Question 14.6: Preliminary Design of a Centrifugal Pump A centrifugal pump ...

For a given flow rate, net head, and dimensions of a centrifugal pump, we are to design the blade shape (leading and trailing edge angles).
We are also to estimate the horsepower required by the pump.
Assumptions 1 The flow is steady. 2 The liquid is incompressible. 3 There are no irreversible losses through the impeller. 4 This is only a preliminary design.
Properties For refrigerant R-134a at T=20^{\circ} C , v_{f}=0.0008157 m ^{3} / kg .
Thus \rho=1 / v_{f}=1226 kg / m ^{3} .
Analysis We calculate the required water horsepower from Eq. 14–3,

\begin{aligned}\dot{W}_{\text {water horsepower }} &=\rho g \dot{V} H \\&=\left(1226 kg / m ^{3}\right)\left(9.81 m / s ^{2}\right)\left(0.25 m ^{3} / s \right)(14.5 m )\left(\frac{ W \cdot s }{ kg \cdot m / s ^{2}}\right) \\&=43,600 W\end{aligned}

Water horsepower:              \dot{W}_{\text {water horsepower }}=\dot{m} g H=\rho g \dot{V} H                       (14–3)

The required brake horsepower will be greater than this in a real pump. However, in keeping with the approximations for this preliminary design, we assume 100 percent efficiency such that bhp is approximately equal to \dot{W}_{\text {water horsepower }},

\text { bhp } \cong \dot{W}_{\text {water horsepower }}=43,600 W \left(\frac{ hp }{745.7 W }\right)=58.5 hp

We report the final result to two significant digits in keeping with the precision of the given quantities; thus, bhp ≈ 59 horsepower.
In all calculations with rotation, we need to convert the rotational speed from \dot{n}( rpm ) \text { to } \omega( rad / s ) , as illustrated in Fig. 14–43,

\omega=1720 \frac{\text { rot }}{\min }\left(\frac{2 \pi rad }{\text { rot }}\right)\left(\frac{1 min }{60 s }\right)=180.1 rad / s                           (1)

We calculate the blade inlet angle using Eq. 14–25,

\beta_{1}=\arctan \left(\frac{\dot{V}}{2 \pi b_{1} \omega r_{1}^{2}}\right)=\arctan \left(\frac{0.25 m ^{3} / s }{2 \pi(0.050 m )(180.1 rad / s )(0.10 m )^{2}}\right)=23.8^{\circ}

\dot{V}=2 \pi b_{1} \omega r_{1}^{2} \tan \beta_{1}                   (14–25)

We find \beta_{2} by utilizing the equations derived earlier for our elementary analysis. First, for the design condition in which V_{1, t}=0 , Eq. 14–17 reduces to

Net head:                H=\frac{1}{g}(\omega r_{2} V_{2, t}-\omega r_{1} \underbrace{\cancel{V_{1, t}}}_{0})=\frac{\omega r_{2} V_{2, t}}{g}

Net head:          H=\frac{1}{g}\left(\omega r_{2} V_{2, t}-\omega r_{1} V_{1, t}\right)                     (14–17)

from which we calculate the tangential velocity component,

V_{2, t}=\frac{g H}{\omega r_{2}}                           (2)

Using Eq. 14–12, we calculate the normal velocity component,

V_{2, n}=\frac{\dot{V}}{2 \pi r_{2} b_{2}}                     (3)

Volume flow rate:        \dot{V}=2 \pi r_{1} b_{1} V_{1, n}=2 \pi r_{2} b_{2} V_{2, n}                       (14–12)

Next, we perform the same trigonometry used to derive Eq. 14–23, but on the trailing edge of the blade rather than the leading edge. The result is

V_{2, t}=\omega r_{2}-\frac{V_{2, n}}{\tan \beta_{2}}

V_{1, t}=\omega r_{1}-\frac{V_{1, n}}{\tan \beta_{1}}                     (14–23)

from which we finally solve for \beta_{2} ,

\beta_{2}=\arctan \left(\frac{V_{2, n}}{\omega r_{2}-V_{2, t}}\right)                     (4)

After substitution of Eqs. 2 and 3 into Eq. 4, and insertion of the numerical values, we obtain

\beta_{2}=14.7^{\circ}

We report the final results to only two significant digits. Thus our preliminary design requires backward-inclined impeller blades with \beta_{1} \cong 24^{\circ} \text { and } \beta_{2} \cong 15^{\circ} .
Once we know the leading and trailing edge blade angles, we design the detailed shape of the impeller blade by smoothly varying blade angle β from \beta_{1} \text { to } \beta_{2} as radius increases from r_{1} \text { to } r_{2} . As sketched in Fig. 14–44, the blade can be of various shapes while still keeping \beta_{1} \cong 24^{\circ} \text { and } \beta_{2} \cong 15^{\circ} , depending on how we vary β with the radius. In the figure, all three blades begin at the same location (zero absolute angle) at radius r_{1} ; the leading edge angle for all three blades is \beta_{1}=24^{\circ} . The medium length blade (the gray one in Fig. 14–44) is constructed by varying β linearly with r. Its trailing edge intercepts radius r_{2} at an absolute angle of approximately 93°. The longer blade (the black one in the figure) is constructed by varying β more rapidly near r_{1} \text { than near } r_{2} . In other words, the blade curvature is more pronounced near its leading edge than near its trailing edge. It intercepts the outer radius at an absolute angle of about 114°. Finally, the shortest blade (the blue blade in Fig. 14–44) has less blade curvature near its leading edge, but more pronounced curvature near its trailing edge. It intercepts r_{2} at an absolute angle of approximately 77°. It is not immediately obvious which blade shape is best.

Discussion Keep in mind that this is a preliminary design in which irreversible losses are ignored. A real pump would have losses, and the required brake horsepower would be higher (perhaps 20 to 30 percent higher) than the value estimated here. In a real pump with losses, a shorter blade has less skin friction drag, but the normal stresses on the blade are larger because the flow is turned more sharply near the trailing edge where the velocities are largest; this may lead to structural problems if the blades are not very thick, especially when pumping dense liquids. A longer blade has higher skin friction drag, but lower normal stresses. In addition, you can see from a simple blade volume estimate in Fig. 14–44 that for the same number of blades, the longer the blades, the more flow blockage, since the blades are of finite thickness. In addition, the displacement thickness effect of boundary layers growing along the blade surfaces (Chap. 10) leads to even more pronounced blockage for the long blades. Obviously some engineering optimization is required to determine the exact shape of the blade.