How Much Dam Energy Can We Get?
By Tom Murphy
21 December, 2011
Do the Math
Having now sorted solar, wind, and tidal power into three “boxes,” let’s keep going and investigate another source of non-fossil energy and put it in a box. Today we’ll look at hydroelectricity. As one of the earliest renewable energy resources to be exploited, hydroelectricity is the low-hanging fruit of the renewable world. It’s steady, self-storing, highly efficient, cost-effective, low-carbon, low-tech, and offers a serious boon to water skiers. I’m sold! Let’s have more of that! How much might we expect to get from hydro, and how important will its role be compared to other renewable resources?
Last week, as soon as I put tidal power into a box labeled “waste of time,” I received some deserved howls of protest. I saw it coming, and had built in words to soften the “waste of time” label. But it was a poor choice from the start. A better set of labels is “abundant,” “potent,” and “niche.” The last could also be thought of as “boutique,” in that it is cute, perhaps decorative, may serve some function, but will never be a heavy lifter. The “potent” label—formerly “useful”— is meant to indicate a source that could supply a healthy fraction (say over a quarter) of our global demand if fully exploited. We will never fully exploit any resource, so the numbers at least need to support ¼-scale before we can believe that it may play a major role.
I should also point out that all along, my approach is to pretend that our goal is to keep up our current energy standards in a post-fossil-fuel world. In the process, we will see just how hard that will be to do. It is by no means impossible, but it’s much more difficult and compromised than most people realize. In the end, it is not clear that we will maintain our current global rate of energy usage: the future is unwritten. On the plus side, some of the approaches I cast into the “niche” box may become “potent” in a scaled-down world. Firewood was once abundant, then moved to potent, and is now a niche. But a reversal of fortunes could change all that.
I may be starting to sound like a broken record, covering the basic physics of hydroelectricity now in several posts. But we learn by repetition, right? So in the spirit of self-containment, here we go…
Hydroelectric dams exploit storage of gravitational potential energy. A mass, m, raised a height, h against gravity, g = 10 m/s², is given a potential energy E = mgh. The result will be in Joules if the input is expressed in meters, kilograms, and seconds (MKS, or SI units). Water has a density of ρ = 1000 kg/m³, so if we know how many cubic meters of water flow through the dam each second (F), the power available to the dam will be P = ηρFgh. We have inserted η to represent the efficiency of the dam—usually around 90% (η≈0.90).
The height of the water behind the dam is the relevant height for the potential energy calculation, even if a given parcel of water is collected at the bottom of the dam. This is because the pressure of the water above provides the motive force. In the absence of turbines or other restrictions, the water would emerge from the penstock at a velocity of v = sqrt(2gh) so that a flow, F, would require an area A = F/v. For example, Hoover Dam, at 222 m high (in the days when Lake Mead was full!) would eject water at a stunning 67 m/s (150 m.p.h.) if a big hole opened up in the bottom. At the nominal flow rate of 1000 m³/s, this corresponds to a hole about 4 m in diameter. I think we should do it.
Now, it is the job of the turbine(s) to extract some of the kinetic energy this water would have if it were allowed to shoot out of the bottom of the dam. As a consequence, it comes out at a much more sedate pace. Some of the 10% inefficiency in hydroelectric dams is due to generator inefficiency, but some is because you can’t take all of the kinetic energy out of the water or it would stop flowing and stall the flow of the next batch. But nature is kind here, since kinetic energy goes as the square of the velocity. The velocity of the energy-sapped water is therefore sqrt(1 − η). So if we pull 96% of the energy out of the water, its flow velocity is 20% of the free-flow value (13 m/s in the foregoing example). Or we can grab 99% at a 10% exit speed (7 m/s, or 15 m.p.h.). This sounds much more reasonable—and seems like a good bargain. The area needed now expands accordingly, but that’s what large turbines/penstocks are for.
Hydroelectricity in Practice
The U.S. has 78 GW of hydroelectric capacity installed. In a year, these plants produce 272 TWh. Divide by 8766 hours in a year, and we find 0.031 TW (31 GW) of average power. This implies a 40% capacity factor. I was surprised by this: I thought dams churned along at a steady rate all the live-long day. Seasonal variations are apparently much larger than I appreciated—snow-melt being one factor. The following table lists all hydroelectric facilities in the U.S. bigger than 1 GW—representing 30% of total installed capacity in just 11 dams. The table shows each dam’s nameplate (peak) capacity, height, implied flow at peak generation capacity (after which spillways must be activated; assumes 90% efficiency), and capacity factor.
Name ..............Capacity (GW) ...........Height (m) ........Peak Flow (m³/s)...... Capacity Factor
Grand Coulee.......... 6.495 ................ 168 ......................4300.................................. 0.36
Chief Joseph............ 2.456 ..................70......................... 3970................................. 0.54
Robert Moses Niagra.. 2.429 .............29.6..................... 9300 .................................0.63
John Day..................... 2.160..................70 .....................3500.................................. 0.47
Hoover........................ 2.079 ................222...................... 1060.............................. 0.22
The Dalles................... 1.820................. 61....................... 338...............................0 0.41
Glen Canyon................ 1.312 ...............216.................... 690................................ 0.30
Rocky Reach................. 1.300.................. 40.................. 3720.............................. 0.55
Bonneville..................... 1.093.................. 60.................. 2060............................ 0.48
Boundary........................ 1.040................ 110................... 1070........................ 0.40
Wanapum....................... 1.039................... 63................... 1870 ......................0.58
Eight of the eleven dams in the table are in the Pacific Northwest. Of possible interest is the fact that the power-capacity-weighted heights of these dams is 113 m, while for the entire U.S. fleet, it is 88 m (much higher than I thought). For reference, the newly completed Three Gorges dam in China is rated at 20.3 GW, has a nominal head height of 81 m, an implied flow of 28,500 m³/s, and a capacity factor of 0.45.
What fraction of our energy currently comes from hydroelectricity? Such a simple question deserves a simple answer. Yet numbers range all over. The hard answer is that 272 TWh of annually delivered electricity in the U.S. corresponds to 0.9% of the primary energy use, or 2.3% of primary energy associated with electricity. Of the delivered electricity, it’s 7.3%. Much of the variation is due to an apples-to-oranges comparison of efficient hydroelectricity to heat engines that convert only 35% of their primary energy into useful energy. For instance, the Department of Energy’s Annual Energy Review (whose numbers are well-depicted in the LLNL graphic) artificially inflates the contribution from hydro to put it on the same footing as the fossil fuel inputs. But however you want to slice it, hydro is on the small side.
Global Hydro Potential
Now the fun part. How much hydro power is theoretically achievable? Hydroelectricity is cashing in on residual potential energy provided by the rain cycle. A look at the Earth’s energy budget shows that a whopping 23% of the solar budget goes into evaporating water! The water cycle is a big deal.
Each gram of water takes 2250 J of energy to vaporize. It takes additional potential energy to lift the water into the atmosphere. Hoisting a gram of water 10,000 m high (to the top of the troposphere) takes a comparatively small 100 J. Since all water caught by the land and used for hydroelectricity had to first be evaporated, we can say that if all the water got lifted to 10 km (it doesn’t, but bear with me for easy math), then 1% of the solar budget goes into lifting water. I love it when numbers like 2250 J align so neatly with 23% of the solar budget!
The lower atmosphere contains far more water than the upper atmosphere—largely because air gets cooler with altitude, and can hold less water. But clouds represent condensed water and can clearly contain significant amounts of water all the way up to 10 km. If water were evenly distributed in the troposphere, the average height of water within it would be about 5 km. Considering the non-uniform profile, I’ll use an average height of 2 km. Therefore, we adjust our hoist factor to 0.2% of the incident solar energy.
Then it rains. And most of that stored energy is wasted as the drops fall irresponsibly through the sky—without a thought to our needs. When water hits the ground, the average height of the land is—I’m guessing—500 m. Since potential energy is linear with height, we can use a simple average in this way. If we could capture all that remains of the potential energy as we return the water to the sea, we get 0.05% of the solar potential. But this only works on land, which is 28% of the Earth’s surface. Now we’re down to 0.014%.
Don’t despair yet. The solar potential is huge. These percentages all relate to the incident energy at the top of the atmosphere: 1370 W/m² over πR² square meters turns into 175,000 TW. So our 0.014% is 25 TW. Our global energy diet is about 13 TW, so we’re in the game.
We can pursue an alternate approach to check that we’re on the right track. If annual rainfall averages 0.5 m (20 inches) on land globally, and typically falls on land 500 m above sea level, we can total up the potential energy and divide by the number of seconds in a year to get a power. I calculate 11 TW by this method, so yes—we’re making sense. My guess of 0.5 m of rain per year may be a little low, but perhaps this compensates a bit for the fact that low areas tend to get more rain than high areas.
Now, nature provides a convenient water collection system, concentrating the water that falls on land into streams and rivers and lakes. This natural concentration is what makes such a diffuse power source usable at all, and is why hydroelectric was the vanguard of modern renewable energy (I’m skipping over firewood as non-modern).
But as convenient as this collection system is, much of the energy is lost en-route to the rivers. Think about the journey a water drop that lands in your yard or on a mountain slope must make before finding a body of water large enough to profitably dam up. Obviously there is friction in the collection process, or water would be screaming along at 100 m/s (220 m.p.h.) at the bottom of a 500 m slope. In the end, we must accept heights of static water collected behind dams: no kinetic harvesting, practically speaking.
I’ll make a rough guess and knock off a factor of two for the energy lost in the collection process leading up to the river/stream. My gut says that I’m probably being generous here. In any case, we’re dealing with something in the neighborhood of 6–12 TW of global potential. For the U.S., with 7% of the world’s land area, this turns into 0.4–0.8 TW.
At present, the U.S. has 78 GW of installed hydro power (out of which we get 31 GW, averaged annually). The world has about 1 TW installed, likely realizing 400 GW on an annual average. The realized capacity therefore undershoots our crude estimate of global potential by a factor of 10 or more. Does this mean we could go nuts and expand hydro to amazing new levels? Should I ask for water skis for Christmas?
I don’t want to discount the top-down approach we did here. After all, if anyone tried to tell me that hydro could deliver much more than 25 TW of power, I would know that the basic physics of the planet does not allow it. But at the same time, the upper limit we established does not account for a whole host of practical considerations, like actual rivers with known flow rates and geographic potential for damming. So I turn to a study that has put some more time into the question than I can afford personally, outside of my day job. Specifically, a report by the Eurelectric group assessed global hydro potential in four cascading steps:
1. Gross potential if all runoff is developed to sea level with no loss;
2. Technical potential, ignoring economic limitations;
3. Economically viable potential, cost competitive with other sources;
4. Exploitable potential, considering environmental and other restrictions.
For the first step, they come away with 5.8 TW—not far at all from my estimate (I’m not cheating I swear: I did not look at any estimates prior to writing the above sections). Other assessments get 4.4 TW, 4.6 TW, and 5.1 TW.
For technical feasibility, these same sources estimate 1.6–2.3 TW globally. Economic feasibility (in today’s economic climate) drops this to 1.0–1.4 TW. Environmental restrictions (in today’s climate) reduce this number further. Thus, having developed 0.4 TW worldwide (using average annual output for proper comparison to studies), the world may be able to expand by a factor of 2–5. This is a large range: a factor of two isn’t that much, while a factor of 5 is a pretty big jump. Where is it, really?
For the U.S., the Idaho National Laboratory estimates a gross potential of 0.3 TW, and a technical potential of 0.17 TW. The latter was determined after a study of 500,000 potential sites, out of which 130,000 made the cut. It is also estimated that existing dams with no hydroelectric capacity could add 0.013 TW (13 GW). So here in the U.S., we could expand by a factor of 5 according to this report—ignoring economic and environmental barriers. Such a boost would bring hydro up to 5% of our gross energy, or 12% if we correct for the heat-engine effect (40% of our electricity). I have seen other reports less optimistic about our expansion potential, coming in closer to a doubling of current capacity—likely factoring in economic and environmental considerations, and consistent with the lower end of the range estimated for global potential.
At a global potential of approximately 10% of our current energy scale, my initial reaction is to throw hydro into the “niche” box with tidal, since my criterion is that a resource be theoretically able to meet a quarter of our demand to be labeled “potent”—which incidentally is in line with what oil, natural gas, and coal each deliver to us today: all are momentarily “potent” sources by this reckoning. If we consider that thermodynamic losses in conventional electricity production do not apply to hydro, we might be tempted to boost it into the “potent” category. But I didn’t need to do this for wind, and certainly not for solar. And even this boost does not put it over the top in the U.S. (at 12%), even if entertaining a 5× increase in hydro development. So I think I’ll leave it in the niche box. It’s a borderline call (and meaningless, really). Hydro beats the pants off of tidal, and is currently used to good effect the world over. But it’s a real stretch to make it a big player in the energy game at today’s rates of usage.
Almost all renewable resources are dilute, and face substantial environmental hurdles for being adopted. Dams are not without controversy. They radically change the landscape and natural ecosystem. They silt up and lose capacity over time. And they can cause long-term threats to settlements downstream in the case of failure (and all dams will someday fail—they really haven’t been around that long). When push comes to shove, we may be willing to ignore aesthetic and environmental concerns. I might rather hike through Glen Canyon than listen to jet skis on Lake Powell (in the fossil fuel crunch, maybe I won’t have to), but weighed against the hardship of energy decline, will we collectively make that choice? Lake Powell already offers one answer. Of course dams require tremendous up-front (energy) investment, and therefore are susceptible to The Energy Trap. So in crunch time, I’m not sure I can predict what we’ll do, or if our choices will be at all rational.
In hydroelectric power, we again face the problem that the impending fossil fuel shortage is not fundamentally an electricity shortage problem. So most of the “solutions” I’ve hit so far do not address the fundamental and most immediate crisis in liquid fuels. Overall, I’m a fan of hydro power, and I’m glad nature does most of the work for us. Nonetheless, my mind is not much eased by the joint facts that it falls far short of our current demand and that it’s yet another way to make electricity. It’s a gift from nature, but much like getting yet another tie for Christmas to add to the pile, I’m not getting excited about the prospect of more dams.
Tom Murphy is an associate professor of physics at the University of California, San Diego. An amateur astronomer in high school, physics major at Georgia Tech, and PhD student in physics at Caltech, Murphy has spent decades reveling in the study of astrophysics. He currently leads a project to test General Relativity by bouncing laser pulses off of the reflectors left on the Moon by the Apollo astronauts, achieving one-millimeter range precision. Murphy’s keen interest in energy topics began with his teaching a course on energy and the environment for non-science majors at UCSD. Motivated by the unprecedented challenges we face, he has applied his instrumentation skills to exploring alternative energy and associated measurement schemes. Following his natural instincts to educate, Murphy is eager to get people thinking about the quantitatively convincing case that our pursuit of an ever-bigger scale of life faces gigantic challenges and carries significant risks.
Thomas Tu contributed research on hydroelectric installations for the table, and rounded up assessments of global hydro potential from a variety of sources.
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