final thermodynamics project report

8/9/2019 Final Thermodynamics Project Report

http://slidepdf.com/reader/full/final-thermodynamics-project-report 1/18

Anil yarlagaddakiran babu

Phani Raghava Yaswanth kasturi

MCE 521- Applied Ther!dynai"s

T#ERM$%Y&AM'C(

PR$)ECT*luid +!w analysis ,!r divergent "hannel



Table !, C!ntents

1 Abstra"t-------------------------------------------------------------------------------------------------

----------------2

2 &!en"lature-----------------------------------------------------------------------------------------

----------------.

. Pr!ble

stateent----------------------------------------------------------------------------------------------

-----/

/ 'ntr!du"ti!n--------------------------------------------------------------------------------------------

----------------5

5 Analysis-------------------------------------------------------------------------------------------------

-----------------0

Result

------------------------------------------------------------------------------------------------------------

------1.

0 C!n"lusi!n---------------------------------------------------------------------------------------------

---------------1.

1



Re,eren"es---------------------------------------------------------------------------------------------

---------------1/

Abstract

The analysis !, the pr!3e"t is t! deterine the diensi!nless ,!r !, the

entr!py generati!n whi"h is a "!binati!n !, heat +u4 +uid ,ri"ti!n ass

di6usi!n and agneti" +u4 irreversibility distributi!n rati! and be3an nuber

with re,eren"e !, the given paper 7steady agnet!hydr!dynai" +!w in a

divergent "hannel with su"ti!n !r bl!wing8 by 9C :ayek (9 ;ry<hevi"h A

( 9upta and M Re<a and re,eren"e e=uati!n given by pr!, 9!rla The entr!py

generati!n !, ea"h "!p!nent is deterined and then it is ade int!

diensi!nless ,!r and "!bined ,!r t!tal diensi!nless entr!py

'rreversibility distributi!n rati! and >e3an nuber are deterined later

2



Nomenclature:

T Teperature

k Theral "!ndu"tivity

% %i6usi!n "!e?"ient

T@ Abient teperature

heat

B %ebye length

&!rali<ed teperature

D 'rreversibility rati!

Electrical conductivity

F ;ineati" vis"!sity

G Theral di,,usivity

h C!n"entrati!n rate

R 9as "!nstant

>@ Magneti" +u4

>e >e3an nuber

C C!n"entrati!nH Abs!lute Iis"!sity

u (u"ti!n vel!"ity

.



Problem Statement

*ind the diensi!nless ,!r !, the e=uati!ns

(JJgen Kk

T 0

2 ( ∂T ∂ x )

2

+( ∂ T ∂ y )

2

Lµ

T 0 ( ∂u∂ y )

2

L R· DC ∞

·( ∂C ∂ y )

2

L R· DT

0

·( ∂ C ∂ y )·( ∂T

∂ y ) L

σ· B0

2

· u2

T 0

the first term is the entropy generation rate due to heat transfer in the axial direction, the second

term is the entropy generation rate due to heat transfer in the normal direction, the third term is

the entropy generation due to fluid friction, the fourth term is the entropy generation due to mass

transfer and the fifth term is the entropy generation rate due to the combined effects of heat and

mass transfer and last tern is due to magnetic effect.

'rreversibility distributi!n rati! K

Entropy due

Entropy due

¿ friction ¿

¿

heat transfer ¿

>e3an nuber K

Entropy due

¿heat transfer ¿Totalentropy generation

*r! the given re,eren"e paper (teady agnet!hydr!dynai" +!w in a

diverging "hannel with su"ti!n !r bl!wing

/



Introduction

'p!rtant e4a"t s!luti!ns !, the &avier-(t!kes e=uati!ns is ass!"iated with

the steady tw!-diensi!nal +!w !, an in"!pressible vis"!us +uid between

tw! n!n-parallel rigid walls #ere the +!w is "aused by a s!ur"e !r sink !,

+uid v!lue at the interse"ti!n !, the walls leading t! either a divergent !r

"!nvergent +!w 't is ,!und that "!nvergent syetri"al +!w is p!ssible ,!r

any "hannel angle and any nite Reyn!lds nuber At large Reyn!lds

nuber this +!w e4hibits tw! identi"al b!undary layers !n the "hannel walls

with an irr!tati!nal +!w !utside these layers 'n the "ase !, divergent +!w

Ns!ur"e +!wO in a "hannel it is !bserved that ,!r a given "hannel angle

purely divergent +!w is p!ssible !nly when the Reyn!lds nuber ,!r the +!w

d!es n!t e4"eed a "riti"al value hen the Reyn!lds nuber e4"eeds this

"riti"al value the +!w be"!es asyetri" in whi"h there is a regi!n !,

in+!w Nie ba"k +!wO near !ne !, the walls As the Reyn!lds nuber

in"reases ,urther and be"!es very large the siilarity s!luti!ns "!ntain

any regi!ns !, in+!w and !ut+!w The width !, ea"h regi!n !, the !ut+!w

then be"!es s! sall that e6e"ts !, vis"!sity are signi"ant everywhere in

the +!w Thus unlike the "ase !, "!nvergent +!w b!undary layer s!luti!ns

Nwith!ut ba"k +!wO are n!t p!ssible in a diverging "hannel at large Reyn!lds

nuber

5



#ere we are studying steady divergent +!wN*igQ 1O !, an

in"!pressible vis"!us ele"tri"ally "!ndu"ting +uid between tw! n!n-parallel

plane p!r!us walls in the presen"e !, a agneti" eld pr!du"ed by an

ele"tri" "urrent al!ng the line !, interse"ti!n !, the walls The walls are

sub3e"ted t! su"ti!n N!r bl!wingO The e4isten"e !, siilarity s!luti!ns with

su"ti!n N!r bl!wingO at b!th the walls is rst e4pl!red ,!r the ,ull M#% &avier-

(t!kes e=uati!ns 't is sh!wn that su"h s!luti!n with e=ual su"ti!n N!r

bl!wingO at b!th the walls is n!t p!ssible #!wever it is ,!und that b!undary

layer type !, s!luti!n ,!r this +!w is p!ssible at high Reyn!lds nuber

pr!vided that the su"ti!n N!r bl!wingO vel!"ity at a p!int !n any !ne !, the

walls is inversely pr!p!rti!nal t! the distan"e !, the p!int ,r! the line !,

interse"ti!n !, the "hannel walls



0



Analysis

Consider Heat Flux componentQ

k

T 0 [(∂ T

∂ x )2

+( ∂T

∂ y )2

]

∂T

∂ x =

∂ ( θ ! x "+1 )

∂ x

K A NBL1O 4B L A 4BL1 ∂θ

∂ x S K

y

x ( #

$· % )1/2

K A NBL1O 4B L A 4BL1 ∂θ

∂& ·

∂ &

∂ x

∂ &

∂ x=− y

x2 ( #

$·% )1 /2

K A NBL1O 4B L !· x

"+1

x J N-SO

∂ &

∂ x=−&

x

K A NBL1O 4B L A 4B J N-SO

∂T

∂ x K A 4B NBL1O US JV

!=1

k · ( #

$· % )−1 /2

∂T

∂ x K1

k ·( #

$· % )−1/ 2

4B NBL1O US JV



∂T

∂ y= !· x

"+1· ∂ θ

∂ y

K A 4BL1 ∂θ∂& · ∂ &∂ y S K

y

x ( #

$· % )1/2

K A 4BL1 J 1

x ·( #

$·% )1/2

∂&

∂ y=

1

x ( #

$·% )1/2

∂T

∂ y K A 4 B J ( #

$·% )1 /2

∂T

∂ y K1

k ·( #

$· % )−1/2

4 B J ( #

$·% )1 /2

∂T

∂ y K1

k 4 B J

Heat Transfer componentQ

'l (l= k

T 0 ( ∂T

∂ x )2

+( ∂T

∂ y )2

'l (l=

k

T 02 ·

[ 1

k 2 ·

(

#

$· % )

−1

· x2 "

· {( "+1 ) · θ−&·θ ˡ }2+ 1

k 2 · x

2 "·(θ ˡ)2

]

'l (l= 1

k·T 0

2· x

2 " ( #

$·% )−1

· {( "+1 ) ·θ−&· θ ˡ }2+(θ ˡ)2 −−−−−−−−−−−−−−−(1)

W



) s='l (l ·k·T 0

2

x2 " =[( #

$· % )−1

· {( "+1 ) · θ−&·θ ˡ }2+(θ ˡ)2]

Consider uid friction component:

µ

T 0 ( ∂u

∂ y )2

u K ( #

$·x ) · f ( & )

S K y

x · ( #

%·$ )1/ 2

∂u

∂ y=

∂ u

∂ &· ∂&

∂ y

∂u

∂&=( #

$·x ) · f l (&)

∂&

∂ y=

1

x ·( #

%·$ )1/2

∂u

∂ y=( #

$·x ) · f l (& ) ·1

x ·( #

%·$ )1/2

∂u

∂ y=(1% ) · f l ( & ) ·

1

x2

·(#

$ )3/ 2

Friction componentQ

Kµ

T 0 ( ∂u

∂ y )2

1@



Kµ

T 0⌈ (1% )· f l ( & ) ·

1

x2 ·( #

$ )3 /2

⌉2

K µT 0

⌈ (1% )

2

·( f l ( & ))2 · 1 x

4 ·( #

$ )3

⌉−−−−−−−−−−−−−−−(2)

Consider mass diusion component:

R· D

C ∞·( ∂C

∂ y )2

C K > h 4BL1

> K1

D ·( #

$· % )−1 /2

∂C

∂ y =B· x

"+1· ∂ h

∂ y

∂C

∂ y =B· x

"+1·

∂ h

∂ &· ∂ &

∂ y

S K y

x · ( #

%·$ )1/ 2

∂&

∂ y=

1

x ·( #

%·$ )1/2

∂C

∂ y =B· x

"+1·h ·l

1

x ·( #

%·$ )1 /2

∂C

∂ y =

1

D · ( #

$· % )−1 /2

· x "

· h ·l( #

%·$ )1 /2

∂C

∂ y =

1

D · x

"· hˡ

11



ass diusion componentQ

¿ R· D

C ∞· ( ∂C

∂ y )2

¿ R· D

C ∞· [ 1 D · x

"· hˡ]

2

C

D·(¿¿∞)2 · x2 "

· (hˡ )2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−(3)

¿ R· 1

¿

Consider an axial ux and friction component:

R· D

T 0

·( ∂ C

∂ y )·( ∂T

∂ y )

R· D

T 0

· 1

D · x

"· h ·l

1

k · x

"· θ ˡ

R· x

2 "· h ·l θ ˡ

T 0· k −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−(4)

Consider ma!netic component :

σ· B0

2

· u2

T 0

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−(5)

"ntropy !enerator per unit #olumeQ

12



(JJgen Kk

T 0

2 ( ∂T

∂ x )2

+( ∂ T

∂ y )2

Lµ

T 0 ( ∂u

∂ y )2

L R · D

C ∞·( ∂C

∂ y )2

L R· D

T 0

·( ∂ C

∂ y )·( ∂T

∂ y ) L

σ· B0

2

· u2

T 0

) s='l (l ·k·T 0

2

*+

2 K ( #

$·% )−1

· {( "+1 ) ·θ−&· θ ˡ }2+(θ ˡ)2 L

k·T 02

*+

2 ·

µ

T 0

⌈ ( 1% )2

·( f l (& ))2· 1

x4 ·(#

$ )3

⌉ L

C

D·(¿¿ ∞)2 · x2 "

· (hˡ )2

k·T 0

2

*+

2 · R·

1

¿

Lk·T 0

2

*+

2 ·

R · D

T 0

·( 1 D · x "

·hˡ)·( 1k , x

",θ ˡ) L

k·T 02

*+

2 ·

σ· B0

2

· u2

T 0

) s='l (l ·k·T 0

2

*+

2 K [( #

$·% )−1

· {( "+1 ) ·θ−&· θ ˡ }2+(θ ˡ)2] L

k · T 0· µ

*+

2 · ⌈

(1

% )

2

·( f l (& ))2

· 1

x4 ·

(#

$ )

3

⌉ L

C

D·(¿¿ ∞)2 · ( hˡ )2

k·T 02

!2

· R· 1¿ L

R· T 0

!2

· (h ·l θ ˡ ) L

k·T 02

*+

2 ·

σ· B0

2

·u2

T 0

=w K A4B

This is as

&s K &# L &, L & L&hy L&M

here

1.



&# K ( #

$· % )−1

· {( "+1 ) ·θ−&· θ ˡ }2+(θ ˡ)2

&, Kk · T

0· µ

* +

2 · ⌈

(

1

% )

2

·( f l (& ))2

· 1

x

4 ·

(

#

$ )

3

⌉

& K

C

D·(¿¿ ∞)2 · (hˡ )2

k·T 0

2

!2

· R· 1

¿

&hy K R· T

0

!2

· (h ·l θ ˡ )

&M Kk·T 0

2

*+

2 ·

σ· B0

2

·u2

T 0

Irre#ersibility ratio $%&:

Entropy due

Entropy due

- =¿ friction ¿¿ heat tansfer ¿=

) f

) .

'rreversibility rati! NDO K

k ·T 0

· µ

*+

2 ⌈ ( 1% )

2

·( f l (& ))2

· 1

x4 ·( #

$ )3

⌉

[( #

$· % )−1

· { ( "+1 ) ·/−&·/ˡ}2+(/ˡ)2]

1/



>e3an nuber K

Entropy due

¿heat transfer ¿Totalentropy generation

= ) .

) s

>e3an nuber K

C

[( #

$· % )−1

· {( "+1 ) ·θ−&· θ ˡ }2

+(θ ˡ)2]+ k · T 0

· µ

*+

2 · ⌈ ( 1% )

2

·( f l (& ))2

· 1

x4 ·( #

$ )3

⌉+k·T

0

2

!2

· R·

D·(¿¿∞)2 · (hˡ )2+ R· T

0

!2

[( #

$· % )−1

· {( "+1 ) · /−&·/ˡ }2+(/ˡ)2]¿

15



'esultQ

) s=¿ [( #

$· % )−1

· {( "+1 ) ·θ−&· θ ˡ }2+(θ ˡ)2] L

k · T 0

· µ

*+

2 · ⌈ (1% )

2

·( f l (& ))2

· 1

x4 ·(#

$ )3

⌉ L

C

D·(¿¿∞)2 · (hˡ )2

k·T 0

2

!2

· R· 1

¿

L R· T

0

!2

· (h ·l θ ˡ ) Lk·T 0

2

* +

2 ·

σ· B0

2

·u2

T 0

'rreversibility rati! NDO K

k · T 0

· µ

*+

2 ⌈ ( 1% )

2

·( f l (& ))2

· 1

x4 ·( #

$ )3

⌉

[( #

$· % )−1

· {( "+1 ) ·θ−&·/ˡ }2+(θ ˡ)2]>e3an nuber K

C

[( #

$· % )−1

· {( "+1 ) ·θ−&· θ ˡ }2

+(θ ˡ)2]+ k · T 0

· µ

*+

2 · ⌈ ( 1% )

2

·( f l (& ))2

· 1

x4 ·( #

$ )3

⌉+k·T

0

2

!2

· R·

D·(¿¿∞)2 · (hˡ )2+ R· T

0

!2

[( #$· % )

−1

· {( "+1 ) · /−&·/ˡ }2+(/ˡ)2]¿

Conclusion:

%iensi!nless ,!r !, entr!py generati!n and irreversibility distributi!n rati!

and be3an nuber are deterined by "!nsidering given re,eren"e papers

1



'eferences:

X1 Raa (ubba Reddy 9!rla ZEntr!py 9enerati!n in Ele"tr!- $s!ti" +!w

in

Mi"r!"hannels[

X2 Raa (ubba Reddy 9!rla Thailselvan &allappan :arry >yrd and %avid M

Pratt ZEntr!py

Minii<ati!n in phase "hange Energy syste[

X. 9C :ayek (9 ;ry<hevi"h A ( 9upta and M Re<a Z(teady

agnet!hydr!dynai" +!w in a

diverging "hannel with su"ti!n !r bl!wing[

10

final thermodynamics project report

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