Ejercicios de Tuberias

 

Cgcrokoki Cgcro koki 437 C` cl skstcfh fistrhmi c` lh ajurh, ouh`mi cl ohumhl mcsmc cl mcpiskti H hl `imi

prk`okphl M cs mc 8=>l/s, lh prcskÿ` c` M cs mc 8,:3ej/of4. _c qukcrc hufc`thr cl ohumhl bhsth 82=l/s, oi` u`h prcskÿ` c` M mc ?,3> ej/of4. ¡Xuc mkhfctri mcdc tcr`cr lh tudcrkh mc 83>>f mc li`jktum, quc bh mc oiliohrsc c`trc D y O c` phrhlcli (mkdughmh c` lk`ch mksoi``uh c` lh ajurh), oi` cxkstc`tc mc ?>of mc mkhfctri, phrh quc sc oufplh` lhs oi`mkoki`cs cxkjkmhs9 Tchlkzhr cl cgcrokoki hplkoh`mi lh couhokÿ` mc Bhzc` & Zkllkhfs. H

Hplkoh`mi Dcr`iullk mcsmc lh supcraokc lkdrc (H) mcl th`quc H bhsth cl pu`ti (M) sc c`c7 4

4

  6 p   + z + Q    + b  ( (  pο    + z + Q  4j ο  4j  M

 M

 H

 H

a   H ∔ M )

 M

 H

C` cl pu`ti (H) cl èrch mcl th`quc cs fuobi fhyir quc lh mc lh tudcræh, pir li th`ti sc pucmc mcsprco mcs prcokhr khr lh vcliok vcliokmhm mhm y oif oifi i cl th`quc th`quc cstè hdkcrt hdkcrti i h lh htfis htfiscrh crh c`ti` c`ti`ocs ocs lh prcskÿ prcskÿ` ` fh`ifëtrkoh c` (H) cs ocri, tifhrcfis mc rccrc`okh cl pu`ti M hsæ quc lh hlturh hllæ cs ocri, qucmh7 4

 z 6 pο    + Q    +b  ( ( ∔ ) 4j  M

  M M

 H

a   H  M

Lh couhokÿ` mc përmkmhs mc Bhzc` & Zkllkhfs csthdlcoc7

b  ( (

a   H ∔ M

X . )6∘ 8> , 1:2. O  m

8 ,234

  L

8 , 234

= ,2:

^hrh cl prkfcr ohsi sc pucmc mctcrfk`hr lh clcvhokÿ` rclhvh mc lh supcraokc mcl th`quc, hsk7

 p

 M

6 8 , :3

  ej

of

4

√

;2>11,3^h 8

  ej

of

Cl pcsi cspcoæaoi mcl hjuh cs7

6;2>1 hjuh

ο 

 @  ?

f

4

68:8181 , ?:3  ^h

 

^ir li th`ti lh hlturh cstèoh cs7

 p ο 

 M

6 8:8181 , ?:3  ^h 6 8: , 3> f   @ 

;2>1 f

hjuh

f

?

Hbirh sc mctcrfk`h lh vcliokmhm mc`tri mc lh tudcræh pir ohmh trhfi, pir oi``ukmhm7

X 6 Q   .  H 6Q   ..

ώ  .

m

4

⇔

=

=.X

Q  6

ώ  .

m

4

^hrh cl ohumhl sc rcmuoc hl _.K7

f

?

f

 l 8 X 6 8=>  .  6 > , 8= s 8>>> l

?

s

^hrh lis mkèfctris kjuhl7

 M 6=> of .   8 f

6 >,= f

8

8>> of   8f 6>,? f  M 6?> of . 8>> of 4

Hbirh lh hlturh mk`èfkoh mcl ÷lfi trhfi (M) cs7

f =√> , 8=

?

s

Q  6  M

ώ .

f

( >,? f )

68 , ;2>3 s

4

Hbirh 7 4

4

Q   6  M

4. j

(

8 , ;2>3  f

 )

s

4√; , 28

 f

s

6 >,4 f

4

^hrh lhs pcrmkmhs cxkstc` mis trhfis mc mkcrc`tcs mkèfctris pcri oifi cstè` c` scrkc cl ohumhl sc fh`c`c oi`sth`tc hsæ quc sc sufh` lhs përmkmhs mc ohmh trhfi7

b

a   (( H ∔ D )

68> , 1:2.

4=>> 8 , 234

8>>

.

> , 8=

8 , 234

= , 2:

>,=

  688 , 383; f

 

b( ∔

83>>

68> , 1:2.

a  D  M )

8?>

8 , 234

.

> , 8= >,?

8 , 234

  68: , ;:4 f

= , 2:

Lucji lhs përmkmhs tithlcs si`.

b

a   (( H ∔ M )

6

b

a   ((  H ∔ D )

+

b

a  ( ( D ∔ O )

6 88 , 38; f + 8: , ;:4 f 6 4; , =22??48 f

Oi` csths përmkmhs pimcfis mctcrfk`hr hbirh lh oith mcl th`quc H7

 z

>,4f f + 4; , =22??48  H 68: , 3> f + >,4

f6=: , 82;= f

Hbirh phrh lh scju`mh oi`mkokÿ` sc c`c u` trhfi c` phrhlcli, phrh cstc ohsi sc c`c7

 p

 M

6 ?,3

  ej

of

4

√;2>11,3^h 6?=?4?4 , :3 ^h 8

  ej

of

4

Lh hlturh cstèoh hbirh cs7

 p ο 

 M

6 ?=?4?4 , :3  ^h 6 ?3 , >>4? f   @ 

;2>1 f

hjuh

f

?

Oifi hbirh bhy mis tudcræhs sc irfh u` `imi c` cl pu`ti M y pir tciræh lh vcliokmhm hllæ cs ocri, pir li quc `i bhy hlturh mk`èfkoh. 4

Q    6>   M M

4. j

^hrh cl `ucvi ohumhl sc rcmuoc hl _.K7 ?

f

?

X 6 82=   l . 8 6 > , 82= s 8>>> l

fs

Oi` cstc ohumhl sc pucmc mctcrfk`hr lhs përmkmhs mcl prkfcr trhfi mc tudcræhs7

b

68> , 1:2.

a   ((  H H ∔ M )

4=>>

8>>

8 , 234

.

> , 82= >,=

8 , 234

= , 2:

  68; , >;? f

Oi` cstis mhtis sc pucmc` mctcrfk`hr lhs përmkmhs phrh lhs tudcræhs c` phrhlcli7 =: , 82;=4 f 6 ?3 , >>4? f + > + 8; , >;? f +

Xucmh7

b

a  ( (  phrhlcli )

 

b  ( (

a   phrhlcli )

6∔1,; f

Oifi lhs përmkmhs si` `cjhvhs `i cs piskdlc oufplkr oi` lhs mis oi`mkoki`cs.

Cgcrokoki 417 S` skstcfh mc hlfhoc`hfkc`ti y rcjulhokÿ` mc hjuh csth oi`irfhmi pir mis

cfdhlscs k`tcroi`cothmis fcmkh`tc u` skstcfh mc tudcrkhs y pir u` rcdishmcri c` cl scju`mi cfdhlsc, cl ouhl cfpkczh h u`oki`hr ouh`mi lh oith mcl cfdhlsc cxocmc =283,3> = 283,3> fctris. Mhtis7 @kvcl mcl cfdhlsc 87 3>>> f @kvcl mc lh orcsth mcl rcdishfkc`ti cfdhlsc 47 =283,3>f @¾ tudcrk h 8 4 ? =

Li`jktu m L, (f) ;>> 3>> 1>> 2>>

Mkhfctri k`tcr`i, m (f) 8,4 >,; >,3 >,:3

es

Ef

>,>43 >,>43 >,>43 >,>43

?,? 4,8 =,1 41

_k cl ohumhl h trhvcs mc lh tudcrkh ? cs X? 6 ? f?/s, mctcrfk`hr sk cl rcdishmcri mcl cfdhlsc 4 cstè u`oki`h`mi ( y lh oith mcl cfdhlsc) y ohloulhr cl ohumhl pir ohmh u`h mc lhs tudcrkhs. Lh ]6 8: ¾O. Mc`tri mc lis oicaokc`tcs mc pcrmkmhs. _iluokÿ`7

8

4

Hplkoh`mi Dcr`iullk mcsmc (8) bhsth (4) sc c`c7 4 8

8

4 4

4

  6   + z + Q   + b  ( ( ∔ )  pο   + z + Q  ο  4 j  p 4j 8

4

a  8 4

 

Oi Oifi fi sc tr trht hth h mc u` cf cfdh dhls lsc c ou ouyh yh supc supcra raok okc c cstè cstè cxpu cxpucs csth th h lh htfi htfis scrh crh lhs lhs pr prcs cski ki`c `css fh`ifëtrkohs c` lis pu`tis (8) y (4) si` ocri (>), mc kjuhl fh`crh lh supcraokc mc hfdis cfdhlscs cs fuy jrh`mc c` oifphrhokÿ` oi` cl mkèfctris mc lhs tudcræhs hsæ quc lh vcliokmhmcs c` sus supcraokcs cs mcsprcokhdlcs v8 y v4 si` ocri (>), hsæ qucmhræh skfplkaohmh lh couhokÿ` mc Dcr`iullk.

 z 6 z + b  ( ( ∔ ) ⇔ z 6 z ∔b  ( ( ∔ ) 8

a  8 4

4

4

a  8 4

8

^hrh cl oèlouli mc lhs përmkmhs sc cfplch cl fimcli mc Mhroy-Zcksdhob k`oluyc`mi lhs pcrmkmhs fc`ircs7 4

 4

b

a   (( 8 ∔ 4 )6

∘ e   Lm . Q  ∘ e  . 4j   +

s

f

Q  4j

_c pucmc ohfdkhr lis Ef mc lis hoocsirkis pir li`jktumcs mc tudcræhs rcoths cqukvhlc`tcs hsk7

4

Q   L  L e   L e  . 4j e  e  e  m   . Q  4j m 4

b

atudcrkh6

b

h

ahoocsirki ⇔

  6

s

f

  ⇔

h

s

  6

f

f⇔

.m

h6

s

e 

^hrh cl ohsi mc Lh tudcræh 8 sc c`c7

 L

?,?√8,4 f h ,8

6

> , >43

  6

832 , 3 f

Hsæ sc bhoc oi` lhs itrhs tudcræhs hsæ sc c`c7 c `c7

 L

h ,4 6

4,8√>,; f   6:3 , 1 f > , >4 >43 3

=,1√>,3 f h ,?6 > , >43   6;4 f

 L

41√> , :3 f :2> >f   6:2 h , =6 > , >4 >43 3

 L

^ir li th`ti sc pucmc k`olukr csth li`jktum mc tudcræh cqukvhlc`tc mc cstis hoocsirkis h lis trhfis rcotis mc tudcræhs

 L  L  L t 6

tr +

h

 L

t , 8 6;>> f + 832 , 3 f68>32 , 3 f

 L

t ,4 63>> f + :3 , 1 f63:3 , 1

 L

t ,? 61>> f +;4 f61;4 f

f

y idtc`cr lh lc`tum tithl sc c`c7

 

 L

t , = 62>> f+ :2> f 6832> f

^ir li quc lhs pcrmkmhs scrkh`7 4

 L Q  b   (( )6∘ e  m   . 4 j

X6Q  .  .  H 6Q  .  .

t 

a  8 ∔ 4

s

ώ  .

m

4

=

⇔

Q  6

=.X

ώ  .

, pir pir it itrh rh phrtc phrtc sc sh shdc dc quc7 quc7

m

4

lucji 4

X Q  681 ώ  . m , sustuyc`mi b 6∘   e s . L  . 2  . X j . ώ  . . m =.

4

4

=

4

lh

couhokÿ`

mc

përmkmhs

sc

c`c7

4

t 

a   (( 8 ∔ 4 )

c`

3

Sshrcfis u`h vhrkhdlc huxklkhr huxklkhr mc`ifk`hmh Tcskstc`okh bkmrèulkoh bkmrèulkoh quc csth mhmh pir7

 T 6

C`ti`ocs lhs përmkmhs si`7 bkmrèulkohs7

b   ((

a  8 ∔4 )6

e s . L . 2 j . ώ  . . m

∘ T . X

t 

4

3

4

, hbirh sc ohloulh` ohmh u`h mc lhs rcskstc`okhs

4

s  T 6   >  f, >43 √8>32 , 3 f√2 6> , 2:2:8>1 f √ ώ  √( 8, 8,4 4f) ; , 28 s 3

4

8

3

4

4

> , >43√3:3, 1 f√2 46

 T

; , 28

 f

s

4

√

3 64

ώ  ( >, >,; ; f) 4

√

s , >8?32? f

3

4

> , >43√1;4f√2

 T

?6

; , 28

 f

s

4

ώ  ( >,3 f ) 4

√

√

s =3 , :=4442 f

36

3

4

s 6 8? , :3?=18  T 6   f f √ ώ  √( > , :3 f ) ; , 28 s  

> , >43 √832> f √2 4

=

4

3

3

 

Lhs tudcræhs 8 y 4 sc c`ouc`trh` c` scrkc y sc oufplc quc cl ohumhl quc phsh pir hfdhs cs cl

X 6X 6X

fksfi rcskstc`okhs7

b

8

4

  y sc pucmc` pucmc` rcsilvcr rcsilvcr oifi u`h u`h tudcræ tudcræh h ÷`koh ÷`koh sufh`m sufh`mi i lhs mis

8,4

 T + T ) . X b + b 6 T . X + T . X 6( T 4

4

6

a  8,4  8,4

a   8 8

a   4 4

8

4

 T

s > , 2:2:8>1 f

8,4 6

4

8

8

4

4

3

+

s 4 , >8?32? f

3

4

4 8,4

 Xucmh7

( T

 T

8,4 6

 T )

8+

4

4

s 4 , 2;44;?1 f

6

3

^hrh cl ohsi mc lh tudcræh ?, yh sc oi`ioc cl ohumhl, pir li ouhl sc pucmc` ohloulhr mkrcothfc`tc lhs pcrmkmhs c` csh tudcræh7 4

b 6 T . X a  ?  ?

?

?

(  )

 f s  . ? f s

4

6 =3 , :=4442

3

?

4

6 =88 , 12>>34 f

^hrh lhs tudcræhs ? y = oifi mcscfdioh` th` ocrohs c` cl cfdhlsc C4, sc pucmc` oi`skmcrhr quc cstè` c` phrhlclis, hsk quc lhs përmkmhs c` hfdis trhycotis sc pucmc` oi`skmcrhr lhs fksfhs7

b 6b a  =

a  ?

6 =88 , 12>>34 f

Oifi yh sc oi`ioc` lhs përmkmhs c` lh tudcræh =, c`ti`ocs sc pucmc mctcrfk`hr cl ohumhl quc phsh h trhvës mcl cl hsk7

b  T . X  = 6 a  =

=

4

=

b X  T

∗

 = a  =

6

6

=6

=

=88 , 12>>34 f 4

s 8? , :3?=18 f

4; , ;?42?

6

f

?

s

3

 Ak`hlfc`tc pir oi``ukmhm lis ohumhlcs sc rclhoki`h` c` cl `imi li quc c`trh cs piskvi y li quc shlc cs `cjhvi, hsæ7 ?

?

X

8,4

6

X + X 6 ? fs   + 4; , ;?42? fs   6 ?4 , ;?42 ?

=

f

?

s

[h tc`kc`mi cl vhlir mc lis ohumhlcs sc pucmc mctcrfk`hr lhs përmkmhs mc lh tudcræh 8 y tudcræh 4 4

b  T X a  8  8 6

8

4 8 6>

, 2:2:8>1

s f

3

.

(

 )

 f ?4 , ;?42 ?4 s

?

4

6 ;3?

, >4433 f

 

4

b  T X a  4  4 6

4

4 4 64

, >8?32?

(

 )

 f s ?4 , ;?42 ?4 f s 3  .

?

4

6 482?

, 2: f

Lucji lhs pcrmkmhs tithlcs si`7

b ( ∔ )6 b + b + b b 6?3=2 , 3:4; f a  8 4

 8 a  8

 4 a  4

 ? a  ?

6 ;3? , >4433 f + 482? , 2: f + =88 , 12>>34 f

a ( 8∔ 4 )

^ir li quc pimcfis mctcrfk`hr quc lh hlturh mcl scju`mi cfdhlsc cs7

 z 6 z ∔ b   (( ∔ ) 63>>> f ∔?3=2 , 3:4; f 4

 z

8

4 68=38

a  8 4

, =4: f

Oifi lh hlturh mcl scju`mi cfdhlsc cs fc`ir quc cl rcdishmcri mcl fksfi `i cstè u`oki`h`mi.