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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
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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
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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 ++=+
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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
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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
+=
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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
=
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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 =
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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
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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
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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
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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
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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
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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=
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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 +=
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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 ' '
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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
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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
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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+&
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Tuberas en Paralelo
El planeam&eno 3e esa =l&ma le &mpl&+a el uso 3e una e+ua+&
7/24/2019 Paralelo y HardyC
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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
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Tuberas en Paralelo
6omo Cu&era Cue el o3o 3e ar3 6ross es un
mo3o &era&)o Cue pare 3e la supos&+&
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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
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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 $onfluyen al nudo i#
3on3e,
' & :Pr3&3a 3e +arga por 'r&++&
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Q0
Q1
QE
( ) ( )ADCABC hfhf =( ) ( ) 0= ADCABC hfhf
8$.4
85.1
L
C
QKhf =
85.1
1QKhf=qQQ += 0
(e3es 3e -&sr&bu+&
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Q0
Q1
QE
85.1
1QKhf= qQQ +=
0
( ) 85.11 qQKhf +=
( ) ( .85.1 185.185.1185.1
1 ++=+
qQQKqQK oo
(e3es 3e -&sr&bu+&
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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+&
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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+&
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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+&
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Q0
Q1
QE
( )( )= 85.01
85.1
1
85.1 QK
QK
q
=
QHf
K
Hf
185.1
(e3es 3e -&sr&bu+&
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1500*1440 gpm
3600gpm
L"1,500m, =12
L"2,500m, =15
4000gpm
(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
7/24/2019 Paralelo y HardyC
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(e3es 3e -&sr&bu+&
7/24/2019 Paralelo y HardyC
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
7/24/2019 Paralelo y HardyC
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(e3es 3e -&sr&bu+&
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(e3es 3e -&sr&bu+&
7/24/2019 Paralelo y HardyC
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FTEA -E (E-E -E TGBE(FA:
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FTEA -E (E-E -E TGBE(FA:
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Qe
Q1
Q2 Qs
H 6
A B 6
- E I
Qe=Qs = Q1+Q2+Q
Q3
Qe
Qe
A(-J 6(K
FTEA -E (E-E -E TGBE(FA:
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H 6
A B 6
- E I
Qe
Qe
A(-J 6(K
( ) 01 ==n
jse QQ
FTEA -E (E-E -E TGBE(FA:
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H 6
A B 6
- E I
Qe
Qe
A(-J 6(K
f
n
jmijij hg
V
kHH += =12
2
FTEA -E (E-E -E TGBE(FA:
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+= = mijmij
ij
n
jmijij
fkg
V
HH 1
2
2
+= = m
mijij
n
j
mijij
ijij fk
gAQHH
12
2
2
FTEA -E (E-E -E TGBE(FA:
1
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mij
mij
ij
n
jmij
ij
ij g
fk
HHQ 2
21
1
+
=
=
FTEA -E (E-E -E TGBE(FA:
1
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T " =mero 3e Tuberas
2
21
1
1
+
=
=
!T
j
mij
mij
ij
n
jmij
ij
ij Ag
fk
HHQ
FTEA -E (E-E -E TGBE(FA:
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6 G1M
( 2/11
2 j
fk
ijijij HHHHAgQ
mij
mijij
n
jmij
=
+
=
FTEA -E (E-E -E TGBE(FA:
!T!T
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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
FTEA -E (E-E -E TGBE(FA:
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( ) 021
1
=
+
=
=ijij
!T
j ij
ij
ijij
n
j
mij
QQgA
fk
i
FTEA -E (E-E -E TGBE(FA:
ETK-K -E A(-J 6(K 6K
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ETK-K -E A(-J 6(K 6K
6K((E66FK -E 6AG-ALE:
( ijmij
mij
ij
n
j
mijmijfijg
Qfkhh
1 2
+
+=+
=
FTEA -E (E-E -E TGBE(FA:
ETK-K -E A(-J 6(K 6K
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ETK-K -E A(-J 6(K 6K
6K((E66FK -E 6AG-ALE:
+= mijmijij hhh(
ij
iijij
mij
mij
ij
n
jmijij gA
QQQ
fkh 2
2
1 2
2 +
+= =
FTEA -E (E-E -E TGBE(FA:
ETK-K -E A(-J 6(K 6K
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ETK-K -E A(-J 6(K 6K
6K((E66FK -E 6AG-ALE:
( 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 +
+= =
FTEA -E (E-E -E TGBE(FA:
ETK-K -E A(-J 6(K 6K
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ETK-K -E A(-J 6(K 6K
6K((E66FK -E 6AG-ALE:
( ) ( ) +
+= ==
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
FTEA -E (E-E -E TGBE(FA:
ETK-K -E A(-J 6(K 6K
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ETK-K -E A(-J 6(K 6K
6K((E66FK -E 6AG-ALE:
( )
+
+=
ij
mijfij
mijfij
i
Q
hh
hhQ
2
FTEA -E (E-E -E TGBE(FA:
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+=
ff (e
51.2
$.%log2
110
2
.9.0
(e
10 $4.5$.%
log
25.0
+
=
!D
f
FTEA -E (E-E -E TGBE(FA:
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2
.9.0
(e
10
$4.5
$.%log
25.0
+
=
!D
f
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FTEA -E (E-E -E TGBE(FA:eNon (apson
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FTEA -E (E-E -E TGBE(FA:eNon (apson
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( )xfxi=( )xfxx ii =+
( )( )xfxfxi O=
FTEA -E (E-E -E TGBE(FA:eNon (apson
( )
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( )
( )xf
xfxi
O
=
( ) ( ) ( ) ( ) ( )
P%P2
OOO2OOO +++=+ iiiiiiii
xxfxxfxxfxfxxf
( ) ( ) ( ) iiiii xxfxfxxf O+=+
FTEA -E (E-E -E TGBE(FA:eNon (apson
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Raz de la
Funcinf
( ) ( ) ( ) iiiii xxfxfxxf O+=+
( ) 0=+ ii xxf
( ) ( ) iii
xxfxf O0
+=( ) ( ) iii xxfxf
O=
FTEA -E (E-E -E TGBE(FA:eNon (apson
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( ) ( ) iii xxfxf O=
( ) =
=
!
j i
i!
x
fxxxf
1
00201 ,,.........,
eNon (apson
fff
.............. 111
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=
!!
!
!!!
!
!
!
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
7/24/2019 Paralelo y HardyC
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=
!!
!
!!!
!
!
!
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
7/24/2019 Paralelo y HardyC
86/111
[ ] [ ]"Dx
Dx
Df=
[ ] [ "Dx
DfDx
Dx
Df
Dx
Df
=
11
[ ] [ ]"Dx
DfDx
=
1
Nu Nmd0=
!u
7/24/2019 Paralelo y HardyC
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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
7/24/2019 Paralelo y HardyC
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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
7/24/2019 Paralelo y HardyC
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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 =
7/24/2019 Paralelo y HardyC
90/111
jj Hx =
i
i
i
i
H
f
x
f
=
( ) !u
ijiHHf
2/1
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( )
=
=
+=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
7/24/2019 Paralelo y HardyC
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( )
=
=
+=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
7/24/2019 Paralelo y HardyC
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= j ii Hx 1
+=
ij
f
ij
ij
ijij
h
gAQ 10
2
5.2
$.%log
22
( )g
Vkh mij
!u
j
mij2
2
1
=
=
( )=g
VkHHhf mijijij
2
2
7/24/2019 Paralelo y HardyC
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g2
21
1
2
=
+= ij
ij
ij
n
jmij
fk
ij
ijij
HHAgQ
n HH
7/24/2019 Paralelo y HardyC
95/111
2
2
1
2
=+
= ij
ijij
ij
ij
ij
n
j
mij
Q
HHgA
fk
eNon (apson
[ ] [DHDH
Df=
H1
7/24/2019 Paralelo y HardyC
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DH
[ ]
=
!uH
H
H
DH
.
.
.
.2
1
inin HH += 1
FTEA -E (E-E -E TGBE(FA:
eNon (apson
7/24/2019 Paralelo y HardyC
97/111
7/24/2019 Paralelo y HardyC
98/111
7/24/2019 Paralelo y HardyC
99/111
Mtodo de la Teo'/a Line
FTEA -E (E-E -E TGBE(FA:
Cal$ula' R'esol1iendo
7/24/2019 Paralelo y HardyC
100/111
'esol1iendo Mediante laen los Nudos
FTEA -E (E-E -E TGBE(FA:
Q1
7/24/2019 Paralelo y HardyC
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
FTEA -E (E-E -E TGBE(FA:
Q
Q1
Q Q ( )L+i!T
QQQ
7/24/2019 Paralelo y HardyC
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Qe Q2 Qs
Q3
NU 3 N.me'o de Nudos( ene'anNU e$ua$iones de las $uales una es'edundante
NC 3 N.me'o de Ci'$uitos( ene'anNU e$ua$iones
( ) L1
=+=j
eiDiij QQQ
11
+==
ii !T
j
mij
!T
j
fij hh
+==
n
jmij
!T
j ij
ij fk
Ag
Q
112
2
2
FTEA -E (E-E -E TGBE(FA:
Q
Q1
Q Q
n!T ij Q
2
7/24/2019 Paralelo y HardyC
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Qe Q2 Qs
Q3
==
j
mij
j ij
ijk
Ag
Q
11
2
2
=
!T
j
ijijQk
1
2
mijij khk =+
ij
ij
ij
ij
n
jmij
ijgA
fk
k2
1
2
+
= =
FTEA -E (E-E -E TGBE(FA:
Q
Q1
Q Q Qkk O
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Qe Q2 Qs
Q3
Qoij =Caudal estimado en la primera Iteracin
Bomba = la Ecuacin se modifica a
#ijijij Qkk =
01
O ==
i!T
j
#ijijQk
B
!T
j
#ijij HQki
==1
O
FTEA -E (E-E -E TGBE(FA:
Qe
Q1
Q Qsij
n fk
+
7/24/2019 Paralelo y HardyC
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Qe Q2 Qs
Q3
ij
ij
ijj
mij
ij gA
fk
k 2
1
2
+
=
=
+=
ff (e
51.2
$.%log2
110
ij
ijQV 4(e ==
#ijijij QkkO=
7/24/2019 Paralelo y HardyC
106/111
7/24/2019 Paralelo y HardyC
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4)"5 4)"6 3 4:#
4)"5 !2% !25 " 0.
)57 !%4 " 0.
)78 !45 " 0.
) ! ! 0
7/24/2019 Paralelo y HardyC
108/111
)59 !45 !5 " 0.
4)96 !1 " 0.
0
0
25
O
2545
O
45%4
O
%42%
O
2%
1
O
15
O
525
O
2512
O
12
=+
=+
QkQkQkQk
QkQkQkQk
7/24/2019 Paralelo y HardyC
109/111
7/24/2019 Paralelo y HardyC
110/111
7/24/2019 Paralelo y HardyC
111/111
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