How Much Electrical Power Can Bagnell Dam Produce? Part 3

Impounded Water and Potential Energy

Part 1
Part 2

Note:  Before we begin any discussion or calculations I will point out the unapologetic use of Imperial (English) units of measure in the following calculations and results.  All given data for Lake of the Ozarks and Bagnell Dam is provided in imperial units, which could easily be converted to metric units, but most Americans intuitively relate to gallons, feet and pounds, especially when conceptualizing very large numbers such as will be discussed.  The use of metric units would make conversions much easier, but any empirical feel for the results by those most interested in the analysis would be lost.  650 billion gallons is a vast number but has some empirical value to Americans, while 2.46 trillion liters just sounds like an impossibly large number of plastic soda bottles.

When asking the question of how much energy a traditional dam is capable of producing, the problem is often presented as a potential energy calculation. Water may be the first thing one thinks of when considering the driving force behind a hydroelectric dam, but it is gravity that is the prime mover. When determining the potential energy for gravity a very simple equation is all that is needed. Any mass (m) raised from rest to a height (h) has potential energy (E) and readily calculated using the formula E=mgh, where g represents the gravity constant (for planet Earth) of  32ft/s2. A more detailed explanation, along with an example, is in the following info graphic.
Figure 1
In the above example of a simple impounding dam, the total potential energy represented by the mass of water, in gallons, raised to height (h) in feet, can quickly be calculated with the result measured in units of “foot-pounds”, abbreviated as “ft-lb” or “ft-lbs”. The equation is very general and the mass could be any material, not just water, and the calculated energy potential value would be the same. Bonus points for anyone who can explain why that would be. Hint: Think Galileo. Think of a feather and a hammer, dropped to the surface of the moon as demonstrated by David Scott on Apollo 15:

It’s important to take a moment and emphasize that mass and weight are not the same thing. Mass represents an intrinsic property of an object, while weight is the measurement of gravitational force upon an object’s mass. Since an objects weight is related to its mass by the standard gravitational constant g, they are often interchanged without much explanation or careful consideration to maintaining proper unit representation. For instance, weight is represented by the familiar “pound” (lb or lbs), while mass, in imperial units, is expressed in the less familiar unit of “slug”. Given the unit of measure for the gravity constant is “ft/s2”, the calculated unit of measure for weight (=mg) is actually expressed as “slug-ft/s2” which has been defined by standards to be a “pound”.

For Bagnell Dam, the amount of impounded water is often quoted as 650 Billion gallons or 2,000,000 acre-feet. Since water gallons convert easily into weight (W), and ignoring minor differences in water density, the energy equation simplifies to mere multiplication of weight times height (see derivation on left panel of Fig.1). Assuming a head height of 102 feet, as our rounded five year hourly average indicates and, for the time being assuming we could convert all that energy with perfect efficiency, we can quickly calculate the total potential energy by the equation:

Substituting our values and converting gallons to pounds,

The result, over 550 Billion foot-pounds, is the total amount of energy potential contained by the impounded water. The foot-pound is typically a measure of torque, but when related to time in units of ft-lb per second (lb/s) it becomes a measurement of kinetic energy. The calculated potential however is better represented by horsepower since we are ultimately evaluating mechanical energy. One horsepower is defined to be 33,000 ft-lb/min, or 550ft-lb/s. Converting our result into horsepower yields,

1 trillion horsepower-sec!  At this point, we can calculate the equivalent electrical energy using the fact that 1 hp = 745.7 watts, therefore,

Since electrical energy production is typically sold in units of Kilowatt-hours, Megawatt-hours, or even Gigawatt-hours, we again change the result using the following unit conversion methodology.

A good reference for how much power this is in terms of consumer electricity comes from dividing out the average annual electricity usage per person of 3MWh, established by government standards. Therefore 207,139MWh is enough power for 69,046 people annually. In Ameren’s annual report it is stated that Bagnell produces enough electricity to power about 40,000 homes per year which is arguably more than 69,000 people. How can it be that the total potential energy contained within all of the water in the lake is less than what is annually generated?

To further illustrate the disconnect between static potential energy and actual production ability, consider a dam made to generate the same, if not more power, using less water but a higher head height. Such is the case with another Ameren operated hydroelectric facility, the Taum Sauk reservoir. Taum Sauk actually consumes energy as well as producing it, being more akin to an energy storage facility then a hydroelectric dam, but for our discussion we will only look at its output capabilities. Despite impounding only about 1.5 billion gallons, the merest fraction of the water impounded by Lake of the Ozarks, the Taum Sauk facility has a stated generating capacity of 440MW, nearly twice Bagnell Dam’s stated output. The increased capacity is not due to the amount of impounded water, but the much higher head height achieved by building the water reservoir atop a mountain over 800 feet above the generating turbines. The resulting calculation for the total potential energy for Taum Sauk is,

And converting to horsepower,

Only 18 billion horsepower, again only a fraction of Bagnell Dam’s potential of 1 trillion horsepower.  How is it that Taum Sauk, with much less overall potential energy is capable of actually providing more electrical energy?  The difference involves time.  In both dam calculations, the potential energy equation gives us a value that is instantaneous, as if all of the water could be converted at once.  For Taum Sauk, the substantially smaller amount of water impounded dramatically decreases the calculated potential energy despite the increase in head height. Although it may be able to make more electricity than Bagnell Dam, it can only do so for relatively brief periods.
As we can see, the potential energy equation E=mgh as a representation of gravity’s total potential to do work, is incomplete.  A more useful calculation for determining energy potential comes from the flow of water from one side of the dam to the other.  Flow is intrinsically related to time as we will see and provides a more accurate picture of a hydroelectric dam’s energy production potential.