Question

Consider the flow of oil with with ρ = 894 kg/m³ and μ = 2.33 kg/m.second and velocity of flow V = 0.5 m/second in a 300 m – long section of the pipeline of diameter 400 mm passes through the icy waters of a lake. Disregarding the entrance effects, what would be the pumping power required to overcome the pressure losses and to maintain the flow of oil in the pipeline if laminar flow exists?

1. 2.095 kW
2. 4.39 kW
3. 8.78 kW
4. 8.78 W
5.

Answer 1

Correct Answer - Option 2 : 4.39 kW

Concept-

loss of head due to friction in pipes could be written as -

\(h_{L}=\frac{flV^{2}}{2gD}\)

Where f is the friction factor 

L = Length of pipe, V = Flow velocity

d = Diameter of pipe

Friction factor can be determined by using Reynold's number.

For Laminar flow, Friction factor, f = 64/R_e

Where R_e = Reynold's Number

Power required, P = γQh_L

Where, Q = Discharge, h_L = Head loss in laminar flow

Calculation:

Given,

ρ = 894 kg/m³, μ = 2.33 kg/m-second

Velocity of flow = 0.5 m/s, Length of pipe (L) = 300 m, Diameter of pipe (d) = 400 mm = 0.4 m

Reynolds number is given by, \({{\mathop{\rm R}\nolimits} _e} = \frac{{\rho VD}}{\mu }\)

⇒ \({{\mathop{\rm R}\nolimits} _e} = \frac{{894 × 0.5 × 0.4}}{{2.33}} = 76.74\)

∵ Re < 2000, so flow is laminar flow and for laminar flow

Friction factor, f = 64/R_e = 64/76.74 = 0.834

In laminar flow the head loss is given by, \(h_{L}=\frac{flV^{2}}{2gD}\)

⇒ \({h_L} = \frac{{0.834 × 300 × {{0.5}^2}}}{{2 × 9.81 × 0.4}} = 7.97\ m\)

Power required, P = γQh_L

Discharge, Q = AV = (π/4) × 0.4² × 0.5 = 0.06283 cumec

Power, P = γQhL = 894 × 9.81 × 0.06283 × 7.97 = 4391.821 W

P = 4.39 kW