Question

1. Find the flow rate of each of the pumps in the pumping system (below) if friction losses in pipe lines A and B between junctions R, and Ry are described by the equations: Ah = 1.0-10'.Q?: Ah, = 2.0.109.02 for Qin (m ) Total length of pipelines C'ic" in the pumping system is / 250m, the diameters are the same and equal d = 250mm, total minor loss coefficient in both elements is equal = 10.2, and Darcy-Weisbach friction factor i = 0.03. Other parameters are given in the figure below. Water density P 1000kg/m', pump characteristics can be found in the additional figure on the back. R™ 3M - 2MP

Answer 1

Provided Informations in problem are as below

$\Delta$h_A=1.0*10³*Q^{​​​​​​2}​​​

$\Delta$h_B=2.0*10³*Q^{​​​​​​2}

For Q in m^{​​​​​​3}/s

Total length of pipe lines = L = 250m

Diameter of pipeline = d = 250mm= 0.25 m

Total minor loss coefficient = $\xi$ = 10.2

Darcy wesbach friction coefficient = $\lambda$ = 0.03

Water density = $\rho$ = 1000 Kg/m^{​​​​​​3}

^{P_{​​​​​​d}= 0.2 MPa= 0.2*1000000= 200000 N/m2}

^{P_{​​​​​​g}= 0.3 MPa= 0.3*1000000= 300000 N/m2}

^{h1=5 m}

^{h2=20+14.8= 34.8 m}

^{g= 9.81 m/s2}

Find out flow rate in each pump

First find out complete piping system flowrate Q then with the correlation given aboveawe can find each pump flow rates

Apply betnaulies equation at water surfaces at suction side and delivery side P_{​​​​​d} and P_{​​​​​​g} surfaces as shown in figure.

Assuming velocity head as zero as it is negligible at these points

Betnaulies equation is

P_{​​​​​​d}+$\rho$*g*h1+ Hf = Pg+$\rho$*g*h2

Where Hf= minor & major frictional loss in piping system

Hf=($\lambda$*L*V^{​​​​​​2}/2*g*d)+($\xi$*V^{​​​​​​2}/2*g)

= (0.03*250*V^{​​​​​​2}/2*9.81*0.25)+(10.2*V^{​​​​​​2}/2*9.81)

= 588.52 V^{​​​​​​2}

Substituting this value along with other parameters in betnaulies equation we get

200000+1000*9.81*5+588.52 V^{​​​​​​2}=300000+1000*9.81*34.8

Solving above equation for V we get

V= 27.1 m/s

Q= A*V=( π*0.25²​​​​​​/4)*27.1= 1.33 m^{​​​​​​3}/s

Substituting this value of Q in provided co relation we can get flow rate of each pump

Q= Q_{​​​​​​A}+Q_{​​​​​​B}

_{V= (2*g*$\Delta$h)^1/2}

_{Q= A*V}

V1= (2*g*10³*Q^{​​​​​​2})^1/2= 140.1Q

V2=(2*g*2*10³*Q^{​​​​​​2})^1/2=198.1Q

Q_{​​​​​​A}= (π* 0.25²/4)*140.1*1.33=9.14 m^{​​​​​​3}/s

Q_{​​​​​​B}=(π*0.25²/4)*198.1*1.33=12.74 m^{​​​​​​3}/s