 # FLUID MECHANICS _________________________________________________...

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FLUID MECHANICS ______________________________________________________ Question 1 This question requires you to develop a simple model for the performance of a wind turbine based on actuator disc theory and use it to make a first estimate of rotor disc diameter in terms power generation requirements and wind speed. Consider the stream tube and actuator disc as shown in Figure 1. Figure 1. a) Assuming a 1-d incompressible, steady and inviscid flow field within the stream-tube; sketch graphs of how you expect the velocity and pressure to vary along the centreline from the controlvolumeinlet,throughthedisc,totheCVoutlet. [2marks] b) Apply the conservation of momentum to the CV to show that the power extracted from the flow by the turbine is given by: 􏰀 􏰁 􏰂􏰃 􏰄 􏰅 􏰆 􏰇 􏰅 􏰈 􏰉 􏰊 􏰋 􏰅 􏰌 . Here 􏰂􏰃 is the mass flow rate within the stream-tube. Explain carefully why pressure and the velocity outside of the stream- tube do not contribute to the power extracted and state any assumptions you make. [6 marks] c) By considering the rate at which kinetic energy within the stream- tube enters and leaves the CV show that an alternative expression for 􏰀 is: 􏰀􏰁􏰂􏰃􏰄􏰅􏰍􏰇􏰅􏰍 􏰋/2 􏰆 􏰈􏰉􏰊 and hence show that the velocity through the disc is given by: 1 􏰅􏰌 􏰁2􏰄􏰅􏰆􏰎􏰅􏰈􏰉􏰊􏰋. It is convenient to define an axial flow induction factor, 􏰏, defined as [9 marks] Show that the power extracted by the turbine is: 􏰒􏰍 􏰀􏰁2􏰏􏰐􏰑􏰌􏰅􏰆􏰄1􏰇􏰏􏰋 . Where 􏰑􏰌 is the actuator disc area. d) Define a power coefficient, 􏰓􏰔, as 􏰀 􏰓􏰔 􏰁 􏰀 , 􏰕 􏰅􏰆 􏰇 􏰅􏰌 􏰏􏰁􏰅. 􏰆 where 􏰀􏰕 􏰁 􏰐􏰑􏰌􏰅􏰆􏰒 /2 is the rate of KE within an unperturbed freestream stream-tube having the same area as the actuator disc. I. Explain why 􏰀􏰕 represents the maximum power available to the wind-turbine. 􏰍 II. Showthat􏰓􏰔 􏰁4􏰏􏰄1􏰇􏰏􏰋 . III. Deduce that the maximum power coefficient is given by 􏰓􏰔,􏰖􏰗􏰘 􏰁 􏰙􏰚 􏰁 0.593. [2 marks] 􏰍􏰛 e) In reality 􏰓􏰔 will be somewhat less than 􏰓􏰔,􏰖􏰗􏰘 and a typical value is around 0.35. Additionally, the electrical power output will be further decreased by drive train and generator inefficiencies with a typical efficiency factor of 0.6. Assuming these values, determine the diameter of rotor required to generate 20kW of electrical power in a steady wind of 8m/s. 3 (You may take the density of are as 1.2kg/m .) [3 marks] [1 mark] [2 marks] Question 2 A Journal Bearing comprises a solid shaft that rotates inside a cylindrical sleeve with a small oil filled gap between them. You made a simple analysis of this problem in CW1 where we assumed that the centrelines of the shaft and sleeve were aligned. By “unwrapping” the annulus you can show that the imposed velocity profile of the oil is linear, (Fig. 1). Figure 1. Flow in the annular space between two concentric cylinders. In practice the Journal bearing must support the weight of the inner shaft. In CW1 you showed that the pressure is constant and the viscous stress acts circumferentially, which implies a concentric arrangement cannot generate a vertical force to support the shaft. One way of overcoming this is to offset the centrelines of the shaft and sleeve slightly. To see how this provides a supporting force consider the simpler slider bearing arrangement shown in Fig.2. This comprises a slider separated from a fixed plate by a thin film of oil. Notice that the gap between the slider and plate varies. Figure 2. Slider Bearing. Assuming that the flow is steady, 2-d, and incompressible, the continuity and Navier-Stokes equations governing the flow are: 􏰜􏰝􏰙 􏰎 􏰜􏰝􏰍 􏰁 0 􏰜􏰞􏰙 􏰜􏰞􏰍 􏰜􏰝􏰙 􏰜􏰝􏰙 1 􏰜􏰟 􏰜􏰍􏰝􏰙 􏰜􏰍􏰝􏰙 􏰝􏰙􏰜􏰞 􏰎􏰝􏰍􏰜􏰞 􏰁􏰇􏰐􏰜􏰞 􏰎􏰠􏰜􏰞􏰍 􏰎􏰠􏰜􏰞􏰍 􏰙􏰍􏰙􏰙􏰍 􏰜􏰝􏰍 􏰜􏰝􏰍 1 􏰜􏰟 􏰜􏰍􏰝􏰍 􏰜􏰍􏰝􏰍 􏰝􏰙􏰜􏰞 􏰎􏰝􏰍􏰜􏰞 􏰁􏰇􏰐􏰜􏰞 􏰎􏰠􏰜􏰞􏰍 􏰎􏰠􏰜􏰞􏰍 􏰙􏰍􏰍􏰙􏰍 To non-dimensionalise these equations put: 􏰞 􏰁 􏰡􏰞̅ , 􏰞 􏰁 􏰢 􏰞̅ 􏰙􏰙􏰍􏰣􏰍 􏰝 􏰁 􏰅 􏰝􏰤 , 􏰝 􏰁 􏰅 􏰝􏰤 􏰙􏰙􏰍􏰣􏰍 􏰟 􏰁 􏰟􏰣􏰟̅ where 􏰢􏰣 􏰁 􏰥􏰄􏰢􏰋 is characteristic of the gap width and 􏰅􏰣 and 􏰟􏰣 are to be determined. 􏰢􏰣 is very small compared to 􏰡 so 􏰦 􏰁 􏰢􏰣/􏰡 is a small parameter. (a) By considering the order of magnitude of terms in the continuityequationdeducethat􏰅􏰣 􏰁􏰧􏰨􏰩 􏰁􏰦􏰅. 􏰪 [2 marks] (b) By considering the order of magnitude of the terms in the first N-S equation show that we must put 􏰟􏰣 􏰁 􏰫􏰧 and that to leading order the equation becomes: 􏰜 􏰍 􏰝􏰤 􏰜 􏰟 ̅ 􏰙 􏰜􏰞̅􏰍 􏰁􏰜􏰞̅ . 􏰍􏰙 􏰪􏰬􏰭 􏰮4 marks􏰯 (c) Now non-dimensionlise the second N-S equation and by considering the order of magnitude of the terms, deduce integrable.) (d) Integrate the equation, and, stating the b/cs you use, show that when written in dimensional units: 􏰞􏰍 1􏰴􏰟􏰍 􏰝􏰙􏰄􏰞􏰙,􏰞􏰍􏰋􏰁􏰱1􏰇 􏰢􏰲􏰅􏰎2􏰳􏰴􏰞 􏰄􏰞􏰍 􏰇􏰞􏰍􏰢􏰋 􏰙 (where you should remember that 􏰢 􏰁 􏰢􏰄􏰞􏰙􏰋􏰋. (e) Use the fact that the volumetric flow rate, 􏰵 􏰁 􏰶􏰨 􏰝􏰙􏰄􏰞􏰙, 􏰞􏰍􏰋􏰴􏰞􏰍 is constant to show that 􏰴􏰟 6􏰳􏰅 12􏰳􏰵 􏰴􏰞 􏰁 􏰢􏰍 􏰇 􏰢􏰒 . 􏰙 that to leading order it becomes 􏰟̅ can only depend on 􏰞̅ and the equation in part (c) is 􏰙 􏰰􏰔̅ 􏰰􏰘̅􏰭 􏰁 0. (This means that [4 marks] 􏰕 [4 marks] Appropriate b/cs for this equation are that 􏰟􏰄0􏰋 􏰁 􏰟􏰄􏰡􏰋 􏰁 􏰟􏰕 where 􏰟􏰕 is the constant (small) pressure away from the slider. By integrating it can be shown we obtain 􏰘􏰸 􏰴􏰞􏰙 􏰘􏰸 􏰴􏰞􏰙 􏰟􏰄􏰞􏰙􏰋􏰇􏰟􏰕 􏰁6􏰳􏰅􏰷 􏰢􏰍 􏰇12􏰳􏰵􏰷 􏰢􏰒 􏰕􏰕 And by noting that 􏰟􏰄􏰡􏰋 􏰇 􏰟􏰕 􏰁 0 it can be shown we obtain [2 marks] the solution is: 􏰞􏰙 􏰢􏰄􏰞􏰙􏰋 􏰁 􏰢􏰙 􏰎 􏰄􏰢􏰍 􏰇 􏰢􏰙􏰋 􏰡 , 6􏰳􏰅􏰡􏰍 􏰢􏰙 2􏰄􏰢􏰙 􏰇 􏰢􏰍􏰋 􏰶􏰪 􏰴􏰞􏰙 􏰕 􏰢􏰍 􏰵􏰁 􏰪􏰴􏰞. 2􏰶􏰙 􏰕 􏰢􏰒 Thus once the gap width, 􏰢􏰄􏰞􏰙􏰋, is specified we can integrate to obtain the pressure and hence find the total upward force on the bearing which is given by 􏰹􏰍 􏰁 􏰶􏰪􏰮􏰟􏰄􏰞􏰙􏰋 􏰇 􏰟􏰕􏰯􏰴􏰞􏰙. For the 􏰕 case where the gap width is a linear function: 􏰹􏰍􏰁􏰄􏰢􏰇􏰢􏰋􏰍􏰺ln􏰢􏰇 􏰢􏰎􏰢 􏰻. 􏰙􏰍􏰍􏰙􏰍 (If you are brave and like complicated integrals you might like to show this!) (f) Assuming that 0 􏰼 􏰄􏰢􏰙 􏰇 􏰢􏰍􏰋 ≪ min􏰄􏰢􏰙, 􏰢􏰍􏰋 deduce that a very large upward pressure force, 􏰹 􏰽 􏰫􏰧􏰪􏰭 􏰄􏰢 􏰇 􏰢 􏰋, 􏰍 􏰍􏰨􏰭􏰾 􏰙 􏰍 results. (Hint: Write 􏰿 􏰁 􏰢􏰙 􏰇 􏰢􏰍 as a small parameter and expand the terms in square brackets as Taylor series as 􏰒 far as 􏰿 .) [4 marks] (g) What happens if the direction of travel of the slider, 􏰅, is in the opposite direction? Discuss qualitatively the problematic implications of this for the load bearing capability of a Journal Bearing where the shaft axis is vertically displaced from the axis of the sleeve. Suggest some design changes to overcome this difficulty. [5 marks] To help you may wish to look up Slider Bearings and Journal Bearings in appropriate textbooks. There is also a wealth of information on both available online. (You will notice that in discussing the vertical force we considered only the pressure. If can be shown that the normal viscous stress component, 􏱀􏰍􏰍, is indeed very small compared to the pressure term.)

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