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Теплопроводность через сферическую оболочку

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Ieienoa?noai iauaai e i?ioanneiiaeueiiai ia?aciaaiey

?inneeneie oaaea?aoeee

OIINEAss AINOAeA?NOAAIIUE OIEAA?NEOAO NENOAI OI?AAEAIEss

E ?AAeEIYEAEO?IIEEE (OONO?)

Eaoaae?a i?iiuoeaiiie yeaeo?iieee (I?Y)

Oaieii?iaiaeiinoue /a?ac noa?e/aneoth iaiei/eo

Iiynieoaeueiay caienea e eo?niaiio i?iaeoo

ii aeenoeeieeia “Oeceea”

Nooaeaio a?.366-4

______Aieiaeeei A.

15.12.1997a.

?oeiaiaeeoaeue

Aeioeaio eaoaae?u oeceee

________I?eianeay E.A.

15.12.1997a.

OIINE – 1997

?aoa?ao

Iauaeoii enneaaeiaaiey yaeyaony noa?e/aneay iaiei/ea caaeaiiie oieueiu n
ia?aiaiiui eiyooeoeeaioii oaieii?iaiaeiinoe e n caaeaiiuie cia/aieyie
oaiia?aoo?u ia aioo?aiiae e aiaoiae iiaa?oiinoyo iaiei/ee.

Oeaeue i?iaeoa — ii?aaeaeeoue ?ani?aaeaeaiea oaiia?aoo?u aioo?e
iaiei/ee.

A i?ioeanna ?aaiou auaaaeaii aeeooa?aioeeaeueiia o?aaiaiea
oaieii?iaiaeiinoe i?eiaieoaeueii e aeaiiui eiie?aoiui oneiaeyi caaea/e e
iieo/aii ?aoaiea yoiai o?aaiaiey a aeaea ooieoeee T(r), aaea T –
oaiia?aoo?a a i?iecaieueiie oi/ea iaiei/ee a r – ?annoiyiea iaaeaeo yoie
oi/eie e aaiiao?e/aneei oeaio?ii iaiei/ee. ?ac?aaioaia i?ia?aiia TSO,
?ann/eouaathuay ooieoeeth T(r) e no?iyuay a? a?aoee aeey ?acee/iuo
caaeaaaaiuo iieueciaaoaeai ia?aiao?ia caaea/e .

?acoeueoaoii enneaaeiaaiey yaeyaony aiaeeoe/aneia ?aoaiea o?aaiaiey
oaieii?iaiaeiinoe T(r) e a?aoe/aneay eeethno?aoeey yoiai ?aoaiey,
ecia?aaeaaiay ia ye?aia eiiiuethoa?a i?ia?aiiie TSO.

Iieo/aiiay a i?iaeoa ooieoeey T(r) e ?ac?aaioaiiay i?ia?aiia TSO iiaoo
auoue iieaciuie aeey ?ac?aaio/eeia oeie/aneeo e yaea?iuo ?aaeoi?ia,
eioeia oaieiauo noaioeee e ?acee/iuo ninoaeia a iaeanoe i?iiuoeaiiie e
auoiaie oaoieee.

Eo?niaie i?iaeo auiieiai a oaenoiaii ?aaeaeoi?a Microsoft WORD 7.0.

Abstract

Object of study is a spherical shell of given thickness with floating
factors heatconduct and with given values of temperature on internal and
external surfaces of shell.

Purpose of project — define a sharing a temperature of inwardly shell.

In the process of work is remove differential equation heatconduct is
aplicable to given concrete conditions of problem and is received
decision of this equation in the manner of functions T(r), where T – a
temperature in the free spot of shell, but r – a distance between this
spot and geometric shell centre. Designed program TSO, calculate
function T(r) and build its graph for different assign by the user of
parameters of task.

Result of studies is an analytical decision of equation heatconduct T(r)
and graphic illustration of this deciding, express on the computer
screen by the program TSO.

Received in the project a function T(r) and developping program TSO are
to be useful for developers of chemical and nucleus reactors, caldrons
of heat stations and different containers in the field of industrial and
home appliances.

Course project is executed in the textual editor Microsoft WORD 7.0.

Caaeaiea

(b=const), aaea r – ?aaeeon io oeaio?a noa?.

Iaeoe caeii ?ani?aaeaeaiey oaiia?aoo?u a yoii aauanoaa O = O(r).

Niaea?aeaiea

1
Aaaaeaiea………………………………………………………
…………………………………. 6 TOC \o “1-3”

2 Iniiaiua iieiaeaiey
oaieii?iaiaeiinoe…………………………………………… 8

2.1 Oaiia?aoo?iia
iiea…………………………………………………………..
………….. 8

2.2 A?aaeeaio
oaiia?aoo?u…………………………………………………….
……………… 10

2.3 Iniiaiie caeii
oaieii?iaiaeiinoe……………………………………………….
… 11

2.4 Aeeooa?aioeeaeueiia o?aaiaiea
oaieii?iaiaeiinoe…………………………….. 13

2.5 E?aaaua
oneiaey………………………………………………………..
………………….. 17

2.6 Oaieii?iaiaeiinoue /a?ac oa?iaoth
noaieo……………………………………….. 18

3
Caeeth/aiea…………………………………………………….
……………………………….. 22

Nienie eniieuecoaiuo
enoi/ieeia……………………………………………………..
… 23

I?eeiaeaiea A I?ia?aiia TSO, ?ann/eouaathuay ooieoeeth
T(r)………….. 24

1 Aaaaeaiea

A o/aiee i oaieiiaiaia ?anniao?eaathony i?ioeannu ?ani?ino?aiaiey
oaieiou a oaa?aeuo, aeeaeeeo e aaciia?aciuo oaeao. Yoe i?ioeannu ii
naiae oeceei-iaoaie/aneie i?e?iaea aanueia iiiaiia?aciu, ioee/athony
aieueoie neiaeiinoueth e iau/ii ?acaeaathony a aeaea oeaeiai eiiieaena
?acii?iaeiuo yaeaiee.

Ia?aiin oaieiou iiaeao inouanoaeyoueny o?aiy niiniaaie:
oaieii?iaiaeiinoueth, eiiaaeoeeae e eceo/aieai, eee ?aaeeaoeeae. Yoe
oi?iu aeoaiei ?acee/iu ii naiae i?e?iaea e oa?aeoa?ecothony ?acee/iuie
caeiiaie.

I?ioeann ia?aiina oaieiou oaieii?iaiaeiinoueth i?ienoiaeeo iaaeaeo
iaiin?aaenoaaiii nii?eeanathueieny oaeaie eee /anoeoeaie oae n ?acee/iie
oaiia?aoo?ie. O/aiea i oaieii?iaiaeiinoe iaeii?iaeiuo e ecio?iiiuo oae
iie?aaony ia aanueia i?i/iue oai?aoe/aneee ooiaeaiaio. Iii iniiaaii ia
i?inouo eiee/anoaaiiuo caeiiao e ?aniieaaaao oi?ioi ?ac?aaioaiiui
iaoaiaoe/aneei aiia?aoii. Oaieii?iaiaeiinoue i?aaenoaaeyao niaie,
niaeanii acaeyaeai nia?aiaiiie oeceee, iieaeoey?iue i?ioeann ia?aaea/e
oaieiou.

Ecaanoii, /oi i?e iaa?aaaiee oaea eeiaoe/aneay yia?aey aai iieaeoe
aic?anoaao. *anoeoeu aieaa iaa?aoie /anoe oaea, noaeeeaaynue i?e naiai
aanii?yaei/iii aeaeaeaiee n ninaaeieie /anoeoeaie, niiauatho ei /anoue
naiae eeiaoe/aneie yia?aee. Yoio i?ioeann iinoaiaiii ?ani?ino?aiyaony ii
anaio oaeo. Ia?aiin oaieiou oaieii?iaiaeiinoueth caaeneo io oece/aneeo
naienoa oaea, io aai aaiiao?e/aneeo ?acia?ao, a oaeaea io ?aciinoe
oaiia?aoo? iaaeaeo ?acee/iuie /anoyie oaea. I?e ii?aaeaeaiee ia?aiina
oaieiou oaieii?iaiaeiinoueth a ?aaeueiuo oaeao ano?a/athony ecaanoiua
o?oaeiinoe, eioi?ua ia i?aeoeea aei neo ii? oaeiaeaoai?eoaeueii ia
?aoaiu. Yoe o?oaeiinoe ninoiyo a oii, /oi oaieiaua i?ioeannu
?acaeaathony a iaiaeii?iaeiie n?aaea, naienoaa eioi?ie caaenyo io
oaiia?aoo?u e eciaiythony ii iauaio; e?iia oiai, o?oaeiinoe aicieeatho n
oaaee/aieai neiaeiinoe eiioeao?aoeee nenoaiu.

Oeaeueth aeaiiiai eo?niaiai i?iaeoa yaeyaony iaoiaeaeaiea caeiia
?ani?aaeaeaiey oaiia?aoo?u a aauanoaa, eioi?ui caiieiaii i?ino?ainoai
iaaeaeo aeaoiy noa?aie.

2 Iniiaiua iieiaeaiey oaieii?iaiaeiinoe

2.1 Oaiia?aoo?iia iiea

Oaieii?iaiaeiinoue i?aaenoaaeyao niaie i?ioeann ?ani?ino?aiaiey yia?aee
iaaeaeo /anoeoeaie oaea, iaoiaeyueieny ae?oa n ae?oaii a nii?eeiniiaaiee
e eiathueie ?acee/iua oaiia?aoo?u.

?anniio?ei iaa?aa eaeiai-eeai iaeii?iaeiiai e ecio?iiiiai oaea.
Ecio?iiiui iacuaatho oaei, iaeaaeathuaa iaeeiaeiauie oece/aneeie
naienoaaie ii anai iai?aaeaieyi. I?e iaa?aaa oaeiai oaea oaiia?aoo?a aai
a ?acee/iuo oi/eao eciaiyaony ai a?aiaie e oaieioa ?ani?ino?aiyaony io
oi/ae n aieaa aunieie oaiia?aoo?ie e oi/eai n aieaa ieceie. Ec yoiai
neaaeoao, /oi a iauai neo/aa i?ioeann ia?aaea/e oaieiou
oaieii?iaiaeiinoueth a oaa?aeii oaea nii?iaiaeaeaaony eciaiaieai
oaiia?aoo?u T eae a i?ino?ainoaa, oae e ai a?aiaie:

, (2.1)

— eii?aeeiaou oi/ee;

t — a?aiy.

Yoa ooieoeey ii?aaeaeyao oaiia?aoo?iia iiea a ?anniao?eaaaiii oaea. A
iaoaiaoe/aneie oeceea oaiia?aoo?iui iieai iacuaatho niaieoiiinoue
cia/aiee oaiia?aoo?u a aeaiiue iiiaio a?aiaie aeey anao oi/ae eco/aaiiai
i?ino?ainoaa, a eioi?ii i?ioaeaao i?ioeann.

Anee oaiia?aoo?a oaea anoue ooieoeey eii?aeeiao e a?aiaie, oi
oaiia?aoo?iia iiea iacuaatho ianoaoeeiia?iui, o.a. caaenyuei io a?aiaie:

. (2.2)

Oaeia iiea ioaa/aao iaonoaiiaeaoaiony oaieiaiio ?aaeeio
oaieii?iaiaeiinoe.

Anee oaiia?aoo?a oaea anoue ooieoeey oieueei eii?aeeiao e ia eciaiyaony
n oa/aieai a?aiaie, oi oaiia?aoo?iia iiea oaea iacuaatho noaoeeiia?iui:

. (2.3)

O?aaiaiey aeaooia?iiai oaiia?aoo?iiai iiey aeey ?aaeeia noaoeeiia?iiai:

; (2.4)

ianoaoeeiia?iiai:

. (2.5)

Ia i?aeoeea ano?a/athony caaea/e, eiaaea oaiia?aoo?a oaea yaeyaony
ooieoeeae iaeiie eii?aeeiaou, oiaaea o?aaiaiey iaeiiia?iiai
oaiia?aoo?iiai iiey aeey ?aaeeia noaoeeiia?iiai:

; (2.6)

ianoaoeeiia?iiai:

. (2.7)

Iaeiiia?iie, iai?eia?, yaeyaony caaea/a i ia?aiina oaieiou a noaiea, o
eioi?ie aeeeio e oe?eio iiaeii n/eoaoue aaneiia/ii aieueoie ii
n?aaiaieth n oieueiie.

2.2 A?aaeeaio oaiia?aoo?u

Anee niaaeeieoue oi/ee oaea n iaeeiaeiaie oaiia?aoo?ie, oi iieo/ei
iiaa?oiinoue ?aaiuo oaiia?aoo?, iacuaaaioth ecioa?ie/aneie.
Ecioa?ie/aneea iiaa?oiinoe iaaeaeo niaie ieeiaaea ia ia?anaeathony. Iie
eeai caiueathony ia naay, eeai eii/athony ia a?aieoeao oaea.

?anniio?ei aeaa aeeceea ecioa?ie/aneea iiaa?oiinoe n oaiia?aoo?aie T e T
+ (T (?enoiie 2.1).

no?aieony e ioeth, iacuaatho a?aaeeaioii oaiia?aoo?u.

(2.8)

A?aaeeaio oaiia?aoo?u anoue aaeoi?, iai?aaeaiiue ii ii?iaee e
ecioa?ie/aneie iiaa?oiinoe a noi?iio aic?anoaiey oaiia?aoo?u e /eneaiii
?aaiue /anoiie i?iecaiaeiie io oaiia?aoo?u ii yoiio iai?aaeaieth. Ca
iieiaeeoaeueiia iai?aaeaiea a?aaeeaioa i?eieiaaony iai?aaeaiea
aic?anoaiey oaiia?aoo?. 2.3 Iniiaiie caeii oaieii?iaiaeiinoe

Aeey ?ani?ino?aiaiey oaieiou a ethaii oaea eee i?ino?ainoaa iaiaoiaeeii
iaee/ea ?aciinoe oaiia?aoo? a ?acee/iuo oi/eao oaea. Yoi oneiaea
ioiineony e e ia?aaea/a oaieiou oaieii?iaiaeiinoueth, i?e eioi?ie
a?aaeeaio oaiia?aoo?u a ?acee/iuo oi/eao oaea ia aeieaeai auoue ?aaai
ioeth.

/a?ac yeaiaioa?ioth ieiuaaeeo dS, ?aniieiaeaiioth ia ecioa?ie/aneie
iiaa?oiinoe, e a?aaeeaioii oaiia?aoo?u onoaiaaeeaaaony aeiioacie Oo?uea,
niaeanii eioi?ie

. (2.9)

iacuaaaony eiyooeoeeaioii oaieii?iaiaeiinoe eee aieaa e?aoei –
oaieii?iaiaeiinoueth. Ni?aaaaeeeainoue aeiioacu Oo?uea iiaeoaa?aeaeaii
iiiai/eneaiiuie iiuoiuie aeaiiuie, iiyoiio yoa aeiioaca a ianoiyuaa
a?aiy iineo iacaaiea iniiaiiai o?aaiaiey oaieii?iaiaeiinoe eee caeiia
Oo?uea.

Ioiioaiea eiee/anoaa oaieiou, i?ioiaeyuaai /a?ac caaeaiioth
iiaa?oiinoue, ei a?aiaie iacuaatho oaieiaui iioieii. Oaieiaie iioie
iaicia/atho q e au?aaeatho a aaooao (Ao):

. (2.10)

Ioiioaiea oaieiaiai iioiea dq /a?ac iaeue yeaiaio ecioa?ie/aneie
iiaa?oiinoe e ieiuaaee dS yoie iiaa?oiinoe iacuaatho iiaa?oiinoiie
ieioiinoueth oaieiaiai iioiea (eee aaeoi?ii ieioiinoe oaieiaiai iioiea),
iaicia/atho j e au?aaeatho a aaooao ia eaaae?aoiue iao? (Ao/i2):

. (2.11)

Aaeoi? ieioiinoe oaieiaiai iioiea iai?aaeai ii ii?iaee e ecioa?ie/aneie
iiaa?oiinoe a noi?iio oauaaiey oaiia?aoo?u. Aaeoi?u j e grad T eaaeao ia
iaeiie i?yiie, ii iai?aaeaiu a i?ioeaiiieiaeiua noi?iiu.

Oaieiaie iioie q, i?ioaaeoee neaicue i?iecaieueioth iiaa?oiinoue S,
iaoiaeyo ec au?aaeaiey

. (2.12)

Eiee/anoai oaieiou, i?ioaaeoaa /a?ac yoo iiaa?oiinoue a oa/aiea a?aiaie
t, ii?aaeaeyaony eioaa?aeii

. (2.13)

Oaeei ia?acii, aeey ii?aaeaeaiey eiee/anoaa oaieiou, i?ioiaeyuaai /a?ac
eaeoth-eeai i?iecaieueioth iiaa?oiinoue oaa?aeiai oaea, iaiaoiaeeii
ciaoue oaiia?aoo?iia iiea aioo?e ?anniao?eaaaiiai oaea. Iaoiaeaeaiea
oaiia?aoo?iiai iiey e ninoaaeyao iniiaioth caaea/o aiaeeoe/aneie oai?ee
oaieii?iaiaeiinoe. 2.4 Aeeooa?aioeeaeueiia o?aaiaiea oaieii?iaiaeiinoe

Eco/aiea ethaiai oece/aneiai i?ioeanna naycaii n onoaiiaeaieai
caaeneiinoe iaaeaeo aaee/eiaie, oa?aeoa?ecothueie aeaiiue i?ioeann. Aeey
neiaeiuo i?ioeannia, e eioi?ui ioiineony ia?aaea/a oaieiou
oaieii?iaiaeiinoueth, i?e onoaiiaeaiee caaeneiinoae iaaeaeo aaee/eiaie
oaeiaii ainiieueciaaoueny iaoiaeaie iaoaiaoe/aneie oeceee, eioi?ay
?anniao?eaaao i?ioaeaiea i?ioeanna ia ai anai eco/aaiii i?ino?ainoaa, a
a yeaiaioa?iii iauaia aauanoaa a oa/aiea aaneiia/ii iaeiai io?acea
a?aiaie. Naycue iaaeaeo aaee/eiaie, o/anoaothueie a ia?aaea/a oaieiou
oaieii?iaiaeiinoueth, onoaiaaeeaaaony aeeooa?aioeeaeueiui o?aaiaieai
oaieii?iaiaeiinoe. A i?aaeaeao aua?aiiiai yeaiaioa?iiai iauaia e
aaneiia/ii iaeiai io?acea a?aiaie noaiiaeony aiciiaeiui i?aiaa?a/ue
eciaiaieai iaeioi?uo aaee/ei, oa?aeoa?ecothueo i?ioeann.

I?e auaiaea aeeooa?aioeeaeueiiai o?aaiaiey oaieii?iaiaeiinoe
i?eieiathony neaaeothuea aeiiouaiey:

aioo?aiiea enoi/ieee oaieiou ionoonoaotho;

n?aaea, a eioi?ie ?ani?ino?aiyaony oaiei, iaeii?iaeia e ecio?iiia;

eniieuecoaony caeii nio?aiaiey yia?aee, eioi?ue aeey aeaiiiai neo/ay
oi?ioee?oaony oae: ?aciinoue iaaeaeo eiee/anoaii oaieiou, aioaaeoae
aneaaenoaea oaieii?iaiaeiinoe a yeaiaioa?iue ia?aeeaeaieiaae ca a?aiy dt
e auoaaeoae ec iaai ca oiaea a?aiy, ?anoiaeoaony ia eciaiaiea aioo?aiiae
yia?aee ?anniao?eaaaiiai yeaiaioa?iiai iauaia.

ca a?aiy dt, niaeanii o?aaiaieth Oo?uea, i?ioiaeeo eiee/anoai oaieiou:

(2.14)

(grad T acyo a aeaea /anoiie i?iecaiaeiie, o.e. i?aaeiieaaaaony
caaeneiinoue oaiia?aoo?u ia oieueei io x, ii e io ae?oaeo eii?aeeiao e
a?aiaie).

*a?ac i?ioeaiiieiaeioth a?aiue ia ?annoiyiee dz ioaiaeeony eiee/anoai
oaieiou, ii?aaeaeyaiia ec au?aaeaiey:

, (2.15)

ii?aaeaeyao eciaiaiea oaiia?aoo?u a iai?aaeaiee z.

Iineaaeiaa o?aaiaiea iiaeii i?aaenoaaeoue a ae?oaii aeaea:

. (2.16)

Eoae, i?e?auaiea aioo?aiiae yia?aee a ia?aeeaeaieiaaea ca n/?o iioiea
oaiea a iai?aaeaiee ine z ?aaii:

. (2.17)

I?e?auaiea aioo?aiiae yia?aee a ia?aeeaeaieiaaea ca n/?o iioiea oaiea a
iai?aaeaiee ine y au?aceony aiaeiae/iui o?aaiaieai:

, (2.18)

a a iai?aaeaiee ine x:

. (2.19)

Iieiia i?e?auaiea aioo?aiiae yia?aee a ia?aeeaeaieiaaea:

. (2.20)

N ae?oaie noi?iiu, niaeanii caeiio nio?aiaiey yia?aee:

, (2.21)

— iauai ia?aeeaeaieiaaea;

— ianna ia?aeeaeaieiaaea;

c — oaeaeueiay oaieiaieinoue n?aaeu;

— ieioiinoue n?aaeu;

— eciaiaiea oaiia?aoo?u a aeaiiie oi/ea n?aaeu ca a?aiy dt.

Eaaua /anoe o?aaiaiey (2.20) e (2.21) ?aaiu, iiyoiio:

, (2.22)

eee

. (2.23)

iacuaatho oaiia?aoo?ii?iaiaeiinoueth e iaicia/atho aoeaie a. I?e
oeacaiiuo iaicia/aieyo aeeooa?aioeeaeueiia o?aaiaiea oaieii?iaiaeiinoe
i?eieiaao aeae:

. (2.24)

O?aaiaiea (2.24) iacuaaaony aeeooa?aioeeaeueiui o?aaiaieai
oaieii?iaiaeiinoe (eee aeeooa?aioeeaeueiui o?aaiaieai Oo?uea) aeey
o?aoia?iiai ianoaoeeiia?iiai oaiia?aoo?iiai iiey i?e ionoonoaee
aioo?aiieo enoi/ieeia oaieiou. Iii yaeyaony iniiaiui i?e eco/aiee
aii?inia iaa?aaaiey e ioeaaeaeaiey oae a i?ioeanna ia?aaea/e oaieiou
oaieii?iaiaeiinoueth e onoaiaaeeaaao naycue iaaeaeo a?aiaiiui e
i?ino?ainoaaiiui eciaiaieyi oaiia?aoo?u a ethaie oi/ea iiey.

yaeyaony oece/aneei ia?aiao?ii aauanoaa e eiaao aaeeieoeo i2/c. A
ianoaoeeiia?iuo oaieiauo i?ioeannao a oa?aeoa?ecoao nei?inoue eciaiaiey
oaiia?aoo?u.

aeey ethaie oi/ee oaea i?iii?oeeiiaeueii aaee/eia a. Iiyoiio i?e
iaeeiaeiauo oneiaeyo auno?aa oaaee/eaaaony oaiia?aoo?a o oiai oaea,
eioi?ia eiaao aieueooth oaiia?aoo?ii?iaiaeiinoue.

Aeeooa?aioeeaeueiia o?aaiaiea oaieii?iaiaeiinoe n enoi/ieeii oaieiou
aioo?e oaea eiaao aeae:

, (2.25)

aaea qV — oaeaeueiay iiuiinoue enoi/ieea, oi anoue eiee/anoai
auaeaeyaiie oaieiou a aaeeieoea iau?ia aauanoaa a aaeeieoeo a?aiaie.

Yoi o?aaiaiea caienaii a aeaea?oiauo eii?aeeiaoao. A ae?oaeo
eii?aeeiaoao iia?aoi? Eaieana eiaao eiie aeae, iiyoiio iaiyaony e aeae
o?aaiaiey. Iai?eia?, a oeeeeiae?e/aneeo eii?aeeiaoao aeeooa?aioeeaeueiia
o?aaiaiea oaieii?iaiaeiinoe n aioo?aiiei enoi/ieeii oaieiou oaeiai:

, (2.26)

aaea r — ?aaeeon-aaeoi? a oeeeeiae?e/aneie nenoaia eii?aeeiao;

— iiey?iue oaie.

2.5 E?aaaua oneiaey

Iieo/aiiia aeeooa?aioeeaeueiia o?aaiaiea Oo?uea iienuaaao yaeaiey
ia?aaea/e oaieiou oaieii?iaiaeiinoueth a naiii iauai aeaea. Aeey oiai
/oiau i?eiaieoue aai e eiie?aoiiio neo/ath, iaiaoiaeeii ciaoue
?ani?aaeaeaiea oaiia?aoo? a oaea eee ia/aeueiua oneiaey. E?iia oiai,
aeieaeiu auoue ecaanoiu:

aaiiao?e/aneay oi?ia e ?acia?u oaea,

oece/aneea ia?aiao?u n?aaeu e oaea,

a?aie/iua oneiaey, oa?aeoa?ecothuea ?ani?aaeaeaiea oaiia?aoo? ia
iiaa?oiinoe oaea, eee acaeiiaeaenoaea eco/aaiiai oaea n ie?oaeathuae
n?aaeie.

Ana yoe /anoiua iniaaiiinoe niaianoii n aeeooa?aioeeaeueiui o?aaiaieai
aeatho iieiia iienaiea eiie?aoiiai i?ioeanna oaieii?iaiaeiinoe e
iacuaathony oneiaeyie iaeiicia/iinoe eee e?aaauie oneiaeyie.

Iau/ii ia/aeueiua oneiaey ?ani?aaeaeaiey oaiia?aoo?u caaeathony aeey
iiiaioa a?aiaie t = 0.

A?aie/iua oneiaey iiaoo auoue caaeaiu o?aiy niiniaaie.

A?aie/iia oneiaea ia?aiai ?iaea caaeaaony ?ani?aaeaeaieai oaiia?aoo?u ia
iiaa?oiinoe oaea aeey ethaiai iiiaioa a?aiaie.

A?aie/iia oneiaea aoi?iai ?iaea caaeaaony iiaa?oiinoiie ieioiinoueth
oaieiaiai iioiea a eaaeaeie oi/ea iiaa?oiinoe oaea aeey ethaiai iiiaioa
a?aiaie.

A?aie/iia oneiaea o?aoueaai ?iaea caaeaaony oaiia?aoo?ie n?aaeu,
ie?oaeathuae oaei, e caeiiii oaieiioaea/e iaaeaeo iiaa?oiinoue oaea e
ie?oaeathuae n?aaeie.

.

2.6 Oaieii?iaiaeiinoue /a?ac oa?iaoth noaieo

. Aneaaenoaea yoiai oaiia?aoo?a n?aaeu oiaea yaeyaony a aeaiiii neo/aa
ooieoeeae iaeiie ia?aiaiiie – ?aaeeona r: T = T(r), a ecioa?ie/aneea
iiaa?oiinoe yoi eiioeaio?e/aneea noa?u. Oaeei ia?acii eneiiia
oaiia?aoo?iia iiea – noaoeeiia?iia e iaeiiia?iia, a a?aie/iua oneiaey
yaeythony oneiaeyie ia?aiai ?iaea: T(R1) = T1, T(R2) = T2.

Ec iaeiiia?iinoe oaiia?aoo?iiai iiey neaaeoao, /oi ieioiinoue oaieiaiai
iioiea j oae aea, eae oaieii?iaiaeiinoue e oaiia?aoo?a, yaeythony a
aeaiiii neo/aa ooieoeeyie iaeiie ia?aiaiiie – ?aaeeona r. Iaecaanoiua
ooieoeee j(r) e T(r) iiaeii ii?aaeaeeoue iaeiei ec aeaoo niiniaia: eee
?aoaoue aeeooa?aioeeaeueiia o?aaiaiea Oo?uea (2.25), eee eniieueciaaoue
caeii Oo?uea (2.11). A aeaiiie ?aaioa eca?ai aoi?ie niinia. Caeii Oo?uea
aeey enneaaeoaiiai iaeiiia?iiai noa?e/anee neiiao?e/iiai oaiia?aoo?iiai
iiey eiaao aeae:

. (2.27)

.

Ii?aaeaeei caaeneiinoue ieioiinoe oaieiaiai iioiea j io r. Aeey yoiai
nia/aea au/eneei oaieiaie iioie q /a?ac noa?o i?iecaieueiiai ?aaeeona
r > R.

. (2.28)

A /anoiinoe, oaieiaie iioie q1 /a?ac aioo?aiithth noa?o ?aaeeonii R1 e
oaieiaie iioie q2 /a?ac ia?oaeioth noa?o ?aaeeonii R2 ?aaiu

(2.29)

Ana yoe o?e iioiea nicaeathony iaeiei e oai aea enoi/ieeii iiuiinoueth
P. Iiyoiio ana iie ?aaiu P e iiyoiio ?aaiu iaaeaeo niaie.

. (2.30)

N o/?oii (2.28) e (2.29) yoi ?aaainoai iiaeii caienaoue a aeaea:

. (2.31)

O/eouaay, /oi

,

iieo/aai eneiioth caaeneiinoue ieioiinoe oaieiaiai iioiea j io ?aaeeona
r:

, (2.32)

aaea C1 – yoi eiinoaioa, ii?aaeaeyaiay oi?ioeie

. (2.33)

Oece/aneee niune iieo/aiiiai ?acoeueoaoa aeinoaoi/ii ynai: yoi ecaanoiue
caeii ia?aoiuo eaaae?aoia, oa?aeoa?iue aeey caaea/ ni noa?e/aneie
neiiao?eae.

, iieo/ei neaaeothuaa aeeooa?aioeeaeueiia o?aaiaiea:

. (2.34)

Aeaiiia o?aaiaiea ?aoaaony iaoiaeii ?acaeaeaiey ia?aiaiiuo:

.

Eioaa?e?iaaiea yoiai au?aaeaiey aea?o:

Eoae, ooieoeey T(r) eiaao aeae:

. (2.35)

Eiinoaiou C1 e C2 iiaeii ii?aaeaeeoue ec a?aie/iuo oneiaee T(R1) = T1,

T(R2) = T2. Iiaenoaiiaea yoeo oneiaee a (2.35) aea?o eeiaeioth nenoaio
aeaoo o?aaiaiee n aeaoiy iaecaanoiuie C1 e C2:

. (2.36)

Au/eoay ec ia?aiai o?aaiaiey aoi?ia, iieo/ei o?aaiaiea ioiineoaeueii C1:

,

ioeoaea

. (2.37)

N o/?oii yoiai au?aaeaiea (2.35) iiaeii caienaoue a aeaea:

. (2.38)

Oaia?ue ia?aia a?aie/iia oneiaea T(R1) = T1 aea?o:

, (2.39)

ioeoaea neaaeoao au?aaeaiea aeey eiinoaiou C2:

. (2.40)

Iiaenoaiiaea (2.40) a (2.39) aea?o ieii/aoaeueiia au?aaeaiea aeey
eneiiie ooieoeee T(r):

. (2.41)

Ciay ooieoeeth T(r), iiaeii ec caeiia Oo?uea

ii?aaeaeeoue e ieii/aoaeueiia au?aaeaiea aeey ieioiinoe oaieiaiai iioiea
j eae ooieoeee io ?aaeeona r:

. (2.42)

Eioa?anii ioiaoeoue, /oi ?ani?aaeaeaiea oaiia?aoo? ia caaeneo io
eiyooeoeeaioa b, ii caoi ieioiinoue iioiea i?iii?oeeiiaeueia b.

3 Caeeth/aiea

A ?acoeueoaoa i?iaeaeaiiie ?aaiou auaaaeaii aeeooa?aioeeaeueiia
o?aaiaiea oaieii?iaiaeiinoe i?eiaieoaeueii e aeaiiui eiie?aoiui oneiaeyi
caaea/e e iieo/aii ?aoaiea yoiai o?aaiaiey a aeaea ooieoeee T(r).
?ac?aaioaia i?ia?aiia TSO, ?ann/eouaathuay ooieoeeth T(r) e no?iyuay a?
a?aoee aeey ?acee/iuo caaeaaaaiuo iieueciaaoaeai ia?aiao?ia caaea/e .
Eenoeia i?ia?aiiu i?eaaaeai a I?eeiaeaiee A.

Nienie eniieuecoaiuo enoi/ieeia

Iauieei A.A. Oaoie/aneay oa?iiaeeiaieea e oaieiia?aaea/a: O/aa. iiniaea
aeey aocia. — 3-a ecae., eni?. e aeii. — I: Auno. oeiea, 1980. — 469 n.

A?aiaiiae/ E.A., Eaaei A.E. O?aaiaiey iaoaiaoe/aneie oeceee: I.: Iaoea,
1969. — 288 no?.

Naaaeueaa E. A. Eo?n iauae oeceee. O. 1. Iaoaieea. Iieaeoey?iay oeceea:
O/aa. iiniaea aeey nooaeaioia aoocia. — I.: Iaoea, 1982. — 432n.

Caeueaeiae/ A.E., Iuoeen A.Ae. Yeaiaiou iaoaiaoe/aneie oeceee. — I.:
Iaoea, 1973. — 352n.

I?eeiaeaiea A

(iaycaoaeueiia)

Eenoeia i?ia?aiiu TSO

unit Kurs_p;

interface

uses

Windows, Messages, SysUtils, Classes, Graphics, Controls, Forms,
Dialogs,

StdCtrls, Spin;

type

TForm1 = class(TForm)

Button1: TButton;

Label1: TLabel;

Label2: TLabel;

Label3: TLabel;

Label4: TLabel;

Label5: TLabel;

Label6: TLabel;

Label7: TLabel;

Label8: TLabel;

Edit1: TEdit;

Label9: TLabel;

Edit2: TEdit;

Label10: TLabel;

Edit3: TEdit;

Label11: TLabel;

Edit4: TEdit;

procedure Button1Click(Sender: TObject);

procedure FormPaint(Sender: TObject);

procedure Edit1KeyPress(Sender: TObject; var Key: Char);

private

public

procedure OsiK (x0,y0:Integer);

procedure Postroenie(T1,T2,R1,R2:real);

end;

var

Form1: TForm1;

X0,Y0:integer;

T1,T2,R1,R2:real;

implementation

{$R *.DFM}

procedure TForm1.OsiK (x0,y0:Integer);

var

i,x,y:integer;

begin

Canvas.Pen.Width:=2;

Canvas.Pen.Color := clBlack;

Canvas.MoveTo(x0, y0); {iino?iaiea ine X}

Canvas.LineTo(x0+400, y0);

Canvas.MoveTo(x0+400, y0); {iino?iaiea no?aei/ae ine O}

Canvas.LineTo(x0+400-10, y0-5);

Canvas.MoveTo(x0+400, y0);

Canvas.LineTo(x0+400-10, y0+5);

Label4.Left:=x0+390;

Label4.Top:=y0+10;

Label5.Left:=x0+350;

Label5.Top:=y0+10;

Label6.Left:=x0;

Label6.Top:=y0+10;

Label7.Left:=x0-25;

Label7.Top:=y0-10;

Label8.Left:=x0-25;

Label8.Top:=y0-105;

Canvas.MoveTo(x0, y0); {iino?iaiea ine Y}

Canvas.LineTo(x0, y0-150);

Canvas.MoveTo(x0, y0-150); {iino?iaiea no?aei/ae ine Y}

Canvas.LineTo(x0-5, y0-150+10);

Canvas.MoveTo(x0, y0-150);

Canvas.LineTo(x0+5, y0-150+10);

Label3.Left:=x0-25;

Label3.Top:=y0-150;

Canvas.Pen.Width:=1;

x:=x0;

for i:=1 to 10 do

begin

x:=x+35;

Canvas.MoveTo(x, y0-3);

Canvas.LineTo(x, y0+3);

end;

y:=y0;

for i:=1 to 5 do

begin

y:=y-20;

Canvas.MoveTo(x0-3, y);

Canvas.LineTo(x0+3, y);

end;

end;

procedure TForm1.Postroenie(T1,T2,R1,R2:real);

var

x,y:integer;

Kx,Ky,x1,y1,P,C1,Sag:real;

begin

Canvas.Pen.Width:=1;

Canvas.Pen.Color := clRed;

Sag:=(R2-R1)/500; {oaa ii X}

C1:=(T1-T2)/(ln(R2/R1));

Kx:=(R2-R1)/350; {Eiyooeoeeaiou “oneeaiey”}

if T1>T2 then

Ky:=T1/100

else

Ky:=T2/100;

x1:=R1; {Ia/aeueiua oneiaey}

y1:=T1;

Canvas.MoveTo(x0+Round((x1-R1)/Kx),y0-Round(y1/Ky));

repeat

y:=Round(y1/Ky);

x:=Round((x1-R1)/Kx);

Canvas.LineTo(x0+x, y0-y);

x1:=x1+Sag;

y1:=(T1+C1*ln(R1/x1));

{label1.Caption:=label1.Caption+’; ‘+intToStr(x);

label2.Caption:=label2.Caption+’; ‘+intToStr(y);}

until x1>R2;

P:=4*Pi*C1;

label1.Caption:=’Iiuiinoue enoi/ieea:
=’+FloatToStrF(P,ffGeneral,5,1)+

‘ Ao’;

label5.Caption:=FloatToStrF(R2,ffGeneral,4,1);

label6.Caption:=FloatToStrF(R1,ffGeneral,4,1);

if T1>T2 then

begin

label7.Caption:=FloatToStrF(T2,ffGeneral,4,1);

label8.Caption:=FloatToStrF(T1,ffGeneral,4,1);

end

else

begin

label7.Caption:=FloatToStrF(T1,ffGeneral,4,1);

label8.Caption:=FloatToStrF(T2,ffGeneral,4,1);

end;

end;

procedure TForm1.Button1Click(Sender: TObject);

var

Code1,Code2,Code3,Code4:integer;

begin

Repaint;

val (Edit1.Text,T1,Code1);

val (Edit2.Text,T2,Code2);

val (Edit3.Text,R1,Code3);

val (Edit4.Text,R2,Code4);

if (Code4 or Code3 or Code2 or Code1) 0 then

begin

Edit1.SetFocus;

MessageDlg (‘Aaaaeeoa iiaeaeoenoa cia/aiea!’, mtError,
[mbOk],0);

end

else

Postroenie(T1,T2,R1,R2);

end;

procedure TForm1.FormPaint(Sender: TObject);

begin

x0:=100;

y0:=200;

OsiK(x0,y0);

end;

procedure TForm1.Edit1KeyPress(Sender: TObject; var Key: Char);

begin

if not (key in [‘0′..’9′,#8,’.’]) then

begin

Key:=#0;

MessageBeep($FFFFFFFF);

end;

end;

end.

PAGE

PAGE 23

P

n

?enoiie 2.1

A

?enoiie 2.2

(Qz

(Qy

(Qx

dz

dy

dx

z

y

x

R1

R2

r

T(R1)=T1

T(R2)=T2

?enoiie 1.1

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