paralelo y hardyc

7/24/2019 Paralelo y HardyC

1/111

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EA EB

fAB hEE = 1fAB hEE =

2fAB hEE =

21 ff hh =

A QQQQ =+= 21


2/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

21 ff hh =

BA QQQQ =+= 21( ) ( ) ( ) ( )BA VAVAVAVA =+= 21

( ) ( ) ( )

( )AA

A

VAVAV 21

+=

1

2

22

=

g

VLf

g

VLf


3/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

( ) ( ) ( )

( )AA

A

VAVAV 21

+=

2

2

1

2

22

=

g

VLf

g

VLf

1fAB hEE =

2fAB hEE =

fBB

BAA

A hp

g

vz

p

g

vz +++=++

22

22

fB

BA

A hp

zp

z ++=+


4/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

fB

BA

A hp

zp

z ++=+ f

AAB

B hp

zzp

+=

( ) fAB hp

zp +=

2

1

2

22

=

g

VLf

g

VLf


5/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

2

2

1

2

22

=

g

VLf

g

VLf

2

2

1

2

2

=

g

Lf

g

VLf

V

( ) ( ) ( )

( )AA

A

VAVAV 21

+=


6/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

( ) ( ) ( )

( )AA

A

VAVAV 21

+=

( ) ( ) ( 2 VAVAVA A =

12 QQQA =

( ) ( ) ( )

1

21

A

VAVAV A

=


7/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

( ) ( ) ( )

1

21

A

VAVAV A

=

1

2

2

2

1

2

2

=

g

Lf

g

VLf

V

( ) ( ) ( )

1

2

1

1A

VA

A

VAV A =


8/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

( ) ( )

1

2

1

1

A

VA

A

VAV A =

2

1

2

22

2

1

2

VV AA =

1

2

2

2

1

2

2

=

g

Lf

g

VLf

V

2

2

1

2

22

2

1

2

=

VV AA


9/111

EA

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2, AB2

EB

1

2

2

2

1

2

2

=g

Lf

g

VLf

V

2

21

2

22

21

2

=

VV AA

2

2

1

2

22

2

1

2

2

=

g

Lf

VV AA

2

2

2

g

VLf


10/111

Tuberas en Paralelo

1

2

2

1

2

22

2

1

2

2

=

g

Lf

VV AA

2

2

2

g

VLf

1

2

2

1

2

22

2

1

2

KVV AA

=

2

2

2

g

VLf

2

2

1

2

2

1

2

22

2

1

2

2

=

g

VLfK

VV AA

0


11/111

Tuberas en Paralelo

2

2

1

2

2

1

2

22

2

1

2

2

=

g

VLfK

VV AA

0

2

2

1

2

2214

1

2

22

2

1

2

2

1

2

12

2

+

=

VLf

VK

VVK

VK AAAA

0

+

= LfVKVVKVK AAAA 2

2

224

1

422

2

14

1

222

2

1

2

2

1

2

1

0

+

=

g

VLf

VVVK

VK AAAA

22 2

2

224

1

4

22

4

1

2

2

2

21

2

2

1

2

1

0


12/111

EA

Tuberas en Paralelo

950m, 8

1200m, 12

EB

2100gpm, 15

0

+

=

g

VLf

VVVK

VK AAAA

22

2

224

1

4

22

4

1

2

2

2

21

2

2

1

2

1

TubLisaPVC=smx /109955.0 2= 1

12

=

g

LfK


13/111

Tuberas en Paralelo

!a"2100gpm, a"15

L2"1200 m, 2"12

L1"950 m, 2"8

+

=

g

VLf

VVVK

VK AAAA

22

2

224

1

4

22

4

1

2

2

2

21

2

2

1

2

1

0

TubLisaPVC=smx /109955.0 2= 1

12 = gLfK

( )

=

gfK

2#8

9501

( )11 28$%.2%8 fK =

2

%2

2

1

2

0%24%.0

/0$5$.0

m

smV AA =

%%4%.2=


14/111

Tuberas en Paralelo

!a"2100gpm, a"15

L2"1200 m, 2"12

L1"950 m, 2"8

( )

+

=

Lf

VVVKf AA2%%4%.228$%.2%8

2

224

1

4

22

4

1

2

2

2

21

2

1

0

( )11 28$%.2%8 fK = 5044.102 41

2

2

2

=

AAV

24

1

4

22 025.5 VV

=

=

g

Vf

g

VLf

2#12

1200

2

22

2

2

22

( %.200 fV=

( ) ([ 2212

1 .200025.55044.1028$.2%8%%4%.228$%.2%80 VVff +=


15/111

Tuberas en Paralelo

Asum&r '1, '2

(esol)er por *2

*er&'&+ar '1, '2

0 ( ) ([ 2212

1 .200025.55044.1028$.2%8%%4%.228$%.2%8 VVff +=

0 ( [ 22211 9851.$80,5%%10.1200$8%.250%41$1.1298 fVVff +=

0 ( ( [ 12

2

121 98.$80,5%%10.1200$8%.250%41$1.1298 fVfVf +=

0 ( )( ) [ ]12212

2.12980$8%.250%9851.$80,5%%10.120 fVffV +=

[ ] [ ] ( ) ( )( )

( )( )21

211

2

11

29851.$80,5%%10.1202

9851.$80,5%%10.12041$1.129840$8%.250%0$8%.250%

ff

fffffV

+

=

As m&r ' '


16/111

Tuberas en ParaleloAsum&r '1, '2

(esol)er por *2

*er&'&+ar '1, '2

! 1200 gpm 0.0$5$ m%/s

8.9$0E0$ m2/s g 9.81 m/s2

-1212 plg

-88 plg 0.0000 mm

'12 '8 *12 *8 (12 (8

0.0120 0.0120 4.1$ 2.%%4 1.419E0 5.288E05

0.0110 0.01%0 4.$%8 2.%%4 1.10E0 5.288E05

0.0110 0.01%0 4.$%8 2.%%4 1.10E0 5.288E05

0 ( )( ) [ ]12212

2.12980$8%.250%9851.$80,5%%10.120 fVffV +=

[ ] [ ] ( ) ( )( )

( )( )21

211

2

11

29851.$80,5%%10.1202

9851.$80,5%%10.12041$1.129840$8%.250%0$8%.250%

ff

fffffV

+

=

Tuberas en Paralelo


17/111

Tuberas en Paralelo

Tub AB

LAB1, AB1

Tub AB

LAB2

,

AB2

EAEB

Tub AB

LAB%, AB%

fAB hEE =

1fAB hEE =

2fAB hEE =

%21 fff

hhh ==

BA QQQQQ =++= %21

%fAB hEE =

o3o ar3 6ross


18/111

o3o ar3 6ross

El o3o 3e Apro7&ma+&ones u+es&)as, 3e

ar3 6ross, es basa3o en el +umpl&m&eno 3e

3os pr&n+&p&os o lees:

Le 3e +on&nu&3a3 3e masa en los nu3os;

Le 3e +onser)a+&


19/111

Tuberas en Paralelo

El planeam&eno 3e esa =l&ma le &mpl&+a el uso 3e una e+ua+&


20/111

Tuberas en Paralelo

La e+ua+&ara+&onal 3e uso un&)ersal, +as& nun+a se a

emplea3o a+opla3a al mo3o 3e ar3 6ross,

porCue &n)olu+ra el +oe'&+&ene 3e 'r&++& 3epen3e 3e la emperaura )&s+os&3a3 3el agua, 3el +au3al 3el 'luo en las

uberas.

Tuberas en Paralelo


21/111

Tuberas en Paralelo

6omo Cu&era Cue el o3o 3e ar3 6ross es un

mo3o &era&)o Cue pare 3e la supos&+&


22/111

Tuberas en Paralelo

Lo aner&or se +ons&ua, asa o, en algopro&b&&)o u obsa+ul&>a3or, no obsane ser la

manera l


23/111

FUNDAMENTOS DEL MTODO DE HARDY CROSS

El mtodo se fundamenta en las dos leyes siuientes!"# Ley de $ontinuidad de masa en los nudos! %La suma ale&'ai$a de los$audales en un nudo de&e se' iual a $e'o%

Donde()i* ! Caudal +ue ,a'te del nudo i o +ue fluye -a$ia di$-o nudo#+i! Caudal $on$ent'ado en el nudo im ! N.me'o de t'amos +ue $onfluyen al nudo i#

3on3e,

' & :Pr3&3a 3e +arga por 'r&++&


24/111

Q0

Q1

QE

( ) ( )ADCABC hfhf =( ) ( ) 0= ADCABC hfhf

8$.4

85.1

L

C

QKhf =

85.1

1QKhf=qQQ += 0

(e3es 3e -&sr&bu+&


25/111

Q0

Q1

QE

85.1

1QKhf= qQQ +=

0

( ) 85.11 qQKhf +=

( ) ( .85.1 185.185.1185.1

1 ++=+

qQQKqQK oo

(e3es 3e -&sr&bu+&


26/111

Q0

Q1

QE

( ) ( .85.1 185.185.1185.1

1 ++=+

qQQKqQK oo

( )qQQKhf oo += 85.085.1

1 85.1

( ) ( ) 0= ADCABC

hfhf

(e3es 3e -&sr&bu+&


27/111

Q0

Q1

QE

( ) ( )85.185.1 85.085.1185.085.1

1 ++ ADCooABCoo qQQKqQQK

( ) ( 85.1 85.085.0185.185.1

1 ADCACBADCABC QQqKQQK

( )( )85.085.0

1

85.185.1

1

85.1ADCACB

ADCABC

QQK

QQKq

=

(e3es 3e -&sr&bu+&


28/111

Q0

Q1

QE

( )( )85.085.01

85.185.1

1

85.1 ADCACB

ADCABC

QQKQQKq =

( )( )

=85.0

1

85.1

1

85.1 QK

QKq

QK

QK

(e3es 3e -&sr&bu+&


29/111

Q0

Q1

QE

( )( )= 85.01

85.1

1

85.1 QK

QK

q

=

QHf

K

Hf

185.1

(e3es 3e -&sr&bu+&


30/111

1500*1440 gpm

3600gpm

L"1,500m, =12

L"2,500m, =15

4000gpm

(e3es 3e -&sr&bu+&


31/111

(e3es 3e -&sr&bu+&


32/111

(e3es 3e -&sr&bu+&


33/111

(e3es 3e -&sr&bu+&


34/111

(e3es 3e -&sr&bu+&


35/111

(e3es 3e -&sr&bu+&


36/111

(e3es 3e -&sr&bu+&


37/111

(e3es 3e -&sr&bu+&


38/111

(e3es 3e -&sr&bu+&


39/111

(e3es 3e -&sr&bu+&


40/111

(e3es 3e -&sr&bu+&


41/111

(e3es 3e -&sr&bu+&


42/111

(e3es 3e -&sr&bu+&


43/111

(e3es 3e -&sr&bu+&


44/111

(e3es 3e -&sr&bu+&


45/111

(e3es 3e -&sr&bu+&


46/111

(e3es 3e -&sr&bu+&


47/111

(e3es 3e -&sr&bu+&


48/111

(e3es 3e -&sr&bu+&


49/111


50/111

FTEA -E (E-E -E TGBE(FA:


51/111



52/111

Qe

Q1

Q2 Qs

H 6

A B 6

- E I

Qe=Qs = Q1+Q2+Q

Q3

Qe

Qe

A(-J 6(K



53/111

H 6

A B 6

- E I

Qe

Qe

A(-J 6(K

( ) 01 ==n

jse QQ



54/111

H 6

A B 6

- E I

Qe

Qe

A(-J 6(K

f

n

jmijij hg

V

kHH += =12

2



55/111

+= = mijmij

ij

n

jmijij

fkg

V

HH 1

2

2

+= = m

mijij

n

j

mijij

ijij fk

gAQHH

12

2

2


1


56/111

mij

mij

ij

n

jmij

ij

ij g

fk

HHQ 2

21

1

+

=

=


1


57/111

T " =mero 3e Tuberas

2

21

1

1

+

=

=

!T

j

mij

mij

ij

n

jmij

ij

ij Ag

fk

HHQ



58/111

6 G1M

( 2/11

2 j

fk

ijijij HHHHAgQ

mij

mijij

n

jmij

=

+

=


!T!T


59/111

011

=+ ==

ji !T

m

fij

!T

j

fij HH

02 11

2

=

+

== ij

ij

ij

n

j

mij

!T

j

fk

g

Vi

02 11

2

2

=

+

== ij

ij

ij

n

j

mij

!T

j ij

ij fk

gA

Qi



60/111

( ) 021

1

=

+

=

=ijij

!T

j ij

ij

ijij

n

j

mij

QQgA

fk

i


ETK-K -E A(-J 6(K 6K


61/111


6K((E66FK -E 6AG-ALE:

( ijmij

mij

ij

n

j

mijmijfijg

Qfkhh

1 2

+

+=+

=




62/111



+= mijmijij hhh(

ij

iijij

mij

mij

ij

n

jmijij gA

QQQ

fkh 2

2

1 2

2 +

+= =




63/111



( ij

iijij

mij

mij

ij

n

j

mijijgA

QQQfkh

2

2

1 2

2 +

+=

=

( ij

ijij

mij

mij

ij

n

j

mijijgA

QQfkh

2

2

1 2

2 +

+= =




64/111



( ) ( ) +

+= ==

n

j

mijij

iij

mij

mijij

n

j

mijij

ij kgA

QQfkgAQ

12

12

2

22

20

( )

( )

+

+

=

=

=

ij

ij

mij

mij

ij

n

j

mij

ij

ij

mij

mij

ij

n

j mij

i

A

Qfk

A

Qfk

Q

21

2

2

1

2




65/111



( )

+

+=

ij

mijfij

mijfij

i

Q

hh

hhQ

2



66/111

+=

ff (e

51.2

$.%log2

110

2

.9.0

(e

10 $4.5$.%

log

25.0

+

=

!D

f



67/111


68/111

2

.9.0

(e

10

$4.5

$.%log

25.0

+

=

!D

f


69/111


70/111


71/111


72/111


73/111


74/111


75/111


76/111


77/111


78/111

FTEA -E (E-E -E TGBE(FA:eNon (apson


79/111



80/111

( )xfxi=( )xfxx ii =+

( )( )xfxfxi O=


( )


81/111

( )

( )xf

xfxi

O

=

( ) ( ) ( ) ( ) ( )

P%P2

OOO2OOO +++=+ iiiiiiii

xxfxxfxxfxfxxf

( ) ( ) ( ) iiiii xxfxfxxf O+=+



82/111

Raz de la

Funcinf

( ) ( ) ( ) iiiii xxfxfxxf O+=+

( ) 0=+ ii xxf

( ) ( ) iii

xxfxf O0

+=( ) ( ) iii xxfxf

O=



83/111

( ) ( ) iii xxfxf O=

( ) =

=

!

j i

i!

x

fxxxf

1

00201 ,,.........,

eNon (apson

fff

.............. 111


84/111

=

!!

!

!!!

!

!

!

f

f

ff

x

x

xx

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

xxx

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

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

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

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

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

%

2

1

0

0%

02

01

21

%

2

2

1

%

2

2

2

1

2

21

eNon (apson

fff

.............. 111[ ]Dx


85/111

=

!!

!

!!!

!

!

!

f

f

ff

x

x

xx

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

x

f

xxx

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

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

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

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

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

%

2

1

0

0%

02

01

21

%

2

2

1

%

2

2

2

1

2

21

Dx

Df

eNon (apson

[ ] [ ]Df


86/111

[ ] [ ]"Dx

Dx

Df=

[ ] [ "Dx

DfDx

Dx

Df

Dx

Df

=

11

[ ] [ ]"Dx

DfDx

=

1

Nu Nmd0=

!u

QQ


87/111

nud!"Q#i#em

nud!

01

==j

Diij QQ

( 2/1

1

2 j

fk

ij

ijij HHH

AgQ

mij

mijij

n

jmij

=

+

=

Nu Nmnud!"0=

!u

QQ


88/111

nud!"Q#i#em

nud!

01

==j

Diij QQ

( )02

12/1

2/1

1

=

=

+

=

Di

!u

j

fk

ij

ij QHH

Ag

ij

ijij

n

jmij

( )02

2/1

2/1

=

Di

!uij

ij QHH

Ag


89/111

12/1

1

=

+

=

Di

j

fk

ij Qg

ij

ijij

n

jmij

( ) ( )2.,.........

12/1

2/1

1

1

= =

+

=

!u

j

fk

ij

ij!u HH

AgHHf

ij

ijij

n

jmij

( ) 0.,.........1 =!uHHf

ii ff =


90/111

jj Hx =

i

i

i

i

H

f

x

f

=

( ) !u

ijiHHf

2/1


91/111

( )

=

=

+=j

fk

ij

ij

jj

i AgHx

f

ij

ij

ij

n

jmij1

2/1

1

2

( )2/11

2

21 ij

fk

ij

j

i HHAg

xf

ij

ijij

n

jmij

=

+

=

( ) !u

ijiHHf

2/1


92/111

( )

=

=

+=j

fk

ij

ij

ji

i AgHx

f

ij

ij

ij

n

jmij1

2/1

1

2

( )

= =

+

=

!u

j

fk

ijij

i

i

ij

ijij

n

jmij

HHAgxf

1/1

/1

1

221

=

!u ii

H

ff


93/111

= j ii Hx 1

+=

ij

f

ij

ij

ijij

h

gAQ 10

2

5.2

$.%log

22

( )g

Vkh mij

!u

j

mij2

2

1

=

=

( )=g

VkHHhf mijijij

2

2


94/111

g2

21

1

2

=

+= ij

ij

ij

n

jmij

fk

ij

ijij

HHAgQ

n HH


95/111

2

2

1

2

=+

= ij

ijij

ij

ij

ij

n

j

mij

Q

HHgA

fk

eNon (apson

[ ] [DHDH

Df=

H1


96/111

DH

[ ]

=

!uH

H

H

DH

.

.

.

.2

1

inin HH += 1


eNon (apson


97/111


98/111


99/111

Mtodo de la Teo'/a Line


Cal$ula' R'esol1iendo


100/111

'esol1iendo Mediante laen los Nudos


Q1


101/111

Qe Q2 Qs

H 6

A B 6

- E I

Qe=Qs = Q1

+Q2

+QQ3

Qe

Qe

A(-J 6(K

( ) 0L1

=+=

i!T

j

eiDiij QQQ


Q

Q1

Q Q ( )L+i!T

QQQ


102/111

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Qoij =Caudal estimado en la primera Iteracin

Bomba = la Ecuacin se modifica a

#ijijij Qkk =

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