Examples Paper Questions

Engineering Tripos Part IB SECOND YEAR

Part IB Paper 8: - ELECTIVE (2)

Mechanical Engineering for Renewable Energy Systems

Examples Paper 1

All questions are of Tripos standard

1. (a) Sketch the form of the Hockey-stick curve used to illustrate the change in mean atmospheric CO2 concentration (see handout for Lecture 1). Look on the web and find a similar curve for the variation of mean global surface temperature (historically stable at around 13.5 oC).

(b) Models for climate change predict the mean global surface temperature and atmospheric CO2 concentration to rise to “crisis” levels of of 16 oC and 500ppm respectively. On your sketches indicate an estimate for time taken to reach crisis point and discuss factors that might influence this estimate.

(c) Explain what is meant by an individual’s Carbon Footprint. The average footprint for the UK is around 12,000 kg CO2 /year per person. How does your own footprint compare with this? You may use any data that you have at hand, or else assume that the average person’s CO2 output is distributed equally between the following 6 categories: i)heating, ii)electricity, iii)petrol, iv)food, v)travel, vi)clothes/furniture/appliances/etc.

(d) Explain briefly but quantitatively why, of all renewable energy options, wind power is probably the best option for the UK. Can the UK provide for itself by renewables alone?

2. (a) Sketch the idealized flow around a horizontal-axis wind turbine as used for deriving the Betz limit for the turbine. On your sketch show the upstream velocity Vo, the velocity u through the rotor plane and the downstream velocity u1.

(b) Define the axial induction factor a .

(d) Explain why the induction factor cannot sensibly take a value above 0.5 for idealized flow.

[Hint for (c): you would be expected to take the following steps:

(i) apply Bernouilli’s equation to the flow upstream and downstream of the turbine to determine the pressure drop across the turbine in terms of Vo – u1 ;

(ii) apply momentum principles to a suitable control volume to determine the turbine’s thrust and hence show that u1 = (1 – 2a)Vo.

(iii) find the power of the turbine and normalize this with the power available to a cross section equal to the swept area of the rotor to obtain the power coefficient CP = 4a(1 – a)2.

(iv) find the maximum value for CP .]

3. The current installed wind power capacity in the UK is around 2.2 GW, with perhaps another 10 GW in the planning stage, far greater than any other renewable energy source. Explain briefly the advantages of wind power that have led to this leading position in comparison with other renewable technologies. What are the main barriers to further development of wind power? Are there wave or tidal power solutions that could realistically approach the same or higher capacity in the UK?

(Note: refer to the Powerpoint slides on this topic at the end of Handout 2, but see what else you can find out from the Websites listed on the renewable energy pages linked from www.hughhunt.com )

4. An engineer replaced the original wooden-framed windows in his 1930s house using double-glazing throughout. One year later, a colleague refers him to an article in a leading newspaper claiming that the energy payback period for double glazing may be 50 years or more. As a user of the Cambridge Engineering Selector, he decides to assess this for himself. His gas bills show an energy consumption of 30,000 and 28,000 kWh for the year before and the year after installation, respectively. He estimates the area of window and the perimeter of window frame in the house, consults the technical specifications of the double glazing manufacturer, and looks up material data in CES. The resulting data are as follows:

Estimates: area of window = 24m2, length of window frame = 110m, thickness of glass = 3mm

Typical window frame construction (cross sectional areas):
PVC extrusion: 1200mm2 Al extrusion: 240mm2 Silicone rubber seal: 60mm2

Data from CES:

Material production energy (MJ/kg) / Density (kg/m3)
Glass / 13 – 14.4 / 2465
PVC / 63.5 – 70.2 / 1440
Al alloy / 184 – 203 / 2700
Silicone rubber / 152 – 168 / 1550

(a) Estimate the total mass of glass, PVC, Al alloy and silicone rubber in the complete set of double glazed windows for the house.

(b) Using mean values for the material production energy per kg, estimate the total material production energy that this embodies. Add an allowance for the manufacturing energy, assuming on average that this is 10% of the material production energy.

(c) Estimate the energy payback period, on the basis of the energy saving apparent in the gas bills. Comment on any underlying assumptions in this analysis. Why might it be unwise to assume that the energy payback period would be the same for an adjoining house of the same size?

(d) The total cost of the double glazing was £7,000. The unit cost of gas is given on the bill as 2.53p/kWh (+5% VAT). What is the financial payback period? Comment on the difference between the energy and financial payback periods.

5. A 5kW wind turbine is proposed for a property where wind conditions over a typical year are expected to be distributed as follows:

Wind speed (ms-1) / Proportion of time spent /

(the tabulated data is shown above as a histogram)
<3.5 / 1%
3.5 – 4.5 / 2%
4.5 – 5.5 / 4%
5.5 – 6.5 / 7%
6.5 – 7.5 / 11%
7.5 – 8.5 / 17%
8.5 – 9.5 / 23%
9.5 – 10.5 / 20%
10.5 – 11.5 / 11%
>11.5 / 4%

The diameter of the swept area is 5m and the inverter is set to deliver a maximum of 5kW to the grid.

(a) In a third column of the table estimate the electrical power delivered by the turbine for each wind-speed range. Assume that the turbine will operate at 75% of the Betz limit, and r = 1.2kg m-3 for air.

(b) The Capacity Factor is defined as the total electrical energy delivered over a year expressed as a proportion of the maximum possible for the installation operating at peak power output continuously. What is the Capacity Factor for the proposed installation ?

(i) less than the expected total energy;

(ii) more than the expected total energy.

(d) The material breakdown for the 5kW wind turbine is given in the Table below, together with the Material Production Energy taken from CES. Using mid-range values for material energy, and again making a 10% allowance for manufacturing energy, estimate the payback period for the turbine.

Mass (kg) / Material Production Energy (MJ/kg)
Concrete / 19,500 / 1.8 – 2.0
Steel / 868 / 22.4 – 24.8
Copper / 9.6 / 63 – 69.7
Composites / 34 / 110 – 120

Questions 6 – 9 concern a horizontal axis wind turbine with an upwind rotor and three blades based on the NACA 0012 symmetric aerofoil. The structural section of the blades will be manufactured from GFRP with a tensile failure stress of = 280 MPa, a shear failure stress of = 60 MPa and a Young’s Modulus E = 50 GPa. For simplicity the blade may be considered to be divided up into three 6m sections with geometry as given in Table 1.

Table 1: Blade details

Radial section / r = 2 to 8 m / r = 8 to 14 m / r = 14 to 20 m
Mid-point r / 5 m / 11 m / 17 m
Chord c / 1.5 m / 1.2 m / 0.6 m
Twist / 18˚ / 4.5˚ / 0.5˚
Structural depth / 0.2 m / 0.16 m / 0.08 m
Structural width b / 0.5 m / 0.4 m / 0.2 m
Mass per unit length / 200 kg/m / 150 kg/m / 50 kg/m

Storm loading

6. In storm conditions the wind turbine will be braked and the blades parked. The design storm loading is a peak wind velocity = 50 m/s which should be assumed to act perpendicular to the chord of the blades.

(a) Find the thickness t of GFRP necessary in the structural part of the blade root (at r = 2m) so that the bending stresses do not exceed the material tensile strength. Neglect the twist of the blade cross-section.

(b) Based on a calculation of longitudinal shear flow, estimate the thickness of the structural webs necessary so that the shear stresses at the blade root do not exceed the material shear strength. Comment on whether this calculation is likely to be the governing factor for the design of the structural webs.

Typical operational aerodynamic loading

7. In typical operational conditions with a wind speed of = 8 m/s the wind turbine rotates with an angular velocity = 30 rpm. Model the blades as the three uniform 6 m long parts given in Table 1.

(a) Use the Blade Element Momentum theory to calculate:

(i) the induction factors a and a’ at the mid-point of each element of the blade;

(ii) the load distribution on the blade (in normal and tangential components).

(b) Hence determine:

(i) the “flapwise” bending moment at the blade root (@ r = 2m);

(ii) the total axial thrust at the turbine hub;

(iii) the tangential bending moment at the blade root (@ r = 2m);

(iv) the total torque at the blade hub (@ r = 0);

(v) the total mechanical shaft power P of the turbine;

(vi) the coefficient of performance .

Data for air: Density =1.225 kg/m3 , Dynamic viscosity = 18 × 10-6 Ns/m2

8. Estimate the “flapwise” tip deflection of the blade due to the aerodynamic normal load distribution determined in Question 7. You should assume that the structural skin thicknesses t and are constant along the length of the blade and are equal to your answers to Question 6(a) and (b). Neglect the contribution of the aerofoil skin to the stiffness. Describe qualitatively how centripetal effects would change this answer.

Operational stresses in blade root: aerodynamic, centripetal and gravitational

9. Calculate the stresses induced at the blade root under the typical operational conditions considered above (= 8 m/s, = 30 rpm).

(a) Find the maximum bending stress due to the “flapwise” bending moment caused by the normal load distribution .

(b) Find the maximum bending stress due to the tangential bending moment caused by the tangential load distribution .

(d) Find the maximum bending stress due to self-weight loading.

(e) Hence find the location in the blade root structural cross section that experiences the highest tensile stress. What is the mean stress and stress range during normal operation at this location ?

Answers

4. (a) 360kg, 190kg, 70kg, 10kg

(b) 5.4GJ, 14GJ, 15GJ, 1.8GJ

5. (a) 0.14, 0.34, 0.65, 1.13, 1.80, 2.68, 3.82, 5, 5, 5.(in kW) (b) 68% (d) 8 months

6 (a) t = 14 mm, (b) = 2 mm. This is rather thin. It is likely that the requirement to prevent buckling of this web under compressive and/or shear stresses will require a greater thickness.

7 (a) Table 2: Results of BEM analysis

Radial section / r = 2 to 8 m / r = 8 to 14 m / r = 14 to 20 m
Mid-point r / 5 m / 11 m / 17 m
Axial induction factor a / 0.114 / 0.203 / 0.165
Angular induction factor a’ / 0.024 / 0.008 / 0.003
Flow angle / 23.8˚ / 10.4˚ / 7.1˚
Angle of attack / 5.8˚ / 5.9˚ / 6.6˚
Coefficient of lift / 0.63 / 0.64 / 0.72
Coefficient of drag / 0.0074 / 0.0075 / 0.008
Normal force / 166 N/m / 585 N/m / 771 N/m
Tangential force / 71 N/m / 100 N/m / 87 N/m

b) i) 104 kNm; ii) 27.4 kN; iii) 14.4 kNm (@ r = 2m); iv) 52.5 kNm; v) P = 165 kW; vi) = 0.42 (71% of the Betz limit of 16/27)

8 Flapwise tip deflection = 2.15m. Centrifugal loading will reduce this deflection.

9 a) 84.5 MPa; b) 10.9 MPa; c) 14.1 MPa; d) 119 MPa; e) The maximum tensile stress occurs in one corner of the structural section of the blade root. This is the upwind and trailing edge corner. When the blade is horizontal to the ground with its leading edge heading upwards the tensile stress has a maximum value of 228 MPa. When the blade has rotated through 180˚ the stress at the same location is a minimum, and has a value of -9.37 MPa (i.e. a compressive stress).

HEMH/HRS/DDS April 2008