Question 16.8: A centrifugal fan running at 1500 rpm has inner and outer di...
A centrifugal fan running at 1500 rpm has inner and outer diameter of the impeller as 0.2 m and 0.24 m. The absolute and relative velocities of air at entry are 21 m/s and 20 m/s respectively and those at exit are 25 m/s and 18 m/s respectively. The flow rate is 0.6 kg/s and the motor efficiency is 80%. Determine (i) the stage pressure rise, (ii) degree of reaction and (iii) the power required to drive the fan. Assuming the flow to be incompressible with the density of air as 1.2 \mathrm{~kg} / \mathrm{m}^{3} .
(i) U_{1}=\frac{\pi \times 0.20 \times 1500}{60}
=15.71 \mathrm{~m} / \mathrm{s}
U_{2}=\frac{\pi \times 0.24 \times 1500}{60}
=18.85 \mathrm{~m} / \mathrm{s}
From Eq. (16.40), the total pressure rise across the stage is
\left(\Delta p_{0}\right)=\frac{1}{2} \rho\left(U_{2}^{2}-U_{1}^{2}\right)+\frac{1}{2} \rho\left(V_{r 1}^{2}-V_{r 2}^{2}\right)+\frac{1}{2} \rho\left(V_{2}^{2}-V_{1}^{2}\right) (16.40)
\left(\Delta p_{t}\right)_{\text {stage }}=\frac{1}{2} \rho\left[\left(V_{2}^{2}-V_{1}^{2}\right)+\left(U_{2}^{2}-U_{1}^{2}\right)+\left(V_{r \mathrm{l}}^{2}-V_{r \mathrm{l}}^{2}\right)\right] =\frac{1}{2} \times 1.2\left[\left(25^{2}-21^{2}\right)+\left(18.85^{2}-15.71^{2}\right)+\left(20^{2}-18^{2}\right)\right]=221.11 \mathrm{~N} / \mathrm{m}^{2}
(ii) The static pressure rise across the stage is
\left(\Delta p_{s}\right)_{\mathrm{stage}}=\frac{1}{2} \rho\left[\left(U_{2}^{2}-U_{1}^{2}\right)+\left(V_{r 1}^{2}-V_{r 2}^{2}\right)\right]=\frac{1}{2} \times 1.2\left[\left(18.85^{2}-15.71^{2}\right)+\left(20^{2}-18^{2}\right)\right]
=110.71 \mathrm{~N} / \mathrm{m}^{2}
The degree of reaction =\frac{110.71}{221.11}
=0.5
(iii) The specific power input to the stage is
w=\frac{\left(\Delta p_{0}\right)_{\text {stage }}}{\rho}=\frac{221.11}{1.2}
=184.26 \mathrm{~J} / \mathrm{kg}
Therefore, the power required to drive the fan is
P=\frac{\dot{m} w}{\eta_{m} }=\frac{0.6 \times 184.26}{0.8}
=138.19 \mathrm{~W}