# Hydro-Electric Dams

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

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 also 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.
• Fast flowing water is extracted under pressure at the base of the dam through turbines that turn generators.

Fig.1. Peace River Dam, Hudson’s Hope BC (author photo)

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 $Q$.

Fig.2. Water flowing from the base of a container.

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

Consider some numbers for the big W.A.C. Bennett Dam in north-eastern British Columbia[1]BC Hydro Peace Region, https://www.bchydro.com/energy-in-bc/operations/our-facilities/peace.html [2019-09-20]..

The catchment area is about 70 000 km2, and the mean rainfall in this area is about 600 mm per year[2]Natural Resources Canada: Water, https://www.nrcan.gc.ca/earth-sciences/geography/atlas-canada/selected-thematic-maps/16888 [2019-09-20]..

Therefore the maximum possible $Q$ (ignoring all evaporation) is given by Eqn.1:

$$$Q = (70 000 \textnormal{ km}^2)(10^6 \textnormal{ m}^2/\textnormal{ km}^2)(0.6 \textnormal{ m/y})/(3.15 × 10^7 \text{ s/y}) = 1300 \text{ m}^3/\text{ s} \tag{1}$$$

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 $\rho$ 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 (Eqn.2).

$$$P=\eta\rho ghQ \tag{2}$$$

Assuming for now that the efficiency of the generators $\eta$ 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 50 000 tonnes of CO­2 will be emitted each year throughout construction and operation[3] Site C Clean Energy Project: Greenhouse Gases Technical Report (Stantec), https://www.ceaa-acee.gc.ca/050/documents_staticpost/63919/85328/Vol2_Appendix_S.pdf [2019-09-19].. 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 (author photo).

Updated 2019-09-20

Footnotes

↑1 BC Hydro Peace Region, https://www.bchydro.com/energy-in-bc/operations/our-facilities/peace.html [2019-09-20]. Natural Resources Canada: Water, https://www.nrcan.gc.ca/earth-sciences/geography/atlas-canada/selected-thematic-maps/16888 [2019-09-20]. Site C Clean Energy Project: Greenhouse Gases Technical Report (Stantec), https://www.ceaa-acee.gc.ca/050/documents_staticpost/63919/85328/Vol2_Appendix_S.pdf [2019-09-19].