## MgZ Z [pAuA pAU mu u Ef

where the pressure force p1A1 acts through a distance u1At, and similarly for p2A2, and Ef is the heat generated by friction.

Figure 2.1 Illustrating conservation of energy: a streamtube rises from height z, to z2.

2.2 Conservation of energy: Bernoulli's equation 31

We neglect fluid friction, Ef, now, but we will examine some of its effects in Section 2.5. In this ideal, frictionless case, (2.1) reduces to p1 + gzi + 1 «2 = - + gZ2 + 1 u2 (2.2)

or, equivalently, so each term has the dimension of height (m)

p u2

--+ z + — = constant along a streamline, with no loss of energy (2.3)

pg 2g

Either of these forms of the equation is called Bernoulli's equation. Equations (8.10) and (9.19) are examples of its application in hydro and wind power respectively.

The sum of the terms on the left of (2.3) is called the total head of fluid (H). It relates to the total energy of a unit mass of fluid, however the constant in (2.3) may vary from streamline to streamline. Moreover, for many situations, the friction losses, Ef, have to be included. Head has the dimensions of length. For hydropower, head is the effective height of the moving water column incident on the turbine - see Section 8.3.

The main limitation of (2.2) and (2.3) is that they apply only to fluids treated as ideal, i.e. with zero viscosity, zero compressibility and zero thermal conductivity. However, this is applicable to wind and hydro turbines with their relatively low-speed movement of air and water, and with no internal heat sources. The energy equation can however be modified to include non-ideal characteristics (see Bibliography), as for combustion engines and many other thermal devices, e.g. high temperature solar collectors.

In solar heating systems and heat exchangers, power Pth is added to the fluid from heat sources (Figure 2.2). Heat E = Pt^Ai is added to the energy inputs on the left hand side of (2.1). The mass m coming into the control volume at temperature T1 has heat content mcT1 (where c is the specific heat capacity of the fluid), and that going out has heat content mcT2. Thus we add to the right hand side of (2.1) the net heat carried out of the control volume in time At, namely mc(T2 — T1). This gives an equation corresponding to (2.2), namely pi + gZ1 + 1 «1 + cT1 + Ph =(-) + gZ2 + 1 «2 + cT2 (2.4)

where the volume flow rate is

Figure 2.2 As for Figure 2.1, but with heat sources present. Thermal power Pth is added to the flow.

In most heating systems, including active solar heating (see Equation 5.2 onwards), thermal contributions dominate the energy balance. So (2.4) reduces to

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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