# Hydro-Electric Dams

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Why is this beautiful valley in north-eastern BC about to be flooded?

Big Ideas:
• The energy flow in the water cycle can be intercepted to produce useful power.

Introduction

Hydro-electricity can be generated in numerous ways: dams, rivers, tides and waves. Here we describe the simplest and oldest method, the dam. However, the analysis can easily be applied to run-of-the-river power generation.

For dams, the energy transformations are as follows:

• Solar radiation evaporates water from the ocean
• Rain falls on mountains, and runs into lakes.
• The runoff is interrupted by a reservoir and dam.
• Water is extracted at the base of the dam to turn generators.

Peace River Dam, Hudson's Hope BC

Consider a reservoir (which may be a natural lake) of surface area A, in which water is extracted at a height h below the water level, and used to turn electricity generating turbines.  The total rate (in m3/s) at which water is extracted is Q.

In principle one can calculate a maximum value of Q from h and the size of the hole using Bernoulli’s principle. However, in real hydro-electric dams Q is largely determined by the amount of rainfall landing on the catchment area A of the lake. One cannot extract more water out of a reservoir than is falling into it as rain.

Example: the W.A.C. Bennett Dam

Let’s look at some numbers for the big W.A.C. Bennett Dam in north-eastern British Columbia1.

The catchment area is about 70 000 km2, and the mean rainfall in this area is about 600 mm per year2.

Therefore the maximum possible Q (ignoring all evaporation) is given by:

Q = (70 000 km2)(106 m2/km2)(0.6 m/y)/(3.15 × 107s/y) = 1300 m3/s

The published height of the dam is 186 m. We will take this to be the height difference between the water level and the turbines, h. Consider a body of water which starts at the surface of the reservoir and eventually moves through the turbines. Its potential energy per unit volume at the surface is ρgh (in J/m3) compared to the level of the turbines, where ρ is the density of the water (1000 kg/m3). The rate at which the water moves through the turbines is Q, and so the rate at which the available potential energy passes the turbines is ρghQ. Now consider that useful electrical energy is generated with an efficiency η and so we can write the power generated P as follows.

P = ηρghQ

Assuming for now that the efficiency of the generators η is 1; we calculate the maximum available power to be 2.4 GW. The maximum power rating of the dam and its 10 turbines is given as 2.73 GW. Plainly this maximum power cannot be sustained as it is more than the number we have come up using all the rainfall and also assuming 100% efficiency and no evaporation. However, the annual average power generated is given to be 13 100 GWh per year. If we divide this number by (24)(365) h/y, we obtain a mean delivered power of 1.5 GW. This is of the same order as, but comfortably less than, our “ideal” maximum value of 2.4 GW.

Greenhouse Gas Emissions

While GHG emissions from hydro-electric projects are small to other means of electricity generation, they are not negligible. They arise from the initial construction and the subsequent decay of biomass in the flooded valley (if that is the way the project is constructed). Take for example the proposed “Site-C” dam which would be just downstream of the Bennett Dam discussed above. This project will have a rated power of about 1 GW, and will cause 10 000 hectares to be flooded. A study predicts that the equivalent of about 150 000 tonnes of CO­2 will be emitted each year until the second decade after completion. About half will come from construction (and the associated levelling of forest) and half will arise from biomass decay in the artificial lake. This number should be compared with about 10 000 000 tonnes per GWy of electricity produced by coal-fired plants.

"Site C", Hudson's Hope BC

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C21: Physics Teaching for the 21st Century
UBC Department of Physics & Astronomy