Question

A light oil is flowing through a commercial steel pipe of diameter D1. The head available to produce flow is independent of the flow rate. It is decided to utilise this head to increase the oil flow rate fourfold by installing a second pipe in parallel with the first pipe and of the same length. What should the diameter of the second pipe be, if:

(i) flow in each of the two pipes is highly turbulent and the same roughness ratio applies to both pipes;

(ii) flow in each of the two pipes is laminar.

Hints: Write the equation that relates the head loss to the friction factor, length, diameter and velocity for one pipe (like the Fanning pressure drop equation, just in head loss form) (2 marks).

Also write down the continuity equation for the volumetric flow rate, the average velocity and the diameter (2 marks).

Write down these equations for the old and new situations. How does the head loss change between the two situations? How does the head loss differ between the two pipes in parallel? (3 marks)

Use these considerations to develop an equation that relates the friction factor, the average velocity and the diameter between the two pipes (or the ratio of the two friction factors in terms of the ratio of the two pipe diameters and the ratio of the two velocities in the two pipes). (3 marks)

Then, using the continuity equation, develop an equation that relates the friction factor and the diameter between the two pipes (or the ratio of the two friction factors in terms of the ratio of the two pipe diameters). (3 marks)

(i) For highly turbulent flow at the same ε/D, then how does the friction factor differ between the two pipes? How does this relationship help to define the ratio of friction factors between the two pipes in terms of the ratios of the two diameters and the two velocities (in the two pipes)? (3 marks)

(ii) For laminar flow, what is the relationship between the friction factor and the Reynolds number? How does this relationship help to define the ratio of friction factors between the two pipes in terms of the ratios of the two diameters and the two velocities (in the two pipes)? (7 marks)

In both cases (i) and (ii), the ratio of friction factors helps to give the ratio of the two pipe diameter

Answer 1

The preesure head avilable for flow was given to be independendent of the flow rate , the head loss is equal in both arrangements ( single pipe and two pipe in parallel arrangement ) . In highly turbulent flow the friction factor is independent of reynolds number so can be assumed constant but in laminar flow the friction factor varies with reynolds number as f = 16/Re where f is the fanning friction factor , these along with the continuity equation yeilds the relation between the diameters of first and second pipe. when flow is turbulent D2 = 1.55D1 and whwn flow is laminar D2 = 1.316D1.