Water at 15°C is to be discharged from a reservoir at a rate of 18 L/s using two horizontal cast iron pipes connected in series and a pump between them. The first pipe is 20 m long and has a 6-cm diameter, while the second pipe is 36 m long and has a 3-cm diameter. The water level in the reservoir is 30 m above the centerline of the pipe. The pipe entrance is sharp-edged, and losses associated with the connection of the pump are negligible. Neglecting the effect of the kinetic energy correction factor, determine the required pumping head and the minimum pumping power to maintain the indicated flow rate. The density and dynamic viscosity of water at 15°C are rho = 999.1 kg/m3 and μ = 1.138 × 10–3 kg/m·s. The loss coefficient is KL = 0.5 for a sharp-edged entrance. The roughness of cast iron pipes is ε = 0.00026 m.

[tex]\frac{P_{1} }{\gamma } +\frac{V_{1}^{2} }{2g} +z_{1} =\frac{P_{2} }{\gamma } +\frac{V_{2}^{2} }{2g} +z_{2}+h_{i} -h_{pump} , if V_{1} =0,z_{2} =0\\h_{pump} =\frac{V_{2}^{2}}{2} +h_{i}-z_{1}[/tex]

For the pipe 1, the flow velocity is:

[tex]V_{1} =\frac{Q}{\frac{\pi D^{2} }{4} }[/tex]

Q = 18 L/s = 0.018 m³/s

D = 6 cm = 0.06 m

[tex]V_{1} =\frac{0.018}{\frac{\pi *0.06^{2} }{4} } =6.366m/s[/tex]

The Reynold´s number is:

[tex]Re=\frac{\rho *V*D}{u} =\frac{999.1*6.366*0.06}{1.138x10^{-3} } =335339.4[/tex]

[tex]\frac{\epsilon }{D} =\frac{0.00026}{0.06} =0.0043[/tex]

Using the graph of Moody, I will select the f value at 0.0043 and 335339.4, as 0.02941

The head of pipe 1 is:

[tex]h_{1} =\frac{V_{1}^{2} }{2g} (k_{L}+\frac{fL}{D} )=\frac{6.366^{2} }{2*9.8} *(0.5+\frac{0.0294*20}{0.06} )=21.3m[/tex]

For the pipe 2, the flow velocity is:

[tex]V_{2} =\frac{0.018}{\frac{\pi *0.03^{2} }{4} } =25.46m/s[/tex]

The Reynold´s number is:

[tex]Re=\frac{\rho *V*D}{u} =\frac{999.1*25.46*0.03}{1.138x10^{-3} } =670573.4[/tex]

[tex]\frac{\epsilon }{D} =\frac{0.00026}{0.03} =0.0087[/tex]

The head of pipe 1 is:

[tex]h_{2} =\frac{V_{2}^{2} }{2g} (k_{L}+\frac{fL}{D} )=\frac{25.46^{2} }{2*9.8} *(0.5+\frac{0.033*36}{0.03} )=1326.18m[/tex]

The total head is:

hi = 1326.18 + 21.3 = 1347.48 m

The required pump head is:

[tex]h_{pump} =\frac{25.46^{2} }{2*9.8} +1347.48-36=1344.55m[/tex]

The required pumping power is:

[tex]P=Q\rho *g*h_{pump} =0.018*999.1*9.8*1344.55=236965.16W=236.96kW[/tex]