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3. When a real incompressible fluid flows through a circular pipe, energy is dissipated due to the viscosity of the fluid. The Moody diagram on page 9 represents this energy loss as a dimensionless friction factor (f) which is a function of the Reynolds number of the flow (Re) for both laminar and turbulent flow and also a function of the relative roughness (=/D) for turbulent flow. a) Explain this dependence of friction factor (f) upon the relative roughness (E/D) for turbulent flow and specifically why the friction factor increases with relative roughness at any given Reynolds number. [5 marks] Water with a density of 1000 kg/m³ and dynamic viscosity of 1.0 x 10³ Pa.s flows under gravity from a reservoir through a cast iron pipe of 75mm internal diameter and an equivalent roughness of 0.26mm at a flow rate of 600 litres per minute into the local atmosphere. The flow path comprises a sharp edged entrance from the reservoir into the pipe (loss factor (KL) of 0.5) and a 100m horizontal length of the cast iron pipe. There is no fitting or restriction at the outlet of the pipe into the local atmosphere and so no additional minor head loss. The liquid surface of the reservoir is exposed to the local atmosphere. b) Calculate the major head loss (hL) and minor head loss (hm) in the flow path and the height of water in the reservoir required above the sharp edged entrance into the pipe to achieve the required flow rate. [8 marks] c) If the height of water in the reservoir above the sharp edged entrance to the pipe and the pipe diameter and length are fixed, propose two other ways to increase the flow rate from the reservoir, evaluate their relative effectiveness and state which is the best option. Steady, uniform, and laminar flow of a fluid with dynamic viscosity (n) occurs between two horizontal, infinite, parallel plates separated by a distance (h) in the vertical direction (y). The lower plate (y=0) is stationary and the upper plate (y=h) moves with velocity (U) in the direction of flow (x). The vertical coordinate (y) where the maximum velocity (u) occurs, (y'), is given by below equation. Assume fluid of dynamic viscosity 0.5 Pa.s passes between the two plates which are 20mm apart with a pressure difference per unit length in the (x) direction of -500 Pa/m. h Undp hdx, 2 [5 marks] d) Calculate what happens to (y') as the upper plate velocity (U) increases from 0 (stationary) to 0.1 m/s and then to 0.2 m/s. With the aid of sketches, provide a physical explanation for this behaviour. [7 marks]