CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
EJERCICIOS DE APLICACIÒN FLUJO DE FLUÌDOS Y LA ECUACIÒN DE BERNOULLI A) Factores de Conversin! Conversin! & L'i! ) & L* ) &+.+7,&-+ '/0 &. '/ ) + L'i! 'i!00 & "l' l'i! ) "* "* ) .712 .712 L'0 L'0 -2 &"l'i! ) +.3 , & '/0 &. *i4 '/ ) 553 "l'i!. "!- C6!i4r8 u! r*id49 d4 :u;6 d4 < "l'i! *i4/'/=
( ) ( ) ( ) 1 pie
gal Q =20 min
3
s gal 449 min
−3 pie
=44.54 x 10
3
.
s
<.- C6!i4r8 u! r*id49 d4 :u;6 d4 52 L'i! = 3
1m
Q = 4500
..-
L s min 60000 L min
=0.075 m3 / s .
C6! C6 !i4 i4rr8 u! r*id *id49 49 d4 :u;6 :u;6 d4 523 523 "l' "l'i i! ! L'i! 'i! = 3.785 L
gal min
Q = 459
min 1 gal min
=1,737.3 L / min .
5.- U! 4!8ild6r *r6duc4 +5 *i4/ 'i! (PC>% d4 ir4. Si l d4!/idd d4l ir4 4/ d4 &.< ?"'@ clcul4 l r*id49 d4 :u;6 d4 / 4! /lu"/'/ l r*id49 d4 :u;6 d4 *4/6 4! l'$. 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#acin de Contin#idad. E! u! /i/84 c4rrd6 d4 c6!ducci! d4 :u;6@ 4l clcul6 d4 l 4l6cidd d4l :ud6 d4*4!d4 d4l *ri!ci*i6 d4
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
c6!8i!uidd. E! l Fi". 4l :ud6 :u4 d4 l /4cci! & l /4cci! < c6! r*id49 c6!/8!84. L c!8idd d4 :uid6 u4 */ *6r cului4r /4cci! 4! u! ci4r86 8i4*6 dd6 4/ c6!/8!84 (/4 l4 ll G:u;6 c6!/8!84%. E! 8ri!6/ d4 l r*id49 d4 / c66= > & ) ><0 8i! >)ρA0 /4 8i4!4= ρ& A& & ) ρ< A< < (Ec. d4 c6!8i!uidd% 4/ álid6 *r 86d6 :uid6 ("/4/@ luid6/%. Si 4l luid6 4/ i!c6*r4/il4@ l6/ 8ri!6/ ρ& ) ρ<0 4!86!c4/ u4d l 4cuci! d4 c6!8i!uidd A& & ) A< <@ u c66 K ) A lu4"6 $" % $& /4 c6!clu4 l Ec. d4 c6!8i!uidd *r u! :u;6 4/8l4 (luid6/%@ l r*id49 d4 :u;6 d 6lu4! 4/ l i/ 4! cului4r /4cci!. Ti! *u4d4 u8ili9r/4 c6! 4rr6r *4u46 *r "/4/ ;/ 4l6cidd4/ 4!6r4/ & '/. &.- El "u :u4 &.< '/ 4! u! c6!duc86 d4 &2 d4 diá48r6. Clcul4 l 4l6cidd d4 :u;6 4! u! c6!duc86 d4 d4 diá48r6 u4 4/8á c6!4c8d6 c6! 4l !84ri6r0 clculr l r*id49 d4 d4 :u;6 d4 6lu4!.- S6luci!.P6r c6!8i!uidd 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 4l6cidd <@ /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
6lu4!=
3
s
.
&! E! u! /4cci! d4 u! /i/84 d4 di/8riuci! d4 ir4@ 4l c6!duc86 8i4!4 u! ár4 d4 & *ul"d/ cudrd/@ 4l ir4@ &5.7 l'*ul"< & MF 8i4!4 u! d4!/idd d4 <.< , & - /lu"/'*i4 *4/6 4/*4cc6 d4 7.3 , &-< l'*i4@ 8i4!4 u! 4l6cidd *r64di6 d4 && *i4/'i!. E! 68r /4cci! d4l i/6 /i/84@ 4l c6!duc86 4/ r4d6!d6 8i4!4 u! diá48r6 d4 &+ *ul"@ l 4l6cidd /4 id4 1 *i4/'i!. Clcul4 l d4!/idd d4l ir4 4! l /4cci! r4d6!d l r*id49 d4 :u;6 d4 *4/6 d4l ir4 4! lir/ *6r $6r. S6luci!.- D4 cu4rd6 c6! l Ec. d4 c6!8i!uidd *r l6/ "/4/@ /4 8i4!4= ρ1 A 1 v 1= ρ2 A 2 v 2 0 / *u4/@ *6d46/ clculr 4l ár4 d4 l/ d6/ /4cci6!4/ r4/6l4r *r A 2=
ρ 2= ρ1
ρ<=
( )( ) 0 A 1
v1
A2
v2
A 1=( 10 pulg )
2
=100 pulg 2 0
π
( D ) 2= π ( 16 pulg )2=201.06 pulg 2 . E!86!c4/@ l d4!/idd d4l ir4 4! 4 2 4
l /4cci! r4d6!d 4/=
(
−3 slugs
ρ2= 2.20 x 10
pie
3
)(
2
100 pulg
2
201.06 pulg
)(
1100 pie / min 800 pie / min
)
=1.504 x 10−3
slugs pie
3
.
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
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
.
'.- D484ri!4 l r*id49 d4 :u;6 á,i *4ri8id@ 4! L'i!@ u4 *u4d4 *r6ducir/4 4! u! 8u6 d4 c4r6 4/8á!dr u4 8i4!4 u! diá48r6 4,84r!6 d4 &.2 *ul" .72 *ul" d4 "r6/6r 4! l/ *r4d4/@ /i l 4l6cidd á,i d44 /4r . '/. S6luci!= U8ili9!d6 l d4!ici! d4 r*id49 d4 :u;6 d4 6lu4!@ /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! 6lu4!=
(
−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
.
(.- D484ri!4 4l 86 r4u4rid6 d4 c6!duc86 d4 c4r6 Clir4 5 4/8á!dr u4 ll44 & L'i! d4 "u c6! u! 4l6cidd á,i d4 2.2 '/. S6luci!.- C66 K /6! c6!6cid6/@ 4l ár4 r4u4rid6 /4 *u4d4 $llr d4= K ) A0 A)K'. Pri4r6 d446/ c6!4r8ir l r*id49 d4 :u;6 d4 6lu4! u!idd4/ d4 '/.
( ) 3
L Q =3100 min
m 1 s
60000
L min
m =0.0516 s
3
0
4!86!c4/
4l
ár4=
3
Q 0.0516 m / s = 9.394 x 10−3 m2 . A = = v 5.50 m / s
E/86 /4 d44 i!84r*r48r c66 4l
ár4 !i di/*6!il4@ *u4/86 u4 cului4r ár4 4!6r *r6ducir u! 4l6cidd 6r u4 2.2 '/. P6r c6!/i"ui4!84@ d446/ u/cr 4! 4l *!dic4 F d4l lir6 u! c6!duc86 4/8á!dr c6! u! ár4 d4 :u;6 ;u/86 6r u4 3.35 , &- < . S4 r4ui4r4 u! c6!duc86 d4 c4r6 Clir4 5 4/8á!dr d4 2 *ul"@ c6! ár4 d4 :u;6 d4 &.<3& , &-< <. L 4l6cidd r4l d4 :u;6 cu!d6 4/84 c6!duc86 ll4 .2 '/ d4 3 0.0533 m / s "u 4/= )K'A ¿ 1.291 x 10−3 m2 = 4.13 m / s 0 /i /4 u8ili9r 4l c6!duc86 /i"ui4!84 / *4u46 (u! c6!duc86 clir4 5 d4 5 *ul"%@ l
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.% 3
4l6cidd *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!84rcid6r d4 cl6r u4 /4 u8ili9 *r 8r!/4rir cl6r d4/d4 4l :uid6 u4 :u4 d4!8r6 d4l 8u6 i!84ri6r l u4 :u4 4! 4l 4/*ci6 c6*r4!did6 4!8r4 l *r84 4,84ri6r d4l 8u6 l *r84 i!84ri6r d4l c/c6 cudrd6 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 8u6. Clcul4 l r*id49 d4 :u;6 d4 6lu4! 4! "l'i! u4 *r6ducir u! 4l6cidd d4 3 *i4/'/@ 8!86 d4!8r6 d4l 8u6 c66 4! 4l c/c6. S6luci!.- U8ili96/ l rul *r l r*id49 d4 :u;6 d4 6lu4!@ K ) A *r cd *r84= % D4!8r6 d4l 8u6 d4 c6r4 8i*6 ? d4 *ul" d4 86 !6i!l= D4l *!dic4 /4 68i4!4= ED).+<2 *ul"0 Diá48r6 i!84ri6r ID ) .2<7 *ul"0 "ru4/6 d4 l *r4d ) .53 *ul"@ 4l ár4 8uulr= 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
6lu4! d4!8r6 d4l 8u6 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 di4r4!ci 4!8r4 4l ár4 d4!8r6 d4l c/c6 cudrd6 l *r84 4,84r! d4l 8u6. 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 r4u4rid 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
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
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 8u6 4/=
gal min
0 El c6ci4!84
Coiente=
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 cr4r !i d4/8ruir@ /i!6 u4 *u4d4 /4r 8r!/6rd d4 u! 8i*6 68r6@ /8 4/ l l4 d4 c6!/4rci! d4 l 4!4r". Cu!d6 /4 !li9 c6!duc86/ c6! :uid6/@ 4,i/84! 8r4/ 6r/ d4 4!4r"@ c66 4;4*l6 4! l Fi". E/8á l6cli9d u! ci4r8 4l4ci! (9%@ 8i4!4 u! 4l6cidd (% u! *r4/i! (*%. L/ rul/ d4 4!4r" /6!= &. E!4r" *684!cil (PE%.- E/ d4id l 4l4ci! 4/ c6! r4/*4c86 u! l!4 d4 r44r4!ci= PE % -!.0 d6!d4 Q 4/ 4l *4/6 d4l 4l44!86. <. E!4r" ci!8ic (E%.- E/ d4id l 4l6cidd@ d4l 4l44!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!8idd d4 8r;6 !4c4/ri6 *r 64r l 4l44!86 d4 :ud6 8r/ d4 u! ci4r8 /4cci! 4! c6!8r l *r4/i!= FE % -!20 Ele$ento del L 4cuci! d4 4!4r" FE@ /4 *u4d4 d4ducir 4! fluido & 6r d4 8r;6 ) *AL ) *0 d6!d4 l u4r9 4/ *A@ pA l *r4/i! 4/ * A 4/ 4l ár4 d4 l /4cci!@ /4 8r/ld 4l 4l44!86 u! di/8!ci L d4!8r6 d4 u! 6lu4! @ 4! 6r d4 *4/6 d4l 4l44!86 Q ) γ 4! l u4 γ 4/ 4l *4/6 4/*4cc6 d4l :uid6@ 4!86!c4/ 4l 6lu4! 4/ ) Q' γ /u/8i8u4!d6 4! 4l 8r;6 ) * Q' γ ) FE 4/ l 4!4r" d4 :u;6.
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$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!8idd 868l d4 4!4r" d4 4/8/ 8r4/ 6r/ u4 *6/44 4l 4l44!86 d4 :uid6 /4rá l /u. ℘ 2 E= !E + "E + #E = + $% + $ v / 2 g 0 cd u!6 d4 4/86/ 8ri!6/ /4 γ
4,*r4/ 4! u!idd4/ d4 4!4r"@ !486!-48r6 (N.% 4! 4l /i/84 i!84r!ci6!l 6 4! *i4/-lir (*i4.l% 4! 4l /i/84 d4 u!idd4/ ri8á!ic/. - C6!/id4r!d6 $6r 4l 4l44!86 d4 :uid6 u4 /4 u4/8r 4! l "ur@ u4 /4 u44 d4 l /4cci!(&% l /4cci! (<%. L6/ l6r4/ d4 *@ 9 /6! di4r4!84/ 4! l/ d6/ /4cci6!4/. E! l /4cci! (&% l 4!4r" 868l 4/= E2=
4/=
$ p2 γ
E1=
$ p1 γ
2
+ $ % 1+
$ v1 2g
. E! l /4cci! (<% l 4!4r" 868l
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!/4rci! d4 l 4!4r" r4ui4r4= E1= E 2 →
$ p 1 γ
2
+$ % 1 +
$ v1 2g
=
$ p2 γ
2
+ $ %2 +
$ v2 2g
El *4/6 d4l 4l44!86 4/ c6! 86d6 l6/ 8ri!6/ /4 l4 *u4d4 c!c4lr@ l 4cuci!@ 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 c66 4cuci! 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!" Inter2retacin de 4a ec#acin de Berno#44i!, A cd 8ri!6 d4 l Ec. d4 B4r!6ulli@ r4/ul8 l
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
4!4r" *6/4d *6r 4l :uid6 *6r u!idd d4 *4/6 d4 :uid6 u4 :u4 4! 4l /i/84. L/ u!idd4/ d4 l6/ 8ri!6/ /4 d! 4! !486!-48r6 *6r !486! (N.'N% 4! 4l /i/84 SI 4! 4l ri8á!ic6 (l.*i4'l%. E/ d4 !68r u4 4l N@ l /4 *u4d4! c!c4lr@ d4;!d6 /6l4!84 u! u!idd d4 l6!"i8ud@ 4l 48r6 (% 6 4l *i4. P6r 8!86@ l6/ 8ri!6/ /4 l4/ c6!6c4 c66 Gc49/= -El 8ri!6= *'γ /4 l4 c6!6c4 c66 c49 d4 *r4/i!. -El 8ri!6= G9 /4 l4 ll c49 d4 4l4ci!. - El 8ri!6= <'<" /4 l4 c6!6c4 c66 c49 d4 4l6cidd. L /u d4 l6/ 8r4/ 8ri!6/ /4 l4/ c6!6c4 c66 Gc49 868l. Cd 84ri! c6rr4/*6!d4 u! l8ur c6! r4/*4c86 u! l!4 d4 r44r4!ci@ /4 c6!/8ru4 u! Di"r c66 /4 6/4r 4! l "ur@ 4/ 8il *r i/uli9r l r4lci! 4!8r4 l6/ 8r4/ 8i*6/ d4 4!4r"@ l c49 868l *4r!4c4 c6!/8!84. E!86!c4/@ l l8ur r4l8i d4 cd 8ri!6 d4 c49 r /4"! l6 4/8l4cid6 *6r l 4cuci! d4 B4r!6ulli. E! l "ur@ l c49 d4 4l6cidd 4! l /4cci! < 4/ 4!6r u4 4! 4l d4 l /4cci! &@ 4/86 /4 d4u4/8r c6! l Ec. d4 c6!8i!uidd= 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
d44 /4r 4!6r
. Y c66 l 4l6cidd 4/8á l cudrd6 4! 4l 8ri!6 2
c6rr4/*6!di4!84 l c49 d4 4l6cidd@
v2 2g
4/ uc$6 4!6r u4
2
v1 2g
.
Ti!@ cu!d6 4l 86 d4 l /4cci! /4 4,*!d4 c66 /4 4 4! l "ur@ l c49 d4 *r4/i! u4!8 d4id6 u4 di/i!u4 l c49 d4 4l6cidd. E! /u@ l Ec. d4 B4r!6ulli 4,*lic 4l ci6 4! l/ c49/ 4!8r4 d6/ *u!86/ d4 u! /i/84 d4 :u;6 d4 :uid6. S4 /u*6!4 8i!@ u4 !6 $ *rdid/ !i "!!ci/ d4 4!4r" *6r l6 u4 l c49 868l *4r!4c4 c6!/8!84. L/ *r4/i6!4/ 4/ c6!4!i4!84 4,*r4/rl6 c66 !68ric. C!& Restricciones a 4a Ec#acin de Berno#44i!, L 4cuci! d4 B4r!6ull /4 *lic uc$6/ *r6l4/@ *4r6 4,i/84! l"u!/ r4/8ricci6!4/ c66= &. Su lid49 4/ /6l6 *r :uid6/ i!c6*r4/il4/ u4 /u *4/6 4/*4cc6 !6 r. <. N6 d44 c6!/id4rr di/*6/i8i6/ 4cá!ic6/ 4! / /4cci6!4/ u4 !6 d44 dir/4 !i ui8r/4 4!4r"@ l Ec. 4/8l4c4 4!4r" 868l c6!/8!84. . N6 *u4d4 $4r 8r!/4r4!ci d4 cl6r d4!8r6 !i u4r d4l :ud6. 5. N6 *u4d4 $4r *rdid d4 4!4r" *6r ricci!. >/ d4l!84 4/8/ r4/8ricci6!4/ /4 c6!/id4rrá! 4! l 4cuci! "4!4rl d4 4!4r". C!' Procedi5iento de a24icacin de 4a ec#acin de Berno#44i!
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
&. D484ri!r u4 4l44!86/ /6! c6!6cid6/ u4 /4 4!c6!8rr. <. D4cidir cuál d4 l/ d6/ /4cci6!4/ d4l /i/84 /4 u8ili9rá! cu!d6 /4 4/cri l Ec. d4 B4r!6ulli. S4 4/c6"4 u! /4cci! d4 l cul /4 c6!6c4! uc$6/ d86/. L /4"u!d 4/@ *6r l6 "4!4rl@ l /4cci! 4! l cul /4 d44 clculr l"6. . E/criir l Ec. d4 B4r!6ulli *r d6/ /4cci6!4/ 4l4"id/ 4! 4l /i/84. E/ r4c64!dl4 u4 l dir4cci! d4l :u;6 4/ d4 i9ui4rd d4r4c$. 5. Si*licr 4! l6 *6/il4 c6! l c!c4lci! d4 l6/ 8ri!6/ cu6 l6r 4/ c4r6 6 d4 u4ll6/ u4 /6! l6/ i/6/ 4! 6/ ld6/ d4 l 4cuci!2. R4/6l4r l 4cuci! l"4ric4!84 *r 4l 8ri!6 d4/4d6. ) 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!8idd4/ c6!6cid/ clculr 4l r4/ul8d6@ 84!4r cuidd6 4! 4l u/6 d4 u!idd4/ c6!/i/84!84/ l6 lr"6 d4l clcul6. EJE>PLO DEL PROCEDI>IENTO (CONSIDERAR LA FIVURA DE C.&% &.E! l "ur *r84 d4 u! 8u4r 4l "u & MC :u4 d4 l /4cci! & l /4cci! <. E! l /4cci! & 4l diá48r6 4/ < l *r4/i! !68ric 4/ ?P l 4l6cidd d4 :u;6 4/ d4 <.1 '/. L /4cci! <@ 8i4!4 5 d4 diá48r6@ 4/8á &.1 /6r4 l /4cci! &. Su*6!i4!d6 u4 !6 $ *rdid/ d4 4!4r" 4! 4l /i/84@ clcul4 l *r4/i! *<. S6luci!.- Li/8d6 d4 d86/= D &)< 0 D<)5 0 &) <.1 '/0 *&) ?P0 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 $llrá *< <0 c6! l Ec. d4 B4r!6ulli= p1 γ
2
+ % 1+
(
p2= γ
v1 2g
p 1 γ
=
p2 γ
2
+ % 2+
v2
2
+ % 1 +
(
v1 2g
p2= p1 + γ % 1− % 2+
2g 2
− % 2− 2
v2 2g
2
v 1−v 2 2g
)0
p2 γ
=
p 1 γ
2
+ % 1 +
v1 2g
2
− % 2−
v2 2g
"ru*!d6
c49/=
) . O8r6/ d86/= " ) 3.1& '/ 0 d4l "u <
γ
) 3.1&
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
?N'. S4 d484ri!rá < d4 l Ec. d4 c6!8i!uidd= 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/8i8u4 l6/ l6r4/ 684!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/ !68ric@ /4 d4/rr6ll 4! u!ci! *& u4 4/ !68ric 8i!.
C!( 6an7#es8 reci2ientes 9 :o7#i44as e;2#estas a 4a at5s
.
) '.0 $
. .1
A '.50 $
.
) o n d u c to 2 0 $ $ d e d i 3 $ e t r o in t e r n o
Flujo
'.0 $
.
E
F
(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! 6uill c6l6cd l !l d4l c6!duc86. E/ d4 6/4rr@ u4 l /u*4rci4 lir4 d4l 8!u4 (*u!86 A% l d4/cr" 4! l 6uill (*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/ c49/ d4 *r4/i! /4rá! c4r6/ /4 *u4d4! c!c4lr. -O8r 6/4rci!@ 4! l /u*4rci4 lir4 d4l 8!u4 /4 /u*6!4 u4 4/8á 8r!uil6 4/ u "r!d4 c6! r4/*4c86 l /u*4rci4 d4l
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
c6!duc86 c4rrd6@ *6r l6 u4 l 4l6cidd /4rá u *4u4 A ) @ d44 4lii!r/4 /u c49 d4 4l6cidd. -Cu!d6 6/ *u!86/ d4 r44r4!ci 4/8á! 4! 4l i/6 c6!duc86.Si"ui4!d6 4l !áli/i/ 4! l "ur l6/ *u!86/= B@ C@ D E /4 4!cu4!8r! d4!8r6 d4l c6!duc86 *6/44! u! ár4 d4 :u;6 u!i6r4 4l :u;6 4/ 4/8l4@ *6r l6 u4 l 4l6cidd /4rá l i/ l6 lr"6 d4l c6!duc86= B)C)D)E. P6r l6 u4@ /u/ c49/ d4 4l6cidd 4! 6/ ld6/ d4 l 4cuci! /4r! i"ul4/ /4 c!c4lr!. -Cu!d6 l 4l4ci! 4/ l i/ 4! 6/ *u!86/ d4 r44r4!ci.- Si l 4l4ci! 4! l6/ d6/ *u!86/ d4 r44r4!ci u/d6/ 4! l Ec. d4 B4r!6ulli 4/ l i/@ 4/86/ 8ri!6/ /4 c!c4l! (9 A@ 9B@ 9D%. L/ 5 6/4rci6!4/ !li9d/ cili8rá! l/ /i*licci6!4/ 4! l /6luci! d4 4;4rcici6/ /iilr4/ l 4;4*l6 c6!/id4rd6.
". E=e524o.- E! l "ur d4 rri 4l /i! /c "u d4l 4/8!u4@ c6! l/ i// di4!/i6!4/ u4 /4 d!. Clcul4 l r*id49 d4 :u;6 d4 6lu4! 8r/ d4l /i! l *r4/i! 4! l6/ *u!86/ B@ C@ D E. S6luci!.- &.- cálcul6 d4 l r*id49 d4 :u;6 d4 6lu4! K@ u8ili9!d6 l Ec. B4r!6ulli@ 4! 4/84 c/6 l6/ *u!86/ 4/c6"id6/ /6! A F. C66 /4 di;6 !84/ 4! C.50 /4 8i4!4!@ 4! A= * A)@ A)0 4! F= * F)@ 9F ) <.2 p A
d4;6 d4 A.
γ
2
+ % A +
v A 2g
=
p ! γ
2
+ % ! +
v ! 2g
0 4! 4/8 Ec@ 4lii!!d6 l/ 2
% A= % ! +
c49/ d4 * u4d= 9.81 m / s
v ! 2g
→ v ! =√ ( % A− % ! ) 2 g
0 d6!d4
2
¿
( 2.5 m ) 2 ¿ 0 % A− % ! =2.5 −0 =2.5 m→ v ! = √ ¿
%C66 K ) A.
K ) A F.F0 ár4 d4 l 6uill A F ) ( π'5%(< %<)
&5.&+ @ c6!ir8i4!d6 <
−4
Q=3.1416 x 10
( )=
m m 7 s 2
A ! =314.16 mm −3 m
2.2 x 10
2
(
1m 6
10
2
)
=3.1416 x 10−4 m2 0 mm 2
3
s
% Cálcul6 d4 l *r4/i! *B0 /4 4/c6"4rá l6/ *u!86/ A B= p A γ
2
+ % A +
v A 2g
=
rr4"l!d6=
p ' γ
2
+ % ' +
[
v ' 2g
4! 4/8 Ec. *A)@ A)0
p' =γ ( % A − % ' )−
2
v ' 2g
]
% A=
p ' γ
2
+ % ' +
v ' 2g
4! 4/8 Ec. ( % A− % ' )= 0 0 u4 l6/
*u!86/ A B 4/8á! l i/6 !i4l. P4r6 !84/ /4 clculrá l 4l6cidd B d4 l Ec.d4 c6!8i!uidd= K)A B.B B ) K'AB0 /4 c6!6c4
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
K *4r6 !6 AB /4 clculrá d4l c6!duc86 A B ) (π'5%( %<) 7+.1+ @
2
A ' =706.86 mm
c6!ir8i4!d6
<
−3 m
2.2 x 10
4l6cidd
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
c49
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/8i8u4!d6 d86/ $lld6/ 4!
2
[ ]
p' =γ
−v 2'
=9.81 k& 3 ( −0.494 m )=−4.846 k& 2 =− 4.846 k"a 0 4l /i"!6 !4"8i6 2g m m
i!dic u! *r4/i! !68ric *6r d4;6 d4 l *r4/i! 86/ric@ d4á/ 4! :uid6/ 4! 6ii4!86 p A ( p ' @ 4l c6!c4*86 d4 u4 *u!86/ 4! 4l i/6 !i4l 8i4!4! l i/ *r4/i! !6 /4 *lic c66 c6! l6/ :uid6/ 4! r4*6/6. c% Cálcul6 d4 l *r4/i! 4! *C0 /iilr4!84 /4 4/c6"4rá l6/ *u!86/ A C=
p A γ
2
+ % A +
v A 2g
0 rr4"l!d6=
=
pC γ
2
+ % C +
v C
0 4! 4/8 Ec. * A)@ A)0
2g
[
2
pC = γ ( % A − %C ) −
v C 2g
]
γ
2
+ % C +
v C 2g
( % A− % C ) =−1 m 0 (!4"8i6@ *6ru4
0
3.1123
9C 9A0 8i! B ) C )
m s
0 ( u4 AC ) AB% 4!
2
c6!/4cu4!ci l/ c49/ /6! i"ul4/ pC =9.81
% A=
p C
v C 2g
2
=
v ' 2g
= 0.494 m l *r4/i! 4!
k& m
k& − − =− =−14.656 k"a m m ( 1 0.494 ) 14.656 3 2 m
d) Cálcul6 d4 l *r4/i! 4! * D0 /iilr4!84 /4 4/c6"4r l6/ *u!86/ A D=
p A γ
2
+ % A +
v A 2g
=
p D γ
2
+ % D +
v D 2g
0 4! 4/8 Ec. * A)@ A). Ti! l
4l4ci! d4 l6/ *u!86/ A@ B D /6! l/ i//@ *6r l6 u4 9 A)9B)9D /4 c!c4lr! 4! l Ec.@ 8i! l/ 4l6cidd4/ 4! B)C)D)E /6! i"ul4/ u4 /4 4!cu4!8r! 4! 4l i/6 ár4 c6!duc86@ 4!86!c4/ 2
D ) B0 lu4"6
v D 2g
2
=
v ' 2g
)
0.494 m
. L Ec. u4d /i*licd=
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$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!clu4 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"ul4/ u4 l 4l4ci! l 4l6cidd /6! i"ul4/. 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
. Ti! l/ 4l6cidd4/ 4! B)C)D)E /6! i"ul4/ 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
l8i6=
2.006 m
=19.678 k"a
RESU>EN DE LOS RESUL6ADOS DEL EJE>PLO RESUEL6O! &.L 4l6cidd d4 :u;6 4! l 6uill l r*id49 d4 :u;6 d4 6lu4! 4! 4l /i!@ d4*4!d4! d4 l di4r4!ci d4 4l4ci! 4!8r4 l /u*4rci4 lir4 d4l :uid6 l /lid d4 l 6uill. <. 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 !i4l u4 A 4/8á 4,*u4/86 l 8/4r. . L 4l6cidd d4 :u;6 4/ l i/ 4! 86d6/ l6/ *u!86/ 4! d6!d4 4l 86 d4l c6!duc86 4/ 4l i/6@ cu!d6 4,i/84 u! :u;6 4/8l4. 5. L *r4/i! 4! 4l *u!86 C 4/ l / ; d4l /i/84@ d4id6 u4 /84 /4 uic 4! 4l /i8i6 á/ 4l4d6. 2. L *r4/i! 4! 4l *u!86 D 4/ l i/ u4 l d4l *u!86 B@ d4id6 u4 6/ 4/8á! l i/ l8ur l c49 d4 4l6cidd 4! 6/ *u!86/ 4/ l i/. +. L *r4/i! 4! 4l *u!86 E 4/ l 6r d4l /i/84 d4id6 u4 4/84 *u!86 /4 4!cu4!8r 4! 4l /i8i6 á/ ;6.
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
'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/i8i6 c6!6cid6 c66 4did6r 4!8uri@ u4 /4 u8ili9 *r 4dir 4l6cidd 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! diá48r6 r4ducid6 6c/i6! u4 l 4l6cidd u4!84 di/i!u l *r4/i!. S4 $ d4 d46/8rr u4 l 4l6cidd d4 :u;6 d4*4!d4 d4 l di4r4!ci d4 *r4/i! 4!8r4 l6/ *u!86/ A B@ *6r l6 u4 /4 u8ili9 u! !48r6 di4r4!cil 4!8r4 6/ *u!86/. E;4*l6 Ilu/8r8i6= ".El 4did6r 4!8uri 6/8rd6 4! l "ur ll4 "u 22 MC. L "r4dd 4/*4cc d4l :uid6 !68ric6 ("lic4ri! /")&.<+%. Clcul4 l 4l6cidd d4 :u;6 4! l /4cci! A l r*id49 d4 :u;6 d4 6lu4! d4 "u. S6luci!.- D86/= /4 c6!6c4 l di4r4!ci d4 4l4ci! 4!8r4 l6/ *u!86/ A B. C6! 4l !48r6 /4 68i4!4 l di4r4!ci d4 *r4/i! 4!8r4 l6/ *u!86/ A B0 8i! /4 c6!6c4 l6/ diá48r6/ d4 l/ /4cci6!4/ A B. L 4l6cidd !6 /4 c6!6c4 4! !i!"! *u!86 d4l /i/84 /4 *id4 4! A . E/ 6i6 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/*4cc6 d4l "u 22 MC 4/ 3.+7
?N'. El rr4"l6 4! c49/=
p A − p ' γ
2
+ ( % A − % ' ) =
2
v ' −v A 2g
0 l di4r4!ci
4! 4l4ci! 4/ % A− % ' =−0.42 m 0 4l l6r 4/ !4"8i6 *6ru4 4l *u!86 A 4/ / l86 u4 B. El *4/6 4/*4cc6 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 4cuci! 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 !68r 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 ) @ diidi4!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 ld6 i9ui4rd6 d4 l Ec. d4 B4r!6ulli /4 4lud6@ $6r /4 *r6/4"uirá c6! 4l i4r6 d4l ld6 d4r4c$6@ c6! l Ec. d4 c6!8i!uidd= AAA)ABB 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 . P6d46/ $6r 86r 4/84 r4/ul8d6@ l di4r4!ci d4 c49/ d4 4l4ci! l c49 d4 di4r4!ci d4 *r4/i! /u/8i8uirl6/ 4! 2 2 p A − p ' v −v + ( % A − % ' ) = ' A . A/= .+31 −0.42 m ) .<71 ) 2g γ 2
6.711 v A 2g
v A =
√
0
r4/6li4!d6
2 g ( 0.278 m ) 6.711
=
√
(
2 9.81
m s
2
)(
v A
*r
/4
0.278 m )
6.711
=0.9015 m / s
L r*id49 d4 :u;6 d4 6lu4! 4/= π ( 0.25 m ) ( 0.9015 m / s )=0 . 04425 m / s % ((!& L2s Q = A A v A = 4
(
2
)
C!? 6EORE>A DE 6ORRICELLI
3
68i4!4=
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
.' (
.
F l u jo d e u n ta n 6 u e
E! l "ur@ 4l :uid6 4/8á :u4!d6 d4l ld6 d4 u! 8!u4 *6r u! 6uill li/ r4d6!d. Pr d484ri!r l 4l6cidd d4 :u;6 u4 /4 68i4!4 4! l 6uill@ /4 4/criirá l 4cuci! d4 B4r!6ulli 4!8r4 u! *u!86 d4 r44r4!ci 4! l /u*4rci4 d4l :uid6 u! *u!86 4! 4l c$6rr6 u4 /4 68i4!4 d4 l 6uill= 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 d4á/ l 4l6cidd & *r6,id4!84 4/ c4r6. E!86!c4/ r4/6li4!d6 *r < /4 68i4!4= v 2= √ 2 g ( % 1− % 2) 0 Hci4!d6 $ ) (9 W 9 %@ /4 68i4!4= & < v 2= √ 2 g h
E/8 4cuci! /4 c6!6c4 c66 846r4 d4 T6rric4lli 4! $6!6r E!"4li/8 T6rric4lli@ ui4! l6 d4/rr6ll 4! 4l 6 &+52. E;4*l6 ilu/8r8i6.- Pr 4l 8!u4 d4 l "ur@ l *r6u!didd d4l :uid6 4/ $ ) 5 @ 4l diá48r6 d4l c$6rr6 4! l 6uill 4/ d4 5 . Hllr l 4l6cidd d4 /lid 4! l 6uill 4l cudl. C64!84 c66 rir l 4l6cidd 4l cudl cu!d6 di/i!u4 4l !i4l d4l luid6 l cir/4. . 4l6cidd d4 /lid=
√(
v 2= √ 2 g h= 2 9.81
. Cudl K= Zr4 d4l c$6rr6=
A =
m s
2
)(
4 m)= 8.858 m / s
.
π
( 0.040 m )2=1.256 x 10−3 m2 0 4l cudl 4
/4rá=
(
m Q = A . v = 1.256 x 10 m 8.858 s −3
.'
2
)
m = 0.011 s
3
=11 Lps
.2
A"ua
.( ) o r ro v e r tic a l
c. L 4l6cidd 4l cudl rirá! di/i!u4!d6 d4 l6r 4did u4 /4 d4/cr" 4l :uid6 *6r l 6uill.
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
O8r *licci! i!84r4/!84 d4l 846r4 d4 T6rric4lli /4 u4/8r 4! l "ur /i"ui4!84@ 4! l u4 4l c$6rr6 d4 luid6 4/ l!9d6 $ci rri. El c$6rr6 lc!9 u! l8ur i"ul l 4l4ci! d4 l /u*4rci4 lir4 d4l :uid6 4! 4l 8!u4. Al !l d4 l l8ur d4 4l4ci! d4l c$6rr6 l p1
4l6cidd 4/ c4r6=
γ
2
+ % 1+
v1 2g
=
p2 γ
2
+ % 2+
v2 2g
0 4/8 /i8uci! 4/ /iilr l
d4/rr6ll6 i!icil d4l 846r4 d4 T6rric4lli= 4/crii4!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$6r < =
0 u *< ) * ) 0 r4/6li4!d6 *r /4 8i4!4=
0 /u/8i8u4!d6 < 9< W 9 ) -$ 0 r4/ul8 *r =
v 3= √ 2 g h + 2 g (−h )
)0 E/84 r4/ul8d6 4ric u4 4l c$6rr6 lc!9 l l8ur d4 l /u*4rci4 lir4 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 (c66 /4 $c4 c6! l"u!/ u4!84/ d4 6r!86@ *6r 4;4*l6%@ /4 *u4d4 d4/rr6llr u! 6r *r4/i! *6r 4!ci d4l :uid6 4! 4l r4ci*i4!84@ 6 /4 *u4d4 u8ili9r u! 6 *r 684!4r u! 6r *r4/i!. E;4*l6 ilu/8r8i6.- U8ili9!d6 u! /i/84 *r4cid6 l u4 /4 u4/8r 4! l "ur@ clcul4 l *r4/i! d4l ir4 r4u4rid *6r 4!ci d4 "u *r $c4r u4 4l c$6rr6 /u 52 *i4/ d4/d4 l 6uill. L *r6u!didd@ $ 4/ d4 1 *i4/. S6luci!.- Pri4r6 /4 u8ili9 l Ec. d4 B4r!6ulli *r 684!4r u! 4,*r4/i! *r l 4l6cidd d4 :u;6 4! l 6uill c66 u!ci! d4 l *r4/i! d4l ir4=
p1 γ
2
+ % 1+
v1 2g
=
p2 γ
2
+ % 2+
v2 2g
0 Au /4 4rá u4 & ) u4
*< ) . R4/6li4!d6 *r < /4 68i4!4=
√ [( ) ( ) ]
v 2= 2 g
p 1 γ
+ % 1− % 2
0 c66 !84/@ $ci4!d6 $ ) (9& W9<%@ 84!46/=
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.% v 2=
√ [( ) ] 2g
p 1 γ
+h
0 4/8 4cuci! 4/ *r4cid l 846r4 d4 T6rric4lli@
8i! !84ri6r4!84 /4 8u6 v =√ 2 g h 0 4l c$6rr6 /4 4l4rá u! l8ur $. *6r !l6"@ 4l /i/84 *r4/uri9d6 $rá u4 4l c$6rr6 /4
( )+ p 1
4l44 u! l8ur=
γ
h
@ 4! 4/84 4;4rcici6@ /i d4/46/ 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 c49 868l *6r 4!ci d4 l 6uill 4/
( ) p 1 γ
+h
.
C! FLUJO DEBIDO A UNA CABEA EN DESCENSO A!84ri6r4!84 /4 d46/8r u4 l 4l6cidd l r*id49 d4 :u;6 d4 6lu4!@ u4 /4 68i4!4 d4 u! 6rici6 4! u! 8!u4@ di/i!u4 d4 !4r !6 li!4l c6!6r4 4l :uid6 :u4 d4l 8!u4 di/i!u4 l *r6u!didd d4l :uid6. E! 4/84 c/6@ /4 d4/rr6llrá 4l 86d6 *r clculr 4l 8i4*6 r4u4rid6 *r cir u! 8!u40 4! 4l 4;4*l6 /4 u4/8r u! 8!u4 u4 *6/44 u! 6uill li/ r4d6!d 4! 4l 6!d6@ 8r/ d4l cul /4 d4/cr" 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
Pr u! *r6u!didd d4 :uid6 dd ($%@ 4l 846r4 d4 T6rric4lli dic4 u4 l 4l6cidd d4 :u;6 4! 4l c$6rr6 4/= v ,=√ 2 gh . L r*id49 d4 :u;6 d4 6lu4! u4 */ *6r l 6uill 4/ Q = A , v , ('/% [ (*i4/ '/%. E! u! *4u4 c!8idd d4 8i4*6 d8@ 4l 6lu4! d4l :uid6 u4 */ *6r l 6uill 6 /l4 4/ - , =Q ( t )= A , v , (t ) .
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
>i4!8r/ 8!86@ d4id6 u4 4l :uid6 4/8 /li4!d6 d4l 8!u4@ 4l !i4l d4 :uid6 di/i!u4@ dur!84 4l *4u46 i!cr44!86 d8 u! *r6u!didd 6 di/8!ci d$0 4!86!c4/ 4l 6lu4! u4 ; d4l 8!u4 4/ 6lu4! r46id6 - t =− A t h 0 4/86/ d6/ 6l4!4/ d44! /4r i"ul4/= A , v , ( t ) =− A t h 0 r4/6li4!d6 *r 4l 8i4*6 d8 /4 8i4!4=
( )
− t =
A t
A ,
v ,
( )
− h
0 c66
v , =√ 2 gh→t =
A t A ,
√ 2 gh
( )
− h=
A t
A ,
√ 2 g
− 1/ 2
h
. El 8i4*6
h
r4u4rid6 *r u4 4l !i4l d4l :uid6 d4/ci4!d d4 u! *r6u!didd $& t 2
68r $< /4 i!84"rrá=
( )∫
−
∫ t = t 1
A t
A ,
√ 2 g
( ) A t
−
h2
h
−1 / 2
h→t 2− t 1=
h1
A ,
√ 2 g
( ) − = ( 2
i!ir8i4!d6 l6/ d6/ 8ri!6/ d4 ($% 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 4cuci! /4 *u4d4 u8ili9r *r clculr 4l 8i4*6 r4u4rid6 *r cir u! 8!u4 d4/d4 $& $/8 $<. E=e524o.- Pr 4l 8!u4 u4 /4 u4/8r 4! l "ur d4 rri@ 4!cu4!8r4 4l 8i4*6 r4u4rid6 *r cirl6 d4/d4 u! !i4l < .2 . l6/ diá48r6/ d4l 8!u4 &.5 l 6uill . S6luci!.- Pr u8ili9r l 4cuci! (&% 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/8i8u4!d6 4! l rul
(&%@ /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
4uil4 &&.212 i!. C.7.& cid6 d4 u! 8!u4 *r4/uri9d6.- Si 4l 8!u4 d4 l "ur d4 rri@ 4/8á /4lld6 c6! u! *r4/i! *6r 4!ci d4 u! :uid6@ /4 d44 p ( ) l *r6u!didd r4l d4l :uid6 "r4"r l c49 *i496]8ric γ
CAPITULO 7. FLUJO DE FLUIDOS Y ECUACIÒN DE BERNOULLI-(Hidráulic. I!". A. #$ui P.%
!84/ d4 ll4r c6 l6/ cálcul6/ u4 /4 d44! r4li9r 4! L 4cuci! (&%. C.7.< E4c86 d4l 8i*6 d4 6uill.- Si l 6uill 8i4!4 u! 6r á/ "ud@ 4l diá48r6 !i6 d4l c$6rr6 4/ /i"!ic8i4!84 4!6r u4 4l diá48r6 d4 l 4r8ur. El ár4 d4cud u4 /4 d44 u8ili9r A ; 4! l 4cuci! (&% 4/ l u4 8i4!4 4l diá48r6 á/ *4u46. E/84 *u!86 c6!6cid6 c66 l 4! c6!8rc8@ /4 *r4/4!8 li"4r4!84 u4r d4l 6rici6- E!86!c4/ *r6,id4!84 A ; ) .+< A6