Homework answers / question archive / Advanced Mechanical Engineering  MSc in Renewable Energy MSc in Offshore Engineering Fluid Mechanics and Loading - Assignment In this assignment you will write a short technical report on the following three topics (A and B)

Advanced Mechanical Engineering  MSc in Renewable Energy MSc in Offshore Engineering Fluid Mechanics and Loading - Assignment In this assignment you will write a short technical report on the following three topics (A and B)

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Advanced Mechanical Engineering  MSc in Renewable Energy MSc in Offshore Engineering Fluid Mechanics and Loading - Assignment
In this assignment you will write a short technical report on the following three topics (A and B). For each topic you will need to answer several questions by using concepts and methodologies you have studied during the module. A. Aerodynamic performance of a HAWT rotor (50% mark) In this section, you will describe a basic procedure of how to estimate the aerodynamic performance of a horizontal-axis wind turbine rotor, using an airfoil performance calculation tool (XFLR5) and the BEM theory. Specifically, here we consider a (simplified) 3-blade horizontal-axis rotor with a rotor radius R = 6m and a rotor diameter D = 2R = 12m. As illustrated in Figure 1, we assume that each blade is a simple rectangular wing (i.e. it has a rectangular planform and is not twisted or tapered). Each blade has a spanwise length of 5m and is positioned between r = 1m and 6m (where r is the radial coordinate with its origin located at the centre of the rotor). Each blade employs “NACA 0020” airfoil and its chord length is c = 1m. The angular velocity of the rotor is Ω = 6 rad/s. Figure 1: Schematic of a 3-blade horizontal axis rotor. You can assume that the natural (undisturbed) wind speed U∞ = 8m/s is uniform across the rotor area and the wind direction is perpendicular to the rotor. You can also assume that the density and viscosity of air are ρ = 1.2kg/m3 and µ = 1.8x10-5kg m-1s-1, respectively. Explain how to estimate the torque and thrust of this rotor by answering the following 4 questions: 1. First, you need to estimate a representative Reynolds number for the rotating blades. Calculate the Reynolds number based on the blade chord length (c = 1m) and the local total velocity Vtotal at the mid-span of the blade (i.e. at r = 3.5m). [6% mark] Hint: At this initial stage, you can assume that both axial and tangential induction factors (a and a’) are zero. Therefore Vtotal can be estimated from U∞, Ω and r. 2/4 2. Describe briefly the definitions of the lift and drag coefficients (CL and CD) of a 2D airfoil. Then, by using XFLR5, calculate CL and CD of NACA 0020 airfoil at Re = 1,500,000 for a wide range of angles of attack (0° < α < 30° with an increment of 1°). Export the calculated CL and CD data to Excel and make appropriate graphs to show the variations of CL and CD with respect to α. [12% mark] Hint: You can follow the XFLR5 tutorial document used during the module. 3. Now you can use this airfoil performance data set (CL and CD) with the BEM theory to calculate the rotor performance. For simplicity, you can assume that the tangential induction factor a’ is always zero. The pitch angle of each blade is fixed at β = 8°. Consider dividing each blade into 5 elements (i.e. dr = 1m) and perform the BEM procedure to obtain the total torque and thrust generated by this rotor. [20% mark] Also, based on the total torque obtained from the BEM procedure, calculate the power generated by this rotor and the power coefficient (CP) of the rotor. [4% mark] Hint: You should use either Excel or MATLAB to answer this question. Start the BEM procedure with the axial induction factor a = 0 for each blade element. Once you have calculated the thrust for each element (dT) using the airfoil data (CL and CD), you can update the value of a using the momentum theory. If necessary, you can assume CL = 1.25 (constant) and CD = 0.24 + 0.015 × (? − 30) for 30° < α < 45°. You will need to repeat the same procedure several times to obtain a sufficiently converged solution for each blade element. See the lecture slides (L4-3) for further details of the BEM procedure. 4. Finally, discuss (with words, using equations and calculations) why the performance of this rotor could be improved by twisting the blades. [8% mark] 3/4 B. Wave Theories: linear and non-linear (40% mark) In this section, you will be comparing the linear and non-linear wave theories, illustrating some key aspects. Make sure, whenever possible, to both write the relevant mathematical formulae and explain their physical meanings, using graphs if relevant. 1. Describe the six hypotheses on which Airy wave theory is based, including the formulae and their physical meanings. To satisfy these conditions, the amplitude of the waves has to be “small” with respect to which parameters? [10 marks] 2. Given the following expressions, relative to the Stokes 3rd order theory: i. Give the definition of each parameter (a, c, d, g, H, k, t, T, x, y, η, λ, Φ, ω), including a brief description and unit of measure [6 marks]. ii. Φ is expanded to the third order using a Fourier cosine expansion: explain which condition can be better satisfied using this approximation and how. [6 marks] iii. If only the terms (1) and (2) in the expression for H, λ, and c are considered, which theory and approximation are we using? Also, if terms (3) are considered, it can be considered the same theory but for which condition? If all the terms (including (4)) are considered, which wave theory is used? [6 marks] 3. Comment on the main differences between the two wave theories identified in (2.iii), considering: wave profile, wave particle orbit, and wave height. [12 marks] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) F F F f f f termsto take into account depth function of kd f kx t a f kx t a a kx t F k y d kx t F k y d kx t F k y d kx t k c , , , , , , cos3 cos2 cos cosh 3 sin 3 3 1 cosh 2 sin 2 2 1 cosh sin 1 2 3 1 2 3 2 3 2 3 2 2 3 2 1 ú ú ú ú ú ú û ù ê ê ê ê ê ê ë é + - + - + - + = ú ú ú ú ú ú û ù ê ê ê ê ê ê ë é + + - + + - + + - + F = w l p w l p w h w w w ! ( ) ( ) ( ) ( ) ( ) ! ( ) ( ) ( ) ú ú ú ú û ù ê ê ê ê ë é ÷ ÷ ø ö ç ç è æ + ÷ ø ö ç è æ = + ú ú ú ú û ù ê ê ê ê ë é ÷ ÷ ø ö ç ç è æ + = + = + %""""$""""# %"$"# %""""$""""# %"$"# %$# %""$""# (4) 4 2 2 (3) (2) 2 (4) 4 2 2 (3) (2) 2 (4) 2 3 2 3 (1) 16sinh 14 4cosh 2 2 tanh 1 16sinh 14 4cosh 2 tanh 1 2 2 2 kd ak kd kd k g c kd kd T kd ak g f kd a H a p l l p 4/4 Important notes: The main part of your report (short introduction, answers to the questions and conclusions) should not be longer than 5000 words in total. The cover page, table of contents, references etc. are excluded from this limitation. A template file is available on Blackboard (together with this document). You should also present an Appendix after the main part of your report. In the Appendix you can present snapshots of your spreadsheets, MATLAB codes, etc. This assignment will be marked based on the quality of your report. Reports presenting only the final answers (without describing the processes to reach those answers) will not receive high marks, even if the answers are correct. You can obtain up to 90% marks by answering the questions A and B. The remaining 10% marks will be given based on the overall quality of your report (layout, written English, correct use of notations, etc.). Please use the nomenclature/symbols used in the present assignment and in the lecture as much as possible. You can copy and use the figures in this document (Figures 1 and 2) in your assignment report. If you use figures from any other sources (such as books and lecture slides) you should cite the relevant references appropriately.