Cooling Tower. Application_ Treybal

Cooling Tower Application, according

Cooling Tower Application, according

1 1 DDaattaa 2 2 TToowweer r hheeiigghhtt 3 3 NNTTU U aannd d HHTTUU 4 4 TToowweer r aarreeaa 5 5 CCoommppeennssaattiioon n wwaatteer  r   6

6 OOppeerraattiinng g ddiiaaggrraamm 7

7 CCoooolliinng g ttoowweer r sscchheemmaattiicc 

 !!""aammpplle e 77##11 $

$ CCoommmmeenntt 1

1%% TTrraappee&&ooiiddaal l rr''llee

(e)# 1 (e)# 1 (e)# 2 (e)# 2 (e)  (e)  Colla*orations Colla*orations

Treybal Treybal [1][1] (e+# c,c# %#11#2%14 (e+# c,c# %#11#2%14 Index Index

Data )or cooling tower application Data )or cooling tower application -ain e.'ations and res'lts

-ain e.'ations and res'lts

Cooling Tower height/ NTU and HTU Cooling Tower height/ NTU and HTU 0reecross sectional s'r)ace o) tower  0reecross sectional s'r)ace o) tower 

Compensation/ elimination/ e+aporation and entrainment )ow rates Compensation/ elimination/ e+aporation and entrainment )ow rates !.'ili*ri'm c'r+e and operation lines

!.'ili*ri'm c'r+e and operation lines chema

chema

Tre*al e"ample o) cooling tower  Tre*al e"ample o) cooling tower 

Data o) thermal power/ not 'sed in e"ample 7#1 Data o) thermal power/ not 'sed in e"ample 7#1 N'merical integration with the trape&oidal r'le N'merical integration with the trape&oidal r'le

ages 274 to 22# Cooling o)

ages 274 to 22# Cooling o) water with air water with air 

acing height and )reecross sectional s'r)ace o) a acing height and )reecross sectional s'r)ace o) a towertower (e)erences

(e)erences

Comments and contri*'tions )rom Comments and contri*'tions )rom

Cooling Tower Application Data Cooling Tower Application Data This application will be realized

This application will be realized with following nuerical data !"ote 1#$with following nuerical data !"ote 1#$

Data for nuerial exaple Data for nuerial exaple

ater )low rate entering the tower 

ater )low rate entering the tower  2 82 8 1155 g g 99ss T

Teemmppeerraatt''rre e oo) ) wwaatteer r eenntteerriinng g tthhe e ttoowweer r aat t tthhe e ttoop p ::22;; 4455 <<CC D

Drr  **''ll* * tteemmppeerraatt''rre e oo) ) aaiir r eenntteerriinng g tthhe e ttoowweerr 33%% <<CC 

eet t **''ll* * tteemmppeerraatt''rre e oo) ) aaiir r eenntteerriinng g tthhe e ttoowweerr 2244 <<CC 

ooccaal l hheeiigghht t aa**oo++e e sseea a llee++eel l H H 88 %% mm##aa##ss##ll## -="im'm cooling temperat're will *e de)ined

-="im'm cooling temperat're will *e de)ined withwith

5

5 >>  ?ir to wat

 ?ir to water )low rate ratier )low rate ratio shall *e @r@ times itso shall *e @r@ times its m

miinniimm''m m ppoossssii**lle e ++aall''ee r r 88 11##55 The compensation water entering the sstem wil ha+e a

The compensation water entering the sstem wil ha+e a

tteemmppeerraatt''rree 11%% AACC

a

annddaahhaarrddnneessss 55%%%% ppppmm

The in the sstem circ'lating water so'ld ha+e a The in the sstem circ'lating water so'ld ha+e a m

maa""iimm''mm hhaarrddnneessss 22%%%%%% ppppmm

--aasss s ttrraannss))eer r ccooee))))iicciieennt t iin n tthhe e aaiirr 66##22!!%%55 T

Toowweer r ee))))eeccttii++e e hheeaat t oor r mmaasss s ttrraannss))eer r ss''rr))aaccee a a 88 55%%%% mmBB99mm 

ii..''iid d ''nniit t mmaasss s ))lloow w rraattee 22##77  ?ir 'nit mass )l

 ?ir 'nit mass )low rateow rate 2#%2#%

 ?ir molec'lar mass

 ?ir molec'lar mass 2#$62#$6 g9molg9mol

Notes Notes

1# This data has

1# This data has *een taen )rom 1E/ *een taen )rom 1E/ e"ample 7#1/ e"ample 7#1/ pages 2721#pages 2721# 2# The data @F@ is not 'sed/ since it wo'ld

2# The data @F@ is not 'sed/ since it wo'ld *e in contradiction with other inp't data :ee sheet $;*e in contradiction with other inp't data :ee sheet $; F F88 2277%%  t t_22 8 8 tdb tdb_%1%1 & & twb twb_%1%1&& a di))erential temperat're

a di))erential temperat're∆∆t a*o+e air wet *'l* temp#t a*o+e air wet *'l* temp# ∆∆t 8t 8

tt_compcomp 8 8 da da _Gc Gc 8 8 da da _G- G- 8 8 

_{GmolGmol} 8 8 _{mol 9 : mmol 9 : m}22_s;s;

 _'' 8 8 _{g9:smg9:sm}22_;; I I_'' 8 8 _{g9:smg9:sm}22_;; --_airair88

'elp (ariables 'elp (ariables C Coooolliinng g TToo tate 1 tate 1

ater lea+ing the tower  ater lea+ing the tower 

24 24 2 82 8 5 5 >> 2 2$$ <<CC tate I1

tate I1 CopensatioCopensation wn water ater 

 ?m*ient air entering

 ?m*ient air entering the tower the tower  11** ++CC

3 3%% <<CC **** pppp 2 244 <<CC H H 88 %%##%% mm h h 8 8 icroG!nthalpGtd*GtwicroG!nthalpGtd*Gtw*GH*GH h h 88 JJKK??UU!!LL MM99gg FF "

" 8 8 icroG?*sol'teH'miicroG?*sol'teH'miditGtd*GtwditGtd*Gtw*GH*GH

"

"88 JJKK??UU!!LL gg99llgg

-ass trans)er coe))icient -ass trans)er coe))icient

-ass trans)er coe))icient in the air -ass trans)er coe))icient in the air

6#2!%5 6#2!%5  ?ir molec'lar mass  ?ir molec'lar mass

2#$6

2#$6 g9molg9mol

-as trans)er per ilogram -as trans)er per ilogram

6#2!%5 6#2!%5 2#$6 2#$6 g9molg9mol %#%%1 %#%%1 >a >a %#%%1 %#%%1 a a 88 55%%%% mmBB99mm %#$% %#$% t t_11 8 8 tw*tw*_I1I1  ∆∆tt tbh tbh_%1%1 ∆ ∆t 8t 8 tt22 8 8 t t_11 8 8 ta

ta_copcop & &

t*s

t*s_I1I1 8 8 dada_cc & &

t*h t*h_I1I1



_{GmolGmol} 8 8 _{mol 9 : mmol 9 : m}22_s;s;

t

t_-1-1 & &

--_airair88



_ 8 8 _{GmolGmol}  -  -_air air  dada_.. & &



_{GmolGmol} 8 8 _{mol 9 : mmol 9 : m}22_s;s;

--_airair88



_ 8 8 _{g 9 : mg 9 : m}22_s;s;

rod'ct > rod'ct >_aa

> >_a 8a 8  _ 8 8 _{g 9 : mg 9 : m}22_s;s; > >_a 8a 8 _{g 9 : mg 9 : m}33_s;s;

 W

 W

Bl

Bl

(e+# c,c# %#11#2%14 er )chea 15 g 9 s 45 <C  ?ir  ater  3% <C 24 <C ater  2$  ?ir  /*** pp tdb_%1 & twb_%1

L1

I1 Cooling tower  I2 2

wdown water: B

G1

L2

ºC

1

Cooling Tower height

Tower pac0ing height [/] Height o) Tra

The pacing height :; o) a tower can *e calc'lated as

1E/ e.# :7#53;/ page 276

HTU 8 with HTU 8 1E/ page 276

Number of

 The numbe

is alulate

and

$heet %&#' N

eamle of

*esult of NT

NTU

NTU

-Tower pacin  8 HTU 8 NTU 8  8

 NTU

HTU

Z

=

⋅

 M   HTU 

 =

 A a k   M  G  HTU   y  B d 

⋅

⋅

⋅

=

 NTU 

 HTU 

 Z 

 Eq

 HTU 

 Z 

 NTU 

 H 

 HTU 

 H 

 Z 

G

 Z 

a

k 

h

h

dh

 N 

 NTU 

tOG tOG d   y h h tOG

⋅

=

=

=

=

⋅

⋅

=

−

=

=

∫ 

)

54

.

7

.

(

' ' '* ' ' 2 ' 1

(e+# c,c# %#11#2%14

s)er Unit @HTU@  P Tower pacing height mE

)low rate o) dr air :is a constant; g9sE

molar mass o) air g9molE

mass trans)er coe))icient in the air mol9:mBQs;E aP e))ecti+e heat or mass trans)er s'r)ace mB9mE  ? P )ree crosssectional s'r)ace o) the tower mBE JK?U!L m enthalp in the air phase 8 enthalp o)  

h'mid air :in the *'l phase; M9gE

ransfer Units

hRP enthalp in the air phase :iP at ther *o'ndar/ M9gE

of transfer units (NTU)

that is/ in sat'rated condition;

 b! numerial inte"tation#

HTUP mE

TU% resents a alulation

NTUP E

he NTU#

'*scripts

S P s'*stance dr air  

U eamle (sheet &# NTU + ,

dP dr air  

2P top o) the tower  

./0LU '

1P *ottom o) the tower  

  height HTU  NTU JK?U!L m JK?U!L  JK?U!L m I_d P -_S P _ P I_d9 :-_S _ a  ?; hR P

Height o) Trans)er Unit :also/ H_tI; N'm*er o) Trans)er Units :also/ N_tI;

(hL3a ' hL3b) 4 N 5

Σf()  A a k  G  y  B d 

⋅

⋅

⋅

Height o) Trans)er Units HTU

IP molar mass )low rate per 'nit area mol :mBs;E IRP mass )low rate per 'nit area g :mBs;E

aP e))ecti+e heat or mass trans)er s'r)ace mB 9 mE  ?P )ree crosssectional s'r)ace o) the tower mBE

HTU 8 IR 8 2 g dr air 9: smB; 2#$6 g9mol 6#2!%5 mol9:mBs; a 8 5%% mB9m HTU 8 2#2 m IR 8 I mol 9 :m;s; ;  -_Sg 9 molE IR 8 I  -_S g 9 :mBs;E

-_SP molar mass o) air g9molE

_P mass trans)er coe))icient in the air mol 9 :mBs;E

IR 9 : -_S  _  a ; -_S 8 _ 8

[ ]

[ ]

m a k   M  G  HTU  m  s kg   A G m  A a k   M  G  HTU  m m m  s m kmol  kmol  kg   s kg   A a k   M  G  HTU   y  B d   y  B d   y  B d 

⋅

⋅

=









⋅

=

⋅

⋅

⋅

=

























⋅

⋅

⋅

⋅

⋅

⋅

⋅

=

' G' G' rate flow mass unit Air te ! "ntro#u$in 2 2 % 2 2

[ ]

m

a

k 

 M 

G

 HTU 

 y  B

⋅

⋅

=

'

"T and 'T calculations

Colun1 Colun /

!.'ili*ri'm c'r+e )or sat'rated air# The c'r+e is drawn 'sing the )'nction

45 <C icroG!nthalpGtd*G)GH

'sing a relati+e h'midit

2$ <C ) 8 1%% 

(angeP and the local heigt

H 8 % m#a#s#l#

N'm*er o) sections ? temperat're range is selected to The range will *e di+ided in a n'm*er @N@ co+er a range ?S/ with

o)sections 25 <C

N8 6 47 <C

Colun 2

Operation line )or the minim'm possi*le air )low rate IRs#min# :i#e#/ r 8 1;

Setween *oth temperat'res/ @N1@ 0or this tpe o) operation/ the operation temperat'res are inserted to de)ine line will ha+e the minim'm slope that the N sections# ?ll section are de)ined wo'ld allow it to to'ch the e.'ili*ri'm with the same temperat're di))erential# c'r+e :will *e tangent to this c'r+e;#

?ir enters at the *ottom o) the tower at Temperat're di))erential contition I1P

<C 3% <C

45 <C 24 <C

2$ <C H8 % m#a#s#l#

16 > JK?U!L M9g

JK?U!L 

ection temperat're increment ?ir lea+es at the top :case r 8 1; heet / shows the calc'lation o) the 16 > enthalp o*tained when accomplishing

N 8 6  with this condition#

2#67 > Th calc'lated +al'e is

JK?U!L M9g Temperat're at point @i1@ at a temperat're

45 <C

Ta

1 2 3 3a 4 5

i.'id !.'ili*ri'm Operation Operation

temperat're c'r+e )or line )or line )or   sat'rated air# r 8 1 r 8 1#5 ater temperat're at inlet o) tower  :top;

t_28 h_air/sat 8

ater temperat're at tower o'tlet :*ottom; t_18

t_2 V t_1

t _? 8 t_S 8 Col'mn 1 starts with temperat're @t_2@

and ends with temperat're @t_1@#

∆t_8 t_2 t_1 t*s_I1 8 t_28 t*h_I1 t_18 ∆t_8 h_I1 ) _I1 8 ∆t_Gect8 ∆t_9 N ∆t_8 ∆t_Gect8 h_IoR 8

t_i1 8 t_i  ∆t_Gsect t_IoR 8

∆h 8 ∆h 8

47 JK?U!L

Top#:2; 45 JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

42#33 JK?U!L JK?U!L 3(A-45 JK?U!L JK?U!L

3$#67 JK?U!L JK?U!L 3(A-45 JK?U!L JK?U!L

37#%% JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

34#33 JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

31#67 JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

 Sottom :1; 2$ JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

25#5 JK?U!L

25 JK?U!L

Note1# 3

The e.'ili*ri'm data 'sed * Tre*al is !nthalp potential taen )rom a graphic :0ig# 7#5a; Col'mns 3a and 5 The e.'ili*ri'm data res'lting )rom the (e)# 4 :2#4; 'se o) the )'nctions/ is similar to the

data )rom ?shrae 0'ndamentals

!"ample ?shrae 0'nctions Tre*al t <C ts M9g ts M9g ts M9g

45 214#16 212#$6 216

2$ $4# $4#41 1%%

Note 2

Wntegration according the the trape&oidal r'le ee sheet 1%#

'eight of tower pac0ing N'm*er o) Trans)er Units @

0rom Trei*al/ !.'ation :75 :7#51a;

:7#51*;

The n'merial integration o) per)ormed * means o) the integration method#

the airwater mi"t're )or the act'al case/ ?ccording this method/ the that is/ in this case )or r 8 1#5# is reali&ed as it is shown in

6/ 7 and # where HR₂ and HR₁ are the enthalpies o) 

(

 _y

_i

 I 

 I 

 I 

 I 

 NTU 

a  y b  y

−

=

∫ 

&

1

& &

∫ 

₋

=

' 2 ' 1 ' ' '  H   H 

 H 

i

H 

dH 

 NTU 

[ ]

m

 H 

 H 

dH 

a

k 

G

 Z 

 H   H  i  y S 

∫ 

−

⋅

⋅

=

' 2 ' 1 ' ' ' '

:7#51c; )ollowing e.'ation :7#51d; NTU 8  ?lso where JK?U!L JK?U!L N 8 6 JK?U!L

N'merical res'lts shown are )rom ne"t NTU 8 JK?U!L calc'lation sheets#

Tre*al 2E res'lt is

NTU 8 3#25

The di))erence comes )rom the icrometric properties#  ?lso/ the n'merical integrat

is not indicated#

Coparison between the exaple calculation table and the table fro Treibal

Calculation table, using )icroetric functions$

1 2 3 4 5 6

C'r+a de Xnea de Xnea de e.'ili*rio para operaciYn operaciYn aire sat'rado para r 8 1 para r 8 1#5

t

M9g M9g M9g M9g 19:M9g;

25 JK?U!L

25#5 JK?U!L

2$ JK?U!L 72 72 JK?U!L JK?U!L

31#67 JK?U!L $6 7#7 JK?U!L JK?U!L

34#33 JK?U!L 11$ 1%3#4 JK?U!L JK?U!L

37#%% JK?U!L 143 11$#1 JK?U!L JK?U!L

3$#67 JK?U!L 166 134#$ JK?U!L JK?U!L

42#33 JK?U!L 1$% 15%#6 JK?U!L JK?U!L

45#%% JK?U!L 213 166#27 JK?U!L JK?U!L

Ta*le )rom Tre*al 1E/ page 2%

1 2 3 4 5 6

C'r+a de Xnea de Xnea de e.'ili*rio para operaciYn operaciYn aire sat'rado para r 8 1 para r 8 1#5

t

M9g M9g M9g M9g 19:M9g; :h_Ga  h_G*; 9:2 h_GinGr81#5 8 h_{Go't8r81#5}8 Σ):"; 8 ∆h8 19∆h

h

_air/sat

h

_operGr81

h

_operGr81#5 h_air/sat h_opGr81#5

∆h8 19∆h

h

_air/sat

h

_operGr81

h

_operGr81#5 h_air/sat h_opGr81#5

⋅

⋅

−

=

2

 ' 

 ' 

 N 

h

h

 NT U 

 L a  L b

Z 

=

 HTU 

⋅

 NTU 

[

m

]

 HTU 

=

G

_S' 

k 

 _y⋅

a

[

m

]

 HTU 

=

G

_S

 M 

_B⋅

k 

 y⋅

a

⋅

 A

2$ 1%%#% 72 72 2#% %#%357 31#67 114#% $6 $2#% 22#% %#%455 34#33 12$# 11$ 1%6#5 23#3 %#%42$ 37#%% 147#% 143 121#% 26#% %#%35 3$#67 166# 166 135#5 31#3 %#%31$ 42#33 1$1#% 1$% 14$#5 41#5 %#%241 45#%% 216#% 213 163#5% 52#5 %#%1$%

icroG!nthalpGtd*G)GH:td*/ )/ H; icroG!nthalpGtd*G)GH:td*/ )/ H;

Operationline Col'mn 3a Col'mn5

ater lea+es the tower at Dri+ing enthalp di))erence at a point @i@ Dri+ing entha 2$ <C

The air properti at Col'mn 4 Colun 6

2$ <C Operation line )or r 8 1#5 (eciproc o) d 24 <C The line starts )rom the same point N Colun 7 H 8 % m#a#s#l# with the properties at the inlet de)ined as Coe))icients )

is thestateI1#

JK?U!L M9g Is 8

The point N in diagram isP r 8 1#5 Colun 8

2$ <C JK?U!L g as9s N'merical int

JK?U!L M9g Is 8 JK?U!L g as9s

The operation line)or r 8 1/ is the lope o) line with r 8 1#5

straight line  8 m 8   Cpw 9 Is :7#54;

lope o) operation line witrh r 8 1  8 15 g9s ?ir )low rate Cpw 8 4#16 M9:g>; 0rom heat *a

JK?U!L M9g Is 8 JK?U!L g as9s

JK?U!L M9g  8 JK?U!L :M9g;9> 45 <C !"it enthalp 2$ <C 8 JK?U!L :M9g;9> :7#54; m8 JK?U!L M9g JK?U!L :M9g;9>  8 JK?U!L :M9g;9> m8 JK?U!L 45 <C 8 15 gag'a9s 2$ <C Cpw 8 4#16 M9:g>; JK?U!L M9g 1E/ !./ :7#54

JK?U!L g as9s Col'mn 4 represents a straight line Col'mn 3 represents a straight line *etween the points N and OR

*etween the points N and O 2$ <C

2$ <C JK?U!L M9g P li.'id )low

JK?U!L M9g 45 <C

45 <C JK?U!L M9g

JK?U!L M9g

le 1$ Tower pac0ing height calculation

6 7 

N'merical ?ir conditions in the tower/)or r 8 1

integration Conditions at the *ottom o) the tower :poi

coe))icient :2; oint @1@

):";

2$#% <C t_18 _∆h i 8 hair/satGi hopGr81Gi ∆hi 8 t*s_I1 8 t*h_I1 C_i 8 h_I18 r I_r81 t_N 8 I_r81 8 h_N 8 ):"_i; 8 conP r 8 I 9 I_min _r81 8 :h_I2R  h_I1; 9 :t_2  t_1; h_I2R 8 h_I1 8 t_2 8 t_N 8 :h_I2  h_I1; 9 :t*s_I2  t*s_I1; _r81 8 h_I2 8 h_I1    :t*s_I2  t*s_I1; I_ 8 I_r81 8 19m    c_pw _r81 h_I1 8 _r81 8 t*s_I2 8 t*s_I1 8  h_I2 8 I_r81 8 I_ P gas )low h_I2 P e"it air e t_N 8 h_I1 P inlet air t_N 8 h_N 8 h_N 8 t_OR 8 c_pw P li.'id sp t_O 8 h_OR 8 t_2 P inlet wate h_O 8 t_1 P e"ir wate 19∆h

C

_i t*s₁ 8

(

(

m c  L G t  t  h h m c  L G h h G bottom to  H   H   ! S   L  L G  ! S  G S  ai# 

⋅

=

−

−

=

⋅

⋅

=

−

⋅

∆

=

∆

1 2 2 2  1  2

Conditions at the top o) the tower :point 2

JK?U!L 1 JK?U!L oint 2R

JK?U!L 2 JK?U!L

JK?U!L 2 JK?U!L 45 <C

JK?U!L 2 JK?U!L JK?U!L M9g

JK?U!L 2 JK?U!L

JK?U!L 2 JK?U!L Height a*o+e sea le+el/ )rom sheet @1# D

JK?U!L 1 JK?U!L H8 % m#a#s#l#

JK?U!L

JK?U!L M9g

2

To be re9iewed

!"planation and

traight line/ d'e e 8 1 (e)# 4 :27#3#1;

TU@ Height o) Trans)er Unit @HTU@ Height o) Tra

1; HTU 8 with Is 8 NTU is trape&oidal HTU 8 2#% g9:mBs; a8 integration %#$ ?8

thecol'mns HTU 8 2#2 m HTU8

Tre*al 2E res'lt is

t*s_2R 8 h_2R 8

Σ):"; 8

!nthalp h_OR/ )rom sheet 6# h_%R 8 -_S 8 IR_ 9: a; _Gmol 8 IR_ P _a P _{g 9 : m}3_s;

)

⋅

dI 

y

 HTU 

=

G

_S' 

k 

 _y⋅

a

 HTU 

=

G

_S

 M 

_B

⋅

k 

 y

⋅

a

⋅

 A

(

)

∫ 

⋅

−

=

a &   & " "   i &

#"

"

"

1

 NTU

Height o) pacing tower  aP  ? P  8 HTU  NTU HTU 8 2#2 m NTU 8 JK?U!L M9g 8 JK?U!L m M9g

Tre*al 2E res'lt is

8 7#22 m

 

 

the +al'es o)  ion method

Trape&oidal n'merical integration r'le

7  N'merical integration coe))icient

):";

1 JK?U!L NTU8 2 JK?U!L where 2 JK?U!L 166#3 M9g 2 JK?U!L 72#% M9g 2 JK?U!L N8 6 2 JK?U!L JK?U!L 1 JK?U!L

JK?U!L NTU 8 JK?U!L 

7 

N'merical integration coe))icient

):";

Tre*al ta*le di))ers )rom the calc'lation ta*le in the +al'es o) the psichrometric

-S P _ P  N;  Σ):";

C

_i  NTU8:h_2  h_1;92N :):"₁;  2):"₂;  2):"₃;  Z# 2):"_N1;  ):"_N; ; :h_Ga  h_G*; 9:2 N;  Σ):"; h_GinGr81#5 8 h_{Go't8r81#5}8 Σ):"; 8 Σ):"; 8

C

_i

∑

 f  

(

 x

)

)

1

(

...

2

2

1

1

2

)

(

1

−

=

=

=

=

⋅

⋅

−

≈

⋅

∑

∫ 

=

 N 

i

 a#a

 g 

 N 

 y

i

 a#a

 g 

 f 

 g 

 N 

a

b

dx

 x

 f 

i i k   N  k  i b a

Z 

=

 HTU 

⋅

 NTU 

1 %#%3575 n'merical integration method/ where 1 %#%4545 the n'merical integration coe))icienst are

1 %#%42$2

1 %#%346 The n'merical integration 'sed is not 1 %#%31$5 indicated and Tre*al gi+es as a

1 %#%241% )inal res'lt a NTU +al'e

1 %#%1$%5

%#23767 NTU8 3#25 

not re.'ired :or C_i 8 1;

(e+# c,c# %#11#2%14 age 1 o) 4

lp di))erence at a point @i@

ri+ing enthalp di))erence

r n'merical integration

1at*othends 1

2 in the other elements

To *e re+iewed

gration elements !"planation

  lance

;/ page 277

ate g9 sE# :ass'med const#;

age 2 o) 4 nt @1@ in diagram; :Col'mn 1; h_air/satGi h_opGr81#5Gi h_I18 C_i  :19∆h_i; rate g as9 sE

nthalp :top; M9gE nthalpa :*ottom; M9gE

ci)ic heat M9:g>;E r temperat're :top; ACE

temperat're :*ottom; ACE

)

(

)

(

)

(

)

)

h h t  t  t  t  c  L G G  L  L  L  L  ! iq"id 

−

−

−

⋅

⋅

=

1 1 2 1 2 1 2 1

R in diagram;

:Col'mn 1; :Col'mn 3;

  ata@

1E/ e.# :7#54;/ page 277

age 3 o) 4 s)er Unit @HTU@

JK?U!L g as9s 2#$6 g9mol 6#2!%5 5%% mB9m JK?U!L mB JK?U!L m I_9 :-_S _ a  ?; mol 9 : m2_s;

 HTU 

=

G

_S

 M 

_B

⋅

k 

 y

⋅

a

⋅

 A

molar mass o) air g9molE mass trans)er coe))icient in the air mol9:mBQs;E e))ecti+e heat or mass trans)er s'r)ace mB9mE )ree crosssectional s'r)ace o) the tower mBE

:ree;cross sectional surface of tower 

 ?rea o) cross sectional s'r)ace eleted area

0rom *orh res'lts/ the smal

 8 sho'ld *e selected/ to ens'

P li.'id )low rate g9sE

the indicated +al'e o)  sectional s'r)ace;P g9 : mBs;E

 ?P area o) cross section %#$%

o

 ? 8  9 ' ? 8 JK?U!L

8 15 g9s

2#7 g9:mBs;

 ? 8 5#56 mB

Using the gas )low rate  ? 8

IP gas rate g9sE

sectional s'r)ace;P g9 : mBs;E  ?P area o) cross section

Is 8 JK?U!L g as9s

2 g9:mBs;

 ? 8 JK?U!L mB

_'  ?

+al'e o) the prod'ct @_a@ _' P 'nit )low rate :)or 'nit o) cross

_a 8

_' 8

I_ 9 I_'

I_' P 'nit )low rate :)or 'nit o ) cross

(e+# c,c# %#11#2%14 lest +al'e re that the mB as at least g 9 : m3_s;

Copensation water 

Cosidering a compensation and a !ntrainment loss rate @@/ water that is contin'o's elimination/ the mass *eing transported with the e"it air/ *alance is lea+ing )rom de top o) the tower as a

:a; loss o) water#

 ?pplication !+aporation rate @!@/ water that is

 ? water hardness *alance is e+apotated in the air )low prod'cing 49aporation the cooling o) the water )low

:*; 0rom e.'ation :d; 1# ?*sol'te h  ?ss'ming tha

and there)ore sat'ratedat

The enthalp

:#c; calc'lationTa

 ?ss'ming init !liminating - )rom :a; and :#c;

the correspon ass'med tem :d;

H 8

- Pcompensationrateg9hE h8

S P elimination rate g9hE Now/ 'sing 

! P e+aporation rate g9hE

 P entrainment loss rate g9hE ith calc'lat

the correspon circ'lating water g9gE or ppmE can *e *e cal

t 8 compensation water g9gE or ppmE

H 8

!limination rate @S@/ re.'ired to replace water with a ma"im'm allowa*le salts

content with )resh water with the in this 2# ?*sol'te h

water e"isting salt content# This is 0rom sheet 2

called the compensation rate#

φ_O = h_O 8 t_O 8 t_O 8 φ_O = temperat're t

da_CP hardness weight )raction o) da_-P hardness weight )raction o)

φ_O = "_a2 8 "_a1 8

$ 

 E 

 B

 M 

=

+

+

(

)

%   M 

 B

$ 

da

da

 M 

⋅

=

+

⋅

(

)

 M  % 

da

da

$ 

 B

 M 

=

+

⋅

(

)

 M  %  da da $   B $   E   B

+

+

=

+

⋅

 E 

 B

=

⋅

$  da da da  E   B da da da  E  $   B da da da  E  $   B da da  E  $   B  E  da da $  da da  B  E  da da $  da da  B  E  da da $  da da $  da da  B da da $   E  da da $  da da  B $   E  da da $  da da  B  B da da $  da da  B $   E   B  M  %   M   M   M  %   M  %   M   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  %   M  % 

−

−

⋅

=

−

+

−

=

−

−

−

=

−

−

−

=

−

 

 

 

 



 

 

−

⋅

−

=

 

 

 

 



 

 

−

⋅

−

 

 

 

 



 

 

−

⋅

=

 

 

 

 



 

 

−

⋅

−

−

⋅

=

 

 

 

 



 

 

−

⋅

−

−

⋅

=

 

 

 

 



 

 

−

⋅

−

−

⋅

=

⋅

−

⋅

+

⋅

=

+

+

1 1 1 1 1 1 1

3#H'miditchange !ntrai g9g :e; JK?U!L g9g JK?U!L g9g JK?U!L g9g ater  ! 8

 rate <4< Dr air )low rate :sheet 2;

Is 8 JK?U!L g as9s 'midit o) e"it air JK?U!L g9g t the lea+ing air is *asicall ! 8 JK?U!L g9s

oint @O@ 4ntrainent loss <=<

1%%  To estimate the entrainment losses/ at this point/ )rom the one ass'mes that these losses are *le 1/ is a percentage [ o) the water )low rate

JK?U!L M9g [ 8 %#2 

iall a temperat're +al'e The water )low rate is

3% AC 8 15 g9s

ding enthalp )or this  8   [

perat're is with  8 15 g9s

4%#% AC [8 %#%%2 

1%%  8 %#%3% g9s

% m#a#s#l# 4liination rate <><

JK?U!L M9g S 8

l+er/ )ind a +al'e o) the ! 8 JK?U!L g9s

 8 %#%3% g9s

d e"it air temperat're 5%% ppm Tre*al res'lts

ding a*sol'te h'midit 2%%% ppm ! 8 %#3465

c'lated S8 JK?U!L g9s  8 %#%3 4%#% AC Copensation rate <.< S 8 %#%55 1%%  - 8 -8 %#462 % m#a#s#l# S 8 JK?U!L g9s JK?U!L g9g  %#%3% g9s 2%%% ppm

'midit o) inlet air 5%% ppm

. & 3(A-45 0g?s JK?U!L g9g ∆"_21 8 "_a2  "_a1 "_a2 8 "_a1 8 ∆"_21 8 I_  ∆"_21 ∆"_21 8 !  : da_- 9 :da_C  da_-; ;   O to o*tain that h 8 hO da _G- 8 da _Gc 8 :S  ;  da_C 9 da -da _Gc 8 da _G- 8

$ 

da

a

da

 M  %   M 

−

−

M, da_M

Comensation

water

(e+# c,c# %#11#2%14 ment water   ?ir  !limination water  g9s g9s g9s g9s 1 I1 Cooling tower  I2 E W, da_C B, da_C

@peration Diagra

1 2 3 3a 4 5 6

C'r+a de Xnea de Xnea de

e.'ili*rio para operaciYn operaciYn aire sat'rado para r 8 1 para r 8 1#5

M9g M9g M9g M9g M9g 19:M9g;

25#% JK?U!L

25#5 JK?U!L

2$#% JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

31#7 JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

34#3 JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

37#% JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

3$#7 JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

42#3 JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

45#% JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L JK?U!L

47#% JK?U!L

h

0rom sheet 1 <C M9g M9g <C 25#% JK?U!L % JK?U!L 25#5 JK?U!L 5% JK?U!L 2$#% JK?U!L 1%% JK?U!L 31#7 JK?U!L 15% JK?U!L

34#3 JK?U!L 2%% JK?U!L 0rom calc'la

37#% JK?U!L 3$#7 JK?U!L 42#3 JK?U!L 45#% JK?U!L 47#% JK?U!L Note

Col'mn 3a shows that the res'lting operation line is n ot e"actl tangent to the sat'ration c'r+e# 0or tangenc/ the minim'm di))erence sho'ld *e %#

 ? negati+e +al'e indicates that the sol'tion line is c'tting the sat'ration c'r+e#

at'ration c'r+e )or air at a height a*o +e sea le+el

H 8 % m#a#s#l#

:)8 1%% ;

∆h 8 ∆h8 19∆h

t

_

h

_air/sat

h

_operGr81 h_air/sat h_opGr81

h

_operGr81#5 h_air/sat h_opGr81#5

t

_ h_T t_T t_N8 h_N8 t_O 8 t_OR 8 t_T 8 h_T 8 oint OR :h_OR/ t_OR 8 h_OR8 oint O :h_O/ ) t_O 8 h_O8 12#6

olnom

<C M9g

25 JK?U!L 77# JK?U!L JK?U!L

26 JK?U!L 7$#6 JK?U!L JK?U!L

2$ JK?U!L $4#5 JK?U!L JK?U!L

32 JK?U!L 1%#5 JK?U!L JK?U!L

34 JK?U!L 125#% JK?U!L JK?U!L

37 JK?U!L 143#7 JK?U!L JK?U!L

4% JK?U!L 164#$ JK?U!L JK?U!L

42 JK?U!L 1#4 JK?U!L JK?U!L

45 JK?U!L 214#2 JK?U!L JK?U!L

47 JK?U!L 235#2 JK?U!L JK?U!L

t

_

h

_air/sat 26 6#6 2#6 8#6 9#6 #6 16#6 Tangent point

Equation of the satutarion curve

h

=

0.1659

⋅

t 2

−

4.7921

⋅

t 

+

92.926

Value of the enthalpy in the saturation curve at the point of tangency

(

t

=

t _T 

)

h

=

0.1659

⋅

t _T 2

−

4 .7921

⋅

t _T 

+

92.926

(

 Eq. a

)

Tangent at any point of the satutarion curve

dh

dt 

=

0.1!

⋅

t 

−

4.7921

Tangent  at the point of tangency

(

T 

)

dh

dt 

 |

t =t T 

=

0.1!

⋅

t T 

−

4.7921

"lope of the operation curve# $hich is tangent to the saturation curve

h_T 

−

h _N  t T 

−

t  N 

=

0.1!

⋅

t  N 

−

4.7921

hT 

−

h N 

=

(

0.1!

⋅

t 

−

4 .7921

)

⋅

(

t 

−

tn

)

h

=

(

0.1!

⋅

t 

−

4.7921

)

⋅

t _T 

−

t  _N 

+

h _N 

(

 Eq. b

)

Equating

(

Eq. a

)

 $ith

(

Eq. %

)

0.1659

⋅

t _T 2

−

4.7921

⋅

t _T 

+

92.926

=

(

0.1!

⋅

t _T 

−

4.7921

)

⋅

(

t _T 

−

tn

)

+

hn 0.1659

⋅

t T  2

₋

4.7921

⋅

t T 

+

92. 926 &hn

=

(

0.1!

⋅

t T 

−

4.7921

)

⋅

t T  &

(

0.1!

⋅

t T 

−

4 .7921

)

⋅

tn 0.1659

⋅

t _T 2

−

4.7921

⋅

t _T 

+

92. 926 &hn

=

0.1!

⋅

t 2_T 

−

4.7921

⋅

t _T 

−

0.1!

⋅

tn

⋅

t _T 

+

4.7921

⋅

tn 0.1!

⋅

t _T 2

−

0.1659

⋅

t _T 2

+

4.7921

⋅

t _T 

−

4.7921

⋅

t _T 

−

0.1!

⋅

tn

⋅

t _T 

+

4.7921

⋅

tn

−

92.926

+

hn

=

0 0.1659

⋅

t _T 2

−

0.1!

⋅

tn

⋅

t _T 

+

4 .7921

⋅

tn

−

92.926

+

hn

=

0 a

=

0.1659 b

=

&0.1!

⋅

tn c

=

4.7921

⋅

tn

−

92.926

+

hn a

⋅

t T  2

₊

b

⋅

t T 

+

c

=

0

HR :gas+apor mi"t're; M9g as 4 O 3 N 0ig're 2 HR₂ HR₂ HR₁ HR₁

i.'id temperat're t_L AC

t_L1 t_L2

*

$

 T

U

(

)

' & H  t  _L ( t  L& H +*)

(

'

)

& i i H  t 

7  N'merical integration coe))icient

):";

1 JK?U!L 2 JK?U!L 2 JK?U!L 2 JK?U!L 2 JK?U!L 2 JK?U!L 1 JK?U!L JK?U!L 2$ <C JK?U!L M9g 45 <C 45 <C ion/ )'rther down

JK?U!L <C JK?U!L M9g 45 <C JK?U!L M9g 45 <C JK?U!L M9g %#165$  t\2  aa 8 %#165$ ** 8 4#7$21

C

_i Σ):"; 8

alc'lated )'rther down;

om sheet 3; h_sat 8 27#6 &6#6 &7#6 86#6 87#6 6#6 2#6 8#6 9#6 :#6 16#6 12#6 f() -*; - 6

Figure 1.- Operation D

Liuid temperature !"C#    E  n   t    $  a    l  p   %   a    i  r   -  &   a   p   o   r    !    '    (    )    '  g    d  a    # =uilibrium ur<e@ for saturated air

?eratin" line with

r -1

?eratin" line with r - 1#7

B 0 N (t_N@ h_N) ? ? ? ?(t_o@h_o) OR:t_OR/h_OR ? t_T O T $_T =uilibrium ur<e 0'B t_O*

t

_O+ ?

cc 8 $3#$3

!nthalp at the tangent point T

h 8 %#165$  t\2  4#7$21  t  $ JK?U!L <C

JK?U!L M9g

oint N

Operation line )or r 8 1

where @r@ is the ratio *etween the act'al mass )low rate and the minim'm )low rate#

The line starts at a point de)ined * the inlet air properties :point N in operating diagram; also called state @I1@

tate I1 :oint N; 0romsheet 3 3% <C tn 8 2$ 24 <C hn 8 JK?U!L H8 % m icroG?*sol'teH'miditGtd*Gtw*GH JK?U!L g9g a8 %#165$ JK?U!L M9g * 8 $#6222 c 8 JK?U!L JK?U!L <C t_T8 h_T 8 t_d*I18 t_w*I18 "_I1 8 t_T8 : *  :*\2  4ac;\%#5 ; 9 : 2a ; "_I1 8 h_I1 8 h_N 8 t_T 8 27 &6 &7 86 87 76 f() -*; - 6

(

)

 N  N  N



t

4.7,21

t

%%1-.



4.7,21

/

%%1-.



+

⋅

=

⋅

+

=

=

c

b

a

a

ac

b

b

⋅

⋅

−

±

−

=

2

4

t

2 T



a

⋅

t 

_T 2

+

b

⋅

t 

_T 

+

c

=

n ,% . ,2 tn 7,21 . 4 $ tn /.%%1-  .105, a

+

−

⋅

=

⋅

=

=

JK?U!L JK?U!L 2$ JK?U!L 45 JK?U!L !nthalp h_OR h_%R 8 h_N  :h_T  h_N; h_N8 h_T8 t_N8 t_T8 t_oR8 h_%R 8

(

)

(

 _N 

)

 N   N   N   N   N   N   N   N   N   N   N 

t 

t 

h

h

t 

t 

h

h

t 

h

t 

h

−

⋅

−

−

+

=

−

⋅

−

−

=

−

−

−

=

−

−

o' T T ' o' T T ' T T o' ' ' '

t

t





t

t





t



t



1

i!ure

rom

t

ture

at tem3era



ntal3

flow.

air

minimum

for

line

3eratin!

(e+# c,c# %#11#2%14

4#7$21  t  $2#$26 76#6

iagram of Cooling Tower

=uilibrium ur<e >ol!nomial (=uilibrium ur<e) Column  ?# L# r - 1#7 h3tan"ent t3tan"ent ;

M9g M9g <C <C <C M9g :t_T  t_N;  :t_oR  t_N;

!"ample 7#1/ )rom 1E/ pages 27 to 21

!"ample 7#11# ? plant re.'ires that 15 g 9 s :1$4 l* 9 min; o) cooling water will )low thro'gh a condensation e.'ipment )or distillation] th's eliminating 27%   :5527% St' 9 min; )rom the condensers# The water lea+es the condenser at 45 A C# To re'se water it is planned to cool it * contact with air in a coo ling tower o)

ind'ced dra)t#

The design conditions are 3% A C inlet air dr *'l* temperat're and 24<C wet *'l* temperat're# The water is to *e cooled to 5 A C *elow the wet *'l* temperat're o) the air :th's/ to 2$ <C; ] a ratio o) air9steam o) 1#5 times the minim'm +al'e will *e 'sed#

ater compensation will come )rom a dam at 1% A C/ with a hardness o) 5%% ppm :parts 9 million; dissol+ed solids#

The circ'lating water cannot ha+e a hardness greater than 2%%% ppm# (egarding the pacage that is to *e 'sed/ it is e"pected that the +al'e :>  a; will *e

%#$% g 9 :m  s; )or a speed o) the li.'id o) at least 2#7 g 9 :m  s; and )or a gas speed o) 2#% g 9 :m  s; :1$$1 and 1474 l*m 9 :hr  )tB; respecti+el#

Calc'late the dimensions o) the paced section and water compensation re.'ired#

SasisP Cross section 1 mB/ Is 8 2#% g9:mBs;# Th dri+ing )orce H1  H1 is o*tained at )re.'ents inter+als o) t in )ig're 7#13/ as it is shown#

the enthalp o) the incoming air is taen )rom the )ig're 7#5a :or 'sing the

The opetaing chart/ 0ig're 7#13/ contains the air enthalp c'r+e at sat'ration# Wn this graph/ the point N represents the condition at the *ottom o) the tower 

 :T1 8 2$ A C and H1 8 72%%% M 9 g dr air;# The operating line will pass thro'gh N and will end in T1 8 45 A C#

0or the minim'm +al'e o) IsR/ the operating line will ha+e the minim'm slope that will to'ch the e.'ili*ri'm c'r+e/ and th's it will pass thro'gh the point O/ where H2 R8 2%$ 5%% M 9 g dr air# There)ore/ the slope o) the line is ORN

where IsRmin 8 7#31 g dr air 9 s# 0or gas )low o) 1#5 times the minim'm/ Is 8 1#5  7#31 8 1%#$7 g dr air 9 s # There)ore/

and H2 R8 163%%% M 9 g dr air/ plotted at point O# There)ore/ the operating line is ON# 0or a li.'id )low o) at least 2#7 :g 9 m  s;/ the cross section

sho'ld *e 15 9 2#7 8 5#56 m# 0or a gas )low o) at least 2#% g 9 :mB  s;/ the cross section is 1%#$7 9 2#% 8 5#5 m# There)ore/ the last +al'e :5#5; is 'sed/ then in this case the minim'm )low o) li.'id will e"ceed the minim'm an so ens'ring that +a 8 %#$% #

ol'tion# 0ig're 7#12 represents the )lowchart o) the operation# The h' midit and appropriate )'nctions;#

The area *elow the c'r+e is 3#25# 0rom e.'ation :7#54;

The paced height  isP  8 7#22 m Wn this case/

3#25 IsR 9 : >a;

2#% 9 %#$ 8 2#22 m  ?lso/ )rom e.'ation :7#54;

8 7#22 m

2#22 m

3#25

Data )rom the last two col'mns are plotted with H as a*scissa#

N_tOI 8 H_tOI 8 H_tOI 8 N_tOI 8 9H_tOI H_tOI 8 N_tOI 8

Data of theral power, not used in exaple 7$1 Wn Tre*al e"ample 7#1/ there is )ollowing dataP the water sho'ld elimnate @ 27%  : 5527% St'9min; )rom the condensers@

S't Tre*al does not 'se this in)ormation#

ea+ing water temperat're/ )or a thermal power o) 

F8 27% 

This inp't data does not agree with the other  inp't dataP

ater inlet temperat're ate o'tlet temperat're ater mass )low rate

W) this data

F8 27% 

is a re.'irement/ then/ the temperat're o) the le+ing water is a calc'lated +al'e and cannot *e an inp't data#

!nthalp o) inlet water !nthalp o) le

ocal height a*o+e sea le+el F 8

H8 % m F928

ocal am*ient pres're

p 8 1%1/325 :1 2/25577!5  H;\5/255

H8 % m#a#s#l# F 8

p8 1#%13 *ar 28

Temperat're o) water entering the tower at the top :2; 45 <C

!nthalp o) entering water peci)ic +ol'

45 <C p8  8 1#%13 *ar   JK?U!L M9g +8 h_1 8 h_2 8 h_1 8 t_2 8 t_2 8 h_1 8 h_2 8

a+ing water Temperat're o) lea+ing water   p 8 1#%13 *ar   + 8 JK?U!L m9g t 8 JK?U!L <C JK?U!L M9g 27%  15 g9s JK?U!L M9g e 1#%13 *ar   JK?U!L M9g JK?U!L m9g 2  :h_2  h_1; h_2  h_1 h_2  F92

"uerical integration with the trapezoidal rule

"uerical ipleentation

Wll'stration o) trape&oidal r'le 'sed on a se.'ence o) samples :in this case/ a non'ni)orm grid;#

nifor grid

"on;unifor grid

hen the grid spacing is non'ni)orm/ one can 'se the )orm'la

0or a domain discreti&ed into N  _{e.'all spaced panels/ or}N _{1 grid points}a ₈ x 

(e

1E Operaciones de trans)erencia de masa 29e (o*ert !# Tre*al

-cIraw Hill/2%%3

1$; ac0ing height

The pacing height o) a tower can *e calc'lated according 2E/ e.'ation :65;

 8 W P Tower pacin )low rate o) d molar mass o

Naming the )irst term as @Height o) Trans)er mass trans)e

Unit:HTU;@ aP e))ecti+eheat

 ? P )ree crossse enthalp in th :in the *'l p enthalp in th and the second term as the N'm*ert o) Trans)er that is/ in sat Units :NTU;

HTU P Height o) Tra NTU P N'm*er o) Tr

'*scripts

S P dr air  

the pacing height *ecomes  P air phase 8 h

a P top o) the tow * P *ottom o) the

i P corresponds ac0ing height and free;cross sectional surface of a tower 2E

F_ 8K P -_S P _ P W_P W_/i P

Z 

=

(

Q

S

 M 

_B

⋅

k 

 _y

⋅

a

⋅

 A

)

⋅

 _I 

∫

 y , b

 I  _{y , a}

1

(

 I 

 _{y, i}

−

 I 

 _y

)

¿

dI 

 y

 A

a

k 

 M 

& 

 HTU 

 y

 B

⋅

⋅

⋅

=

(

)

y

 y

i

 y

 I 

 I 

dI 

 I 

 I 

 NTU 

a  y b  y

⋅

−

=

∫ 

&

1

& &

Z 

=

 HTU 

⋅

 NTU 

  height mE  air :is a constat; g9sE

) air g9molE

coe))icient in the air mol9:mBQs;E or mass trans)er s'r)ace mB9mE tional s'r)ace o) the tower mBE e air phase 8 enthalp o) h'mid air M9gE

  ase;

e air phase :iP at ther *o'ndar/ rated condition; s)er Unit ans)er Units mid air  er    tower 

Indian institute of technology

eson 27# schrometr traightline law :27#3#1; eson 2# !nthalp potential

httpP99www#iitgp#ac#in9

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ir2%Cond9pd)9([?C2%ect're2%27#pd)  ir2%Cond9pd)9([?C2%ect're2%27#pd) 

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1E Operaciones de trans)erencia de masa 29e (o*ert !# Tre*al -cIraw Hill/ 2%%3 2E 3E httpP99li*rar#)'pm#ed'#sa9WW92%%6952%%6#pd)  4E httpP99nptel#ac#in9co'rses9e*co'rsecontents9WWT2%>haragp'r9(e)2%and2%? httpP99nptel#ac#in9co'rses91121%512$9pd)9([?C2%ect're2%27#pd)  httpP99nptel#ac#in9co'rses9e*co'rsecontents9WWT2%>haragp'r9(e)2%and2%?

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