Курбатов Ю.Л., Шелудченко В.И., Кравцов В.В. Механика жидкости и газа: Учебное пособие

dp

=

p·dU.

(13)

<l>YHKll,1U1

nOTeHll,HaJIa

~Bn~eTC~

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CnOC06HOM

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B

paBHOBeCHH.

K

TaKHM

CHnaM

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n)l(ecTli,

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ll,eHTP06e)l(HhIe,

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2.3.

PasHosecue

Hec)l(uNlaeNloii

)l(UOKOCmu

noo

oeiicmsueNl

cun

mR)I(ecmu

PaccMOTPHM

npocTeMIIIHM

cnyqaM

paBHoBecH~

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)l(H)1,KOCTH,

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no)1, B03)1,eMCTBHeM

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CHn

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HanpaBneHO

BepTHKaJIhHO

BHH3.

EcnH

KOOp)1,HHaTHYIO OCh

HanpaBHTh

BHH3,

TO B

ypaBHeHHH

(12)

KOMnOHeHThI

yCKopeHH~

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CHn

Gx=Gy=O,

a

Gz=g

H

)1,H¢¢e-

peHll,HaJIhHOe

ypaBHeHHe

CTaTHKH

npHMeT

BH)1,

dp

=

p·g·dz.

(14)

EcnH

nonaraTh,

qTO

cB060)1,Ha~

nOBepXHOCTh

(pHC.

5)

HMeeT

KOOP)1,H-

Hary

Zo H

Ha

3TOM

nOBepxHocTH

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paBHO

Po

(B

qaCT-

HOM

cnyqae

3TO )1,aBneHHe

MO)l(eT

6hITh

paBHO

aTMoc¢epHOMY),

TO,

HHTerpHpy~

ypaBHeHHe

(14) B

npe)1,enax

OT Zo

)1,0

Z H OT Po

)1,0

P

nony-

qHM,

npH

ycnoBHH

p=const

p z

J

dp

P J

gdz

Po

Zo

p =

Po+p·g·(z-zo).

(15)

o

l

P,

EcnH

HaqaJIO

KOOp)1,HHaTHOM OCH

Oz

COBMeCTHTh C

ypOBHeM

CB060)1,-

HOM

nOBepxHOCTH,

TOr)1,a

zo=O

H

ypaBHeHHe

(15)

npHMeT

BH)1,:

P =

Po+p·g·z.

(16)

3TO

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Ha3hIBaeTC~

0 C -

HOBHhIM

3aKOHOM

rH)1,pOCTa-

THKH:

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B

nlO60M

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paBHO

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(Po),

cnO)l(eHHOMY

C

BeCOM

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H~

0 T

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H C

nno-

20

Il(a~blO

OCHOBaHHH,

paBHOH

e~HHH~e.

ECJIH

B

BblpmKeHHH

(12)

nOJIO)l(HTb

p=const,

HJIH

dp=O,

TO

no-

JIyqHM

ypaBHeHHe

nOBepxHocTH

YPOBHH

Gx·dx+Gy"dy+Gz·dz

= 0, HJIH

dU=O,

CJIe)],OBaTeJIbHO

nOBepXHOCTb

ypOBHH

HBJIHeTCH H

nOBepXHOCTblO

paB-

Horo

nOTeH~HaJIa,

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nOT

e H

~

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JI

h

HOM

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KaKoBa

IPOpMa

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no)],

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qTofibl

OTBeTHTb

Ha

3TOT

BO-

npoc

nOJIO)l(HM

p=const,

Tor)],a

ypaBHel{He

nOBepxHocTH

(15)

npHMeT

\

BH)],:

const-po

z = =

const

pg

,

(17)

T.e.

nOBepxHocTblO

ypOBHH

HBJIHeTCH

ropH30HTaJIbHaH

nJIOCKOCTb.

2.4. PasHosecue

HeCJKUMaeMOu

JKUOKOCmu

npu

HanU'IUU

He-

apasumaLCUOHH&IX

MaCCOS&IX

cun

2.4.1.

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ABI1>KeHl1e

>KI1AKOCTI1

B

ropl130HTanbHoM

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PaccMorpHM

)],BH)I(eHHe )l(H)],KOCTH B

HeKoTopoM

cocy)],e

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B

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C

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a;

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HaXO)],HTCH

no)],

B03~eHcTBHeM

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a

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no~

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a 0

9

z

PHCYHOK

6 -

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jKH.llKOCTH

"PH

paBHOYCKopeHHOM

ropH30H-

CTBHeM

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yCKO-

peHHeM

g

(pHC.

6).

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OCb

Ox

HanpaBHTb

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no

Xo)]'y

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a OCb

Oz

-

BepTHKaJIbHO

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TO B

npoeK~HH

YCKopeHHH

MaCCOBbIX

CHJI B

ypaBHeHHH

(12)

fiY)],YT

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Gx=a;

Gy=O;

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a

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ypaBHeHHe

rH~pOCTaTHKH

npH-

TaJlbHOM

.llBHjKeHHH

MeT

BH,lJ,:

dp

=

p·(a·dx+g·dz).

HHTcrpHpYH

3TO

ypaBHcHHc,

nOJIyqHM

21

p =

p·a·x+p·g·z+c.

KOHCTaHTa

HHTerpHpOBamUI

Onpe,ll,enjleTCjI

H3

ycnoBHjI:

X=XO;

Z=Zo;

p=PO,

Cne,ll,OBaTenbHO, 3aBHCHMOCTb

,ll,aBneHHjI

OT KOOp,ll,HHaT 6Y,ll,eT

HMeTb

BH,ll,:

p =

po+p·a(x-xo)+p·g·(z-zo).

(18)

,l1,llj{

TOro,

QT06bI

onpe,ll,enHTb

IPOPMY

nOBepxHocTM

YPOBHjI,

nOnO:>KMM

p=const.

Tor,ll,a

ypaBHeHHe

(18)

MO:>KeT

6bITb

npeo6pa30BaHo

K

Bblpa-

:>KeHHlO

THna

z=-A

+ B·x, (19)

T.e.

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nHHHH.

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B

,ll,aHHOM

npHMepe

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jlBnjleTCjI

HaKllOHHajl

nnoc-

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2.4.2. PaBHOBeCl-1e

>Kl-1AKOCTl-1,

nOKos:llllelilcs:I OTHOCl-1TenbHO

cocYAa

l-1

apalllalOlllelilcs:I

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OCl-1

nYCTb

:>KH,ll,KOCTb,

HaXOM~ajlCjI

B

cocy,ll,e,

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C

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C

yrnoBoH

CKOPOCTblO

0)

(PHC.

7).

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H

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pacnono-

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OX

H

Oy

B

ropH30HTallbHOH

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a

OCb

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TO

npOeKl(HH

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l(eHTpO-

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paBHbI

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2 2

ro

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Gx=O)

x,

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y,

a

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TjI:>KeCTH

Gz=g.

Tor,ll,a

OCHOBHoe

,ll,MIPIPepeHl(MallbHOe

ypaBHeHMe

fH,ll,pOCTaTHKH (12)

9

) (

w

i!

npMMeT

BH,ll,:

dp

=

p.(0)2xdx+0)2ydy+gdz).

nocne

MHTerpHpOBaHHjI

nony-

QHM

p=

pul

(x

2 +

y2)+

pgz+

C.

2

KOHCTaHTa

HHTerpHpOBaHHJf

onpe,ll,enjleTCjI

H3

ycnoBHsr:

,ll,nsr

x=O,

y=O,

Z=Zo,

P=Po.

Tor,ll,a

PHCYHOK

7 -

PaBHOBeCHe

iKH)J,KOCTH,

C=Po-p·g·z,

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BOKpyr

BepTHKaJIbHOii

a 3aBHCMMOCTb

,ll,aBneHHjI

OT

OCD KOOp,ll,HHaT 6Y,ll,eT

HMeTb

BH,ll,:

22

(20)

nOBCpXHOCTb

ypOBml

(p=const)

HaXO)J,HTC}I nOCJIC

npc06pa30BaHH}I

(20) K BH)J,y:

2gz

X

2

+y2=-W

2

+

C

) (21)

(B

KOHcTaHry

C

1

BOlliJIH

BCC

nOCTOSlHHblC

BCJIHqHHbI).

Bblpa)l(CHHC

(21) npc.D.CTaBJISICT

c060u

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napa60J10H.D.a

BpalllCHHSI.

2.5. YpasHeHuR 2uiJpocmamuKu

iJnR

C)KUMaeMblX

cpeiJ

PaccMOTPHM

paBHoBccHC

C)I(HMaCMOu

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(ra3a),

Haxo-

)J,sUI(cik}l

no)J,

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KaqeCTBC

npHMcpa

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)J,aBJ1CHH}I

B03)J,yxa

no

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aTMoccPbl.

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OCb

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BBCpX.

Tor)J,a

OCHOBHOC

)J,HCPCHl(HallbHOC

ypaBHcHHc

rH)J,poCTaTHKH

(12) 6Y)J,CT

HMcn

BH)J,

dp =

-p·g·dz.

(22)

T.K.

B03)J,yX

-

C)I(HMaCMa}l

cpc)J,a,

Hco6xo)J,HMO

yqHTblBaTb

3aBHCH-

MOCTb

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OT

TCMncpaTypbI

H )J,aBJ1CHHSI,

qTO

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C)J,CJ1aTb,

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KJIanciipOHa

- MCH)J,CJ1CCBa:

P

P.

=RT'

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yqHTblBa}l

3aBHCHMOCTb

TCMncpaTypbI

OT

BbICOTbl,

H3

(22)

no-

JIyqHM:

dp

dz

-=-g--

P

RT{z)"

nOCJIC

HHTcrpHpOBaHHSI

C

yqCTOM

Toro,

qTO

Ha

ypOBHC

z=zo

)J,aBJICHHC

P=Po,

nOJIyqHM

P

~

Po

ex{

-g 1

R~Z»)

(23)

3Ta

$opMYJ1a

HOCHT

Ha3BaHHC

6 a

po

M C T

pH

q C C K 0 U.

ECJIH

CqHTaTb,

qTO

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H30TCPMHqCCKOM

paBHoBccHH,

T.C.

T(z)=const

TO

p =

Po

exp(

-

:T

(z

-

Zo

) )

23

(24)

EcnH

3a~aHO

nHHenHOe

H3MeHeHHe

TeMnepaTyphl

no

BhlCOTe

aTMO-

cephI T=To-b(z-Zo),

'ITO

cnpaBeMHBO

,lJ,0

z=11

KM,

6apOMcrpH'Ie-

CKaj{

<popMyna

npHMeT

BH,lJ,:

g

(

b(z

-

Zo

)

JbRT

O

P =

Po

l---'----'=-'-

To

(25)

,D,pyron

BapHaHT

6apoMcrpH'IecKoH

<l>0pMynhI

MO)l(HO

nony-

'IHTh,

npHHSlB

TeMnepaTYPY

o,lJ,HHaKoBoH

no

BbICOTC H HCnOJlh3YSI

3a-

P

KOH

bonnSl

-

MapHorra.

nO~cTaBmISI

B

ypaBHcHHe

(22)

P=Po·-,

po

pa3,D,CJUlj{

ncpCMCHHhlC

H

HHTCrp"pySl,

IlOJly'lHM

z::::~lnPo

.

gpo

P

(26)

h

Po

BBOM

0603Ha'IeHHe

0 =

--

H

pernaSl

OTHOCHTenbHO

p,

nOJIY'IHM

gpo

e~e

O,lJ,HH

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6apoMerpH'IeCKOn

<l>0PMYJIhI:

z

P =

poe

ho.

ECJIH

3anHcaTh

(26)

~ml

,lJ,ByX BhICOT

ZI

H Z2:

zi = ho

In

Po

;z2

=

hO

In.E!L

PI

P2

nOJIY'IHM

6apOMeTpH'IecKYlO

<l>OpMyJIY B

BH,lJ,e:

-h

1 P2

zl-

z

2-

o-n-,

PI

(27)

(28)

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no

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npHBe~eHhl

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(MCA).

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npHH.sIThI

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MaJIhHhle

aTMocepHhle

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T=288,

p=

760

MM.pT.CT.

(101300

lla).

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11

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1

KM).

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He-

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(,lJ,hIMOBhle

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-

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300-400

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24

P.

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600

~-:--+--~:---+---+--t-----+-

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JI.

tr

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278

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•

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H

nJIOTHOCTH

no

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aTMoccIlepbl

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p = Po-(z-zo)·g·p.

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BblCOTa

06'heKTa

He

60JIee

400

M,

rrorpeuIHocTb

:noH

<P0PMYJIbI

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rrpeBblllIaeT

0,5 %, '-ITO BrrOJIHe

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,

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=

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HO

npH

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25

2000

0,156

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pa3)J,eJIeHHbIX

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HBJIHeTCH CJIe)J,CTBH-

eM

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(PHC.

9).

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Po=

PO+ZgPB

H

a6COJIIOTHOrO

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neqHblX

ra30B

..

--

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to'

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.

~

•...

:

..

':.

Pr=

PO+zgPr

~

&P~O

PHCYHOK

9 -

CTaTHKa

)J,ByX

ra30B

26

rrpOBe,n:eM

qepe3

HaqaJIO

KOOp,n:HHaT.

(3,n:eCh

Po

- a6COmOTHOe ,n:aBJIe-

HHe

Ha

ypOBHe

Z=

0).

OTMeTHM,

qTO

BTOpa~

JIHHH~

npOll,n:eT

60JIee

KPYTO

qeM

nepBaH,

T.K.

neqHble

ra3hI

HMelOT nJIOTHOCTh

MeHhIIIe

qeM

aTMocifJepa.

1136hlTo4Hoe

,n:aBJIeHHe

ra30B

L\.p

=

Pr-Po

6y,n:eT nOJIO)l(HTeJIhHhIM

BhIllie

HYJIeBOH JIHHHH H OTpHlJ:aTeJIhHhIM -

HH)I(e. TIpoBe,n:eM MhICJIeHHhIH

3KcnepHMeHT

no

H3MepeHHIO

H36hI-

T04HOro

,n:aBJIeHHH

raJOB C nOMOlll,hlO ,n:HifJifJepeHlI,HaJIhHOrO

U-06paJHoro

MaHOMerpa

no,n:

CBO,n:OM

neqH

Ha

ypOBHe

ZI<O,

Ha

HY-

JIeBOH

JIHHHH (Zo=O) H B ,n:hIMoxo,n:e

Ha

ypOBHe Z2>

O.

1136hlTo4Hoe

,n:aBJIeHHe

Ha

ypOBHe

Zl:

L\.p

=

PlrPol

=

(PO-ZIPrg)-(PO-ZIPsg)=-ZIPrg+Z1PBg=Zlg(Pr-Ps).

T.K.

PB>Pr

TO

L\.p>O. 1136b1T04HOe ,n:aBJIeHHe

Ha

ypOBHe

Z2:

L\.p

=

Pr2-PB2=(PO+Z2Prg)-(PO+Z2Pog)=Z2Prg-Z2PBg=Z2g(Pr-PB).

T.K.

PB>Pr

TO

L\.p<O.

TaKHM

06PaJOM,

MaHoMerphI

nOKa)l(YT nOJIO)l(H-

TeJIhHOe

H36hITo4Hoe

,n:aBJIeHHe

ne4HhIX

raJOB no,n:

CBO,n:OM

neqH

H

oTpHlJ:aTeJIhHOe H36hITOqHOe ,n:aBJIeHHe (pa3pH)I(eHHe)

- B ,n:hIMoxo,n:e.

Ha

HYJIeBOll JIHHHH

ypoBeHh

)l(H,n:KOCTH

B JIeBOH H npaBOH

BeTB~X

Ma-

HOMerpa

6y,n:YT o,n:HHaKOBhIMH. ECJIH TOnJIHBHaH rOpeJIKa BKJII04eHa,

IIIH6ep

OTKphIT H npo,n:yKTbI

ropeHHH

y,n:a.JUIIOTCH

4epe3

,n:hIMOXO,n:

B

,n:hIMOBYIO

rpy6y,

TO

nOJIO)l(eHHe HYJIeBOH JIHHHH B

ne4H

MO)l(HO

OT-

perYJIHpOBaTh ,n:hIMOBhlM

IIIH6epOM:

3aKphlBaH

wH6ep

6y~eM

onyc-

KaTh HYJIeBYIO JIHHHIO, a

CJIe,n:OBaTeJIhHO

nOBhlIIIaTh

H36hIT04HOe

,n:aB-

JIeHHe

Ha

ypOBHe

rre4Horo

OKHa H BepmlTHOCTh BbI6HBaHHH

neqHhIX

raJOB a

aTMocifJepy;

OTKPbIBa~

llIH6ep

6y,n:eM no,n:HHMaTb HYJIeBYIO

JIHHHIO.

n p H M e

p.

PaCC4HTaeM 3Ha4eHHe H36b1T04HOro

,n:aBJIeHHH

no,n:

CBO,n:OM

MapTeHOBCKOH

rre4H:

z=

4 M,

Pr=0,17

Kr/M3 rrpH

tr=

1900

°C,

PB=

1,20

rrpH

t=

20°C,

,n:aBJIeHHe Ha ypOBHe

nopora

3aBaJI04HbiX

OKOH

Po= 1 MM.

BO,n:.

CT.

PeIIIeHHe.

L\.pz=po+zg(Po-Pr)=

1·9,8+4·9

,8(

1,20-0,17)=50,8

Oa.

C'raTHKa

,n:bIMOBOH

Tpy6hl.

~hlMoBaH

Tpy6a

-

3TO

BepTH-

KaJIhHhlH KaHaJI 60JIhIIIOH BhICOThl,

4epe3

KOTOPhIH

OTBO,n:HTCH

roPH-

4He npo,n:yKThl

cropaHHH

C

I1JIOTHOCThlO

3Ha4HTeJIhHO

MeHhweH,

4eM

fIJIOTHOCTh oKpY)I(alOllI,erO

aTMoccPepHoro

B03,n:yxa.

~hlMoBaH

Tpy6a

-

3TO

T~rOBoe

cpe,n:CTBO,

npe,n:Ha3Ha4eHHoe

,n:JIH

C03,n:aHHH

pa3pH-

27

)KeHIDI

(THrH),

no,lJ, ,lJ,eMCTBHeM

KOToporo

npO,lJ,YKThI

cropaHHH

0 T C

a-

ChI

B a 10 T C H

H3

KaMephI

npOMTenJIO:)HepreTfitleCKOM

YCTaHOBKH,

npe-

O,lJ,OJIeBaH

conpOTHBJIeHMe

CMCTeMbI ,lJ,bIMOXO,lJ,OB.

(npM

llCnOJIb30Ba-

HHH

,nblMococa

,nbIMOBa51

TPy6a

BbInOJIH5IeT B OCHOBHOM <PYHKlJ,lllO

pacceMBaHliH

BbJ6pOCOB).

MexaHH3M

C03,lJ,aHHH

pa3pe)KeHH5I

MO)l(-

HO

YHCHMTb,

paCCMOTpeB

CTaTMKY ,nblMOBOM

Tpy6bl

(pMc.l

0).

npe,nno-

.--

___

o-~-

__

~p

..

PHCYHOK

10

-

CTaTHKa

~"'MOBOii

Tpy6b1

JIO)KHM,

qTO

,nbIMOBOM

llIM6ep

onYIlJ,eH

H

,nBll)KeHMe

ra30B

B

TPy6e

npeKpaTllJIOch.

B nJIOCKOCTH YCTbH

TPy6bI

,naBJIeHHe

ra30B

H

aTMO-

c<pepbI

paBHbI

Me)K,ny

C060M

H COCTaBJUlIOT

BeJIHqHHY,

KOJIHqeCTBeH-

HO

paBHYIO

,naBJIeHHIO

BceM

aTMoc<pepbI,

JIe)KaIlJ,eM

BbIIlIe

,nblMOBOM

Tpy6bI.

no

Mepe

nepeMeIlJ,eHM5I

BHM3

01'

YCTb51

,naBJIeHMe

ra30B

H

aT-

Moc<pepbI

6y,neT

YBeJIMqHBaTbC5I H y OCHOBaHMJl

TPy6bl

COCTaBHT

,lJ,JIH

ra30B:

Prl=PO+HgPn

,lJ,JIH

aTMOC<pepbI:

PBl=Po+HgpB'

fhObTTOqHOe

,lJ,aBJIeHHe

raJOB

Y OCHOBaHHH

TPy6hI

~Pl=Prl-PBl=Hg(Pr-PR)'

a

T.K.

Pr<PB

TO

~Pl<O'

T.e.

Y OCHOBaHMH

TPy6bl

6y,neT

Ha6JIIO,lJ,aTbCH

OTPHlJ,aTeJIbHOe

H36bITOQHOe

,lJ,aBJIeHHe, HJIH

paJpe)KeHHe,

qTO

MO)KHO

Onpe,lJ,eJIHTb M

3KCnepHMeHTaJIbHO,

npHMeHHB

,lJ,H<p<pepeHIJ,HaJIbHhIM

U-06paJHbIH

MaHOMeTP

(pHc.l

0).

,ll,eMcTBHe

,lJ,blMOBOH

TPy6bI

MO)KeT

6bITb

TaK)Ke

npOHJIJIIOCTPll-

pOBaHO

Ha

pHC

11.

nYCTb

,nblMOBaK

TPy6a

o6eCneQHBaeT

THrolO

Ka-

28

Koe-JIu60

rrpOMTerrJIO:mepreTUqeCKOe

YCTPOHCTBO,

HarrpUMep

KOTeJI.

CrrpaBa

OT KOTJIa HaxOAUTCSl AhlMOBaSl

TPy6a,

a CJIeBa -

Boo6pa:>Kae-

MaSl

Tpy6a,

TaKOH

:>Ke

BhICOThI, HarrOJIHeHHaSl

aTMoc<\lepHhIM

B03AY-

XOM.

TaKaSl

CUCTeMa

C006IQaIOIQHXCSl

COCYAOB

6YAeT

HaXOAHThCSl B

PUCYHOK t t -

npHHQHD

~eii:CTBuH ~LlMOBOii:

TPYOhi

COCTOSlHHH rrOKOs[:, eCJIH KOTeJI

He

pa60TaeT.

Ho

eCJIH

npH

pa60Te

KOT-

JIa AhIMOBaSl

TPy6a

HarrOJIHSleTCSl

TOnOqHhIMH

ra3aMU,

TeMnepaTypa

KOTOPhIX

3HaqHTeJIhHO

BhIllie

TeMnepaTyphI

Hap)':>KHoro

B03Ayxa,

a

nJIOTHOCTh

3HaQHTenhHO

MeHhllIe,

TO

paBHOBeCHe

B

CHCTeMe

HapyllIa-

eTCSl

B CTOPOHY

AhIMOBOH

TPy6hl.

CHna,

C KOTOPOH HapY:>KHhIH

B03AYX

6YAeT

AaBHTb

Ha

HU:>KHee

CeqeHHe

AhIMOBOH

TPy6hI,

paBHSleTCSl

pa3-

HOCTH BeCOB

cTon6oB

HapY:>KHOrO

B03Ayxa

H

AbIMOBhlX

ra30B.

3Ta

CH-

na

H

o6ycnaBnHBaeT

TeOpeTHqeCKYIO

TSlry

TPy6bI.

Pa3pe:>KeHUe, UJIU

3HepruSl

OTcoca,

KOTopoe

c03AaeT

AhIMOBaSl

Tpy6a

BhICOTOH

H,

pacxoAyeTcSl

Ha

rrpeOAOJIeHUe

corrpOTUBJIeHUSl

AbIMOBoro

TPaKTa,

UJIH

CHCTeMbI

AbIMOXOAOB,

COeAHHSlIOIQUX

pa6o-

qylO

KaMepy

YCTaHOBKH C

TPy60H,

CJIeAOBaTenhHO

L\PPaJp. =

Hg(PB

- Pr)

~

L

L\P

conp

' (29)

rAe

LAp

conp -

cyMMapHoe

fHApaBnHqeCKOe

corrpOTHBneHHe

29

Курбатов Ю.Л., Шелудченко В.И., Кравцов В.В. Механика жидкости и газа: Учебное пособие

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