Fluid Mechanics and Thermodynamics — June/July 2024

Question 1(a)

Define the following terms as applied to reciprocating pumps: (i) co-efficient of discharge; (ii) slip.

(4 marks)

Answer

Question 1(b)

With the aid of a sketch, explain the operation of the reciprocating pump.

(6 marks)

Answer

The rotating crank moves the piston back and forth. On the suction stroke, cylinder volume increases and pressure falls: the suction valve opens and the delivery valve remains closed, drawing water into the cylinder. On the return delivery stroke, cylinder volume decreases and pressure rises: the suction valve closes and the delivery valve opens to discharge water. The valves prevent reverse flow, so the reciprocating motion produces a net delivery.

Question 1(c)

A single acting reciprocating pump has a piston diameter of 300 mm and a stroke of 400 mm. The pump centre is 4.5 m above the water surface in the borehole and delivers water 25 m above. Both the suction and delivery pipes have the same diameter of 150 mm and are 8 m and 25 m long respectively. If the pump is working at 40 rpm, determine: (i) the pressure head at the beginning, middle and end of the suction stroke; (ii) the power required to drive the pump. Take density of water ρ = 1000 kg/m³.

(10 marks)

Answer

For the formal examination calculation, assume simple harmonic piston motion, negligible friction, no air vessel, negligible slip and g = 9.81 m/s². Pressure heads below are gauge heads at the pump cylinder, with the standard acceleration-head approximation.

Physical check: the calculated beginning-of-stroke gauge head is below the vacuum limit for an ordinary atmospheric suction supply. It would imply an impossible negative absolute pressure at normal atmospheric pressure, so the assumed continuous water column would separate or cavitate. The figures are formal acceleration-head results, not feasible operating pressures for this arrangement at 40 rpm. The delivery-height datum is not stated clearly, and motor input additionally requires mechanical efficiency and friction data.

Question 2(a)

(i) State two types of losses in a pipeline. (ii) Water is discharged from a reservoir into the atmosphere through a pipe of length 45 m. The entrance into the pipe is sharp and the pipe is 15 m long and 100 mm diameter. The pipe then suddenly enlarges to 225 mm diameter for the rest of its length. Taking the frictional coefficient f as 0.0048 and 0.0058 for the smaller and larger pipes respectively, determine the difference of height between the surface of the reservoir and the pipe exit which will maintain a flow of 0.0065 m³/s.

(10 marks)

Answer

(i) Major loss: wall-friction loss along a straight pipe. Minor/local losses: entrance, exit, bends, valves and changes of section.

(ii) Take the first section as 15 m and the second as 45 − 15 = 30 m. Adopt the traditional examination convention hf = 4f(L/D)V²/(2g). A sharp entrance is represented by K = 0.5. Apply Bernoulli from the reservoir surface to the free jet at the pipe exit.

The exit kinetic head is included once; do not also add an exit loss for the same free-jet boundary. If f denotes a Darcy factor instead, omit the factor 4 in both friction terms: H = 0.067486 m. State the friction-factor convention.

Question 2(b)

Show by dimensional analysis that the equation relating the variables in the rotation of similar discs in a fluid in which the fluid motion is turbulent is given by T = ρD⁵N² φ{μ/(ND²)}. Where T = frictional torque; D = disc diameter; N = rotational speed; ρ = mass density; μ = fluid viscosity.

(10 marks)

Answer

The printed function argument μ/(ND²) is not dimensionless if μ is dynamic viscosity: it has density units. The missing density must be included, as shown. If the symbol instead denotes kinematic viscosity ν = μdynamic/ρ, then ν/(ND²) is dimensionless and the printed form is valid with that different definition.

Question 3(a)

Show from first principles that the pressure loss due to laminar flow in a pipe is given by ΔP = 32μL V̄/D². Where μ = fluid viscosity; L = length of the pipe; V̄ = mean velocity of flow; D = pipe diameter.

(16 marks)

Answer

Assume steady, incompressible, fully developed laminar flow of a Newtonian fluid in a horizontal circular pipe, with no slip at the wall and constant viscosity. Let the radius be R = D/2 and the positive pressure drop over length L be ΔP.

Question 3(b)

A pipe of 110 mm diameter and 120 m long conveys oil of viscosity 0.45 kg/(m s). If the pressure difference at the ends of the pipeline is 65 kN/m², calculate the quantity of oil flowing.

(4 marks)

Answer

Oil density is not supplied, so a numerical Reynolds-number check cannot be completed from the stated data alone.

Question 4(a)

A rigid vessel contains 0.8 m³ of dry saturated steam at 3.5 bar. The vessel is cooled until the pressure is reduced to 2 bar. Calculate: (i) the mass of steam in the vessel; (ii) the final dryness fraction of the steam; (iii) the amount of heat transferred during the cooling process.

(8 marks)

Answer

Treat pressures as absolute. The closed rigid vessel has constant mass and volume; boundary work is zero. Use internal energy, not enthalpy, for its heat balance.

Question 4(b)

Two rows of a velocity compounded impulse turbine have a mean blade speed of 150 m/s and a nozzle steam velocity of 675 m/s. The nozzle angle is 20°. The exit angles of the first moving row, fixed row and second row of moving blades are 25°, 25° and 30° respectively. There is 10% loss of velocity due to friction in all blades. The steam flow rate is 5.2 kg/s. Determine the: (i) power output; (ii) diagram efficiency.

(12 marks)

Answer

Measure nozzle and blade angles from the wheel tangent. Let C be absolute velocity, W relative velocity and U = 150 m/s blade speed. Positive whirl is in the direction of blade motion. Moving blades reduce relative speed by a factor 0.9; the fixed row reduces absolute speed by 0.9.

Question 5(a)

A quantity of gas has an initial pressure, volume and temperature of 1.3 bar, 0.224 m³ and 21° respectively. The gas is compressed until the final volume is 0.028 m³ according to the law PV¹·³ = constant. Calculate the change in entropy. Take Cv = 0.718 kJ/(kg K), Cp = 1.005 kJ/(kg K), R = 0.287 kJ/(kg K).

(4 marks)

Answer

Question 5(b)

In a generating set of a gas turbine, the low pressure turbine drives the low pressure compressor and an alternator, while the high pressure turbine drives the high pressure compressor. The gas temperature at entry to both turbine stages is 600°C and the ambient air pressure is 20°C. The pressure ratio of each compressor is 3:1. The mass flow rate is 115 kg/s and the isentropic efficiencies of the compressors and turbines are 0.82 and 0.9 respectively. A heat exchanger of effectiveness 0.7 is fitted in the system. Take γ = 1.332 and Cp = 1.14 kJ/(kg K) for gases and γ = 1.4 and Cp = 1.005 kJ/(kg K) for air. Calculate: (i) net power output; (ii) thermal efficiency.

(16 marks)

Answer

The printed “ambient air pressure is 20°C” is interpreted as ambient temperature. Assume negligible fuel mass and pressure losses, equal stream mass flow, mechanical efficiency 1, reheat to 873.15 K, and no intercooler because none is stated. A separate perfect-intercooling result is supplied below. The high-pressure turbine expansion ratio is set by its compressor work balance; it is not automatically 3.

The second line is conditional: perfect intercooling is extra information, not stated in the printed question.

Question 6(a)

During an engine trial, the volumetric analysis of the dry flue gas gave the following results: CO₂ = 13%; CO = 0.3%; O₂ = 6%; N₂ = 80.7%. The ultimate analysis of the flue gases was reported as follows: C = 62.4%; H₂ = 4.2%; O₂ = 4.5%; moisture = 15%; ash = 13.9%. Calculate the: (i) minimum air required to burn 1 kg of fuel; (ii) actual mass of air supplied per kg of fuel; (iii) amount of excess air supplied per kg of fuel. Take atomic mass as C = 12, O₂ = 16 and H₂ = 1.

(10 marks)

Answer

Interpret the C/H/O/moisture/ash percentages as the ultimate fuel analysis: ash and fuel moisture do not describe a dry-gas composition. Use conventional dry air with 23% oxygen and 77% nitrogen by mass; neglect fuel nitrogen. The given 16 and 1 are atomic masses, so molecular masses O₂ and H₂ are 32 and 2.

The nitrogen balance uses all supplied dry-gas readings without assuming CO is absent. Different declared standard air-composition constants give slightly different numerical values.

Question 6(b)

A two stage single acting reciprocating air compressor is to be designed to take air at 0.95 bar and 22°C, and deliver at 13 bar. It is anticipated that the volumetric efficiencies of both the high and low pressure stages will be 83% and the mechanical efficiency will be 85%. If the air mass flow rate is to be 0.017 kg/s and the compressor runs at 400 rev/min, calculate: (i) power required to drive the compressor; (ii) high pressure and low pressure cylinder swept volumes. Take the polytropic index as 1.4.

(10 marks)

Answer

Use absolute pressures, Rair = 0.287 kJ/(kg K), perfect intercooling back to 22°C and equal stage pressure ratios for the minimum-work design. Intercooler losses are neglected. These standard design assumptions are necessary because the intermediate pressure and intercooler performance are not otherwise specified.

Question 7(a)

A steel pipe carrying a process fluid has a bore of 150 mm and thickness 5 mm. The pipe is covered with a 50 mm layer of insulation of thermal conductivity 0.05 W/(m K). The thermal conductivity of the steel may be taken as 45 W/(m K) and the surface heat transfer coefficient as 35 W/(m² K) for the inner surface of the pipe and 7 W/(m² K) for the outside surface of the insulation. Assuming the process fluid temperature to be constant at 300°C and that of the ambient as 16°C, determine: (i) heat loss per length of the pipe; (ii) outside surface temperature.

(10 marks)

Answer

Question 7(b)

At the beginning of compression of an ideal diesel cycle, the gas temperature and pressure are 25°C and 1 bar. The volume ratio of compression is 16:1. The maximum cycle temperature is 1500°C. Determine the thermal efficiency of the cycle. Take Cp = 1.004 kJ/(kg K) and γ = 1.4.

(10 marks)

Answer