Question 14.4: The impeller on the radial-flow pump in Fig. 14–8a has an av......
The impeller on the radial-flow pump in Fig. 14–8a has an average inlet radius of 50 mm and outlet radius of 150 mm, and an average width of 30 mm. If the blade angles are β_1 = 20° and β_2 = 10°, determine the flow through the pump, and the ideal pump head when the impeller is rotating at 400 rev/min. The flow onto the impeller is in the radial direction.
Fluid Description. We will assume steady, incompressible flow and use average velocities.
Kinematics. To find the flow, we must first determine the speed of the fluid as it moves onto the blades. We will also need the speed of the blades at its entrance and its exit.
The kinematic diagram for the flow onto the impeller is shown in Fig. 14–8b. Since V_1 is in the radial direction (\alpha_1 = 90°),
V_{1}=V_{r}=U_{1}\tan\beta_{1}=(2.094\mathrm{\,m/s)\tan20^{\circ}=0.7623\,m/s}Flow. The flow into as well as out of the pump is
Q=\,V_{1}A_{1}\,=\,V_{1}(2\pi r_{1}b_{1})\\=\;0.7623\;\mathrm{m/s}\left[2\pi(0.05\;\mathrm{m)}(0.03\;\mathrm{m})\right]\\=\;0.007184\;\mathrm{m^{3}/s}=0.00718\;\mathrm{m^{3}/s}Ideal Pump Head. The ideal pump head is
h_{\mathrm{pump}}={\frac{U_{2}{}^{2}}{g}}-{\frac{U_{2}Q\;\mathrm{cot}\,\beta_{2}}{2\pi\;r_{2}b g}}\\=\frac{(6.283\,{\mathrm{m/s}})^{2}}{9.81\:{\mathrm{m/s}}^{2}}-\frac{(6.283\,{\mathrm{m/s}})(0.00718\,{\mathrm{m}}^{3}/{\mathrm{s}})\,{\mathrm{cot}}\,10^{\circ}}{2\pi(0.150\,{\mathrm{m}})(0.03 m )(9.81\,{\mathrm{m/s}}^{2})}\\=3.10\,\mathrm{m}