Output prediction for wind turbines in Weibull-distributed wind regimes

Citation for published version (APA):

Lysen, E. H. (1981). Output prediction for wind turbines in Weibull-distributed wind regimes. (TU Eindhoven. Vakgr. Transportfysica : rapport; Vol. R-475-D). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981 Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

providing details and we will investigate your claim.

III1II1

IIIII1

*9305811* EINDHOVEN oo'fwikke#ingssamenwerkin1

f..H.

iEiilfKi/rroven -

gebouw

0

DOCUMENTATIECENTRUM 8.0.S. - T.H.E. class.

!AIJJ/f)I. II

dv. J datum

I

OUTPUT PREDICTION FOR WIND TURBINES IN WEIBULL-DISTRIBUTED WIND REGIMES

by E.H. LYSEN

April 1981

WIND ENERGY GROUP Department of Physics

University Technology Eindhoven

CONTENTS Page .List of symbols ii

-.

I. INTRODUCTION 2. OUTPUT CHARACTERISTICS 2 3. WIND REGIMES 5 4. OUTPUT PREDICTION 10

4. 1 Ideal wind turbine (cubic output) 11

4.2 Wind turbine with quadratic P(V) curve 13

4.3 Wind turbine with linear P(V) curve 14

4.4 Constant torque 16

4.5 Comparison of the four P(V) curves 19

5. EXAMPLE 20

6. REFERENCES 22

GRAPHS 23 to 34

A swept rotor area m

C power coefficient

p

c Weibull scale factor mls

e_system dimensionless energy output

F(V) cumulative distribution function

F

fey) velocity frequency distribution function slm

G gamma function

k Weibull shape factor

L ratio A_maxlAd

p p r p ref T

v

v

v

c power rated power reference power: (C n) !p A V 3 p max time wind speed

average wind speed

cut-in wind speed

design wind speed

furling wind speed

rated wind speed

W W W s mls mls mls mls mls mls

x dimensionless wind speed (= VIV)

x dimensionless cut-in wind speed (= V IV)

c r

x

d dimensionless design wind speed (= V/V)

x

f dimensionless furling wind speed (=Vf/V)

x dimensionless rated wind speed (=V IV)

r r

n efficiency

Ad tip speed ratio at design speed

A_max max. tip speed ratio

3

1. INTRODUCTION

Predicting the energy output of a wind energy convers~on system via

mathematical representation of both wind system and the wind regime becomes a well-established procedure.

In this paper we have chosen a Weibull distributed wind regime and a linear output curve for the wind system. The latter is questionable, although many of the small and large wind systems recently measured 'show this more or less linear behaviour between the cut-in wind speed and the rated speed. This is why we also compared the linear output with three other types of output curves, culminating in the cubic response, the ultimate goal of every

wind system designer.

The energy outputs are presented in graphs 4.5 - 4.10 at the end of the paper, showing the .dimensionless energy output as a function of the relative design wind speed for different values of the rated wind wind speed. The influence of the furling wind speed is shown in a

separate set of graphs 4.11 - 4.16. An example of how to manipulate

with the dimensionless quantities is given in section 5. The values

of e t are calculated by means of a programmable calculator

sys em

2. OUTPUT CHARACTERISTICS

The output characte'ristic of a windmill can _he determined by measuring the energy output and the average windspeed during a short time interval of say 5 or 10 minutes. Plotting the average power in this interval as a function of windspeed will result in a cloud of points, such as in fig 2.1.

kW Generated power 60 50 3 20 10 4 5 6 :...

..

' .-

_..

..

_.

_~

_.

7 8

..

.

..

;. '

.

.

,

.

windspeed 9 10 11 12 13 m/s

Fig 2.1 Typical example of a power output curve of a wind turbine,in this case a Swedish 60 kW turbine [1] .

The shape of this output curve between V_cut · and V t d can

~n ra e

be described by various analytical functions, dependent on the

type of wind turbine. For the ideal windturbine the shape is a cube, but in practice a square or linear curve is a better approximation for the measured data (fig 2.2).

a c

v

_c b I I I

v

V r r V_ d

pi

/I

pi

II I I I I / I I / / _/ "t. / I ... _I

-

I V_c V_f V_c V_r V_f

y

-fig 2.2 Four different shapes of the output curve of a wind

turbine between V and V

c r

a ideal, i.e. cubic

b quadratic

c linear

d derived from constant

Algebraically the shapes of the output curves lietween V and V

c r

can be written as follows:

cubic : quadratic linear : "constant torque load": 3 P(V}= a V 2 2 P(V}= a(V -V_{. c} } P(V}= a(V-V } V4 c 2 P(Vl= a(V + b VcV 1 c (V =O} c (2.1 } (2.2)

(2.3}

(2.4}

Given the fact that in many cases the output curves are nearly linear, most graphs of section 4 are devoted to this linear output. For comparison purposes we have calculated the output with four different output curves in one wind regime

(figs 4.3 and 4.4)

To introduce the idea of the design speed we have to realize

that for all non~cubic output curves the conversion efficiency

C .Q of the wind system will vary as a function of V. This

p

means that at one windspeed V

d the system reaches its maximum

conversion efficiency (C n) • This windspeed is called the

p max

design windspeed of the system. It turns out that for the three non-cubic output curves a unique relation exists between V

d and V :c quadratic: linear :. "constant torque load": (2.S} (2.6) (2.7)

These values can be found t k 'a ~ng t eh d 'er~vat~ve. dPdV = 0 of the P(V) curves mentioned.

3. WIND REGIMES

There is a growing evidence that the velocity frequency dis-tribution of a wind regime quite satisfactorily can be approxi-mated by means of so-called Weibull functions [2], shown in

fig 3. I : -.

with: k: Weibull shape factor [-]

c: Weibull scale factor

[m/s]

[s/m]

(3. I)

From a given set of winddata the Weibull shape factor can be determined by plotting the cumulative distributed data on a

so-called Weibull graph paper (fig 3.31 of which the

verti-cal axis bears the cumulative distribution function (fig 3.21

The scale factor c is a function of the average windspeed V and the shape factor k via:

I

V = c

r

(1+ -)

k

By introducing a dimensionless windspeed

V x = -V (3.2) (3.3) (3.4)

I. 4 , -_ _- , .--_ _-,- , -_ _----, ---,- _ jf(x)

I

I.2f---+---+--+--\----t---+----+---j~--__j

o.

8f--~-__+_-.,f--_I_-+--+-+---__+---__+_---+_--___j

o.

6f---,~_Pfl:_f_----'~+_----3oc---\\<+---+_----+----t---j 0 0.5 1.5 2 2.5 3 3.5 V x = - + V I. k 1.0 4. k = 2.5 2. k 1.5 5. k = 4.0 3. k 2.0

Fig 3.1 Wind velocity distribution functions, represented by Weibull functions for different values of the Weibull shape factor k

I.0

,---r----:::;;;;;o"'-..,..--=_-::::::=---,

r

F(x) O. 8 f---f---I---I-1'--+.~---!---__j 0.6 f . - - - + - 1 - R - - - + - - - j 0.4 0.2 f--I--I:-J.--I-I---+---+---j 4. k = 2.5

o

1. k = 1.0 2. k = J.5

LO

3. k 2.0 2.0 5. k = 4.0 x ₎ 3.0

Fig. 3.2 The cummulative distribution function F(x)

as a function of the reduced wind speed; the shape factor k is used as parameter.

4 3 2 1

I

I I I I I I I I I

I

I I I I I I I I I

I

I I I I I I I I

+-1

k·axis 95 1ttttt 90 80 70 ~ 60 50 40

-

>

-1.1.. 30 c .2

...

25 ~ _1-." .0 ₂₀ 'C

...

en "U _1S OJ > ·z ctI ₁₀

-

~

E

6 ~ u 6 5 4 3 2

+

. .

- point c-estimation 2 3 5 6 8 10 15 20 25 1)

Weibull probability paper for wind energy studies.

Cumulative distribution functionFlv)versus

winds~v.

Wind Energy Group, Dept. of Physics, Uni•• of Technology, Eindho..n, Nl1herlands.

Fig. 3.3 Weibull paper

f(:xl=

k

k-l -Gx

G.k.x • e (3.5)

with the gammafunction: G(k} = rk(l+

t )

A reasonable approximation of this gammafunction for 1.5<k<3.5 is given by:

G(k)

=

0.2869 . k- 1 + 0.6880 . k- 0.1

(3.6)

(3.7)

In coastal areas the shape factor of the Weibull distribution

generally is around k = 2. In this situation the velocity

distri-bution takes a relatively simple form:

k

=

2 (3.81

4. OUTPUT PREDICTION

The average output E of a wind turbine with an output curve P(V) in a wind regime characterized by a velocity distribution f(V} in a period T is given by:

co

E T

J

P(V}, f(Vl d V

o

In the case of a constant output power P

r between Vr and Vf

(4.1)

the expression becomes:

V r E =

T~P(Vl

f(Vl d V+ V c (4.21

For dimensionless windspeeds one becomes:

x x

r f

E = TJP(X

1

f(xl dx + _TPr

f

_{f(xl dx} (4.31

Xc _{x r}

As E(xl is still in joules (or watt hoursl preferably a

di-mensionless energy output e is utilized:

system (4.41 E P_refT E e = - - - = - - - = -system ( ) 1 2 PAV· 3_T cpn max11

The result is:

e_system

=

x r pi jP(X

1

f(xl ref x c x f dx +

;r

j(X}

dx ref x r (4.5}

This expression will be worked out for a number of output curves P(Vl, or better P(xl.

4.1 Ideal wind tur1ll.ne: cubic p(Vl curve

The ideal wind turbine possesses a cubic output curve between

V_c and V r: or: or:

-.

P(Vl = C_{p max} Tl

!p

V3 A P(x) = C n

!p

x3

ti

A (4.6) p max P(xl P 3 ref x

Above V and below V_{r f} the output is assumed to remain constant:

P(V) = P r (4.71 P(xl P 3 or: = x ref r

With Weibull distribution as descrihed in section 3 the

dimen-sionless energy output becomes:

{X

k+2

e = G

system

with (3.61

This expression for e t is shown in fig 4.1 for different

sys em

values of k, with x

=

0 and x = 00

c f

1T 2

- 7; x

e dx +

For k

=

2 the expression becomes:

J

r 'IT 4 esystem=

2"

x x c 'IT 2 'IT 2 3( - -4 x - -4 _{x f \} x e r - e J r

k=l .5 .k=2 k=2.5 k=3 k=3.5 x 0 c ind turbines x r Output ideal 2.0 1.5

o

1.0 k=3.5 k=3

~

k=2.5 k=2 k=I.5 I

o

.5 I k=1

r

I

e system

o '

2 3 V r ) x r

Fig. 4.1 The value of e as a function of the

system

relative velocity x, for several values of the Weibull shape factor (related to a wind turbine with an ideal output characteristic).

4.2 Wind turbine with quadratic P(V} curve

(4.10) V 2_ V 2

r c

P(V}

=

P_r

The quadratic P(V} curve chosen is [4]: V 2 _ V 2 c (4.11) p(V) = P r %

This is a special case of a gemeral formula adopted by Powell [4] }:

Vk _ V k c

This formula is chosen because it yields an analytical expression for the dimensionless energy output:

-Gx k r ) _ (x k _ x k) r c e-GX /

J

(4.12}

The design windspeed of the windturbine given by (4.111 is:

or, for k = 2: V_d

=~.

V_c (4.13)

In the quadratic case, with k = 2, the energy output becomes:

'IT 2 - - x (e 4 c 'IT 2 - - x e 4 r} (x 2- x 2} r c

-2!.x

2]

e 4 f (4.14)

~} NOTE: This general formula is rather questionable, because

P(Vl now becomes a function of k. Here we will limit

4.3 Wind turbine with linear P(V} curve

The linear P(V} curve is given by 2.1 and possesses the following typical features (3):

v

= 1.5 V d c V 3 _V c r _1} (C n)V = (Cpn)max 6.75 ;-3 ( -V

-P r _r c V 3 ( 3Vr ) or (C n)V (C n) d

?

- - -1 p r p max V_d r (4.15) (4.16) (4.17)

The energy output is found to be: xr TP

j

e-Gx k dx -Gx k r E = - -_{x -x} - TP_r e f (4.18} r c x c

The dimensionless energy output is found with (4.161 and (4.41:

2 -Gx k - x (3x - 2x d} e f d r 6.75 e ;:: system or in terms of x d: e = system xr xcje-GX k dx x x c r 2 ( -Gxk 3Xd) e dx 2

'3

x_d 2 -Gx k - 6.75 x (x -x } e f (4.19) c r c (4.20)

The latter function is shown in figs 4.5- 4.10 for a number of k-values and for x

f

=

00.

The reduction in output caused by the choice of x_f is shown

in figs 4.11- 4.16. The method of reading the latter graphs is

explained below in fig. 4.2. The function x

d 2

(3xr-2xd1 is plotted in the third quadrant as a help-function and is afterwards

-Gx k

multiplied by e f to yield the l e .

o

l

helP

function

k=2

Fig. 4.2 The influence of the furling windspeed x

f = Vf/V on the

dimensionless energy output of a wind turbine can be calculated via the help function in the third quadrant of this graph.

With x_f

=

00 the output of the turbine chosen (x

d

=

1.1 and

x

=

1.8) should have been e

=

1.21 but the introduction

r system

of the furling windspeed x

f

=

2.2 reduces this value to

4.4 Constant torque load

The power outout of an ideal waterpumping windmill, consisting of

a pump with a constant torque and a rotor with quadratic C - A

p curve [3] is given by:

P(V2

=[~

_V .L -

"t

d (L - 1)] (C n)

!p

d V P max with V c= Vd

F

A and L

=

max A d V 3 A d (4.21l . k -Gx e

The dimensionless energy output is found to he

x r ·system -

X/

Gk

J

[~d·L

xl

[L

::x~

(L-I) ::].

[.-Gx/ _

.-Gx/]

dx -(4.222

k=2 3 2

>

V3 r---1

,'"====1

_{/,. 'V}V2 ---I-constant torque load

L

~

a

2.0 p

-.

1

1.6 V x

=

-

-;. V x d 1.4 x_{f '"} 1• 2 0.4

rYst~.

8 f---+--II-U---l---I

x r V r V

Fig. 4.3 The value of e as a function

system

of the rated windspeed x_r'

The value of x

0.8 0.4

o

1.0 x r V r = -v 2.0 ) 3.0

Fig. 4.4 _{The value of e as a function of}

system the rated windspeed x •

r The value of x

4.5 Comparison of the four P(VI curves

For comparison purposes the four P(V} curves mentioned before were plotted in the following conditions:

(11 (Ill. k

=

2, k

=

2, and x

=

(Xl f and x

=

(Xl f

The design windspeed of x

d

=

1.4 has been chosen because at

k

=

2 the three non-cuEic P(VI curves produce the highest

out-put at tliis value.

Tlie value x_d = 1.0 has been chosen because tliis is a more

rea-listic value in a practical situation, when some power during more liours is worth more than a lot of power during a small numEer of Bours. The curves are shown in figs. 4.3 and 4.4.

5. EXAMPLE

Suppose one wishes to know the annual output of a wind turbine with the following (linear) characteristics:

v

= 4 mls c V 9 mls r V f =15 mls D =10 m p _{= 8kW} r V d = 6 mls (5.11 in a windregime with V = 5mls

(5.2}

k = 2.5

The first step is to derive the dimensionless quantities:

x = 0.8 x d = 1.2 c x = 1.8 r x f = 3

(5.3)

In fig 4.8 one finds: e t

=

1.2 for x

f

=

00. The correction

sys em

for x

f = 3 can be found in fig 4.14 and shows that the correction

is negligible.

The definition of e gives the total energy output in a

system period T: E e (C n)

.!p

A V_3 T syst. p max (5.4) x) With formula (4.17): (C n) p max p r 3V (~- I)

!p

A V 3 V_d d

For a year, i.e. T

=

8760 hours, the total energy output becomes with these data:

E

=

19467 kWh/year (5.5)

For a water pumping windmill with a linear output characteristic the procedure is similar. We assume the following data:

H

₌

10 m V d

=

5 m/s xd 1.25 V_r

₌

8 m/s _{x 2} V f 10 mls r

=

x_f

=

2.5 D 5 m (CpT))max

=

0.2 A = 19.6 m2

V

= 4 mls k = I.75

In fig. 4.6 one finds e

=

1.38 for x

f

=

00. The correction for

system

x_f

=

2.5 can be found in fig. 4.12 and is estimated at~e_{sys em}t

=

-0.10.

The resulting e becomes 1.28.

system

The annual energy output is calculated with (5.4):

E

=

1.28

*

0.2

*

0.6

*

19.6

*

43

*

8.760

=

1688 kWh

For a head of

E

=

1688

Note:

10m this corresponds to:

3.6 x 106 3

x 9.81 x 10

=

61944 m of water per year.

Exact calculation with a "constant torque load" characteristic yields an annual otuput of 1600 kWh in this example.

6. REFERENCES [1] Gustavsson, B. Tornkvist, G. [2] Stevens, M.J.M. Smulders, P.T. [3] Lysen, E.H. [4] Powell, W.R. [5] Lysen, E.H.

Test results from the Swedish 60 kW experimented wind power unit

Second Int. Symposium on Wind Energy Systems

Oct. 1978, Amsterdam.

The estimation of the Parameters of the Weibull Wind Speed Distribution for Wind Energy Utilization Purposes

Wind Engineering, Vol. 3, nr. 2,

pp. 132-145, 1979

The output of water pumping windmills in non-Weibull distributed wind regimes

(it?- Dutch)

Internal Report, Eindhoven University

of Technology, ~ 396 D, March 1980.

An analytical expression for the

average output power of a wind machine. Solar Energy, Vol. 26, nr. 1,

pp. 77-80, 1981.

Comments on: lIAnalytical expression for the average power of a wind machine". Internal Report, Eindhoven University of Technology, R 474 D, April 1981.

!

'I_,

I

,·

i;,.

'i~ Ilii l'fT~I'- .'.', I:,".c' 'II II! I' "'J,II.'I i I' I!1+11[1:!1 I I i i " IIi itl![t'j111111111111 iII1111IillN<J.III',IIIII'11 il'II'!i il III! 1'llr1j

Ilillllll

ji II"

HH+I' II III i ' , ' ,: .... ~~.rr' II

":! '

'i!':/!i ~: II it' I , ; , , ' III!;

,4+

'111 ,I Illf II i IIIII~ :1; Ii +: ,Iii

H.:I

II I:

,;11:::

Ii

i+

:1, I, III I, ,1'i'jif ,.ji. ' IT fiJlI:. fiY 'i:.!, I, " "

.,1 "

l'hLII " IWI III ' , "Io.i.H '111 I . } '

:ill

11'1 II 1\l11"11 :1 "

;! Ii II)" t :'~'i/! tHf' " J jil~:/:::,I':: if ~~t_-r,- --: ! j •. p".-~-ft' ~t dii<' til I ',j~ - I ' !~-I 1+ _1-11-1 lj"+ I,~" _ IJNJ Jij :1 :!I~I I

fIT

ii':1

I II. 1 :,il ::' lill I·

id

l

~.:

!y,! II! i : 111

1

! I

IfF

I

~~.IIII!!11

!II

ill :

II

~

'

IIi!

'I II

I~II

Ililll!!IIIlH

'i!

liill,:1

~

H+H_I:;":~HHh+:'HiH+-++lj . ,. 1 ," ",I •

H, .

.,

j'

tn

'It!r~ I:I IIt III' i "I ' J nI II, 'Iii T'iitn

t+ll'

1\ 1I:

II

I

,Ii: ;) ':-11 toi\ 1

r'l

'

i ,'1 HJ' t: ~ : , '"J 1't ~i::t t-,._:f-iqj~tt~·_ln4_t_ tJlf :i f-H 11l"U'· l-i i._III :₁ I l·j!!lll-1 j fi: .

i!,' 1':,'I! 'I',d I !,.;I Ii 1 1 / : ' 1' lilt, i '1II,,!,:1 II 1I1I1

i

'11111/'~",.,r H . l ' ,'I"Ii I ,Illillli,t , ,I',',!I,!,

1

11 :. Ii II ~ I

il,rl'lllil

1:,

C : · ' 4 ' . I' " I'! ' . j . ' , I , Ih1

+H

+, 1t+f'+lJ. 'rM II I! It ' '" II r~I~ +"+1

• ITHI₁''',Hilc'':::','H1;1 i . :11'

1i1'

',11 1:' . ,

!I·

III ,! lin,·'dr·! Iill I!l I 111₁ Ii I Ill\:n

liii

'1,

" , :1, .:... : ,",,, " '11 ,I ., jl.;.[' !~I f I 1" " 1I , i i , It , i I!IIN.. !,

I! :[

ill,W

Illi

II: Ii: ! Ii I

IlfliVJ",I,IJllkfill,.i'u'llll

,iii

[I,i"II",

;'li!PT~, ,,!

hIIHilil'!

~ll

l'l!!ltt'IHl

I

r{j'i',,!JI: 1!!lili~I,jll!1

UIWWU

" I i i I: ,1,1rLi !ILHIII ,' : ' 1 :tIl !I, f /'1'1 I , 11\11111'1. I

ilqr

!T

_ll

I I Ilrrf I IlilillIi'll

nn

!d ,.. "

' i ; - ; , i r:1: I,~Ji; J t " -i I: i iii" to!: ! 1 /t-·1--l~:--lrH-· i-f-·!;)t Il,- ~ j 11 I!I.. ' I / ; . ! .~!N '

WI'

"r:"'''l''

']'Ylj'

'Ir~"

1I"'jll"jll'lj'!'III'I"tll"I"nl'it'i

I!I~'

;'1

11

1

'1

, I: ,.1' iI:111 ::! ' ,Ii:. II Iii i., :.':,. !1.1 Iii I' , I" "[ I I'

,11.1 i;.j":"I",+~t~ L~.).I ! il!I'Itt'1':1' f'4

i,ll,

1.).1 'tH',IJ'~' HIJHI

I i :I i iji tit,i ' I ' : . , - - -n 1- tI I ,I . , , !I i i i I! II; , . , ' I ' 'tI

!,I' t'!! ",I' :.I, I" ~ t -t--ItII- n;.. i t ' I III i : ' ! ,-Iir i l lIi: !,_1' ; ] 1rI

iI " i ' , I,t '.'" ~' ,ft ' I .'t f -I I iII i i 'L ' , ' i.' -I1r l ' • j ' I I I

... 1Ii" : ' " .... ,.,1 " , ' "i, ,t",.,!.! "I' ,,1: Lit+", "i' . IIII j", oil,

11/11 1

ii

II

tl

i:

,~!ril

i:

f[li,1I,'fi,!

i!rnililill

ii~1

i

il'~11

,···lH·,ht,.. ,·jH" ,4

t+!t

t!+,.w ittl +,.). ·H+t+H+Fl;+

L . , I L, . ., "'j ;:1 :;__illl-lH·1 'il:-tIJi-j'11\

,- IrI liT! I'-Iti-Ijti+-,-I j111- -~d

\Jl

o

iil~:iIJi~llii]!lfI1I':!1II

IJti

I.• ,'

·"llill'!lj·'nljl+ij.

I.l! I II' j l l l l l : 1 filii

11.' 'fll'lllllT!

CIT "lilll,!!lll ffil

If,ii*ITf

III.

III!

111,:f~iflli'!lllll

l'lli

" iii' iI:,

,,,.til' "

Ii I II 1,; ~Ii 1 + Iii/ i ' d . ' , 1" Ii II 'i H I, , e+U'

Ii

II t "!IN,, I " II ,~"'THH

TIT,

m

t::TIT]; iii Ii Ii: il IF ;~'/,:/: ' i'I, . Ii{ j jl " ,II: . ,I I t ' .. III' I'!' i ,'Iill fllli]1t! III :11 'i

In

Trl"~tj

":1 :Ii! :::1 i:tl i:1 tH I". II : ' i r r , / , ! . '~

Til

j !1,[!! I

d "

" i l l t . , f I f i l ! " Il i t H ! T., hl:i i 11')'1'111, I ITl! "

m

;1 d~ ~r~ I : ' V~

[If

II

It,'

j

r'I

II

:'I!IIIIW!

J

11

iii It' li~I . I '

11

'!

Ii ~lllUI ~l!il r I

U

I'H:~H+i+ft+,Clfril+l

:it' -I I'! "I' ~1<!'·Htj!

I'

I I ' I tIt Ii II I , t · 1" 1,1-1 i p i l l -I , j , - , I' i 1 I ! ' r tffi,!ft~

,.'

I Il!1 i t

.. ',!,

' I :" " ' , I I ' t .r 11 "+' '," 1 " , . ;I) ' I " ' , ,I 1 ' ' ' ' , , ' . ,.

'I

,,"

"I' I'" ".IiI" ," 'I'

;Ii Iii !!",i I) I d; 1"1: .d·ii- :j t d 1 - _ ';.H;j : , 11 I- I 11 L - .i ' ill - II i! j j \;'1 11,1~:r I I I ! - ' .:: _ 'iii ;:! Ii 1 :;.! I: I

,ut:

J"ii,'

ii"

III

I!I" "

[If

I I,I!

III

I

llllfjitl,

ill,

.ill,:j 'I ,II

Ii

ii

II ,

i il'il

fillil

!

Ii

I

Ill!, ,

jjll 11

IIIXI

1111

",i,lllll'

'N!

ii ii

'HI

t?-I'i"'!' ,. '!Ii '~ 'litl::1 iil.i mll:1

II

"i

11-

r n"'1Il II IJ I Ii III' , ,. :11' I ,',

ill

"n"

III

n

Irfl,t,h

1

':\1 't'

lliii ill;1 ;;;i • ",i I1IIH':: i i i ! :II}i;.11

iili

Iii. t l t . n IIli, !'; i, I , l l . ' f , III

'd

i t l llii lit

III

Inl

rr

II!. 1,:1; ,Ii.! II ill'i i' !!I: •

""

N ... -2' -- ...,-:::::':.: ::c.,. =;:;-I=t.--~:~

--:

~;i:::

--~=::

::--=;:::::tt

.----. ---.-. !-'----fc:':- t::::::-r----;;ii = :.=;:r--::.-::::;.::::-.:: +c---e---:::., >---. _4--:..-;;. '--'::S:.:. ----::-'F=:--71"" .:.---,,-.-:. 'r. -..:~. '"7 ;=='-fc-.

I---=

~ z

c;;.

~::+-. . -=:::H":~:' ---:c-:- f-... ---!-+-..:t=-. ~-l e:=t+ ---::-: -+- ~r',/!--f-"-' ' +_....::t- ---::·::r-c

::':1

:::::::::E;t

--

'-T -t·---__ 'fc-+'-'-=. - r::.-'-- 1=7'.:- --. , ,,---+---., .--''-'''." -~ =-=:=l=;.i'r-::---.

=1/:--._. '---- .~-:-=::-c

_.

=:~

--.--.

c_ ,D. ~ -r:- :\-. i:¥fc-- --·C---. 'I- ~'=7.'-.. -.r···---~. -~-:;t::=r

'-.

c-+.:.:.')

.:.=-:~ ----Fe -~ + 'c:-: r:=:=:' .--i ~: __ ~ -.iT ~ -24-<0 <0 co _", <0 -:-'~ --:T::~i ... -,

--l:=::

--=,

::

-U-f r--. cf.'::::-'--'::' -,--:;J

-_:..

=,

:::c:::::-::

_.:::[::;

..C_. . - ---,--. =F:~cE::

r::::;::==

=~j-3: -~:: .'-'-.-, -_. ~~r-- . - ---~

..

.• ,-r:r::;:: t~ ---~- -:.-.. c. --. !-:c-'-",,' r---+~;::: :=I::::-::g: --:::': r-:-' -- .--I"~-~ ...::~~, " Id-.:-. ,; -_=---r::-:C+fc L ::-~ I:jr:~,'~'::::~;F

--"';

l".:::.:.::~::::41~,,,:

.

.::::i:,," "~*3.f=.:d~j,:

-- c-• _ '--.-~:--.::!::~--=:~:.:-~5t:~.

'\.

:\.:'

:~:::;: ~\F;:~:: .",f-:: _;~_ .. _. . •.

----fR:.

::':::::--,==:."':: --f ~~:'5..i""I::: ::\ :-:f::::: :::\'::._=1=-'='::-'::.-.'. ·c---_:.:--c I::--:::·:-r:: _:::',_':7. --_ --:4,·--, ,-. ~=~'.: -. +-'--"::::.. ;_ ct:±£h..~~ ~;:'\,\:'-: ~ R":~

...:

~. "-~ r--~~~~ilit"':",'0~

.-=+

=~_--,-::+::: . . ..'"

;:~r:---,...

:

-=---. 1:=:' F::'=:i::'

-e----:::-~~~~~~:~~~=

--=t_c: ." -.~:"X .. :t~: f-:-: --:"liIIl x1';;---~f::::, "--:--':::':': :: ... -~ :; :-,=' ::='1-::.. -- ,--:F: :i:J:=-i ~l, ·."t .' .' __.:, __ ----:C' . '-_n _. ---~_.--

f---=:":

-=--~~ ;;:.==--=;~

-- --::t::~-=:~'-_c':::

":

::::-:l..:'.

--__

~?: :::~:'

___

-t-·"

::::..+:::

--:,:t'::.

-_

'-FO

-,--~~=

--=-:-.:::---'..:f':

='f_:-'=:::~::J-::

:::'-t:-:~r~-- i----.:.-+ -: l' .-."--:,-..• :::::::.:.:0. ----::::. ::=:::;:

=

-:-'--'-- fc

-

--.:4=:-..-:..:=t==

---

_.- -25-~-=t---.::j::;=t ~+ + -+-::.~ _.. j. :::s:: . Fig. 4.7 -=t;.; ~':'=:j::=-~ ~

..

:.~~ -=.~~ I~~ : • -:-;+-E:~ '::;-'t::::: ±::=~ -3. "=-~ ~ "-~I---~-: ~. t=".:::--..:: !=$~ :::=::t:= --:n=t~T " +,-+ J~: -r ~ .•-... ",-,l-:.,.;...u • -~~ ~:t--. -d:.;E' -_

:r:::--+ ~~~~ I---+=-~:c:: ~: -!-T~~ __ E.::.: -:t '--.:.:±.::::- -=- -'-=---:~=-~f,' .

~

-~-·-~1~-~~-~ ~;t~--~= ~-: -r'--~t=--~~~~~ f----. -: i-+; .- ..,.-~-' 1=:;1': ="i--X· :: f?--:-cE~F j:-~

.

-=t=~ ~~F'=3~c .~~~~::

'-:y

~.

-

, - ::--ii:i t1!-~~ :=t-!==~ -''i-=~j::::-J~+ of:::: -'---'-JE~ '~:,h;::~-I::c:<J :71:: --J:;:~~1:::

S2?E

f- t--=--'--_-·~~..·~:c::=t<-t--~ ---r--A::::c.::t;::-;.\..,.,: -~=+ 1---, -f--'"---l --. '-. :7---::+=:-- J-.--±=7

=

.

'-A' -, _ ._.c-. -::+=;: -~ 'r;-~ j"' -- !-,.-- f.e;+~I:::::::::::i·- --'---·,,--t;, .. ' ;: -,---", . ... r= ± :iFI:::::::::::i=2 L --r:=±:7:= '.' .• 1=::' F::=i-Tt f,-- f'--:~~.:::± r::: ~, -' i--- >--t ==t:+'~ 1---1.""' .." -c:.-~~ ~ __._ =.~

=

=

.

...

, I--- 1--'=---c: ~~~ 1-'---1 :fr 1-::,,_ ·+/··.- .1---fi§ c:.- ~-,"'1-;::::' -: ---:~f'--l.f:;~ -=-;..[:::-':;

Jz

:~=§:: :~: --t7~:c;-__ '-'C_,~ -~---J

i---=

f,"-· ' . :J3£1---'·I--'. -_ . :A' ::;:t-c '.,._ I t-_... _-;-+ --I'. t'

,-f--.-:-f:t::

=:

-:-'7

~]§:~.

'.:;

-:·-~::tf-=:-;

:

--.-:+~ : ,-;:=:; ::.::' ~t!±t =:~~ ::: ~::::: ~:t~: ---T, ,..,...-r+ ... ' r-+---T _p::l =~~~;. -.'=tC-0= ~:::'i:.

'---C._ 1'-+~. 1--.-, ,t -::: .:\:: -,. ~:~ =-:t:-:-::i ,"._- -...---;-i + ... - --~Ilt== _:~! ::::±..--=.¢=-' -:..-_J..~L ~ ... p=:~:=" ..":::' ~: ~'\f:=

\:

-c:\=: :-:c

.__

._

_

~_-~:+K:X:C:\;~:: .C' -.'!s,:?'c!-· ... \--.. -~~ .=l=~-::: -.-:1---:: ~-!--'-~ .. c. _ ---, '1 -=+~,

>6:

=:=

1-::::. ~:~:: ::~F:

--=

c..c: -:...: - + -C-:~l F~

--.=:l:t't

=12:: IO::;'-l-'-~~ I--·--e---;c., '--',....;.' =---,;-=,. -,~: ,-!----, _.'

-26-.... t:::.:=:''':::, .= ·c-· -+'-~. f~ _ . • T~~" ,,, ... ;.'It'::. .'0_':" :~::'E~:h=

-'.

f===t:::·::

•

;,<,/ •. i--. ~"'-. :: :;;:': ;<. l_+ f-.--....:L;.~: --.. , ...

-'::=+-:"'\0:'

l=::' ::B

-...

~~:r .=1= :.a{:";·, :....:: • ~.::j: .. _~. -f= •;,:-c __ Jc:=:·C: _ .• ::::::;:

...

;~;;

;i.o=f:=

::-:.::~ s ., ... ...

_,

:-~ --"--~ .+:: • ,+ --:-+ ;.:t -:' ':...c _ ... _ , •• _ • q ,-,,::~.-... '1::

'=

..

~f-'-- t7-'--H---~-, ·+:cgE.~: ", <__.t ..

,4;---.

~. ", .. -."~--' .:c,. .. '~. ". --t .. ' '-.

=.

~;.,'

'"

::-:=2

==:: '. ~-;::;: ... -'n". T .. ~--:-c ~. .::=-===c; . 2 ,.... "", = 7" .. -4

..

,~~ . ··iX ~+ ; =:C' ... ... T h'-+ "'+:

=tt;

~; -:::. I-'C.

..tsi.

..

.-,,<. _ •• ~;'S:p;o;~;:; Xc . c .... ~ "¢' C"-.cy'l. ~ '::., , f-~ --:::"; .. -::ti r'~, ~.=t: +~ .. ~ :-:.: -++

-<+

"c'r"c .=:·1··· f-. f-::::! -=~=.

=-I:S.; • t-::~~=:::­ =:~:~

=.;Jf:1

~ :;~: ';~

"

. :,'~ ... :~ T .~ f--';"'-i=:== .. ~ C--.. ,_. _"

-==:i:.

~:£:~ .-:-:-~~=*L ', . ..,. -'---, ~

.._=:

--=," .~~ ::=;:t -l...:"F-,,::.±'-::.

.i::2.

=;' .; .. .=..;. ~~ .. " ~ ,.r-, .... .:~: ~:.:;::.:.'. .~;:t :=:t::E:f--:::-r::::If-:.I~c~:

S:=:'11=:E:-:=

:-t:C't

':"''''c'-- -27-c.t::::::::~ .. :'-L.. _. .!-'-~+--~ ':::;1:.:_: t::=:: .. :~ ~, ::::: cb J ·':;:--=;=2 P:: . ""r' ~:= ,,' . H -"+ C::-':. ~~. =!=:~ :r,:::'_c: ;:::ct .. ·' l==::;.t: =~:--t:;=~~E~

,

:c::,: '~ ~=~;t: ~t'--,---~ I=:J~h ':"'t---; _lcL;"'T" : _~-',

.;::::t::

:E::~ I--:~: +-'·F"'--' -h-c"1---:::::." -~t;··-'--t:::ti= : -~:t~ 1=.-;- ---'-' 1--.• = ~lt7 ~=+::;;. -.,/1 .. '. 1-'--. t---+ --[:X--;.;;:-; =+;;;;i f--.. "~ "-::.I=::;r:::= :. :r::.-F=: '-0-- ,1-.-h-e:p -~·p,:::F .. ~ -.-=~+ .t::: '.' f-b~~·--·

:=

=

._::c +', ' +++ "'~, -d-~++-.

·"It

, -i--H ... i.. -~

,0£='

~t=::+;::'. +-:::,:-~:;,;'E ~ .+;-w ,___-~~ __ -,--~:-::-. +4-, ;-+- 1- -'-'-I-'--~,' +.--,.~:: .~, .

'

.. :: . --,... ~-T .,::. 'c::: ~":: .. ~:::c:: .'F-:-s -,:::.I=;~ lIii.;:~:; . ''i h-", . •-, .. _ ..

,- !--::=:~c:; ~*:::: f-::::: f--.:-:' rr=,. , ~. f'----~ 1='-. -,)::):':: -=~'X:,I-C. --i.e-_=::..I. ::~~~ F~:l=~:' :1-=::: . :-t:::: .~ •. -h :--~:'=::=t;:-i 1-==:::1'2 1=':::': " .. :. _.--' .. :.;:t.:::: -+-r---:-t-h---1-:~ ~f:::! 1::::-' 1=.' :21 """':"""'j -~;-l t--L -:.i-.~~­ I:~ .::' ~--+ ::'S ~": -~:;I':'t:c r:.:c --~'-. 1~ r+--~ ,_ •. : -~ -.-~~ ~~ .': :;.. ~.' .• 1-" --:1::: 1 :::r -.--r----: ...- ;_-=--.:=.-':::d):::::. '" ·t';'·

28 -±td __: =:=:'2 __ ~~~S '1-=:: "': ---:;, ---, ,'-q-i2...-' --:-:: :J:::::-::+-_: , , ::±,. ,, ,+.== =::::-

---=

,--b-'::---:-~.:2' :.::r---:f::::.=7~::. :::::: ~~ , 1--~ __ :t,E:';:

---c

-,:;

...

::l:=-:.-::-"

.===

--"i~~:F~' ::=

--==.""t=

t· -::-r-: :o£::----: ,~--'::-0L::.. ~-: ---" "'::::v' ~=:.~~-: ',-i:7'.,'-,,;.A":, ,:T"-~; f'~;:": ,::: ::::--..;;;~~ 1,.--,+"" _: .. ~ . ::=..'jt!":. ,=1=' _ ' :~=:t.:-, .;.... .... ~.~ -+~--, " ---.c-i--~~.-+~: _-T

-,

=

r'_ t-:C:-::=, Y' --1---"': ,= ,':S:r=:£~.:..:;..,

:

-E=:

-=---g:'

E=:=E---+£-:~

_.

=t:::,~ --~:+: e;-: '1=.;::: ,':-'~F-:'-E¥:= Fig. 4. 10

-,

--:+ _=t~:',:::;,

'-'

---=--, i=:~

-"g

-=:--f :-' 2==S~: _':";~~ -'-~~:+---~cc­ .-:t:~..;.'~ ,.:--'::"=--$I=':.±:-~ -, .~ -'"=-:: ::::.:::E""t'

t'-.:

~,:±-:::: _:=~::J =-~' ~_ =+=---:.:+,,--,~:.t _ ' . _ ~..:t= -, --1---,--, __ +,:-::t +" ,-"," ="::'J:: '-. -' ',' =:::' I-~: ,~t:.:-_ -,r,' --, -- '::j:::,:l::+tc-"t ---r--' ta=:lQ3 ' '/:,': -'-C__::; ---:-,~ -:¥~

-::--.±=-'=-~?~ -:::¥:~T 'l~::f::]

=:l

--,

&

__ .~_ ,_"-::~ __ -=--§.-~ " ,"',t~:i!::1Ll1~'~=:-=~~ f----.:

_:-=1=:

~:k::

:s.z::- ""'':::f---f:=::t: _:t.::. ::\C ':", ::j~:; =~

-t---",:

-;-:~I::::+: ::::.: ':: ,1-,': -, , ":r-:::;t=,: _.-':-,-~=:t:_: , --1-...;... --f--,,-~""""7 -~~ _.-t--~, '.:-, ~' ,::=t?.:::::

...c,:l,:=-,

~'.:---r~ -:=t=:: r=-=c: " --+-t-~. I==t::::, "" ~- ,t---'\C:~l_ ~=~~:-:' t,:'" ,+-,- :~-r--~ f- St~~'Iii::?-:¥~ 'r---: _...=:.-I==" ,,:'''::.;.:::: '::;:;-, '-::.-J-: _. ::~: t== I C': " ••. I L:.' :~:;..::,-=-/.: ~~-=;

:.::::::::-1-:-::-':-':-:

-::::~.:

=:r=-+-'--; .. +-i--,

,,--.---=

'--'-- --,"-~,:I:: , em',: ... ~

---,

"':::-i&=: .:c=

"t::::---:±~T ,

--'-~::' ,-=- _,-,-=~R::=r--"~ "'"' ", "--~--. ---:'=~~ ::-~,~

"=:t-'

='" :.::. 1---,

-.:.n

~-::-=:~::= ~.:..."--,,,,,-', ---~+--,

,~=-:::

l- , ---... , 'E_'-::::~~f

,'---'-:::=: ';"":£:' ~...:t=:=:- e:--=-'T;~ ,C , :~_---:­ :J'oI-+ -:_, i! -. SoT -.. , , , .. , 1--- --+---;::... ~-• "

-::.;

,';

~::. ~ S;;=-~. ! ': ... :: :~. :: '-1'--'~ -~r-:.::--HP-'~±: ~ L~..:::'=:~:

::-=,..,.

,..;. ':~-::-:-,: ., ~-,-,--:c= :~~c::=~;-_ : .. , ",,' ~-=l-, c,3r-:, ' --' --, te:: I:::~+::, ::i::'-=: _c~

f

=:- Lee:-:::J:~~ :-_:~It-c--= :-=1f£=',:iS~~ " -~ --I-::'

II

tH.I,Illi111InmnnnIW!11I'ltl

I.JtPiF(f

Inl;

III

tiT',ITlltlmm!!.mt~jT!q.H

ijj!!':.jiHf

ItlEI+UHH-J:r8lli++iJl

Tn IT!

I;I

TiTIrrrmrmtnlm

j

Ii:.

I

lITh

I I

L jill

i II .111 I Ii

lrmnrniffirrmtl

tmlTIh

m+f+; l++if! ,+++1 ++f t+t-tti

:_::::;=§

j::c:::::rl'F

.

-32--+Fig.4.J4I +. -:::p:' +: ~. =1.+'-"_ ~ ,...;::::: +;:;",.::: ::::::t:= fG'. ;;'1---++ +t ';'-.J. ,-:; ._ . ...; ~ ~+ .... '~+ ,[-+ t--. ~' : ,:;;:if ~3jofj~ .1.: ;tE~~ r,,~ , ~,,..:t + ... _.;~:.w..,. ~Jf-.... -~ ~-~,~-;­ .::-:'i:l . ~ -::=-F-'-~ IE::

;S

b

ca

':'::

:::~ .~ _~r . ._---. -4'

.:',

;::/""'+.,

-,err

.~...:;;.:,:::r:::.' :l\:~:f:~ C-:-.c t::-. ::::=l? ·:·t::: r '.) ':~~.:xf::=:c~:t:==-F: ·I==t=l.c ~ :;=..~.

,,'$

:i:~

::;n

::~., 1':'\ ::~ .,.:.:±~ ,,·::f,S~· >;-11+§l::.:+.=

..,.,

-'

t:.=

,..

- .+-4-, +--.~J..+-... -.§ff.=;~ ~.~ :..• ,' p':!:: ~

:s.-'

rC-. ~ -~=== t::::: , _i ~ .. ,---ci -.:..:.' ~ =:-~_ "';:;Cr--• .. ~4 :::=i-::'::,,:, ::=:::= +: -.- "00+-• --~+.:.: _ .... ~ :;=: ...: ;,--;:ep-r2;:~:::+ ~ :'"~: -t-;~--'

-33-• :c::.;: = ~ 1--::::: '... \---' -=t:~ :e-$ ~",: ::q rc-c :r±+

==

=

·~.s:F~---!::: =.= -~.~ l:¢' :Lf=-~ = ~-+-" I':: t;:E ~>

2'

~ !="'::;:; i="h' __ 1--.

,=

""

.=

i= '~ p.--=-1:- 1---" ~li ::j', ~l."; • \--- , ,_.,

_.

--,.;±;.,= -' '":: !:!

:'tl

~L ·::::~:il :IL' " '~_:cIL 'i-'1: Ir' ;::I--CC' _.'~ :;-.. .' T, ;::tl== '~ .,...,;.-J." .. ...,., --+~i=,+ ,PIt= ~::~= I-C:::to.": .::c: \--' ~, ~" .-+ =t~ ,~--, =="~ iC; rt :rt"

.~,iT

~,."

'"',' <.~ -W-L+; ,;..~C:, 1-M-+ ,+-:'+ ,-r-+-->-.,....:-.., ,-..:...

-,

-~+

,=,

i:::r ~'-+ -"" .;. .;.... .:.: .~ ~,,= .-~::::' -t-++ -!:':; ::+;:, r=t;:: E-:-~-? = r=; ''--, 1--'-'---:::~::-_. I---' :.::::".~ FIE:;:::: r=:;-= i' E

-t:::--

:=:-- I -.~lil':. 7" '" e;:L:: - --r::: 1== ';-:E:'-,F: '~ -" ... 'r" ,~;;--"., =,:' -... F: :. _"~!=-= l::,:"

:=

::~tE

--,

~ '"+f-t ,+ :.... "

~-6::

'::

'.',

---,

'::~ ~~:E2" ~ :::.: -S::-' ~_ ," -'---4= •• IT'· i=

-=~

~:~?"-'--':::::=--""':::~-+-r~:~£

~:.:;:::

~ ;.... .. -t::tr+ H, -'-+-, t±, I::; C L?=±F=:tI-"-. F-, ; x::

-,

:4

.~ i ;::=-=:::+: .f:= ~:::- :': --+ t'rT ~-:

-34-.::1=3

=

-:~

- ,;:::'c:::~b:::::-'=:':t:~: --ry \Fl.g. ::t;,--;::-.' .::.,.::.. . .,;= . ~ "+::.'. -->-=: + >-=:: i--'--~ .•. - ;'c;-:--:::::;:t -:. ~'~r~ .=,tc= . ...,.

_.

-+' ~±L;: ~~ ~"'-~~~C'-~=:::;:::-'::::C r-=+ .. .~- r--"=;:t J: :.. ---;::;:'~r:,Fc-,

~7-;-.

=---: r-= ~~~ ~.

==

.

'~r:c-: ; • --~ --. ~. ~.~,.--. ;:.l:::::::: = _-==-.i+ --;::C:'=' ;:::;.;. ..L-~ ±":,,,~.:.: -_. :::t:::f.' ::E-'l . _:c.-.;-, f--: ..: ~. ." ~.l t:::: ';::l.I--.::t::: i-L' ~ ,.t~+ 1-'+" ~+' h-,.. t---++ ' :~: :~ r-+ "'+' :i~~~~' V ~i~-±~~_ ... --::.': !P'C" ~ .~. -I¥+ ... -: c:t:;::! • - t-c.-:-~K'jf:·~ci;·fl~ §~:.·22~ ~ ~:-: -.. :.: c;<~: .:~ ;:::¥':'t::-~1,i "_:~;7':7: ;:.,~;:-:;:~ ·::,,:-:::g::-+~~~I;--,-··

.;:

.~-=}:::

51'.:;:-==.

'+ +-t-. --' . . , 1--·'·_·_'

::t:=-i=s

-'-m.i:

==--:;:-iT, I--=_r=::;:: --.: ::f.f=-t: . :-.+: ~ ~:::-: -' ,."t=:::::t~: t:?=.!: .. ;~-= --_::

-I·· ..;:.: :::::: -+.' ·e:::-y .. ."~::~~ ."5-' +. _~-: r---:-''t t:++-'-.. ,.---+~' ~t'~ c: : =-1 " -:=- .c:;--;±~i ~~ -. ,:C~ :f-··~flC: .-r=r- ~":i~;E·-~~~.:

.:-=

j~ R§~~: =±::±t _ .::t. ~: -.

f·,-

"-r--2:j==3~; .;;; ;~;. F:'.0: -=~; c---:::. -2; ;..:.. ::j .-:-~4-L r--=-.:::1 ;...,... tJ= :: ~~~ :j-~-b::' r--f--,--1:::--+ ···r-··-.:::::. --: :: •..ct---.. ,

APPENDIX Calculator programme e

system

The graphs 4.5 - 4.10 of e versus x

d for different values of x

system r

are calculated with a TI59 programmable calculator. The programme calculates the value of the integral (see formula 4.20):

x -G k 2 r e = 3x d f e x dx system 2 1~

The value of G is approximated with:

G

!

*

0.2869 + k-O•I

*

0.688

k

which is exact for k=2 and k=3 and accurate within 2% for 1.5 < k< 3.5.

The programme prints the values of k and the x_r concerned and subsequently

prints the values of e for xd=O until 3 in steps of 0.1. If x_d=3

system

is reached, or e has become negative beforehand, the procedure is

system

repeated for a higher value of x (x + 0.2) until the required number

r r

of xr's has been finalized. The integral is calculated with Simpson's rule, available in the calculator as PGM 09.

The programme is initialized by entering the following data via the labels B-E:

B value of first x minus 0.2

r

C required number of x 's

r

D Weibull shape factor k; after entering k, G ~s calculated and

displayed

E number of intervals for numerical integration (usually n=IO

is sufficient).

APPENDIX ~. t.: .-, .::. "! .-, i-' 1.':: 1_" -:' ..- ' ! tIi :' C: LC:L Ci( (17 t:S .:'" 42 ~:~TD 15 !J2 iJ7 elf ::.1·5 \.'>:: ~?El LE:L ~2 E:

42 STD

Cl_{2 [12} ,.I ,,; -OJ 11. { 11i:; 1. J.2 1Ci 1 1:~ C~ i '1 .: l . .L 1 126 121 122 12:3 1:3 1 1~35 1:::7 140 12C~ 1

(ie,

114 115 11

t:

124 l[I;::: 1O:~: 1c~ E! 4:3 i?C:L.. 1 Cl 1 Ci 45 \0'::';: 1 C! E5 .~i F~...~::; Cil 01 3t: F:!~t:1 Ci"3 Ci9 Ci2 2

24 CE

22 It·~!:/

77

GE

C!13 [1:3

00

00

55

05:3 Ot:Ci !-i::::C 1_1._:l_~ C194 0::; Ci;:: 0:95 ::-~.:.t 7 / ' -C~57 ei7e1 C!? 1 [;72 CitJ4 Citl5 Cit:t: [i67 (i54 Cl55 4 4 6:3 OF: Ci4 C!l~

04

04

05

00

02

2

Cit: Cit: '3 ::: A.D '/ 1[1 1C! 7E: L.E~L 4:3 F:CL 01 1 ,-,:-:.-: :_;!_i~

oel

CiC14 CICl5 c~1 7 j-I'; ~ I_I,;" .i. (121 C!22 (j24 [125

D2e

Ci:32 Cl::;::;

(;42