Question

Title: Mathematics in Wind Energy Production Wind power is defined as the use of air flow through wind turbines to provide the mechanical force to generate electricity. Wind power is an alternative to burning fossil fuels, and is renewable and produces no greenhouse gas emissions during operation. Modern horizontal-axis wind turbines often use three blades. Theoretically, the maximum power P (unit: watts) that a three-blade wind turbine can extract from the wind power can be calculated as: P = 1/2 ? A v^3 C_p .......................... (Equation 1) where ? is the air density (kg/m^3), A is the sweep area of the turbine (m^2) and can be calculated from the length of the turbine blades, and v is the wind speed (m/s). C_p is the power coefficient that is unique to each turbine type. This coefficient represents the amount of kinetic energy from the wind that is captured by the turbine. From the Betz's limit law we know that the best power conversion possible is C_p,max ? 0.59. Part 1 Given the following data: Blade length l = 52 m Air density ? = 1.23 kg/m^3 Power coefficient C_p = C_p,max = 0.59 Plot the power output P as a function of wind speed v on the interval [0 m/s, 30 m/s]. Part 2 Many factors impact the final output of wind power. The rotor-blade friction and drag, gearbox losses, generator and converter losses, all reduce the power delivered by a wind turbine. In 2001, commercial utility-connected turbines deliver 75% to 80% of the Betz limit C_p,max of power extractable from the wind, at rated operating speed, that is, C_p = 0.40. Create a second plot of power output as a function of windspeed using the original value of C_p,max = 0.59 and the value of C_p = 0.40 (68% of the previous value) together over the interval of velocities [0 m/s, 30 m/s]. Part 3 In the above parts we have considered C_p as a constant quantity. This is not, however, entirely accurate. In reality power conversion will be more or less efficient and depend on wind velocity. Consider a power coefficient curve C_p(v) = 0.0026 v^3 - 14 v^2 + 180 v - 20000. At what velocity does C_p reach its maximum value and what is the maximum value of C_p? Part 4 Using the power coefficient curve C_p(v) from part 3 find the power P of a turbine (using the other constants from Part 1) as a function of velocity. Plot this power function over the interval [0 m/s, 30 m/s]. Part 5 At what velocity is the power function from part 4 maximized? 

          Title: Mathematics in Wind Energy Production

Wind power is defined as the use of air flow through wind turbines to provide the mechanical force to generate electricity. Wind power is an alternative to burning fossil fuels, and is renewable and produces no greenhouse gas emissions during operation.

Modern horizontal-axis wind turbines often use three blades. Theoretically, the maximum power P (unit: watts) that a three-blade wind turbine can extract from the wind power can be calculated as:

P = 1/2 ? A v^3 C_p .......................... (Equation 1)

where ? is the air density (kg/m^3), A is the sweep area of the turbine (m^2) and can be calculated from the length of the turbine blades, and v is the wind speed (m/s). C_p is the power coefficient that is unique to each turbine type. This coefficient represents the amount of kinetic energy from the wind that is captured by the turbine. From the Betz's limit law we know that the best power conversion possible is C_p,max ? 0.59.

Part 1
Given the following data:
Blade length l = 52 m
Air density ? = 1.23 kg/m^3
Power coefficient C_p = C_p,max = 0.59
Plot the power output P as a function of wind speed v on the interval [0 m/s, 30 m/s].

Part 2
Many factors impact the final output of wind power. The rotor-blade friction and drag, gearbox losses, generator and converter losses, all reduce the power delivered by a wind turbine. In 2001, commercial utility-connected turbines deliver 75% to 80% of the Betz limit C_p,max of power extractable from the wind, at rated operating speed, that is, C_p = 0.40.

Create a second plot of power output as a function of windspeed using the original value of C_p,max = 0.59 and the value of C_p = 0.40 (68% of the previous value) together over the interval of velocities [0 m/s, 30 m/s].

Part 3
In the above parts we have considered C_p as a constant quantity. This is not, however, entirely accurate. In reality power conversion will be more or less efficient and depend on wind velocity.

Consider a power coefficient curve C_p(v) = 0.0026 v^3 - 14 v^2 + 180 v - 20000. At what velocity does C_p reach its maximum value and what is the maximum value of C_p?

Part 4
Using the power coefficient curve C_p(v) from part 3 find the power P of a turbine (using the other constants from Part 1) as a function of velocity. Plot this power function over the interval [0 m/s, 30 m/s].

Part 5
At what velocity is the power function from part 4 maximized?
Show more…
Title: Mathematics in Wind Energy Production Wind power is defined as the use of air flow through wind turbines to provide the mechanical force to generate electricity. Wind power is an alternative to burning fossil fuels, and is renewable and produces no greenhouse gas emissions during operation. Modern horizontal-axis wind turbines often use three blades. Theoretically, the maximum power P (unit: watts) that a three-blade wind turbine can extract from the wind power can be calculated as: P = 1/2 ? A v^3 Cp .......................... (Equation 1) where ? is the air density (kg/m^3), A is the sweep area of the turbine (m^2) and can be calculated from the length of the turbine blades, and v is the wind speed (m/s). Cp is the power coefficient that is unique to each turbine type. This coefficient represents the amount of kinetic energy from the wind that is captured by the turbine. From the Betz's limit law we know that the best power conversion possible is Cp,max ? 0.59. Part 1 Given the following data: Blade length l = 52 m Air density ? = 1.23 kg/m^3 Power coefficient Cp = Cp,max = 0.59 Plot the power output P as a function of wind speed v on the interval [0 m/s, 30 m/s]. Part 2 Many factors impact the final output of wind power. The rotor-blade friction and drag, gearbox losses, generator and converter losses, all reduce the power delivered by a wind turbine. In 2001, commercial utility-connected turbines deliver 75% to 80% of the Betz limit Cp,max of power extractable from the wind, at rated operating speed, that is, Cp = 0.40. Create a second plot of power output as a function of windspeed using the original value of Cp,max = 0.59 and the value of Cp = 0.40 (68% of the previous value) together over the interval of velocities [0 m/s, 30 m/s]. Part 3 In the above parts we have considered Cp as a constant quantity. This is not, however, entirely accurate. In reality power conversion will be more or less efficient and depend on wind velocity. Consider a power coefficient curve Cp(v) = 0.0026 v^3 - 14 v^2 + 180 v - 20000. At what velocity does Cp reach its maximum value and what is the maximum value of Cp? Part 4 Using the power coefficient curve Cp(v) from part 3 find the power P of a turbine (using the other constants from Part 1) as a function of velocity. Plot this power function over the interval [0 m/s, 30 m/s]. Part 5 At what velocity is the power function from part 4 maximized?

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Calculus: Early Transcendentals
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James Stewart 8th Edition
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Title: Mathematics in Wind Energy Production Wind power is defined as the use of air flow through wind turbines to provide the mechanical force to generate electricity. Wind power is an alternative to burning fossil fuels and can help reduce greenhouse gas emissions during operation. Modern horizontal-axis wind turbines often use three blades. The maximum power (in watts) that a three-blade wind turbine can extract from the wind can be calculated using the equation: P = 0.5 * ρ * A * v^3 * Cp Where ρ is the air density (in kg/m^3), A is the sweep area of the turbine (in m^2), v is the wind speed (in m/s), and Cp is the power coefficient specific to the turbine type. Cp represents the amount of kinetic energy from the wind that is captured by the turbine. According to the Betz limit law, the maximum possible power coefficient is 0.59. Given the following data: Blade length = 52 m, Air density = 1.23 kg/m^3, Cpmax = 0.59, plot the power output function of wind speed over the interval [0 m/s, 30 m/s]. Many factors impact the final output of wind power, including rotor-blade friction, drag, gearbox losses, generator and converter losses, which all reduce the power delivered by the wind turbine. In 2001, commercial utility-connected turbines delivered 75% to 80% of the Betz limit Cpmax of power extractable from the wind at the rated operating speed, Cp = 0.40. Create a second plot of power output as a function of wind speed using the original value of Cpmax = 0.59 and the value of Cp = 0.40 (68% of the previous value) over the interval of velocities [0 m/s, 30 m/s]. In the above parts, we have considered Cp as a constant quantity. However, in reality, power conversion will be more or less efficient depending on wind velocity. Consider the power coefficient curve Cp(v) = 0.0026 * v^3 - 1463 * v^2 + 180 * v - 20000. At what velocity does Cp reach its maximum value and what is the maximum value of Cp? Using the power coefficient curve Cp(v) from the previous part, find the power of the turbine (using the other constants from Part 1) as a function of velocity. Plot this power function over the interval [0 m/s, 30 m/s]. At what velocity is the power function from the previous part maximized?
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Given the following data for a multi-blade wind turbine: Blade length, l = 52 m Wind speed, v = 12 m/sec Air density, ̑ = 1.23 kg/m3 Power Coefficient, Cp = 0.4 calculate the power converted from the wind into rotational energy in the turbine (ii) Marine turbines are designed using the same principles as wind turbines. However, they are used in the different conditions and the variables used in the power equation are also slightly different. As the marine turbine works in water rather than air, we will use density of water instead of air: Density of water, ̑w = 1000 kg/m3 Power Coefficient Marine Turbine, Cpm = 0.35. Given these information, calculate the length of blade that would be needed to produce the same power by marine turbine as produced by the wind turbine in part (i). Assume velocity v = 2.5m/s, which is the typical rated marine (tidal) flow speed. (iii) Comment on the sizes of the wind and marine turbine for the same power out/ which one du you think is suitable from maintenance point of view. (iv) Define the tip speed ratio (̑) of the turbine and write the mathematical equation n that relate it to the other relevant parameters. (v) Given that the rotational speed of the turbine in part (i) as 15 rpm (Blade length, L = 52 m, wind speed =12 m/s), calculate ̑. Based on the value of Selecting the value ̑, select Cp from the graph below (Fig Q2) and calculate the daily energy produced by this turbine in KWh if the wind turbine works for 10 hrs per day.

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Derivation of wind turbine formula A derivation of the function $R$ in Exercise $69,$ based on three equations from physics, is outlined here. Consider again the figure given in Exercise $69,$ where $v_{1}$ equals the upstream velocity of the wind just before the wind stream encounters the wind turbine, and $v_{2}$ equals the downstream velocity of the wind just after the wind stream passes through the area swept out by the turbine blades. An equation for the power extracted by the rotor blades, based on conservation of momentum, is $P=v^{2} \rho A\left(v_{1}-v_{2}\right),$ where $v$ is the velocity of the wind (in $\mathrm{m} / \mathrm{s}$ ) as it passes through the turbine blades, $\rho$ is the density of air (in $\mathrm{kg} / \mathrm{m}^{3}$ ), and $A$ is the area (in $\mathrm{m}^{2}$ ) of the circular region swept out by the rotor blades. a. Another expression for the power extracted by the rotor blades, based on conservation of energy, is $P=\frac{1}{2} \rho w\left(v_{1}^{2}-v_{2}^{2}\right)$ Equate the two power equations and solve for $v$. b. Show that $P=\frac{\rho A}{4}\left(v_{1}+v_{2}\right)\left(v_{1}^{2}-v_{2}^{2}\right)$ c. If the wind were to pass through the same area $A$ without being disturbed by rotor blades, the amount of available power would be $P_{0}=\frac{\rho A v_{1}^{3}}{2} .$ Let $r=\frac{v_{2}}{v_{1}}$ and simplify the ratio $\frac{P}{P_{0}}$ to obtain the function $R(r)$ given in Exercise $69 .$ (Source: Journal of Applied Physics, 105,2009 )

Calculus: Early Transcendentals

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47. F = (yz co 48. Find poten Example 8. 49. Show that that any constar 50. Let φ = ln radial vector er 51. For P = (a FIGURE 11 21. A river 200 meters wide is modeled by the region in the xy-plane given by -100 ≤ x ≤ 100. The velocity vector field on the surface of the river is given by F = (-0.05x, 20 - 0.0001x²) in meters per second. Determine the coordinates of those points that have the maximum speed. 22. The velocity vectors in kilometers per hour for the wind speed of a tornado near the ground are given by the vector field F = (-y / e^(x²+y²-1)², x / e^(x²+y²-1)²). Determine the coordinates of those points where the wind speed is the highest. In Exercises 23-30, calculate div(F) and curl(F). (a) Verify that (b) Calculate er (c) Find a poter 52. Which of function for the dicular to the lev 23. F = ⟨x, y, z⟩ 24. F = ⟨y, z, x⟩ 25. F = ⟨x - 2zx², z - xy, z²x²⟩ 26. sin(x + z)i - ye^(xz)k 27. F = ⟨yz, xz, xy⟩ 28. F = ⟨y/x, y/z, z/x⟩ 29. F = ⟨e^y, sin x, cos x⟩ 30. F = ⟨x / (x² + y²), y / (x² + y²), 0⟩ In Exercises 31-37, prove the identities assuming that the appropriate partial derivatives exist and are continuous. 31. div(F + G) = div(F) + div(G) 32. curl(F + G) = curl(F) + curl(G) 33. div curl(F) = 0 34. div(F × G) = G · curl(F) - F · curl(G)

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Transcript

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00:01 The mass flow of air passing 3d1, top section.
00:05 And there should also go to av.
00:07 And the kinetic energy associated with the green is spreading it.
00:11 The kinetic energy is just a half, and we spread.
00:15 That's half.
00:17 Low a .v...
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