of an unsymmetrical blade will be approximately $\beta_{1}-\left(5^{\circ}\right.$ to $\left.15^{\circ}\right)$. Equation (6-134) also indicates that $\eta_{b}$ improves as $a_{1}$ is reduced.

Design values of $k_{b}$ vary from 0.80 to 0.90 . Design values of $\eta_{b}$ range from 0.7 to 0.92 .

Referring to figure 6-56, the radial height at the rotor inlet, $h_{b}$, is usually slightly larger ( 5 to 10 percent) than the nozzle radial height $h_{n}$. This height, together with the blade peripheral speed $U$, will determine the centrifugal stress in the blades. The mean diameter of the rotor blades is defined as $d_{m}=d_{t}-h_{b}$, where $d_{t}$ is the rotor tip diameter. Pitch or blade spacing, $P_{b}$, is measured at the mean diameter $d_{m}$. There is no critical relationship between blade pitch $P_{b}$ and nozzle pitch $P_{n}$. There just should be a sufficient number of blades in the rotor to direct the gas flow. The number of blades $z_{b}$ to be employed is established by the blade aspect ratio, $h_{b} / C_{b}$ and the solidity $C_{b} / P_{b}$, where $C_{b}$ is the chord length of the rotor blades. The magnitude of the blade aspect ratio ranges from 1.3 to 2.5. Design values of blade solidity vary from 1.4 to 2 . Best results will be determined by experiment. The number of rotor blades should have no common factor with the number of nozzles or of stator blades.

The blade face is concave, with radius $r_{f}$. The back is convex, with a circular arc of small radius $\mathbf{r}_{\boldsymbol{r}}$ concentric with the face of the adjoining blade ahead. Two tangents to this arc to form the inlet and outlet blade angles $\theta_{b_{1}}$ and $\theta_{b_{2}}$ complete the blade back. The leading and trailing edges may have a small thickness $t_{b}$.

The inlet blade angle $\theta_{b_{1}}$ should be slightly larger than the inlet relative flow angle $\beta_{1}$. If $\theta_{b_{1}}<\beta_{1}$, the gas stream will strike the backs of the blades at the inlet, exerting a retarding effect on the blades and causing losses. If $\theta_{b_{1}}>\beta_{1}$, the stream will strike the concave faces of the blades and tend to increase the impulse. The outlet blade angle $\theta_{b_{2}}$ is generally made equal to the outlet relative flow angle $\beta_{2}$.

The mass flow rate $\dot{w}_{t}$ through the various nozzle and blade sections of a turbine is assumed constant. The required blade flow areas can be calculated by the following correlations. Note that the temperature values used in calculating the gas densities at various sections must be corrected for reheating effects from friction and turbulence.

$$\begin{equation*} \dot{w}_{t}=\frac{\rho_{1} V_{1} A_{b_{1} \epsilon b_{1}}}{144}=\frac{\rho_{2} V_{2} A_{b_{2}} \epsilon b_{2}}{144} \tag{6-138} \end{equation*}$$

Total blade inlet area, in ${ }^{2}$ :

$$\begin{equation*} A_{b_{1}}=z_{b} b_{b_{1}} h_{b_{1}}=z_{b} h_{b_{1}}\left(P_{b} \sin \theta_{b_{1}}-t_{b}\right) \tag{6-139} \end{equation*}$$

Total blade exit area, in ${ }^{2}$ :

$$\begin{equation*} A_{b_{2}}=z_{b} b_{b_{2}} h_{b_{2}}=z_{b} h_{b_{2}}\left(P_{b} \sin \theta_{b_{2}}-t_{b}\right) \tag{6-140} \end{equation*}$$

where

$$\begin{aligned} & P_{b} \quad=\text { pitch or rotor blade spacing } \\ & =\pi d_{m} / z_{b} \text {, in } \\ & \begin{aligned} \rho_{1}, \rho_{2} \quad & \text { density of the gases at the inlet and } \\ & \text { outlet of the rotor blades, } \mathrm{lb} / \mathrm{ft}^{3} \end{aligned} \\ & V_{1}, V_{2}=\text { relative gas flow velocities at the } \\ & \text { inlet and outlet of the rotor blades, } \\ & \text { ft/sec } \end{aligned} \quad \begin{aligned} \epsilon_{b_{1}}, \epsilon_{b_{2}}= & \text { area coefficients at inlet and outlet } \\ & \text { of the rotor blades } \\ = & \text { number of blades } \\ z_{b} \quad & \\ h_{b_{1}}, h_{b_{2}}= & \text { radial height at the inlet and outlet } \\ & \text { of the rotor blades, in } \\ b_{b_{1}}, b_{b_{2}}= & \begin{array}{c} \text { passage widths (normal to flow) at } \\ \text { the inlet and outlet of the rotor } \\ \text { blades, in } \end{array} \\ \theta_{b_{1}}, \theta_{b_{2}}= & \text { rotor blade angles at inlet and out- } \\ & \text { let, deg } \\ t_{b} & =\text { thickness of blade edge at inlet and } \\ & \text { outlet, in } \end{aligned}$$

blades and disks are shown in figures 6-53, 6-55, 6-56, and 6-57. Usually, blades are designed with a shroud, to prevent leakage over the blade tips and to reduce turbulence and thus improve efficiency. Frequently the shroud forms an integral portion of the blade, the shroud sections fitting closely together when assembled. In other designs the shroud may form a continuous ring (fig. $6-55$ ) which is attached to the blades by means of tongues at the blade tip, by rivets, or is welded to the shrouds. The blades may be either welded to the disk, or attached to it using "fir-tree" or other dovetail shapes.

The main loads to which a rotor blade is exposed can be divided into three types:

1. Tension and bending due to centrifugal forces. -The radial component of the centrifugal forces acting on the blade body produces a centrifugal tensile stress which is a maximum at the root section. As a remedy, blades are often tapered, with the thinner section at the tip, for lower centrifugal root stresses. The centroids of various blade sections at different radii generally do not fall on a true radial line. Thus the centrifugal forces acting upon the offset centroids will produce bending stresses which also are a maximum at the root section.
2. Bending due to gas loading.-The tangential driving force and the axial thrust produced by the momentum change of the gases passing over the blades may be treated as acting at the midheight of the blade to determine the amount of bending induced.
3. Bending due to vibration loads.-The gas flow in the blade passages is not a uniform flow as assumed in theory, but varies cyclically from minimum to maximum. The resultant loads represent a dynamic force on the blades, having a corresponding cyclic variation. If the frequency of this force should become equal to the natural frequency of the blades, deflections may result which will induce bending stresses of considerable magnitude.

Detail stress analyses for rotor blades can be rather complex. A basic approach is to counteract a major portion of the bending moments from gas loading with the bending moments induced by the centrifugal forces at nominal operating speeds. This can be accomplished by careful

Figure 6-57.-Typical rotor blade constructions.

blade design. Thus the centrifugal tensile stresses become a first consideration in blade design, while other details such as centroid location and root configuration are established later to fulfill design requirements. The following correlations are established at the blade root section where stresses are most critical.

Centrifugal tensile stress at the root section of blade of uniform cross section, psi:

$$\begin{equation*} S_{c}=0.0004572 \frac{1}{g} \rho_{b} h_{b} d_{m} N^{2} \tag{6-141} \end{equation*}$$

Centrifugal tensile stress at the root section of a tapered blade, psi:

$$\begin{align*} S_{\mathrm{ct}}=0.0004572 \frac{1}{g} \rho_{b} h_{b} d_{m} N^{2} & \\ & {\left[1-\frac{1-\left(\frac{a_{t}}{a_{t}}\right)}{2}\left(1+\frac{h_{b}}{3 d_{m}}\right)\right] } \tag{6-142} \end{align*}$$

Bending moment due to gas loading at the root section, in-lb:

$$\begin{equation*} M_{g}=\frac{h_{b} \dot{w}_{t}}{2 z_{b}} \sqrt{F_{t}^{2}+F_{a}^{2}} \tag{6-143} \end{equation*}$$

where $\rho_{b}=$ density of the blade material, $\mathrm{lb} / \mathrm{in}^{3}$ $h_{b}=$ average blade height, in $d_{m}=$ mean diameter of the rotor, in $N=$ turbine speed, rpm $\mathrm{a}_{r}=$ sectional area at the blade root, in ${ }^{2}$ $a_{t}=$ sectional area at the blade tip, in ${ }^{2}$ $\dot{w}_{t}=$ turbine gas flow rate, $\mathrm{lb} / \mathrm{sec}$ $z_{b}=$ number of blades $F_{t}=$ tangential force acting on the blades, $\mathrm{lb} / \mathrm{lb} / \mathrm{sec}$ (eq. (6-127)) $F_{\mathrm{a}}=\mathrm{axial}$ thrust acting on the blades, $\mathrm{lb} / \mathrm{lb} /$ sec (eq. (6-131)) The bending stresses at the root can be calculated from the resultant bending moment. The vibration stresses can be estimated from past design data. If the blade is fitted with a separate shroud, its centrifugal force produces additional stresses at the root. The total stress at the root section is obtained by adding these stresses to those caused by the centrifugal forces acting on the blades.

The stresses in a turbine rotor disk are induced by (1) the blades, and (2) the centrifugal forces acting on the disk material itself. In addition, there will be shear stresses resulting from the torque. As seen in figure 6-55, turbine disks are generally held quite thick at the axis, but taper off to a thinner disk rim to which the blades are attached. In single-rotor applications, it is possible to design a disk so that both radial and tangential stresses are uniform at all points, shear being neglected. In multirotor applications, it is difficult to do this because of the greatly increased axial length and the resulting large gaps between rotor and stator disks.

Equation (6-144) may be used to estimate the stresses in a uniform stress turbine disk, neglecting rotor blade effects: where

$$\begin{equation*} S_{d}=0.000114 \frac{1}{g} \rho_{d} \frac{d_{d}^{2} N^{2}}{\log _{e}\left(\frac{t_{0}}{t_{r}}\right)} \tag{6-144} \end{equation*}$$ $$\begin{aligned} S_{d}= & \text { centrifugal tensile stress of a constant } \\ & \text { stress turbine disk, psi } \\ \rho_{d}= & \text { density of the disk material, } \mathrm{lb} / \mathrm{in}^{3} \\ d_{d}= & \text { diameter of the disk, in } \\ N= & \text { turbine speed, rpm } \\ t_{0}= & \text { thickness of the disk at the axis, in } \\ t_{r}= & \text { thickness of the disk rim at } d_{d}, \text { in } \end{aligned}$$

Equation (6-144a) permits estimation of the stresses in any turbine disk, neglecting effects of the rotor blades:

$$\begin{equation*} S_{d}=0.0004425 \frac{1}{g} W_{d} r_{i} \frac{N^{2}}{a_{d}} \tag{6-144a} \end{equation*}$$

where

$$\begin{aligned} S_{d}= & \text { centrifugal tensile stress of the turbine } \\ & \text { disk, psi } \\ W_{d}= & \text { weight of the disk, lb } \\ r_{i}= & \text { distance of the center of gravity of the half } \\ & \quad \text { disk from the axis, in } \\ a_{d}= & \text { disk cross-sectional area, } i n^{2} \\ N= & \text { turbine speed, rpm } \end{aligned}$$

For good turbine design, it is recommended that at maximum allowable design rotating speed, the $S_{d}$ calculated by equation (6-144a) should be about 0.75 to 0.8 material yield strength.

Turbine rotor blades and disks are made of high-temperature alloys of three different base materials: iron, nickel, and cobalt, with chromium forming one of the major alloying elements. Tensile yield strength of 30000 psi minimum at a working temperature of $1800^{\circ} \mathrm{F}$ is an important criterion for selection. Other required properties include low creep rate, oxidation and erosion resistance, and endurance under fluctuating loads. Haynes Stellite, Vascojet, and Inconel X are alloys frequently used. The rotor blades are fabricated either by precision casting or by precision forging methods. Rotor disks are best made of forgings for optimum strength.

Design of Single-Stage, Two-Rotor VelocityCompounded Impulse Turbines (figs. 6-9, 6-55, and 6-58)

In most impulse turbines, the number of rotors is limited to two. It is assumed that in a singlestage, two-rotor, velocity-compounded impulse turbine, expansion of the gases is completed in the nozzle, and that no further pressure change occurs during gas flow through the moving blades. As mentioned earlier, the two-rotor, velocity-compounded arrangement is best suited for low-speed turbines. In this case, the gases ejected from the first rotor blades still possess considerable kinetic energy. They are, therefore, redirected by a row of stationary blades into a second row of rotor blades, where additional work is extracted from the gases, which usually leave the second rotor blade row at a moderate velocity and in a direction close to the axial.

The velocity diagrams of a single-stage, tworotor, velocity-compounded impulse turbine are shown in figure 6-58, based on the mean rotor diameter. The peripheral speed of the rotor blades at this diameter is represented by $U$. The gases leave the nozzles and enter the first rotor blades with an absolute velocity $C_{1}$, at an angle $a_{1}$ with the plane of rotation. $V_{1}$ and $V_{2}$ are the relative flow velocities in $\mathrm{ft} / \mathrm{sec}$ at the inlet and outlet of the first rotor blades. The gases leave the first rotor blades and enter the stationary blades at an absolute flow velocity $C_{2}$, and at an angle $a_{2}$. After passing over the stationary blades, the gases depart and enter the second rotor blades at an absolute flow velocity $C_{3}$, and at an angle $\alpha_{3} . V_{3}$ and $V_{4}$ are the relative inlet and outlet flow velocities at the second rotor blades. Angles $\beta_{1}, \beta_{2}, \beta_{3}$, and $\beta_{4}$ represent the flow directions of $V_{1}, V_{2}, V_{3}$, and $V_{4}$.

As with single-rotor turbines, the exit velocity from any row of blades (rotary or stationary) is less than the inlet velocity, because of friction losses. It can be assumed that the blade velocity coefficient $k_{b}$ has the same value for any row of blades:

$$\begin{equation*} k_{b}=\frac{V_{2}}{V_{1}}=\frac{C_{3}}{C_{2}}=\frac{V_{4}}{V_{3}} \tag{6-145} \end{equation*}$$

In a multirotor turbine, the total work transferred is the sum of that of the individual rotors:

Figure 6-58.-Velocity diagrams of a typical single-stage, two-rotor, velocity-compounded impulse turbine.

Total work transferred to the blades of a tworotor turbine, $\mathrm{ft}-\mathrm{lb} / \mathrm{lb}$ of gas flow/sec

$$\begin{align*} E_{2 b}=\frac{U}{g}\left(C_{1} \cos a_{1}\right. & +C_{2} \cos a_{2} \\ & \left.+C_{3} \cos a_{3}+C_{4} \cos a_{4}\right) \\ = & \frac{U}{g}\left(V_{1} \cos \beta_{1}+V_{2} \cos \beta_{2}\right. \\ & \left.+V_{3} \cos \beta_{3}+V_{4} \cos \beta_{4}\right) \tag{6-146} \end{align*}$$

Combined nozzle and blade efficiency of a tworotor turbine:

$$\begin{equation*} \eta_{\mathrm{nb}}=\frac{E_{2 b}}{J \Delta H} \tag{6-147} \end{equation*}$$

where

$$\begin{aligned} & \Delta H=\text { overall isentropic enthalpy drop of the } \\ & \quad \text { turbine gases, Btu/lb } \\ & \quad=\text { total available energy content of the tur- } \\ & \quad \text { bine gases (eq. } 6-17) \\ & \text { Equation }(6-137) \text { can be rewritten for the tur- } \end{aligned}$$

bine overall efficiency $\eta_{t}$ of a two-rotor turbine as

$$\begin{equation*} \eta_{\mathrm{t}}=\eta_{\mathrm{nb}} \eta_{\mathrm{m}} \tag{6-148} \end{equation*}$$

Ideally, $\eta_{\mathrm{nb}}$ is a maximum for the singlestage, two-rotor, velocity-compounded impulse turbine velocity ratio

$$\frac{U}{C_{1}}=\frac{\cos a_{1}}{4}$$

i.e., when $U=\frac{1}{4} C_{1} t$. The workload for the second rotor of a two-rotor, velocity-compounded turbine is designed at about one-fourth of the total work.

The design procedures for the gas flow passages of the rotor and stationary blades of a single-stage, two-rotor turbine are exactly the same as those for a single-rotor turbine. However, velocities and angles of now change with each row of blades. As a result, the radial height of symmetrical blades increases with each row, roughly as shown in figure 6-55. The effects of reheating (increase of gas specific volume) in the flow passages must be taken into account when calculating the gas densities at various sections. Equation (6-136) may be used to estimate the amount of reheat at each row of blades. Also see sample calculation (6-11) and figure 6-60 for additional detail.

In the calculations for multirow unsymmetrical blades, the radial heights at the exit side of each row are determined first by equation (6-140). The radial heights at the blade inlets are then made slightly larger, approximately 8 percent, than those at the exit of the preceding row.

Design of Two-Stage, Two-Rotor PressureCompounded Impulse Turbines (figs. 6-10, 6-14 and 6-59)

An operational schematic of a typical twostage, two-rotor, pressure-compounded impulse turbine and its velocity diagrams at the mean diameter are shown in figures 6-10 and 6-59. Each stage of a pressure-compounded impulse turbine may be regarded as a single-stage impulse turbine rotating in its own individual housing. Most of the design characteristics of a single-stage turbine are applicable to the individual stages. The gas-spouting velocities $C_{1}$ and $C_{3}$, at flow angles $a_{1}$ and $a_{3}$, of the firstand second-stage nozzles, are designed to be approximately the same. $V_{1}, V_{2}, V_{3}$, and $V_{4}$ represent the relative flow velocities at inlets and outlets of the rotor blades. $\beta_{1}, \beta_{2}, \beta_{3}$, and $\beta_{4}$ are the corresponding flow angles for $V_{1}, V_{2}$, $V_{3}$, and $V_{4}$. The second-stage nozzles are designed to receive the gas flow discharged from the first-stage rotor blades at an absolute velocity $C_{2}$, and to turn it efficiently to a desired angle $a_{3}$. Simultaneously, the gases are accelerated to a desired velocity $C_{3}$, through expansion to a lower pressure. The flow at the outlet of the second rotor has an absolute velocity $C_{4}$ and a flow angle $a_{4}$. $U$ is the rotor peripheral speed at the mean effective diameter $d_{m}$.

The total work performed in the turbine is the sum of that of the separate stages. These may be designed to divide the load equally (i.e., the

Figure 6-59.-Velocity diagrams of a typical twostage, two-rotor, pressure-compounded impulse turbine.

velocity diagrams of each stage are identical or $C_{1}=C_{3}, C_{2}=C_{4}, a_{1}=a_{3}, a_{2}=a_{4}$, etc.). The friction losses occurring in the first stage is passed on in the gas stream as additional enthalpy and increases the available energy for the second stage. Also, the kinetic energy of the gases leaving the first stage is largely used and not entirely lost as with a single-stage turbine. The carryover ratio $I_{c}$, i.e., the ratio of the kinetic energy actually utilized as inlet energy by the second-stage nozzles to the total kinetic energy of the gases leaving the first stage, can vary from 0.4 to close to unity. The axial distance between the first-stage rotor and the second-stage nozzle, as well as the leakages through the sealing diaphragm between stages, should be minimized for optimum carryover.

The determination of the right enthalpy drop resulting in equal work for each stage may require a trial-and-error approach, in view of the effects of reheating. Or, the proper enthalpy drop may be estimated from previous designs and test data. With the velocity coefficients for nozzles and blades given by past or concurrent experiments, equations (6-122) and (6-136) can be used to estimate the amount of reheating.

Most equations established for the singlestage turbines may be employed in the design calculations for two-stage turbines. The following additional correlations are available for the design of second stage nozzles:

$$\begin{gather*} T_{2 t}=T_{2}+r_{C} \frac{C_{2}^{2}}{2 g J C_{p}} \tag{6-149}\\ P_{2 t}=P_{2}\left(\frac{T_{2 t}}{T_{2}}\right)^{\frac{\gamma}{\gamma-1}} \tag{6-150}\\ C_{3}=k_{n} \sqrt{2 g J C_{p} T_{2 t}\left[1-\left(\frac{p_{3}}{p_{2 t}}\right)^{\frac{\gamma-1}{\gamma}}\right]} \\ =k_{n} \sqrt{r_{C} C_{2}^{2}+2 g J \Delta H_{2-3}} \tag{6-151}\\ \left(A_{n t}\right)_{2}=\frac{\dot{w}_{t}}{\left[\sqrt{\frac{g \gamma\left[\frac{2}{\gamma+1}\right.}{R T_{2 t}}}\right]^{\frac{\gamma+1}{\gamma-1}}} \tag{6-152} \end{gather*}$$

where

$T_{2 t} \quad=$ turbine gas total (stagnation) temper-
    ature at second-stage nozzle inlet,
    ${ }^{\circ} \mathrm{R}$
$T_{2} \quad=$ turbine gas static temperature at
    second-stage nozzle inlet, ${ }^{\circ} \mathrm{R}$
$p_{2 t} \quad$ = turbine gas total pressure at second-
    stage nozzle inlet, psia
$p_{2} \quad=$ turbine gas static pressure at second-
stage nozzle inlet, psia
$C_{2}=$ absolute gas flow velocity at first-
    stage rotor blade outlet, ft/sec
$C_{3} \quad$ = gas-spouting velocity at second-stage
    nozzle exit, ft/sec
$I_{c} \quad=$ second-stage carryover ratio of kinetic
    energy
$C_{p} \quad=$ turbine gas specific heat at constant
    pressure, Btu/lb-deg F
$\gamma \quad=$ turbine gas specific heat ratio
$\Delta H_{2-3^{\prime}}=$ isentropic enthalpy drop of the gases
    flowing through the second-stage
    nozzles due to expansion, Btu/lb
$\left(A_{\mathrm{nt}}\right)_{2}=$ required total second-stage nozzle
    area, in ${ }^{2}$
$k_{n} \quad=$ nozzle velocity coefficient
$\epsilon_{\mathrm{nt}} \quad=$ nozzle throat area coefficient

Sample Calculation (6-11)

From sample calculation (6-5), the following data have been obtained for the turbine of the A-1 stage engine turbopump.

Turbine gas mixture ratio, $\mathrm{LO}_{2} / \mathrm{RP}-1=0.408$ Turbine gas specific heat at constant pressure, $C_{p}=0.653 \mathrm{Btu} / \mathrm{lb}-\operatorname{deg} \mathrm{F}$ Turbine gas specific heat ratio, $y=1.124$ Turbine gas constant, $R=53.6 \mathrm{ft} /{ }^{\circ} \mathrm{R}$ Gas total temperature at turbine inlet, $T_{0} =1860^{\circ} \mathrm{R}$ Gas total pressure at turbine inlet, $\mathrm{p}_{0}=640$ psia Gas static pressure at turbine exhaust, $p_{e}=27$ psia Total available energy content of the turbine gases, $\Delta H=359 \mathrm{Btu} / \mathrm{lb}$ Turbine gas flow rate, $\dot{w}_{t}=92 \mathrm{lb} / \mathrm{sec}$ Turbine shaft speed, $N=7000 \mathrm{rpm}$ Overall turbine efficiency (when using velocitycompounded wheels), $\eta_{t}=58.2$ percent In addition, the following design data are set forth:

Nozzle aspect ratio $=9.7$ Nozzle velocity coefficient, $k_{n}=0.96$ Nozzle throat area coefficient, $\epsilon_{\mathrm{nt}}=0.97$ Nozzle exit area coefficient, $\epsilon_{\text {ne }}=0.95$ Rotor and stator blade velocity coefficient, $k_{b}=0.89$ Rotor and stator blade exit area coefficient, ${ }_{\epsilon_{b 2}}=0.95$ Chord length of rotor and stator blades, $C_{b}=1.4 \mathrm{in}$ Partition thickness at the exit of nozzles and blades, $t_{n}=t_{b}=0.05 \mathrm{in}$ Solidity of first rotor blades $=1.82$ Solidity of stator blades $=1.94$ Solidity of second rotor blades $=1.67$ (a) Determine the velocity diagrams and principal dimensions of the single-stage, two-rotor, velocity-compounded, impulse-type turbine for the A-1 stage engine turbopump, with about 6 percent reaction in rotor and stator blades downstream of the nozzles. (b) Determine the velocity diagrams of an alternate two-stage, two-rotor, pressurecompounded, impulse-type turbine for the A-1 stage engine turbopump, with equal work in each stage and about 3 percent reaction in the rotor blades downstream of the nozzles of each stage.

Figure 6-60.-Temperature-entropy-enthalpy diagram of the gas processes in a single-stage, two-rotor, velocity-compounded impulse turbine with small amount of reactions downstream of the nozzles.

Solution

(a) Single-stage, two-rotor, velocitycompounded impulse turbine.

A representative velocity diagram for this turbine is shown in figure 6-58. Figure 6-60 represents the temperature-entropy-enthalpy diagram for the gas processes involved in the operation of this turbine. The following subscripts denote the various points and processes listed: $0,1,2,3,4=$ Points representing inlet conditions at the nozzles; first rotor blades; stator blades; second rotor blades; and the exit conditions of the second rotor blades. $1^{\prime}, 2^{\prime}, 3^{\prime}, 4^{\prime}=$ Points representing exit conditions at the nozzles; first rotor blades; stator blades; and second rotor blades, for an ideal isentropic expansion process. $0-1^{\prime}, 1-2^{\prime}, 2-3^{\prime}, 3-4^{\prime}=$ Path of an ideal isentropic expansion process in the nozzles; first rotor blades; stator blades; and second rotor blades. $0-1,1-2,2-3,3-4=$ Path of actual processes in the nozzles; first rotor blades; stator blades; and second rotor blades. $1^{\prime}-1,2^{\prime}-2,3^{\prime}-3,4^{\prime}-4=$ Differences along constant pressure lines, between ideal isentropic expansion processes and actual processes, due to friction losses and reheating in the nozzles, first rotor blades, stator blades, and second rotor blades

Point "0"-Nozzle Inlet

$T_{0}=$ nozzle inlet total temperature = turbine inlet total temperature $=1860^{\circ} \mathrm{R}$ $p_{0}=$ nozzle inlet total pressure $=$ turbine inlet total pressure $=640 \mathrm{psia}$ $\Delta H=$ overall isentropic enthalpy drop of the turbine gases = total available energy content of the turbine gases $=359 \mathrm{Btu} / \mathrm{lb}$ $\eta_{n}=$ nozzle efficiency $=k_{n}{ }^{2}=(0.96)^{2}=0.92$

Point "1"-Nozzle exit = First Rotor Blade Inlet

Since about 6 percent of the overall isentropic enthalpy drop $\Delta H$ is assumed to occur in the rotor and stator blades, the isentropic enthalpy drop in the nozzles

$$\Delta H_{0-1^{\prime}}=\Delta H(1-0.06)=359 \times 0.94=337.5 \mathrm{Btu} / \mathrm{lb}$$

We can write:

$$\Delta H_{0-1}=C_{p} T_{0}\left[1-\left(\frac{p_{1}}{p_{0}}\right)^{\frac{\gamma-1}{\gamma}}\right]$$

From this, the gas static pressure at the nozzle exit

$$\begin{aligned} p_{1} & =p_{0}\left[1-\frac{\Delta H_{0-1}}{C_{p} T_{0}}\right]^{\frac{\gamma}{\gamma-1}} \\ & =640 \times\left[1-\frac{337.5}{0.653 \times 1860}\right]^{\frac{1.124}{0.124}} \\ & =640 \times(0.722)^{9.06}=640 \times 0.053=33.94 \mathrm{psia} \end{aligned}$$

From equation (6-121), the gas spouting velocity at the nozzle exit

$$C_{1}=k_{n} \sqrt{2 g J \Delta H_{0-1^{\prime}}}=0.96 \sqrt{16.9 \times 10^{6}}=3940 \mathrm{fps}$$

From equation (6-122), the amount of reheat in the nozzles

$$q_{\mathrm{nr}}=\frac{\left(1-k_{\mathrm{n}}{ }^{2}\right) C_{1}{ }^{2}}{k_{n}{ }^{2} 2 g J}=\frac{0.08 \times 15524000}{0.92 \times 64.4 \times 778}=27 \mathrm{Btu} / \mathrm{lb}$$

Referring to figure 6-60, the gas temperature at the nozzle exit, following an isentropic expansion

$$T_{1^{\prime}}=T_{0}-\frac{\Delta H_{0-1^{\prime}}}{\mathrm{C}_{p}}=1860-\frac{337.5}{0.653}=1344^{\circ} \mathrm{R}$$

The actual gas static temperature at the nozzle exit

$$T_{1}=T_{1^{\prime}}+\frac{q_{\mathrm{nr}}}{C_{p}}=1344+\frac{27}{0.653}=1385^{\circ} \mathrm{R}$$

The gas density at the nozzle exit

$$\rho_{1}=\frac{p_{1}}{T_{1}} \times \frac{144}{R}=\frac{33.94 \times 144}{1385.4 \times 53.6}=0.0658 \mathrm{lb} / \mathrm{ft}^{3}$$

We will use an angle $a_{1}$ of $25^{\circ}$ for the spouting-gas-flow direction at the nozzle exit.

Ideally, the efficiency $\eta_{\mathrm{nb}}$ of a two-rotor, velocity-compounded impulse turbine is a maximum when the turbine velocity ratio

$$\frac{U}{C_{1}}=\frac{\cos a_{1}}{4}$$

From this, the peripheral speed at the mean diameter of the rotor

$$\begin{aligned} U=C_{1} \frac{\cos a_{1}}{4}=3940 \times \frac{\cos 25^{\circ}}{4} & \\ & =3940 \times 0.226=890 \mathrm{fps} \end{aligned}$$

From equation (1-129), the turbine rotor mean diameter

$$d_{m}=\frac{720 U}{\pi N}=\frac{720 \times 890}{\pi \times 7000}=29.1 \mathrm{in}$$

From equation (1-130), the relative gas flow angle $\beta_{1}$ at the inlet to the first rotor blade can be calculated:

$$\begin{aligned} \tan \beta_{1} & =\frac{C_{1} \sin a_{1}}{C_{1} \cos a_{1}-U}=\frac{3940 \times 0.423}{3940 \times 0.906-890}=0.622 \\ \beta_{1} & =31^{\circ} 53^{\prime} \end{aligned}$$

Referring to figure 6-58, the relative gas flow velocity at the first rotor blade inlet

$$\begin{aligned} V_{1}=\frac{C_{1} \sin a_{1}}{\sin \beta_{1}}=\frac{3940 \times \sin 25^{\circ}}{\sin 31^{\circ} 53^{\prime}} & \\ & =\frac{3940 \times 0.423}{0.528}=3156 \mathrm{fps} \end{aligned}$$

Point "2"-First Rotor Blade Exit = Stator Blade Inlet

Assume that the given 6 percent reaction downstream of the nozzles is equally divided between the two rotors and the stator. Then the isentropic enthalpy drop in the first rotor blade can be approximated as

$$\Delta H_{1-2^{\prime}}=\frac{0.06}{3} \times 359=7.18 \mathrm{Btu} / \mathrm{lb}$$

Using equation (6-135), the relative gas flow velocity at the exit of the first rotor blades

DESIGN OF TURBOPUMP PROPELLANT-FEED SYSTEMS

$$\begin{aligned} V_{2} & =\sqrt{k_{b}^{2} V_{1}^{2}+2 g J \eta_{n} \Delta H_{1-2^{\prime}}} \\ & =\sqrt{(0.89 \times 3156)^{2}+64.4 \times 778 \times 0.92 \times 7.18} \\ & =2866 \mathrm{fps} \end{aligned}$$

From equation (6-136), the amount of reheat in the first rotor blades,

$$\begin{aligned} q_{\mathrm{brl}} & =\left(1-k_{b}^{2}\right) \frac{V_{1}^{2}}{2 g J}+\left(1-\eta_{n}\right) \Delta H_{1-2^{\prime}} \\ & =\left[1-(0.89)^{2}\right] \times \frac{(3156)^{2}}{64.4 \times 778}+(1-0.92) \times 7.18 \\ & =41.975 \mathrm{Btu} / \mathrm{lb} \end{aligned}$$

The static gas pressure at the first rotor blade exit

$$\begin{aligned} p_{2} & =p_{1}\left[1-\frac{\Delta H_{1-2^{\prime}}}{C_{p} T_{1}}\right]^{\frac{\gamma}{\gamma-1}} \\ & =33.94 \times\left[1-\frac{7.18}{0.653 \times 1385}\right]^{9.06} \\ & =33.94 \times 0.93=31.6 \mathrm{psia} \end{aligned}$$

The gas static temperature at the exit of the first rotor blade row following an isentropic expansion

$$T_{2^{\prime}}=T_{1}-\Delta H_{1-2^{\prime}} / C_{p}=1385-7.18 / 0.653=1374^{\circ} \mathrm{R}$$

The actual static gas temperature at the first rotor blade row exit

$$T_{2}=T_{2^{\prime}}+\frac{q_{\mathrm{br} 2}}{C_{\mathrm{p}}}=1374+\frac{41.975}{0.653}=1438^{\circ} \mathrm{R}$$

Gas density at the first rotor blade exit

$$\rho_{2}=\frac{144 p_{2}}{R T_{2}}=\frac{144 \times 31.6}{53.6 \times 1438}=0.059 \mathrm{lb} / \mathrm{ft}^{3}$$

We use an angle $\beta_{2}$ of $25^{\circ}$ for the relative gas flow direction at the first rotor blade exits (unsymmetrical blades). The absolute flow angle $\alpha_{2}$ at the first rotor blade exits can be calculated from

$$\begin{aligned} \tan a_{2} & =\frac{V_{2} \sin \beta_{2}}{V \cos \beta_{2}-U}=\frac{2866 \times \sin 25^{\circ}}{2866 \times \cos 25^{\circ}-890}=0.707 \\ a_{2} & =35^{\circ} 15^{\prime} \end{aligned}$$

The absolute flow velocity at the first rotor blade exit

$$C_{2}=\frac{V_{2} \sin \beta_{2}}{\sin \alpha_{2}}=\frac{2866 \times \sin 25^{\circ}}{\sin 35^{\circ} 15^{\prime}}=\frac{1210}{0.577}=2080 \mathrm{fps}$$

Point "3"-Stator Blade Exit $=$ Second Rotor Blade Inlet

The isentropic enthalpy drop in the stator blades

$$\Delta H_{2-3^{\prime}}=\Delta H_{1-2^{\prime}}=7.18 \mathrm{Btu} / \mathrm{lb}$$

Analogous to equation (6-135), the absolute gas flow velocity at the stator blade inlets

$$\begin{aligned} C_{3} & =\sqrt{k_{b}{ }^{2} C_{2}{ }^{2}+2 g J \eta_{n} \Delta H_{2-3^{\prime}}} \\ & =\sqrt{(0.89 \times 2080)^{2}+64.4 \times 778 \times 0.92 \times 7.18} \\ & =1938 \mathrm{fps} \end{aligned}$$

Reheat in the stator blades

$$q_{\mathrm{bs}}=\left(1-k_{b}^{2}\right) \frac{C_{2}^{2}}{2 g J}+\left(1-\eta_{n}\right) \Delta H_{2-3^{\prime}}$$

(Analogous to eq. (6-136))

$$\begin{aligned} & =\left[1-(0.89)^{2}\right] \times \frac{(2080)^{2}}{64.4 \times 778}+(1-0.92) \times 7.18 \\ & =18.53 \mathrm{Btu} / \mathrm{lb} \end{aligned}$$

The static gas pressure at the stator blade exits

$$\begin{aligned} p_{3} & =p_{2}\left[1-\frac{H_{2-3^{\prime}}}{C_{p} T_{2}}\right]^{\frac{\gamma}{\gamma-1}}=31.6 \times\left[1-\frac{7.18}{0.653 \times 1438}\right]^{9.06} \\ & =29.42 \mathrm{psia} \end{aligned}$$

Gas static temperature at the stator blade exits following an isentropic expansion

$$T_{3^{\prime}}=T_{2}-\Delta H_{2-3^{\prime}} / C_{p}=1438-7.18 / 0.653=1427^{\circ} \mathrm{R}$$

Actual static gas temperature at the stator blade exits

$$T_{3}=T_{3^{\prime}}+\frac{q_{\mathrm{bs}}}{C_{p}}=1427+\frac{18.53}{0.653}=1456^{\circ} \mathrm{R}$$

Gas density at the stator blade exit

$$\rho_{3}=\frac{144 p_{3}}{R T_{3}}=\frac{144 \times 29.42}{53.6 \times 1456}=0.0544 \mathrm{lb} / \mathrm{ft}^{3}$$

We use an angle $a_{3}$ of $35^{\circ}$ for the absolute gas flow direction at the stator blade exit ( $\alpha_{3} \cong a_{2}$ ). The relative flow angle $\beta_{3}$ at the stator blade exit can be calculated from

$$\begin{aligned} \tan \beta_{3} & =\frac{C_{3} \sin a_{3}}{C_{3} \cos a_{3}-U}=\frac{1938 \times 0.574}{1938 \times 0.819-890}=1.596 \\ \beta_{3} & =57^{\circ} 56^{\prime} \end{aligned}$$

The relative flow velocity at the stator blade exit

$$V_{3}=\frac{C_{3} \sin a_{3}}{\sin \beta_{3}}=\frac{1938 \times 0.574}{0.847}=1312 \mathrm{fps}$$

Point "4"-Second Rotor Blade Exit

The isentropic enthalpy drop in the second rotor blades

$$\Delta H_{3-4^{\prime}}=\Delta H_{1-2^{\prime}}=7.18 \mathrm{Btu} / \mathrm{lb}$$

The relative gas flow velocity at the second rotor blade exit

$$\begin{aligned} V_{4} & =\sqrt{k_{b}^{2} V_{3}^{2}+2 g J \eta_{n} \Delta H_{3-4^{\prime}}} \\ & =\sqrt{(0.89 \times 1312)^{2}+64.4 \times 778 \times 0.92 \times 7.18} \\ & =1306 \mathrm{fps} \end{aligned}$$

The amount of reheat in the second rotor blades

$$\begin{aligned} q_{\mathrm{br} 2} & =\left(1-k_{b}^{2}\right) \frac{V_{3}^{2}}{2 g J}+\left(1-\eta_{n}\right) \Delta H_{3-4^{\prime}} \\ & =\left[1-(0.89)^{2}\right] \times \frac{(1312)^{2}}{64.4 \times 778}+(1-0.92) \times 7.18 \\ & =7.73 \mathrm{Btu} / \mathrm{sec} \end{aligned}$$

Gas static pressure at the second rotor blade exit

$$\begin{aligned} p_{4} & =p_{3}\left[1-\frac{\Delta H_{3-4^{\prime}}}{C_{p} T_{3}}\right]^{\frac{\gamma}{\gamma-1}} \\ & =29.42 \times\left[1-\frac{7.18}{0.653 \times 1456}\right]^{9.06} \\ & =27.46 \mathrm{psia}>27 \mathrm{psia}\left(p_{e}\right) \end{aligned}$$

$p_{4}$ is slightly higher than the turbine exit pressure (underexpansion), because of the reheating effects.

The gas static temperature at the second rotor blade exits following an isentropic expansion

$$T_{4^{\prime}}=T_{3}-\Delta H_{3-4^{\prime}} / C_{p}=1456-\frac{7.18}{0.653}=1445 \mathrm{Btu} / \mathrm{lb}$$

The actual gas static temperature at the second rotor blade exit

$$T_{4}=T_{4^{\prime}}+\frac{q_{\mathrm{br} 2}}{C_{p}}=1445+\frac{7.73}{0.653}=1457^{\circ} \mathrm{R}$$

Gas density at the second rotor blade exits

$$\rho_{4}=\frac{144 p_{4}}{R T_{4}}=\frac{144 \times 27.46}{53.6 \times 1457}=0.0506 \mathrm{lb} / \mathrm{ft}^{3}$$

We use an angle $\beta_{4}$ of $44^{\circ}$ for the relative gas flow direction at the second rotor blade exits (unsymmetrical blades). The absolute flow angle $a_{4}$ at the second rotor blade exits can be calculated from

$$\begin{aligned} \tan \alpha_{4} & =\frac{V_{4} \sin \beta_{4}}{V_{4} \cos \beta_{4}-U}=\frac{1306 \times 0.695}{1306 \times 0.719-890}=18.5 \\ a_{4} & =86^{\circ} 55^{\prime} \end{aligned}$$

The absolute flow velocity at the second rotor blade exits

$$C_{4}=\frac{V_{4} \sin \beta_{4}}{\sin \alpha_{4}}=\frac{1306 \times 0.695}{0.9985}=908 \mathrm{fps}$$

Nozzle Dimensions

From equation (6-123), the required total nozzle throat area

$$\begin{aligned} A_{\mathrm{nt}} & =\frac{\dot{w}_{t}}{\epsilon_{\mathrm{nt}} p_{0} \sqrt{\frac{g \gamma\left[\frac{2}{\gamma+1}\right]^{\frac{\gamma+1}{\gamma-1}}}{R T_{0}}}} \\ & =\frac{92}{0.97 \times 640 \sqrt{\frac{32.2 \times 1.124(0.94)^{17.15}}{53.6 \times 1860}}} \\ & =13.22 \mathrm{in}^{2} \end{aligned}$$

We use a radial height $h_{\mathrm{nt}}$ of 1.5 inches at the nozzle throat. Thus the nozzle width at the throat

$$b_{\mathrm{nt}}=\frac{h_{\mathrm{nt}}}{\text { Nozzle aspect ratio }}=\frac{1.5}{9.7}=0.1548 \mathrm{in}$$

The number of nozzles

$$z_{n}=\frac{A_{\mathrm{nt}}}{b_{\mathrm{nt}} h_{\mathrm{nt}}}=\frac{13.22}{0.1548 \times 1.5}=57$$

Pitch or nozzle spacing

$$P_{n}=\frac{\pi d_{m}}{z_{n}}=\frac{\pi \times 29.1}{57}=1.604 \mathrm{in}$$

We allow $2^{\circ}$ between nozzle exit angle $\theta_{n}$ and nozzle spouting-gas flow angle $a_{1}$; thus

$$\theta_{n}=a_{1}-2=25-2=23^{\circ}$$

From equation (6-125), the required total nozzle exit area,

$$A_{\mathrm{ne}}=\frac{144 \dot{w}_{t}}{\rho_{1} C_{1} \epsilon_{\mathrm{ne}}}=\frac{144 \times 92}{0.0658 \times 3940 \times 0.95}=53.75 \mathrm{in}^{2}$$

Combining equations (6-125) and (6-126), we obtain radial height and width at the nozzle exit:

$$\begin{aligned} h_{\mathrm{ne}} & =\frac{A_{\mathrm{ne}}}{\pi d_{m} \sin \theta_{n}-z_{n} t_{n}}=\frac{53.75}{\pi \times 29.1 \times 0.391-57 \times 0.05} \\ & =1.64 \mathrm{in} \end{aligned}$$

$b_{\mathrm{ne}}=\frac{\frac{A_{\mathrm{ne}}}{Z_{\mathrm{n}}}}{h_{\mathrm{ne}}}=\frac{53.75}{57 \times 1.64}=0.576 \mathrm{in}$

First Rotor Blade Dimensions (at $d_{m}$ ) The pitch or blade spacing

$$p_{\mathrm{br} 1}=\frac{\text { Blade chord length } C_{b}}{\text { Blade solidity }}=\frac{1.4}{1.82}=0.769 \mathrm{in}$$

From equation (6-140a), the number of blades

$$z_{\mathrm{br}_{1}}=\frac{\pi d_{m}}{P_{\mathrm{br}_{1}}}=\frac{\pi \times 29.1}{0.769}=119$$

Allow $2^{\circ} 7^{\prime}$ between inlet blade angle $\theta_{b_{1 r 1}}$ and inlet relative flow angle $\beta_{1}$; thus

$$\theta_{b_{1 \Gamma 1}}=\beta_{1}+2^{\circ} 7^{\prime}=31^{\circ} 53^{\prime}+2^{\circ} 7^{\prime}=34^{\circ}$$

Make exit blade angle $\theta_{b_{2 r 1}}$ equal to exit relative flow angle $\beta_{2}$

$$\theta_{b_{2 r 1}}=\beta_{2}=25^{\circ}$$

We select a blade radial height at the inlet

$$h_{b_{1 \mathrm{r} 1}}=h_{\mathrm{ne}}(1 \times 0.08)=1.64 \times 1.08=1.77 \mathrm{in}$$

The blade passage width at the inlet

$$\begin{aligned} b_{b_{1} r_{1}} & =p_{b r_{1}} \sin \theta_{b_{1} r_{1}}-t_{b}=0.769 \times 0.559-0.05 \\ & =0.379 \mathrm{in} \end{aligned}$$

From equation (6-138), the required total blade exit area

$$A_{b_{2} r_{1}}=\frac{144 \dot{w}_{t}}{\rho_{2} V_{2} \epsilon_{b_{2}}}=\frac{144 \times 92}{0.059 \times 2866 \times 0.95}=82.5 \mathrm{in}^{2}$$

Combining equations (1-139) and (1-140a), we obtain the blade radial height at the exit

$$\begin{aligned} h_{b_{2} r_{1}} & =\frac{A_{b_{2} r_{1}}}{\pi d_{m} \sin \theta_{b_{2} r_{1}}-z_{b} t_{b}} \\ & =\frac{82.5}{\pi \times 29.1 \times 0.423-119 \times 0.05}=2.52 \mathrm{in} \end{aligned}$$

The blade passage width at the exit

$$\begin{aligned} b_{b_{2} r_{1}} & =P_{b r_{1}} \sin \theta_{b_{2} r_{1}}-t_{b}=0.769 \times 0.443-0.05 \\ & =0.291 \mathrm{in} \end{aligned}$$

The mean blade radial height

$$h_{b r_{1}}=\frac{1.77+2.52}{2}=2.145 \mathrm{in}$$

Assume a tapered blade with shroud, and that it is subject to approximately the same tensile stresses from centrifugal forces, as would be a uniform blade without shroud. The blades shall be made of Timken alloy, with a density $\rho_{b}=0.3 \mathrm{lb} / \mathrm{in}^{3}$. Check the centrifugal tensile stresses at the root section using equation (6-141).

$$\begin{aligned} S_{\mathrm{cr} 1} & =0.0004572 \frac{1}{g} \rho_{b} h_{b r 1} d_{m} N^{2} \\ & =0.0004572 \times \frac{0.3}{32.2} \times 2.145 \times 29.1 \times(7000)^{2} \\ & =13050 \mathrm{psi} \end{aligned}$$

Stator Blade Dimensions

Pitch or blade spacing

$$P_{\mathrm{bs}}=\frac{\text { Blade chord length } C_{b}}{\text { Blade solidity }}=\frac{1.4}{1.94}=0.721 \mathrm{in}$$

From equation (6-140a), the number of blades

$$z_{\mathrm{bs}}=\frac{\pi d_{m}}{P_{\mathrm{bs}}}=\frac{\pi \times 29.1}{0.721}=127$$

Allowing $2^{\circ} 24^{\prime}$ between inlet blade angle $\theta_{b_{1 S}}$ and inlet absolute flow angle $a_{2}$

$$\theta_{b_{1 S}}=a_{2}+2^{\circ} 24^{\prime}=34^{\circ} 36^{\prime}+2^{\circ} 24^{\prime}=37^{\circ}$$

We hold exit blade angle $\theta_{b_{2} s}$ equal to exit absolute flow angle $a_{3}$ :

$$\theta_{b_{2 S}}=a_{3}=35^{\circ}$$

From equation (6-149), blade radial height at the inlet

$$h_{b_{1 s}}=1.08 \times 2.52=2.72 \mathrm{in}$$

The blade passage width at the inlet

$$\begin{aligned} b_{b_{1 \mathrm{~s}}} & =P_{\mathrm{bs}} \sin \theta_{b_{1 \mathrm{~s}}}-t_{b}=0.721 \times 0.602-0.05 \\ & =0.384 \mathrm{in} \end{aligned}$$

Using equation (6-138), we obtain the required total blade exit area

$$A_{b_{2 S}}=\frac{144 \dot{w}_{t}}{\rho_{3} C_{3} \epsilon_{b_{2}}}=\frac{144 \times 92}{0.0544 \times 1938 \times 0.95}=132.5 \mathrm{in}^{2}$$

Combining equations (6-139) and (6-140a), we calculate the blade radial height at the exit

$$\begin{aligned} h_{b_{2} s} & =\frac{A_{b_{2} s}}{\pi d_{m} \sin \theta_{b_{2} s}-z_{\mathrm{bs}} t_{b}} \\ & =\frac{132.5}{\pi \times 29.1 \times 0.574-127 \times 0.05}=2.87 \mathrm{in} \end{aligned}$$

The blade passage width at the exit

$$\begin{aligned} b_{b_{2 S}} & =P_{\mathrm{bs}} \times \sin \theta_{b_{2 S}}-t_{b}=0.721 \times 0.574-0.05 \\ & =0.364 \mathrm{in} \end{aligned}$$

Second Rotor Blade Dimensions Pitch or blade spacing

$$P_{b r_{2}}=\frac{\text { Blade chord length } C_{b}}{\text { Blade solidity }}=\frac{1.4}{1.67}=0.838 \mathrm{in}$$

From equation (6-140a), the number of the blades

$$z_{b r 2}=\frac{\pi d_{m}}{P_{b r 2}}=\frac{\pi \times 29.1}{0.838}=109$$

Allow $2^{\circ} 4^{\prime}$ between the inlet blade angle $\theta_{b_{1} r_{2}}$ and the inlet relative flow angle $\beta_{3}$; thus

$$\theta_{b_{1} r_{2}}=\beta_{3}+2^{\circ} 4^{\prime}=57^{\circ} 56^{\prime}+2^{\circ} 4^{\prime}=60^{\circ}$$

We make the exit blade angle $\theta_{b_{212}}$ equal to the exit relative flow angle $\beta_{4}$

$$\theta_{b_{2} r 2}=\beta_{4}=44^{\circ}$$

From equation (6-149), the blade radial height at the inlet is

$$h_{b_{1 T 2}}=1.08 \times 2.87=3.10 \mathrm{in}$$

The blade passage width at the inlet

$$\begin{aligned} b_{b_{1} T_{2}} & =P_{b_{12}} \sin \theta_{b_{1} r_{2}}-t_{b}=0.838 \times 0.866-0.05 \\ & =0.677 \mathrm{in} \end{aligned}$$

From equation (6-138), the required total blade exit area

$$A_{b_{2} r_{2}}=\frac{144 \dot{w}_{t}}{\rho_{4} V_{4} \epsilon_{b 2}}=\frac{144 \times 92}{0.0506 \times 1306 \times 0.95}=211 \mathrm{in}^{2}$$

Combining equations (1-139) and (1-140a), we obtain the blade radial height at the exit

$$\begin{aligned} h_{b_{2} r_{2}} & =\frac{A_{b_{2} r_{2}}}{\pi d_{m} \sin \theta_{b_{2} r_{2}}-z_{b} t_{b}} \\ & =\frac{211}{\pi \times 29.1 \times 0.695-119 \times 0.05}=3.66 \mathrm{in} \end{aligned}$$

The blade exit passage width

$$\begin{aligned} b_{b_{2 I 2}} & =P_{b r_{2}} \sin \theta_{b_{2 I}}-t_{b}=0.838 \times 0.695-0.05 \\ & =0.533 \mathrm{in} \end{aligned}$$

The mean blade radial height

$$h_{b r_{2}}=\frac{3.10+3.66}{2}=3.38 \mathrm{in}$$

Check the centrifugal tensile stress at the root section using equation (6-141)

$$\begin{aligned} S_{\mathrm{cr} 2} & =0.0004572 \frac{1}{g} \rho_{b} h_{\mathrm{br} 2} d_{m} N^{2} \\ & =0.0004572 \times \frac{0.3}{32.2} \times 3.38 \times 29.1 \times(7000)^{2} \\ & =20550 \mathrm{psi} \end{aligned}$$

Turbine Efficiencies

From equations (6-146) and (6-147), the combined nozzle and blade efficiency

$$\begin{aligned} \eta_{\mathrm{nb}} & =\frac{\begin{array}{c} U\left(C_{1} \cos a_{1}+C_{2} \cos a_{2}\right. \\ \left.+C_{3} \cos a_{3}+C_{4} \cos a_{4}\right) \end{array}}{g J \Delta H} \\ & =\frac{\begin{array}{c} 890(3940 \times 0.906+2080 \times 0.817 \\ +1938 \times 0.819+908 \times 0.055) \end{array}}{32.2 \times 778 \times 359} \\ & =0.683 \end{aligned}$$

From equation (6-148), the turbine machine efficiency

$$\eta_{m}=\frac{\eta_{t}}{\eta_{\mathrm{nb}}}=\frac{0.582}{0.683}=0.852$$

A-1 Stage Engine Turbine (Single-Stage, TwoRotor, Velocity-Compounded Impulse Type) Design Summary
For velocity diagrams at mean diameter $d_{m}$,

see figure 6-58.

$$\begin{aligned} & U=890: \\ & \alpha_{1}=25^{\circ} ; \beta_{1}=31^{\circ} 53^{\prime} ; C_{1}=3940 \mathrm{fps} ; V_{1} \\ & =3156 \mathrm{fps} ; \alpha_{2}=35^{\circ} 15^{\prime} ; \beta_{2}=25^{\circ} ; C_{2}=2080 \\ & \mathrm{fps} ; V_{2}=2866 \mathrm{fps} ; \alpha_{3}=35^{\circ} ; \beta_{3}=57^{\circ} 56^{\prime} ; \\ & C_{3}=1938 \mathrm{fps} ; V_{3}=1312 \mathrm{fps} ; \alpha_{4}=86^{\circ} 55^{\prime} ; \\ & \beta_{4}=44^{\circ} ; C_{4}=908 \mathrm{fps} ; V_{4}=1306 \mathrm{fps} \end{aligned}$$

Isentropic enthalpy drops:

$$\text { Nozzles, } \Delta H_{0-1^{\prime}}=337.5 \mathrm{Btu} / \mathrm{lb}$$

First rotor blades, $\Delta H_{1-2^{\prime}}=7.18 \mathrm{Btu} / \mathrm{lb}$ Stator blades, $\Delta H_{2-3^{\prime}}=7.18 \mathrm{Btu} / \mathrm{lb}$ Second rotor, $\Delta H_{3-4^{\prime}}=7.18 \mathrm{Btu} / \mathrm{lb}$

$$\text { Total } \Delta H=359 \mathrm{Btu} / \mathrm{lb}$$

Working efficiencies:

$$\eta_{t}=58.2 \% ; \eta_{n}=92 \% ; \eta_{\mathrm{nb}}=68.3 \% ; \eta_{m}=85.2 \%$$

Mean diameter of nozzles and blades:

$$d_{m}=29.1 \mathrm{in}$$

Nozzle dimensions (at $d_{m}$ ):

$$\begin{aligned} & \text { Aspect ratio }=9.7 ; z_{n}=57 ; P_{n}=1.604 \mathrm{in} ; \\ & \theta_{n}=23^{\circ} ; h_{\mathrm{nt}}=1.5 \mathrm{in} ; a_{\mathrm{ne}}=1.64 \mathrm{in} ; b_{\mathrm{nt}} \\ & =0.1548 \mathrm{in} ; b_{\mathrm{ne}}=0.576 \mathrm{in} \end{aligned}$$

First rotor blade dimensions (at $d_{m}$ ):

$$\begin{aligned} & \text { Solidity }=1.82 ; C_{b}=1.4 \mathrm{in} ; z_{b r_{1}}=119 ; \\ & P_{b r_{1}}=0.769 \mathrm{in} ; \theta_{b_{1 r 1}}=34^{\circ} ; \theta_{b_{2 r 1}}=25^{\circ} ; \\ & h_{b_{1 r 1}}=1.77 \mathrm{in} ; h_{b_{2 r 1}}=2.52 \mathrm{in} ; b_{b_{1 r 1}}=0.379 \\ & \text { in; } b_{b_{2 r 1}}=0.291 \mathrm{in} \end{aligned}$$

Stator blade dimensions (at $d_{m}$ ):

$$\begin{aligned} & \text { Solidity }=1.94 ; C_{b}=1.4 \mathrm{in} ; z_{b s}=1.27 \\ & P_{b s}=0.721 \mathrm{in} ; \theta_{b_{1 s}}=37^{\circ} ; \theta_{b_{2 s}}=35^{\circ} \\ & h_{b_{1 s}}=2.72 \mathrm{in} ; h_{b_{2 s}}=2.87 \mathrm{in} ; b_{b_{1 s}}=0.384 \\ & \text { in; } b_{b_{2 s}}=0.364 \mathrm{in} \end{aligned}$$

Second rotor blade dimensions (at $d_{m}$ ):

$$\begin{aligned} & \text { Solidity }=1.67 ; C_{b}=1.4 \mathrm{in} ; z_{b r_{2}}=109 ; \\ & P_{b r_{2}}=0.838 \mathrm{in} ; \theta_{b_{1} r_{2}}=60^{\circ} ; \theta_{b_{2} r_{2}}=44^{\circ} \\ & h_{b_{1} r_{2}}=3.10 \mathrm{in} ; h_{b_{2} b_{2}}=3.66 \mathrm{in} ; b_{b_{1} r_{2}}=0.677 \\ & \text { in; } b_{b_{2} r_{2}}=0.533 \mathrm{in} \end{aligned}$$

(b) Two-stage, two-rotor, equal-work, pressurecompounded impulse turbine. (For velocity diagrams, see fig. 6-59.)

From prior trial-and-error calculations, the following isentropic enthalpy drops resulting in (approximately) equal work for each stage were obtained. We assume a stage carryover ratio $\mathrm{r}_{c}=0.91$.

First-stage nozzles:

$$\Delta H_{0-1^{1}}=50 \% ; \quad \Delta H=0.5 \times 359=179.5 \mathrm{Btu} / \mathrm{lb}$$

First-stage rotor blades:

$$\Delta H_{1-2^{\prime}}=3 \% ; \quad \Delta H=0.03 \times 359=10.75 \mathrm{Btu} / \mathrm{lb}$$

Second-stage nozzles:

$$\Delta H_{2-3^{\prime}}=44 \% ; \quad \Delta H=0.44 \times 359=158 \mathrm{Btu} / \mathrm{lb}$$

Second-stage rotor blades:

$$\Delta H_{3-4^{\prime}}=3 \% ; \quad \Delta H=0.03 \times 359=10.75 \mathrm{Btu} / \mathrm{lb}$$

Point "0"-First-Stage Nozzle Inlet $T_{0}=1860^{\circ} \mathrm{R}$ $p_{0}=640 \mathrm{psia}$

Point "1"-First-Stage Nozzle Exit=Rotor Blade Inlet

From equation (6-121), the gas-spouting velocity at the first-stage nozzle exit

$$\begin{aligned} C_{1} & =k_{n} \sqrt{2 g J \Delta H_{0-1^{\prime}}}=0.96 \times 223.8 \times \sqrt{179.5} \\ & =2880 \mathrm{fps} \end{aligned}$$

We use a value of $25^{\circ}$ for the spouting-gas flow angle $a_{1}$. For optimum efficiency, the peripheral speed at the rotor mean diameter

$$U=\frac{\cos a_{1} C_{1}}{2}=\frac{0.906 \times 2880}{2}=1308 \mathrm{fps}$$

Using equation (6-130), the relative gas flow $\beta_{1}$ at the first-stage rotor blade inlet can be calculated as $\tan \beta_{1}=\frac{C_{1} \sin \alpha_{1}}{C_{1} \cos \alpha_{1}-U}=\frac{2880 \times 0.423}{2880 \times 0.906-1308}=0.936$

$$\beta_{1}=43^{\circ} 8^{\prime}$$

The relative gas flow velocity at first-stage rotor blade inlet

$$V_{1}=\frac{C_{1} \sin a_{1}}{\sin \beta_{1}}=\frac{2880 \times 0.423}{0.683}=1784 \mathrm{fps}$$

Point "2"-First-Stage Rotor Blade Exit=SecondStage Nozzle Inlet

From equation (6-135), the relative gas flow velocity at the first-stage rotor blade exit

$$\begin{aligned} V_{2} & =\sqrt{k_{b}^{2} V_{1}^{2}+2 g J \eta_{n} \Delta H_{1-2^{\prime}}} \\ & =\sqrt{(0.89 \times 1784)^{2}+64.4 \times 778 \times 0.92 \times 10.75} \\ & =1736 \mathrm{fps} \end{aligned}$$

We chose a relative exit gas flow angle $\beta_{2}=38^{\circ}$ for the first-stage rotor blades. The absolute gas flow angle, $a_{2}$, can then be calculated as $\tan \alpha_{2}=\frac{V_{2} \sin \beta_{2}}{V_{2} \cos \beta_{2}-U}=\frac{1736 \times 0.616}{1736 \times 0.788-1308}=17.25$

$$a_{2}=86^{\circ} 40^{\prime}$$

The absolute gas flow velocity at the firststage rotor blade exits

$$C_{2}=\frac{V_{2} \sin \beta_{2}}{\sin a_{2}}=\frac{1736 \times 0.616}{0.998}=1070 \mathrm{fps}$$

Point "3"-Second-Stage Nozzle Exit $=$ Second

Rotor Blade Inlet

From equation (6-151), the second-stage nozzle gas-spouting velocity

$$\begin{aligned} C_{3} & =k_{n} \sqrt{r_{c} C_{2}^{2}+2 g J \Delta H_{2-3^{\prime}}} \\ & =0.96 \sqrt{0.91 \times(1070)^{2}+64.4 \times 778 \times 158} \\ & =2880 \mathrm{fps} \end{aligned}$$

Since $C_{3}=C_{1}$, the remainder of the secondstage velocity diagram is the same as that of the first stage, i.e., $a_{3}=a_{1}=25^{\circ} ; \beta_{3}=\beta_{1}=43^{\circ} 8^{\prime}$; $V_{3}=V_{1}=1784 \mathrm{fps} ; a_{4}=a_{2}=86^{\circ} 40^{\prime} ; \beta_{4}=\beta_{2}=38^{\circ}$; $C_{4}=C_{2}=1070 \mathrm{fps} ; V_{4}=V_{2}=1736 \mathrm{fps}$.

From equation (6-129), the turbine rotor mean diameter

$$d_{m}=\frac{720 U}{\pi N}=\frac{720 \times 1308}{\pi \times 7000}=42.7 \mathrm{in}$$

From equation (6-147), the combined nozzle and blade efficiency

$$\begin{aligned} \eta_{\mathrm{nb}} & =\frac{U\left(C_{1} \cos a_{1}+C_{2} \cos a_{2}\right.}{\left.+C_{3} \cos a_{3}+C_{4} \cos a_{4}\right)} \\ & =\frac{1308(2880 \times 0.906+1070 \times 0.058}{+2880 \times 0.906+1070 \times 0.058)} \\ & =0.78 \end{aligned}$$

The turbine machine efficiency is assumed to be the same as that used in design (a):

$$\eta_{m}=0.852$$

From equation (6-148), the overall turbine efficiency

$$\eta_{t}=\eta_{\mathrm{nb}} \eta_{m}=0.78 \times 0.852=0.664$$

A-1 Stage Engine Alternate Turbine Design Summary (Two-Stage, Two-Rotor, PressureCompounded, Impulse Type)

For velocity diagrams at mean diameter $d_{m}$, see figure 6-59.

$$\begin{aligned} & U=1308 \mathrm{fps}: \\ & a_{1}=25^{\circ} ; \beta_{1}=43^{\circ} 8^{\prime} ; C_{1}=2880 \mathrm{fps} ; V_{1}=1784 \\ & \mathrm{fps} ; a_{2}=86^{\circ} 40^{\prime} ; \beta_{2}=38^{\circ} ; C_{2}=1070 \mathrm{fps} ; \\ & V_{2}=1736 \mathrm{fps} ; a_{3}=25^{\circ} ; \beta_{3}=43^{\circ} 8^{\prime} ; C_{3}=2880 \\ & \mathrm{fps} ; V_{3}=1784 \mathrm{fps} ; a_{4}=86^{\circ} 40^{\prime} ; \beta_{4}=38^{\circ} ; \\ & C_{4}=1070 \mathrm{fps} ; V_{4}=1736 \mathrm{fps} \end{aligned}$$

Isentropic enthalpy drops: First-stage nozzles, $\Delta H_{0-1^{\prime}}=179.5 \mathrm{Btu} / \mathrm{lb}$ First-stage rotor blades, $\Delta H_{1-2^{\prime}}=10.75 \mathrm{Btu} / \mathrm{lb}$ Second-stage nozzles, $\Delta H_{2-3^{\prime}}=158 \mathrm{Btu} / \mathrm{lb}$ Second-stage rotor blades, $\Delta H_{3-4^{\prime}}=10.75$ Btu/lb Working efficiencies:

$$\eta_{t}=66.4 \% ; \eta_{n}=92 \% ; \eta_{\mathrm{nb}}=78 \% ; \eta_{m}=85.2 \%$$

Mean diameter of nozzles and blades:

$$d_{m}=42.7 \mathrm{in}$$

Comment: The overall efficiency of the pressure compounded turbine is higher than that of design (a). However, a relatively large $d_{m}$ is required (weight, size).