Question 14.2: The blades on an impeller of the axial-flow pump in Fig. 14–......
The blades on an impeller of the axial-flow pump in Fig. 14–5a have a mean radius of r_m = 125 mm and rotate at 1000 rev/min. If the pump is required to produce a flow of 0.2 m³/s, determine the initial blade angle β_1 so that the pump runs efficiently. Also, find the average torque that must be applied to the shaft of the impeller, and the average power output of the pump if the fluid is water. The average open cross-sectional area through the impellers is 0.03 m².
Fluid Description. We assume steady ideal flow through the pump and will use average velocities. ρ = 1000 kg/m³.
Kinematics. The midpoint velocity of the blades has a magnitude of
And the axial velocity of the flow through the impeller is
Q\,=\,V_{a}A;\qquad\quad0.2\,\mathrm{m^{3}/s}\,=\,V_{a}\bigl(0.03\,\mathrm{m^{2}}\bigr);\qquad V_{a}=6.667\,\mathrm{m/s}The pump will run efficiently when the velocity of the water onto the blades is V_1 = V_a = 6.667 m/s as shown in Fig. 14–5b. Here \alpha_1 = 90° and so
\tan\beta_{1}=\frac{6.667\,\mathrm{m/s}}{13.09\,\mathrm{m/s}}\qquad\qquad\beta_{1}=27.0^{\circ}Also, because \alpha_1 = 90°,
V_{t1}=0At the exit, Fig. 14–5c,
V_{t2}=13.09\,\mathrm{m/s}-(6.667\,\mathrm{m/s})\,\mathrm{cot}\,70^{\circ}=10.664\,\mathrm{m/s}Torque and Power. Since the tangential components of velocity are known, Eq. 14–2 can be applied to determine the torque.
From Eq. 14–4, the power supplied to the water by the pump is
Note: If \alpha_1 < 90^\circ , \,V_1 would have a component V_{t1} as in the previous example and this would decrease the power.