Can we Overcome the Betz Limit in Windpower Extraction?
October 23, 2009 by Windpower Engineering
Filed under Turbine Design, Wind Watch
Editor’s note:
Inventor Horia Nica has applied for a patent on a device that couples a vertical-axis wind turbine with a device that captures heat from passing wind, a capability conceived by early 20th century inventor Nicola Tesla. Furthermore, the captured thermal energy is transferred to the wind turbine as rotational kinetic energy, according to Nica. The Tesnic Inc website (tesnic.com) shows a few early concepts for such a device intended only for vertical-axis turbines. In a brief interview, Nica said he is now building a working model of the wind turbine and thermal-capture device that will allow extracting more energy from the wind than a turbine alone. The following white paper provides some mathematical validity for the idea.
–Paul Dvorak
On the possibility of overcoming Betz limit in wind-power extraction
Horia Nica,
Tesnic Inc.
Laval, Quebec, Canada
Tesnic.com
The energy in the wind has components of kinetic and heat energy. Existing wind technologies can extract only a fraction of the kinetic energy. The maximum theoretical value of kinetic energy extractable from the wind was demonstrated in 1919 by Albert Betz and it is known as Betz’s Law. According to it, the maximum coefficient of performance (Cp) in kinetic-energy extraction is 59.3%, which is also called the Betz Limit. Existing wind turbines actually have a lower Cp than the Betz Limit. What if the wind turbines could extract a portion of the heat energy from the wind in addition to its kinetic energy?
Assume an ideal wind turbine can extract the wind’s kinetic energy at the Betz Limit, 59.3% of the kinetic and that this ideal turbine has a frontal surface area of 100 m2 (10 m by 10 m). In a wind of 10 m/s and a temperature of 15 C, the energy extracted by such an ideal wind turbine is:
Ek= 0.5ρAV3Cp
Where Ek = kinetic energy in the wind, W; ρ = air density, kg/m3; A = area, m2; and V = wind speed, m/s. Then,
Ek = 0.5 * 1.225 * 100 * 1,000 * 0.593
Ek = 36,321 W , or
= 36.32 kW
After operating for one hour at these conditions, the turbine will produce:
Pk = 36.32 kWh
Now assume there exists a device that, if combined with the above ideal turbine, can extract a portion of the thermal energy in addition to the above calculated kinetic energy. Assume that with the device, a portion of the airflow exiting the turbine is slightly cooler than the input airflow. Assume 50% of the input airflow exits at 0.1° C lower.
In such case, the thermal power calculation is:
Pt = ρQair/hr ΔT Hair
Where Pt = thermal energy captured, kj; Qair/hr = volume of air flow per hour, m3/hr; ΔT = temperature difference, °C; and Hair = specific heat for air; kj/kg. Then
Pt =1.225 *( 100 m2 * 10 m/s * 3,600 s * 50%)* 0.1 * 1.005kj/kg
Pt = 221,602.5 kj
Knowing that 1 kilojoule (kj) = 0.0002777 kWh we obtain:
Pt = 61.55 kWh
The corresponding thermal energy is transferred to the wind turbine as rotational kinetic energy. Consequently the turbine in the above theoretical example, having a device that lowers the temperature of the exit airflow by only 0.1° C will be able to produce a total of:
Ptotal = Pk + Pt
= 97.87 kWh, which is 2.69 times more than the Betz Limit.
Therefore, in theory, wind power extraction can go beyond the Betz Limit and without contradicting Betz’s Law. The capability will result in future powerful wind turbines having smaller dimensions than current designs with the same capacity. Our recent patent application discloses a device able to capture the thermal energy as described above.
For more information, view Tesnic’s white paper here.
WindPower Profitability and Break Even Point Calculations
September 14, 2009 by Taylor Johnson
Filed under Wind Power Projects
I have become increasingly tired of finding comments and discussion around the web, where random people make even more random claims concerning the numbers/money aspects of WindPower Generation. Due to this annoyance, I have researched and provided links for every piece of data you will find in this article. All data provided is backed up by a national/government resource and can be substantiated.
According to EIA (Electricity Information Administration) the average wholesale cost of electricity for 2007 was 5.72 cents per kilowatt hour (2007 is their most recent data). However, according to PacifiCorp annual reports (a Mid-American Subsidiary) the average revenue per kilowatt hour is 7.2 cents, this is the information necessary for calculation not the wholesale value. This statistic however is not constant. It varies by region, state, regulated vs non-regulated, and a number of other things. Some areas of the country have an average cost as high as 25 cents per kilowatt hour. However, for this calculation I will only use 7.2 because first off, this is in a regulated (conservative side of the numbers) area of the country. Secondly because I know these numbers to be factual, not estimated (look up PacifiCorp’s Annual reports for verification).
Calculations
This all being said, let’s get into the calculations.
According to NREL (National Renewable Energy Labratories) the formula for calculating profitability of a
wind turbine/farm is as follows:
Where:
FCR is equal to fixed charge rate
ICC is equal to initial capital cost (cost of turbines, installation, balance of station)
LRC is equal to levelized replacement cost (yearly sinking fund for overhauls and replacements)
O&M is equal to operations and maintenance cost (annual turbine maintenance)
LLC is equal to land lease cost
AEP is equal to net annual energy production in kWh.
This formula will return a net profit, not revenue, not expenses, but total net profit
FCR So, lets assume we are a utility company who is building a 1 MW wind powered plant rather than building another coal powered plant. Since we are a utility company we expect our revenue per kilowatt hour (kWh) to be 7.2 cents as per above (not the wholesale price).
ICC Initial cost of capital is the total cost of the entire installation which according to AWEA (American Wind Energy Association) is $1.3 million dollars per MW or 1000 kW.
AEP From here we get our annual energy production (multiply 1000 kW by the number of hours the energy is produced (9.3 hrs/day * 365 days/yr)3,394,500 kWh’s. This is our estimted annual kwh’s produced or AEP. (9.3 hours per day stat found here)
LRC Next is the Replacement cost, well that is simple enough, if your turbine has a life of 20 years and a cost of $1.3 million. Divide $1,300,000 by 20 years to get your annual levelized replacement cost of $65,000 LRC.
O & M Operations and Maintenance cost simply run 8% of annual gross revenue (AEP * average revenue/kWh) $244,404 * 8% = $19,552
LLC Land lease cost of course is a variable as well but according to our AWEA statistics they run 5% of annual revenue = 12,220
Results
Now that we have actual numbers, not variables, lets calculate
= $0.05608 or 5.6 cents per kilowatt hour.
The next step we should take is to multiply this cost/kwh X total kWh’s produced or AEP
=.05608 * 3,394,500
= $190,363.56
This is the your total annual expenses. From here subtract expenses from Gross Income of (.072 * 3,394,500) 244,404
This is your annual profit.
Now that we know how much we make each year, we need to plug that into a formula to find our ROI (Return on Investment)
If your turbine cost $1.3 million and you are getting a return of $54,041 that gives you an ROI = .04157 or 4.15 %. This is a fairly low number for ROI. Generally companies will require an ROI of 8% or higher if they are to invest in an idea/product.
Another very important figure is the Break Even Point or how long until your investment is paid for.
To figure this, we need the annual profits found above of $54,041 and the total cost found above of $1.3 million.
Therefore, the Break Even Point in our example is 24.05 years. As you can see here, when you have a product life of 20 years, your product will have paid for itself after 24.05 years.
Summary
What do we learn from this? Well at the current costs of turbines, turbine installation, and maintainance, along with the current price of 7.2 cents per kWh, WindPower is not the best financial decision a power company could make. This explains the exuberant amounts of federal grants and stimulus money being pumped into the renewables market. This being said, I still believe that WindPower is the future of energy production around the world, it just takes some time to integrate new ideas into the market place and for prices to become competitive. It is very rare to find a new product/technology that competes economically with an existing product/technology. If we do not start building and expanding now, when it becomes necessary to do so, we will not have the resources and/or infrastructure to make the necessary changes.
