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5Wschodnioeuropejskie Czasopismo Nankowe (East European Scientific Journal) #12, 2016 üüy
14. M. Iskandarova, F. Atabaev B., Mironyuk N. A. Kadyrova F. D. New technological solution to the problem of producing clinker and the additive cements with a complex manmade mineral ingredients //Materials of international scientific practical conference. Bukhara, On 10-12 November 2015. - P. 255-258.
Bozhko V.S.
postgraduate student of Dnipropetrovsk National University named by Oles Gonchar
THE NEW EMPIRICAL APPROACH TO THE MODELLING OF LOCAL WIND
CHARACTERISTICS
Summary: The study of wind characteristics with the emphasis on the efficient wind energy extraction is performed. The new analytical expression for the wind speed frequency distribution function , which is based on systematic experimental data, is suggested. It is shown that the use of such empirical expression in the wind characteristics analysis can provide more close to reality estimate of local wind energy potential and the energy output of wind turbines which are installed at a particular site area. It can also be used in the procedure of optimal parametric design of wind turbines.
Key words: wind speed, frequency distribution, mathematical description, power density, wind potential, wind turbine, energy output.
Nomenclature
V speed (velocity);
Vw wind speed;
O rotational speed of wind turbine
P power;
Pw power of wind flow;
Pn nominal power of wind turbine;
Pt power of wind turbine;
Pw = P 1 w ■ /S power density;
R radius of wind turbine rotor;
S area;
St swept area of the rotor;
Cp = PT/ P P w power coefficient;
CP =Pe/ Pn capacity factor;
1 = oR/Vw tip speed ratio;
p probability;
t time;
T annual time (8760 hours);
E energy;
E = ■ E/S energy density;
m
c
N
WS
FD
PD
WT
W
T
Index and abbreviations
denotes maximum values; denotes average quantities; denotes nominal parameter; wind speed; frequency distribution; power density; wind turbine; denotes wind parameters; denotes wind turbine parameters.
1. Introductory remarks. Wind energy transformation into electricity over the last few decades experienced significant progress in the world. The substantial growth of wind energy production requires the further study and analysis of wind
characteristics. Obtaining the reliable wind data, primarily on the local level, is relevant to the following topics:
- Selection of a particular site which is suitable for deploying wind turbines (WT);
- Decision of now many and what types of wind turbines can be installed at the particular site area;
- Planning and performance evalution of the energy output and cost effectiveness of the wind energy system.
For this purpose it is important to have detailed knowledge of the site-specific wind data and their analytical description. Wind characteristics were studied by many researchers. Some of the relevant publications are directly devoted to the problem of wind energy extraction [1-9]. At the same time, the modelling and rational mathematical description of wind characteristics, especially oriented on the problem of wind turbine site selection, still needs the further studies. The some new approach to the solution of such problem is presented in a given paper.
2. Methodical background. Wind characteristics and other principal atmospheric parameters are recorded and analysed at the numerous meteorological stations which are scattered around the world. The meteorological statistics includes two types of characteristics - measured and modelled. Among the measured atmospheric parameters are: pressure, density, temperature, humidity and others. The main parameter which is systematically measured is wind speed (WS). There are also several modelled characteristics which are directly related to wind energy extraction. They are as follows: mean annual WS; WS frequency distribution; WS variability; mean annual power density (PD); WS which produces maximum energy; power curve; energy output of wind turbine (WT) and others. The main modelled characteristics and their mathematical description will be considered below.
3. Wind characteristics. Mean wind speed Vc
is determined for a proper period of time (usually for one year, when T = 8760 hours) and given by
1 n
Vc =1 tV
n
i =1
1)
of acceptable WT sites is recommended to be 10 minutes (for the open terrain). Most of the measurements are made about 10m above the ground. But knowledge of WS at the heights up to the hub of WT, i.e. at elevations of 20 to 120m is very desirable. It is known that wind power output is increasing with elevation over the ground and this fact is necessary to count in WT design. In order to estimate WS frequency distribution they are normally measured in integer
values, including zero, i.e. Vj =0; 1; 2... m/s. In this
case mean value of WS should be given by
1 *
Vc = -t mV .
ni=1
2)
Where mj is the number occurrence of Vj, X
is the total number of different (digital) values of WS observed and n is total number of observations.
Wind speed variability is another important modelled characteristic. The deviation of each wind speed
Vj from mean value vc is characterized by so called
variance J, which is presented as
n
=—л t (Vi - Vc )2.
n -1 i=i
3)
The standard deviation is defined as square root of variance.
Wind parameters vary in random way and that is why their analysis is based on the use of probability theory and mathematical statistics. The probability
p(Vj) of the discrete wind speed Vj being observed
is defined as
pV )=m
n
4)
Here Vj is a set of measured wind speeds, and
n is total number of observation. WS is disposed to the strong temporal, spatial and directional variations. That is why, is order to get reliable result it is necessary to follow generally accepted procedure of measurements. It means the use commonly agreed averaging time interval, standard reference height of anemometers over the ground etc. In most cases, the wind characteristics data are typically recorded once an hour (usually, every third hour). At the same time, it is necessary to note, that according to IEC Standard the most common averaging interval used in selection
Here mj is the number of occurrences of a given wind speed Vj, and n is total number of measurements. The cumulative distribution function f (vj ) is WS characteristics, which is the probability that a measured discrete WS will be greater or equal to Vj and is expressed by
F(Vi )= tp(Vj ).
j=1
5)
The discrete WS frequency distribution is usually presented by tables or histograms. The typical histogram is shown in Fig.1. Here and later WS frequency distribution is considered on the annual basis.
p(Vi) 0,02
0,015
0,01
0,005
0
5
10
15
20
25
Fig. 1. Wind speed histogram
For a number of theoretical reasons it is more convenient to model WS frequency distribution by
continuous mathematical function f (Vj). In this
case the function f (V) represents the probability of the WS being within the small interval AV = 1m/s , which is centered on a given V. The continuous cumulative function F(V) represents the probability that given WS is greater or equal to V and is related to frequency distribution function according to relation
f v )=^
dV
(7)
V
F V ) = f f (x)dx, 0
or by the other way
(6)
F(V)
0,5
The normalization of f (v) and F(V) can be presented as
J7 f(v)dV = F (o)- F(«) = 1, (8)
The mean (expectation) value of WS is given by
m(V)= Vc =J0°vf(v)dv, 9)
In Fig. 2 is shown the graphical representation of typical cumulative distribution function F(V) and density function f (v).
f(V) 0,2 0,15 0,1 0,05 0
Fig. 2. Typical curves of F(V), and f (v)
(in this case Vm < Vc ). Such features of f (V) It can be seen that f (v) either equals zero or depend on its mathematical representation. The each greater than zero when V=0, and the peak value of value of f (j ) in Fig. 2 can be considered as rela-f (V) either corresponds to V = Vc or V = Vm
1
0
when Vj is real-
(10)
tive number of hourse per year tj ized and
tj = tj/T,
Here T=8760 house.
There exist several different analytical representations of wind speed density function f (V).
Among them are the most common in contemporary use Weibull and Rayleigh function. The Weibull two-
v B ,
f (v)=A f V
Here A and B are shape and scale parameter respectively. In case of wind characteristics modelling the constants A and B will depend on measured local wind data and can be obtained in the process of fitting the observed WS frequency distribution. The data collected at many location around the world can be reasonably well described by Weibull density function. The substitution of (12) into (9) and integration leads to expression
Vc = ВГ
1+-
v Ay
(13)
{
Where f
1
\
1 + -
v Ay
is gamma function. Expres-
sion (13) contains three parameters - vc, B and A.
Each of them can be determined, when two other are known. Mean annual WS is usually obtained at the most meteorological stations and this fact is used in the process of determination of A and B at the given location.
The one-parametric Rayleigh distribution function can be considered as subset of Weibull, in which the shape parameter A=2.
F (V )
= exp
л 4
/ Л 2
V
V vv c y
(14)
f (V )==Л2 exp
л! V
4 V
4 VVcy
(15)
parametric statistics is borrowed from the probability theory and is actually the generalized gamma distribution. It is presented in the form
F (V) = exp
V
v В y
(11)
exp
(V V В y
(12)
Here vc is mean value of wind speed. It should
be noted that that Weibull function is somewhat versatile and Rayleigh function is somewhat simpler to use. The comparative quality at fitting the wind characteristics data with the use of Weibull and Rayleigh statistics can be estimated only on the basis of addition analysis. One of the difficulties in application of Weibull statistics is quite cumbersome procedure of selection the correct parameters A and B in the analytical description of local wind speed frequency distribution.
It is also necessary to note, that both Weibull and Rayleigh distributions cannot fit a WS frequency curve at zero speed. At the same time it is known that in
reality frequency of calm (when V = 0) is always
greater then zero. The peak value of f (v) in both
distributions does not correspond to mean value of
wind speed vc and it is difficult to explain.
4. Analytical description of experimental data.
To avoid the indicated negative patterns of the above mentioned functions it is necessary to employ more substantiated approach to analytical description of WS frequency distribution. Such approach which is based on the use of systematic experimental data is suggested in this paper. One of the most detailed representation of the large amount of experimental data has been given by Pomortzev in [4]. He analyzed and summarized wind characteristic data for five years period (1887-1891) which have been collected at 19 meteorological stations at the territories of Russia and several European countries. The data are presented in Fig. 3
ÎS
-<^Vc=3 m/s -o~Vc=4 m/s a Vc=5 m/s ♦ Vc=6 m/s
-"-Vc=7 m/s -cr-Vc=8 m/s
f
^........f::4
0 2 4 6 8 10
Fig. 3. Experimental representation of measured wind frequency distribution
12
V, m/s
On the basis of these experimental data, as well as data given in [1-2] and others works, the new em-
f (V ) = fmexp[- g(V - Vc )
In this relation f m is maximum value of the function f (v) which correspond to the mean annual value of each wind speed vc . The value of f m and vc are considered to be known from experimental data. The constant g in (16) is supposed to be different for the left (g = gi) and right (g = g2)
branches of the curves which is shown in Fig. 4. From (16) we will have
ln(fl /fm )
pirical relation for the WS frequency distribution function is suggested in the form
2
(16)
g 2
H. f 2 /fm ) V2 - V )2
Here fi = f (Vi) when j = 2m/s, and fz = f (Vz ), when V2 = 15Vc . The values of fl and fz are taken from the graphical representations of experimental data in Fig.3. In order to widen the range of application of (16) we will use the statistical properties of frequency distribution functions (8) and (9).
gl
Vl - V )
2
(17)
f(V) 0,2
0,15
0,1
0,05
2
f —S 3_
1 я / 1 1 ■ f
1 u ■ 1—■—1—■—
V2
10 11 12 13
14
15 16
17 V
1
1
0
1
2
3
4
5
6
7
8
9
Fig. 4. The graphical representation of empirical relation (16). The numbers 1, 2 and 3 correspond to mean speeds Vc=4 m/s; 6 m/s, 8 m/s respectively
This can help us to determine more correctly the constants gi and g2. From (8) we will have
jfVv = jVCf(v)dV + ÇcAF)jF = 1 (!8)
By substitution of (16) into (18) we obtain
fm
4л
2
1
4g
erf (VcJg! )-
1
= 1
(19)
Where
2
erf (x) = i,
x -t 1 e t dt
(20)
4ni0
The second requirement (9) can be written in the form
Vc = fm
m
л V
2 g £(1- e-gV
2 Л
+ fm
m
4Л Vc
1
As far as vc and f M (as a function of vc ) are known, the constants gj and g2 can be obtained by solving transcendent equation of the type
Q(x) = a (22)
where
g1 =
Л Л 2
x V
V V c y
2 Vg2 2 g 2 g1
(21)
; g 2
,-gVc
(24)
Q(x)
_ erf(x)+y[\-
- x
1 - e
The equation (22) can be solved with the use of successive approximation method, having in mind that
initial approximate value of g1 is given by (17).
The comparison of suggested empirical relation
(16) for f (v) as well as Weibull and Rayleigh func-
2
a =
fmVcJx
x
; and x
= Vcyg
(23) tions with available experimental data for vc =6 is shown in Fig 5.
m/s
When x is determined, we can obtain
f(V) 0,25
2
2 4
1_
0,20 0,15 0,10 0,05
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
V
Fig. 5. The comparison of different WS density distribution function with experimental data 1 - suggested empirical relation; 2 - Weibull function; 3 - Raleigh function; 4 - Pomortzev data
The shape parameter A in Weibull function (12) was determined with the use of relation recommended by [5]
A = dy^V
(
25)
Here d is site-specific constant with average value of 0.94 (its value is between 0.79 and 1.05 for 80% of sites). The scale parameter B was determined from (13). Rayleigh function (15) is written for vc =6 m/s. The presented comparison shows that suggested
in this paper relation (16) reasonably well correlates with available experimental data. The function (16) further will be used for the description of several modelled wind characteristics.
5. Modelled wind characteristics. Power of the wind flow passing through an area S perpendicular to wind direction is given by
PW = 0.5,pV 3S
Here p is air density.
(26)
The important modelling characteristic is wind power density
Pw = 0.5 pV:
(27)
Vc =f V 3f (V )dV, We can write
(29)
r3
(30)
The mean annual wind power density, which can characterize the expected wind energy yield at the selected wind turbine site, is given by
Pwc = 0.5pf^V3 f (V)dV (28)
The upper limit in the integral can be replaced by large but finite value without noticeable loss of correctness. As far as a mean value of a cubed WS will be
Pwc = 0.5 pVc
The substitution of suggested empirical expression for f (v) in (28) and (29) makes PWC easily
obtainable for each value of Vc . It can be shown that
a cubed wind speed frequency distribution is equal to zero at V=0, reaches a peak value at some wind speed
VMc , and finally returns to zero at large values of V . It means that wind speed VMc produces more energy than any other speed. The curve @(V) for different vc is shown in Fig. 6.
V
<p(V) 100 90 80 70 60 50 40 30 20 10
X
\
\
Лу \
w \ ч
4
—o-o—с
9 10 11 12 13 14 15 16 17
V
mc
0
1
2
3
4
5
6
7
8
Fig. 6. The cubed WS frequency distribution
The design of wind turbine should be oriented on The function f (v), given by expression (16),
mc, which is close to the nominal (rat-
the value of VM
ed) wind speed Vn . Wind potential at a particular wind turbine site can be estimated by the value of wind energy density in kWh/m2 which is given by
Pwc • t
Ewc =
1000
(31)
can be used in description of several wind turbine parameters and in its design procedure.
The important characteristic of any wind turbine is its power curve, which is shown in Fig. 7.
P
500 400 300 200 100
0
Fig. 7. Power curve of wind turbine
5
10
15 V
The nominal power of WT corresponds to V = Vn and given by
PN = 0.5pVN • Si (32)
At the power curve are usually fixed: Pn -nominal power, vs - cut-in (starting) wind speed, Vn - nominal (rated) wind speed, vfc - cut-out
Pn = MOTV V3CpV)f V)dV + Pn V f (V)dV
Here Si is swept area of the rotor, Cp is power coefficient of WT, which is expressed as
P (
wind speed. For contemporary wind turbines Vs ~ 3 m/s and Vfc ~ 25 m/s. It can be observed that starting
from V = Vn the power of the turbine is constant
and does not depend on wind speed. The rotational speed m is also constant and is equal to its nominal value. The mathematical description of the power curve on the annual basis is given by
(33)
À = û)R/V,
35)
Cp - P
T/ PW,
34)
In case when rotor radius R and angular speed m ixed, Cp will be the function of V alone. Mean
Where Pw is the power of wind fl°w and PT annual energy density of WT Ec = Ec / S1 can be wer given by
is power of WT. Cp is the function of tip-speed ratio obtained with the use of expression (32) and (33)
E - pT
V V 3Cp (v)f(v)dvft/Wv,
2000J Vs _
Here T=8760 hours, and E is presented in
kWh/m2.
(36)
On this stage the objective function which should be maximized, is power coefficient Cp ,
by
The widely used capacity factor of WT is given where A is given by (35) and a is solidity factor
Cp — P / PA
F = rc' rN, (37)
Where Pc is mean annual power of WT given by
(33). The value of CF shows how WT parameters
are adapted to the local wind characteristics.
Expression (36) can be used in the procedure of optimal parametric design of WT. On the first stage of this procedure design of WT rotor is performed.
<7 —
bjA
R
(38)
Here b is chord of the rotor blade airfoil, l is number of blades. The function C p (A,a) is subject to proper constraints (Cp > 0; Vs < Vn ^ Vfc etc). As a result of optimization we
obtain Cp = Cpm and X = ^, which corre- main PurPose of this PaPer The more information
p pm m related to the problem of wind turbine parametric op-
sponds to this value. On the second stage of optimiza- timization can be found in [2]. tion the objective function^, which should be maxim- in conclusion should be denoted that presented in
ized, is energy density E, which is given by (36). this paper detailed analy^ of suggested description of
local wind characteristics and wind turbine parameters The mam variation parameter is nominal speed vn , shows that such empirical approach can be considered which has the specific value for a selected wind tur- as useful in its practical applications. bine site. The corresponding constraints are used, but they are not described in detail because it is not the
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ДК: 677.027.04
Волков В.А.1, Агеев А.А.2 Volkov Viktor Anatolievitch.1, Ageev Andrey Anreevitch2
1.Московский государственный университет дизайна и технологии.
2.Российский новый университет.
1Federal state-financed higher educational organization Moscow state university of design and technology
(MSUDT)
2Non-state educational institution of higher education Russian new university (RosNOU)
ГРАВИТОМЕТРИЧЕСКИЙ КИНЕТИЧЕСКИЙ МЕТОД ОПРЕДЕЛЕНИЯ ПАРАМЕТОРОВ КАПИЛЛЯРНО-ПОРИСТЫХ ТЕЛ И НОВЫЙ МЕТОД РАСЧЕТА РАСПРЕДЕЛЕНИЯ КАПИЛЛЯРНОГО ПРОСТРАНСТВА ПО РАЗМЕРАМ КАПИЛЛЯРОВ ( НА ПРИМЕРЕ ТЕКСТИЛЬНЫХ МАТЕРИАЛОВ) PARAMETERS CAPILLARY-POROUS BODIES AND THE NEW METHOD OF CALCULATING DISTRIBUTION CAPILLARY SPACE ON THE SIZE OF CAPILLARIES (ON THE EXAMPLE OF TEXTILE MATERIALS)
Аннотация. Предлагается новый метод определения капиллярных параметров капиллярно-пористых тел и расчета интегральных и дифференциальных кривых распределения капиллярного пространства текстильных материалов по размерам капилляров, основанный на сочетании метода измерения линейной скорости подъема жидкости по образцу ткани с определением скорости поглощения жидкости гравитационным методом.
Annotation. A new method for determining the parameters of capillary capillary-porous bodies and the calculation of the integral and differential distribution curves of the capillary space textile size capillaries based on a combination of the method of measurement of the linear velocity of the liquid lift on a sample of tissue from the definition of liquid is absorbed by the gravitational velocity
Ключевые слова. Кинетический гравитационный метод, размер капилляров, распределение капилляров по размеру.
Keywords. Kinetic gravity method, the size of the capillaries, capillary distribution by size
