ede - equações diferenciais de estado

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).

2

RELAES

TERMODINMICAS

PARA EQUAES DE

ESTADO

DEQ/UEM

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


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


)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

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


)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

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


)(

)(

)(

)(

)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

8

RELAES TERMODINMICAS PARA A EQUAO VIRIAL

(TRUNCADA AT O 2O TRMO)

)(;1,

TBBondeBxxBRT

BPZ ijijijj

ji

i


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

9

RELAES TERMODINMICAS PARA A EQUAO DE LEE-KESLER

3978,0;/)(; )()()0()()1()1()0( rrr ZZZZZZ


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

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 .

ede - equações diferenciais de estado

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