The solution is: Can someone please show me how to arrive at this solution? Than

Question

The solution is:

Can someone please show me how to arrive at this solution? Thanks!

A centrifugal outward-flow water pump has a rotor that rotates clockwise at an angular speed omega = 6 rpm. The inlet and exit rotor diameters are Di = 6 cm and Dc = 24 cm, respectively, and the blade height h = 4 cm. The inlet and exit blade angles measured with respect to the radial direction are betai = 45 degree and beta c = 6 degree, respectively. If the inlet flow is purely radial, draw velocity triangles for a single blade. Then find the mass flow rate through the rotor m, the magnitude of the shaft torque T, and the pump power W. Velocity triangle at inlet: Velocity triangle at exit: m = rho Vi Ai = 14.2 kg/s T = 11.5 N m Wp = 72 W

Explanation / Answer

Ui = ri*w = .03*600*2*pi/60 = 188.5 cm/s (? )

Wi*cos(45)=188.5

also, Vi = Wicos(45) = 188.5

since the flow is radial , Vi = 188.5 cm/s r

and from fig. , it is backward vanes

Wi = 188.5 r -188.5 ? cm/s

Ue = re*w ? = 12*600*2*pi/60 = 754 ?

now, mass flow rate is same.

2*pi*ri*Vr_i*b = 2*pi*re*Vr_e*b

3*188.5 = 12*Vr_e

Vr_e = 47.12 cm/s [ component of V in r direction ]

hence , Wr_e = 47.12 cm/s [ no component of U in r direction]

component of W in theta direction = 47.12*tan(60) =81.62 cm/s

hence , We = 47.12 r -81.62 theta

also , 81.62 = Ue - V_theta_e

V_theta_e = 754 - 81.62 = 672.38 cm/s

Ve = 47.12 r + 672.4 theta cm/s

mass flow rate, m_dot =rho* 2*pi*ri*Vr_i*b = 1000*2*pi*0.03*1.885*0.04 = 14.2 kg/s

T = m_dot*( Ve_theta*re-Vi_theta*ri) = 14.2 ( 6.72*0.12-0 ) =11.45 N.m

Power = T*omega = 719.5 ~ 720 watts

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