Бутковский А.Г. Методы управления системами с распределенными параметрами

498

tIHCJIEHHbIA

MEC;'

'COnpH:>KeHHOH: CHCTeMbl

ypaBHeHHH::

(a

21jJ

C

a1jJ

"A,

~,

a1jJ

) _

at..

a1jJ

_ 0

(Q)

l'

at + (

Q)

ap2

+p

ap ap ap

- ,

"A(Q)

(jaW

I +

~n

(4CXIQ~+CX2)-'I'0(4cx8Q~+CX4)=

p

/p=r

,.,

=2[Qn(t)-Q~(t)],

,po

+

'1'0

(4cx~R3

+CX

4

+CX

5

)-

!n

(4a

1

R3

+CX

2

)

= '

j'

, = 2 [R

(t)-

R*(t)J,

1jJ

(p,

T)

=

'1'0

(T) =

O.

' _

BblBO,lt

epOPMYJI (2), (3)

cOBeplUeHHo

aHaJIOrHtIeH

BblBO,lty

epop-

MyJI,

,(13)-(18)

§ 1

3T0l1

rJIaBbJ.

,

Ha

Ka:>K,ltOM

lUare

rpa){HeHTHOH:

npOue,ltypbl

onTHMH3aUHH'

Bbl-

nOJIH

HIOTCH CJIe,ltyIOI1.(He

BbltIHCJIeHH

51:

1.

flpH

3a,ltaHHOM

U (t), 0

~

1

~

T, H

HatIaJIbHbIX

3HatIeHHHX

Q (p, 0). R

o

pelUaeTcH

CHCTeMa

(1)-(4)

§ 1 ::nOH:

rJIaBbl

H

paCCtIH-

TblBaeTcH

3HatIeHHe

epyHKUHOHaJIa

(1).

2.

BbltIHCJIHeTCH

rpa,ltHeHT

epYHKUHlf'

11

(CX),

,ltJIH

tIero

np~,ltBa

pHTeJIbHO

HHTerpifpyeTcH

B

06paTHOM

BpeMeHH

CHCTeMa

ypaBHe-

HHH: (3).

3.

HOBble

YJIytIlUeHHble

3HatIeHHH

CXi

onpe,lteJIHIOTCH

no

epop-

MyJIaM

499

30

8

lf

9KcnEPHMEHTAJIbHA~

npOBEP~A

1

9

,2515,10

4

1

0

,2178,104

I 0,1912.10

4

2

/1

N

(}:,

G~

/1

I0,5251:

10

4

0,2911.10

4

I

20

6501

'~/h

I

"/0,

I

~"l

.750

,

800

I

-'-

--paC'IemllblO epaq}liK

~

I I

D 3Kl'nepliMellm. epttl/!tLK

'J(

-

.-

aCXoollttR Kpu8dR

R(t)

.npli

npol!3/J017hHOM

Ha'ld17bH~

'~

I I

/Jb/l50pe

'IliCe17

eti

. 0 A '

PI

NIl

I 5 I,

10

I

15

1.

MOIUHOCTb

3JIeKTpHqeCKH

H3MeflHeTcH

B COOTBeTCTBHH C

pac-

QeTHbIMynpaBJI5IIOIUHM'

B03,ltei.fcTBHeM.

2. B

KaqeCTBe

3a.n;aHHH B

nporp'aMMHblH

perYJIHTOp

BBO.n;HTCH

paCCqHTaHHa51

epa30Ba51

KOOp,ltHHaTa

OTI1HMaJIbHOrO

I1pouecca

R (t)'

JIH60

Qn

(I).

50iJ~,

T~"

Ll

; I ! ! I ! I

8 m

~

~ m

ffl

W

BpeMR,l1lill

PHC.

9.3.2.

80

BTOpOM

cJIytIae

3KcnepHMeHT

I1pOXO,ltHJI.

KaK

I1paBHJIO,

60JIee

yCl1eWHO,

H60,

HCI10JIb3YH

06paTHyIO

CB5I3b

110

epa30BOH:

KOOp,ltH-

HaTe,

y,ltaJIOCb

KOMl1eHCHpOBaTb

CJIYllaH:Hble

B03MyIUeHH5I,

epJIIOKTya-

UHHI10,ltBo.n;HMoro

Hal1p5l:>KeHHH H T.

.n;.

Bce

paCtIeTbl

OI1THMaJIbHblX

pe:>KHMOBI1POH3BO,llHJIHCb

C

HCI10JIb30BaHHeM

«HaHJIylllUHX»

K03CpepH-

UHeHTOB,

onpe,lteJIeHHblX

110

OITHCaHHOH

Bblwe

MeTo.n;HKe.

T a 6

JJ

H U a 9.5

(3)

(4)

rrJI.

9

all

aa{

•

a'

~

a'-'-h'[~

(

~~

tr

r,lte

k-HoMep

HTepaUHH.

J1TepaUHH

Be,ltyTcH

.n;o

Tex

nop,

nOKa

He

6y,lteT

BbmOJIHeHO

yCJIOBHe

I

~<'),

r,lte

<')

3a,ltaHO.

'

Ha

pHC.

9.3.2

npHBe,lteHbl

pe3yJIbTaTbl

O,ltHOro

H3

tIl:\.CJIeHHbIX

paCtIeTOB

HaHJIytIWHX

~

CMblCJIe

epyHKUHOHaJIa

(1)

K03epepHlIHeHToB

C

nOMoIUbIO

onHCaHHOH:

BbllUe

npoue.n;ypbI.

YpaBHeHHH

(1)-(4)

§ 1

3TOH:

rJIaBbl,

a

TaK)f{e

COnpH)f{eHHaH

CHCTeMa (2)

pelUaJIHCb

MeTo-

,ltOM

rrporoHKH

npH

N =

120,

/'il = 0,005,

~p

= 0,3

.10-

3

M.

TIpouecc

CXOJlHMOCTH

HTepaUHl1

rrOKa3aH

B

Ta6JIHue

9.5.

MaKCHMaJIbHaH

HeB5I3Ka

Me:>K,lty

paCqeTHblMH

H

3KcnepHMeHTaJIb-

HblMH

Kpl'lBblMH

cOCTaBHJIa,

KaK

BH,ltHO

H3

pHC.

9.3.2,

'He

60JIee

400.

.LI.ocTHrHyTaH

TOtIHOCTb

arrnpoKcHMauHH

3KcnepHMeHTaJIbHblX

KpHBblX

C

ndMOIUbIO

«HacTpOeHHOH:»

MaTeMaTHqeCKoH'

MO,lteJIH,

npo-

uecca

HarpeBa

HBJIHeTCH .n;ocTaTOQHoH ,ltJIH

rrpaKTHqeCKHx

paCqeTOB

OnTH¥aJIbHbIX

pe:>KHMOB.

PeaJIH3aUHH

paCCtIHTaHHblx

Ha

3BM

OnTHMaJIbHblX

pe:>KHMOB

oCyl1.(ecTBJIHJIaCb

.n;ByMH

cnoc06aMH:

'~

.....

F-

--

~

r--

C'

500

lJHCJlEHHblff

METO~

CD

ITJ};

:9'

SJJ

3KCnEPIiMEHTA.iJiJiiA5I

npbBEPKA

501'

OK

Ha:

pHC.

9.3·.3,

9.3.4

npe.n;cfak.iI'eAapeii;iIH3aUH:51

«Tpa.n;Hu.HOH~·

Harm> pe)KHMa. HarpeBa,XOTopbIH,

3aKJlIOqaeTC5I

II

IIo.n;beMe

T~Mne:

....l..

/

R(t)

700

patypbI

nOBepXHbcTH

TeJIa

.n;o

3a.zt~~IiOH

t

'nocJre:u.yIbrn.eH

BbI.n;ep)KKOH

1"-.

'-

npH

3TOH

TeMnepaType.

Ha

)l,HarpaMMeplic.

9.3A, nOKa3aHbI nepe-

Q(R,t)"'>

"

xo.n;Hble

npdueCCbi perY.HHtlOBami5l npH

3a.n;aHHH

perYJl5ITOpy 3Haqe-

800

.......

HH5I

Q

(r,

T)

= 680

OK.

..

.. . .

(J(o,t)

"

~

.......

liIo..

Bo

BTOPOM

cJlytiae (pHC.

9.3.3)

nOCT05lHCTBO

TeMnepaTypbI

no~

r---

500

BepXHOCTH

o6ecnettHBaJlOcb. ja.n;aimeM perYJl5lTOpy Teldrrepa:Typbi

.'

KJla.n;KH

CbrJlaCHO

paCqeTHoH.

KaK

BH.n;HO

H3

.n;RarpaMM,

TOqHOCTb

2ft.

22

iO

18

16

14

f2---..1fL_ 8

6 II

2 o

11

-11

f8

18

11;

12

10

8

~

8

It

2,

o

900

U(t)

1

o 0

5

10

15

20

i,HUH

!JpetfR,

HaH

PRe. 9.3.6.

PHC.

9.3.4.

9KCne\mMellTa.

Ha

pHC.

9.3.5

npe.n;cTaBJleHblpe3YJlbTaTbI peaJIH3a-

uim

onTHMaJibHoro peROiMa

HarpeBa.

OrpaIlHqeHH5I R

(t)

H Qn

(t)

BBIOOJIH5IIOTpl

npH

3TOM.

e

TOqHOCTbio

l)

<

15

0.

PeaJIH3aUH5I

pac"

c4IITaHHOro pe)KHMa otyrn.ecTBJI5IJIatb C nOMortiblo

a'a.n;aHH5I

peryJI5I-

TOP~

paCCqHTafIHOHKpUBOifR (t).

EC.tiH

Cy.n;~Tb

6pe~¥JIbTaTe

Harp~a,a

3a

BpeM5I

T =

20

Mart

no

nOJIyqeHHoMy nepena.n;y TeMnepaTyp

Me)K.n;y

OK

nOBepxHocTbFO

H ueHTp'oM

3'arO'TOBKH,

TO

HMeeM.

tJ.Q.

40° B cJIyqae

Tpa.n;HUHoHHoro pe)KHMa (pHC.

9.3.3)

H

I1Q

~

10°

I:Il:lH

OrInlMaJIbHOM

pe)KHMe, Kor.n;a

.n;onycKaeTc5I:

neperpeB

nOBepxll::6tTH

#a

20".

Ha

700

800

:.:

I

pRC.

9.3.6

trpe.n;CTaBJIeH'll.

peaJIH3a:U;H5I

OIlTHMaJIbHOrO

pe)KHM8

.n;JI5I

~

CJIyqaH pa3JIHtiHbIX

Haqi'lJIb'H~ix

TeMnepaTypneqH

H

KJIa.nKH.

TI1'J1

500

p'accMdrpemiH

.31:01'6

npRMepa

CJIe.n;yeT

06paTHTb

BHHMaHHe

'Ha

Z4

22

20

18

lOa

14

12

10

8

G

4

2 o

oc66~HH:bc'i:b

Haq'aJIbHOH

eta;ztHH

nponecca

Ha:rpeBa 06pa3Ua. YI3-3a

UpeHR,tfaH

"'

66JIbtlloro

TennbBoronoTOKa

qepea

nOBepXHOCTb

TeJIa

TeMnepaTypa

PHC.

9.3.5.

CTeHKH

neqH HeKoTOpoe

BpeMH

n-a'.n;aei

. .9Tot xapaKTepHbIH

.n;JUI

TaKHX

pe)l{HMOB

cna.n;

Te1>mepaTypbI

neqH

np"

noca.n;e

XOJIO.n;HbIX

3arOTOBOK·

BpeHR,

tfaH

PHC.

9.3.3.

OK

t

..

R(t)

(J(r,t)

~

"-

....

~

.......

(J(O,tJ

800

400

f-

R(t)

--

~

K

-

--

{l(r;tJ

~

r--

-

/Il(o,t

~

......

oTpaoDTKH

3a.n;aHH5I

B nepBoM cJlyqae

COCTaBJl5IJla

15--20°,

BO

BTO-

I

pOM--1O--25°.

3TH

3HaqeHH5I

Jle)KaT B npe.n;eJIax

TOqHOCTH

.n;aHHoro

I

pdC'/emH6le

it'

~

fin

{t},

Q(o;t),

R(t)

---

Kp11061e

____

3KCnepl1HeH-

man6Hble

G,

..:....:.:...

....

'

tIHCJlEHHblR

METO~

[rJl,

II

502

~OCTaTo.qHO

TOtIHOepHKcHpyeTcH

B npHHHTOA

MaTeM<1THtIeCKOA

MO,neJlH.

Pe3YJlbTaT~npOBe,neHH~X

Jla60paTOpH~X

9KCnepHMeHTOB

n03BO~

Jl5IIOT

C,neJlaTb cJle,nYIOm:He BbIBO,nbI. "

1)

npHHHTaH

MaTeMaTHtIeCKaH Mo,neJlb

npoll;ecca

HarpeBa

C ,no-

CTaTOtIHOH

TOqHOCTblO

onHCbIBaeT

npoueccbI

B

06beKTe.

2)

PeaJlH3aUHH OnTHMaJlbHbIX

pe:>KHMOB

Ha

06beKTe

MO:>KeTocy-

m:eCTBJlHTbCH

KaK

C nOMom:blO

nporpaMMHoro

3(i,naHHH ynpaBJlHIO-

m:ero

B03,neHcTBHH, TaK H C

no-

BUQ

cDe/lx!I

MOmblO

OTCJIe:>KHBaHHH

O,nHOH

H3

~

Koop,nHHaT paCCqHTaHHOro onTH-

MaJIbHOrO

pe:>KHMa.

npH

9TOM

npe,nnOtITHTeJlbHee Bbl60p TeMne-

paTypbI

neqH R

(t)

B KaqeCTBe

3a,naHHH

perYJIHTopy.

3)

3KcnepHMeHTaJIbHO ,nOKa-

3aHa

9tPq>eKTHBHoCTb

BbI60pa

on-

THMaJIbHOrO

pe:>KHMa

HarpeBa

C

nOMom:blO

H3JIOJReHHOH

3,neCb

Me-

TO,nHKH.

l{JIH

npoBepKH

9<p<peKTHBHo-

CTH

npe,nJIaraeMOH B

J3a60Te

Me-

TO,nHKH

onTHMH3aUHH

pe:>KHMa

HarpeBa

MaCCHBHbIX H3,neJIHH B

yCJIOBHHX, 6J1H3KHX K peaJlbHbIM

npoH3Bo,ncTBeuHbIM, 6bIJI

ocyme-

CTBJIeH

9KcnepHMeHT Ha orHeBOM

CTeH,ne

pereHepaTHBHoro

Harpe-

BaTeJlbHOrO KOJlo,nua (Mo,neJIb

Pac.

9.3.7.

1:

10), npHHa,nJIe:>Kam:eM

JIa60pa-

TOpHH MeTaJIJlypmqeCKOH TenJIO-

TeXHHKH

MarHHToropcKoro

MeTaJlJIypmqeCKOrO KOM6HHaTa.

Bo

Bpe-

MH

9KcnepHMeHTa anp06HpOBaJIHCb

pe:>KHMbI

HaHcKopeHIUeroHarpeBa

H3,neJIHH

,nJlH

rex

OC06blX

CJIyqaeB,Kor,na

BO

BpeMH

HarpeBa

,no-

nycTHM

JIHIUb BpeMeHHbIH

neperpeB

nOBepXHOCTH

H3,neJIHH

BblIUe

TeMnepaTypbI

BbI,naqH. 3KcnepHMeHTaJlbHbIH KOJlo,neu HCnOJIb30BaJICH

KaK

06~tIHaH

nJlaMeHHaH KaMepHaH

neqb

(peBepCHpoBaHHe nJIaMeHH

He ocym:eCTBJIHJIOCb).

B

pa60lfeA

KaMepe KOJIo,nua HarpeBaJIaCb ca,nKa

H3

Tpex

3aro-

TOBOK

MarHC3HTOBoro

KHpnHqa.

Ha

pHC.

9.3.7

npe,ncTaBJIeHbI nJlaH

pa.aMemeHHH 9KcnepMMeHTaJlbHbIX

CJlHTKOB

B KaMepe KOJlo,nua H

no-

JlO:>KeHHe

TOlfeK 3aMepoB

TeMnepaTypbI

B cpe,nHeM UHJIHH,npHlfecKoM

CJIHTKe H

~

,KaMepe KOJIo,nua. B UHJIHH,npHqeCKOM CJlHTKe

TeMllepa-

Typa

H3MepHJlaCb B

Tpex

TOqKaX (,nBe TOlfKH Ha nOBepXHOCTH (1, 2)

H o.uHa

Ha

OCH

I1,HJIHH.upa).

llJecTb

TepMonap

«(4-9),

eM. pHC.

9.3.7)

CJIy:>KaT

.uJIH

H3MepeHHH

TeMnepaTypbI

B

pa60qeM

npocTpaHcTBe

nelfH.

/.'

II

r,

C',-

SKCI1EPMMEHTAJIbHA5I npOBEPKA

503

-'

.3a

TeMliepaTypy

BpeHR.

!'faJl

Pac.

9.3.8.

,

m~pa':'!lpa

new/

r

noDepXJlOCmb

"

\

F<:"

n08eIJXIIOCmb

\,

'"

\'\

-

r--

""

me!'fnepamf/pa

ne'lu

\\'

r-.

tl,eJlmp

.............

0-

\

ljeJlmp

~

,

"

,

I'..

-....

.......

............

28

2lf

..

20

15

..

12

8

0

nelfH

R (t) 6blJIO lIpHHHTO cpe.u

Hee

3HaqeHHe lIOKa,-

'",

3aHHA 9THX

TepMonap.

B KaqeCTBe

TepMonap

HcnOJlb30BoClJlHCi>

lIJIaTHHo-po.uHA-nJiaTHHOBble

(nn)

TepMOnapbI.

Pa60·me

cnaH

He

9KpaHHpOBaJlHCb.

3anHCb

TeMliepaTyp

npOH3Bo,nHJlaCb Ha

Tpex

H

rnecnf

TOlfeqHblX nOTeHII,HOMeTpliX

THna

3nn-09

C rpa,nyHpoBKOH

nn.

/

Pa3MemeHHe

CJlHTKOB

B KaMepe KOJIo,nua 06eCneqHBliJlO paBHO-

MepHOCTb

TeMliepaTypHoro

nOJlH

BOKpyr

UHJIHH,npHqeCKOrO

06pasua

°C

1200

1000

800

500

400

200

(CM. pHC.

9.3.7).

MaKCHMaJIbHaH

pa3HocTb

noKasaHHH

TepMoliap

,

(4~9)

He lIpeBblIUaJIa

20°.

BKaqeCTBe

MaTeMaTHqeCKOH

Mo,neJIH

npou

ecca

HarpeBa

6bIJIH lIpHHHTbI ypaBHeHHH

(1)-(4)

§

1.

Ha-

cTpoHKa

K09<p<PHUHeHToB

Mo,neJIH

H paCqeT OIITHMaJibHoro lIO 6bICTpO-

,neHcTBHIO

pe:>KHMa

6bIJIH lIpoH3BeAeHbI lIO olIHcaHHoH

Bblllle

MeTo,nHKe.

nOJIYQeHHbIH OnTHMaJIbHbIH

pe:>KHM

peaJIHSOBaJICH

OTCJIe:>KHBaHHeM

TeMnepaTypbI

nOBepxHocTH

HarpeBaeMoro

CJIHTKa. PeryJIHpOBaHHe

lIo,nalfH

ra3a

ocymeCTBJIHJIOCb

BpyqHylO.

Ha

pHc.9,3.8

lIpe.u

CTaB

-

JIeHbI pe3YJIbTaTbI o,nHoro 9KclIepHMeHTa lIO HaHcKopeHIUeMy

HarpeBy

UHJIHH.upHqeCKOH 3arOTOBKH. Tpe60BaJIOCb lIOJIyqHTb

TeMlIepaTypy

1080

0

C

npH

,nOlIyCTHMOM MaKCHMaJIbHOM

OTKJIOHeHHH

+30°C.

B

lIep-

BOM

OlIbITe

HarpeB

II

pOBO,nHJICH

IIO 06bIlfHOMY

clIoc06y,

IlpHHHTOMy

lIpaKTHKaMH:

HarpeB

nOBepXHOCTH

.uo

1080°C

C 1I0cJIe.uY

lO

m

eH

B~,nep:>KKOH

IIpH

9TOH

TeMlIepaType.

Bo

BTOpOM

cJIyqae

.uolIycKaJICH

neperpeB

1I0BepXHocTH Ha

·100

°C

H

HarpeB

.uo 3a.uaHHoro

lIepe

lIa

.u

a

ocymeCTBJIHJICH

sa

HaHMeHbIUee BpeMH.

B pe3YJIbTare peaJIH3aUHH

BToporo

clIoc06a,

HarpeBa

3arOTOBKH

y,naJIOCb B

1,5

pa3a

cOKpaTHTb BpeMH

HarpeBa

,no

sa,naHHbIX

yeJIOBliH.

"I

504

LItfCJIEHHbIf!

l'j:',':,A

.r:'-

,

"-'/

[r

TaK

"Q(x" y)

,

B

HHe

JI.

'

§ 4. 0 K:0p,peK:Tl;f,ocTH sa,AalJ

,AJUI

pacnp'e~~eHlu;>I~

cl'!cTe~

Ba)I{He~llIHM

Tpe60BaHHeM,

3a,Aa'laM

KOTopoe

rrpe,Ab5JBJUIeTC5J K

ypaBHeHHH

MaTeMaTj.r~ecKOli

epH3HKH

KaK

){HepepepenUHaJIbHbIX,

H

HHTerpaJIbHbIX,

51BJI5IeTC5I

Tpe60BaHHe

HeK,oTopOH

rp

y60CTH3THX

ypaBHeHHH,

B

'onpe){eJIeUHOM

CMbICJIe

3Toro

CJIOBa.

.lJ.eHcTBHTeJIbHO, eCJIH

KaKa5l-JIH60

epYHKUH5I,

CKa)I(eM

Q{x,

y),

onHCbIBaeT

COCT05lHHe

HeKOTopOH

epH3H'leCKOH

peaJIbHOH

CHCTeMbI

B

06JIaCTH

D (x,

y),

CKa)I(eM

3TO,

pacnpe){eJIeHHe

TeMrrepaTypbI

B

TeJIe

D,

TO

eCTeCTBeHHO

nOTpe60BaTb,

'lT06bI

COOTBeTCTBYIOIqa51

MaTeMaTH'leCKa51

3a){a'la,

peweHHe

KOTOpOH COCTOHT B

onpe){eJIeHHH

HCKOMOH

epyHKUHH

Q

(X,

y),

6bIJIa

6bI

B

onpe){eJIeHHOM

CMblCJIe

rpy60H.

HanpHMep,

eCJIH MbI

HHTepecyeMc5I

CTaUHOHapHbIM

pacnpe){e-

JIeHHeM

TeMnepaTypbI

B

06JIaCTH

D

(X,

y),

TO

Mbl

3HaeM,

'lTO

TeM-

nepaTypa

Q

(X,

y)

){OJI)I(H;;t

y){OBJIeTBOp5lTb

ypaBHeHHIO

JIanJIBCa

8

2

Q 8

2

Q

!1Q

=

8x

2

+8

y

2

=0.

(1)

O){HaKO

){JI5I

0){H03Ha'lHOCTH

peWeHH5I

ypaBHeHH5I

(1)

Hy)I{HO

3a){aTb

eIqe

H

KpaeBble

yCJIOBH5I,

T.

e.

3Ha'leHH5I

epyHKUHH

Q(x,

y)

Ha

rpaHHue

r

06JIaCTJi

D

(X,

y).

TIpaKTHqeCKH

3TO

«3a){aHHe»

rpaHHq-

HblX

yCJIOBHH ){eJIaeTC5I

nyTeM

HenOCpe){CTBeHHbIX

H3MepeHHH

TeM-

nepaTypbl

Ha

rpaHHue

r

06JIaCTH

D

(X,

y).

Ho

BCHKoe

H3MepeHHe

npaKTH'leCKH

MO)I(eT

6bITb

BbIllOJIHeHO

JIHWb

C

orrpe){eJIeHHOH

CTe-

neHbIO

TOqHOCTH,

OHO

BCer){a

CO){ep)I(HT

norpeWHOCTb

H MO)f(eT

6bITb

BbIllOJIHeHO

JIHWb

B

KOHe'lHOM

qHCJIe

TO'leK

I;'paHHUbI

r.Ta-

KHM

06pa30M,

npH

MaTeMaTH'leCKOM

peweHHH

3a){a'lH

(1) Mbl

BCer){a

){O~OJIbCTByeMc5I

HeKOTopOH

npH6JIH)I(eHHOH

annpOKCHMaQHeH

rpa-

HHqHbIX

yCJIOBHH. B

npe){eJIax

TO'lHOCTH

H3MepeHHH

y

pa3HbIX

HCCJIe){OBaTeJIeH

(){a

H y O){HOrO H

Toro

me

HCCJIe){OB8TeJI5J B

pa3-

Hoe

BpeM5I)

){JI5I O){HOrO H

Toro

)I(e

COCT05lHHSJ

06beKTa

6Y){yT,

Bo06Iqe

rOBop5l,

nOJIY'laTbC5Ipa3Hble

rpaHH'lHble

yCJIOBH5I,

KOTopbIe

B

onpe){eJIeHHOM

'CMbICJIe

(B

onpe){eJIeHHOH

HopMe)

MaJlO OTJIH'laIOTCSJ

,){pyr

OT

){pyra.

K

C'laCTbIO,

npHpO){a

){aHHOro

rrpHMepa

TaKOBa,

'lTO

pe3YJIbTaTbI

paCLIeTa

nOJI5I Q (x, y)

BO

BceH

06JIaCTH

D

TaK)I{e

(ly){yT

MaJIO

(8

onpe){eJIeHHOH

HOp

Me)

OTJIHqaTbC5I

){pyr

01'

){pyra,

eCJIJi

rpaHH'lHble

yCJIOmiH

MaJIO OTJIH'laIOTC5I

){pyr

01'

){pyra.

ITpn

TaKOM

nOJIO)I(eHHH ){eJl MeTO,ll

ypaBHeHHH

npHHOCHT

OrpoMHyIO

nOJIb3Y

HayKe

H

npaKTHKe.

Pe3YJIbTaTbI

3KcnepHMeHTOB

cTaHoBHTCSJ

0){H03Ha'lHbIMH

H BOC-

npoH3BO,llHMbIMH

(KOHe'lHO,

C

onpe){eJIeHHOH

H3BecTHOH H

){OCTaTO'l-

HO

MaJIOH

norpeWHOCTbIO).

.lJ.,e~,CTBH1:eJIbHO,

HanpJi~ep,

B

cJIy'lae.

Kot){a

06JIaCTb

D

SJBJI5IeTcSJ

KpyroM

pa.ztHyca

R.

c

ueHTpbM

B

Ha'laJIe

KOOp){HHaT,

TeMnepaTypa

c:

,~

w

~-~':

o KOPPEI(THO,CTH

3;A.l1Aq~

ooac

-I

BlIyTpH

D

BbIqHCJISJeTC5Ino

CPOPMYJI&

TIyaccoHa,

KOTopaH

H

'B

,nOJISIpHbIX

KOOp,AHHaT8X

HMeeT'

BHJl

2"

I

" " 1

('"

(R2_

p

2)

d<p

" '

i'

'~'<x,

y) = Q

(p

cos

a,

p SID

a~=2n

~

f

(rp)

RI.,..;.2Rp

cos

(<p_a)+p2'

(2),

r,Ae f

(rp)'--TO

caMoe

rpaHHQHOe

yCJIOBHe.

TeMnepaTypa

Ha

rpaHHQe:

Kpyra

D.

113

cpOpMyJIbI

(2)

BHJlIUI

HenpepbIBllaSI

saBHCHMOCTb

pewe-

HHSIQ

(x,

y) 01' f

(rp).

OWH6KH

B 1:13MepeHHH f

(rp)

CKa)I(YTc5I·

Ha

Q(x,y),

HO,

'lTO

Ql.leHb

8'3)I(HO,

3TH

OIllIIOKH

MO)I(HO

OQeHHTb

H;

CJIe:n.OB'aTeJIbHO,

pe3YJIbTaTbI'

pa3HblX

$I{CnepUMeHTOlJ

MoryT

6bI1'b:

p

cpaBHHMbI

Me)I(,Ay

C060H.

3AeCb

6bIJIa

paC~MOTpeHa

B KaLIeCTBei

",

II

npHMepa

KpaeBaSJ ,3a,AaQa

)lHpflXJIe,AJISJ

ypaBHeHHSJ

JIanJIac,a'

Kpyre.

3Ty.

3a,AaQy

MO)I(HO, B

,CHJIy

CKa3aHHoro

BbIwe,

Ha3BaTb'

II

KoppeKTHoH.

Boo6Iqe,'3<8.)!(a'la

Ha3bIBaeTCSJ

KoppeKTHOH,

eCJIH

pewe-

cyIqeCTByeT

H eA'ltUc.1'tleHHO

npH

JII06b1x

Ha'laJIbHbIX

HJIH

rpa-

~I

!j

HH'lHbIX

){aHHbIXH3

aeg6l\O,pOrQAOCTaTO'lHO

itmpoKoro

K,1IaCCa H

31'0'

peweHHe

uenpepbIBHO

(i'r

onpeJte.1lellHOH

MeTpHKe)

3aBHCHT

01' HCXO,l{-'

j

HbIX

YCJIOBHH

3a){a'lH.

, .lJ.OJIroe

BpeMH

CQHTaJIOCb,

~ToBce

paccMaTpHBaeMbje

cpH3H'leCKHe

•

3'a:n.a'lH ){OJI)I(HbI

6blTb

Kopp'eKTHbIMH

B,

Yl<a3aHHOM

Bblwe

CMbICJIe, I

I!

II,

H

ypaBHeHHSJ,

oTo6'pallialQill,He

peaJIbHble

epu3UQeCKHe

npOQeCCbl,

He

MoryT

6bITb

HeKOppf'KTHbIMH.

TaKHM

06pa30M,

6blJI

npOB03rJIaWeH

Ii

':

npHHQHn

KoppeKTHOCTH

3a.n.aLI){JISJ

ypaBHeHHH

MaTeMaTH'leCKOH

cpH-

3Him.

O.n.HaKO B 1904

r.

AllaMap

flQCTpOHJI

npHMep,

He

TaKOH

y)I{

3K30-

i

1

TH'leCKHH,

HeKoppeKTHOH

aa.n.a'lH,

OCHQ~Hblli

Ha

TOM

)I(eypaBHe-

,

HI-lH

JIaUJIaCa

(1),

HO

,B:JIH

KOToporo

pemaJmet.B;,e",

3a){aQa

.lJ.HpIlXJIe,

r

a

3a.n.a'la

KOWH.

3TOt

npmJep

nOKa3bIBaeT;'trTo

){JIH

O){HOrO

H

Toro

)I(e

OCHOBHoro

ypaBHeHHH,

CKa)I(eM

JIanJIaca,

3a,llatIa

MO)I(eT

6bn'b'

~

KoppeKTHoH

HJIH

IieKoppeKTHoH

B 3aBHCHMOCTH 01'

TOro,

KaKHe

rpa-

I

HH'lHbIe

yCJIOBHSJ 3a,llaIOTCSJ.

~

JIerKo

HaHTH

H

Henocpe.n.cTBeHHoH

no.n.cTaHoBKoH

npc:>BepHTb,'

I

t

qTO

epyHKUHH

'

Q (x, y)

=e-

vt1

eng

sin

nx, n = 0,1,;2,

...

,

;{3~

n

I

SJBJISJIOTCSJ

peweHHSJMH

yp

aBHeHHH

(1)

npH

JII060M

n.

3TH

peWeHM}I'

npR'

y = 0 y){OBJIeTBopSJIOT CJIe.n.yIOI1\HM

Ha'laJIbHbIM

){aHHbIM:

!

i

Qn

(x,

0)

=e-Y'nsin

nx,

n=O,

1

•...

,

(4) ,

;

i

.

iJQ

y-

ai'(x.O)=ne-

n'sInnx.

n-O.

1,

...

(5) , "

1

I

3TH

cpyHKUHH

MO)I(HQ

CqHTaTh

HaqWJtbHbIMH

yCJIOBHHMH 3a.n.aqH

K9WI:i,

KOr){a

"PH

!!

-=

0 3a.n.aHa

~KUHSJ

H

ee

npOH3BOAH8SJ

(01'-

KIIOHeHHe

H CKOpOCTb OTKJIOHeHHSf)" 'JlJIH

ypaBHeHHSlJlanJI8ca.

.

17

A.

r.

5YTKOBCKBI

,

CKOJIb

lIHCJIEi.HbIA

MfIZ~

506

HPH

n _

00

9TH

HallaJIl>Hble

J1.aHHbIe CTpeMHTCH K HyJIIO

H,

oOJIee

Toro,

Bce

npOHSBo.n;Hble

no

x

(npt!

!I

= 0)

JII()ooro

<pHKcHpoBaHHoro

nopH.n;Ka

p,

KaK

JIerKO

npOBepHTb,

TaK}f{e

Oy.n;yT CTpeMHTbCH K

HyJIIO.

C

.n;pyroH

CTOpOHbl, H B 9TOM

.n;eJIO,

<pyHKUHH Q

n

(x, y)

=e-Vn+nY

sin

nx

C . pOCTOM

tl.

HeorpaHHtIeHHo

BospacTaIOT

.n;JIH

JIIOOOrO .n;a}f{e

yro.n;Ho

MaJIOrO y >

0.

3TOT

npHMep

nOKaSbIBaeT,

liTO HS MaJIOCTH

HOpMbl

B

HallaJIbHblH

MOMeHT

BpeMeHH

He

CJIe.n;yeT MaJIOCTb

perne-

HHH B nOCJIe.n;yrow.He MOMeHTbI

BpeMeHH.

no

npe.n;JIO}f{eHHIO

A.n;aMapa

TaKHe

sa.n;allH HaSbIBaIOTCH

HeKOp-

peKmHblMu.

Booow.e,

sa.n;alla HaSblBaeTCH

HeKOppeKmHOU;

eCJIH OHa

HeHMeeT

perneHHH

npH

HeKOTopblX

HallaJIbHbIX

.n;aHHbIX HJIH 9TO

perneHHe

Hee.n;HHCTBeijHO

H~H

eCJIH HeJIbSH

BblopaTb

TaKHe

HOpMbI,

lITOObl B

HHX

HMeJIa

MeCTO

HenpepblBHaH

saBHCHMOCTb

perneHHH

OT

HCXO.n;HblX .n;aHHblX sa.n;allH.

TaKHM

oopaSOM,

sa.n;alla

KornH

.n;JIH

ypaBHeHHH

JIanJIaca

HBJIHeTCH

HeKoppeKTHoH.

TaK}f{e

HeKoppeKTHOH

HBJIHeTCH sa.n;alla

KornH

.n;JIH

TaK

Ha3blBaeMoro

06pamH020

ypa8l-teHUR..

JIanAaca.

HeKoppeKTHOH

HBJIHeTCH H 3a.n;alla

KornH

.n;JIH

TaK

HaSblBaeMoro

06pamHo2o

ypa8-

HeHUR.

menAonp0800Hocmu:

iJQ

iJlQ

(6)

7fi=

-]iT'

eCJIH sa.n;aHO

HallaJIbHOe

yCJIOBHe

"PH

t = 0,

CKa}f{eM

Q(x, 0)

=e-vnsin

nx, n

=0,1,

...

,

(7)

TOllHO

TaKoe,

KaK

If

B npe.n;bl.z(yw.eM

npHMepe

A.n;aMapa.

PerneHHe

Qn

(x,

t),

COOTBeTcTByIOm.ee

HallaJIbHOMY

yCJIOBHIO (7),

"PH

t >°

oy.n;eT HMeTb BH.n

Qn

(x,

t)

=

e-

Vii e

n

•

t

sin

nx,

n = 0,

1,2,

..

. (8)

I1s

(8)

BH.n;HO,

liTO

,Aa}f{e

npH

CKOJIb yro.n;HO MaJIOM

(B

CMblCJIe

KaKOH-TO

. onpe.n;eJIeHHOH

HOpMbl)

HallaJIbHOM

YCJIOBIHI *) C pOCTOM

BpeMeHH

t

perneHHe

(8) oy.n;eT CKOJIb

yro.n;Ho

OOJIbrnHM.

3T0

H

.no-

Ka3bIBaeT

HeKoppeKTHOCTb

3a.n;allH

KeirnH

.n;JIH

oopaTHoro

ypaBHeHHH

TenJIOnpoBo.n;HOCTH (6).

OopaTHoe

ypaBHeHHe

TenJIOnpoBo.n;HOCTH (6)

He

CTOJIb 9KSOTHllHO,

KaK

MO}f{eT

nOKa3aTbCH

C

nepBoro

B3fJIH.n;a. C 9THM

ypaBHeHHeM

Mhl CTaJIKHBaeMCH

npH

perneHHH

.n;OBOJIbHO

OOblqHblX 3a.n;aq

onTH-

MaJIbHOrO

ynpaBJIeHHH.

O.n;HHM

HS

npHMepOB

3a.n;aqH

OnTHMaJIbHOrO

yhpaBJIeHHH,

r.n;e MbI

npHxo.n;HM

K

oopaTHoMy

ypaBHeHHIO

TenJIOnpo-

BO./tHOCTH, HBJIHeTCH CJIe.n;yIOw.aH 3a.n;aqa.

HaHTH

TaKOe-HaqaJIbHoe

*)

Manoli

BeJlH'IHHbI

Ha'laJIbHOrO

yc:.nOilHR

MOX<HO

A0(iHTbCli

BbiOOPOM

AOCTa-

TO'lHO

(iOJibworo

n B (7). .

\ -l

C

..

,

501

,oj

o KOPPEKTHOCTH

-3A.n;AQ

":'~>'

~

pacnpe.n;eJIeHHe

TeMnepaTypbl

I

(9}

O~x~l,

I

Q (x, 0) =

Qo

(x),

l

:qTOObl B CHJIy

ypaBHeHHH

TenJIOnpOBo.n;HOCTH

iJQ

(}2Q

(10)

,',

..

/

ar=

iJx

2

fl<:lJIyqHJIOCb

TaKoe

pacnpe.n;eJIeHHe

Q (x,

T)

B MOMeHT t =

T,

npH

..\\

KOTOpOM

HHTerpaJI

I

(1

U

~

[Q.(x)-Q

(x,

T)]2dx

!I

o

.n;oCTHraeT

MHHHMyMa.

YpaBHeHHH

(9) H (10)

Hy}f{HO

.n;onOJIHHTb

Ii

rpaHHqHblMH

yCJIOBHHMH,

Ha

KOTOpbIX Mbl

He

oy.n;eM OCTaHaBJHlBaTbC5f·,

Tl!K

KaK

::ITO

B .n;aHHOM

CJIyqae

Hecyw.eCTBeHHO.

ECJIH ObI OOpaTI-IOe

ypaBHeHHe

TenJIOnpOBo.n;HOCTH OblJIO

KoppeKTHbIM,

TO

C<pOpMyJIHpO'

I

BaHHaH

TOJIbKO

liTO

3a.n;alla

onTHMaJIbHOrO

ynpaBJIeHHH

ObIJIa ObI

TpHBHaJIbHOiL

B

::ITOM

CJIyllae

MO}f{HOObIJIO

ObI

HaHTH

TaKoe

Q0 (x),

\

npH

KOTOpOM

HHTerpaJI

(11) .n;OcT}-\r

ObI

aOCOJIIOTHOrO MHH}-\MyM8,'

paBHoro

HyJIIO,

T.

e.

Q*

(x) = Q(x,

T)

. .

)leHCTBHTeJIbHO,

.n;JI5I

TOro.

l

\

lJTOObl HaHT}-\

TaKoe

Qo

(x),

npH

KOTOpOM

Q*

(x) = Q(x,

T),

.n;ocTa

\

I

TOllHO OblJ10 Obi

pernHTb

oopaTHoe

ypaBHeHHe

TenJIOnpOBo.n;HOCTH

, I

(12)

(}Ql iJ2Ql'

O~x~l,

(ft=

-

iJx

2

,

1

npR.HallaJIbHOM

yeJIOBRH

(13)

j

Ql(X,

0)=

Q.

(x),

O~x~l.

PerneHHe

ypaBHeHH5I

(12) C

HaqaJIbHblM

YCJIOB}-\eM (13) B MOMeHT

BpeMeHH

t = T .n;aJIo ObI HCKOMoe

HaqaJIbHoe

pacnpe.n;eJIeHHe

Q0 (x):

Qo(X)=Ql(X,

T),

O~x~l.

(14)

HeKoppeKTHOCTH

9TOH 3a.n;allH OHa lI.aJIeKO

He

Bcer.n;a

O.n;HaKO B CHJIy

ECJIH,

CKa}f{eM,

pacnpe.n;eJIeHHe

Q*

(x) onpe.n;eJISleTCSI

HMeeT

perneHHe.

PSI.n;OM

.QD

(

15)

.Q. (x) =

~

e-

Vii

sin

nx

n=l

npH

1=

1T.

H

rpaHHlIHblX

YCJIOBHSlX Q1 (0, t) = Q1

(1T.,

t) = 0,

TO

HCKcr

Moe

Qll

(x) B CHJIy (14)

.n;OJIJKHO

onpe.n;eJIHTbCSI

PSl.n;OM

QD.

(16)

Q0 (x) =

~

e-

vii

e

n

'/

sin

nx,

11

..

1

,,*

509

508

lJHCJJE~Hbrn

C1~

KOTOpbIH

pacXO~HTCSI

rrpH

f > 0

H,

CJIe~OBaTeJJbHO,

HHKaKoH

<PYHK-

lJ:Hlf

orrpe~eJJHTb

lie

MOJKeT.

3a){alJH

yrrpaBJJeHHSI

TaKoro

.po){a·

pac-.,.

'CMOTpeHbI

B [120,229]. .

06paTHoe

ypaBHeHHe

TeUJJ'orrpOBO){HOCTH MOJKeT

BOSHHKHyTb

H .B

){pyrHx

sa){alJaXOrrTHMHSalJ:uH,

KBK

ypaBHeHHe,

«corrpSIJKeHHoe»

06bIlJHoMy.ypaBHeHHIO

TeUJJOrrpOBO){HOCTH.

nycTb,

HarrpHMep,

Sa){aHO

ypaBHeHHe

TerrJJOrrpOBO){HOCTH

iJQ

iJ2Q

. ,

7F=

8x2

+U (x, f), (17)

r){e

Q (x,

f)-pacrrpe){eJJeHHe

TeMrrepaTypbI

B TeJJe, 0

~

x

~

t, f >0

If

U (x,

f)-yrrpaBJJSIIOiu:ee

BOS){eHCTBlie,

CTeCHeHHoe

yCJJOBHeM

IU (x, f) I

~

N.

(18)

nYCTb

Sa){aHM

HeKoTopMe

rpaHHlJHMe

H

HalJaJIbHMe

YCJJOBHSI:

Q (0, f) = Q

(1,

f) =

0'(19)

Q (x, 0) = Q0 (x). (20)

3a){alJa

OIITHMaJJbHOro

ynpaBJJeHHSI

COCTOHT B TOM,

lJTo6bI

HaHTH

.TaKoe

yrrpaBJJeHHe

U (x, f),

CTeCHeHHoe

yCJJOBHeM (18),

lJTo6bI

pac-

rrpe~eJJeHHe

TeMrrepaTypl>I

Q(x,'

f) B 'CHJJy

ypaBHeHHSI

(17) H yCJJO-

BlfH

(18), (19) CTaJJO

Sa){aHHbIM,

T.

e.

Q*

(x),

sa

MHHHMaJIbHoe

BpeMSI T i :

T.

e.

lJT06bI

yCJJOBHe

Q (x, T)--:.Q'* (x)

(21)

BMrrOJJHHJJOCbrrpH

HaHMeHbilleM

T.

ECJJH CJJe){OBaTb <popMaJJbHO

o606ru:eHHIO

rrpUHlJ:Hrra

MaKCH-

MyMa

JI.

C.

nOHTpSIrHHa

Ha

sa){alJY

(17)-(21),

TO

HyJKHO

COCTa-

BHTb

<PYHKU:HIO

nOHTpSIrHHa

[49]:

iJ2Q

II (Q,

U,

1jJ)=- 1+1jJ

iJx

2

+

1jJu.

(22)

OTCIO){a

CpBSy

BH){HO, lJTO

OnTHMaJIbHOe

ynpaBJIeHHe,

npH

KOTOpOM

){OCTHraeTCSI

MaKcHMyM

<PyHKU:HH (22),

HMeeT

BH){

, U (x, f) = N sign

1jJ

(x; f),

(23)

r){e

<PYHKU:HSI

1jJ

(x, f)

y){OBJIeTBOp

SIeT

conp

SIJKeHHoMy

yp

aBHeHHIO

OW

0

2

1/1

ar=-aJ2'

(24)

.KOTopoe

B

){aHHOM-CJIylJae

SIBJISIeTCSI

06paTHMM

ypaBHeHHeM

TenJIO-

npOBO~HOCTH

c

rpaHHlJHMMH

yCJIOBHSIMH

1jJ

(O,f)

=

1jJ

(I, f) =

O.

(25)

Ta'KHM

06pasoM,

S){eCb MbI CHOBa CTaJIKIiBaeMCSI C

HeKoppeKTHOH

Sa){alJeH

){JISI

ypaBHeHHSI

C

lJaCTHbIMH'

npOJfSBO){HMMH.

:::.-

.

.-:

~'.;

,

C

o I<OPPEI<TliOCTH 3A,l{AlJ

HeCMOTpH

Ha

TO, lJTO

CO

BpeMeH

A,~aM;a:pa

6bIJI

rrpOBOSrJIaIlleH

npHHII.Hrr

KoppeKTHOCTH

3a){alJ

Jr)IH .

ypaBHeHHH

MaTeMaTHlJeCKOH

,qlH3HKU,

yqeHbIMnpHxo.zr;HTCSI

CT8JIKHBaTbC$I'C

HeKoppeKTHO

rrOCTaB-

JIeHHbIMH

Sa){atiaMH,

B03HHKaIOllJ:HMHBrrOJIHe

e:CTeCTBeHHbIM

06pa30M*).

qacTbIMHCTOlJHHKOM

HeKoppeKTHblx

sa){alJ

SIBJIHIOTCH

Sa){alJH

0 BOC-

CTaHOBJIeHHH TeqeHHSI

KaKOrO-JIH(5o

npou:ecca

B

rrpOIIIJIOM,

T,

e.

sa.zr;alJH 0 BOCCTaHOBJIeHHH

HCTOpHH

npou:ecca

no

pesyJIbTaTaM

Ha-

6JIIO){eHHH B HaCTOSIllJ:HH H 6Y){YllJ:He MOMeHTbI

BpeMeHH.

ECJIH

rOBO-

pHTb

He

0

BpeMeHHblx,a

0

npOCTpaHCTBeHHbIX

HCTOllHHKax;

HeKOp-

peKTHblx

3a){alJ,

TO

OHM

CBSI3aHbI C

npOOJIeMOHUOJIylJeHHSI

cy>K){eHHH

o

pacrrpe){eJIeHHH

KaKHX-JIH6oBeJIlllJHH

B O){HOM MeCTe

(HarrpHMep,

HaM

He){OcTyrrHoMrro

KaKHM-JIH60

'npHlJHHaM)

rro

H3MepeHHHMHe~

KOTOpMX

pacrrpe){eJIeHHH

B

.zr;pyrOM(.zr;ocTyrrHoM

HaM)

MeCTe.

TUrrHq-

HbIM

rrpHMepoM

TaKoH

CH'tyaU:HH

SIBJIHeTCSI

rpaBHMeTpHlJeCKaSIp(i3~

Be){Ka.

H3MepeHHe

nOTeHII.HaJIa

'Matc npOHSBO){HTCSI OKOJIO

rrollepx-

HOCTH SeMJIH

(B

){OcTyrrHoM

HaM

MeCTe),

a

S8KJIIOqeHHe

MbI ){OJI>KHbI

C){eJIaTb

0

pacrrpe){eJIeHHH

Macc

B

JlpyroM

MeCTe

(cKa1KeM,

no){

seM-

JIeH,

r){e

SaJIeraIOT

HHTepecyIOllJ:HeHaC

JKeJIeSHMe HJIH

){pyme

py){bI)

.

*)

ECTb

ueJibIA

pH,!{

paSJlHlIHblXCnoco6011

pelIIeHHH

HeKOppeKTHbiX

sa,!{atl.

O,!{HaKo

Ka>K,!{bIH

pas,

II

CJlOlla

«perneHHe

HeKoppeJ<;THoA

sa,!{allH»

IIKJla,!{b1l1alOT

paSHbIH

CMblCJI. MO>l<HO

CAeJlaTb

S3,!{ally

KoppeKTHoA,

pezyJlJlpU3Up06amb

ee

pas-

JlHlIHblMH

cnoco6aMH:

CYSHTb 06J1l1CTb

,!{onycTHMblX

HCXO,!{HblX ,!{aHHblX, BIieCTH

cneUH<pH'IeCKHe

HOpMbI,

cHeMHoro»

HSMeHHTb

ypaBHeHHll

sa,!{allH

HJlH

paCCMaTpH-

lI11eMbie

MHHHMHsHpyeMbie

<PYHKUHOHaJlbl H

T.

A.

[347, 348].

y

i

BeJlHqHHa

(pHC.

llpH

3TOM

nOJle

PHqHbIMH,

ITO

--,:

0.'>

rJlABA

W'

MATEMATMllECKOE

MO~EJIMPOBAHME

TEnJIOBOrO

npO<l>MJISIBAJIKOB

,npOKATHbIX

CTAHOB

KAK

ynPABJISIEMblX 06bEKTOB

CPACnPE~EJIEHHbIMM

nAPAMETPAMM

§

t.

nOCTaHOBKa

sa,Aal.JU

PeIlIeHHe

np06J1eMbI

'C03,1(aHHH CHCTeMbI aBTOMaTHI.JeCKOrO

ynpaB-

JleHHH TenJlOBOH npO<pHJlHpOBKOH BaJlKOB TOHKOJlHCTOBbIX

npOKaTHbIX

ci'aHoB

BbI,I(BHraeT B KaI.JeCTBe

ueHTpaJlbHOrO

BOnpOCa 3a,l(aI.Jy

no-

CTpOeHHH MaTeMaTHI.JeCKOH MO,l(eJiR

BaJlKa

KaK

06beKTa

ynpaBJleHHH

[284, 352, 353, 380].

YnpaBJleHHe

TaKoro

pO,l(a

06beKTaMH

06yCJIOBJlHBaeTcH

TeXHO-

JlOrHl.JeCKOH He06xo,l(HMOCTbIO

KOMneHcaUHH

HCKa)l{eHHH <P0PMbI

flO-

BepxHocTH

BaJlKa

BCJle,l(CTBHe

HepaBHoMepHOrO

pa30rpeBa

H

H3Hbca.

ero

npH

npoKaTKe

H

peaJlH3yeTcH

nyTeM

H3MeHeHHH TenJlOBOrO nOJlH

BaJlKa.

flOCJle,l(Hee ,l(OCTHraeTCH

nyTeM

npHJlO)l{eHHH

pacnpe,l(eJleHHbIX

ynpaSJlHIOIlI.HX

B03,1(eHCTBHH K

BaJlKaM

B BH,I(e H3MeHeHHH

pacxo,l(a

oxJIa)l{,I(aIOIU,erO

(nO,l(OrpeBaIOIU,ero)

areHTa

no

HX ,l(JlHHe.

flp06J1eMa

ynpaBJleHHH

TenJlOBOH npo<pHJlHpOBKOH BaJlKOB

npH

npoKaTKe'

KaK

3a,l(aI.Ja

OnTHMaJlbHOrO

ynpaBJleHHH

06'beKToM

C

pac-

npe,ll,eJreHHbIMH

napaMeTpaf.1H

paccMaTpHBaJlaCb

B

pa60Te

[79].

naJlb-

HeHIlIee

H3J10)l{eHHe

6y,l(eT

CJle,l(OBaTb

TepMHHaM

H 0603HaI.JeHHHM

9TOH

pa60Tbi.

B HeH no,l(XO,l( K

nocTpoeHHIO

CHCTeMbI

ynpaBJleHHH

OCHOBaH

Ha

nocTpoeHHH

,l(eTepMHHHpoBaHHoH

a,l(anTHBHoH MO,l(eJlH

pa60I.Jero

BaJlKa

H

opraHH3aUHH

ynpaBJleHHH

no

3TQHMO,l(eJlH.

110,1(

MO,l(eJlbIO

BaJlKa

nOHHMaeTCH

cOBoKynHocTb

,l(ByX nOJleH:

TeMnepa-

TypHoro

nOJlH

T(r,

Z,

e,

t),

r,l(e

(r,

z,

e)-UHJlHH,I(pill.JeCKHe

Koop,l(HHaTbI

TOItKH

BaJlKa,

t--"":BpeMH,

H

BbI3BaHHoro

HM

TepMoyn-

pyroro

nOJlH

HanpH)I{eHHH

a(r,

Z,

e,

t) H nepeMeIU,eHHH W (r,

z,

e,

t)

B

BaJlKe.

TenJlOBaH

np0<pHJlHpoBKa

BaJlKOB

paccMaTpHBaeTcH

KaK

ynpaBJlHeMblH

npouecc<popMHpoBaHHH

nOJlH TenJlOBOrO

pacIlIHpeHHH

,I(~aMeTpa

BaJlKanO

ero

,l(JlHHe

BO

BpeMeHH

no,l(

,l(eHCTBHeM

TeMne-

paTypHoro

nOJlH B

Macce

BaJlKa.

B

TepMHHax

TeopHH

TepMoynpy-

rOCTH

nOJleTenJlOBOrO

pacIlIHpeHHH

,l(HaMeTpa

HBJlHeTCH pa,l(HaJlbHOH

COCTaBJlHIOIU,eH Wr (R,

Z,

e,

t)

nOJlH

TepMonepeMeIU,eHHH

Ha

nOBepx-

HOCTR

BaJlKaW

(R. z,

e,

t),

.r,l(e

R-pa,l(Hyc

BaJlKa,

Z-KOOp,l(U-

HaTa

ero

,l(JlHHbI.

Tor,l(a

TenJlOBaH COCTaBJlHIOIU,aH

reoMeTpHH

60KO-

BOH

nOBepxHocTH

BaJlKa

B

<pHKcHpoBaHHbIH

MOMeHT

BpeMeHH

t*

onpe,l(eJlHTCH <P'yHKUHeH

pacnpe,l(eJleHHH

Wr (R,

z,

e,

t*).

3Ta

<PYHK-

UHH

OOhlI.JHO

Ha3bIBaeTCH menA08blM npo¢uAeM 8aAKa.

llJlH

oueHKH

BeJlHI.JHHbI

HepaBHoMepHocTH

TenJlOBOrO npo<pHJlH

npHHHMaeTCH

·'·',·

511

C

'.

nOCTAHOBKA

3A;D.A'·m

10.1.1)

f(t'")=

max

\Wr(R,

z,

t

e

) -

min

W(R,

z,

t*)1·

(1)

ze[-Lt.

L.l

ze[-L

•.

Ltl·

TeMnepaTypHoe

nOJle,

a CJle,l(OBaTeJlbHO, H

BbI3BaHHoe

HM

nepeMeIlI.eHHH

H

TepMOHanpH)I{eHHH

npHHHMalOTcH

ocecHMMeT-

T.

e.

He

3aBHCHIlI.HMH

OT

yrJlOBOM KOop,l(HHaTbI

e.

B

cJlyqae

cHMMeTpHH

Wr OTHOCHTeJlbHO

cepeAHHbI

AJlHHbI

BaJlKa

Bblpa)l{eHHe

npHMet

BHA

f

(t'")

= IWr (R,

Z,

t

e

)

11:=0-Wr (R,

Z,

t")

I,,==L.I·

(2)

llpH

Wr (R, 0, t

e

)

> Wr (R, L

t

,

t

e

)

BeJlHqHlta

f (t*)

S·

BbIpa)l{eHHH

(2)

B

TeopHH

npoKaTKH

06bIqHO

HOCHT

Ha3BaHHe

menAo80t1

8btnY~Aocmu

8aAKa.

TaKHM

o6pa30M,

C

TOI.JKH

3peHHH

TeopHH

ynpaBJleHHH

CHCTeMaMH

c

paCnpeAeJleHHbIMH

napaMeTpaMH

B

npouecce

ynpaBJleHHH

TenJlOBOH

r

J{f.(R,z,t)

__

-I--..,~[{t)

.

.,..-

I

__

L_

.,

-L

f

z

PHc.lO.l.l.

npO<pHJlHpOBKOH BaJlKOB B KaI.Jec'rBe

ynpaBJIHeMoH

KoopAHHaTbIBbIC'iY-

.

naeT

JlH60

HenOcpeAcTBeHHo

TenJlOBOH

npo<p

HJlb

BaJiKa

(B

HeKoTo~

pOM

<PYHKuHOHaJlbHOM

npOCTpaHCTBe),

JlHOO

onpeAeJlHeMaH

HM

<pyIlK-

U,HH

ero

HepaBHOMepHOCTH.

llOCKOJlbKy

npH

BbI60pe

ynpaBJlHIOIlI.HX

B03AeHCTBHH, onpeAeJl51IO-

IlI.HX

pe)l{HM

OXJla)l{AeHHH

(nOAOrpeBa)

BaJlKOB,

He06xo,l(HMO

COOJlIOCTH

TeXHOJlOrH'I.JeCKHe

orpaHHI.JeHHH

Ha

AonycTHMyIO

BeJlHI.JHHY

TeMoepa-

1'ypbl

nOBepXHOCTH pa60I.JeH

l.JaCTH

60I.JKH

BaJlKa

H ,l(OnycTHMyIO'

BEfJlHI.JHHY

TepMOHanpH)I{eHH

H

B

HeM,

B KaI.JeCTBe BbIXO,l(HbIX KOOPAH-

HaT

OObeKTa

ynpaBJleHHH

B <pa30BOM <PYHKu,HOHaJlbHOM

npOCTpaHCTBe

HeboxO,l(HMO

HMeTb

HenOl;peACTBeHHO

TeMnepaTypiIoe

nOJle

Ha

nOBepx-·

BOCTH

BaJlKa

T

(R,

z,

e,

t) H

nOJle

ero

TepMOHanpH)I{eHHH

a

(r,

z,

e,

t).

XapaKTepHoH

3aAaI.JeH

ynpaMeHHH,

Kor,l(a

B KaI.JeCTBe

ynpaBJlHe-

MOB

KOopAHHaTbi

BblcTynaeT

<PYHKU,HH

HepaBHOMepHocTH

TenJlOBOrO

npo<p

HJlH

BaJlKa,

HBJlSleTCH

aaAaqa

CTaOHJlH3aUH

H

TenJlOBorO

npo-

<p

BaJlKa.

llpou

ecc

ynpaBJleHHH

CTpOHTCH

TaK,

I.JTOObI

HaHJlyq-

HJlH

IlIHM

06pa30M

no

BbiopaHHoMy

KpHTepmo

ontHMaJlbHOCTH

AOOHTbC~

. I

.

"j;

.1

\

I

!

8:

~~>

1512

MO,lu!nHPOBAHHE

1'EnnOBoro

npOcJ>HJJSlBAnI<OB

[rJJ. Ill.

HaHny1J:IIiero

npH6.iIH2KeHHJt"

l(t)

K f· UPH

co6nio,lleHHH

eCTeCTseHHblX

OrpSHH1J:eHHA

iIa

ynpaBneHHe

HCOCTOHHHe

06'beKTa.

. B

CBOIO

01J:epe,llb

THriH1J:HOH

3MatJ:eH

ynpaBJleHHSI" B KOTOpO!l

B .

KS1J:eCTBe

ynpsBJlSleMoA.

KOOp,llHHaTbl .

npon;ecca

BblcTynaeT

Heno-

Cpe,llCTBeHHO TeriJlOBOA

npo<pHJlbBEVIKa,

SlBJlSleTCSI

3a,llatJ:'a

onTHM8nb-

HOH

KOMneHCan;HH

HSHOCa

BanKOB.

B 3'rOM

cJlY1J:ae

TenJlOBOA

npolPHJlb

CJle,llHT

3a

HSMeHeHHeM HaHaCa

BaJlKa,

38,llaBaeMblM

ero

MaTeM81Wlec-,

KoA

MO,lleJIblO f· (z,

'.

t),

TaK,

tJ:To6bl HCKycCTaeHHO

«saKpbITh»

era,

-,/rz,ti

..

- 1

Bnok

onlll(

arz,t}

meHhepa-

Trr,z,l,f)

x-rS;WJ,

mepHO-

f11V/lNIl2O.

~-

UI1P1/8tlX

hOAel1

PHC.

10.1.2.

a OnTHMaJlbHOCTb ynpaBJleHHSI, nOHHMaeTcSI B CMblCJle MHHHMHsan;HH

cyMMapHoro

HCKa2KeH'HH

OOKOBOH

nOBepxHoCTH

BaJiKa

OT

,lleHcTBHJI

8THX ,llBYX «npoTHBoepa3HblX»

npOn;eccoB

C ytJ:eToM orpaHHtJ:eHHA.

CTpyKTypHO

onpe,lleJleHHeonHcaHHblx

KOOpJI:HHaT

npon;ecca

SIB-

JlSleTCSI

OOIll;HM' ,llJlSl

BceXTHnOBCTaHOB;

YtJ:HTbIBaSl CYIll;eCTBYIOru;yro

B3aHMOCBSl3b

TeMnepaTypHoro

nOJiSl H nOJleH TepMOHanpSl2KeHHH

H TepMonepeMeIll;eHHR B BaJiKe (pHC.

10.1.2).

OrJlHtlHTeJlbtIOH

tJ:epToH

TaKOH

CTpyKTypbl

MO,lleJiH

,BaJiKa

.

SlBJlSleTCSI

TOT

<paKT,

tJ:TO

BoaMY-

ru;eHHe

1)

(z, t) H

ynpaBJleHHe

u (z,

t)

nepe,llalOTCH Ha BXO,ll 6J10Ka

TepMoynpyrHx

nOJleH He HenOCpe,llCTBeHHO. a tJ:epe3 HeKOTopblH

«WHJlbTP» B aH,lle

6J10KaTeMnepaTYPHoro

nOJIR T

(r,

z,

e,

t)

=

= A (1)

(Z,

t),

u (z, t»,

r,lle

A-HeKOTopblH

oneparop.

npaKTHtJ:ecKH

STOT

onepaTOp

B 06Il1;eM cJlytJ:ae npe,llcTaBJIHeT

c060itpemeHHe

CHC-

TeMbl ,llH<P<PepeHUifaJlbHblX

ypaBHeHHR

.B,

tJ:acTHbiX

npoH3BO,llH~IX,;

OnHcbIBalOIlI;Hx

npon;ecc

TenJlOnpOBO,llHOCTH H

TenJlo06MeHSB

cHcTeMe

conpHKaCHIO!4HxcSI

H asaHMonepeMeIll;alOIlI;HXCSI

reJl:

npOKaThIBaeMblH

MeTaJin-paOOtJ:HH

BaJlOK-orioPHbIH

DaJlOK (,llJlH tJ:eTblpeXDaJlKOBbIX

KJleTeA).

. BeJlH1J:HHa T (r,

z,

e,

t)HBJlSleTCH

BXO,llHOH ,llJlSl 6J10Ka TepMO-

ynpyrux

noneR,

D KOroPOM:

no,ll

JJ;eHCTBReM

onepaTopa

F

<POpMH-

pyeTcR,

BblXO,llHaSl BeJlHtJ:HHa OJlOKa B BH,lle· BeKTop-<pyHKn;HH

X-(W(r,

z,

e,

t),

a(r,.

z,

e,

t))=F(T(r,

z,

e,

t».

HSX02KJJ;eHHe

onepsTopOB

A R F MO,lleJlH

BaJlKa

JJ;JISI

onHca,HHSI

J{HHaMRKR nCKOMblX Koop,nHHaT

npon;ecca

ynpaDJleHHH

TenJlOBOA

np'o<pHJlHPOBKOH BaJiKOB

HCOCT8BlISle:r

3aJ{atJ:y HacTOHmeA rJlaBhf..

····:'

",.'"",

c

UHcJ>POBOE

MOJlEJJHPOBAHHB

-IH3

BaJiKa

JJ;eHHH

,UJl51

YCJlOBHH

B

nOJIH

§ 2.

UHlPpoBoe

MOJJ;eJlHpOBaIfHe,

TeMnepaTypHoro

nOJIH

BaJJKa

c nOMOIll;blO 9KOHoMW1lfbiX

paaHocTHblX

cxeM

OOHcaHHe

TpexMepHoro

HecTaUI:IOHapHOro

TeMnepaTypHoro

nOJin

T

(r,

Z,

e,

t),

npHBe,lleHHOe

B.

rJlaBe,

1,

OTpa}l{aeT

06ru;He

xapaKTepHble

lIepTbl

npouecca

nOB~pxHocTHoro

TenJlo06MeHa

BaJlKa,

npHcyU1,Re

BceMy

KJlac,~y

paCCM8TpHDaeMQlX OO'beKTOB

ynpaBJleHHSI:'

TenJlo06MeH

C

nOJlocoii'

MeTaJlJla.Ha

ee

mHpHHe

no

,llyre

3aXBaTa

B otJ:are Jle<popMan;HH,' C oXJla}l{.nalOIlI;eH }l{H,llKOCThlO B

ceKTope

oXJI.a}l{~

no

,llJlHHe

KOJlJleKTOpHoro

HCnOJIllRTeJlbHOrO

MexaHH3Ma,

C

OKPY~

~alOU1,HM

B03,llyXOM

Ha

OCTaJlbHOH lIaCTH,60KOBOH

nOBepxHocTH

BaJlKa;

onHcaHHH

TenJloo6MeHa

YK~3a1iHblx

BH,llOB

npHHHTbl

rpaHHlIHble

.III

po,lla;

tJ:TO

06ycJloBJlHBaeT

He06xo,llHMOCTb aa,llaHH)]

K03<P<pHUH6HTOB

TenJlo06MeHa

HT,eMnepaTypbl

oKpy}l{alOU1,eH Cpe,llbl.

3TOM

cJlYllae

Be~b

npouecc

MOJJ;eJlHpOBaHHSI

TeMnepaTypHoro

MO}l{eT 6bITb

pa36Wf

YCJlOBHO

Ha

.n.Ba

nana.

.

Ha

nepBoM

3Tane

(3Tan

onpe.neJleHHH

rpaHHlIHblX

yCJlOBHH)

onpe-

.n.eJl51eTCH

,llJlHTeJlbHOCTb BpeMeHH TenJlo06MeHa. B Ka}l{,llOH

30He,

TeM-

nepaTypa

Ha

nOBepxHocTH

nOJlOCbl

no

.n.JlHHe

Ollafa

JJ;e<popMaUHH,.

pacclIHTbl,BaeTC51 K03<p<pHUHeHT

KOHBeKTHBHoro

TenJlo06MeHa

C

oXJla}l{-

.n;alOIlI;eH

:>KH,llKOCTblO

no

KpHTepHaJIhHoMy

ypa,BHeHHIO.

I1cxo,llHblMH

.n;aHHbl.MH

,llJlSl

paClIeTa

Ha

3TOM

.3Tane

cJI.y}l{ai;: rpa<pHK

npoKaTKH

H pe}l{HM 06}1{aTHH,

K9HcTpyKTHBHble.xapaKTepH<STHKR

CTaHa H TenJlO-

<pH3HlIecKHe

napaMeTpbl

BaJlKS, nO.1l0Cbl, oXJla}l{AalOU1,eH

}l{HJJ;KOCTH.

TaKHM

o6pa30M,

nepBblH

3TSnueJiliKOM

onpe,lleJineTCH

cneuH<pHKoH

KOHKpeTHoro

CTaHa, a

cnoco6bl

onpe,lleJleHHH

YKa3aHHbix

napaMeT-

pOB

Jle}l{aT

Bc<pepe

TeXHOJlOrHlIecKH~

p,aclIeTHblx

MeTo.n.HK

Toro

HJlK

I'IHoro

CTaHa

ropHlIeH

HJlH XOJlo,llHdA

npoKaTKH.

·OcHoBHoe

B03MYIll;alOU1,ee

D03,D;eHCTBHe

Ha

06beRT

ynpaBJleHHSI'

B BH,lle

TeMnepaTypbl

nOBepxHocTH

QOi!OCbI

no

,llJlHHe otJ:ara

Jlel}lop~'

MaUHH

<popMHpyet

TenJlO, BHOCHMoe B KJleTb C nOJlOCOH, Bbl,lleJiHIO-

IlI;eecSI

H3

nbBepxHocTH

COOpHKocHoBeHHH BaJiKOB H MeTaJlJJa

npH

TpeHHH

H B

Macce

nOJlOChl B

pe3yJlhTaTe

ee

nJiaCTHlIecKoH

JJ;e¢op-

Man;HH

no,ll

,lleHCTBHeM CHJJOBMXH KHHeMaTHlIeCKHX

$aKTopoB

(YCH~

JlHH

npoKaTKH,

HaT51mellH51 nOJlOCbl, CKOpOCTH

npOKaTKH),

HrpalO1l\Hx

POJIb B03MYIll;alOIlI;Hx B03,lleHC-TBHH

aa

06beKT

yilpaBJleHHH.

CJle,llOBa·' .

TeJlbHO, Bce B03MYIlI;eHJI;?,

npouecca

(co

CTOpOHbl KOMnOHeHTbl TenJlO'-

nO.llBo.n.a) «HHTerpHpyIOTCH»

B,Ollare

.n.e<popMauHH H

n~pe,llaIOTCH

BaJlIvy

3a

Bp'eMH

conpH~'OCHOBeIfHH

C

npOKaTblBaeMOH'

nOJlo.coft.

no-;

~TOMY

paclIeT

TeMnepaTypbI

nOBepxHoCTH

nOJloCbl,KOJlHlIeCTBf

TenJla;

OT,ll8BaeMOro HaJiKY oT,:,nOJlOCbl,'

,TeMnepaTypbl

nOJlOCbI Ha', Bblxo.n.e

H3 KJleTH,

OT

KOTOpOft:

3aBHcHt

TenJlOnepe.n.alla Me}l{,llY KJleTHMfI:

CTaHa,

51BJl51eTCH

ueHTpaJlbHbJMBOnpOCOM

3Toro

:nana

pemeHHH;

3a,llalla.

,,:

. ,

"',',

..

npH6J1H}I{eHHbIH

paclleT

OD,HC8.HHbIX

BeJlHlIHHMO}l{eT6blTb

:npo-;

H3aeJ{eif

6e3

yqeTa

pacn~e,lleJlemHI:.

TeMnepaTypbl

no

TOJlIll;HHe no.nQCbI:

······

MO~EJIHPOBAHHE

TEnJIO:Oro

n£~JI5I

BAJII(OB

514

(,I\JIH TOHKOJIHCTOBOM

npOKaTKH

,l\Hana30H

T0:1III.J.HH

0,5-.,-4

MAt)

C

no-

'

MOlII.bID

BeJIHqHHblCpe,l\HeMaCCOBOM

TeMnepaTypbl

Bbl,l\eJIeHHOrO

3JIe-

MeHTapHo.rO

06'beMa

nOJIOCbl B

Oqare

,l\el!>0pMaI.I.HH B

yCJIOBHHX

a,l\Ha6aTHqeCKOrO

HarpeBa.

TIPH

3TOM

npeHe6peraeM

np0,l\OJIbHOH

TenJIOnpOBO,l\HOCTbIO nOJIOCbI H

npflHHMaeM

KyCOqHO-JIHHeHHbIH

3aKOH

H3MeHeHHSI

TeMnepaTypbI

nOJIOCbI

no

,l\yre

3aXBaTa.

TIPH

,l\BH>KeHHH

3JIeMeHTapHoro.

06'beMa

no

,l\JIHHe

oqara

,l\el!>opMaI.I.HH

ero

Cpe,l\He-

MaccoByIO

TeMnepaTypy

lia

Ka>K,I\OM ,l\aHHOM

mare

KoppeKTHpyeM

Ha

BeJIHqHHY,

COOTBeTcTByIOII.J.yIO KOJIHqeCTBY

TenJIa,

nepe,l\aHHOro

BaJIKy

Ha

3TOM

mare,

H

npHHHMaeM

31'0

CKoppeKTHpOBaHHoe

3HaqeHHe

B

KaqeCTBe

HaqaJIbHOH

TeMnepaTypbI

Ha

nOCJIe,l\YIOII.J.eM

mare.

TeMne-

paTypa

3JIeMeHTapHOrO

06'beMa

Ha

BbIXO,l\e

HS

oqara

,l\el!>opMaI.I.HH

npHHHMaeTCSI

sa

TeMnepaTypy

nOJIOCbI

Ha

BbIXO,l\e

H3

KJIetH.

TIPH

Heo6XO,l\HMOCTH.60JIee

TOqHbIH

yqeT

Te"mepaTypbl

nOBepx-

HOCTH nO.JIOCbI MO>KeT

6bITb

npOBe,l\eH

peI1IeHHeM

CHCTeMbI ,l\Hl!>l!>epeH-

UHaJIbHbIX

ypaBHeHHM

TenJIOnpOBO,l\HOCTH ,l\JISI

BaJIKa

H nOJIOCbI

npH

3a,l\aHHOM

HaqaJIbHOM

pacnpe,l\eJIeHHH

H C

rpaHHqHblMH

YCJIOBHSIMH

III H

IV

P0,l\OB

Ha

nOBepxHocTH

conpHKocHoBeHHSI

C riOMOII.J.bIO

Me-

TO,l\a

KOHeqHbIX

pa3HocTeH.

30Ha

TenJIo06MeHa

C oXJIa>K,I\aIOII.J.eH >KH,I\KOCTblO - 31'0

ynpaB-

JISIIOII.J.aSI

30Ha,

nOCKOJIbKY

paCXO,l\ >KH,I\KOCTH

no

,l\J1I'lHe

BaJIKa

(06bIq-

HO

B

3-5

OT,I\eJIbHbIX

ceKUHSIx)

SIBJISIeTCSI

ynpaBJISIIOII.J.HM

B03,1\eii·

CTBHeM,

a K03l!>l!>HI.I.HeHT

TenJIo06MeHa

H Cpe,l\HSISI

TeMnepaTypa

}J{H,I\KOCTH 3aBHCSIT .01'

3Toro

pacxo.z(a

. .l1JISI

KOHKpeTHblx

yCJIOBHM

ox-

JIa>K,I\eHHSI

Ha

CTaHe

06bIqHO

3KcnepHMeHTaJIbHO

onpe,l\eJISIIOT

KpHTe-

pUaJIbHOe

ypaBHeHHe

KOHBeKTHBHoro

TenJIo06MeHa

B BH,I\eCX

3

= c

y!u;

r,l\e

CX

3

-K03<pl!>HI.I.HeHT

TenJIo06MeHa,

u-pacxo,l\

>KH,I\KOCTH B

ceK-

UHH

KOJIJIeKTOpa

(ynpaBJIeHHe),

C H

m-HeKoTopble

nOCTOSIHHble,

qHCJI.eHHble

3HaqeHHSI

KOTOpbIX

onpe,l\eJISIIOTCSI

reoMeTpHeH

BaJIKa,

KOHcTpyKUHeH

6pbI3raJIbI:iOrO

KOJIJIeKTOpa,

TenJIo<pH3HqeCKHMH

HKH~

HeMaTHqeCKHMHnapaMeTpaMH

OXJIa,l\HTeJISI [353].

TIOCKOJIbKY

npH

onpe,l\eJIeHHH

napaMeTpoB

rpaHHqHblX

yCJIOBHM

PSI,I\

qHCJIOBblXKOHCTaHT

H3BeCTeH

JIHmb

opHeHTHpOBOqHO,

T.

e.

cy-

II.J.eCTB'yeT

anpHopHaSI

HeOnpe,l\eJIeHHOCTb

HX

BeJIHqHH,

B03HHKaeT

He06xo,l\HMOCTb

peaJIH3aI.I.HH MO,l\eJIH B KJIaCCe

a,l\anTHBHbIX

MO,l\eJIeH

(HJIH

ynpaBJISIeMblX

MO,l\eJIeH C

caMoHacTpoHKOH).

A,I\anT?UHH

MO,l\eJIH

n03BOJIHT

npoBecTH

m~pBHqHYIO

HacTpoiiKy

H

caMoHacTpoHKy

ee

B

npoI.I.ecce

ynpaBJIeHHSI

no

peaJIH3aI.I.HSIM

npo-

uecca,

nOJIyqeHHblM

B

pe3yJI.bTaTe

KOHTPOJIbHblX

H3MepeHHH,

C

no-

MOII.J.bIO

npOue,l\ypbl

caMoHacTpoHKH

no

3a,l\aHHoMY

KpHTepHIO

Ka-

tiecTBa

MO,l\eJIH.

BBH,I\y

HeKOHTpOJIHpyeMocTH

BeJIHqHHbI

TenJIOBOrO

npol!>I1JISI B

KaqeCTBe

peaJIH3aUHH

npoI.I.ecca

ueJIeC006pa3Ho

npHHSITb

BeJIHqHHY

TeMnepaTypHoro

nOJISI

Ha

nOBepxHocTH

BaJIKa

T

(R,

Z, t) H

TeMnepaTypy

nOJIOCbl

Ha

BblXO,l\e

H3

KJIeTR,

a B

KaqeCTBe

napaMeT-

poB

caMoHacTpoMKH

(ynpaBJISIeMbIX

napaMeTpoB

MO,l\eJIH)

npHHSITb,

. •

.'

I

c·

'.'-//

515

UH~POBOE

MO,l\EJIHPOBAHHE

f

2]

HimpHMep,

BeJIHqHHbI

HeKoHTPOJlHpyeMbIxB03MYII.J.eHHH

B BH,I\e K034>-

lPfiUHeHTOB

TenJIo06MeHa

C rtOJlOCOHCX" oI<py>KaIOII.J.HM B03,1\YXOM

~,.

,--:"QapaMeTpbI

c H

m,

yCJIOBlIi;>IH

npe,l\eJI

teKyqeCTH

nOJIOCbl O'n'l'

.'

A,I\anTHBHaSI

MO,l\eJIb

n03BOJI51eTn}hiHSITb

,l\OnYII.J.eHHe

0.

B03MO)l(~

> HOCTH

npaBHJIbHOrO

OT06pa>KeHH5I,I\HHaMHKH

TeMnepaTypHoro

nOJl51

1',

pa60qero

BaJIKa

(cKa>KeM,

Ha

OTpesKe

BpeMeHH

npoKaTKH

O,l\HOH

,~nOJIOcbI)

6e3

yqeTa

tenJIOOTBO,l\a

K OITOpH'OMy

BaJIKy,

a

npH

He06"

r.XOJJ;HMOCTH

yqeT

3TOM COCTaBJISIIOn(eH

'tenJIOBOrO

6aJIaHCa

KJIeTH

npO-

BO,l\HTb

HeKoTOpblM

YBeJIHqeHHeM

,l\pymx

KOMnOHeHT

TenJIOOTBo.n.:a

.'(HanpHMep,

napaMeTpa

c).

OCHOBal!HeM

,l\Ji51

TaKoro

nO,l\XO,l\a MO)l(eT

CJIy>KHTb Me,l\JleHHblH ,l\peHl!>

TeMnepaTyphl

nOBepxHbcTH

onopHoro

BaJIKa,

a

TaK>KeerO

60JIbma51

TenJlOBasi

HHepUH5I.

3aMeTHM,

qTO

piaJlbHblH

BaJIOK

HMeeT l!>OpMy

CTyneHqaTOrO

UH-

JlHH,I\pa,

COCTaBJleHHOrO

H3

pa60qeM

qaCTH-60qKH-BaJIKa

Hera

IlleeK.·

YtleT

TenJlOOTBO,l\a B

llieMI<H

MO>KeT

6bITb

Ba>KeH C

TOtlKH

spe-

HH5I ,l\OCTOBepHOCTH

MO,l\eJIHpOBaHH51TeMnepaTYPHbIX

pacnpe,l\eJIeHHH

.

Ha

Topue

60tlKH,

l!>OpMHpyIOII.J.HX

ero

TepMOpaCI1IlIpeHHe,

01' KOTOPO"

1'0 npOHCXO,l\HT OTCqeT BeJI'HqHHbI

HepaBHOM~pHOCTH

TenJIOBOrO

npo-

$HJIH

BaJIKa

KaK

ynpaBJl5leMOM

Kbbp,l\HHatbI

npouecca.

CYII.J.eCTBeHHoe npeHMYII.J.eCTBO

MeTO,l\aKOHetlHbIX

pa3HOCTeH;

npH.

H5ITOrO ,l\JI5I

pemeHlI5I

Sa,l\aqH,

COCTOHT B

OTCytCTBHH

OrpaHHtleHHri

Ha

reOMeTpHIO

MO,l\eJIHpyeMoH

06JIaCTH.

BY,I\eM

npHHHMaTb,

cJle,l\y51 [353],

Ha

nOBepxHocTH

meHKH

no

,l\lIHHeCOnpHKocHoBeHH5I

C

nO,l\uiHnHHKoM

rpaHHqHble

YCJlOBH5I III

pO,l\a,

B

KOTOpbIX

B

KaqeCTBe

3HaqeHHH

TeMnepaTypbI

cpe,l\bI

npH-

MeM

TeMnepaTypy

CMa3KH,

a

Ha

OCTaJlbHOH

ee

qaCTH-yCJlOBH5I

TenJIOH30JISII.I.HH.

B OTJIHqHe 01'

nepBoro

9Tana

BTOpOH

3Tan

MO,l\eJIHpOBaHHSI'

SIBJI5IeTCH .06II.J.HM

,l\JISIBCerO

KJIaCCa

06'beKTOB

ynpaBJIeHHSI

BCHJIy

06II.J.HOCTH

TenJIol!>H3HqeCKHX

3aKoHoMepHocTeM

npou.

ecca

TenJIOnpo-

BO,II.HOCTH

,l\JI5I TeJI UHJIHH,I\pHqeCKOM epOpMbI H COCTO"T B

Henocpek

cTBeHHoMpemeHHH

ypaBHeHIfSI

TenJIOnpoBO,l\HOCTH

npH

3a,l\aHHOM

HaqaJIbHOM

pacnpe,l\eJIeHHli

H

npoCTpaH<;TBeHHo~BpeMeHHblx

rpaHHq-

HblX

YCJIOBH51X III

'p0,l\a

Ita

nOBepxHocTH

BaJIKa,

cl!>opMHpoBaHHblx

aa

nepBoM

3Tane.

PaccMOTpelUie

aJIrOpHTMa

pemeHH5I:noH

3a.llaql'l

npoBe,l\eHO

Bblme.

3aMeTHM,

qTO

onHcaHHbIM

xapaKTep

rpaHHqHblX

YCJI01:mM

HarneR

sa,l\aqH

H PSI,I\ HCCJIe,l\OBaHHM n03BOJISIIOT

anpHopH

COCTaBI-ITb

'HeKO-

Topoe

npe,l\CTaBJIeHHe

0

KaqeCTBeliHoM

cTpyKType

HCKOMoro

TeMne-

paTypHoro

nOJISI T

(r,

i,e,

t)(CM. [323, 284, 237]).

OTJIHqHTeJIbHOH

oc06eHHocTbiO

3tOM

CTpyKt'ypbI

SIBJI5IeTC5I TO,

qTO B HeM MO>KeT 6b1Tb lJ:eTKO Bbl,l\eJteHa

HeocecHMMeTpHqHa

SI

BbIeo-

KOqaCTOTHa51

COCTaBJI5IIOII.J.a51

9Toro

nOJI5I B TOHKOM

noaepXHoc'tHbM

CJIQe

BaJIKa{nopSI,I\Ka

1

~2

% '01'

ero

paAl-lyca),

KOTOphl

H

I-laSOBeM

«b6oJIOqKOH»,

Ii

HH3KoqaCToTHi'i51 cb¢'taBJI5IIOII.J.i'i51

31'oro

nOJI5I B OCTa.nb-

'DB

Macce.

BaJIKa,

KOTOpylO.

H8.S0BeM C5I,I\p0M»

II

B KOTOpOH

TIOJle

,

~,

...

~

c

','

-

c.:'

'.

:

,~

, "

.:

UH~POBOE

MO,lJ;EJlHPOBAHHE

..

TEI;IJlOBor,O

np'o~~Jl5!

BA.nKOB

517,

.516

HMeeT

SlpK:O 8b1pa:H<eHHylO OCeCltMMeTpHqHOCTb.

nOCJleAHee

OOCTOSI;:

TeJlbCTBO H nOSaOJlSleT

BA~JlbHefnIleMnPOBOAHTb

MOAeJJHpOBaHHe

TepMoynpyrlix

nOJleR

C

HcnOJlbSOBaH;HeMMaTeMaTHqeCKOrO

annapaT&

peIlleHHSloceCHMMeTpHqHOR

S'3.Aaqn

TepMoynpyrocTH

;AJlSI

TeJI

UHJIHH-

APHqeCKOR

lj>OPMbI. ,

.

nOCKOJIbKY

HHSKOqaCTOTHaSl.

'oceCHMMeTpHqHaH

COCTaBJISlIOlII,a~

nOJIH

Ha.nOBepXHOCTH

«HApa»

lj>i?pMHpyeTcH

B

pesyJIbTaTe

pacnpQ;

CTpaI!eHH5I

C nOBepXHOCTH B

rJIyob

BaJIKa

BbICOKOqaCTOTHOR

COCTaB,:,

JI5IIOiu,eR

3Toro

nOJIH,

TO

eCTb OCHOB.aHHe

CqHTaTb

«OOOJIOqKy»

CBoero

pOAa

«<l>HJIbTpOM» , BblAeJISlIOlII,H'M

oceCHMMeTpHqHylO

COCTaBJI5IIOIIJ,YIO

TeMnepaTypHoro

nOJIH.

(naHHKHTeMnepaTypbI

nOBepXHOCTH

BaJIKa,

YCTaHOBJIeHHble

Ha

era

HOBepXHOCTH,

Ha

npaKTHKe

HSMepHIOT HMeHHO

3TY

lj>yHKUHIO.)

,

'.

.

OOJIaCTb

npHMeHHMOCTH

3Tora'

cnoc06a

OrpaHHqeHa,

OAHaKO,

06'beMoM

BblqHCJIHTeJIbHOR

paooTbrnpH

He06xoAHMOCTH

MOAeJIHpOBaHH5I

9Toro

nOJI5I

Ha

OTpeSKax

BpeMeQH,

COHSMepHMbIX C

peaJIbHblM

Bpe~

MeHeM

npOKaTKH

(nQpHAKa

5-8

MUH

Ha

npoKaTKy

pyJIOHa

npH

XOJIOAHOR

npoKaTKe),

HeCMOTp}I

Ha

,nOCTOHHCTBO

3Toro

MeTOAa

B CMbICJIe

YMeHbIlleHH5I

qHCJIa apHlj>MeTHqeCKHX

onepaUHR

0

(1/h~)

npH

nepexoAe

OT

CJI051

K

CJIOIOC

IlIarOM

h

Ha

KBaApaTHoH

ceTKe

a

npocTpaHcTBe

H C

IlIarOM

no

BpeMeHH

'to

06yCJIoBJIeHo

3TO CBH-

saHHOCTblO

IlIarOB

h H

't

COOTHOIlleHHeM YCTOHqHBOCTH

(AJI5I

HepaB-

.

1

HOMepHoR

ceTKH

AOJI:H<HO

BbIllOJIH5ITbCH yCJIOBHe

't

<2

min

hih

i

+

1

\

. a i )

KOTopoe

TpeoyeT

(.u.JIH

OTOOpa:H<eHH5I BblCOKHX

rpaAHeHToB

TeMnepa-

TypbI

Ha

nOBepxHocTH)

HSMeJJbtJeHH5I

IlIara

ceTKH

B OOOJIOqKe

H,

HCKJIlOqHTeJIbHO .MeJIKOrO

IlIara

't.

(nopHAK:a

0,1 +0,01

MeeK).

npOCTOTa

me

BbltJHCJIHTeJIbHOR

npoueAypbI

3THX

cxeM.

H

0003pH-

MOCTb

pesyJIbTaTOB

AeJIaJOT

HX

MeT9AHqeCKH

Ba:H<HbIMH

npH

paCqeTaX

HeCKOJIbKHX

nepBbIX

OOOpOTOB

Ba.IIKa,

Ha

KOTOpbIX BOSHHKalOT

MaKCH-

MaJIbHble

rpaAHeHTbI

TeMnepaTypbI

no

paAHYCY

H B OKpy:H<HOM

HanpaB~

,/IeHHH

Ha

noBepXHOCTH

Ba:7lKa.

,MaKcHMaJIbHble

aOCOJIIOTHble

TeMrre-

paTypbI

era

H,

CJIe.u.OBaTeJIbHO,.

MaKCHMaJIbHble

BeJIHqHHbI

TepMO-

Hanp5l:H<eHHii B OOOJIOqKe BaJIKCj., Ba:H<HbI, C TOtJKH SpeHH5I

yqeTa

OrpaHHqeHPR

Ha

lj>aSOBble

KoopAHHaTbI

npll

BbIoope

ynpaBJIeHHH,

MeTOA#qeCKHnpHMeHeHHe

3THX

~xeM

MO'H<eT

BXO)l.HTb B

Ty

qaCTb

aJIrOpHTMA

Mo.u.eJIHpOBaHH5I, KOTOpa51 lj>OpM::HpyeT

MHO:H<eCTBO

ySJIOB

.u.Jl51

He.paBHOMepHoR

ceTKH

B

npocrpaH~TBe.

3.u.ec,:>

~OJIyqeHHO~

C .

nOMO~blO

3THX

CX~M

peIlleHHe'sa.u.aqH

C MHHHMaJIbHO BOSMO}KHbIM

(c

TOqI<'li

9peHH5I

npHMeH5IeMbIX

BblqHCJIHTeJi'bHbIX

cpeACTB)

IlIarOM

h

paBHOM;epHOH .CeTKH

(nor

H z)

npHHHMaeTCH

sa

«3TaJIOHHoe»

pe-

IlIeHlie.·

.' ,

)1.aJIee,

ynpaBJI5I5IpaCnOJIO:H<eH'HeM

YS.lIOB

HepaBHOMepHOH

CeTKH,

Opr.BHH3yeTcH

npoue,nypa

annpoKCHMaUHH

MHO:H<eCTBa

ySJIOBpaBHO"

M~HO.H

ceTKH

TaKijM,MHO:H<~qBOM

Y3JI.0~

HepaBI!OM~pHOR

CerKH,

~q1opoe

,oOecneqHBaeT

MHHHMaJIbHOe OTKJIOHeHHe

nOJIyqaeMOrO

npH

3.TOM

peIlleHH51 01'

ero

3TaJIOHHOrO

SHaqeHH5InpH

sa.u.aHHOM

KpHTepfIH

.

KaqeCTBa

OTKJIOHeHH5I.

"PaceMOTpHM'

npHMeHeHHe

K

pellieHHIO

HailieR

sa.u.aqH

COBpeMeH-

HblX

MeTo.u.OB

KOHCTpyHpoBaHH5I

3KOHOMHqHbIX

paSHOCTHbIX

cxeM

.u.JI51

pelleHHH

MHorOMepHblX

sa.u.aq

MaTeMaTHqeCKOH

lj>HSHKH, OTJIHqHTeJIb-·

.HOR

qeptoR

KOTopbDe

·51B.1I5IeTC51

COqeT'aHHe B

ceoe

JIytJllHX

KaqeCTB'

~~HbIX

(OO'beM BblqHCJIHTeJIbHOR

paOOTbI

Q(1/h

2

)

H He51BHbIX

(oesyc-,.

,'~OBHa51

YCTORqmioCTb)

deeM.

""

.,'

,.

OCHOlmaH

H.u.e51

3KOHOtdHqHbIX'

cxeM

COCtOHT B TOM, tJTOObI,

npe.u.~:

. CTaBJIHH

ceTKy

KaK

COBo.KynHocTb

ySJIOB;

pacnOJIO}KeiIHbIX

Ha

«CTPO-'

Kax»

H «CTOJIouax>)» (.u.Jisi

.u.BYMepHoH'3a)(aqH);

CBeCTH

pelleHHe

saAaqii

K.

nOCJIe.u.OBaTeJIbHOMY

pelleHHIO

O.u.HOMepHbIX 'sa.u.atJ

B.u.O

JIb

CTpOKH'

B)J.OJIb

CTOJIOUOB. •

npHMeHHM

K

pelleHHIO

HalleR

sa,natJH pasHoBH.u.HOCTb 3KOHOMHti-'

HbIX

cxeM-JIOKaJIbHO-OAHoMepiIylO

'ci'eMy,

KOTOpa51 OTHOCHTC5I

K

"KJIaccy'

aMHTHBHbIX

3KOHOMHtJHbIX paSHOCTHbIX

cxeM.

nJI51 TaKJ4K

CXeM

norpeillHocTb

annpOKCIiMaUHH

'IjJ

perneHH51;HanpHMep,

.u.ByMep-

IiOR

sa.u.aqH

nOfIHMaeTc5I

KaK

CyMMa'IjJl

+'IjJ'2

norpeillHOCTeR

annpoK-

CHMaUHH

.u.JI51

npOMe:H<yTOqHblX

o.u.HoMepHblx

cxeM;

flpe}K)l.e

Bcero,

saMeTHM', tJTb 3TOT

npHHUlin

MO.u.e.lIHpOBaHfHI

CBO.u.

HT

pelleHHe

TpeXMepHoR

sa.u.aqH onpeAeJIeHH51 T

(r,

z,

e,

t)

K'

pelleHHIO

.u.ByMepHoR

sa.u.aqIi·

T

(r,

z,

t)

.u.

JIH

KJIHHOOOpa3HOR

nJIaCTHHbI C

a.u.HaoaTHqeCKHMH

.n;oBepXlJOCT51MH B

oceBoM

CeqeHHH

UHJIHHApa,

npH

sa.u.aHHoM

HaqaJIbHOM

pacnpe.u.eJIeHHH

To

=

const

a

H HSMeH51IOlII,HXC5I

BO

BpeMeHH

rpaHHqHbIX

YCJIOBH51X

II

I po.u.

Ha

rpaHH

r =

8,

OTpa:H<alOlII,HX

cMeRy

SOH

.TeITJIOOOMeHa

npHBpalII,eHHH

BaJIKa.

B 3TOM

CJIyqae

HCKOMoe'

nOJIe T

(r,

t,

e,

·t)

annpOKCHMH-

pyeTC51

nOJIyqeHHblMH

B

.u.HCKpeTHbI~

MOMeHTbI

BpeMeHH

.u.ByMepHbIMH

nOJI51MH

T (r,

z,

t), 'C.u.BHHYTbIMH

O.u.HO

OTfiOCHTeJIbHO

.u.pyroro

Ha

BpeMH

sana3.u.bIBaHH51, onpe.u.eJIHeMOe paCCT051HHeM

Bw6paHHoR

lj>HK-

cHpoBaHHoH

TOqKH

BaJIKa

e

==const,

.u.JI51

KOTOpOR

nOJIyqeHbI

<pyHKUItH

T

(r~

z, t),

.u.o

TOqeK

BblopaHHoro

.u.HcKpeTHoro

MHO}KeCTBa

no

OKPY}K-

HOCTH

BaJIKa.

.

fIpHHUHn

:H<e

rrOJIyqeHHH

pelle~H51

T(r,

z,

t)

COCTOHT

B nOCJIe-

.u.oBll.TeJIbHOM'

pelleHli

H

3a.u.atJB'

·m~R.'

sOHeTenJIOOOMeHa

B'

coOTBeT-

CTBHH C sa.u.aHHbIM nOp5l.u.KoM

npoxo~.u.eHH51S0H,

«CKJIeeHHbIX»

Me:H<.u.y

C066R

no

npaBHJIy:

KOHeqHOe

pacnpe.u.eneHHe

B

m-R

SOHe HBJIHeTC5I

Ha'lIaJIbHbIM

pacnpe.u.eJIeHHeM

(m+

1

)-R

sa.u.aqH. .

PacCMQTpHM

aJIrOpHTM

MO.u.eJIHpOBaHHSl'

HCKOMoro

TeMnepaTYi)Horo

nOJI5I.

eT

PelleHHe

onHcaHHoR

sa.u.aqH

.u.JIH

O.u.Horo

ooopoTa

BaJIKa

oy.u.

Coc1'051Tb

npH

3TOM B

pelleHHH

m sa.u.atJ. B oOlII,eM

cJIyqa~

m=

1,

2, 3, 4,

CqHTa5I,'

qTO

nocJie.u.oBaTeJIbHbIMHsoHaMH

TenJIOOOMeHa

SlBJI,SJIOTCH

MeTaJIJI, oKpy:H<alOlII,HR B03:n;yX, OXJIa}J{AalOlII,a51 }KH.u.KOCTb.

nOCJie,aylOllI,He

,OOOPOTbI

'IJpH

9TOM BbIpa351TC51

~

U.HKJIHtJ~CKOM

J

Бутковский А.Г. Методы управления системами с распределенными параметрами

Подождите немного. Документ загружается.