Question

Problem 4 The pumping system shown in the figure below to deliver 3 cfs of 60°F water. The total elevation difference between is 84 ft. The suction line is a 6-inch diameter, 300-ft long cast-iron pipe. The discharge line is 6-inch diameter in an 860-ft long plastic pipe. The suction inlet for the pump is 20 ft above the reservoir. The atmospheric pressure is 14.70 psia. The required NPSH is 7.0 ft. Minor losses include entrance: k, = 0.5, open gate valve: k, = 0.23; and elbow: kp = 0.67. Determine whether the system will have a cavitation problem. [10 pts]

Answer 1

Ans) Given,

Discharge = 3 cfs

Elevation difference ($\dpi{100} \Delta$Z) = 84 ft

Diameter of pipe at inlet and outlet , D1 = D2 = 6 in or 0.5 ft

Length of pipe 1 = 300 ft

Length of pipe 2 = 860 ft

Suction inlet height = 20 ft

NPSH = 7 ft

We know,

NPSH = (P_atm / $\dpi{100} \rho$g) - z - H_L - (P_v/$\dpi{100} \rho$g)

where, z = maximum height that pump can be placed without cavitation

P_v = Vapor pressure of water at 60 F = 0.256 psia

H_L = head loss due to friction , valves and elbows

=> z_max = (P_atm / $\dpi{100} \rho$g) - NPSH - H_L - (P_v/$\dpi{100} \rho$g)

We know,  

Q = A x V

=> V = Q/A = 3 / ($\dpi{100} \pi$/4) (0.5)² = 15.28 ft/s

Reynold number , Re = $\dpi{100} \rho$VD/$\dpi{100} \mu$

where, $\dpi{100} \mu$ = dynamic viscosity of water at 60 F = 2.3 x 10^-5

=> Re = 1.94 x 15.28 x 0.5 / 2.3 x 10^-5

=> Re = 6.4 x 10⁵

Roughness of cast iron pipe,e = 0.01 in or 0.00085 ft

Relative roughness = e/D = 0.00085/0.5 = 0.0017

According to Moody diagram, for Re = 6.4 x 10⁵ and e/D = 0.0017. friction factor (f) = 0.023

=> Head loss due to friction = f L V² / 2gD = 0.023(300)(15.28)² / ( 2 x 32.2 x 0.5)

=> Head loss due to friction = 50.02 ft

Minor loss = $\dpi{50} \sum$K V² / 2g = (0.5 + 0.23 + 0.67) (15.28)² / ( 2 x 32.2)

Minor loss = 1.4 x 233.47 / 64.4 = 5.07 ft

=> H_L = 50.02 + 5.07 = 55.09 ft

Putting values in above equation,

z_max = (14.7 x 144 / 62.2) - 7 - 55.09 - (0.256 x 144/ 62.2)

=> z_max = 34.08 - 7 - 55.09 - 0.592

=> z_max = 28.6 ft > 20 ft

Since, maximum allowable height of pump is larger than 20 ft, pump will not suffer cavitation