apendice 1

Appendix 1

A.I.I THE TREBBLE-BISHNOI EQUATION OF STATE

We consider the Trebble-Bishnoi EoS for a fluid mixture that consist ofNc components. The equation was given in Chapter 14.

A. 1.2 Derivation of the Fugacity Expression

The Trebble-Bishnoi EoS (Trebble and Bishnoi, 1988a;b) for a fluidmixture is given by equation 14.6. The fugacity, fj, of component j in a mixture isgiven by the following expression for case 1 (i > 0)

bd _ _ _ I K

where

j = ln( X j p)+ —— (Z-l)-\nZ-Bm\ + ykX-V> — (A.I .I)U Ir, / Y

(A.I.2)'

403

Copyright © 2001 by Taylor & Francis Group, LLC

404 Appendix 1

X =a d / n bd r|6d

a m bm

Z 2 + ( B m + C m ) Z - ( B m C m + D 2 J

1 = 10 5 =Qand X = In 2Z + B m ( u - 0 )2 Z + B m ( u + 0 )

(A. 1.3)

(A. 1.4)

(A. 1.5)

i f i > 0 (A.1.6a)

B nmy

u=l+cm/bm

c m b m +c2m +4d2

m

-n i f i < 0 (A.1.6b)

(A. 1.7)

(A. 1.8)

6bmc, -6b H c m +2cmc, +8d,,,d, -^^m_m d - d m m d m d

a m PR 2 T 2

RT

'm RT

RT

(A. 1.9)

(A.l.lOa)

(A. 1.1 Ob)

(A.l.lOc)

(A.l.lOd)

where am, bm, cm and dm are given by Equations 14.7a, b, c and d respectively. Thequantities ad, bd, c d , dd and u^ are given by the following equations


Appendix 1 405

Also

_ a ( a m n 2 L - (A. 1.1 la)

(A.I. l ib)

(A.I . l ie)

dn j(A.I . l id)

aeon.

(A.I . l ie)

_a(u)(A. l . l l f )

(A.I.12)

A.I.3 Derivation of the Expression for

Differentiation of Equation A 1.1. with respect to Xj at constant T, P and x,for i=l,2,...,Nc and i^j gives

51nf: bd 9Z , (Z-l) dbd bd 5bm - F X , + F X 2 (A.I.13)


406

where

Appendix 1

F X , = -RT

(A. 1.14)

RT

FX -, =A j 9x j I ox j ox j J A I <9x j A ox j

(A.I.15)

A number of partial derivatives are needed and they are given next,

5X:(A.I.16)

5X 1 n na m oxj

5d bd 5bm

(A.I.17)

!MzBm+0,B^ + un0 d B m - enu d B m )—

B m n6 dOX ,-

.58 n9du „ 9nu,

- -nu, aed T~~ u ^——~ u u dox j ox j

(A.I.18)

ox

-c,j

SB,,

oX OX;-OX,

(A.I.19)


Appendix 1 407

SAm P Sam 3Bm P 5bm

R 2 T 2 <9x j RT ox j

5Cm P 9cm 5Dm P Sdm

9x RT ox 3x RT <9x

(A.1.20a)

(A.1.20b)

If T>0 then A, is given by equation A. 1.6a and the derivative is given by

oA 1g-h

1 9g 9h (g-h) Sg , 9hox h)2

vg + h

where g = 2Z + Bmu, h = Bra0 and

+ R 5 u+ , , 5 B m- + ts,_ -— + U ———

ox j ox j ox j ox j

oh 59= B,^ "m

OTj

(A. 1.2 la)

If T is negative then A. is given by equation A.I .6b and the derivative is given by

2h Sg g Sh

In addition to the above derivatives we need the following

OU

Sx"

SI C"Vdcm cm

bm I oXj bm 9xj

(A. 1.2 Id)

(A. 1.22)

Cm/ f l f C m /

/ b m j | 2 C m

OX, bm OXj(A. 1.23)


408 Appendix 1

nu A.

where

SQ oQOc <5xj

QA = 6 b m c d -6 bdcm + 2 c m c d + 8d m d c

QB =

) 5°d +c 5bm b a°m c'in "^—— Cd ~T—— Dd ~—— crOX j OX j OX j

(A.1.25a)

(A.1.25b)

(A.1.25c)

(A.1.25d)

(A.1.25e)

- + d,

5c

+ 16—3-d, 5d

(A.1.25f)

^OX

-+2b2m —

OX


Appendix 1 409

56 1 9r .—— = ———— i f t>0 (A.1.25h)oX 26 SX

or

59 1 5r—— = - — —— it T is negative (A.1.251)OX : 29 OX ;


apendice 1

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