Lecture+2A+Bernoulli E10 Mech E

Lecture+2A+Bernoulli E10 Mech E - Engineering 10:...

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Unformatted text preview: Engineering 10: Engineering Engineering Design & Analysis Mechanical Engineering Module Fall, 2010 Lecture 2 Power in the Wind Logistics Logistics Labs are going … Labs Wednesday: 2109 Etcheverry Wednesday: Thursday: 39 Evans Thursday: Logistics Logistics Download Chapters 1 and 3 Download of “Wind Energy Handbook” from the UC Library “EBooks” collection collection (instructions on bSpace in Resources/ME Module). Resources/ME Read Chapter 1. Read Today’s Lecture Today Energy & Power Energy A derivation of (one form of) Bernoulli’s derivation equation The maximum power available from a turbine The Characteristics of turbine blades Characteristics Energy & Power Energy Power = Energy per unit time Power Energy = Power * Time Energy Electrical generation is usually given in terms Electrical of the amount of power that can be produced power We pay for the amount of energy we use! We energy Energy & Power Energy Power: Watts (kW, MW, GW, …, BTU/h) Power: Energy = Joules (J), calories, kiloWatt hours Energy kiloWatt (kWh), British Thermal Units (BTU) Energy & Power Energy BTU: Energy required to raise the BTU: temperature of 1 pound of water 1 degree F calorie: Energy required to raise the calorie: temperature of 1 gram of water 1 degree C joule: The work done by a 1 Newton force joule: moving through a distance of 1 meter J=Nm watt: joule per second (W = N m/s) watt: Power from the Wind Power The usual expression for power generated by a wind turbine is (p. 6 of Wind Energy Handbook): Power Factor Area Power 1 3 Π = C p ρ Au 2 Where does this come from??? Where Density Wind Speed Maximum Power Available: Maximum Derived from Bernoulli’s Eqn Eqn We will derive Bernoulli’s equation in the We simplest case using: Newton’s second law for an element of air Newton Quantities involved: Quantities Wind speed u(x,t) (assumed to be “steady”) Wind u(x,t (assumed Air density ρ (assumed to be constant) Air (assumed Pressure p(x,t) Pressure p(x,t Cross sectional area A(x) Cross A(x Force, mass and acceleration Force, Maximum Power Available: Maximum Derived from Bernoulli’s Eqn Eqn By “steady flow” we mean that the velocity By does not change in time at a particular position, but the velocity may be different at different locations. u ( x, t ) = u ( x(t )) Derivation of Bernoulli’s Eqn Derivation Eqn Let F be the force acting on an element of air Let in a tube of cross sectional area A. Newton’s second law states that this force is the mass (m) times the acceleration (a) F A x ∂u ( x, t ) F = ma = m ∂t Derivation of Bernoulli’s Eqn Derivation Eqn F A x ∂u ( x, t ) F = ma = m ∂t Force = (Pressure Difference)*Area p1 x1 p2 dx x2 F = − Adp dp = p2 − p1 Derivation of Bernoulli’s Eqn Derivation Eqn F A x ∂u ( x, t ) F = ma = m ∂t Mass = Volume * Density p1 x1 p2 dx x2 m = ρ Adx V = Adx Derivation of Bernoulli’s Eqn Derivation Eqn F A x ∂u ( x, t ) F = ma = m ∂t Acceleration in steady flow ∂u ( x(t )) ∂u ∂x ∂u a= = = u ∂t ∂x ∂t ∂x Derivation of Bernoulli’s Eqn Derivation Eqn F A x F = ma F = − Adp m = ρ Adx ∂u a=u ∂x ⎛ ∂u ⎞ − Adp = ρ Adx ⎜ u ⎟ ⎝ ∂x ⎠ ∂p ∂u − = ρu ∂x ∂x Bernoulli’s Eqn Bernoulli F A x ∂p ∂u − = ρu ∂x ∂x ∂ ⎛1 2 ⎜ ρu + ∂x ⎝ 2 ⎞ p⎟ = 0 ⎠ 12 ρu + p = C 2 Assume that turbine stops the wind Assume stops Maximum Power Available Maximum 1 2 ρu1 + p1 = C 2 0 1 2 ρu2 + p2 = C 2 0 1 2 F2 = p2 A = ρ Au1 2 Maximum Power Available Maximum 1 2 F2 = ρAu1 2 Power = Force * velocity Π max 1 3 = ρ Au 2 Quick Calculation Quick Π actual = C p Π max 1 3 = C p ρ Au 2 Diameter =90 m Area =6360 m2 Density (ρ)=1.22 kg/m3 Quick Calculation Quick u (m/s) 4 5 10 15 25 30 Pmax (kW) Pactual (kW) Efficiency 244 480 3,800 13,900 59,600 103,000 0 250 1,250 3,000 3,000 0 0% 52% 33% 23% 5% 0% Wind Turbine Blades Wind What are the general characteristics of wind What turbine blades? How will you choose a blade shape for your designs in the next few weeks? Get together with a few people around you Get and identify as many characteristics of wind turbine blades as you can think of in the next few minutes. Wind Turbine Blades Wind Your Thoughts Here! Energy Flow: Sources & Uses Energy Source: Annual Energy Review, Department of Energy, Report Number DOE/EIA-0384(2008), June 2009. Electricity Flow: Sources & Uses Electricity Source: Annual Energy Review, Department of Energy, Report Number DOE/EIA-0384(2008), June 2009. Electricity Prices (US) Electricity Source: Annual Energy Review, Department of Energy, Report Number DOE/EIA-0384(2008), June 2009. Renewable Energy Consumption Consumption Source: Annual Energy Review, Department of Energy, Report Number DOE/EIA-0384(2008), June 2009. Renewable Energy Consumption Consumption Source: Annual Energy Review, Department of Energy, Report Number DOE/EIA-0384(2008), June 2009. Renewable Energy Consumption Consumption Source: Annual Energy Review, Department of Energy, Report Number DOE/EIA-0384(2008), June 2009. Predicted Electricity Use Predicted Source: Annual Energy Review, Department of Energy, Report Number DOE/EIA-0384(2008), June 2009. ...
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