How Much Electrical Power Can Bagnell Dam Produce? Part 5

Efficiency

Part 1

Part 2

Part 3

Part 4

With a refined equation for calculating the potential energy capacity for a hydroelectric dam, we now turn our attention the final parameter that dictates maximum generating capacity, efficiency. The calculated mechanical energy potential for Bagnell Dam was found to be over 426,000 hp, at maximum rated flow, far more than the current design output of 268,000 hp (eight turbines producing 33,500 hp each).  The major difference in mechanical power output from 1931 and now is primarily due to improvements in turbine design, but also due to efficiency. Kinetic energy from flowing water must first be converted into the mechanical energy of spinning turbines, and then converted again into electricity by mechanically coupling those turbines to generators, and each conversion introduces unrecoverable energy losses, mostly in the form of water turbulence, heat due to friction, and even losses due to the inherent properties of electromagnetic devices such as generators.  Despite these unavoidable losses, most large scale dams are quite efficient, often as high as 95%.

Given we now know the potential energy for Bagnell Dam for a given flow, and how much is electrical energy is actually produced, we can determine the overall efficiency of the system.  Hydroelectric dams are basically giant energy converters and therefore subject to the realities of efficiency.  Efficiency of any energy conversion system is simply a representation of the difference between the available energy input, and the usable energy output in another form.  For example, gasoline engines convert the chemical energy in fuel into mechanical energy with about 20% efficiency, while hydroelectric dams typically operate at inefficiencies well into the 90% range.  But how efficient is Bagnell Dam?

Strangely enough, insight into a hydroelectric dam’s overall efficiency lies in the relationship between flow and final water velocity (efflux). In general, for any object in free-fall, a velocity can be calculated simply by knowing the height from which it starts. For fluids, the equation is a special case of Bernoulli’s equation as outlined in the info-graphic below.

Figure 2

The maximum theoretical flow velocity (v) is provided by a derivation of Bernoulli’s equation (see left panel of Figure 2). Notice this velocity equation is exactly the same as it is for calculating the final velocity for any mass dropped vertically from any height h.

Supplying our parameters yields,

Consider a molecule of water that essentially starts at rest, then falls the distance of the head height through the penstocks. No matter the size of the opening at the bottom, the water molecule will have a velocity of nearly 81 ft/sec. Notice that the equation does not take into account the amount of mass dropped, as it is irrelevant (famously derived by Galileo and demonstrated on the moon by Apollo 17). The only parameter is the height to which the mass is raised and the fact it is unhindered by friction or other restrictive forces. At 81ft/s, this means the unrestricted velocity of water at discharge, for any diameter hole, at Bagnell Dam would be about 56 mph!

Photo Courtesy of Missouri State Archives

The above photo is of an open sluice way during the final stages of Bagnell Dam’s construction, a temporary hole later plugged which allowed the Osage River to flow through the dam as it was being built.  In the photo we can easily see the extreme velocity at the opening.  Although the maximum velocity of water for our given head height may be independent of opening size, flow (F) is not.  Bagnell Dam’s penstocks are 19 feet in diameter and therefore the maximum flow possible for the given head height, without restriction of any kind, can be calculated from the relationship of velocity and flow to any opening size of area (A).

First we determine the area of the penstock given it is 19 ft. in diameter. 19 ft. diameter = 9.5 ft. radius

And using our previously calculated maximum velocity based on head height and unrestricted flow,

Therefore,

and solving for F

This is the maximum flow for an unrestricted opening 19 feet in diameter with a head height of 102ft and represents maximum inefficiency (zero energy converted).  23,000cfs is over four times the amount of flow Ameren is allowed to release through a single penstock.  But all penstocks are restricted, by the turbines themselves, and not all turbines even operate at the same flow rate. There are four main 172inch turbine-generators rated at 4210cfs, two 168inch turbine-generators at 5000cfs, and two additional 165inch turbine-generators at 4,556cfs each.  Two auxiliary turbine generators rated at 2.1MW each at 450cfs. 

While some flow is restricted simply due to the shape of the penstocks (the path the water takes to impart energy to the turbines), the predominant restriction is the conversion of kinetic energy into mechanical energy.  How much resistance to flow the turbines actually provide is directly related to efficiency.

At this point a very relevant analysis regarding efficiency can be made, the ratio of restricted to unrestricted flow. When compared to the maximum flow of an operational turbine at 5,000cfs we determine the following,

As we will see, this ratio represents the velocity reduction of unconverted flow, and directly relates to the efficiency of the turbine. Capturing the kinetic energy from flowing water and converting into the mechanical energy of a spinning turbine is the first and primary energy conversion a hydroelectric dam performs.  A turbine is essentially a water-wheel, and Bagnell Dam utilizes one of the most efficient types of turbines available, called a Francis turbine, named after its inventor James Francis.  Unlike the classic water wheels these turbines rotate about a vertical shaft, which keeps water away from the mechanically coupled generator mounted above.  Water is directed at the turbine blades and exits out the bottom through the tailgate.

Bagnell Dam Francis Turbine

Restricting water flow is intuitive concept.  We have all placed our hand into a flowing stream of water and felt the force against it.  The effort to keep our hand in place also requires force and, as Sir Isaac Newton so aptly pointed out, this causes an equal and opposite reactionary force upon the flow of water.  In reaction to our resistance, flow around our hand becomes more turbulent and flow velocity slows down.  Likewise, a turbine spins due to the guided flow of water through it, and its own resistance to movement, being connected to a load (generator), creates an opposing reaction to the water, causing turbulence and velocity reduction in the flow.

Although this interaction between flow and restricting flow may be intuitive, the implications to the energy equation are less so.  We can illustrate the resulting forces on flow using a simple child’s toy, the pinwheel.   The act of blowing air into the wheel (turbine) has two effects; one, the pinwheel begins to turn, and two, the remaining unconverted airflow velocity is reduced and turbulent.  The moving pinwheel accounts for the part of the kinetic energy of the air flow that is converted into mechanical rotational energy while the remaining airflow represents unconverted energy. What happens to the airflow is directly linked to the efficiency of the conversion process.  For a simple toy, the conversion is not very efficient, but for something like a commercial windmill, which is precisely machined and involves smooth bearings, the conversion is relatively efficient.

Replace the air with water, and the pinwheel for a turbine and the same process effectively occurs with a hydroelectric generator.  Water begins to flow through the penstock under the influence of gravity (providing force) into a specially designed channel called a “scroll case” to impart the kinetic energy onto the turbine blade at an optimum angle to make it rotate.  This conversion process dramatically reduces the velocity of the water as it exits the tailgate in comparison to unrestricted flow.  This remaining reduced flow, represents unconverted kinetic energy, and exits tailgate at a specific velocity relative to efficiency.

Looking at it another way, the final velocity is the sum of two velocities, one equaling zero and represents converted energy from mass flow, and the remaining representing unconverted energy remaining as flow.

Since efficiency can never be greater than 100% (unity), and therefore a unity value of less than or equal to 1, substituting our velocity equation, allows the general summation above to be rewritten in following equivalent form.

Since the velocity of the converted flow is zero, our equation reduces to

This allows us to calculate the expected flow ratio if the turbine efficiency is known. For example, a turbine 96% efficient,

The result states that a hydroelectric dam that is 96% efficient will expel water at a velocity that is only 0.2 (20%) of the theoretical free flow maximum.

Since for Bagnell Dam we have already calculated the flow ratio of restricted to unrestricted flow to be .2177 for turbines rated at 5000cfs, we can represent efficiency as a percentage by squaring the term,

Therefore the high flow turbines, rated at 5000cfs, convert kinetic energy to mechanical energy at 95.26% efficiency. But as was stated earlier, despite all penstocks being of the same size, not all of the turbines operate with the same flow rate due to differences in turbines.   How does less flow affect efficiency?  For the lowest turbine flow at 4210cfs we calculate the following flow ratio

Again,

Maximum efflux velocity (v=F/A)

Therefore we see, that greater efficiency is reflected in lower flow velocity.

Now, for turbines operating at 4556cfs we calculate the flow ratio as,

Therefore efficiency for this turbine is 

With a tailgate velocity of,

And finally, calculated again for 5000cfs

for a turbine efficiency of 

And a tailgate velocity of,

Dimensions for the two service generator turbines are not available, but we can assume similar efficiency. We can easily see that with decreasing efficiency comes increasing tailgate flow velocity as less energy is converted into mechanical energy. Less velocity at the tailgate is the result of higher efficiency as more kinetic energy is converted into mechanical energy.

Now armed with the actual kinetic-mechanical energy conversion efficiency for each of the three types of turbines, we can go back and calculate our maximum mechanical energy production.

Recalling,

First we calculate the kinetic energy of the 5000cfs turbines

Using our now confirmed efficiency at 95.23% ( =0.9523), we can now calculate the actual mechanical power of the turbine

Recalling the equivalent horsepower unit conversion,

Allows us to calculate the actual produced mechanical power,

Repeating the calculation for turbines operating at 4210cfs

Using our confirmed efficiency at 96.64% ( =0.9664), we calculate the kinetic energy 

Converting to horsepower

And finally for turbines operating at 4556cfs

Using our confirmed efficiency at 96.06% ( =.9606), we calculate the kinetic energy

Converting to horsepower

With all turbines individually evaluated we can now sum up the total horsepower for the entire dam.

Over 400,000 horsepower is the total maximum mechanical power Bagnell Dam is capable of producing within licensed flow rates that accurately reflects energy conversion losses unique to each penstock.  This mechanical power output of the turbines must now be converted into electrical energy by coupling each turbine to a generator.  Most generators are 98-99% efficient and we will assume uniformity across all the turbines.  Using a 98% conversion efficiency and recalling 745.7W is equivalent to 1hp, we can now calculate the total wattage output for the entire generating plant.

Or about 292.34MW, but does not include the two service generators.

Comparatively speaking, for Ameren's stated capacities for all the turbines, the total power output is 253MW.  Our results indicate a capacity 13% higher.