Download - EDE - Equações Diferenciais de Estado
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UNIVERSIDADE FEDERAL DO PARAN
SETOR DE TECNOLOGIA
DEPARTAMENTO DE ENGENHARIA
QUMICA
GRADUAO EM ENGENHARIA QUMICA
Termodinmica Aplicada
MATERIAL DE APOIO
Relaes Termodinmica
para EDEs
Prof. Marcos L. Corazza
OBS: Este material foi obtido a partir de partes do Material Didtico
produzido originalmente e disponibilizados pelos profs. Lcio Cardozo
Filho e Fred Wolff (DEQ-UEM).
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2
RELAES
TERMODINMICAS
PARA EQUAES DE
ESTADO
DEQ/UEM
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3
RELAES TERMODINMICAS PARA EQUAES DE
ESTADO CBICAS
I- EQUAO GENRICA
ZRT
bP
V
bh
whuh
h
bRT
a
hz
wBABZuBuBwBAZuBBZ
wbVubV
a
bV
RTP
;)1(1
1
0)()()1(
2
22223
2
RT
bPB
RT
aPATrm
Pc
RTc
aaaa
Pc
RTc
bb
kkkkaaxxabxb
n
iii
i
i
ijii
i
i
i
ijijii
ji
ijjiji
i
ii
CC
;)(
;)1(1
)(;;
;0;)1()(;
2
2/
2
2
,
2/1
ENTALPIA RESIDUAL, ENTROPIA RESIDUAL E COEFICIENTE DE FUGACIDADE
][
][
][
][
)4(2
)4(2ln
4
)/()ln()1(ln
)4(2
)4(2ln
4)ln()1(ln
)4(2
)4(2ln
4)ln(/
)4(2
)4(2ln
4
)(
)1(/
2
2
2
2
2
2
2
2
2
2
2
2
wuuh
wuuh
wubRT
bbaZhZZ
b
b
wuuh
wuuh
wubRT
aZhZZ
wuuh
wuuh
wubRT
dT
daT
ZhZRS
wuuh
wuuh
wubRT
dT
daTa
ZRTH
iiii
R
R
-
4
2/2/1
2/12/122/1
2/1
2/1
,
2/1
;)/(2
)(;)(~0
/
/;)1(2;)1()(
n
iiiiC
i
ii
i
n
iiCii
i
iiij
j
jjj
iii
j
ijjj
i
i
ji
ijn
jjCijji
TrmandT
daTaa
TramxandT
daTaxaoaentkse
PcTcx
PcTc
b
bkax
a
akTraamxxn
dT
daT
obs.: i) para substncias puras tem-se: a = ai ; b = bi ; da/dT = dai /dT ; i = 2 ;
A = ai P/(RT)2 ; B = bi P/RT ; 0 iiij kk
ii) no havendo informaes para ijk ento faz-se 0ijk ; consulte Sandler-
"Chemical and Engineering Thermodynamics",2nd
edition, Table 7.4-1 (verifique que a fonte citada nessa fonte existe na BCE/UEM); consulte tambm Reid et al. -"The Properties of Gases and Liquids",4
th edition, Table 4-2;
EQUAO u w b a n mi
VAN DER WAALS 0 0 1/8 27/64 xxxxx 0 REDLICH-KWONG 1 0 0,08664 0,42748 -1/2 -1
SOAVE 1 0 0,08664 0,42748 1 0,48 + 1,574i - 0,176i2
PENG-ROBINSON 2 -1 0,07780 0.45724 1 0,37464 + 1,54226i - 0,26992i2
II- EQUAO DE VAN DER WAALS
ZRT
bP
V
bhh
bRT
a
hZABAZZBZ
V
a
bV
RTP
;
1
1;0)1(; 23
2
RT
bPB
RT
aPA
Pc
RTca
Pc
RTcb
kkkkaaxxabxb
i
i
i
i
i
i
jiij
i ji
iiijjijiii
;)(
;)(
64
27;
8
;0;)1()(;
2
2
,
2/1
ENTALPIA RESIDUAL, ENTROPIA RESIDUAL E COEFICIENTE DE FUGACIDADE
hbRT
bbaZhZZ
b
bh
bRT
aZhZZ
ZhZRShbRT
aZRTH
iiii
RR
)/()ln()1(ln;)ln()1(ln
)ln(/;)1(/
-
5
2/122/1
2/12/1
)()( 2;:~0
)/(
/;)1(2
a
aaxaoaentkse
PcTcx
PcTc
b
bkax
a
a
i
i
i
iiij
j
jjj
ii
j
iijjj
i
i
III- EQUAO DE REDLICH-KWONG
ZRT
bP
V
bh
h
h
bRT
a
hZ
ABZBBAZZbVVT
a
bV
RTP
o
o
;11
1
0)(;)(
2/3
223
2/1
RT
bPB
TR
PaA
Pc
TcRa
Pc
RTcb
kkkkaaxxabxb
o
i
ioi
i
ii
jiijii
ji
ijojiojio
i
ii
;;42748,0;08664,0
;0;)1()(;
5,22
5,22
,
2/1
'
obs.: a relao entre (ao ; aoi ) da equao. de Redlich-Kwong e (a ; ai ) da equao.
Genrica tratada anteriormente : ao = aT1/2
; aoi = aiT1/2
ENTALPIA RESIDUAL, ENTROPIA RESIDUAL E COEFICIENTE DE FUGACIDADE
)1ln(
)(
)ln()1(ln
)1ln()ln()1(ln
)1ln(2
)ln(/;)1ln(2
3)1(/
2/3
2/3
2/32/3
hbRT
b
ba
ZhZZb
b
hbRT
aZhZZ
hbRT
aZhZRSh
bRT
aZRTH
iio
ii
o
oRoR
2/12
2/1
2/1
2/1
)/(2;)(0
/
/;)1()(
)(2
)( ooiii
oiioij
j
jjj
iiiijoi
j
j
o
oi
i
aaaxakse
PcTcx
PcTc
b
bkax
a
a
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6
IV- EQUAO DE SOAVE
ZRT
bP
V
bh
h
h
bRT
a
hZ
ABZBBAZZbVV
a
bV
RTP
;)1(1
1
0)(;)(
223
RT
bPB
RT
aPA
mTrmPc
RTca
aaPc
RTcbkkkkaaxxabxb
iiiiii
i
i
Ci
iCii
i
i
ijiijii
i ji
ijjijiii
;)(
176,0574,148,0;)1(1;)(
42748,0
;08664,0;;0;)1()(;
2
22/1
2
2
,
2/1
ENTALPIA RESIDUAL, ENTROPIA RESIDUAL E COEFICIENTE DE FUGACIDADE
)1ln()/(
)ln()1(ln
)1ln()ln()1(ln
)1ln()ln(/;)1ln(
)(
)1(/
hbRT
bbaZhZZ
b
b
hbRT
aZhZZ
hbRT
dT
daT
ZhZRShbRT
dT
daTa
ZRTH
iiii
RR
2/12/1
2/12/122/1
2/1
2/1
,
2/1
;)/(2
)(;)(:~0
/
/;)1(2;)1()(
iiiiCi
ii
i
iiCii
i
iiij
j
jjj
iii
j
ijjji
i
ji
ijjjCijji
TrmadT
daTaa
TramxadT
daTaxaoaentkse
PcTcx
PcTc
b
bkax
a
akTraamxx
dT
daT
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7
V- EQUAO DE PENG-ROBINSON
ZRT
bP
V
bh
hhh
h
bRT
a
hZ
BBABZBBAZBZbVbbVV
a
bV
RTP
;)1()1(1
1
0)()32()1(;)()(
32223
RT
bPB
RT
aPAm
TrmPc
RTcaaa
Pc
RTcbkkkkaaxxabxb
iii
iii
i
i
CiiCii
i i
i
ijiij
ji
iiijjijiii
;)(
;26992,054226,137464,0
)1(1;)(
45724,0;
07780,0;;0;)1()(;
2
2
2/12
2
,
2/1
ENTALPIA RESIDUAL, ENTROPIA RESIDUAL E COEFICIENTE DE FUGACIDADE
)(
)(
)(
)(
)21(1
)21(1ln
22
)/()ln()1(ln
)21(1
)21(1ln
22)ln()1(ln
)21(1
)21(1ln
22)ln(/
)21(1
)21(1ln
22
)(
)1(/
h
h
bRT
bbaZhZZ
b
b
h
h
bRT
aZhZZ
h
h
bRT
dT
daT
ZhZRS
h
h
bRT
dT
daTa
ZRTH
iiii
R
R
2/12/1
2/12/122/1
2/1
2/1
,
2/1
;)/(2
)(;)(:~0
/
/;)1(2;)1()(
iiiiCi
ii
i
iiCii
i
iiij
j
jjj
iii
j
ijjji
i
ji
ijjjCijji
TrmadT
daTaa
TramxadT
daTaxaoaentkse
PcTcx
PcTc
b
bkax
a
akTraamxx
dT
daT
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8
RELAES TERMODINMICAS PARA A EQUAO VIRIAL
(TRUNCADA AT O 2O TRMO)
)(;1,
TBBondeBxxBRT
BPZ ijijijj
ji
i
ENTALPIA RESIDUAL, ENTROPIA RESIDUAL E COEFICIENTE DE FUGACIDADE
j
ijji
RR
BBxRT
P
RT
BP
dT
dB
R
PRS
dT
dB
T
B
R
PRTH
)(
)(
2ln;ln
/;/
dT
dBxx
dT
dB ijj
ji
i,
Para usar as equaes acima necessrio ter disponvel expresses Bij(T) para todos os pares (i,j) possveis. Na falta de expresses especficas para os pares (i,j), podem ser usadas correlaes generalizadas. Um exemplo tpico de correlao generalizada para o 2
o coeficiente
virial dada a seguir:
ij
ij
ijij
ij
ijij
ij
ij
ij
ij
ij
ij
ji
ijji
ji
ijijij
ij
ij
ji
ij
ii
ji
ijii
ji
ij
ji
ij
ji
ij
ij
ijij
ijijjiij
ij
ij
ij
ij
ij
ijijijij
ij
ij
ij
Pc
P
TrdTr
dB
TrdTr
dB
dTr
dB
dTr
dBxx
dT
dB
R
PBB
Trxx
T
B
R
P
dadosdefaltanakoujisecVcV
cVcVkk
cVcVcV
ZcZcZc
cV
RTcZcPckTcTcTc
Tc
TTr
TrB
TrBBB
Pc
RTcB
Pr;722,0
;675,0
Pr;)(Pr
)0(;)(
)(81;0
2;
8
3)(
2;;1()(;
172,0139,0;
422,0083,0;)(
2,5
)1(
6,2
)0(
)1()0(
,,
)1()0(
33/13/1
2/1
3/13/1
)
2/1
2,4
)1(
6,1
)0()1()0(
)(
obs.: para substncias puras tem-se: B=Bii e dB/dT = dBii /dT
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9
RELAES TERMODINMICAS PARA A EQUAO DE LEE-KESLER
3978,0;/)(; )()()0()()1()1()0( rrr ZZZZZZ
ENTALPIA RESIDUAL, ENTROPIA RESIDUAL E COEFICIENTE DE FUGACIDADE
jikij j
pc
j
jikij j
pc
j
pcjik
ij j
pc
j
pc
R
i
rr
rRrRRRRR
rRrRRRRR
xx
x
xx
Px
P
Z
xx
Tx
TRT
H
SSSSSS
HHHHHH
,
)1(
,,
)()0()()1()1()0(
)()0()()1()1()0(
)()0()()1()1()0(
ln)1(
)(lnln
/)ln(lnln;lnlnln
/)(;
/)(;
(i) O clculo de Z (0)
("simple fluid") e Z(r)
("reference fluid") realizado com as equaes
abaixo, nas quais so usadas as constantes fornecidas na tabela em (v). As equaes so
usadas uma vez para "simple fluid" e outra para "reference fluid".
Tr
ddD
Tr
c
Tr
ccC
Tr
b
Tr
b
Tr
bbBTTTrPP
VrouVrVrTrVrZVrVrVrTr
c
Vr
D
Vr
C
Vr
BZ
pcpc2
12
3213
4
2
321
)1()0()(
2223
4
52
)(
;;;/;/Pr
;/Pr;)exp()(1
obs.: para dados T e P note que Z( )
e Vr so obtidos pela soluo simultnea das
equaes acima (Vr no o volume residual definido por V / Vc ).
(ii) Para os casos em que M = { HR , SR , ln} , o clculo de M(0) e M(r) realizado com as equaes abaixo, de modo similar ao explicado em (i).
][
][
)exp()1(12
52ln1ln
252
)/2()/2/(ln/
352
)/3()/3/2(1/
223
4
52
)()()(
5
1
2
3
31
3
4
2
31)()(
5
2
2
2
32
2
432)()(
VrVrTr
cE
EVr
D
Vr
C
Vr
BZZ
EVr
d
Vr
Trcc
Vr
TrbTrbbZRS
ETrVr
d
TrVr
Trcc
TrVr
TrbTrbbZTRTrH
R
pc
R
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10
(iii) No caso de misturas:
33/13/1,2/1
,
4/1
,
4/1
)(8
1;)(;085,02905,0
/;;;
cjcicijijcjcicijpc
pcpcpcpc
i
iipccijj
ji
ipccijcijj
ji
ipcpc
VVVkTTTZ
VRTZPxVxxVTVxxVT
obs.: para k ij, h dados em Reid et al. - "The Properties of Gases and Liquids", 4
th
edition, 1987-Tabela 4-4; na falta de informaes, use-se k ij, =1.
][
][
)(085,011
)(2
25,0)(2
,,,
,
,
4/34/14/14/1
,
pc
ij
jikj
pc
pcjik
j
pc
pc
pc
jikj
pc
cni
n
cnjn
jikj
pc
jikj
pc
pcpccnicni
n
cnjcnjnpc
jikj
pc
Zx
x
V
Vx
x
T
TP
xx
P
VVx
xx
V
xx
VTVTVTVxV
xx
T
(iv) No caso de substncias puras:
cpccpc PPTT ;
(v) Constantes da equao de Lee-Kesler:
simple fluid reference fluid simple fluid reference fluid
b1 0,1181193 0,2026579 c3 0,0 0,016901
b2 0,265728 0,331511 c4 0,042724 0,041577
b3 0,154790 0,027655 104d1 0,155488 0,48736
b4 0,030323 0,203488 104d2 0,623689 0,0740336
c1 0,0236744 0,0313385 0,65392 1,226
c2 0,0186984 0,0503618 0,060167 0,03754
(vi) As equaes. anteriores so geradoras das tabelas e grficos que tm substitudo o mtodo
generalizado original de Pitzer. Ver, por exemplo (a) Reid et al.-"The Properties of Gases and
Liquids"-McGraw-Hill - 4th
edition e (b) Smith&Van Ness - "Introduction to Chemical
Engineering Thermodynamics" - McGraw-Hill - 4th
edition .