1) Una TH de reacción ( se desprecian las pérdidas las pérdidas ) ) tiene ti ene las siguientes características: n = rpm,, ( β1 = 90º, α1 = 10º ), ( c 1m = cm = m s ), " # = ( 0$5 • #1 ), %1 = 100mm 375rpm 375 m! s 100mm &$ 'l agua sale od ete e sin componente periférica$ del rodet periférica$ 'l espesor de de ls álabes resta un * al área útil a a la entrada del rodete$ +alcular: Salto neto ( neto ( ) - β - ( Diámetros ( Diámetros )( #1, # ) - Potencia - Potencia ( .e/e ) desarrllada pr la H
2 de tan α1 = ( c1 m ! u1 ) → u1 = ( c1 m ! tan α1 ) = ( ! tan 10º ) = 11.3426 m ! s = c1 u entrada m ! s ( w1 = c1m= m ! s ) c1 = c1 u + c1m = 11$3 + = 11.5176 m ! s & " ( u ! u1 ) = ( # ! #1 ) = 0$5 & ⇒ " u2 = ( 0$5 ⋅ u1 ) = ( 0$5 ⋅11$3 ) = 5.6713 = → = = = β 19.43 " tan β ( c ! u ) & arctan ( c ! u ) arctan ( ! 5$713 ) º 2 m 2 de salida m ! s = c + u = + 5$713 w2 = ( u ! cs β ) = ( 5$713 ! cs 19$3º ) = 6.0138 m u1 = " ( ⋅ #1⋅ n ) ! 0 & → D1 = " ( 0 ⋅ u1 ) ! ( ⋅ n ) & = " ( 0 ⋅ 11$3 ) ! ( ⋅ 375 ) & = 0.5777 D = " ( 0$5 ⋅ # ) = ( 0$5 ⋅ 0$5777 ) = 0.2889 m & = "( 0 ⋅ u ) ! ( ⋅ n )& = " ( 0 ⋅ 5$713 ) ! ( ⋅ 375 ) & 2 1 (u )1 = ( 81⋅ 61⋅ c1 m ) = ( ⋅ #1⋅ %1⋅ 61⋅ c1 m ) ( u )1 =( u ) b2 = %1⋅ ( #1 ! # ) ⋅ ( 61 ! 6 ) = c → = 0$1 ⋅ ⋅ "( 1 − 0$0 ) ! 1 & = 0.192 ( ) ( 8 6 c ) ( # % 6 c ) = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ m = c m m u 1m m 3 = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ − ⋅ = ( ) ( # % 6 c ) " 0$5777 0 $ 1 ( 1 0$0 ) & 0 $ 35 m ! s Caudales ú tiles u 1 1 1 1 1m iguales en 1 4 3 = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ( ) ( # % 6 c ) ( 0$9 0$19 1 ) 0 $ 35 m ! s u m 3 e desprecian las Caudal útil ( u ) = ( ⋅ < ) = Caudal abs orbido ( ) = 0$35m ! s pérdidas pérdidas en la ⇒ Altura de Euler ( u ) = " ( u1⋅ c1 u ) − ( u ⋅ c u ) & ! g = ( u1 ! g ) = = = = = 1 < ; m t = ( 11$3 ! 9$1 ) = 13.1146 m = ( ⋅ ; ) = H( Altura net a ) Peje= ( .⋅ t ) = . = ( =⋅ ⋅ ) = ( .i ⋅ m ) = .i = ( =⋅ u ⋅ u ) = " ρ ⋅ g⋅ u ⋅ ( u1 ! g ) & = = ( ρ ⋅ u ⋅ u1 ) = ( 1000 ⋅ 0$35 ⋅ 11$3 ) = 44836.1193 W ⋅ ( 1 CV ! 735$5 W ) & = 0$9 CV
( c )( va%sluta )salida del agua agua del del rodete n tiene
componente periférica ( c u = 0 ) ( α=90º ) u1 = 11$3m 11$3 m! s, s, c1 = 11$517m 11$517m! s, s, >1 = m m! s s s, c = cm =m s, > = $013m s u = 5$713m 5$713m! s, =m! s, $013m! s
Francis de eje vertical eje vertical se cncen ls siguientes dats: Diámetro ) #e una TH Francis dats: Diámetro de entrada al entrada al rodete 5cm 5cm,, anco del anco del rodete a la entrada 5cm 5 cm,, diámetro de salida del salida del rodete 30cm 30cm,, anco del rodete a la salida 7cm 7 cm?? ls álabes restan (cupan) un * al (del) área útil en en la entrada (a la salida del rodete ls álabes pueden supnerse a@ilads)? ángulo a la salida del distribuidor , º, ángulo de ángulo de entrada de ls álabes del rodete, 5º, ángulo de salida de salida de ls álabes del rodete 30º? pérdidas 30º? pérdidas idráulicas idráulicas en en el interir de la TH eAui
+alcular: Velocidad de rotaci"n ( rotaci"n ( n ) - Altura - Altura neta ( ) - Altura - Altura útil ( ( u ) - Caudal útil ( ( u ) #endimiento idráulico ( idráulico ( ; ) 4 total 4 total ( ( t ) - Potencia - Potencia interna ( interna ( .i ) - Potencia al freno ( freno ( .e/e ) - ( n s )
( Caudal abs orbido = Caudal út il ) : = u = ( ⋅ 0$5 ⋅ 0$05 ⋅ 0$9 ⋅ $9 ) = 0.5786 m3 ! s = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ = P 250831.600 2 Potencia i nterna : ( ρ g ) ( 1000 9 $ 1 0$57 $ 1911 ) W i u u P = ( . ⋅ ) = ( 5031 $00 ⋅ 0$9 ) = 235781.704 2W ⋅ ( 1CV ! 735$5W ) ≅ 30$ CV eje i m $% espec&f ico de revoluciones 1 ! − 5 ! 1 ! − 5 ! n! = ( n ⋅ .e/e ⋅ ⋅ 50$1911 ) ≅ ( 1 ⋅ 30$ ) ≅ 118rpm ⇒ Irancis nrmal
3) Una TH Francis de fujo radial tiene un rodete de diámetro e'terior 3cm 3cm 4 4 diámetro interior cm cm$$ Bs ancos Bs ancos de de ls álabes del rodete en las seccines de entrada 4 salida sn, respecti
s sale del distribuidor de una TH Francis Francis cn velocidad 0m s, ) Un caudal de 15$7m 15$7m3! s sale 0m! s, @rmand un Kngul de 9º cn el radio$ radio$ 'l diámetro a diámetro a la salida del distribuidor es es $1m $1m 4 a la entrada del rodete es m m$ +alcular: Componentes ( Componentes ( periférica, periférica, radial ) de la velocidad absoluta del absoluta del agua a agua a la salida del distribuidor , si se supne Aue el espesor el espesor de de ls álabes directrices cupa el 5* de la secciGn teGrica$ i el rodete rta a 00rpm 00rpm,, calcular: +omento calcular: +omento a a la entrada ( L1 ) 4 a la salida ( L ) del rodete, si la descarga en el tubo de aspiración tiene direcciGn aMial - Velocidad absoluta de entrada ( c1 ) en el rodete - Par - Par motor ( ( Lm ) e/ercid s%re el rodete - Altura - Altura útil ( u ) apr
4
pr el rodete - Potencia - Potencia interior ( ( .i ) en el eje de la TH supniend Aue n ;a4 pérdidas ;a4 pérdidas volumétricas ( volumétricas ( < = 1 ) - #endimiento - #endimiento idráulico ( idráulico ( ; ) sa%iend Aue el salto el salto neto es neto es 0$5m 0$5m #endimiento mecánico ( mecánico ( m ) 4 rendimiento total ( ( t ) si la potencia la potencia al freno ( freno ( potencia desarrllada potencia desarrllada pr la TH) es 7355CV 7355CV i el diámetro a diámetro a la entrada del tubo de aspiración es 1$m 1$m, calcular la velocidad de descarga ( descarga ( cd ) upniend Aue c d cincide cn la velocidad absoluta de salida ( salida ( cd = c 4! pd N p ), calcular la 'cinEtica ( 'c ) 4 la ' presiGn ( ' pr ) ) entregadas al rodete - Grado de reacción ( O ) de la TH
DATOS : .e/e =7355CV =7355CV , n =00rpm =00rpm,, =0$5m =0$5 m, #1 =m =m, 0 =15$7m =15$7m3! s, s, c0 = 0m 0m! s, s, #0 =
c0 r = ( c0 ⋅ sen α0 ) = ( 0 ⋅ sen 1º ) = 1$5 m ! s = c0 m c = ( c ⋅ cs α ) = ( 0 ⋅ cs 1º ) = 5$01 m ! s ≅ 5 m ! s 0 0 0 u = ( ⋅ # ⋅ % ⋅ 6 ⋅ c ) → % = " ! ( ⋅ # ⋅ 6 ⋅ c )& = 0 0 0 0m 0 0 0 0 0m 0 = "15$7 ! ( ⋅ $1 ⋅ 0$95 ⋅ 1$5 )& = 0$115m ⇒ %0 = 11$5mm ( < = 1 ) ⇒ u = ( ⋅ < ) = = 0 = 15$7 3 m ! s s
En el vórtice libre formado libre formado entre la salida del distribuidor y y la entrada en el rodete, rodete, el momento ( momento ( 1 ! no var"a#
'cuaciGn de v"rtice libre : " ( c0 u ⋅ r 0 ) = ( c1 u ⋅ r 1 ) & c = " c ⋅ ( # ! # ) & = " 5 ⋅ ( $1 ! ) & = 0$ m ! s 0u 0 1 1u L = " ( ρ ⋅ ) ⋅ ( c ⋅ r ) & = " ( ρ ⋅ ) ⋅ ( c ⋅ r ) & = L u 1u 1 u 0u 0 0 1 L1 = ( 1000 ⋅ 15$7 ) ⋅ " 0$ ⋅ ( ! ) & ≅ 959$ ( $ ⋅ m
Ba ecuaciGn del par del par puede puede aplicarse a partes @i/as de la T$ 'ntre seccines ( 1P-P ) cualesAuiera: Lm
= ( L1′ − L ′ ) = ( ρ ⋅ ) ⋅ " ( c1′ u ⋅ r 1′ ) − ( c′ u ⋅ r ′ ) & Q i ;a4 álabes ( palas) Aue
mdi@iAuen la v@lu/ 4! su cantidad de movimiento, movimiento, cm curre en el distribuidor Fink de de una TH de reacción, el momento respect momento respect al eje de rotación
( c1′ u ⋅ r 1′ ) = ( c ′ u ⋅ r ′ )
in componente tangencial en la descarga V"rtice libre entre ( salida del rodete−descarga ) ⇒ " ( c u ⋅ r ) = ( c3 u ⋅ r d ) = 0 & ⇒ ( c u = 0 ) ( c3 u = 0 ) ⇒ ( c3 = c3 a ) = ⋅ ⋅ ⋅ = L ( ρ ) ( c r ) 0 S ;a4 momento a l a salida del rodete $ 'l flu,o sale u u → L3 = ( ρ ⋅ u ) ⋅ ( c3 u ⋅ r d ) = 0 de ls álabes del rodete sin su@rir rtaciGn alguna Par motor Aue Aue e/erce el agua s%re el rodete : #m = ( L1 − L ) = L1 = 959.82 ($ ⋅ m 0 = ( ⋅ #0 ⋅ %0 ⋅ 60 ⋅ c0 m ) = ( #0 ⋅ %0 ⋅ 60 ⋅ c0 m ) = ( #1⋅ %1⋅ 61⋅ c1 m ) ⇒ ( ) = ( ⋅ # ⋅ % ⋅ 6 ⋅ c ) = ⇒ ipGtesis : ( % ⋅ 6 ) = ( % ⋅ 6 ) 1 1 1 1m 0 0 1 1 u 1 Componente meridiona l (radial ) : c = " c ⋅ ( # ! # ) & = " 1$5 ⋅ ( $1 ! ) & = 3$ m ! s 1m 0m 0 1 Velocidad absoluta a la entrada del rodete : c = c + c = 0$ + 3$ = 64.78m ! s 1 1u 1m
$
Velocidad periférica a la entrada : u1 = " ( ⋅ #1⋅ n ) ! 0 & = " ( ⋅ ⋅ 00 ) ! 0 & = $3 m ! s ⋅ − ⋅ = " ( u c ) ( u c ) & ! g Altura úti l 1 1 u u m = 387.35 Altura de Euler : Hu = = " ( u1⋅ c1 u ) ! g & = " ( $3 ⋅ 0$ ) ! 9$1 & #endimient o idráuli co : $% = ( u ! ) = ( 37$35 ! 0$5 ) = 0.9624 Potencia interna : Pi = ( ρ ⋅ g⋅ u ⋅ u ) = ( 1000 ⋅ 9$1 ⋅ 15$7 ⋅ 37$35 ) = 60304468.5 5W . = ( L ⋅ T ) = L ⋅ " ( ⋅ ⋅ n ) ! 0 & = 95917$ ⋅ " ( ⋅ ⋅ 00 ) ! 0 & = 030711$ W m m i Potencia absorbida : . = ( ρ ⋅ g⋅ ⋅ ) = ( 1000 ⋅ 9$1 ⋅ 15$7 ⋅ 0$5 ) = 3091$75 W #endimient o mecánico : $m = ( .e/e ! .i ) = " ( 7355 ⋅ 735$5 ) ! 030$55 & = 0.9557 #endimient o total : $ = ( ⋅ ⋅ ) = ( 1 ⋅ 0$9 ⋅ 0$9557 ) = 0.9197 = ( .e/e ! . ) & < ; m
Caudal de descarga : d = ( 8d ⋅ cd ) = " ( ! ) ⋅ #d & ⋅ cd = Velocidad de descarg a : cd = " ( ⋅ ) ! ( ⋅ #d ) & = " ( ⋅ 15$7 ) ! ( ⋅ 1$ ) & = 12.73m-s = c2 Altura dinámica : Hd = " ( c1 − c ) ! ( ⋅ g ) & = " ( $7 − 1$73 ) ! ( ⋅ 9$1 ) & = 205.63 m m ≅ " ( p1 − p ) ! = & Altura de presi"n : H( = ( u − d ) = ( 37$35 − 05$3 ) = 181.72 .rado de r eacci"n : " '= ( ! ) = ( 11$7 ! 37$35 ) = 0.469 &<1 p u
5) #e una TH Francis de fujo radial de eje vertical se cncen ls dats siguientes: Potencia en el eje 000(W %a/ un salto neto de 50m 4 rtand a 50rpm$ Bs diámetros interior 4 e'terior sn 0 4 10cm, respecti
DATOS : ( .e/e = 000(W , = 50m, n = 50rpm )? " ( rodete radial ) ( #1 J # )? ( #1 =
Pérdidas Cámara esp iral 'n el tubo internas Distribuid or difusor + rodete 'n el = + = + + + Altura net a : ( ) " ; ; ; & u r int u r (' −1) r (1− ) r (−) Pérdidas en rodete : % m r1−2-= " − u − ; r (' −1) − ; r (−) & = (50 − $70 − 0$9 − 1) = 3.3914
*
Pérdidas 'n la tuber&a Pérdidas a e'ternas for!ada la salida ; r (8 − ') = " % − − ; r (− W) & = = − = − − → ( ) " ; ; & % r eMt % r (8 − ') r (− W) = ( 5 − 50 − 0$015 ) = 3$935 m +ntinuidad: +audal Aue sale del rodete = Caudal Aue sale del tubo Pérdidas : ; r (− W) ≅ " c ! ( ⋅ g )& = = " ( ! ) ⋅ # ⋅ c & = = " ( ! ) ⋅ # ⋅ c & = " 0$59 ! ( ⋅ 9$1 ) & = 0$015 m c = c ⋅ ( # ! # ) = ⋅ ( 0$ ! 3 ) = 0$59 m ! s = ⋅ ⋅ = ⇒ = ⋅ ⋅ = ⋅ ⋅ = " ( ! ) # v & v "( ) ! ( # )& ( 9$591 ) ! ( ) 3$0539 m ! s t t t t t ⋅ g⋅ # t ⋅ ; r (8 − ') ⋅ 9$1 ⋅ ⋅ 3$935 /actor de fricci"n B t vt ; = 0.0305 = ⋅ ⋅ g ⇒ f = r (8 − ') = @ ⋅ # t ⋅ ⋅ B v 550 3$0539 t t
) 'l rodete de una TH Francis , cn un diámetro de 1$5m 4 rtand a 30rpm, desarrlla una potencia de 1500(W cn un caudal de 1$3m3! s$ 'l agua entra en el rodete sin co*ue, cn una componente meridiana de velocidad de 9$m! s, 4 sale ;acia el tubo diusor cn velocidad 7$m! s, sin componente acimutal $ Ba di@erencia entre las cotas pie!ométricas a la entrada 4 salida del rodete es de 0m$ • +alcular: Velocidad absoluta ( c1 ) 4 direcci"n ( α1 ) del agua a la entrada del rodete - )ngulo de entrada ( β1 ) de ls álabes del rodete - Altura de pérdidas ( ;r(1C) ) en el rodete 3 #1 =1$5m, n =30rpm, .e/e = 1500(W , =1$3m ! s, c1m = 9$m! s, c = 7$m! s, cu = 0, 1C = 0m
Velocidad periférica a la entrada : u1 = " ( ⋅ #1⋅ n ) ! 0 & = " ( ⋅ 1$5 ⋅ 30 ) ! 0 & = 33$771 m ! s Potencia útil %tenida en el e,e : . = ( . ⋅ ) = ( =⋅ ⋅ ) ⋅ = " =⋅ ( ⋅ ) ⋅ & ⋅ i m e/e u u m < u m = = ⇒ = ⋅ = ⋅ ⋅ ⋅ = ( < m 1 ) u " .e/e ! ( = ) & " 1$5 10 ! ( 1000 9$1 1$3 ) & 103$593m Ecuaci"n d e Euler : ( g⋅ ) = " ( u ⋅ c ) − ( u ⋅ c ) & = ( u ⋅ c ) ⇒ c = " ( g⋅ ) ! u & u 1 1u u 1 1u 1u u 1 c1 u = " ( g⋅ u ) ! u1 & = " ( 9$1 ⋅ 103$593 ) ! 33$771 & = 30$0917 m ! s
c1 =
c1u + c1m
=
30$0917
+ 9$ = 31$559 m ! s
º "1 = arctan ( c1 m ! c1 u ) = arctan ( 9$ ! 30$0917 ) = 17.69 β = arctan " c ! ( u − c )& = arctan " 9$ ! (33$771 − 30$0917)& ≅ 69º 1m 1 1u 1
7) Xa/ ciertas cndicines una TH Francis de fujo radial cn un salto neto de 50m, caudal de 10m3! s, 4 rtand a 70rpm$ Bs diámetros crrespndientes a las seccines de entrada 4 salida del rodete sn, respecti
Velocidade s u1 = " ( ⋅ #1⋅ n ) ! 0 & = " ( ⋅ 1$5 ⋅ 70 ) ! 0 & = 1$05 m ! s periférica s : u = " ( ⋅ # ⋅ n ) ! 0 & = " ( ⋅ 0$ ⋅ 70 ) ! 0 & = 11$3097 m ! s Caudal útil : u = ( ⋅ < ) = ( u )1 = " ( ⋅ #1⋅ %1⋅ 61 ) ⋅ c1 m & = ( u ) = " ( ⋅ # ⋅ % ⋅ 6 ) ⋅ c m & c = " ( ⋅ ) ! ( ⋅ # ⋅ % ⋅ 6 ) & = " ( 10 ⋅ 0$9 ) ! ( ⋅ 1$5 ⋅ 0$3 ⋅ 1 ) & = 6.9321 m ! s < 1 1 1 1m Coef1 de r educci"n : 6 = " ( ⋅ < ) ! ( ⋅ # ⋅ % ⋅ c m ) & = " ( 10 ⋅ 0$9 ) ! ( ⋅ 0$ ⋅ 0$5 ⋅ ) & = 0$97 c1 u = u1− ( c1 m ! tan β1 ) = 1$05 − ( $931 ! tan 0º ) = 19$935 m ! s c1 = c1 u + c1 m = 19$935 + $931 = 21.1517 m ! s w1 = c1m + (u1 − c1 u ) = $931 + (1$05 − 19$935) = 7.04m ! s º "1 = arctan ( c1 m ! c1 u ) = arctan ( $931 ! 19$935 ) = 19.13 w = u + c = 11$3097 + = 13.8531 m ! s c2 = c m = 8m ! s 2 β2 = arctan ( c ! u ) = arctan ( ! 11$3097) = 35.27 º "2 = 90º
Ecuaci"n d e Euler : ( g ⋅ u ) = " ( u1⋅ c1 u ) − ( u ⋅ c u ) & = ( u1⋅ c1 u ) H = ( u ⋅ c ) ! g = ( 1$05 ⋅ 19$935 ) ! 9$1 = 43.1974 m 1 1u u #endimient o idráuli co : $% = ( u ! ) = ( 3$197 ! 50 ) = 0.86 Bas velocidade s periféri cas se mantienen : ( u1′ = u1 = 1$05 m ! s )( u′ = u = 11$3097 m ! s ) ′ ′ ′ ′ = → = = ⋅ = ⋅ = ( α 19$13º ) ( α 1$13º ) c ( c sen α ) ( 1$1517 sen 1$13º ) $51 9 m ! s 1 1 1 m 1 1 → c′ = ( c′ ⋅ cs α′ ) = ( 1$1517 ⋅ cs 1$13º ) = 0$101 m ! s c1 = c1′ = 1$1517 m ! s 1 1 1u uper@icie s de entrada ! salida ( c′ m ! c1′ m ) = ( c m ! c1 m ) = " ( #1⋅ %1⋅ 61 ) ! ( # ⋅ % ⋅ 6 ) & ⇒ c′ = c′ ⋅ ( c ! c ) = $519 ⋅ ( ! $931) = 7$5959 m ! s del rodete n se mdi@ican m 1m m 1m c′ = " u − ( c′ ! tan β′ ) & = " 11$3097 − ( 7$5959 ! tan 35$7º ) & = 0$597 m ! s β camb*a u m 1 β′1 = arctan " c1′ m ! ( u1′ − c1′ u ) & = arctan " $519 ! ( 1$05 − 0$101 ) & = 80.48 º β2 no m H′u = " ( u1⋅ c1′ u ) − ( u ⋅ c′ u ) & ! g = " ( 1$05 ⋅ 0$101 ) − ( 11$3097 ⋅ 0$597 ) & ! 9$1 = 42.7958 #endimient o idráuli co : $′ = ( ′ ! ) = ( $795 ! 50 ) = 0.8559" < ( = 0$39 ) & % u ; ( i = ′i ) ⇒ "( < ⋅ ; ) = ( ′< ⋅ ′; )& → ′< = " ( < ⋅ ; ) ! ′; & = "( 0$9 ⋅ 0$39 ) ! 0$559& = 0$99 m3 ! s ′ = "( ′u )1 ! ′< & = "( ⋅ #1⋅ %1⋅ 61 ⋅ c1′ m ) ! ′< & = "( ⋅ 1$5 ⋅ 0$3 ⋅ 1 ⋅ $519) ! 0$99& = 9.4065 "( ⋅ < ) ! ( ′⋅ ′< )& = ( c1 m ! c1′ m ) → ′ = ( c1 m ! c1′ m ) ⋅ ( < ! ′< ) ⋅ = i n dice Aue i = cte = ( $519 ! $931 ) ⋅ ( 0$9 ! 0$91 ) ⋅ 10 = 9$05 m3 ! s ′ = supnems ( ) < < ) Una TH Francis de eje vertical rta a 375rpm desarrlland una potencia en el eje de 9(W , en cndicines nminales$ 'n las cndicines anterires, un manómetro cnectad en la secciGn de entrada ( E), antes de la cámara espiral 4 despuEs de la válvula, mide una presi"n de 10mca, siend el diámetro de la tubería orzada en ese punt de 500mm, 4 llegand a la TH un caudal de m3! s$ %re un plan de la instalaciGn en el Aue se representa la TH en crte meridinal, se ;an medid las siguientes dimensiones: #1 =1$m? # =0$m? %1 = 50mm? % = 0$m? D' = D1 = m? D = 0m (salida (S) de la TH en el ni
/
(entrada en el tubo de aspiración , en el punt )$ 8sumir "( < = 0$95, m = 0$9 )? ( 6 1 = 1, 6 = 0$9 )? ( cu =0 )&, 4 Aue las pérdidas idráulicas se reparten pr igual entre el rodete, el tubo de aspiración 4 el cn/unt ( cámara espiral distribuidor Fink )$ e cnsidera la secciGn de salida de la TH en la super@icie del canal de desagüe aguas aba,o, despreciKndse la energ&a de velocidad en esa secciGn$ +alcular: Altura de Euler ( u ) - Altura neta ( ) - #endimiento idráulico ( ; ) 4 total ( t ) - ( T+ )entrada 4 ( T+ )salida del rodete? ( α1, β1 )( α, β ) - Presi"n a la entrada ( p1 ! ) 4 salida ( p ! ) del rodete
0ernoulli entre ( entrada rodete−salida ) " Ss dicen Aue ( ≡ W )⇒( p ≡ p W = patm )( D ≡ D W =0 )( c ≡ c W ≅ 0 ) & X1 − ( u + r(1−) ) = X Pérdidas entre ( 1 − ) : r(1−) = " ; r(1− ) + ; r( −) & ( p1! = ) man + D1 + " c1 ! ( ⋅ g ) & − ( u + ; r(1− ) ) = Pérdidas t ubo de aspiraci"n = ( p ! = ) − 5$7 ; r( −) = en f 1 ( p man al rodete ) = + + ⋅ − + ( p ! = ) $5 " 3$ ! ( 9$1 )& ( 109$31 3$03 ) entrada 1 man " X' − ( u + r(' −) ) = X & → X' = X + u + " ; r(' −1) + ; r(1− ) + ; r( −) & ⇒ " X' − ; r(' −1) = X1 & → X' = ( X1+ ; r(' −1) ) = X + u + " ; r(' −1) + ; r(1−) + ; r( −) & ; r( −) = " ( X1 − X ) − ( u + ; r(1− ) ) & → Z%tenems la misma ecuaciGn Aue arri%a = ( u + r int ) = u + " ; r(' −1) + ; r(1−) + ; r( −) & = u + ; r(' −1) + ; r(1− ) + ( p1! = ) man − 5$7 = − − − + = − − − + = ( / 57.2 = ( ) ; ; 5$7 (11 109$31) 1$ 3$03 5$7 mca 1 man u r(' −1) r(1− ) " ( p1 − p ) ! = &man = 0 → (2 / = -man= " ( p1! = ) man − 0 & = ( 57$ − 0 ) = −2.8mca Pérdidas en tubo de as piraci"n : %r2−-= ( p1! = ) man − 5$7 = ( 57$ − 5$7 ) = 2.46m ( ≡ W ) p " ( p1− p ) ! = & + ( D1− D ) 0 + 0$5 .rado d e r eacci"n = 0.5535 '= " ( < 1 ) , para una +Hreacc*& = 109$31 = u u
12
" X1 − ( u + ; r(1 − ) ) = X & → u = ( p1! = ) man + ( D1 − D ) + " ( c1 − c ) ! ( ⋅ g ) & − ; r(1 − ) in tubo de as piraci"n ⇒ ( ≡ ) ⇒ " ( p ≡ p = p atm )( D ≡ D )( c ≡ c ) & Ba altura útil cam%ia : u = 57$ + 0$5 + " ( 3$ − ) ! ( ⋅ 9$1 ) & − 3$03 = 10$51 m − 3 Peje= " ( =⋅ ⋅ u ) ⋅ ( < ⋅ m ) & = " ( 910 ⋅17$3 ⋅10$51) ⋅ ( 0$9 ⋅ 0$9 ) & ⋅10 = 17065 (W
10) Una TH Francis de eje vertical estK instalada en una central idroel!ctrica Aue tiene una altura neta de m 4 estK acplada a un alternador Aue ;a de rtar a 500rpm ( f = 50 2! )$ Ba TH desarrlla una potencia en el eje de $5 +W 4 cnsume un caudal de 1m3! s cuand tra%a/a en su punt de dise[ "entrada sin co*ues en el rodete 4 salida radial (c u = 0)&, estimKndse en estas cndicines uns rendimientos ( < = 0$95, m = 0$9 )$ 'n el rodete se ;an medid las siguientes dimensines: Diámetro de entrada, #1 = 1$5m? anco de entrada, %1 = 0$5m? diámetro de salida, # =1m? anco de salida, % = 0$5m? coeficientes de obstrucci"n en la entrada 4 salida del rodete, ( 61 = 6 = 0$9 ) Ba TH dispne de tubo de aspiración 4 estK instalada cn las siguientes cotas geodésicas, re@eridas al ni
Tri&n'los de velocidades a la entrada!salida del rodete
;
Ec1 de Eul er : ( g⋅ u ) = ( u1⋅ c1 u ) − ( u ⋅ c u ) = ( u1⋅ c1 u ) c = " ( g⋅ ) ! u & = "( 9$1 ⋅ 7$ ) ! 5$13 & = 34.27 m ! s u 1 1u
c = c + c = + 3$7 = 35.19 m ! s 1 1 m 1 u w1 = c1 m + ( c1 u − u1 ) = + ( 3$7 − 5$13 ) = 12.15m ! s = = = " 13.14 arctan ( c ! c ) arctan ( ! 3 $ 7 ) º 1 1 m 1 u β1′ = arctan "c1 m ! ( c1 u − u1 )& = arctan " ! ( 3$7 − 5$13 )& = 1$19º º β1 = ( 10º − β1′ ) = ( 10º −1$19º ) = 138.81 º c = c β2 = arctan ( c ! u ) = arctan ( ! 9$ ) = 40.34 2 m = m ! s w2 = c + u = + 9$ = 12.36m ! s "2 = 90º
" X − ; r(− W) = XW & "( ≡ W ) ⇒ ( p ≡ p W = patm )( D ≡ D W = 0 )( c ≡ c W ≅ 0 ) & (2 / = -man= − D − " c ! ( ⋅ g ) & + ; r(− W) = − $5 − " ! ( ⋅ 9$1 ) & + 0$5 = −5.26mca "X − − ; = X & → (1 / = -man= ( p ! = ) man + (D − D1) + "(c − c1 ) !( ⋅ g )& + u + ; r(1− ) = 1 u r( 1 − ) = −5$ + ( $5 − 3 ) + " ( − 35$19 ) ! ( ⋅ 9$1 ) & + 7$ + 3$ = 25.786 mca ( p al rodete ) entrada p = p W + " =⋅ ( D W − D ) & → " ( p − patm ) ! = & = ( p ! = ) man = ( D W − D ) ( p W = patm )(Ver fig $) " X − ; r( −) = X & → %r2−-= " ( p − p ) ! = &man + ( D − D ) + " ( c − c ) ! ( ⋅ g ) & = =0 2 = " ( p ! = ) man + D + ( c ! ( ⋅ g ) ) & − D W − " c ! ( ⋅ g ) & = %r2−-− c / 2⋅ - = = 0$5 − " 1$5 ! ( ⋅ 9$1 ) & = 0.39m ( Pérdidas en el tubo de as piraci"n )( W ≠ )
1)
Una TH de reacción de eje vertical cn las siguientes dimensines: Diámetro de entrada del rodete, 30mm, diámetro de salida, 390mm, anco a la entrada, 95mm, anco a la salida, 100mm, ( α1 = º, β1 = 70º )$ 'l coeciente de obstrucción de ls álabes a la entrada del rodete es 0$5 4 a la salida aprMimadamente igual 1$ Un manómetro situad detrKs de la válvula de admisión de la TH marca una presi"n eAui
• Altura neta ( ) - $% de revoluciones ( n ) - Caudal útil (u ) - Potencia útil ( .e/e ) - $% espec&fico de revoluciones ( ns ) - Pérdidas en el tubo de aspiración (inclu4end las de salida del mism)
1$
• *altura útil ( u ) Aue se perdería si se Auitara el tubo de aspiración, supniend Aue la
energ&a del agua a la entrada del rodete permaneciera constante en am%s cass, así cm la energ&a cinética a la salida del rodete 4 la fricci"n en el mism$ DATOS : ( #1 =0$3m, %1 =0$095m, 61 =0$5 ), ( # = 0$39m, % = 0$1m, 6 = 1 ), ( α 1 =º, β1 =70º ), " c ( cu = 0 ), ' ! ( ⋅ g ) & ≅ 0 , ( p' ! )man = 5mca? ( cotas de altura )" D' = D1 = D = m, DW = 0 &, (
rendimientos ) " ; = 0$9, m = 0$9, < = 1 &? ( pérdidas entre '- ) " ;r('C) = 5 ⋅ ( c m ! g ) &
" X' − ( u + r int ) = X & "( ≡ W ) ⇒ ( p ≡ p W = patm )( D ≡ D W = 0 )( c ≡ c W ≅ 0 ) & r int = ; r (' −1) + ; r (1− ) + ; r (− W) Altura pie !ométrica en ' #esprecia%le Altura net a : H= ( u + r int ) = ( p' ! = )man + D ' − " c ' ! ( ⋅ g ) & = ( 5 + + 0 ) = 29m ( Altura útil )( Altura de Euler ) : Hu = ( ⋅ ; ) = ( 9 ⋅ 0$9 ) = 25.81m Ec1Euler : ( g⋅ u ) = "( u1⋅ c1 u ) − ( u ⋅ c u )& = ( u1⋅ c1 u ) = "( ⋅ #1⋅ n ) ! 0 & ⋅ ( c1 m ! tan α1 ) ⋅ = ⋅ ⋅ ⋅ ⋅ → ( c n ) " ( 0 g tan α ) ! ( # ) & ( 1 ) u 1 1 1 m ( 2 ) : c1 m = " ( u1 − c1 u ) ⋅ tan β1 & = " ( ( ⋅ #1⋅ n ) ! 0 ) − ( c1 m ! tan α1 ) & ⋅ tan β1 entrada ( c1 m ! n ) = " ( ⋅ #1⋅ tan β1⋅ tan α1 ) ! ( 0 ⋅ ( tan α1 + tan β1 ) ) & → ( ) " 300 ⋅9$1⋅5$1 ⋅( tan º + tan 70º ) & ! ( ⋅0$3 ⋅ tan 70º ) ≅ 500 rpm rpm Velocidad de rotaci"n : n = " 0 ⋅ g⋅ u ⋅ ( tan α1 + tan β1)& ! ( ⋅ #1 ⋅ tan β1 ) = 494.5637 ⋅9$1⋅5$1 ⋅ tan º ) ! ( ⋅0$3 ⋅9$537 ) ⇒ Componente meridiona l de c : c = ( 0 ⋅ ⋅ ⋅ ⋅ ⋅ = " ( 0 g tan α ) ! ( # n ) & $ 11 m ! s 1 1m u 1 1 Caudal útil : u = ( ⋅ < ) = ( u )1 = "( ⋅ #1⋅ %1⋅ 61 ) ⋅ c1 m & = ( u ) = "( ⋅ # ⋅ % ⋅ 6 ) ⋅ c m & 3 0.3486 = ⋅ = = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ( ) " ( # % 6 ) c & " ( 0$3 0$095 0$5 ) $11 & m ! s u < 1 1 1 1m = ⋅ ⋅ = ⋅ ⋅ = ( c !c ) ( # ! # ) ( % ! % ) ( 6 ! 6 ) ( 0$3 ! 0$39 ) ( 0$095 ! 0$1 ) (0$5 ! 1) 1$30 1 1 1 m 1m c m = ( 1$30 ⋅ c1 m ) = (1$30 ⋅ $11) = $5 m ! s ( c u = 0 ) ⇒ c2 = c m = 2.8452 m ! s Peje= ( .i ⋅ m ) = " ( =⋅ u ⋅ u ) ⋅ m & = " ( 910 ⋅ 0$3 ⋅ 5$1) ⋅ 0$9 & = 103$07 W 1! − 5! 1! − 5! n! = ( n ⋅ .e/e ⋅ ⋅ 9 ≅ 78rpm ⇒ Irancis lenta ) = 500 ⋅ ( 103$07 ! 735$5 )
" X' − ( u + r int ) = X & " ( ≡ W ) ⇒ ( p ≡ p W = patm )( D ≡ D W = 0 )( c ≡ c W ≅ 0 ) & = ( u + r int ) = ( u + ; r(' −) + ; r(− W) ) = " ( p ' ! = ) man + D ' & "c' ! ( ⋅ g )& desprecia% le 5+ − 5$1 −5⋅" $5 ! ( ⋅9$1 ) & Pérdidas en tubo deaspiraci"n , m inclu4end la de%ida a la c : %r2−-= ( p ' ! = ) man + D ' − u − 5 ⋅ " c m ! ( ⋅ g )& = 1.127
1)
" X − ( + ) = X & → u = ( p' ! = ) manométrica + ⋅ " c m ! ( ⋅ g ) & − ' u r(' ) altura de p = D c =c m #esprecia% le =5⋅" c m ! ( ⋅g ) & = ( p atm ! = ) = D ' + − ⋅ − + = + − ⋅ ( p ! = ) D " c ! ( g ) & ( ; ) ( p ! = ) D " c ! ( g ) & ' ' u r(' − ) ' ( u )sin tubo = ( p ' ! = )man + ⋅ " c m ! ( ⋅ g ) & = 5 − ⋅ " $5 ! ( ⋅ 9$1 ) & = $5 m = + − ⋅ ⋅ − #e la ecuaciGn de ; : ( ) " ( p ! = ) D & 5 " c ! ( g ) & ; r(− W) u cn tubo ' man ' m r(− W) + " $5 ! ( ⋅9$1 ) & − 1$17 ( 5$1− $5 ) " ( u )cn tubo − ( u )sin tubo & = D ' + " c m ! ( ⋅ g ) & − ; r(− W) = 3$5m 5$1 → 100* ( u )cn tubo − ( u )sin tubo 3$5 ⋅100 = 1$73* → = ⋅ = * 100 H 12.73 3$5 → * u ( ) 5 $ 1 u u cn tubo 13) Una TH Francis de fujo radial tiene un rodete de diámetro e'terior 1m 4 diámetro interior 0$75m$ Bs ancos de ls álabes del rodete en las seccines de entrada 4 salida sn, respecti
c1 u = u1− ( c1 m ! tan β1 ) tan α1 + tan β1 tan 10º+ tan 0º = ⋅ = ⋅ = ⋅ u c c ( 5$7 c ) c = ( c ! tan α ) 1 tan α ⋅ tan β 1 m tan 10º⋅ tan 0º 1 m 1m 1m 1 1 1 1u Velocidade s u1 = " ( ⋅ #1⋅ n ) ! 0 & u = ( # ! #1 ) ⋅ u1 = ( 0$75 ⋅ u1 ) → : periférica s u = " ( ⋅ # ⋅ n ) ! 0 & u = " 0$75 ⋅ ( 5$7 ⋅ c1 m ) & = ( $357 ⋅ c1 m ) Caudal útil : u = ( ⋅ < ) = ( u )1 = " ( ⋅ #1⋅ %1⋅ 61 ) ⋅ c1 m & = ( u ) = " ( ⋅ # ⋅ % ⋅ 6 ) ⋅ c m & c = "( # ! # ) ⋅ ( % ! % ) ⋅ ( 6 ! 6 )& ⋅ c = "( 1 ! 0$75 ) ⋅ ( 0$1 ! 0$7 ) ⋅ 1 & ⋅ c = ( 0$93 ⋅ c ) m 1 1 1 1m 1m 1m c u = u − ( c m ! tan β ) = ( $357 ⋅ c1 m ) − " ( 0$93 ⋅ c1 m ) ! tan 15º & = ( $5 ⋅ c1 m ) → " c = ( c u + c m ) & → c u = c − c m = ( 3$ ⋅ ⋅ 9$1 ) − ( 0$93 ⋅ c1 m ) "( $5 ⋅ c1 m ) = ( 3$ ⋅ ⋅ 9$1 ) − ( 0$93 ⋅ c1 m ) & → Componente radial : c1m= 3.1624 m ! s ( Altura útil )( Altura de Euler ) : u = " ( u1⋅ c1 u ) − ( u ⋅ c u ) & ! g = ( $3 ⋅ c1m ) = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ = ( 1 ! 9$1 ) " ( 5$7 c ) ( c ! tan 10º ) ( $357 c ) ( $5 c ) & m H 22.4397 u 1m 1m 1m 1m
" ( X − ; r (−) ) = X ≡ XW & "( ≡ W ) ⇒ ( p ≡ p W = patm )( D ≡ D W = 0 )( c ≡ c W ≅ 0 ) & ( X − XW ) = ( p ! = ) man + D + " c ! ( ⋅ g ) & = ; r (− W) = = 1$7 m 3 = ( − 0 ⋅ 10 ! 910 ) man + $5 + " $07 ! ( ⋅ 9$1 ) & Pérdidas de%idas a la vsalida : %r− -= " c ! ( ⋅ g ) & = " 3$1513 ! ( ⋅ 9$1 ) & = 0.5062 m − = → = − + − + − ⋅ " ( X ; ) X & ; " ( p p ) ! = & ( D D ) " ( c c ) ! ( g ) & r ( −) r (−) p = p + " =⋅ ( D − D ) & → " ( p ! = ) + D & = D → ustitu4en d en ; W man W r (−) atm ≡ %r2−-= ( p ! = ) man + ( D − D W ) + " ( c − c ) ! ( ⋅ g ) & = %rd*fu!or ⇒ D W =0 =; m r (− W) − D W − " c ! ( ⋅ g ) & = ( %r2−-− %r−-) = ( 1$7 − 0$50 ) = 1.1938 = ( u + r int ) = u + ( ; r (' −1) + ; r (1−) + ; r (−) ) = u + " ( 0$03 ⋅ % ) + ; r (1− ) + ; r (−) & − − ⋅ + = ( ) " ( 0$03 ) ; & u % r (−) ≡ = = m % % 1.8782 rrode&er 1 − 2 = ( 5$1 − 0$0 ) − " ( 0$03 ⋅ ) + 1$193 &
c1 u = " ( g⋅ u ) ! u1 & = 19$37 m ! s c1 m = ( c1 u ⋅ tan α1 ) = ( 19$37 ⋅ tan 0º ) = 7$0 m ! s c1 = c1u + c1 m = 19$37 + 7$0 = 0$595 m ! s > = c + ( u − c ) = 7$0 + ( 0$59 − 19$37 ) = 7$150 m ! s = ( c ! sen β ) 1m 1 1u 1m 1 1 > c u $07 1$05 17$995 m ! s β arctan ( c ! u ) $3º = + = + = = = \ @rma de u = " ( u1 − u ) − ( >1 − > ) + ( c1 − c ) & ! ( ⋅ g ) = " (0$59 − 1$05 ) − la Ec1Euler − − + − ⋅ = ( 7$150 17$995 ) ( 0$595 $07 ) & ! ( 9 $ 1 ) 0 $ 0 m
15) Una TH Francis %a/ un salto de 5m 4 rtand a 30rpm da una potencia de 109000CV cn un rendimiento total de 0$93$ 'l rendimiento idráulico es 0$95$ 'l coeficiente de la velocidad absoluta a la entrada del rodete es 0$$ 'l ángulo de salida de ls álabes del distribuidor es 1º 4 el ángulo de salida de la corriente absoluta del rodete es 90º (el agua sale sin componente acimutal ) (cu = 0) +alcular: Diámetro ( #1 ) del rodete - Altura ( %1 ) del álabe a la entrada del rodete
" ( α = 90º ) ⇒ ( c u = 0 ) & ⇒ ( g⋅ u ) = " ( u1⋅ c1 u ) − ( u ⋅ c u ) & = ( u1⋅ c1 u ) = ⋅ = ⋅ ⋅ = ⋅ ⋅ u " ( g ) ! c & " ( g ) ! c & ( # n ) ! 0 u 1u ; 1u 1 1 #el ( 2 ) = ⋅ = ⋅ ⋅ ⋅ ⋅ : c ( c cs α ) ( ] g ) cs α entrada 1 u 1 1 c 1 1 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 g 0 9$1 5 0$95 D = ; = = 2.4635 m 1 ⋅ n ⋅ ] c ⋅ ⋅ g⋅ ⋅ cs α1 ⋅ 30 ⋅ 0$ ⋅ ⋅ 9$1 ⋅ 5 ⋅ cs 1º 1 Potencia útil %tenida en el e,e de : .e/e = ( .⋅ t ) = ( =⋅ ⋅ ) ⋅ t → = " .e/e ! ( =⋅ ⋅ t ) & Caudal útil Aue atra
1) Una TH cn rodete de @lu,o radial de 150cm de diámetro 4 un ángulo de ls álabes en la secciGn de entrada de 75º, apr
DATOS : ( % = 5m, = 11m3! s ), " #1 = 1$5m, ( α1 = 1º, β1 = 75º ), ( c u = 0 ) &, ( tubo diusor )" #ent = # =1$m, #sal = # =$1m, Dent = D =$m, ( p )man = -7(Pa&? (rendimientos)
( ; = 0$9, m = 0$97, < = 0$9 )? pérdidas " ;r(8C') = ( 0$05 • % ), ;r(c$e) = ( 0$015 • % ), ;r(dis) = ( 0$01 • % ) & 050.r%lemas 4 eMKmenes HHHC3L8Y+ ! pag$11 17) Bs diámetros e'terior e interior del rodete de una TH de flu,o radial sn, respecti
20 ( c ! g ) &
050.r%lemas 4 eMKmenes HHHC3L8Y+ ! pag$7
1) Una TH Francis de eje vertical rta a 1000rpm 4 se alimenta a tra
• ( 8 ) #urante su @uncinamient cn caudal nominal , calcular: Altura neta ( ) - Caudal ( u ) Aue circula a tra
• ( X ) +nsiderand Aue el rodete tiene en su entrada un diámetro de 1$m 4 un anco de 0$1m cn ( 61 = 1 ), 4 en su salida, un diámetro de 0$5m 4 un anco de 0$m 4 ( 6 = 1 ), calcular, para su @uncinamient cn el caudal nominal 4 ( cu = 0 ): "riángulo de velocidades en la entrada 4 salida del rodete - Presi"n en la entrada ( p 1 ! ) 4 a la salida del ( p ! ) del rodete
Ba TH estK situada a determinada cota de altura respect al ni
ademKs, el agua sale del rodete cn (es una pérdida) Q", ( 'c )& pueden cn un tubo de aspiración , recge el agua Aue sale del rodete 4 la lle
22 ( A ) in T#A ( ) +.7 rec&o c*l8ndr*co
