How Much Electrical Power Can Bagnell Dam Produce? Part 4

Energy and Flow

Part 1
Part 2
Part 3

Since a hydroelectric dam must be capable of harnessing usable work energy, the question of how much energy is available must shift from the amount of impounded water to a discussion of the flow of water.  Water flow, when expressed as mass moving passed a fixed point in space and time, introduces a key element in the discussion of power generating potential, namely time.  By knowing the density of water, and the volume amount of water flowing in time, we can represent flow rate as “mass flow”, which can be substituted for mass (m) in the original potential energy equation.  First we define flow as the volume of water passing through a fixed point over time,

The density of water (designated by p), can change slightly with varying temperatures and salinity but can be considered constant in our discussions given we are talking about a fresh water lake of modest depth.  Density is simply the ratio of any mass to its volume expressed as

Together these two equations allow us to calculate a “rate of flow of mass” representing the amount of mass moving past a fixed point over time rather than volume. Representing mass (m) as a product of density and flow.

Rewriting the potential energy equation (E=mgh) now with the substitution provides a potential energy equation in respect to the rate of mass flow which changes the expression of potential energy to a rate over time and rewritten as follows.

With a constant water density of , 1.9403 slug/cu.ft, an average head height of 102 feet, and a total operational flow of 36,852cfs (adding up the individual penstock flows rather than licensed flow), the total potential energy can now be calculated as

Noting that 1 slug-ft/sec-sec is the mass expression for a single imperial pound (lb) of force and substituting in the above calculation converts the units to the more familiar unit of measure “foot-pounds per second”.

Again converting this kinetic energy, into the mechanical energy equivalent using the following unit conversion 

Therefore,

From this number we can convert to watts again using a unit conversion of 745.7W for each horsepower.

Or about 318MW. Notice that the amount of energy that can be theoretically produced is dependent on only two parameters, head height and flow, not the amount of water impounded as the original potential energy equation would lead us to believe.

If Ameren released 36,852cfs continuously, and subtracting the average inflow of 10,500cfs for a net outflow of 26,352cfs, how long would it take to drain the lake? Since 1cu.ft = 7.48 Gallons, at maximum generating flow the Lake of the Ozarks loses 197,113 gallons every second. At 650 billion gallons it would take only 38 days to empty. Comparatively speaking, it took over a year for the lake to reach full pool after Bagnell was completed.

318MW is the theoretical generating capacity, based strictly on calculated energy available from the amount of water flow passing through the penstocks, but it is not the actual output.  Because no energy conversion is ever completely efficient, there will always be unrecoverable energy losses during the process. For hydroelectric dams kinetic to mechanical losses are encountered in the form of flow turbulence and friction, but most relevant and somewhat counter-intuitive is the fact that a 100% efficient hydroelectric dam would not work at all as will be explained.

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