ejerci f1

CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%

EJERCICIOS DE APLICACIÒN FLUJO DE FLUÌDOS Y LA ECUACIÒN DE BERNOULLI A) Factores de Conversin! Conversin! & L'i! ) & L* ) &+.+7,&-+ '/0 &. '/ ) +  L'i! 'i!00 & "l' l'i! ) "* "* ) .712 .712 L'0 L'0 -2   &"l'i! ) +.3 , &  '/0 &. *i4 '/ ) 553 "l'i!. "!- C6!i4r8 u! r*id49 d4 :u;6 d4 < "l'i!  *i4/'/=

( ) ( ) ( ) 1 pie

gal Q =20 min

3

s gal 449 min

−3 pie

=44.54 x 10

3

.

s

<.- C6!i4r8 u! r*id49 d4 :u;6 d4 52 L'i!  = 3

1m

Q = 4500

..-

L s min 60000 L min

=0.075 m3 / s .

C6! C6 !i4 i4rr8 u! r*id *id49 49 d4 :u;6 :u;6 d4 523 523 "l' "l'i i! !  L'i! 'i! = 3.785 L

gal min

Q = 459

min 1 gal min

=1,737.3 L / min .

5.- U! 4!8ild6r *r6duc4 +5 *i4/ 'i! (PC>% d4 ir4. Si l d4!/idd d4l ir4 4/ d4 &.< ?"'@ clcul4 l r*id49 d4 :u;6 d4 / 4! /lu"/'/  l r*id49 d4 :u;6 d4 *4/6 4! l'$. S6luci!.-

( )( ( ) = ( 3

pie 1 min Q =640 min 60 s ρ=

1.20 kg 3

m

0

)

3

slug m

3

3

=0.302 m s

slug

=0.08225

14.59 kg

m 3

m 0.302 s

)

3

0

0 l r*id *id49 49 d4 :u;6 u;6 d4 / /  4/=

=0.0248

slug s

0  l r*id49 d4 *4/6 4/=

)

ρgQ =0.08225

W = 0.8

1 pie

1 slug

M = ρQ 0.08225 W = γ . Q

0.3048 m

slug m

3

(

32.2

pies s

(

2

)(

3

m 0.302 s

lb lb 3600 s →W = 0.8 s s 1h

)

=2880

)

( )( )( 2

lb.s =0.08225 pies

1

m

3

32.2

pies 2

s

)(

)

m 0.302 s

3

lb h

B) Ec#acin de Contin#idad. E! u! /i/84 c4rrd6 d4 c6!ducci! d4 :u;6@ 4l clcul6 d4 l 4l6cidd d4l :ud6 d4*4!d4 d4l *ri!ci*i6 d4


c6!8i!uidd. E! l Fi". 4l :ud6 :u4 d4 l /4cci! &  l /4cci! < c6! r*id49 c6!/8!84. L c!8idd d4 :uid6 u4 */ *6r cului4r /4cci! 4! u! ci4r86 8i4*6 dd6 4/ c6!/8!84 (/4 l4 ll G:u;6 c6!/8!84%. E! 8ri!6/ d4 l r*id49 d4 / c66= > & ) ><0 8i! >)ρA0 /4 8i4!4= ρ& A& & ) ρ< A< < (Ec. d4 c6!8i!uidd% 4/ álid6 *r 86d6 :uid6 ("/4/@ luid6/%. Si 4l luid6 4/ i!c6*r4/il4@ l6/ 8ri!6/ ρ& ) ρ<0 4!86!c4/ u4d l 4cuci! d4 c6!8i!uidd A&  & ) A<  <@ u c66 K ) A lu4"6 $" % $& /4 c6!clu4 l Ec. d4 c6!8i!uidd *r u! :u;6 4/8l4 (luid6/%@ l r*id49 d4 :u;6 d 6lu4! 4/ l i/ 4! cului4r /4cci!. Ti! *u4d4 u8ili9r/4 c6! 4rr6r *4u46 *r "/4/  ;/ 4l6cidd4/ 4!6r4/  & '/. &.- El "u :u4  &.< '/ 4! u! c6!duc86 d4 &2  d4 diá48r6. Clcul4 l 4l6cidd d4 :u;6 4! u! c6!duc86 d4   d4 diá48r6 u4 4/8á c6!4c8d6 c6! 4l !84ri6r0 clculr l r*id49 d4 d4 :u;6 d4 6lu4!.- S6luci!.P6r c6!8i!uidd d4l :u;6 4! 6/ c6!duc86/= A& & ) A< <0 l A m π ( 150 mm ) / 4 m m ( v =v = = 1.20 1.20 0.25 )=0.30 4l6cidd <@ /4rá= A s π ( 300 mm ) / 4 s s

(

( ) 1

2

1

2

0

 Q 1= A1 v 1=

l π (150 mm )

2

4

(

r*id49

d4

)(

)

1.20

m s

1m 1000 mm

2

)

2

2

:u;6 =0.212 m

d4

6lu4!=

3

s

.

&! E! u! /4cci! d4 u! /i/84 d4 di/8riuci! d4 ir4@ 4l c6!duc86 8i4!4 u! ár4 d4 & *ul"d/ cudrd/@  4l ir4@  &5.7 l'*ul"<  & MF 8i4!4 u! d4!/idd d4 <.< , & - /lu"/'*i4  *4/6 4/*4cc6 d4 7.3 , &-< l'*i4@ 8i4!4 u! 4l6cidd *r64di6 d4 && *i4/'i!. E! 68r /4cci! d4l i/6 /i/84@ 4l c6!duc86 4/ r4d6!d6  8i4!4 u! diá48r6 d4 &+ *ul"@  l 4l6cidd /4 id4 1 *i4/'i!. Clcul4 l d4!/idd d4l ir4 4! l /4cci! r4d6!d  l r*id49 d4 :u;6 d4 *4/6 d4l ir4 4! lir/ *6r $6r. S6luci!.- D4 cu4rd6 c6! l Ec. d4 c6!8i!uidd *r l6/ "/4/@ /4 8i4!4= ρ1 A 1 v 1= ρ2 A 2 v 2 0 / *u4/@ *6d46/ clculr 4l ár4 d4 l/ d6/ /4cci6!4/  r4/6l4r *r A 2=

ρ 2= ρ1

ρ<=

( )( ) 0 A 1

v1

A2

v2

A 1=( 10 pulg )

2

=100 pulg 2 0

π

( D ) 2= π ( 16 pulg )2=201.06 pulg 2 . E!86!c4/@ l d4!/idd d4l ir4 4! 4 2 4

l /4cci! r4d6!d 4/=

(

−3 slugs

ρ2= 2.20 x 10

pie

3

)(

2

100 pulg

2

201.06 pulg

)(

1100 pie / min 800 pie / min

)

=1.504 x 10−3

slugs pie

3

.


L

r*id49

d4

(

)

−2

W = γ 1 A 1 v 1= 7.09 x 10

lb pie

3

:u;6

( 100 pulg ) 2

(

d4

1100 pies

min

)(

*4/6

1 pie

2

144 pulg

2

)(

60 min

h

4/=

)=

3,249.6 lb / h

.

'.- D484ri!4 l r*id49 d4 :u;6 á,i *4ri8id@ 4! L'i!@ u4 *u4d4 *r6ducir/4 4! u! 8u6 d4 c4r6 4/8á!dr u4 8i4!4 u! diá48r6 4,84r!6 d4 &.2 *ul"  .72 *ul" d4 "r6/6r 4! l/ *r4d4/@ /i l 4l6cidd á,i d44 /4r . '/. S6luci!= U8ili9!d6 l d4!ici! d4 r*id49 d4 :u;6 d4 6lu4!@ /4 8i4!4= K ) A@ 4l ár4 4/= 2 π −4 2 A = [ ( 1.50 −2 x 0.075 ) x 0.0254 ] x =9.2347 x 10 m 0 4!86!c4/ l r*id49 d4 4

Q=( 9.2347 x 10 m −4

:u;6 4! 6lu4!=

(

−3 m

Q =2.77 x 10

 L'i!@ /4 8i4!4=

s

3

2

)

( )= m 3 s

2.77 x 10

60000 L / min 3

1m /s

−3 m

)=

3

s

0 c6!ir8i4!d6

166.22 L / min

.

(.- D484ri!4 4l 86 r4u4rid6 d4 c6!duc86 d4 c4r6 Clir4 5 4/8á!dr u4 ll44 & L'i! d4 "u c6! u! 4l6cidd á,i d4 2.2 '/. S6luci!.- C66 K   /6! c6!6cid6/@ 4l ár4 r4u4rid6 /4 *u4d4 $llr d4= K ) A0 A)K'. Pri4r6 d446/ c6!4r8ir l r*id49 d4 :u;6 d4 6lu4!  u!idd4/ d4  '/.

( ) 3

L Q =3100 min

m 1 s

60000

L min

m =0.0516 s

3

0

4!86!c4/

4l

ár4=

3

Q 0.0516 m / s = 9.394 x 10−3 m2 . A = = v 5.50 m / s

E/86 /4 d44 i!84r*r48r c66 4l

ár4 !i di/*6!il4@ *u4/86 u4 cului4r ár4 4!6r *r6ducir u! 4l6cidd 6r u4 2.2 '/. P6r c6!/i"ui4!84@ d446/ u/cr 4! 4l *!dic4 F d4l lir6 u! c6!duc86 4/8á!dr c6! u! ár4 d4 :u;6 ;u/86 6r u4 3.35 , &- < . S4 r4ui4r4 u! c6!duc86 d4 c4r6 Clir4 5 4/8á!dr d4 2 *ul"@ c6! ár4 d4 :u;6 d4 &.<3& , &-< <. L 4l6cidd r4l d4 :u;6 cu!d6 4/84 c6!duc86 ll4 .2 '/ d4 3 0.0533 m / s "u 4/= )K'A ¿ 1.291 x 10−3 m2 = 4.13 m / s 0 /i /4 u8ili9r 4l c6!duc86 /i"ui4!84 / *4u46 (u! c6!duc86 clir4 5 d4 5 *ul"%@ l

CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.% 3

4l6cidd *r6ducid /4r= )K'A0

v=

0.0533 m / s −3

8.213 x 10 m

2

=6.50 m / s @ u

l8.

.- E! l "ur@ /4 u4/8r u! i!84rcid6r d4 cl6r u4 /4 u8ili9 *r 8r!/4rir cl6r d4/d4 4l :uid6 u4 :u4 d4!8r6 d4l 8u6 i!84ri6r l u4 :u4 4! 4l 4/*ci6 c6*r4!did6 4!8r4 l *r84 4,84ri6r d4l 8u6  l *r84 i!84ri6r d4l c/c6 cudrd6 u4 E n tra d a d e flu ì d o a l c a s c o

F lu j o e n el casc o F lu jo e n e l tu b o

A

S=0.85” t

As S

T u b o d e c o b r e t ip o  d e ! p u l" a d a S a l id a d e flu ì d o d e l c a s c o

Secciòn transversal

In t e rc a m b i a d o r d e c a l o r d e c a s c o y t u b o

r6d4 l 8u6. Clcul4 l r*id49 d4 :u;6 d4 6lu4! 4! "l'i! u4 *r6ducir u! 4l6cidd d4 3 *i4/'/@ 8!86 d4!8r6 d4l 8u6 c66 4! 4l c/c6. S6luci!.- U8ili96/ l rul *r l r*id49 d4 :u;6 d4 6lu4!@ K ) A *r cd *r84= % D4!8r6 d4l 8u6 d4 c6r4 8i*6 ? d4  *ul" d4 86 !6i!l= D4l *!dic4 /4 68i4!4= ED).+<2 *ul"0 Diá48r6 i!84ri6r ID ) .2<7 *ul"0 "ru4/6 d4 l *r4d ) .53 *ul"@ 4l ár4 8uulr= A t =

(

π 0.527 4

12

)

2

pies

=1.515 x 10−3 pie 2 . E!86!c4/@ l r*id49 d4 :u;6 d4

6lu4! d4!8r6 d4l 8u6 4/= 2 3 −3 Q t = A t v =( 1.515 x 10 pie ) ( 9 pies / s )= 0.013635 pies / s 0 c6!ir8i4!d6  "*= Q t =0.013635

pie s

3

( ) 449

gal min

pie 1 s

3

=6.122

gal min

% D4!8r6 d4l c/c6= 4l ár4 d4l :u;6 !48 4/ l di4r4!ci 4!8r4 4l ár4 d4!8r6 d4l c/c6 cudrd6  l *r84 4,84r! d4l 8u6. E!86!c4/= π π 2 2 2 2 2 A s = S − ED =( 0.85 pulg ) − ( 0.625 pulg ) = 0.4157 pulg 0 c6!ir8i4!d6  4

4

*i4/< /4 8i4!4= A s =0.4157 pulg

2

1 pie

2

=2.8868 x 10−3 pie 2 . L r*id49 d4 :u;6 r4u4rid 144 pulg 2

3

4/

4!86!c4/=

pies − Qs= A s v = ( 2.8868 x 10 pie ) ( 9 pies / s )=0.0259 0 s 3

2


c6!ir8i4!d6  "*=

Qs= 0.0259

pies s

3

( ) 449

gal min

pie 1 s

3

=11.665

d4l :u;6 4! 4l c/c6 4!8r4 4l :u;6 4! 4l 8u6 4/=

gal min

0 El c6ci4!84

Coiente=

Qs Qt

)

gal min =1.905 gal 6.122 min

11.665

C! CONSER*ACIÒN DE LA ENER+ÌA,ECUACIÒN DE BERNOULLI E le $ e n to d e l f lu i d o

p

v

% # i v e l d e r e f e r e n c ia

F i g . E l e m e n t o d e f lu i d o e n u n c o n d u c to

S4 *r4!di u4 l 4!4r" d4 u! /i/84@ !6 /4 *u4d4 cr4r !i d4/8ruir@ /i!6 u4 *u4d4 /4r 8r!/6rd d4 u! 8i*6  68r6@ /8 4/ l l4 d4 c6!/4rci! d4 l 4!4r". Cu!d6 /4 !li9 c6!duc86/ c6! :uid6/@ 4,i/84! 8r4/ 6r/ d4 4!4r"@ c66 4;4*l6 4! l Fi". E/8á l6cli9d  u! ci4r8 4l4ci! (9%@ 8i4!4 u! 4l6cidd (%  u! *r4/i! (*%. L/ rul/ d4 4!4r" /6!= &. E!4r" *684!cil (PE%.- E/ d4id  l 4l4ci!  4/ c6! r4/*4c86  u! l!4 d4 r44r4!ci= PE % -!.0 d6!d4 Q 4/ 4l *4/6 d4l 4l44!86. <. E!4r" ci!8ic (E%.- E/ d4id  l 4l6cidd@ d4l 4l44!86= /E % -!v&0&1 . E!4r" d4 :u;6 (FE)Fl6 E!4r"%.- E/ l 4!4r" d4 *r4/i! 6 4l 8r;6 d4l :u;6@ 4/ l c!8idd d4 8r;6 !4c4/ri6 *r 64r l 4l44!86 d4 :ud6  8r/ d4 u! ci4r8 /4cci! 4! c6!8r  l *r4/i!= FE % -!20 Ele$ento del L 4cuci! d4 4!4r" FE@ /4 *u4d4 d4ducir 4! fluido & 6r d4 8r;6 ) *AL ) *0 d6!d4 l u4r9 4/ *A@ pA l *r4/i! 4/ *  A 4/ 4l ár4 d4 l /4cci!@ /4 8r/ld 4l 4l44!86 u! di/8!ci L d4!8r6 d4 u! 6lu4! @ 4! 6r d4 *4/6 d4l 4l44!86 Q ) γ  4! l u4 γ 4/ 4l *4/6 4/*4cc6 d4l :uid6@ 4!86!c4/ 4l 6lu4! 4/  ) Q' γ /u/8i8u4!d6 4! 4l 8r;6 ) * Q' γ ) FE 4/ l 4!4r" d4 :u;6.

CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.% E le $ e n to d e l fluido

( p ( . % ( .v

E le $ e n to d e l fluido

' p ' .% ' .v

(

'

-L c!8idd 868l d4 4!4r" d4 4/8/ 8r4/ 6r/ u4 *6/44 4l 4l44!86 d4 :uid6 /4rá l /u. ℘ 2 E= !E + "E + #E = + $% + $ v / 2 g 0 cd u!6 d4 4/86/ 8ri!6/ /4 γ

4,*r4/ 4! u!idd4/ d4 4!4r"@ !486!-48r6 (N.% 4! 4l /i/84 i!84r!ci6!l 6 4! *i4/-lir (*i4.l% 4! 4l /i/84 d4 u!idd4/ ri8á!ic/. - C6!/id4r!d6 $6r 4l 4l44!86 d4 :uid6 u4 /4 u4/8r 4! l "ur@ u4 /4 u44 d4 l /4cci!(&%  l /4cci! (<%. L6/ l6r4/ d4 *@ 9   /6! di4r4!84/ 4! l/ d6/ /4cci6!4/. E! l /4cci! (&% l 4!4r" 868l 4/= E2=

4/=

$ p2 γ

E1=

$ p1 γ

2

+ $ % 1+

$ v1 2g

. E! l /4cci! (<% l 4!4r" 868l

2

+ $ % 2+

$ v 2

. Si !6 /4 "r4" 4!4r" l :uid6 6 /4 *i4rd4

2g

4!8r4 l/ /4cci6!4/ &  <@ 4!86!c4/ 4l *ri!ci*i6 d4 c6!/4rci! d4 l 4!4r" r4ui4r4= E1= E 2 →

$ p 1 γ

2

+$ % 1 +

$ v1 2g

=

$ p2 γ

2

+ $ %2 +

$ v2 2g

El *4/6 d4l 4l44!86  4/ c6!  86d6 l6/ 8ri!6/  /4 l4 *u4d4 c!c4lr@ l 4cuci!@ 4!86!c4/@ /4 c6!i4r84 4!3 p1 γ

2

+ % 1+

v1 2g

=

p2 γ

2

+ % 2+

v2 2g

A /8 /4 c6!6c4 c66 4cuci! d4 B4r!6ulli. ) a b e % a T o ta l

*v *v

( (+ , ("

( '+ , ("

p

) a b e % a d e v e l o c id a d

( , γ

v( ( )abe%a de presiòn

v' F lu jo

%'

%

(

' )abe%a de e l e v a c i ò n

# i v e l d e r e f e r e n c ia

F i " . ) a b e % a s - d e e l e v a c i ò n  p r e s i ò n  v e l o c id a d / t o t a l

C!" Inter2retacin de 4a ec#acin de Berno#44i!, A cd 8ri!6 d4 l Ec. d4 B4r!6ulli@ r4/ul8 l


4!4r" *6/4d *6r 4l :uid6 *6r u!idd d4 *4/6 d4 :uid6 u4 :u4 4! 4l /i/84. L/ u!idd4/ d4 l6/ 8ri!6/ /4 d! 4! !486!-48r6 *6r !486! (N.'N% 4! 4l /i/84 SI  4! 4l ri8á!ic6 (l.*i4'l%. E/ d4 !68r u4 4l N@ l /4 *u4d4! c!c4lr@ d4;!d6 /6l4!84  u! u!idd d4 l6!"i8ud@ 4l 48r6 (% 6 4l *i4. P6r 8!86@  l6/ 8ri!6/ /4 l4/ c6!6c4 c66 Gc49/= -El 8ri!6= *'γ /4 l4 c6!6c4 c66 c49 d4 *r4/i!. -El 8ri!6= G9 /4 l4 ll c49 d4 4l4ci!. - El 8ri!6= <'<" /4 l4 c6!6c4 c66 c49 d4 4l6cidd. L /u d4 l6/ 8r4/ 8ri!6/ /4 l4/ c6!6c4 c66 Gc49 868l. Cd 84ri! c6rr4/*6!d4 u! l8ur c6! r4/*4c86  u! l!4 d4 r44r4!ci@ /4 c6!/8ru4 u! Di"r c66 /4 6/4r 4! l "ur@ 4/ 8il *r i/uli9r l r4lci! 4!8r4 l6/ 8r4/ 8i*6/ d4 4!4r"@ l c49 868l *4r!4c4 c6!/8!84. E!86!c4/@ l l8ur r4l8i d4 cd 8ri!6 d4 c49 r /4"! l6 4/8l4cid6 *6r l 4cuci! d4 B4r!6ulli. E! l "ur@ l c49 d4 4l6cidd 4! l /4cci! < 4/ 4!6r u4 4! 4l d4 l /4cci! &@ 4/86 /4 d4u4/8r c6! l Ec. d4 c6!8i!uidd= A 1 v 1= A2 v 2 → v 2=v 1 v1

u4

A 1 A 2

0 *u4/86 u4

A 1 < A2

0 

v2

d44 /4r 4!6r

. Y c66 l 4l6cidd 4/8á l cudrd6 4! 4l 8ri!6 2

c6rr4/*6!di4!84  l c49 d4 4l6cidd@

v2 2g

4/ uc$6 4!6r u4

2

v1 2g

.

Ti!@ cu!d6 4l 86 d4 l /4cci! /4 4,*!d4 c66 /4 4 4! l "ur@ l c49 d4 *r4/i! u4!8 d4id6  u4 di/i!u4 l c49 d4 4l6cidd. E! /u@ l Ec. d4 B4r!6ulli 4,*lic 4l ci6 4! l/ c49/ 4!8r4 d6/ *u!86/ d4 u! /i/84 d4 :u;6 d4 :uid6. S4 /u*6!4 8i!@ u4 !6 $ *rdid/ !i "!!ci/ d4 4!4r" *6r l6 u4 l c49 868l *4r!4c4 c6!/8!84. L/ *r4/i6!4/ 4/ c6!4!i4!84 4,*r4/rl6 c66 !68ric. C!& Restricciones a 4a Ec#acin de Berno#44i!, L 4cuci! d4 B4r!6ull /4 *lic  uc$6/ *r6l4/@ *4r6 4,i/84! l"u!/ r4/8ricci6!4/ c66= &. Su lid49 4/ /6l6 *r :uid6/ i!c6*r4/il4/  u4 /u *4/6 4/*4cc6 !6 r. <. N6 d44 c6!/id4rr di/*6/i8i6/ 4cá!ic6/ 4! / /4cci6!4/  u4 !6 d44 dir/4 !i ui8r/4 4!4r"@ l Ec. 4/8l4c4 4!4r" 868l c6!/8!84. . N6 *u4d4 $4r 8r!/4r4!ci d4 cl6r d4!8r6 !i u4r d4l :ud6. 5. N6 *u4d4 $4r *rdid d4 4!4r" *6r ricci!. >/ d4l!84 4/8/ r4/8ricci6!4/ /4 c6!/id4rrá! 4! l 4cuci! "4!4rl d4 4!4r". C!' Procedi5iento de a24icacin de 4a ec#acin de Berno#44i!


&. D484ri!r u4 4l44!86/ /6! c6!6cid6/  u4 /4   4!c6!8rr. <. D4cidir cuál d4 l/ d6/ /4cci6!4/ d4l /i/84 /4 u8ili9rá! cu!d6 /4 4/cri l Ec. d4 B4r!6ulli. S4 4/c6"4 u! /4cci! d4 l cul /4 c6!6c4! uc$6/ d86/. L /4"u!d 4/@ *6r l6 "4!4rl@ l /4cci! 4! l cul /4 d44 clculr l"6. . E/criir l Ec. d4 B4r!6ulli *r d6/ /4cci6!4/ 4l4"id/ 4! 4l /i/84. E/ r4c64!dl4 u4 l dir4cci! d4l :u;6 4/ d4 i9ui4rd  d4r4c$. 5. Si*licr 4! l6 *6/il4 c6! l c!c4lci! d4 l6/ 8ri!6/ cu6 l6r 4/ c4r6 6 d4 u4ll6/ u4 /6! l6/ i/6/ 4! 6/ ld6/ d4 l 4cuci!2. R4/6l4r l 4cuci! l"4ric4!84 *r 4l 8ri!6 d4/4d6. ) a b e % a T o ta l (

*v (+ *v '+

(

, ("

, ("

p

) a b e % a d e v e l o c id a d

( , γ

v( ( )abe%a de presiòn

v'

%

F lu jo

%'

(

' )abe%a de e le v a c i ò n

# ivel de referen cia

F i " . ) a b e % a s - d e e le v a c iò n  p r e s i ò n  v e lo c i d a d / t o t a l

+. Su/8i8uir l/ c!8idd4/ c6!6cid/  clculr 4l r4/ul8d6@ 84!4r cuidd6 4! 4l u/6 d4 u!idd4/ c6!/i/84!84/  l6 lr"6 d4l clcul6. EJE>PLO DEL PROCEDI>IENTO (CONSIDERAR LA FIVURA DE C.&% &.E! l "ur *r84 d4 u! 8u4r 4l "u  & MC :u4 d4 l /4cci! &  l /4cci! <. E! l /4cci! & 4l diá48r6 4/ <   l *r4/i! !68ric 4/  ?P  l 4l6cidd d4 :u;6 4/ d4 <.1 '/. L /4cci! <@ 8i4!4 5  d4 diá48r6@ 4/8á  &.1  /6r4 l /4cci! &. Su*6!i4!d6 u4 !6 $ *rdid/ d4 4!4r" 4! 4l /i/84@ clcul4 l *r4/i! *<. S6luci!.- Li/8d6 d4 d86/= D &)< 0 D<)5 0 &) <.1 '/0 *&) ?P0 9&)0 9<)&.1  lu4"6 9< W9& ) &.1 0 * < )X E! l "ur l/ /4cci6!4/ /6! &  <= 4! & /4 c6!6c4 * &0  &  9&0 4! l /4cci! < /4 8i4!4 9<@ /4 $llrá *<  <0 c6! l Ec. d4 B4r!6ulli= p1 γ

2

+ % 1+

(

p2= γ

v1 2g

p 1 γ

=

p2 γ

2

+ % 2+

v2

2

+ % 1 +

(

v1 2g

p2= p1 + γ % 1− % 2+



2g 2

− % 2− 2

v2 2g

2

v 1−v 2 2g

)0

p2 γ

=

p 1 γ

2

+ % 1 +

v1 2g

2

− % 2−

v2 2g

"ru*!d6

c49/=

) . O8r6/ d86/= " ) 3.1& '/ 0 d4l "u <

γ

) 3.1&


?N'. S4 d484ri!rá < d4 l Ec. d4 c6!8i!uidd= A&&)A<<

( )()

< )



2

π D 1

v 2= v 1

&(A&'A<%0

2

D 1

m =v 1 2 =2.8 s D2

4

2

π D2

(

20 mm 40 mm

)

2



4

v 2=

1 4

m s

( 2.8 )=0.7

m s

0 A$6r /4 /u/8i8u4 l6/ l6r4/ 684!id6/ 4! l

Ec. d4 B4r!6ulli= p2=300 k"a + 9.81

p2=300 k"a + 9.81

k& m

3

(

−1.80 m

(+ ) −( ) m 2.8 s

2

m 0.7 s

2 ( 9.81

m s

2

)

2

)

k&

k&

m

m

( −1.80 m+ 0.3746 m )=300 k"a + 9.81 3

p2=300 k"a −13.983

k& m

2

3

(−1.4254 m)

=300 k"a−13.983 k"a→p2=286 k"a

0 E/8 *r4/i!

4/ !68ric@ /4 d4/rr6ll 4! u!ci!  *& u4 4/ !68ric 8i!.

C!( 6an7#es8 reci2ientes 9 :o7#i44as e;2#estas a 4a at5s
.

) '.0 $

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.

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(0 $$ de d i3 $ e t ro

F i " . S if ò n p a r a e l e je $ p l o

-E! l "ur /4 u4/8r u! /i! u4 /c :uid6 d4l r4ci*i4!84  l6 rr6;  8r/ d4 u! 6uill c6l6cd l !l d4l c6!duc86. E/ d4 6/4rr@ u4 l /u*4rci4 lir4 d4l 8!u4 (*u!86 A%  l d4/cr" 4! l 6uill (*u!86 F% !6 4/8á! c6!!d/  r6!84r/ /lid/ /i!6  l 8/4r@ *6r u4 l/ *r4/i6!4/ P A  P F /4rá! c4r6/@ *6r c6!/i"ui4!84 /u/ c49/ d4 *r4/i! /4rá! c4r6/  /4 *u4d4! c!c4lr. -O8r 6/4rci!@ 4! l /u*4rci4 lir4 d4l 8!u4 /4 /u*6!4 u4 4/8á 8r!uil6  4/ u "r!d4 c6! r4/*4c86  l /u*4rci4 d4l


c6!duc86 c4rrd6@ *6r l6 u4 l 4l6cidd /4rá u *4u4  A ) @  d44 4lii!r/4 /u c49 d4 4l6cidd. -Cu!d6 6/ *u!86/ d4 r44r4!ci 4/8á! 4! 4l i/6 c6!duc86.Si"ui4!d6 4l !áli/i/ 4! l "ur l6/ *u!86/= B@ C@ D  E /4 4!cu4!8r! d4!8r6 d4l c6!duc86  *6/44! u! ár4 d4 :u;6 u!i6r4  4l :u;6 4/ 4/8l4@ *6r l6 u4 l 4l6cidd /4rá l i/  l6 lr"6 d4l c6!duc86= B)C)D)E. P6r l6 u4@ /u/ c49/ d4 4l6cidd 4! 6/ ld6/ d4 l 4cuci! /4r! i"ul4/  /4 c!c4lr!. -Cu!d6 l 4l4ci! 4/ l i/ 4! 6/ *u!86/ d4 r44r4!ci.- Si l 4l4ci! 4! l6/ d6/ *u!86/ d4 r44r4!ci u/d6/ 4! l Ec. d4 B4r!6ulli 4/ l i/@ 4/86/ 8ri!6/ /4 c!c4l! (9 A@ 9B@ 9D%. L/ 5 6/4rci6!4/ !li9d/ cili8rá! l/ /i*licci6!4/ 4! l /6luci! d4 4;4rcici6/ /iilr4/ l 4;4*l6 c6!/id4rd6.

". E=e524o.- E! l "ur d4 rri 4l /i! /c "u d4l 4/8!u4@ c6! l/ i// di4!/i6!4/ u4 /4 d!. Clcul4 l r*id49 d4 :u;6 d4 6lu4!  8r/ d4l /i!  l *r4/i! 4! l6/ *u!86/ B@ C@ D  E. S6luci!.- &.- cálcul6 d4 l r*id49 d4 :u;6 d4 6lu4! K@ u8ili9!d6 l Ec. B4r!6ulli@ 4! 4/84 c/6 l6/ *u!86/ 4/c6"id6/ /6! A  F. C66 /4 di;6 !84/ 4! C.50 /4 8i4!4!@ 4! A= * A)@ A)0 4! F= * F)@ 9F ) <.2  p A

d4;6 d4 A.

γ

2

+ % A +

v A 2g

=

p ! γ

2

+ % ! +

v ! 2g

0 4! 4/8 Ec@ 4lii!!d6 l/ 2

% A= % ! +

c49/ d4 *   u4d= 9.81 m / s

v ! 2g

→ v ! =√ ( % A− % ! ) 2 g

0 d6!d4

2

¿

( 2.5 m ) 2 ¿ 0 % A− % ! =2.5 −0 =2.5 m→ v ! = √ ¿

%C66 K ) A.

K ) A F.F0 ár4 d4 l 6uill A F ) ( π'5%(< %<)



&5.&+  @ c6!ir8i4!d6 <

−4

Q=3.1416 x 10

( )=

m m 7 s 2

A ! =314.16 mm −3 m

2.2 x 10

2

(

1m 6

10

2

)

=3.1416 x 10−4 m2 0 mm 2

3

s

% Cálcul6 d4 l *r4/i! *B0 /4 4/c6"4rá l6/ *u!86/ A  B= p A γ

2

+ % A +

v A 2g

=

rr4"l!d6=

p ' γ

2

+ % ' +

[

v ' 2g

4! 4/8 Ec. *A)@ A)0

p' =γ ( % A − % ' )−

2

v ' 2g

]

% A=

p ' γ

2

+ % ' +

v ' 2g

4! 4/8 Ec. ( % A− % ' )= 0 0  u4 l6/

*u!86/ A  B 4/8á! l i/6 !i4l. P4r6 !84/ /4 clculrá l 4l6cidd B d4 l Ec.d4 c6!8i!uidd= K)A B.B B ) K'AB0 /4 c6!6c4 


K *4r6 !6 AB /4 clculrá d4l c6!duc86 A B ) (π'5%( %<) 7+.1+  @

2

A ' =706.86 mm

c6!ir8i4!d6

<

−3 m

2.2 x 10

4l6cidd

2

v' 2g

(=

v '=

)=

m s

10

)

=7.0686 x 10−4 m2 0 mm 2

l

m

2

=3.1123

m s

0

l

c49

4/=

2

m 3.1123 s 2 x 9.81

7.0686 x 10

6

2

3

s −4

(

1m

0.494 m

. A$6r /i /u/8i8u4!d6 d86/ $lld6/ 4!

2

[ ]

p' =γ

−v 2'

=9.81 k& 3 ( −0.494 m )=−4.846 k& 2 =− 4.846 k"a 0 4l /i"!6 !4"8i6 2g m m

i!dic u! *r4/i! !68ric *6r d4;6 d4 l *r4/i! 86/ric@ d4á/ 4! :uid6/ 4! 6ii4!86 p A ( p ' @ 4l c6!c4*86 d4 u4 *u!86/ 4! 4l i/6 !i4l 8i4!4! l i/ *r4/i! !6 /4 *lic c66 c6! l6/ :uid6/ 4! r4*6/6. c% Cálcul6 d4 l *r4/i! 4! *C0 /iilr4!84 /4 4/c6"4rá l6/ *u!86/ A  C=

p A γ

2

+ % A +

v A 2g

0 rr4"l!d6=

=

pC γ

2

+ % C +

v C

0 4! 4/8 Ec. * A)@ A)0

2g

[

2

pC = γ ( % A − %C ) −

v C 2g

]

γ

2

+ % C +

v C 2g

( % A− % C ) =−1 m 0 (!4"8i6@ *6ru4

0

3.1123

9C  9A0 8i! B ) C )

m s

0 ( u4 AC ) AB% 4!

2

c6!/4cu4!ci l/ c49/ /6! i"ul4/ pC =9.81

% A=

p C

v C 2g

2

=

v ' 2g

= 0.494 m l *r4/i! 4!

k& m

k& − − =− =−14.656 k"a m m ( 1 0.494 ) 14.656 3 2 m

d) Cálcul6 d4 l *r4/i! 4! * D0 /iilr4!84 /4 4/c6"4r l6/ *u!86/ A  D=

p A γ

2

+ % A +

v A 2g

=

p D γ

2

+ % D +

v D 2g

0 4! 4/8 Ec. * A)@ A). Ti! l

4l4ci! d4 l6/ *u!86/ A@ B  D /6! l/ i//@ *6r l6 u4 9 A)9B)9D /4 c!c4lr! 4! l Ec.@ 8i! l/ 4l6cidd4/ 4! B)C)D)E /6! i"ul4/  u4 /4 4!cu4!8r! 4! 4l i/6 ár4  c6!duc86@ 4!86!c4/ 2

D ) B0 lu4"6

v D 2g

2

=

v ' 2g

)

0.494 m

. L Ec. u4d /i*licd=

CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.% 2

0=

p D v D γ

+

p D

2g

=

γ



−v D2 2g

− v 2'

)

2g

p D = γ (−0.494 m ) =9.81

*r4/i! 4/

) −0.494 m 0 /4 c6!clu4 u4 l

k& 3

m

( −0.494 m )=−4.846 k& 2 =−4.846 k"a= p' m

! L/ *r4/i6!4/ 4! B  D /6! i"ul4/  u4 l 4l4ci!  l 4l6cidd /6! i"ul4/. 4% Cálcul6 d4 l *r4/i! 4! E0 4/c6"i4!d6 l6/ *u!86/ A  E= p A γ

2

+ % A +

% A=

p E γ

v A 2g

=

p E γ

2

+ % E +

v E 2g

0

4!

4/8

[

2

+ % E +

v E

0

2g

rr4"l!d6=

Ec.

*A)@ 2

p E = γ ( % A− % E ) −

v E 2g

]

0

A)0 d4

l

u4d= "ur

% A− % E=2.50 m

. Ti! l/ 4l6cidd4/ 4!  B)C)D)E /6! i"ul4/  u4 /4 4!cu4!8r! 4! 4l i/6 ár4  c6!duc86@ 4!86!c4/ E ) B0 2

v E

lu4"6 p E =9.81

R4/ul8=

2g

2

=

v '

0.494 m

)

2g

k&

k&

m

m

( 2.50 m −0.494 m )=9.81 3

p E =19.678

k& m

2

3

.

P6r

l8i6=

2.006 m

=19.678 k"a

RESU>EN DE LOS RESUL6ADOS DEL EJE>PLO RESUEL6O! &.L 4l6cidd d4 :u;6 4! l 6uill  l r*id49 d4 :u;6 d4 6lu4! 4! 4l /i!@ d4*4!d4! d4 l di4r4!ci d4 4l4ci! 4!8r4 l /u*4rci4 lir4 d4l :uid6  l /lid d4 l 6uill. <. L *r4/i! 4! 4l *u!86 B 4/8á *6r d4;6 d4 l *r4/i! 8/4r@ *4/4  u4 l6/ *u!86/ A  B 4/8á! l i/6 !i4l  u4 A 4/8á 4,*u4/86  l 8/4r. . L 4l6cidd d4 :u;6 4/ l i/ 4! 86d6/ l6/ *u!86/ 4! d6!d4 4l 86 d4l c6!duc86 4/ 4l i/6@ cu!d6 4,i/84 u! :u;6 4/8l4. 5. L *r4/i! 4! 4l *u!86 C 4/ l / ; d4l /i/84@ d4id6  u4 /84 /4 uic 4! 4l /i8i6 á/ 4l4d6. 2. L *r4/i! 4! 4l *u!86 D 4/ l i/ u4 l d4l *u!86 B@ d4id6  u4 6/ 4/8á!  l i/ l8ur  l c49 d4 4l6cidd 4! 6/ *u!86/ 4/ l i/. +. L *r4/i! 4! 4l *u!86 E 4/ l 6r d4l /i/84 d4id6  u4 4/84 *u!86 /4 4!cu4!8r 4! 4l /i8i6 á/ ;6.


'50 $ $ φ 1 Flujo 0.4( $ A

/ == == = = ( 50 $ $ φ = == == ' $ == = == == = == == = = = =

C! >edidores *ent#ri 9 otros siste5as cerrados con ve4ocidades desconocidas!, E! l "ur /4 u4/8r u! di/*6/i8i6 c6!6cid6 c66 4did6r 4!8uri@ u4 /4 u8ili9 *r 4dir 4l6cidd d4 :u;6 4! u! /i/84 d4 :u;6 d4 :uid6. El !áli/i/ /4 $rá c6! l Ec. d4 B4r!6ulli. E! l /4cci! B c6! diá48r6 r4ducid6 6c/i6!  u4 l 4l6cidd u4!84  di/i!u l *r4/i!. S4 $ d4 d46/8rr u4 l 4l6cidd d4 :u;6 d4*4!d4 d4 l di4r4!ci d4 *r4/i! 4!8r4 l6/ *u!86/ A  B@ *6r l6 u4 /4 u8ili9 u! !48r6 di4r4!cil 4!8r4 6/ *u!86/. E;4*l6 Ilu/8r8i6= ".El 4did6r 4!8uri 6/8rd6 4! l "ur ll4 "u  22 MC. L "r4dd 4/*4cc d4l :uid6 !68ric6 ("lic4ri! /")&.<+%. Clcul4 l 4l6cidd d4 :u;6 4! l /4cci! A  l r*id49 d4 :u;6 d4 6lu4! d4 "u. S6luci!.- D86/= /4 c6!6c4 l di4r4!ci d4 4l4ci! 4!8r4 l6/ *u!86/ A  B. C6! 4l !48r6 /4 68i4!4 l di4r4!ci d4 *r4/i! 4!8r4 l6/ *u!86/ A  B0 8i! /4 c6!6c4 l6/ diá48r6/ d4 l/ /4cci6!4/ A  B. L 4l6cidd !6 /4 c6!6c4 4! !i!"! *u!86 d4l /i/84  /4 *id4 4! A . E/ 6i6 u4 l6/ *u!86/ d4 i!84r/ /6! A  B. L Ec. d4 B4r!6ulli= p A γ

2

+ % A +

v A 2g

=

p ' γ

2

+ % ' +

v ' 2g

0 4l *4/6 4/*4cc6 d4l "u  22 MC 4/ 3.+7

?N'. El rr4"l6 4! c49/=

p A − p ' γ

2

+ ( % A − % ' ) =

2

v ' −v A 2g

0 l di4r4!ci

4! 4l4ci! 4/ % A− % ' =−0.42 m 0 4l l6r 4/ !4"8i6 *6ru4 4l *u!86 A 4/ / l86 u4 B. El *4/6 4/*4cc6 d4 l "lic4ri! k& 3 γ g =1.26 ( γ $ a 4 )C )=1.26 ( 9.81 k& / m ) =12.36 3 . E! l "ur !6 /4 8i4!4 l m

di/8!ci (% 4! 4l !48r6. E! l 4cuci! d4l !48r6 4*49!d6 d4l *u!86 A $/8 ll4"r l *u!86 B=

CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.% p A + γ $ ( * ) + γ $ ( 1 m )− γ g ( 1 m ) −γ $ ( * ) −γ $ ( 0.42 m )= p '

E/ d4 !68r u4 /4 c!c4l l *r4/i! 4! (%@ rr4"l!d6= p A − p '= γ $ ( 0.42 m− 1 m )+ γ g (1 m )= γ $ ( −0.58 m ) + γ g ( 1 m ) @ diidi4!d6= p A − p ' γ $

=−0.58 m+

γ g ( 1 m) γ $

12.36

k& 3

( 1 m )

m k&

=−0.58 m+

9.67

m

=−0.58 m+ 1.278 m →

p A− p ' γ $

=0.698 m

3

. El ld6 i9ui4rd6 d4 l Ec. d4 B4r!6ulli  /4 4lud6@ $6r /4 *r6/4"uirá c6! 4l i4r6 d4l ld6 d4r4c$6@ c6! l Ec. d4 c6!8i!uidd= AAA)ABB B ) A(AA'AB%0 

( ) 2

π D A

v '= v A

4

2

πD'

= v A

( ) (

2

2

2

D A

= v A

2

D '

250 mm 150 mm

)

2 2

2

→ v '=2.777 v A → v ' =7.711 v A

0

/4

4

2

2

2

2

2

2

v '− v A=7.711 v A− v A =6.711 v A → v ' − v A = 6.711 v A !4c4/i8 . P6d46/ $6r 86r 4/84 r4/ul8d6@ l di4r4!ci d4 c49/ d4 4l4ci!  l c49 d4 di4r4!ci d4 *r4/i!  /u/8i8uirl6/ 4! 2 2 p A − p ' v −v + ( % A − % ' ) = ' A . A/= .+31  −0.42 m ) .<71  ) 2g γ 2

6.711 v A 2g

v A =

√

0

r4/6li4!d6

2 g ( 0.278 m ) 6.711

=

√

(

2 9.81

m s

2

)(

v A

*r

/4

0.278 m )

6.711

=0.9015 m / s

L r*id49 d4 :u;6 d4 6lu4! 4/= π ( 0.25 m ) ( 0.9015 m / s )=0 . 04425 m / s % ((!& L2s Q = A A v A = 4

(

2

)

C!? 6EORE>A DE 6ORRICELLI

3

68i4!4=


.'  (

.

F l u jo d e u n ta n 6 u e

E! l "ur@ 4l :uid6 4/8á :u4!d6 d4l ld6 d4 u! 8!u4 *6r u! 6uill li/  r4d6!d. Pr d484ri!r l 4l6cidd d4 :u;6 u4 /4 68i4!4 4! l 6uill@ /4 4/criirá l 4cuci! d4 B4r!6ulli 4!8r4 u! *u!86 d4 r44r4!ci 4! l /u*4rci4 d4l :uid6  u! *u!86 4! 4l c$6rr6 u4 /4 68i4!4 d4 l 6uill= p1 γ

2

+ % 1+

v1 2g

=

p2 γ

2

+ % 2+

v2

0 L/ *r4/i6!4/ 4! &  < /6! !ul/ 4/8á!

2g

4,*u4/8/  l 8/4r *& ) *<0 d4á/ l 4l6cidd & *r6,id4!84 4/ c4r6. E!86!c4/ r4/6li4!d6 *r  < /4 68i4!4= v 2= √ 2 g ( % 1− % 2) 0 Hci4!d6 $ ) (9 W 9 %@ /4 68i4!4= & < v 2= √ 2 g h

E/8 4cuci! /4 c6!6c4 c66 846r4 d4 T6rric4lli 4! $6!6r  E!"4li/8 T6rric4lli@ ui4! l6 d4/rr6ll 4! 4l 6 &+52. E;4*l6 ilu/8r8i6.- Pr 4l 8!u4 d4 l "ur@ l *r6u!didd d4l :uid6 4/ $ ) 5 @ 4l diá48r6 d4l c$6rr6 4! l 6uill 4/ d4 5 . Hllr l 4l6cidd d4 /lid 4! l 6uill  4l cudl. C64!84 c66 rir l 4l6cidd  4l cudl cu!d6 di/i!u4 4l !i4l d4l luid6 l cir/4. . 4l6cidd d4 /lid=

√(

v 2= √ 2 g h= 2 9.81

. Cudl K= Zr4 d4l c$6rr6=

A =

m s

2

)(

4 m)= 8.858 m / s

.

π

( 0.040 m )2=1.256 x 10−3 m2 0 4l cudl 4

/4rá=

(

m Q = A . v = 1.256 x 10 m 8.858 s −3

.'

2

)

m = 0.011 s

3

=11 Lps

.2 

A"ua

.( )  o r ro v e r tic a l

c. L 4l6cidd  4l cudl rirá! di/i!u4!d6 d4 l6r  4did u4 /4 d4/cr" 4l :uid6 *6r l 6uill.


O8r *licci! i!84r4/!84 d4l 846r4 d4 T6rric4lli /4 u4/8r 4! l "ur /i"ui4!84@ 4! l u4 4l c$6rr6 d4 luid6 4/ l!9d6 $ci rri. El c$6rr6 lc!9 u! l8ur i"ul  l 4l4ci! d4 l /u*4rci4 lir4 d4l :uid6 4! 4l 8!u4. Al !l d4 l l8ur d4 4l4ci! d4l c$6rr6 l p1

4l6cidd 4/ c4r6=

γ

2

+ % 1+

v1 2g

=

p2 γ

2

+ % 2+

v2 2g

0 4/8 /i8uci! 4/ /iilr l

d4/rr6ll6 i!icil d4l 846r4 d4 T6rric4lli= 4/crii4!d6 l Ec. d4 B4r!6ulli 4!8r4 l6/ p2 γ

+ % 2+

v

2 2

2g

=

p3 γ

+ % 3+

v 3= √ v 2+ 2 g ( % 2− % 3 ) 2

v

2 3

2g

v 2= √ 2 g h

*u!86/

0 A$6r <  =

0 u *< ) * ) 0 r4/6li4!d6 *r   /4 8i4!4=

0 /u/8i8u4!d6 <  9< W 9 ) -$ 0 r4/ul8 *r =

v 3= √ 2 g h + 2 g (−h )

)0 E/84 r4/ul8d6 4ric u4 4l c$6rr6 lc!9 l l8ur d4 l /u*4rci4 lir4 d4l 8!u4.

.2 7resiòn d e a i re

.'

 A"ua

.(

t a n 6 u e p r e s u r i% a d o 6 u e p r o d u c e c  o r r o v e r tic a l

Para @acer 7#e #n c@orro a4cance 5a9or a4t#ra (c66 /4 $c4 c6! l"u!/ u4!84/ d4 6r!86@ *6r 4;4*l6%@ /4 *u4d4 d4/rr6llr u! 6r *r4/i! *6r 4!ci d4l :uid6 4! 4l r4ci*i4!84@ 6 /4 *u4d4 u8ili9r u! 6 *r 684!4r u! 6r *r4/i!. E;4*l6 ilu/8r8i6.- U8ili9!d6 u! /i/84 *r4cid6 l u4 /4 u4/8r 4! l "ur@ clcul4 l *r4/i! d4l ir4 r4u4rid *6r 4!ci d4 "u *r $c4r u4 4l c$6rr6 /u 52 *i4/ d4/d4 l 6uill. L *r6u!didd@ $ 4/ d4 1 *i4/. S6luci!.- Pri4r6 /4 u8ili9 l Ec. d4 B4r!6ulli *r 684!4r u! 4,*r4/i! *r l 4l6cidd d4 :u;6 4! l 6uill c66 u!ci! d4 l *r4/i! d4l ir4=

p1 γ

2

+ % 1+

v1 2g

=

p2 γ

2

+ % 2+

v2 2g

0 Au /4 4rá u4  & )   u4

*< ) . R4/6li4!d6 *r < /4 68i4!4=

√ [( ) ( ) ]

v 2= 2 g

p 1 γ

+ % 1− % 2

0 c66 !84/@ $ci4!d6 $ ) (9& W9<%@ 84!46/=

CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.% v 2=

√ [( ) ] 2g

p 1 γ

+h

0 4/8 4cuci! 4/ *r4cid l 846r4 d4 T6rric4lli@

8i! !84ri6r4!84 /4 8u6 v =√ 2 g h 0 4l c$6rr6 /4 4l4rá  u! l8ur $. *6r !l6"@ 4l /i/84 *r4/uri9d6 $rá u4 4l c$6rr6 /4

( )+ p 1

4l44  u! l8ur=

γ

h

@ 4! 4/84 4;4rcici6@ /i d4/46/ u!

l8ur d4 52 *i4/  $ ) 1 *i4/= p1 γ

(

p1= γ ( 37 pies ) = 62.4

lb pie

3

)

(37 pies )

1 pie

2

144 pulg

2

= 45 pies −8 pies =37 pies 0

=16.03 lb 2 +elat . pulg

P6r c6!/i"ui4!84 l c49 868l *6r 4!ci d4 l 6uill 4/

( ) p 1 γ

+h

.

C! FLUJO DEBIDO A UNA CABEA EN DESCENSO A!84ri6r4!84 /4 d46/8r u4 l 4l6cidd  l r*id49 d4 :u;6 d4 6lu4!@ u4 /4 68i4!4 d4 u! 6rici6 4! u! 8!u4@ di/i!u4 d4 !4r !6 li!4l c6!6r4 4l :uid6 :u4 d4l 8!u4  di/i!u4 l *r6u!didd d4l :uid6. E! 4/84 c/6@ /4 d4/rr6llrá 4l 86d6 *r clculr 4l 8i4*6 r4u4rid6 *r cir u! 8!u40 4! 4l 4;4*l6 /4 u4/8r u! 8!u4 u4 *6/44 u! 6uill li/  r4d6!d 4! 4l 6!d6@  8r/ d4l cul /4 d4/cr" 4l :uid6. d



'



 t



(

 j vj F l ujo 6 u e s a le d e u n ta n 6 u e c o n u n a c a b e % a de descenso

Pr u! *r6u!didd d4 :uid6 dd ($%@ 4l 846r4 d4 T6rric4lli dic4 u4 l 4l6cidd d4 :u;6 4! 4l c$6rr6 4/= v ,=√ 2 gh . L r*id49 d4 :u;6 d4 6lu4! u4 */ *6r l 6uill 4/ Q = A , v , ('/% [ (*i4/ '/%. E! u! *4u4 c!8idd d4 8i4*6 d8@ 4l 6lu4! d4l :uid6 u4 */ *6r l 6uill 6 /l4 4/ - , =Q ( t )= A , v , (t ) .


>i4!8r/ 8!86@ d4id6  u4 4l :uid6 4/8 /li4!d6 d4l 8!u4@ 4l !i4l d4 :uid6 di/i!u4@ dur!84 4l *4u46 i!cr44!86 d8 u! *r6u!didd 6 di/8!ci d$0 4!86!c4/ 4l 6lu4! u4 ; d4l 8!u4 4/ 6lu4! r46id6 - t =− A t h 0 4/86/ d6/ 6l4!4/ d44! /4r i"ul4/= A , v , ( t ) =− A t h 0 r4/6li4!d6 *r 4l 8i4*6 d8 /4 8i4!4=

( )

− t =

A t

A ,

v ,

( )

− h

0 c66

v , =√ 2 gh→t =

A t A ,

√ 2 gh

( )

− h=

A t

A ,

√ 2 g

− 1/ 2

h

. El 8i4*6

h

r4u4rid6 *r u4 4l !i4l d4l :uid6 d4/ci4!d d4 u! *r6u!didd $& t 2

 68r $< /4 i!84"rrá=

( )∫

−

∫ t = t 1

A t

A ,

√ 2 g

( ) A t

−

h2

h

−1 / 2

h→t 2− t 1=

h1

A ,

√ 2 g

( ) − = ( 2

i!ir8i4!d6 l6/ d6/ 8ri!6/ d4 ($%  6*4r!d6=

[

t 2 t 1

1/2

1 /2

h 2 − h1 1 /2

]

A t A ,

√ 2 g

1/ 2

h1

−h12 /2 )

\\\(&% E/8 4cuci! /4 *u4d4 u8ili9r *r clculr 4l 8i4*6 r4u4rid6 *r cir u! 8!u4 d4/d4 $& $/8 $<. E=e524o.- Pr 4l 8!u4 u4 /4 u4/8r 4! l "ur d4 rri@ 4!cu4!8r4 4l 8i4*6 r4u4rid6 *r cirl6 d4/d4 u! !i4l <   .2 . l6/ diá48r6/ d4l 8!u4 &.5   l 6uill  . S6luci!.- Pr u8ili9r l 4cuci! (&% l/ ár4/ /6!= π π −3 2 2 2 2 A t = (1.40 m ) =1.539 m → A , = ( 0.03 m ) = 0.70686 x 10 m 0 4l c6ci4!84 d4 4

4

At

l/ ár4/=

A ,

=

2

1.539 m

−3

0.70686 x 10

2

m

=2177.24 /u/8i8u4!d6 4! l rul

(&%@ /4 8i4!4=



 o =  i 3 $ e tr o de l orificio

j =  i3 $ e tr o e n la v e n a c o n t r a c ta

F l u j o a t ra v  s d e u n o r i f ic i o con bordes a"udos

t 2 −t 1=

(

2 2177.24

)

(( 2 m) / − ( 0.50 m) / ) =695.14 s 1 2

√ 2 ( 9.81 m / s ) 2

1 2

0

E/86

4uil4  &&.212 i!. C.7.& cid6 d4 u! 8!u4 *r4/uri9d6.- Si 4l 8!u4 d4 l "ur d4 rri@ 4/8á /4lld6 c6! u! *r4/i! *6r 4!ci d4 u! :uid6@ /4 d44 p ( )  l *r6u!didd r4l d4l :uid6 "r4"r l c49 *i496]8ric γ


!84/ d4 ll4r  c6 l6/ cálcul6/ u4 /4 d44! r4li9r 4! L 4cuci! (&%. C.7.< E4c86 d4l 8i*6 d4 6uill.- Si l 6uill 8i4!4 u! 6r á/ "ud@ 4l diá48r6 !i6 d4l c$6rr6 4/ /i"!ic8i4!84 4!6r u4 4l diá48r6 d4 l 4r8ur. El ár4 d4cud u4 /4 d44 u8ili9r A ; 4! l 4cuci! (&% 4/ l u4 8i4!4 4l diá48r6 á/ *4u46. E/84 *u!86 c6!6cid6 c66 l 4! c6!8rc8@ /4 *r4/4!8 li"4r4!84 u4r d4l 6rici6- E!86!c4/ *r6,id4!84 A ; ) .+< A6

ejerci f1

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