Fuel oil of dynamic viscosity 0.8kg/ms flows at 110litres/second through a 20cm diametercast iron pipe. The density of the fluid is 850kg/m3.(i) Calculate the Reynolds number of the flow (8 marks)(ii) State whether the flow is laminar or turbulent. (1 mark)(iii) Determine the friction factor. (4 marks)(iv) Estimate the head loss due to friction in a 100m length of the pipe taking into accountthat the system includes a valve. (10 marks)Take the loss coefficient K for the valve as 10.

(i) The Reynolds number (Re) is a dimensionless quantity that is used to predict the onset of turbulence in fluid flow. It is defined as the ratio of inertial forces to viscous forces and can be calculated using the formula:

Re = (ρvD)/μ

where: ρ = density of the fluid (kg/m³) v = velocity of the fluid (m/s) D = diameter of the pipe (m) μ = dynamic viscosity of the fluid (kg/m.s)

First, we need to convert the flow rate from litres/second to m³/s:

110 litres/second = 110 x 10^-3 m³/s = 0.11 m³/s

Next, we calculate the velocity of the fluid:

v = flow rate / cross-sectional area of the pipe v = 0.11 m³/s / (π/4 x (0.2 m)²) v = 3.5 m/s

Now we can calculate the Reynolds number:

Re = (850 kg/m³ x 3.5 m/s x 0.2 m) / 0.8 kg/m.s Re = 747500

(ii) The flow is considered to be laminar if Re < 2000, transitional if 2000 < Re < 4000, and turbulent if Re > 4000. Therefore, with a Reynolds number of 747500, the flow is turbulent.

(iii) For turbulent flow in a smooth pipe, we can use the Blasius equation to estimate the friction factor (f):

f = 0.316/Re^0.25 f = 0.316/(747500^0.25) f = 0.015

(iv) The head loss due to friction (hf) in a pipe can be calculated using the Darcy-Weisbach equation:

hf = f (L/D) (v²/2g)

where: L = length of the pipe (m) D = diameter of the pipe (m) v = velocity of the fluid (m/s) g = acceleration due to gravity (9.81 m/s²)

hf = 0.015 (100 m / 0.2 m) (3.5 m/s)² / 2(9.81 m/s²) hf = 26.6 m

The head loss due to the valve can be calculated using the formula:

h_valve = K (v²/2g)

h_valve = 10 (3.5 m/s)² / 2(9.81 m/s²) h_valve = 6.2 m

Therefore, the total head loss is:

hf_total = hf + h_valve hf_total = 26.6 m + 6.2 m hf_total = 32.8 m