# 5.4: The Energy of Rivers

- Page ID
- 13477

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The content of this section is a little less grandiose than the title would suggest. I would like to introduce some basic ideas about the energetics of rivers and then supply a calculation to show how much energy is actually expended by a representative large river.

Remember the “home experiment” on dropping a lump of modeling clay to demonstrate the nature of energy, back in Chapter 1? A river is a falling body too, in a very real sense; its fall is just constrained to be at a very low angle, by the gently sloping bed of the river.

When viewed as an energy system, a river is a converter of mechanical (potential) energy to thermal energy. The potential energy of the river water is converted to thermal energy by internal friction within the water. The kinetic energy of the river, however, remains nearly constant, because the flow isn’t changing its speed much downstream. The nature of the internal friction is actually very complicated, because it depends on the details of the turbulence in the river.

When we obtain hydroelectric power from rivers, what we’re doing is locally arranging the river, by building a dam and making a lake, so that the conversion of potential energy to heat energy is suppressed along some stretch of the river, and we convert the potential energy directly to electrical energy by turbines and generators instead.

It might interest you to think about the power expended by a large river. Let’s make a very crude calculation of the rate of energy release by the lower Mississippi River, per square meter of the bed, as it flows downhill. One way of doing this is to think about a column of water above one square meter of the bed of the river, and how fast that column of water loses its potential energy as the river flows downslope. That loss of potential energy shows up as heat, via friction within the water column, owing to shear of the water, and at the bed of the river, as bottom friction. Think of this as the continuing degradation of the mechanical energy of the river into the thermal energy of the water. (Of course, the river doesn’t keep on heating up: it’s losing heat to its surroundings all the time at about the same rate that the heat is being produced by friction.)

You’re likely to get confused about units here. In the mks (meter– kilogram–second) system of units in physics, the unit of force (including weight, which, remember, is a force) is the **newton **(N). The unit of energy is the **joule **(J), which is equal to one newton-meter.

What comes to your mind when I mention work? Maybe what you do for a living, or things you have to do that are the opposite of fun. In physics, however, work has a very specific meaning: when a body of matter is acted upon and thereby moved by a force, the * work *done by the force on the body is equal to

*the product of the component of the force in the direction of movement, and the distance the body moves*.

In physics, work is equivalent to energy. You probably have heard of Newton’s second law of motion, mentioned in the background section on energy. It’s not difficult to show, with some math, that Newton’s second law can be recast into an equivalent form that says that *the work done on a body is equal to the change in kinetic energy of the body*. That’s why the joule, the unit of energy in the mks system of units, is equal to one newton-meter.

In its lower reaches, the Mississippi is about ten meters deep, as a very round number, and its mean velocity is as much as a few meters per second. Let’s assume, conservatively, one meter per second. The slope of the river is something like 10-4 (meaning that it drops about a tenth of a meter in one kilometer of downstream travel).

If our column of water is moving at one meter per second and drops a tenth of a meter in one kilometer of travel, it is losing elevation at a speed of 10-4 meters per second. (Think about that for a while, to convince yourself.) The weight of the unit-area column of water is equal to the weight of a cubic meter of water, times its height of ten meters. The mass of a cubic meter of water is (basically by definition!) one thousand kilograms. We have to multiply that by the value of the acceleration of gravity, about ten meters per second per second, to find its weight. Then we have to multiply by the height of the column, ten meters. The result is 105 newtons. That mass, with a weight of 105 newtons, is losing elevation at 10-4 meters per second, so the rate of loss of potential energy is ten newton-meters per second—or 10 joules per second, as per the definition of the joule in the background section above. That’s the rate at which the unit-area column of water in the river loses its mechanical energy. One joule per second is called a * watt *(abbreviation: W). The grand final result is

*ten watts per square meter of river bottom*. That doesn’t sound like a lot (a ten-watt bulb is even dimmer than the classic dim bulb), but think of how many square meters there are on the bed of the Mississippi River (a few kilometers wide, and hundreds of kilometers long, even in just its lower reaches).

That long and involved computation above has relevance to hydropower. What a hydroelectric station does is convert the mechanical energy of the river directly into electrical energy. The falling water turns turbines connected to electrical generators, with minimal friction involved, instead of slowly losing its potential energy to heat by friction as it flows downstream.