- Email: [email protected]
Simulation of binary vapor condensation in the presence of an inert gas
INT. COMM. HEAT MASS TRANSFER 0735-1933~83 $3.00 + .00 Vol. I0, pp. 555-561, 1983 ©Pergamon Press Ltd. Printed intheUnitedStates
S I M U L A T I O N OF BINARY V A P O R C O N D E N S A T I O N IN THE PRESENCE OF AN INERT GAS* /,
Andrzej Burghardt Polish A c a d e m y of Sciences R e s e a r c h Centre of Chemical E n g i n e e r i n g and E q u i p m e n t Design Gliwice, Baltycka 5, Poland (Cxilil~rLicatedby E. Hahne)
in a reoent publleatlom [1] R.T&ylor and M.K.Neah have e~apamed several me%hods of ~kleulatlng mass transfer rates in a vazAe%y of mm/tlemaponent ooadensatlon design problems. The aim ef their study was to evaluate the explicit methods of detezttning eomposltlon profiles in a ver%leal tube, in whieh a binary vapor i s c o n d e n s i n g i n t h e p r e s e n c e o f noneondensing g a s . Anong e%her explioit methods they amalymed the approximate solution of MaxwellStefan equations for mass transfer in nultieompenen% mixtures eontalnlng one or more inert epeelee given by Burghardt and F~upiezka[2] As a result of their analysis they stated that ,The method of Burghardt and Kruploska [2] is not in exact agreement with the more rigorous medele even if all the binary eoeffielents are equal". Obviously t it Is very difflc~Ll% to reallse at a first glanoe, that the formulae def~-1-~ molam fluxes given by Burghazdt and Krupleska converge %o the exact solution of the Maxwell-Stefan equations for all equal binax7 diffusion coefficients. Therefore it seems useful to prove this convergence in a more detaAled manner [~]. For the steady - state one - dimensional mass transfer in an n..eomponent system at eonstant t e m p e r a t u r e and pressure the MaxwellStefan equations o a ~ be written as :
*Comment to the paper by R. Taylor and M.K. N o a h
555
[i]
556
A. Burghardt
Vol. i0, No. 6
n
d z
e
;~-1
~'~
~¢t
I - 1,2,...n
Oons£dezLa~ a 8 y s t a ,
i n whJ.oh t h e r e
Ls o 1 1 y one L n o r t s p o -
oles,
t h e n o l s : f l u x o f v b t e h 18 equa~ nmaSkt s a d d o l o t L a ~ t h i s speeJ.es "by t h e L z d e x I we os,u ] ~ r : - L t e equa,t:l.o~ ( 1 ) sa f o l l w a
"
Yl
~
" Nt
- t,2,...a-I
t
Sad far~ 1;he t n e x - t s p e e l e 8
:
d YI
n
ds
Let us am~me nov ~mt
o
-31 blaazy dlfl~a£Gm eoeffieisats
are
i.e.
(,) I - tp2t...n - 1,2,...n
:. ~..,. rednoe8 to
~...~.
o, ~ , . ~ , . , ~
.~,~o.
(2) u (3)
:
d s
o
o;B I " 1,2,o.oa,-1
(5)
Vol. i0, No. 6
~I~SATION
.
IN THE PRESENCE OF AN INERT GAS
~tz
557
o2~
where
is the net total
atxt~e
flux. rofJpeo~
to the variables
y£
and 8o e a ~
dAfferentlal the l t a l t m :
ln~e~-ated meperstely v l t h l n •
0
"
"'~
e q u a l : t o n oa~ b e
Yl
"
7:Lo
'
YI
"
YIo
(8)
~'~.
"
~'~.,~ ,
~'i
"
~
(9)
g:L~J..ug : Ylc~ ,. yto ex.p(N~<~'~\o~, / + N%]I~" [1
\ o:~/J
(10)
t - 1,2,.,.n-1 and f o r t h e l n e z ~ m p e o l e s :
\e~J E q u = t l o n (11) 8 1 1 o v s %o e l l n l n a % e t h e n e t t o l ~ l H~ ~ a n e q u s ~ l e n s (10) b y n e a a s o f t h e h o l e / ~ o t t e a s "r31e l n e z ~ s p e o l e s . ? b u s we o b t ~ t = :
7z~ Y : ~ " YAo
+
YIo I - 1,2,...n-1
Fron t h l s equa~£on the n o l a r f l a x
Nl
esm be 4etex'~Lze4 as :
of
A. Burghardt
558
Vol. I0, No. 6
YI# Ylo " Y!o Y$~ Ni
,,
~' Ylm i - 1,2,...n-1
where
yI=
"
yI~
04)
no
In Ylo ~e
set of equatlons
~9
oembe pceHntedlnan
equlvalent form
by InCrodno%ien of the axlthme%Ic mean values of mole fraetAens :
2 i
-
2
1,2,...n-1
glvAng
NI
=
Ylm
i = 1,2,..o~-1
where
i
B e e a u e there is : n
AYl ""
"~he t o n , - Z a
(16)
k' j=1
~"~,.,stoz',,s
iato
/kYS
•
= 1,2,...n-1
Vol. 10, No. 6
CONDenSATION IN THE P
~
OF AN I ~
CoAS
559
;}=1 N1
S
Ylm t - 1,2~...a-1
~he na~elx no~atlon o f ~heee f e ~
(.).
is z
° [~1. ( , o - , ~ )
(2o)
~rlm ~here the elene~%8 ef the dlfl~usAon eeeffAelen% n a ~ I x
ID] axe as
follows :
D . . ~ C~ +
%)
(~)
D,.~ . ~ %
(,,)
Im a s ~ u t l a r h a r m e r ~he f o z s n ~ a e ~ e f J ~ J ~ n o l a ~ f l u z e e i n a 8 y s % a o f n . o o u p o n e n % e ~ t h / n . n / :tnez~ s p e c i e s oan be olx~84Jtod[4] .
i~
~ . = ~ ~0 . = . . . ~ , ~ ,,~ , i . ~ ~,.!,~.o,,,,~, (4) "- 0 9
°~lm '"
(2o) + (~2) = , l,=~i[I]
~0,, k~.~. ~ , , -
k
~he &pp~oxlna%e 8olu%Ion derived by Bu~gha~% aa~L F~eupioska fer the general ease of dlffe~en% b l ~ d £ 1 ~ l e n eoeffAelen%s d e t e z s t l z e s t h e m o l a r ~vZuxe8 a s :
o[DI
(,, - ,~,)
(,,)
~here ~ae e l e a e n t o o f t h e ~uveree o f t h e d ~ f f n e l o n e o e ~ . c i e n t
b ] " [DI"~
(")
560
A. Burqhardt
Vol. i0, No. 6
1"I
All "
AiJ
=
iz
-.~Sij
t -
1,2p...
"--1
For all equal binary dlff~lon eoeffioieato the inverse diffaslon coefftotent -~trlx l~ke8 the fo~a :
[D]" .
1
-
~2
I -
~2
......
of the
" ~2
e s e e e e e e g o o e e o e e e e o e e e o e e e e o o e e
-
whleh u ~
~-~
-
~-1
......
' " ~-1
J.wver81on given :
b] "
ThAs matrix Is £dea~Aeal wi~h ~atrlx [D1 ~eflned by equations (21) a ~ d (22) VtLteh h a e b e e n obtaAne
VOl. i0, NO. 6
COND~SATICN IN THE PRES~XKIE OF AN INERT G%S
Nomonol&%~LTo o
Mol~ de]~ity
knole/n~ •
of fluJ.d LtX~e
~ % ~ i x of m m l ~ i o ~ o n ~ dii~8£~ ooefflelen~e Diff~sie~ o o o f f ~ o i O ~ o f bi~KE~ pa~T i-~
,.2/, m2/s
Nt
Mola~ flux of spe~Io8 1
~mole/nt2a
N%
Ne% ~o~al nAx~n'e flux
~-mole/n28
Mole ~£'8~%ien of spe~io8 i Ari~hme~io nneam Talue o f mole i ~ i ~ o f speoles i Dis~moe w i ~ t n ~he film
ox-o~ ~ o ~ e n c~
Lo,w~ ot ~t:'u,,:ton p~'~
Na4~/.~ Wol;a~lm
( )
Colmm ~m~
o:t ~m,na,om
[ t. 1
S q u a ~ nm~eix o t ~l.me'malo',,-me..~lz InToz-so o f 8 ~
(...,)
Suboori~8
i t ~ e-
Denotes oomponon~
Z
Denotes iltez~ speo£os
o
I n ~ e r phamo
[1].
. u A MoK. Noah t ~. t T&ylooe 46~p 119621.
[2]
A. Bux-~uued.t amd. R.K~pieLkm.t I.~oC'hom., ~. • 487, 717,
[4]
I.e~'~s. Hea% amd Mama; Tzmmfo~,
1t 97~1. A. ~ n - ~ a r d t
mLct a.T,.z'up:Lom, I j . C ' h e n . , ~. , 43, / t ~ 3 / o
A. I k n . ~ t ;
~
R.T,.z~p£os]m, Ini.(~nem., 4: , 13, / 1 9 7 4 / .
561
S I M U L A T I O N OF BINARY V A P O R C O N D E N S A T I O N IN THE PRESENCE OF AN INERT GAS* /,
Andrzej Burghardt Polish A c a d e m y of Sciences R e s e a r c h Centre of Chemical E n g i n e e r i n g and E q u i p m e n t Design Gliwice, Baltycka 5, Poland (Cxilil~rLicatedby E. Hahne)
in a reoent publleatlom [1] R.T&ylor and M.K.Neah have e~apamed several me%hods of ~kleulatlng mass transfer rates in a vazAe%y of mm/tlemaponent ooadensatlon design problems. The aim ef their study was to evaluate the explicit methods of detezttning eomposltlon profiles in a ver%leal tube, in whieh a binary vapor i s c o n d e n s i n g i n t h e p r e s e n c e o f noneondensing g a s . Anong e%her explioit methods they amalymed the approximate solution of MaxwellStefan equations for mass transfer in nultieompenen% mixtures eontalnlng one or more inert epeelee given by Burghardt and F~upiezka[2] As a result of their analysis they stated that ,The method of Burghardt and Kruploska [2] is not in exact agreement with the more rigorous medele even if all the binary eoeffielents are equal". Obviously t it Is very difflc~Ll% to reallse at a first glanoe, that the formulae def~-1-~ molam fluxes given by Burghazdt and Krupleska converge %o the exact solution of the Maxwell-Stefan equations for all equal binax7 diffusion coefficients. Therefore it seems useful to prove this convergence in a more detaAled manner [~]. For the steady - state one - dimensional mass transfer in an n..eomponent system at eonstant t e m p e r a t u r e and pressure the MaxwellStefan equations o a ~ be written as :
*Comment to the paper by R. Taylor and M.K. N o a h
555
[i]
556
A. Burghardt
Vol. i0, No. 6
n
d z
e
;~-1
~'~
~¢t
I - 1,2,...n
Oons£dezLa~ a 8 y s t a ,
i n whJ.oh t h e r e
Ls o 1 1 y one L n o r t s p o -
oles,
t h e n o l s : f l u x o f v b t e h 18 equa~ nmaSkt s a d d o l o t L a ~ t h i s speeJ.es "by t h e L z d e x I we os,u ] ~ r : - L t e equa,t:l.o~ ( 1 ) sa f o l l w a
"
Yl
~
" Nt
- t,2,...a-I
t
Sad far~ 1;he t n e x - t s p e e l e 8
:
d YI
n
ds
Let us am~me nov ~mt
o
-31 blaazy dlfl~a£Gm eoeffieisats
are
i.e.
(,) I - tp2t...n - 1,2,...n
:. ~..,. rednoe8 to
~...~.
o, ~ , . ~ , . , ~
.~,~o.
(2) u (3)
:
d s
o
o;B I " 1,2,o.oa,-1
(5)
Vol. i0, No. 6
~I~SATION
.
IN THE PRESENCE OF AN INERT GAS
~tz
557
o2~
where
is the net total
atxt~e
flux. rofJpeo~
to the variables
y£
and 8o e a ~
dAfferentlal the l t a l t m :
ln~e~-ated meperstely v l t h l n •
0
"
"'~
e q u a l : t o n oa~ b e
Yl
"
7:Lo
'
YI
"
YIo
(8)
~'~.
"
~'~.,~ ,
~'i
"
~
(9)
g:L~J..ug : Ylc~ ,. yto ex.p(N~<~'~\o~, / + N%]I~" [1
\ o:~/J
(10)
t - 1,2,.,.n-1 and f o r t h e l n e z ~ m p e o l e s :
\e~J E q u = t l o n (11) 8 1 1 o v s %o e l l n l n a % e t h e n e t t o l ~ l H~ ~ a n e q u s ~ l e n s (10) b y n e a a s o f t h e h o l e / ~ o t t e a s "r31e l n e z ~ s p e o l e s . ? b u s we o b t ~ t = :
7z~ Y : ~ " YAo
+
YIo I - 1,2,...n-1
Fron t h l s equa~£on the n o l a r f l a x
Nl
esm be 4etex'~Lze4 as :
of
A. Burghardt
558
Vol. I0, No. 6
YI# Ylo " Y!o Y$~ Ni
,,
~' Ylm i - 1,2,...n-1
where
yI=
"
yI~
04)
no
In Ylo ~e
set of equatlons
~9
oembe pceHntedlnan
equlvalent form
by InCrodno%ien of the axlthme%Ic mean values of mole fraetAens :
2 i
-
2
1,2,...n-1
glvAng
NI
=
Ylm
i = 1,2,..o~-1
where
i
B e e a u e there is : n
AYl ""
"~he t o n , - Z a
(16)
k' j=1
~"~,.,stoz',,s
iato
/kYS
•
= 1,2,...n-1
Vol. 10, No. 6
CONDenSATION IN THE P
~
OF AN I ~
CoAS
559
;}=1 N1
S
Ylm t - 1,2~...a-1
~he na~elx no~atlon o f ~heee f e ~
(.).
is z
° [~1. ( , o - , ~ )
(2o)
~rlm ~here the elene~%8 ef the dlfl~usAon eeeffAelen% n a ~ I x
ID] axe as
follows :
D . . ~ C~ +
%)
(~)
D,.~ . ~ %
(,,)
Im a s ~ u t l a r h a r m e r ~he f o z s n ~ a e ~ e f J ~ J ~ n o l a ~ f l u z e e i n a 8 y s % a o f n . o o u p o n e n % e ~ t h / n . n / :tnez~ s p e c i e s oan be olx~84Jtod[4] .
i~
~ . = ~ ~0 . = . . . ~ , ~ ,,~ , i . ~ ~,.!,~.o,,,,~, (4) "- 0 9
°~lm '"
(2o) + (~2) = , l,=~i[I]
~0,, k~.~. ~ , , -
k
~he &pp~oxlna%e 8olu%Ion derived by Bu~gha~% aa~L F~eupioska fer the general ease of dlffe~en% b l ~ d £ 1 ~ l e n eoeffAelen%s d e t e z s t l z e s t h e m o l a r ~vZuxe8 a s :
o[DI
(,, - ,~,)
(,,)
~here ~ae e l e a e n t o o f t h e ~uveree o f t h e d ~ f f n e l o n e o e ~ . c i e n t
b ] " [DI"~
(")
560
A. Burqhardt
Vol. i0, No. 6
1"I
All "
AiJ
=
iz
-.~Sij
t -
1,2p...
"--1
For all equal binary dlff~lon eoeffioieato the inverse diffaslon coefftotent -~trlx l~ke8 the fo~a :
[D]" .
1
-
~2
I -
~2
......
of the
" ~2
e s e e e e e e g o o e e o e e e e o e e e o e e e e o o e e
-
whleh u ~
~-~
-
~-1
......
' " ~-1
J.wver81on given :
b] "
ThAs matrix Is £dea~Aeal wi~h ~atrlx [D1 ~eflned by equations (21) a ~ d (22) VtLteh h a e b e e n obtaAne
VOl. i0, NO. 6
COND~SATICN IN THE PRES~XKIE OF AN INERT G%S
Nomonol&%~LTo o
Mol~ de]~ity
knole/n~ •
of fluJ.d LtX~e
~ % ~ i x of m m l ~ i o ~ o n ~ dii~8£~ ooefflelen~e Diff~sie~ o o o f f ~ o i O ~ o f bi~KE~ pa~T i-~
,.2/, m2/s
Nt
Mola~ flux of spe~Io8 1
~mole/nt2a
N%
Ne% ~o~al nAx~n'e flux
~-mole/n28
Mole ~£'8~%ien of spe~io8 i Ari~hme~io nneam Talue o f mole i ~ i ~ o f speoles i Dis~moe w i ~ t n ~he film
ox-o~ ~ o ~ e n c~
Lo,w~ ot ~t:'u,,:ton p~'~
Na4~/.~ Wol;a~lm
( )
Colmm ~m~
o:t ~m,na,om
[ t. 1
S q u a ~ nm~eix o t ~l.me'malo',,-me..~lz InToz-so o f 8 ~
(...,)
Suboori~8
i t ~ e-
Denotes oomponon~
Z
Denotes iltez~ speo£os
o
I n ~ e r phamo
[1].
. u A MoK. Noah t ~. t T&ylooe 46~p 119621.
[2]
A. Bux-~uued.t amd. R.K~pieLkm.t I.~oC'hom., ~. • 487, 717,
[4]
I.e~'~s. Hea% amd Mama; Tzmmfo~,
1t 97~1. A. ~ n - ~ a r d t
mLct a.T,.z'up:Lom, I j . C ' h e n . , ~. , 43, / t ~ 3 / o
A. I k n . ~ t ;
~
R.T,.z~p£os]m, Ini.(~nem., 4: , 13, / 1 9 7 4 / .
561