The future of the world wind power.
(The part 2, is written in June, 2008.)
About the first part
Article has been written in July, 2006
and corrected in March, 2007.
It became result of long-term supervision
of tendencies in wind
and also result of comparison of these tendencies with real aerodynamic calculations.
The aerodynamic calculations applied to article, are based on the theory of the Soviet scientist – professors G.H.Sabinin (the pupil of professor
N.E.Zhukovsky). In 1929 there were no computers,
and calculations were fulfilling with the help of slide rules. For this purpose the formulas were
as possible reduced with introduction allowable then errors. I have restored their full kind.
Besides I have counted G.H.Sabinin's
formulas under modern aerodynamic factors Cy, Cx and
k = Cy / Cx. Diagrams of family of aerodynamic characteristics Cy = f (α); Cx = f (α)
for airfoil Espero
I have presented as formulas of approximation, and function å = f (À) (where A = å/(1+å)/(1-å)2)
as the equation of the third degree using the formula of the Kardano.
The main merit of prof. G.H.Sabinin in wind power for ever remain the presence proved to him so-called
« the affixed weight » as a result of
which the greatest possible part of energy which can be taken from an ideal rotor makes 68,6 % (instead of 59,3 % on A.Betz). As appeared, almost all
world does not know about it and counts aerodynamics of rotors using the formulas A.Betz. A limit of 59,3 % name a limit and even law A.Betz.
It is time to remember, that the law or limit A.Betz does not
There is more exact limit – G.Sabinin's limit.
Analyzing dependences of aerodynamic losses from various parameters and factors, I managed to find
a variant of a design of a rotor which has much
smaller losses and the best efficiency. Side benefits
of a design were found out also. I have checked up the received preliminary conclusions
calculations. Preliminary conclusions have completely proved to be true. Results of all work and calculations are submitted in the first part of article.
Also corresponding applications for inventions have been sent.
Taking into account importance of article and calculations, I have decided to publish them in one
of known wind power editions or on a website of
anyone known wind power the organizations
or associations. Since October 2006 till January 2007 I have directed article with calculations
addresses of 136 known wind power firms and the organizations. However in the common stream of the information the article about prospective following
generation of wind turbines appeared underrated and unnoticed. Probably, the insufficient analysis of a stream of the information
also is the reason
why till now aerodynamic calculations fulfilling using the formulas A.Betz.
The full version of the first part of article has been placed on this website since August, 2007.
What would I change in the first part of article now?
Strangely enough, for last 2 years of qualitative changes in designs of rotors has not taken place almost. Therefore, the first part of my article
remains actual, as well as earlier.
However there are the aspects deserving more attentive studying.
For example, in the first part of article, at my present sight, the kind of towers as trellised designs
(the example of such towers is made by
company SeeBa) is completely unduly forgotten. Besides,
now I think this kind of towers the most perspective for the superbig turbines with
diameters of a rotor up to 250 – 300 m. The offshore variant of such designs is convenient also when for each " foot"
of a tower all over again in
water the steel or concrete support is put, then on them already above water the tower fastens in parts.
One more advantage of the superbig turbines about which it has not been told in the first part of article, the increase in their service life down to
50 years (in addition decreases the cost price of energy) is.
As the common sizes of elements of a rotor is increase, the degree of influence on them
of external factors (the sun, humidity, etc.) is decreases. Besides to reduce this degree of influence,
due to the collected experience, development
of technologies allows. To prolong service life of a plenty of less perspective and less powerful existing designs – it is economically less justified.
Important I think also a variant of a new design of a rotor at which along with an external ring
of the aerodynamic form there is also at least
one intermediate. This variant means division of blades
into internal and external parts. Such division considerably simplifies manufacture and
delivery of long blades. The opportunity of separate regulation of adjusting corners of external parts of blades also becomes simpler. Thus there is an
opportunity of providing of the maximal unloading of external parts and an external ring at speeds of a wind is higher rated, that is desirable in view
of narrower and thin external parts of blades. Connection of shaft of blades is carried out with the help of an intermediate ring and can be both rigid,
and adjustable. All rings will consist of the separate segments connected
at installation. The quantity of segments for unification is multiple to
number of blades. Appointment
of an intermediate ring is stabilization of the middle of long blades, additional removal from blades
external ring of a part of loading and simplification of connection of parts of blades among themselves. In such variant of a design the loading for
blades decreases and at installation of long blades. For rotors with diameters 150 – 240 m. sufficient will be presence of one intermediate ring,
diameters 250 – 300 m. – two.
One of lacks of the superbig turbines is reduction of speed of rotation of a shaft of the turbine, demanding increase in transfer number of a gearbox.
Therefore, for the superbig turbines
will be perspective most likely, application of 2 step planetary gearboxes, together with the multipolar
In the calculations applied to the first part of article, I did not consider losses from braking of rotation
of a rotor due to friction of air
about an external ring. These losses will lower the general Ñð to size
56 – 58 % even at use of aerodynamic airfoiles with high quality and by good
optimization of the sizes
of blades and rings. In the calculations applied below, these losses are taken into account.
Some experts criticize the design of a rotor described in article, speaking about substantial growth
of its weight, complexity of manufacturing
and installation, and also about obligatory application
for its installation of the high powerful crane. In my opinion, first, the further
development of rotors
of wind turbines with 2 – 3 blades practically has exhausted itself, as already there is no opportunity considerably to
increase diameter and efficiency of a rotor. The increase in the sizes will cause to increase of losses because of increase in torsion speed (especially
at 2-blades rotors), to sharp increase in weight and in cost of blades, complication of delivery of components of the turbine, is especial blades.
Second, the criticism is justified but not completely.
The weight of a rotor (at transition from a rotor with 3 blades and at preservation of the sizes)
for a rotor in diameter of 120 m will
really increase, however no more than in 2 times. The reason -
each blade is much easier and cheaper, as they narrower, thin and have more thin
Rings of a rotor for the same reasons very heavy will not be. It is easier than the blade,
than in traditional turbines to do built-up,
that will reduce the price of their manufacture and delivery. The weight and cost of a rotor of a new design in diameter 200 - 250 m will be comparable
with weight and cost of 3 -blades a rotor of the same diameter at 2-3 multiple increase in gathering of energy.
The main advantage there is an
opportunity of substantial growth of the sizes of a rotor without deterioration of aerodynamic quality as torsion speed decreases, there are no ended
and there is an opportunity to increase rated speed of a wind because of reduction of loading by blades.
Complexity of manufacturing of elements of a rotor in diameter more than 150 – 200 m will be less, than at traditional 3 –blades the same
diameter, and cost of manufacturing due to quantity of elements will increase not on many as manufacturing of separate components and their delivery
It is not necessary to forget thus, that on total gathering energy for a year, one new turbine replaces
2 – 3 traditional the same
Installation of a rotor of a new design not necessarily demands the expensive equipment.
It can be carried out in more perspective ways,
for example, with the help of the small crane
on a platform which fastens to tower. In process of construction of a tower such platform moves above.
After construction of a tower from same platform the cabin of the turbine with the generator, a reducer and other equipment is established. Instead of
usual further fastenings 2 – 3 blades, are exact
as from same platform there is a fastening 8-9 internal parts of blades. Then the platform moves
below for fastening segments of an intermediate ring to the ends of internal blades and among themselves. Then, external parts of blades and an external
ring similarly fasten. Complexities any are not present, though time for installation is spent more. At that the sizes and capacity of a rotor increase
All above reasons prove that the suggested design of a rotor undoubtedly is more perspective, and to it all manufacturers of powerful turbines
sooner or later will come. Those people
and the companies who it will understand before others and will start to introduce the first
technology, those appear ahead of the others. All others will be late.
Taking into account small popularity of the theory of G.H.Sabinin, I have decided to add to article
the chapter about its difference from theory
A.Betz and about technique of aerodynamic calculation corresponding to it. The full enunciating of the theory of G.H.Sabinin occupies a lot of place and
will be hardly interesting to the majority of readers. For those who wants to familiarize with the theory more in detail, besides the primary
source which name is given on page F.A.Q., there is a source
with its partial enunciating (in Russian) is
E.M.Fateev's book “Wind engine and wind installation” (1948).
Aerodynamics of a wind rotor under G.H.Sabinin's theory.
The classical theory of an ideal wind rotor has been developed A.Betz simultaneously and independently with professor N.E.Zhukovsky in 1920 and used
for calculations till now. G.H.Sabinin's more exact theory has appeared in 1929 and is published in 1931. Its difference from former theories consists
that at definition of axial force of pressure of a stream on a wind wheel the impulse of forces is counted up on the vortical solenoid in that place
where it has accepted already established cylindrical form, instead of at the moment of its formation as it was done with former theories. G.H.Sabinin's
for the first time has proved presence of the additional (affixed) weight of air participating in formation of the total twisting moment of a
rotor. Consequence of it became the increase in coefficient
of a use of a wind power of an ideal rotor shown on the chart.
||2V1 / (1 + V1 / V)|
|Â =||4å (1 + å)||4å / (1 + å)|
|Ñði =||4å (1 - å)2||4å (1 - å) / (1 + å)|
|å at Ñði max =||0,333||0,414|
|Â at Ñði max =||0,888||1,172|
|Ñði max =||0,593||0,686|| ||
Before and further reductions are accepted:
||- Auxiliary function, À = å / (1 + å) / (1 - å)2|
|b||- Width of the blade, m|
|Â||- Coefficient of loading on the disk area, Â = 4å / (1 + å)|
|Ñð||- Efficiency of a wind power, Ñð = P / P0|
||- Efficiency of a wind power of an ideal rotor, Ñði = 4å (1 - å) / (1 + å)|
|Cx||- Profile drag coefficient of the airfoil|
|Cy||- Lift coefficient of the airfoil|
|c||- Thickness of the airfoil, m|
|c_||- Relative thickness of the airfoil, c_ = ñ / b|
||- Velocity drop coefficient in the plane of the rotor, å = V1 / V|
|Fð||- Force of pressure upon a rotor, n|
|i||- Number of blades|
|k||- Coefficient of quality of the airfoil, k = Cy / Cx = 1/ μ|
|n||- Number of elements (of segments) of the blade|
|nc||- Number of cycle of a rotor in a second, turnover/s|
||- Number of cycle of a rotor in one minute, rpm, Nm = 60 nc|
|P||- Power of a real rotor, W|
|P0||- Full power of the stream in a plane of a rotor, W|
|Pj||- Loss in power due to induced drag of blades (trailer losses), W|
|Pm||- Loss in power due to twisting of the stream, W|
|Pp||- Loss in power due to profile drag of the blades, W|
|r||- Average radius of an element of the blade, m|
|r0||- Inside radius of a rotor, m|
|R||- Outside radius of a rotor, m|
|S||- Disc area of the rotor, m2, S = πR2|
||- The area of a separate ring of a rotor for the segment of the blade, m2|
||- Circumferential velocity of rotation of the rotor, m/s, u = ωr = 2πrnc|
||- Circumferential velocity of rotation of the stream in the plane of the rotor, m/s|
||- Circumferential velocity of rotation of the stream behind the rotor, m/s|
|V||- Velocity of flow far ahead of rotor (at height of its axis), m/s|
|V1||- Change in velocity of flow in the plane of the rotor, m/s|
||- The full lost velocity of the stream far behind the rotor, m/s|
|W||- Relative velocity of a stream, m/s|
|z||- Number of modules at radius r, z = ωr / V|
|Z||- Number of modules on the end of the blade, Z = ωR / V|
||- Number of relative modules, zu = (ωr + u1) / (V - V1)|
|α||- The angle of incidence – the angle between a chord of an element
of the blade and relative velocity, deg., α = β - φ - γ|
|β||- The angle of the relative velocity W with the plane
of rotation of the rotor, deg., β = arcctg zu|
|γ||- Twist of the blade – the angle between projections
of chords of an initial and current element |
of the blade, deg.
||- Coefficient of inverse quality of the airfoil, μ = Cx / Cy = 1/ k|
|ρ||- Density of air, kg/m3|
||- The angle between a chord of an initial element of the blade and a plane of rotation, deg.|
|ω||- Angular velocity of rotation of a rotor, 1/s|
Technique of aerodynamic calculation.
The technique of the calculation, as well as in other theories, is based on splitting of the area of a rotor into separate narrow identical rings on
width. These rings as if cut the blades onto the separate elements (segments), for each of which independent calculation is carried out. Parameters of
elements of blades in a ring are accepted identical and summarize. Then the forces and powers calculated
for everyone ring are summarized in final
result. The quantity of rings gets out of reasons of sufficiency at preservation concerning a small difference in initial parameters of the next elements.
The quantity of rings is usual is in limits from 7 up to 20 and defines an error and complexity of calculations. An example can be
calculation of a rotor in diameter of 240 m, and also calculations to the first part of article.
More often the purpose of calculations is the deriving of characteristics of powers and forces,
and also adjustment of the sizes of each element
of the blade (width, thickness, corners of the twist and installations) at the preset sizes of a rotor and known aerodynamic parameters of its elements.
During calculations such parameters as rated speed of a wind, width and thickness of elements
of blades, speeds of rotation of a rotor, corners of
the twist of the blades and others, can be corrected for improvement of the general result.
After the preset of radius of a rotor and number of segments, calculations for each element begin
with definition of their average radiuses and
their width. Δr = (R - r0) / n; r = (rmax - rmin) / 2.
Then the square of a ring for each segment of the blade and full wind power before a ring
for each of speeds of a wind is calculated Sr
= 2πr Δr; ΔP0 = ρ Sr V3 / 2.
After the preset of a preliminary range of speeds of rotation of a rotor for each of speeds of a wind
for all segments calculate is a number of
modules z = 2πr nc / V.
Before to continue calculations, it is necessary to preset number of blades, their preliminary sizes,
i.e. width and thickness of each element
(segment). Aerodynamic characteristics Cy = f (α)
and Cx = f (α), corresponding to concrete relative thickness of blades,
are represented as formulas of approximation. For this purpose section-nonlinear approximation is usually used. This artful name means reception of the
formula of the function consisting of sites of known nonlinear functions (powermode, exponential, logarithmic and others), carefully connected among
themselves. The schedule of resulting function should coincide with the schedule of the corresponding aerodynamic characteristic.
The following phase consists in calculation for every wind speed and for each element of the blade
of concrete values Cx, Cy, å, À,
zu, α, β, at selection optimum φ and γ. Apparently from formulas
¹¹ 1 – 3 and formulas of
calculation Cx, Cy, α and β, all these factors are connected among themselves so, that the slightest change of one of them
(for example, at change φ and γ) is given
in respective alteration of all others. Therefore, the criterion of optimality
is necessary for definition
of an optimality of selection φ and γ. As such criterion the maximum of resulting power and a
of the sum of capacities of losses usually serves. Taking into account it, the cells of table Excel containing calculation of powers and
other results (formulas 4 - 11), it is necessary to fill before calculation of factors and corners.
Formulas on which calculations are made are below submitted.
Before filling of cells with factors and corners it is necessary to check up adjustment of table Excel for actuation in it of cyclic references (it is
a regime of calculations when the calculated parameter depends on other parameter which calculation depends on value of the first parameter). For this
purpose in open table Excel in the menu "Service" it is necessary to press "Parameters" and to choose "Calculations", where to establish a badge of
"Iteration" with limiting number 100 and a relative error 0,000001.
Calculations are more convenient for carrying out separately for each segment of the blade
(a column of the table), selecting optimum corners
φ and γ, other parameters of an element
of the blade and summarizing final results for each of the chosen speeds of a wind.
of powers for speeds above than nominal owing to the nonideal of the twist of the blades for different speeds of a wind can appear
negative values å, À, Ñð and ΔÐ, testifying about braking an element
of the blade concerning all blade.
After the preset of preliminary values b, nc, φ and γ, at filled other cells, most likely the majority of cells
of the table will show a error of calculations. It is allowable, as optimization was not carried out yet. Optimization is carried out separately for
each element of the blade and for each chosen speed
of a wind. It is the best way to begin with cells with γ = 0 and φ = 0,
choosing optimum b and nc.
If in corresponding cells a error it can be cleaned, having preset in a corresponding cell with
the formula of calculation zu instead of the formula the concrete number close to expected. After that it is necessary to replace the
substituted number the formula from the next cell. After optimization b and nc for γ = 0 and φ = 0,
it is necessary to optimize φ. Similar operations are carried out for other cells, choosing optimum b, nc, φ and
γ. After optimization of all cells it is necessary to check up some times still carefulness of a choice b, nc, φ
and γ, changing the corresponding parameter on small size.
The part of above mentioned formulas differs from G.Sabinin's formulas that here they is outcome
of the discrete summation of concrete values for
concrete elements of blades that usually is more exact, as against the integrated summation of the values average for all blade used by G.Sabinin.
Aerodynamic calculation of a rotor of the wind turbine
in diameter of 240 m with
8 blades, external both
intermediate rings and power 120 MW.
Calculation is carried out by the mentioned above technique and exhibit as table Excel.
Difference consists that in this rotor additional regulation of adjusting corners of an external part
of the blade is used. Therefore, instead of
corners φ are calculated φext, φint and Δφ (φext corresponds
φ to an external part of the blade, φint – internal, and Δφ – a corner of adjustment between
them), corners γ are calculated for each part of the blade separately. The coordinate of a chord of an external ring on radius of a rotor
corresponds to a point of 120 m, and intermediate – 60 m.
Existing theories and techniques of aerodynamic calculations of rotors of wind turbines do not provide presence in rotor external and intermediate
rings, and, means, do not describe influence of these rings on aerodynamic result. However after some analysis this influence becomes clear. It is
carried out through elimination of ended losses (more precisely speaking, of the inductive slope a stream from blades, leaving the slope a stream in
front of the blades, inherent in an ideal rotor), and also
to emergence of losses of braking of a rotor owing to friction of air about walls of
it is insignificant, but frontal pressure upon a rotor raises due to pressure upon rings. Calculation of force of such pressure and
power of losses of braking is carried out in formulas 10 and 11. This calculation
is based that air bends around an airfoil of a ring in a corner
dependent on number of modules z.
The projection of force of friction of air to a chord of a ring along a stream will give force of pressure,
and the projection of force of friction in a plane of rotation will give force of braking.
Results of calculations are shown in Tab. 1.
|Wind speed, m/s||5||6||8||10
|Hub power, MW
|Force of pressure, Mn
The general parameters of a rotor the following: rated power 120 MW, speeds of a wind – incipient, nominal, maximal and allowable accordingly – 5, 20,
40 and 90 m/s. Speeds corresponding to them at height of 10 m – 3,2; 12,7; 25,4 and 57,1 m/s. External and intermediate rings have a symmetric airfoil
with relative thickness of 20 %, in the width 2,5 m and 4 m accordingly, Cx = 0,01. The external part
of the blade has length of 59,4 m, width 2,3
– 5 m. The internal part of the blade has length of 54,6 m, width – 5 – 9 m. Speed of rotation from 3,3 up to 9,9 rpm. Calculation of durability was not
carried out, however it is supposed, that due to redistribution of loadings of the blade it is possible to make even narrower, a little having increased
speed of rotation.
On the second sheet of table Excel calculation of mid-annual manufacture of the energy received
such rotor, and its comparison with similar calculation of manufacture of energy for already existing rotors power 5 MW for districts with various wind
classes and parameters Weibull K is shown.
Results of calculations are shown in Tab. 2.
Tab. 2. Gross annual production of wind energy, GWh / year|
|K = 1,5, for classes||
K = 2,0, for classes||K = 2,5, for classes|
|Rotor 5 MW
|Rotor 120 MW
Calculations show, that the suggested rotor on total energy for a year is similar 8 – to 16 rotors power 5 MW and a benefit is it more,
than more mid-annual speed of a wind and parameter Weibull K
of district of installation of the turbine. They also show, that the new rotor though
has rated power
120 MW, but on summarized energy is similar to a classical wind power station total power 40 - 80 MW. At more careful optimization
of blades in view of calculations of durability of a design and,
applying the dual generator, the general benefit will increase in addition.
The author of article: Izosimov Evgeny, Ukraine, Belaya Cerkov
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