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BblpaR<eHHSlX

(3) H

, (4),

MOR<HO

)l.06HTbC51

XOpOIlIHX

pe3YJIbTaTOB

ynpaBJ1eHH51,

HanpH-

Mep,

nocJIe

YBeJIHlIeHHH

CKOpOCTH B 1,4

pa3a

CJISl6bI C

Si=

100'MAt

MapKH

CT.

70

He)l.OrpeTbI

Ha

4°

npH

is

= 3 H /(r, = 0,5.

BBe)l.eHHe

B

a.rIrOpHTMbl

ynpaBJIeHHSl

3HalleHHH

3a)l.aHHoro

nepHo)l.a

Bbl)l.allH

C.[I~60B,

HaXo)l.SlIlJ,HXCSlem

e

B

neliH

(HJIH,

Ha

npaKTHKe,

3H3-

l1eHl1H

npe)l.nOJJaraeMoro

nepHO)l.a),

)l.aeT B03MOR<HOCTb

nOJ1YllHTb

,J}yllIlIl1e

pe3YJJbTaTbI

ynpaBJJeHH51

He

TOJIbKO B CMbICJ1e

MHHHMyMa

<pyHKuHOHaJIa

J

2'

HO

H B CMbICJIe

MeHbIlIero

KOJJe6aHHSl

TeMnepaTypbl

"B

30Hax

neliH.

PaCCMOTpHM

eme

OAHE peR<HM

HarpeBa

MeTaJJJ1a,

KOTOpbIH,

no

, cyIlJ,eCTB

y,

CBH33HC

HenOCTOSlHHOH

CKbpOCTbIO

npO)l.BHR<eHHSl MeTaJJJIa

no

)l.JJHHe

nellH.

PaccMotPHM

nepeXOJI.bl

OT

HeHyJ1eBOH CKOpOCTH npOABHR<eHI.f51

MeTaJJJJa

'K

HyJ1eBOH CKOpOCTH H

Ha060pOT,

T.

e.

np<;>lv!O)l.eJJHpyeM

ynpaBJieHHe

nellbIO

npH

npOCTOSlX

neliH.

Pe3YJJbTaTbl

MO)l.eJIHpOB3.- ,

'HHSl

(pHC.'

7.6;2)

nOKa3biBalOT,

'lTO"

ynpaOJJ5ISl

netIblO

BO

Bpe-

MR

npocToeB

no

UJ1fOpHTMaM(8)npe)l.bI)l.yIlJ,erO

naparpaq>a,

npH

npaBHJIbHOM

B.bI6qpe

3HalleHH5I

h = sC

m'

(He3aBHcHMO

OT,

)l.eHCT-

;

BI-ITe~bHblx

3H3'leHHH

hi

=siC~,)],V1Sl3aroToBoK,

HaxO)l.SlIlJ,I1XCSl B

CBa-

POliHOH

3<me)

MOR<HO;l1.o6H;tbCSl

XOpOIlIHX

pe3YJIb;raTOB

ynpaBJIe-

, HHH, B CMbICJ1e

MHH'I1MYMaepYHKU.HoHaJia

J 2 H B CMbICJIe

3KOHOMIHI

rrOnJIlloa.

\

'J

428

OnTHM~cJibHbE

~nPAi3JIEH~B~MHO£,JHHbIMH

ArPEfATAMH

§ 7.

AJIrOpHTMbf,

on'THMaJIbHoro

ynpaBJIeHHJJ

JJ.JUI

ynpaBJIeHHH

HarpeBoM

MeTaJIJIa

B

TpexsoHHOH

MeTo,QIfqe-

CK&H

neqIf

Heooxo~IfMO

peaJIIfSOBaTb

CJIe,QYIOmIfe

aJIrOpIfTMbI:

~l(I)=Qgo+Kl

V~

V

s

(Qgi'_Q~'),

(1)

!!.2

(I) =

(Q

gll_

Q~')

b h/(a/r:

+K.,

o

(2)

!!.~

(I) =

~2

(I)

+K

e

,

r,l{e f

....

(Si·+Sl.+

1

+

+slo)/(/,_I.+

I),

I,~la,

(3)

(hl;+hl;+l +

+h

1

·)1(i8-

/

7+

1

)

npH

hl;~ht"

B=O,

h...

hi;

npH

hi;

> hi, HJIH B = I,

{

~

npH

B=-2,

i

8

~

1

7

,

h'

=

s'C:n,

B = 0 HBJUIeTCH

npIf3I:1aKoM

Toro,

'ITO

nepe,Q

BbI,Qa-

qeH

CJIHOa i = i

8

He

OY,QeT

npOCTOH

neqIf;

B =

1-

npIfsHaKoM

Toro,

'ITO

nepe,!{

ald~aqeHCJUI6a

i:oR

i,

6Y,QeT

npOCTOH

neqIf;

E!.

= 2_

npIfS-

H8KOM

Tero,

'ITO

y}Ke

H8cTynIfJI

npOCTOH

neqIf;

~",+

(I-K,):r.

eCJIH

I!

- 0

MIl

e~H

I!

~

I H

.:':

>

".

r:

= Vl. eCJIH B = 1 H

r:

~

r;O,

[

."-

_.

J,

eCJIH

l!.

= 2,

f,!{e

r;O=(tn-1n-i.-l)l(ie+

I),

ie~O,

•

~

=

~j.

+::,.+1

+... +:!.6)I(i

6

-i

6

+I), i

~

la'

e

3HaqeHIfH

BeJI,HqIfH i

k

(k = 3, 4, 5, 6, 7, 8) onpe,QeJIHIOTCH

H3

HepaBeHcTB

lk-

I

lk

~

b'~lk<

~b',

k=3,4,5,6,7,8,

l=O

'=0

r~e

I.

~

la,

Ie

~

1

6

,

la

~

17-nocToHHHbIe

BeJIIfqIfHbI,

KOTopble

onpe-

,QeJIHIOT

npQMe}KYTKIf

no

~JIIfHe

neqH

l~-la,

18-1a, 1

-1

,

B

npe-

8

7

~eJIax

KOTOpbIX

npOIf3BO,QIfTCH COOTBeTCTBeHHO

ycpe~HeHIfe

SHaqeHIfH

napaMeTpOB

s', r:', hi.

3HaqeHIfe

BeJIIfqIfHbI

i = I,

2,

'"

yKasbIBaeT,

e

no

CKOJIbKIfM

nOCJIe~HIfM

BbI~aqaM

CJIHOOB onpe,QeJIHeTCH ,QeHcTBIf-

TeJIbHbIH

(ycpe~HeHHbIH)

nepHo~

~bI,QaqH

'CJIH60B

To.

Pe3YJIbTaThI

MO~eJIIfpOBaHHH

nOKaSbIB8IOT,

'ITO ,!{JIH

ynpaBJIeHHH

HarpeBOM

MeT4,IIJIB B

'Iel'bIpeXSOHJioA

MeTO,!{IfqeCKoil:

nelm

MO}KHO

,QJIH

TOMIfJIbHoA:

If

CBapOqHOA

30RbI

,.1

I

HCnOJIbSOB8Tb,

TIt

JKeaJIfO-

pHTMbI

(1) H (2)

If,

KpoMe

Toro,

~JIH

yrrpaBJIeHIfH

CBapO'lHOH

SOH08

rI

c.

~~>

onTHMH3AUHH

~BYX30HHbIX

ArPEfATOB

;.

MO>KRO

HCnOJIh30BaTb

aJIrOpIfTM,

C TOqHOCTbIO

,QO

SHaqeHHH

'tOB,

COBna,llaIom.Iii,

C

aJIrOpHTMOM

(2), B KOTOpOM

SHaqeHHH

4~~

HH,QeK-

K03<!>-

<!>HUHeHToB

K

3

H K.

orrpe,QeJIeHhI

TaK,

qTOOhI

npH

CTaUHOHapHhIX

peJKHMaX

ObIJIO

BhmOJIHeHO

yCJIOBHe

u'-u

2

=

l1u'

=Ku'

(r:).

§ 8.

OnTHMH3aUHH

riapaMeTpoB

,QBYX30HHbIX.

arperaToB

C .IJ.BHlKymHMcSI CJIOeM

MaTepHaJIa

KOHBeAepHhle

neqH

H

cyuIHJIa

UIHpOKO

HCnOJIhsYIOTCH B

npoMhIIlI-

JIeHHOCTH

,QJIH

TepMooopaooTKH,

CYlliKH

H

oOMHra

paSJIHqHhIXMaTe-

pHaJIOB

(MeTaJIJIhI,

cTpOHTeJIhHhle

MaTepHaJIhI,

cBapollHble

3JIeKTpo,QhI

H

T.

,ll.).

npH

Bhloope

H

paclleTe

KOHCTpyKUHH

TaKHX

arperaToB"

HapH,lly

C

3a,llaHHQH

npoHSBO,llHTeJIhHOCThIO,

npI1XO,llHTCH

YllIfThIBaTh

pasHooopa3HbIe

TeXHOJIOrHlleCKIfe

o/;,paHHqeHIfH,

,llHKTyeMhIe

ocoaeH-

HOCTHMH

oopaoaTblBaeMorb

MaTepIfana

H

TpeooBaHHHMIf

K

KallecTBY

rOTOBoro

npO,llyKTa.

npH

3TOM

MoryT

OhITh,

HanpHMep,

orpaHIflleHhI

CKOpOCTH

HarpeBa

IfJIH

CYlliKH

HS,lleJIHH

BO

IfsoeMaHHe

pacTpecKH-

BaHHH

IfS-3a

TepMo-

IfJIH

BJIarOHanpHMeHHH,

HaSHalleHhI

MHHHMaJIhHhle

TeXHOJIOrHqeCKHe

BhI,llepMKH

npH

paSJIHl(HhIX

TeMnepaTypax,

ooec.:

nellHBaIO/UHe

npoTeKaHHe

Bcex

XHMHl(eCKHX

If

cTpyKTypHblxnpe-

Bpa/UeHIfH

B

MaTepHaJIe,

OrpaHHQeHhI

TeMnepaTyphI

'rpelOmeH

Cpe,llh~.

3THM

B

3HallHTeJIbHOHMepe

onpe,lleJIHeTCH

BhIOOP TenJIOBOH

cxeMhI

npoeKTHpyeMoro

arperaTa:

KOJIHlleCTBO

TeXHOJIOmlfeCKHX

SOu,

l(HCJIO,

H

THn

HCnOJIbsyeMblx

THrO,llyTheBhlx

yCTpOHCTB,

TeMnepaTyphI

TenJIO.

HOCHTeJISI

B'

30Hax

H

xapaKTep

ero

,llBHMeHHSI OTHOCHTeJIhHO

oopa-

OaThIBaeMOro

MaTepHaJIa

(npHMoT9K

HJIH

npoTHBoTOK),

nJIOTHOCTb

yKJIa,llKH

IlITYllHhIX

HS,lleJIHH,

napaMeTp~I

HCnOJIb3yeMOH: CHCTeMhI

aBTOMaTHtIeCKOrO

peryJIHpOBaHHSI.

,

B HaCTOH/UeM

naparpa<!>e,CJIe,llyH

IO.

H.

AH,llpeeBY

H E.

n.lIleJIe-

riOBy

[13],

MhI

paccMaTpHBaeM

3a,llallH

Bhloopa

HeKoTophIX

OnTHMaJIb-

JibIX

napa,MeTpoB

KOHBeHepHhlx

yCTaHoBoK

C

,llBHMyllI,HMCH

CJIOeM

Ha

npHMepe

yCTaHoBKH

,llJIH

CYlliKH

H

npOKaJIHBaHHH.

B

3THX

YCTa-

HOBKax

TenJIOHOCHTeJIb nO,llBO,llHTCSI

H.

0'i6HpaeTcH

B

HeCKOJIbKmC

MeCTax,

oopasYH

COBMeCTHO C

,llBHMymeHcH

HarpeBaeMoH

HaCa,llKoii:'

PH~

npHMOTOqHbIX

H

npOTHBOTOllHbIX

SOH.

npHtIHHhI

TaKoro

HCnOJI-'

HeHHH

CJIe,llYIO/UHe: '

- OOJIbllIOe

rH,llpaBJIHqeCKOe

conpOTHBJIeHHe

O,llHOSOuHoro

arperaTa;

,

-

OOJIbllIaH

pa3HOCTb

TeMnepaTyp

CJIOH

no

BbICOTe O,llHOSOHuOii: •

paOOlleH

KaMepbI.

B

"KaqeCTBeOnTHMaJIhHOH

npHHHMaeTCH

KOHCTpyKUHH,

KOTopaSl

ooecnellHBaeT

npH

3a,llaHHbIX

orpaHHlleHHHX

MaKCHMaJIhHOe

SHatIeHHe

TeMnepaTypbI'MaTepHaJIa

Ha

BhIXO,lle

H3

arperaTa.

PeMHMHble

H

KOHCTpyKTHBHhle

napaMeTphI,

BbIOOP

KOTOPbIX'

ft':BJIHeTCH CYIl,I,eCTBeHHhIM

npH

peweHHH

3a,llallH

onTHMH3aUHH

KORell-

,

Hoil:

TeMnepaTypbI

MaTepHaJIa,

MOLKHO

pa36HTb

Ha

l(eThlpe

rpynnbl:

431

(4)

,lip}

io t

t (0) =

to;

1(0) =

to,

°

t

rz

C'l.

--

o

t

rz

If

Pile. 7.8.1.

-·--!!..-c----ll

I

t

r

(lo)=t~,

o

to

t

1

t

z

--

----.-----

I-

___

to

If

r

z

to

t

1

t

z

--

-~.~__e------

~

,

0

t

rz

to

'1

to

If

I I , )0

II

to

1.

.-1

o

t'1

lY

G OnTI1MH3AUH5I

.llBYX30HlibIX

AfPEfATOB

c_-OO'beMHhIH

paq,o,lJ.

ra30B

B

rrOrrepeqHOM

CeqeHHH

KaMepbI

.us/if.,

K,KaJl/(.Ms,0C);

iI"

t-TeMrrepaTypa

ra30B

H

MaTepHaJIa,

0c.

3HaK

rrJIroc

rrepe,lJ.

VI'

COOTBeTcTByeT

np5IMotc'KY.

Mlmyc

-

rrpoTHBoToKy

.

..

fpaHHqHble

yCJI()BHH ,l{JI5I

rrp5IMOToKa:

.

tr(O)=t~,

,l{JI5I

rrpOTHBoToKa:

;'.,

:f"

[fJI',

7-~

~'.

OnTHMAJJbHOE

ynPABJJEHHE

M.HOf~~:6'HHbIMH

AfPErATAMH

430

. /

.~.

-

teMnepaTypa

TenJIOHOCHTeJHI,

rrO,l{aBaeMorO

B30HbI

arperaTa;

OO'bel.f

terrJIOHOCHTeJIH

B

30HaX;

,l{JIl:JHa H KOJIHtIeCTBO

30H;

HanpaBJIe

He

,l{BH)KeHHH

MaTepHaJIa

H TenJIOHOCHTeJIH B

30Hax.

By,l{eT

paCCMOTpeH

xapaKTep.

3aBHCHMOCTH KOHetIHOH

TeMrrepa~

TypbI

MaTepHaJJa

OT

COOTHoweHHH

,l{JIHH

30H

Ha

npOCTeHweH

MaTe-

MaTHtIeCKOH

MO,l{eJIH

HarpeBaTOHKHX

TeJI 6

,l{BYX30HHOM'

arperaTe.

fIpH

ElTOM

He

6y,l{eT

yqHTbIBaTbCH

fH,l{paBJIHqeCKHH

pe)KHMpaOOTbl

arperaTa,

a OTHOWeHH5J

HallaJIbHbIX

TeMrrepaTyp

TeMnJIOHOCHTeJI5I H

OO'beMOB

ero

B

30Hax

rrpHHHMaroTCH

H3BeCTHbIMH.

[IOJIYtIaeMbIe

3,l{eCb

3<p<peKTbI

coxpaHHIOTCH

Ii

BCJIyllae

rrOJIHOro

'paCtIeTa

arperaTac

yqe-

'

TOM

rH,l{paBJIl1qeCKoH

cHcTeMbI,

O,l{HaKo

paCCMOTpeHJ.fe

IiX

Ha

ynpo-

nteHHOH

MO,l{em'I

nOMoraet

nOHHTb

<PH3HKy

npou.ecca

H ou.eHH'Th

npe-

,l{eJIhHhIe

3HaqeHHH

3<P<P

e

KTOB;

,l{OcTHraeMhIX

B CHCTeMe

3a

CtIeT

OntH-

MH3au.HH

..

KpoMe

Toro,

B

,l{aHHoM

CJIyqae

3TO

paccMoTpeHHe

MO)KeT

OMTh

npoBe,l{eHo

,l{O

KOHu.a

aHaJIHTHqeCKH.

'

I10cTaBJIeHHaH

3a,l{aqa

,npH

YKa3aHHhIX

yCJIOBHH-X

6y,l{eT

Bcer,l{a

HMeTb

CMbICJI.

1J.

eH

cTBHTeJIhHO,

eCJIH

HMeeTcH,

HanpHMep,

,l{BYX30If~

it:aH

npoTHBOTOqHO-npHMOTOqHaH

KaMepa

onpe,l{eJIeHHoH

,l{JIHHhI,'

C

onpe,l{eJIeHHhIMH

TeMnepaTypaMH

H OO'beMaMH

nO,l{aBaeMoro

B

30HhI

tenJIOHOCHTeJIH,

TO,

HCnOJIb3yH'

B

60JIhWeH

CTeneHH

npOTHBOTOtIHyro

30Hy,

nOJIytIHM

YBeJIHqeHHe

TeMnepaTyphI

MaTepHilJIa

Ha

BbIXO,l{e

H3

arperaTa.

BMeCTe

C TeM

BOCnOJIh30BaThCH

tOJIhKO

npOTHBOTOqHOH

30HO

H

HeBhIrO,l{HO,

TaK

K3K'

qaCTh

npe,l{Ha3HatIeHHoro

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npHMO-

TOtIHOH

30HhI

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He

6Y,L\eT,HCnOJIh30BaThCH COBceM.

TaKHM

06pa30M,

Bcer,l{a

MO)Kf/O

BMopaTh

TaKoe

COOTHoweHHe

,l{JIHH

npOTH-

BOTOtIHOH H

npHMOTOtIHOH

30H,

npH'

KOTOpOM ,l{OCTHraeTCH

MaKCHMYM

TeMrrepaTYPhIMaTepHaJIa.

"

PaCCMOTpHM

tIeThIpe

CXeMb(

,l{BHlKeHH5l

TenJIOHOCHTeJIH

(pHC.

7.S.I}

11

,l{BYX30HHOM

arperaTe,

HMeroru.eM

nOCTOHHHyro

6e3pa3MepHYro

,l{JIHHy,

F

_

at/o

0-

Vc

I\Qe

CX-K03<p<pHu.HeHT

TenJIOOT,l{atIH,

KKfl/l/('l .

.u

2

•

°C);

f-nOBepXHOCTh

1ienJIOOOMeHa

Ha

e,l{HHHU.Y ,l{JIHHbI

arperaTfI,

.u

2

/.u;

10-06ru.aH ,l{JIHHa'

arperaTa,

.u;

V,

C-OO'beMHbIH

pacXo,l{

H

y,l{eJIhHa5l

TenJIOeMKOCTh

MaTepHaJIa,

.u

3

/'l,'

K1CQA/(.u

3

• 0c) COOTBeTCTBeHHO.

1J.JI5l Ka)K,l{OH

H3

cxeM

Tpe6yeTcH

BhIHCHHTh,

npH

KaKOM3HatIe_

HHH

F

1

=

Qt~l

(Fa

<PHI{CHpOBaHo)

MaTepHaJIOM

6Y,l{eT

yCBoeHo

MaKCH-

'MaJIhHOe

ROJIHlIeCTBO

TerrJIa,

T.

e.

6y,l{eT

nOJIyqeHa

M3KCHMaJIbijil5l.

TeMnepaTypa

MaTePHaJIa,

f,l{e

1

-

,l{JIHHa

nepBOH

no

XO,l{y

MatepHaJIa

SOHbI (.u).

YpaBIleHH5I

KOHBeKTffBHoro

TeUJIo06MeHa

,l{JI5I

paCCMaTpHBaeMoro

CJIytIa5I

3anHChIBaroTC5l

B CJIe;Qyroru.eM BH,l{e:

+

Vrcrdti-=cxf(t-tr)dL,

1('1)'

cxf(tr-t)dL=Vcdt,

'(2)\

432

OUTI1MAJJbHOE .YUPABJJEHHE

MHCJOHHbIMH

ArPErATAMH

[rJJ,}

BbIliHCJIIiM

npOH3BO){HyIO

e~,

IIMeSi B

BH){y,

liTO FI = F

oYH

e;

=

rAe

F.=Fo(l-y):

(I-e.)

F

o

exp

[-(1

+Wi) Foy]':-

-(p-a

t

)

Foexp

[-(I

+W.)Fo(l-y»).

(10)

HHTerpHpYH

CHcTeMy

JIHHeAHbIX

){Hl}>l}>epeHQH8JIbHbIX

ypaBHeHHA

~:

(1), (2) C

rpaHHliHbIMH

yOJIOBHHMH (3), (4),

UOJIyliHM

BbIpa)f{eHge

){JIH

KOHeliHOA

TeMnepaTypbI

MaTepllaJIa

){JIH

npoTIiBOToKa:

(Jc

..

~=

I-exp

[-(I-Wo)F

o

] .

13-/

0

I-Wo'exp

[-.(I-W

)

F ] ,

(5)

o

){JI5I

np5lMOTOK8:

1-/

0

J

a=O---=-{I_exp

[-(1

+

Wo)F

]},

Ip-/O

1+

W .

o

(6)

~ .

W

o

r){e

=

y--OTHOIueHlIe

OO'heMHbIX TenJIOeMKOCTeA

MaTepllaJIa

H

rCr

ra3a.

<l>YHKQIIIi (5), (6) HBJI5IIOTC5I

HenpephIBHbIMIi

II

HenpepbIBHO

){Hl}>l}>epeHQlIpyeMbIMIi.

PeIUeHlIe

nOCTaBJIeHHoA

3a){alill

HBJI5IeTC5I,

no

cYIQecTBY,

IICCJIe){OB8HHeM

3KCTpeMaJIhHhIX

CBoAcTB

3THX

l}>YHKQIIA

no

pa3JIHliHbIM

rpynrraM

napaMeTpoB.

CxeMa

1-

){Be

np5lMOTOliHbIe

:':lOHhI.

DapaMeTpbI

nepBOH

30HbI:

J

W.=~=

W

o

F

=a/loy

. to

1 Vrlc

r

x'

I

Vc'

n,

r){e

x =

VrI/Vr-OTHOCIITeJIbHbIH

OO'heM

ra3a,

nOCTyrraIOIQero

B

nep-

ByIO

30HY;

Y = 1i/

l

o-oTHOCHTeJIbHa51

llJIHHa

nepBoH

30HbI.

Dapa-

MeTpbI

BTOpOH

30HbI:

W -

Vc

- W

o

F

_allo(l-y)

to

2 -

Vr.c

r

-

I-x'

1-

Vc

'

rl'

TeMnepaTypa

B KOHQe

nepBOH

30HbI

paccliIiTbIBae'I'C5I

no

l}>op-

MyJIe

(6):

1

~-~

J

[W

J

a =

I

rl

0

-to

= I + W

1

{I-exp

-

(1

+

1)

F

}.

I

(7)

TeMrrepaTypa

B KQHQe

arperaTa

COrJIaCHo

TOR

)f{e

lflOpMYJIe

(6)

paBHa

e

1

l

1:-t

=I+JW

{1-exp[-(l+WlI)F,,]}.

trl-/

I

. 2 .

(8)

. DO){CTaBJISISI (7) B (8),

nOJIyliliM

1,,-/0

II

a

=

t

n

0

-to

="1

+

(~-aI)

all,

(9)

rAe

~

=

1~2

--:-

to

tl!t-t

O

•

P

~~j~~1

onTHMH3AUHSI

,ltBYX30HHbIX

ArPErATOB

433

TeMnepaTypa

MaTepllaJIa

B KOHQe

KaMepbI

6YAeT

MaKCHMaJIbHOH

npH

Y=

Ym'

YAOBJIeTBOpsIIOIU.eM

ypaBHeHIIIO

a;

===0

O.

3'1'0

rpaBHeHlIe

MO)f{HO

rrpeAcTaBHTb

B BHAe

p=e

l

+

(1-8.)

exp

{-

[(1

+WI)

yo-(I

+

W.)

(I-yo)]

F

o

}.

(11)

OHO

HMeeT

peIUeHlIe

JIIIIUb

npH

BbInOJIHeHHII

HepaBeHcTBa

AJISI

napaMeTpa

Ym:

0 <

Ym

<

1,

II,

CJIe){OBaTeJIbHO, AJISI.

napaMeTpa

p:

Pi

<p

<

P.,

(12)

1

PI

= 1+ W 2

{I

+

W.

exp

[:-

(1

+W.) F

o

]},

1

P.=

1-

WI

{1

+

WI

exp

[-

(l

+

WI)

F

o

]}·

<l>OpMyJIy

(11)

MO)f{HO

rrpeoopa30BaTb

K BW,lly,

paclieTa

BeJIWliHHbIYm:

I ( I W 1 1

1-6

1

)

Ym=2+W

I

+W.

+ .+ F

o

n~-6I

.

Dpw

paBeHCTBe

nOTOKOB

rasa

B

ooewx

30Hax,W1 =Wt

<l>opMyJIa

(13)

rrpHHHMaeT

BH,ll

2

[.

2

(~-I)]

F

o

(1-2Ym)=-2+W

1n

~+

W

•

o

i

.

I

y,llOOHOMY ,llJIH

r

I

i

(13)

Ii

,I

il

II

=;

W

0'

II

H

(

14)

!.

3aBWCHMOCTb' (14) npe.n.CTaBJIeHa rpa<l>HKaMH

Ha

pHC.

7.8.2, KO-

Topble

MoryT

OblTb

HCnOJIb30BaHbI

,llJISI

paclieTa

OnTHMaJIbHOH ,llJIWHbI

nepBoH

30HbI

Ym

no

SMaHHbIM

F

o

,

W

o

'

p.

DPH

p=l,

x=0,5

OrrTIiMaJIbHOH oy.n.eT KOHCTPYKQHSI C

paBHbIMH

30HaMH,

T.

e.

Ym

=

D,S.

TaKIiM

OOpa30:'vl,

npll

paBeHcTBe

TeMrrepaTyp

H

oObeMoBTa30B

B

30Hax

HymHo

BbloHpaTb

paBHbIe

no

,llJIWHe

rrpSiMoToliHbIe

30HbI

C

TeM,

liTOObI rrOJIYliHTb MaKCHMaJIbHYIO

TeMrrepaTypy

HarpeBa

Ma-

TepHaJIa.

HCHO, liTO

rrpH

JII060M

liHCJIe

npSlMoTolillbIX

30H

C

paBHbIMH

TeMnepaTypaMw

H

oObeMaMH

raSOB

HX HeooxO,llHMO

BbI6l1paTh

paB-

HbIMW

no

.n.JIHHe.

CxeMa

II-){Be

rrpoTHBoToliHbIe

30HbI.

DpwpaBHW-

BaSI

Hymo

rrpOH3Bo.n.HYIO KOHeliHOH

T.eMrrepaTypbI

MaTepHaJIa

nOCJIe

BTOPOH

30HbI,

nOJIyliHM

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paCqeTa

OrrTHMaJIbHOH

.n.JIH"

HbI

nepBOH

30HbI

Ym:

.

(1-

W

1)2

(1

-

W.lJl2)2

lJl1

~

=;

8

1

+

(1-

(

2

)

(1-

W

2)2

(1-

WI

lJlI)2'CP;

,

(15)

435

4.14

OnTI1MAJlbI-lOE ynPABJlEHI1E

~HOCOIHHbIMH

ArPErATAMH

f){e

__

(--<Pi

._.

8

i

--

I

W'

t - I,

2,

--

iCf!i

CPi

=

exp

J-

(1-

W;) F

o

y.17l

J

,

i =

I,

2.

CxeM8

III-nepBBH

SOHa

npHMOTOtm8H,

BTop8H-npOTHBOTOqHa51.

"

Fo>O;

YCJIOBHe

OnTHMaJIhHoqH

samUlleTC~

B SH){e

1

rpaHHJ..J,bl

no

(3

IJOJlY'II1M, n0-!lcTaBJl5HI B

npaBylo

'laCTb

epOpMyJlbl

(IS)

3Ha'IeHHH

Ym=='OoHYm=l: ' 0 0

I 0

00'

~l=I--W2{I-W2exp[-(1-W2)Fo]}>I

npH

W

2

>O,

~

=1-

W

l

~l,--

WI)

{l--e-xp

(--

(1-117

1

)

Fol}

exp

(-

(1-

WI)

Fol

0 _

2 0

{1-W

1

exp;(-:-(l.,.o-W

l

)F

o

J}2

KaK

H

npH

paccMoTpeHHH

cxeMbI

I,

MbI

He·

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o3.i{eCb

Ha

){OK3SaTeJlbCTSe

e){HJICTB~HI:WCTH

MBKCHMyMB.

80

scex

pBCCMBTpHBBeMbIx CJIyqB51X :iTO npOBep51eTCH HenOCpe){CTSeHHO.

1,8

/3

01,4

PHC.

7.8.2.

. 1

B

CJIyqae

paBHbIX

nOToKOB,

WI

= W2

="2

W

0'

nOJIyqaeTCH 5lBHB5i

4>opMyJIa

.){JI51

paCtIeTa

Ym:

--I

. {I

__

~

o[

Ym=(I--W)F

In·

T=j3W 1+

o

, /

(I--~Wl{W(I--W)expr-(l--W)Fol

+V I + W

(l

_.~)2

I+

M

J},

0,6)

1 0

-f){e,Wo=2

Wo

'

DpH

~=I

H W

1

=W

2

,KaK

H B

cxeMe

I,

yCJIOBHe

MaKcHMyMa

TeMnepaTypbI

Tpe6yeT

paBHbIX

){JIHH SOH:

3TOT'

BbIBO.i{

CnpaBe){JIHB H"){JIH MHorOSOHHbIX J<OHCTpyKJ..J,Hll.

:

OOO

:

" "

0/

C

OnTHMI13AUH5I ,ll,BYX30HHbIX

ArPErAToB

~=I+lt'\

[I-exp

[-(1

+

WI)

FoYm]]

+

o

+exPl-~1+1V~I)FoYrnj{I-W2exp[(1~W2)Fo(l:"'-Ym)]}'

(17)

Ypa8HeHHe

(17) HMeeT e,ll,HHCTBeHHoe

peweHHe

no

Ym'

eCJIH

~.

y){OBJleTBopHeT

Hep3BeHcTsy.

(12),

f){c

1 0

~l=

l-W

{I-W2exp[-(l-~'V2)Fo]}

>

I,

2

~2=

I~W2

{I

+W

1

exp[:-(1

+W

1

)F

o

]}

<

I.

l.lT06bI

sbi'!HCJIHTb 3KCTpeMaJIbHOe

SHaqeHHe

Ym'

){OCT8TOQHO

pewHTb

ipaHcJ..J,eH){eHTHOe

ypasHeHHe

(17),

peWeHije

KOToporo

npH

SbIIlOJl-

H,e/-lHH

yCJIOBH5I

(12)

cy1ll,eCTByeT H e){HHCTBeHHO.

DPH

O,ll,HHBKOBbIX

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TenJIOHOCHTeJIH,

nO){aBaeMOro

B

30HbI,

peweHHe

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KBI<

3TO

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cxe~

Max

I H

II.

YpaBHeHHe

(17) npHBO){HTCH B 3TOM

cnyqae

K

TpaHc;

J..J,eH){eHTHoMy

ypaBHeHH~

2-(1

~

W

2)

exp

[-

(1

+W

I)

Foy",]

+

+(I+W

1

)exp[-(I-W

2

)F

o

(I-Ym)]=O,

(18)

peweHHe

KOToporo

nOJIyqHTb

B SlBHblX

epyHKJ..J,HHX

He y){8eTCH.

CxeMa

IV

-nepBaSl

SOHanpOTHBOTOqHaH,

BTopaH-npHMOTOqHaH.

YpaBHeHHe

){J151

3KctpeMBJIbHOrO

3Ha4eHHH

Ym

~MeeT

BH){

, .

y

6

1

~=I+

(I-W

1

)2

Il+

W

2

_

--

W

lYIJ

=0

(19)

.

(I-W,y)2

_1+W~

I-Wi

'

l')i.e

'y

=

exp

[-

(1-

WI)

FoYm],

6=

exp

[(1

+

W~)

F

o

(1

~Ym)]'

fpaHHJ..J,bl

no

.~

06JIaCTH

cyu~ecTBoBaHHH

0

3KcTpeMyMa

HMe~T

SH){

o

1

~1

=

I+W

{l

+W

2

exp

[(1

+ W

2

)F

o

])

>

I,

2

R=I

Wdl-Wl)(l-eSjlf

< 1

1"2

(1'---W

1

1f)2

'

r){e

s=-(l-W

1

)F

o

'

fIpH

~

= 1

ypaBI-teHHe

(19)

npe06pa3yeTcH

K BH){Y

2-

(1

~

WI)

ev

..

-(I

+W

2}

eVa

=

0,

(20)

Yl

=-

(I

+W l) F

o

(1-

Ym),

.

1'2

=....:..

(1-

WI)

FoYm'

YpaBHeHHe

(20).

npe06pa3yeTcH

B YP;;(BHeHHe 0 (18),

eCJIl.I

Ym

H

WI

3aMeHHTb

Ha

(l-y,;.)

H W

2

HJIH

Ha060pOT,

T.

e.

nOMeH51Tb

oi>

....

(~

.

436

OnTHMAJlbHOE

ynPABJlEHHE

MHOro~Ii~:HbIMH

ArPErATAMH

(rJl.

7,1

MeCTaMH

nepBylO H

BTOPYIO~

30HbI. TaKHM o(5pa30M,

arperaTbI

KOH-

CTpyKIUfH III H IV

npH

fl

=

I,

B

KOTOpbIX

napaMeTp Y BbI6paH

H3

YCJlOBHSI

MaKcHMy"Ma

TeMnepaTypbI HarpeBa MaTepHaJla,

HMelOT

O)(H-

HaKoBylO

npOTSl1KeHHoCTb

npSlMOTOllHbIX

,H

npOTHBOTOtIHbIX

30H,

YpaBHeHHe

(20)

6bIJIO

peIIIeHo C

nOMOW;blO

UBM

«HaHpH»

)(JISI

x =

0,

I;

0,2;

.••

;

0,9

BCJIe)(YIOW;HX

06JIaCTSlx

H3MeHeHHR

HCXO)(HbIX

)(aHHbIX:

o~

W

~

I;

0,2

~

o

o,8~

nn

J I I

I---+-....,...;~ll-O'Z

I

'-+--.=:J

0,51

I

.',

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o 0,8

Wo

1,0

'

Ys

I - I I

~

F

o

~

10.

Ha

pHC.

7.8.3

npHBe)(eH rpaqlHK,

n03BOJISI-

IOW;HH

paCCtIHTaTb OnTHMaJlb-

HbIe

COOTHOIIIeHHSI

)(JlHH

30H

arperaTa,

pa60TalOw;ero

no

cxeMe IV,

npH

x =

0,5.

Bo

Bcex lIeTbIpex

cxeMax

)(BYX30HHbIX

arperaToB

nOJIy-

tIeHbI !P0PMYJlbI

)(JlSI

BbISlCHe-

HHSI

cyw;eCTBoBaHHSI

3KcTpe-

MyMa H paCtIeTa

Ym'

nOpSl)(OK

paCtIeTa

cJIe)(ylOW;HH.

npH

3a-

)(aHHbIX

W

o

,

x,

~,

F BbItIHCJlH-

o

torcSI

3HatIeHHSI

~1

H

~2'

3aTeM

0,2

at,

O.B

npoBepSleTCSI

BbInOJIHeHHe

He-

PRe. 7.8.3.

paBeHcTBa

(12).

ECJIH

9'1'0

He-

paBeHCTBO

He

BbInOJIHSleTCH,'

TO

MaKCHMaJIbHaR

TeMnepaTy-

pa

6y)(eT

)(OCTHraTbCR

B 0)(H030HHOM

arperaTe

(HeCMOTpSl

Ha

TO,

11'1'0

lIaCTb ropSltIero

TenJIOHOCHTeJISI

Heoy)(eT

HCnOJIb30BaHa

Bo06w;e). B CJIyllae

BbInOJIHeHHSI

HepaBeHcTBa (12) napaMeTp onTH-

MaJIbHoro

)(eJIeHHSI

KaMepbI

no

aOHaM'

BbItIHCJIReTCSI

no

COOTBeTCT-

BYIOw;eH

!popMYJIe. •

n

pH

Me p

I.

PaCCMOTpHM

)(BYX30HHbIH

arperaT,

pa60TalOW;HH

no

cxeMe IV.

napaMeTpbI

arperaTa:

F =

4,0;

W =

0,3;

x =

0;5;

o

fl

= I. CpaBHHM

KOHetIHbIe

TeMnepaTypbI HarpeBa MaTepHaJIa B

arpe-

raTe C paBHbIMH

)(JIHHaMH

30H

H B

arperaTe

C

OnTHMaJIbHbIM

)(e-

JIeHHeM

aOH.

I.

ArperaT

C paBHbIMH

)(JIHHaMH

aOH:

. I

y=0,5;

Pi

=F,=F

o

y=2;

Wi = W

2

=-

W

=0,6.

o

X

no

!popMyJIe(5)

HMeeM

l-e-

Cl

-

O

,6).2

8

1

0,6e-Cl-O,6)

••

=

0,75.

no

!popMyJIe

(6)

HMeeM

e - --L.-

[I-e-

C1

+

o

,8).,]

==

°6

,-

1

+0,6

, •

CnTHMH3AUlUI

,nB;"X80HHbIX ArPBrATOB

431

HX

HaMH

.

Hy

9Toro

F

o

JIaM

F

o

=

tIeM

.

maeT

KOr)(a

,KOHetIHaSl 'l'eMnepaTypa

paCClIHTbIBaeTCR

j.......... • I

10,

78'1

, ,

? I

--=

I

no

!popMyJIe

(9)1

8

=0,75+

(1-0,7~).0,6

....

0,9.

2.

ArperaT

C OnTHMaJIbHbIM

)(eJIeHHeM

30H.

no

pHC.

7.8.3

Ha-

XO):{HM

Ym=0,79.

To'r)(a

Fi=FoYm

=4,0·0,79

=3,16;

F.

=F

o

(1-Ym)

=4,0.0,21

=0,84;

1 .

l_e-(l-O,6).8,16

WI =

WI

=X

W

o

c-O,6;

8

1

= t n

,,

__

"_0

.\.g

..

=

0,87;

1

e =

---

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C1

+

o

,6).0,S,]

= °

46'

I

1+0,6

',,'

,

H=

0,87

+

(1-0,87)·0,46

=

0,93.

TaKHM 00pa30M, B

arperaTe

OnTHMaJIbHOH

KOHCTpyKII.HH

6y,neT

yCBoeHo TenJIa npHMepHO Ha 3

% 60JIbIIIe, tIeM B

CHMMeTpHtIHOM.

n p H

Me

p II.

.lJ.JISI

yCJIOBHH

npHMepa I cpaBHHM

,nJIHHy

arpe-

0,98;

raTa

C

paBHbIMH

30HaMH H )(JIH- e

Hy

arperaTa

C OnTHMaJIbHblM )(e-

JIeHHeM

30H. npHMeM KOHetIHylO 0.91

TeMnepaTypy MaTepHaJIa

.D.JISI

000-'

arperaToB

8=0,9.

1.

ArperaT

C paBHblMH )(JIH-

30H

6Y.D.eT

HMeTb,

KaK

HB

. npHMepe I, 6e3pa3MepHylO

.D.JIH-

0.

81

I I

10.

7

F

o

=

4,0.

'

3,0

3,38

4,0

5,0'

2 . .lJ.JIlwy

arperaTa

C onTH-

Fo

MaJIbHhIM

)(eJIeHHeM

30H

HaXO)(HM

PHe.

7.8.4.

rpa!pHtIeCKH (pHC.

7.8.4)

.

.lJ.JISI

no

pHC.

7.8.3

onpe.D.eJISleM

3HatIeHHSI

Ym

)(JISlW

o

":':'

0,3

H

=

3,0;

4,0;

5,0

H

CTpOHM

KPHBYIO

Ym

= f (F

o

)'

3aTeM

no

epOPMY-

(5),

(6)

H

(9)

,paCCtIHThIBaeM

3HatIeHHSI

8

)(JISI

W 0 =

0,3

H

3,0;

4,0;

5,0.

C

nOMOW;blO

nocTpoeHHoH

KpHBOH

8 =

cp

(F

o

)

.D.JISI

8 =

0,9

onpe-

):{eJIReM

6eapa3MepHyro

.D.JIHHy

arperaTa

OnTHMaJIbHOH

KOHCTpyKII.HH:

F

o

=3,38.

naJIee

)(JISI

F

o

=3,38

no

KpHBOH

Ym=f(Fo)

onpe.D.eJISleM

9KCTpeMaJIbHOe

SHatIeHHe 6eapa3MepHoH

,nJIHHbl

nepBoH 30HbI

Ym=0,764.

PaclIeT nOKa3blBaeT,

tITO

arperaT

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Бутковский А.Г. Методы управления системами с распределенными параметрами

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