Theory of Wind Energy

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The principles concerned with converting the potential energy of fluids into useful power relies on three basic fundamentals: conservation of mass, energy and momentum, so it is useful to discuss these before examining the operation of wind turbines.

Conservation of Mass:

The continuity equation applies the principle of conservation of mass to fluid flow. Consider a fluid flowing through a fixed conduit having one inlet and one outlet as shown in Figure 3.1

Figure 3.1 Conservation of mass of a fluid flowing in a duct/pipe

If the flow is steady i.e no accumulation of fluid within the control volume, then the rate of fluid flow at entry must be equal to the rate of fluid flow at exit for mass conservation. If the flow cross-sectional area A (m2), and the fluid parcel travels a distance dL in time dt, then the volume flow rate (Vf, m3/s) is given by:
V_f = A.dL / dt
but since dL/dt is the fluid velocity (V, m/s) we can write: Vf = V × A
The mass flow rate (m, kg/s) is given by the product of density and volume flow rate. Between any two points within the control volume, the fluid mass flow rate can be shown to remain constant:
or   ρ₁A₁V₁ = ρ₂A₂V₂             (1)
Conservation of Energy:

Conservation of energy necessitates that the total energy of the fluid remains constant, however, there can be transformation from one form to another.

There are three forms of non-thermal energy for a fluid at any given point:-

The kinetic energy due to the motion of the fluid.

The potential energy due to the positional elevation above a datum.

The pressure energy, due to the absolute pressure of the fluid at that point.

If all energy terms are written in the form of the head (potential energy), ie in metres of the fluid, then conservation of energy principle requires that:

This equation is known as the Bernoulli equation and is valid if the two points of interest 1 & 2 are very close to each other and there is no loss of energy.

In a real situation, the flow will suffer a loss of energy due to friction (hL) and obstruction between stations 1 & 2, hence

Conservation of Momentum:
Consider a duct of length L, cross-sectional area A_c, surface area A_s, in which a fluid of density ρ, is flowing at mean velocity V. The forces acting on a segment of the duct are that due to pressure difference and that due to friction at the walls in contact with the fluid.
If the acceleration of the fluid is zero, the net forces acting on the element must be zero, hence

This is known as Darcy formula.

The value of the friction factor (f) depends mainly on two parameters namely the value of the Reynolds number and the surface roughness.

The Reynolds number is defined in terms of the density, velocity of flow, diameter and the dynamic viscosity as follows:

For laminar flow (ie Re < 2000),

While for a smooth pipe with turbulent (i.e. Re > 4000) flow,

For Re > 2000 and Re < 4000, this region is known as the critical zone and the value of the friction factor is certain.

In the turbulent zone, if the surface of the pipe is not perfectly smooth, then the value of the friction factor has to be determined from the Moody diagram . The relative roughness is the ratio of the average height of the surface projections on the inside of the pipe (k) to the pipe diameter (D). In common with Reynolds number and friction factor this parameter is dimensionless.

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