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P re d ic ti on o f t he r mo dy n am ic p r op er t ie s o f n a tu ra l g a s m ix tu re s

u si ng 1 0 e qu at io ns o f s ta te i nc lu di ng a n ew c ub ic t wo -c on st an t

e qu a ti on o f s ta te

K h . N a s ri f a r , O . B ol la nd

Dep a rtment o f E n ergy a n d P ro cess E n g in eering , No rweg ian Un iversity o f S cien ce a n d Tech n o lo gy ( NTNU) , NO- 7 4 9 1 Tro n d h eim, No rwa y

R e ce i ve d 2 7 A p ri l 2 0 05 ; r e ce i ve d i n r e vi s ed f o r m 2 9 S e pt e mb e r 2 0 0 5 ; a c ce p te d 1 4 J a nu a ry 2 0 06

Abstract

I n t h i s c o n t r i b u t i o n , 1 0 e q u a t i o n s o f s t a t e ( E o S s ) a r e u s e d t o p r e d i c t t h e t h e r m o - p h y s i c a l p r o p e r t i e s o f n a t u r a l g a s m i x t u r e s . O n e

o f t h e E o S s i s p r op os e d i n t h is w o rk . T h is E o S i s o b ta i ne d b y m a t c hi n g t h e c r it i ca l f u ga c it y c o ef f ic i en t o f t h e E o S t o t h e c r i t ic a l

f u g ac i t y c o e f f i ci e n t o f m e t ha n e . S p e c i al a t t e nt i o n i s g i v en t o t h e s u p er c r i t ic a l b e h a v i or o f m e t h a n e a s i t i s t h e m a j or c o m po n e nt o f

n a t ur a l g a s m i x t u r es a n d a l m o s t a l w a y s s u p er c r i t ic a l a t r e s e rv o i r a n d s u r f ac e c o n di t i o ns . A s a r e s ul t , t h e p r o p o s ed E o S a c c u r a t el y

p red icts th e su p ercritical fu g acity o f meth an e fo r wid e ran g es o f temp eratu re an d p ressu re. Usin g th e v an d er Waals mix in g ru les

w i t h z e r o b i n ar y i n t e ra c t i on p a r am e t er s , t h e p r o po s e d E o S p r e di c t s t h e c o m pr e s s ib i l i ty f a c t or s a n d s p e e d s o f s o u nd d a t a o f n a t ur a l

g as m i xt u re s w i th b es t a c cu r ac y a m on g t h e o t h e r E o Ss . T h e a v e r ag e a b so l ut e e r ro r w a s f o u n d t o b e 0 .4 7 % f o r p r ed i ct i ng t h e

c o m p r e s s i b i l i t y f a c t o r s a n d 0 . 7 0 % f o r t h e s p e e d s o f s o u n d d a t a . T h e p r o p o s e d E o S w a s a l s o u s e d t o p r e d i c t t h e r m a l a n d e q u i l i b r i u mp ro p erties. Fo r p red ictin g iso b aric h eat cap acity, Jo u leT h o ms o n c o e f f i c ie n t , d e w p o i nt s a n d f l a s h y i e ld s o f n a t ur a l g a s m i x t u r e s,

t h e p r e d i c t i v e a c c u r a c y o f t h e E o S i s c o m p a r a b l e t o t h e p r e d i c t i v e a c c u r a c y o f t h e R e d l i c hKwongS o a v e ( R K S ) E o S o r o n e o f i t s

v a r ia n t s . F o r p r e di c t i ng s a t ur a t e d l i q u i d d e n s i t y o f L N G m i x t u r e s, h o w ev e r, t h e a c c u r a c y o f p r e di c t i on s i s b e t we e n t h e R K S a n d

PengR o bi n s on ( P R ) E o S s.

2 0 0 6 E l s e v i er B . V. A l l r i g h ts r e s e rv e d .

Keywo rd s: N a tu r a l g a s ; T h e r m od y n a mi c p r o p e rt y ; C o m p r e s s ib i l i t y f a c t o r ; S p e e d s o f s o u n d; P h a s e e q u i l i b ri u m ; E q u a t i o n o f s t a t e

1. Introduction

P r ed i c ti n g t h e t h er m od y n am i c p r op e rt i e s o f n a t ur a l

g as m ix t ur es a re i mp o rt an t i n g a s i nd u st ryt ha t i s i n

production, processing, storage and transportation. Ac-

c u ra t e v a l ue s o f n a t ur a l g a s c o mp r e ss i b il i t y f a ct o r s a n d

s pe ed s o f s ou nd d at a a re c ru ci al i n c us to dy t ra ns fe r

o p er a ti o ns . O t he r t h er m od y na m ic p r op e rt i es , e . g. ,

s a tu r at e d l i q ui d d e n si t y o f l i q ue f ie d n a t ur a l g a s ( L NG )m ix tu re s, a re u se d i n t he d es ig n o f l iq ue fa ct io n p ro -

c es se s a nd s to ra g e f ac i li t ie s; J ou l eT h o ms o n c o e ff i -

c i e n t s a r e u s e d i n t h r o t t l i n g p r o c e s s e s a n d d e w p o i n t s a r e

u s ed i n p i p el i n e d e s i g n .

T he re a re a cc ur at e c or re l at io n s/ eq u at i on s o f s t at e

( E oS s ) f o r c a lc u l at i n g n a t ur a l g a s p r o pe r ti e s . McCarty

(1982) r e p or t ed a n a c cu ra t e e xt e nd e d c or re sp o nd i ng

s ta te s (E CS ) mo de l fo r L NG sy st ems . U si ng EC S

models, E s te l a -U r ib e a n d T ru s l er ( 2 0 01 ) and Estela-

U r i b e e t a l . ( 2 0 0 4 ) p r e d i c t e d t h e c o m p r e s s i b i l i t y f a c t o r s ,

J o ur n a l o f P e tr o l eu m S c ie n ce a n d E n gi n ee r in g 5 1 ( 2 00 6 ) 2 5 3266

www. els evier. com/locate/petrol

C o r r e sp o n d i ng a u t h o r. Te l . : + 4 7 7 3 5 9 8 4 6 2 ; f a x : + 4 7 7 3 5 9 8 3 9 0 .

E - ma il a d d ress: khas hayar. nas rifar@ntnu. no ( K h . N a s r if a r ) .

0920 - 41 0 5/ $ - s e e f r o n t m a t te r 2 0 0 6 E l s e v i e r B . V. A l l r i g h t s r e s e r v e d .doi:10. 1016/j. petrol. 2006. 01. 004

mailto:khashayar.nasrifar@ntnu.nomailto:khashayar.nasrifar@ntnu.nohttp://dx.doi.org/10.1016/j.petrol.2006.01.004http://dx.doi.org/10.1016/j.petrol.2006.01.004http://dx.doi.org/10.1016/j.petrol.2006.01.004mailto:khashayar.nasrifar@ntnu.no

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d e n s i t i e s , s p e e d s o f s o u n d a n d b u b b l e p o i n t p r e s s u r e s o f

n a tu r a l g a s m i x tu r es q u i te a c c ur a t el y. A c cu r a te m o de l s,

f o r i n st a n ce , A GA N X- 19 ( 19 78 ) a n d M G ER G- 88

(J ae sc h ke e t a l. , 1 9 89) a re u se d i n c u st o dy t ra n sf er f or

c a lc u l at i n g c o mp r e ss i b il i t y f a ct o r s o f n a t ur a l g a s m i x-

t u r e s . T h e B e n e d i c t e t a l . ( 1 9 4 0 ) ( B W R ) E o S , m o d i f ie dR e dl i c h a n d K w o ng ( 1 94 9 ) ( R K ) E o S b y S o a v e ( 1 9 7 2 )

( RK S) a n d P e ng a nd R ob i ns on ( 19 76 ) ( PR ) E oS a re

o f te n u s ed i n t h e g a s i n d us t ry f o r p r e d i ct i n g n a t u r al g a s

e q u i l i br i u m p r o p e r ti e s .

E xc ep t t he R KS a nd P R E o Ss , t he o t he r m od e ls a re

e it he r c om pl ex o r r eq ui re m an y p ur e c om po ne nt

c o ns t a nt s a n d /o r b i n ar y p a ra m et e r s (E s t el a -U r ib e e t a l .,

2004 ) . F o r i n s t a n c e , t h e B W R E o S h a s 8 c o n s t a n t s . T h e

M G E R G (J a e s c h k e e t a l . , 1 9 8 9 ) m o d e l i s n o t s u i t a b l e f o r

t he rm al p ro p er ti e s c al cu l at i on s. T he E CS m od e ls o f

E s t e l a -U r i b e a n d T r us l e r ( 2 0 0 1 ) and E s t el a -U r ib e e t a l .(2004) t ak e t he a d va nt a ge o f b in a ry p a ra me t er s, a n d

t he re fo re c an n ot b e e x te nd e d t o n at u ra l g as m ix t ur es

w i th h e a vy f r ac t i on s .

T he R KS a nd P R E oSs a re o ft en u se d i n t he g as

i nd u st ry a s p re di c ti v e t oo l s. W he n t h es e t wo E oS s a re

c om pa re d w it h t he m en t io n ed m od el s a bo v e, t he y a re

r a t h e r a c c u r a t e . M o r e o v e r , b o t h E o S s t a k e t h e a d v a n t a g e

o f s im pl ic it y. T he y a re r el ia bl e a nd p re di ct t he

t h er m od y na m i c p r op e r ti e s o f n a tu r a l g a s m i xt u re s w i th

r e a s o n a b l e a c c u r a c y . I n a d d i t i o n , t h e s e t w o E o S s c a n b e

u s e d f o r p r e d i c t i n g t h e p r o p e r t i e s o f n a t u r a l g a s m i x t u r e s

c o nt a i ni n g h e av y f r ac t i on s (Nasrifar et al., 2005 ).Natural gas mixtures comprise supercritical methane

a s t he m aj or c om po ne nt . W he n t he n ew f in di ng s i n

s up er cr it i ca l b e ha vi o r o f E oS s a re t a ke n i nt o a cc o un t

(M a t h i a s , 2 0 0 3 ) ; a n a c c u r a t e E o S c a n b e c o n s t r u c t e d f o r

a p pl i ca ti o n i n t h e g a s i n du st ry. T he o bj e ct iv e o f t h is

w o rk i s t o o b ta i n a p re di c ti v e t wo -c o ns t an t E oS . T h is

E o S s h o ul d e x h ib i t a n a c cu r at e d e sc r ip t i on o f t h er m o-

d yn am ic p ro pe rt ie s o f n at ur al g as m ix tu re s w hi le

preserving the outstanding characteristics of the RKS

a n d P R e q ua t i on s .

2. Model development

Natural gas mixtures comprise of many hydrocarbon

a n d n o n -h y d ro c a rb o n c o n st i t ue n ts w i t h m e t ha n e a s t h e

m aj or c om po ne nt . H ea vy h yd ro ca rb on s u p t o C40s om et im es e xi st i n n at ur al g as m ix tu re s. N it ro ge n,

c a rb o n d io x id e a nd h yd ro g en s ul fi d e a re u su al l y t h e

n o n- h yd ro c ar bo n c o mp on e nt s. W h il e t h e g a s p h as e

properties of natural gas mixtures, to a large extent,

r es ul t f ro m t he p re se n ce o f m et h an e, t h e e q ui l ib ri u mpropertie s are affected by the presence of heavier

h y d r o c a rb o n s (Nasrifar et al., 2005 ) . T h us , t h e E oS o f

n a tu ra l g a s m ix tu re s m us t b e a cc ur at e a t r ep re se n ti n g

m e t h a n e a n d h e a v i e r h y d r o c a r b o n p r o p e r t i e s a s l i m i t i n g

c a s e s o f t h e m i x t u r e t r u e b e h a v i o r . H o w e v e r , t h i s i s o n l y

a n e c e s s a r y c o n d i t i o n .

T he p re ss ur e a nd t em pe ra t ur e o f m os t n at u ra l g as

m ix t ur es , a t r es er vo i r a nd s ur fa ce c o nd it i on s , c a n b e

f ou n d u p t o 1 5 0 M Pa a n d 5 00 K , r es pe ct i ve l y. A t t h es e

c o nd i ti o ns , n i tr o ge n , m et h an e a nd e t ha n e a re a l mo st

a l w ay s s u p er c ri t i ca l w h i le t h e h e av i e r h y d ro c a rb o n s a r e

s u bc ri t ic al . I n o t he r w or d s, t o a cc u ra t el y d e sc ri b e t h eproperties of natural gas mixtures, the supercritical

behavior of methane and to a less extent nitrogen and

e t ha ne , a n d t he s ub c ri t ic a l b eh a vi o r o f h e av y h y dr o-

c a rb o n s s h o u l d b e a c c ur a t el y d e sc r ib e d . I n a n E o S, t h e

Ta b l e 1

A c c u r a c y ( % A A D ) o f t h e E o S s i n p r e d i c t i n g t h e v a p o r p r e s s u r e o f s o m e n a t u r a l g a s c o n s t i t u e n t s ( e x p e r i m e n t a l d a t a f r o m D a u b e r t a n d D a n n e r , 1 9 9 2)

Component Tr r ange This work RKS/RKSNB RKTCC PR PRGGPR MNM 3M

H2S 0.501 3. 04 2.30 3. 75 3. 91 3.42 3.75 3. 52

CO2 0. 711 0. 99 0.44 0. 11 0. 52 0.14 1.27 2. 67

C1 0. 481 1. 66 2.02 0. 26 0. 50 1.06 0.79 3. 14C2 0. 301 2. 28 3.60 1. 83 3. 46 1.45 7.26 3. 54

C3 0. 231 2. 33 3.08 7. 37 9. 37 3.53 13.72 4. 06

i-C4 0. 291 8. 55 7.17 4. 40 16. 33 9.09 3.40 8. 86

n-C4 0. 321 1. 80 2.30 3. 00 4. 33 0.85 5.85 6. 49

i-C5 0. 251 2. 33 2.50 7. 78 11. 81 1.25 12.73 7. 89

n-C5 0. 301 6. 76 6.04 2. 35 14. 79 8.39 4.26 10. 40

n-C6 0. 351 4. 80 4.88 2. 32 8. 44 5.03 2.47 11. 73

n-C8 0. 381 2. 51 2.11 3. 50 3. 53 1.79 5.02 11. 93

n-C12 0. 401 2. 08 1.69 2. 34 9. 00 3.65 4.13 16. 34

n-C16 0. 401 4. 36 3.54 4. 00 11. 73 2.18 8.36 18. 48

n-C20 0. 401 3. 26 4.03 4. 35 25. 90 7.04 9.89 26. 22

C6H6 0. 491 1. 63 0.78 1. 11 1. 59 1.67 1.27 5. 05

cyc-C5 0. 351 1. 79 1.15 0. 87 5. 10 2.67 2.60 4. 69

Average 3. 14 2.98 3. 08 8. 14 3.33 5.42 9. 06

254 K h . N a s ri f a r, O . B o l la n d / J o u rn a l o f P e t ro l e um S c i e nc e a n d E n gi n e e ri n g 5 1 ( 2 00 6 ) 2 5 32 6 6

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s u bc r i ti c a l a n d s u pe r cr i t ic a l b e h av i o r o f f l u id s n o t o n l y

a tt ri b ut e s t o t h e p re ss u revolumet e m pe r at u re (PVT)

r el at io ns hi p o f t he E oS b ut a ls o t o t he t em pe ra tu re

d e pe n d en c y o f t h e f u nc t i on . A g e ne r al t w o c o ns t an t

EoS may b e defin ed b y (M ic h el se n a n d M ol l er up ,

2004 ):

P RTv b

aCaTr v d1 bv d2 b 1

with

b Xb RTCPC

2

aC Xa R2 T2C

PC3

where P i s t h e p r es s ur e , T i s t h e t e mp e ra t u re , v i s t he

s p e c i f ic v o l u m e , R i s t h e g as c o ns ta nt , b i s t he m ol ar

covolume, a i s t h e a t t ra c t iv e p a ra m et e r, a n db and aa re t wo c o ef fi ci e nt s w h ic h d ep en d o n t h e c on st a nt s1

and2. T h e s u b s c r i p t s r a n d C s t a n d f o r t h e r e d u c e d a n dc ri ti ca l p ro pe rt ie s. A t t he c ri ti ca l t em pe ra tu re , t he

s e co n d v i ra l c o ef f ic i en t o f E q . ( 1 ) i s e x p re s se d b y :

B2;C PC

RTC Xb Xa 4

where B2,C i s t h e s e c o n d v i r i a l c o e f f ic i en t a t t h e c r it i c al

t e m p e r a t u r e . M a t hi as ( 20 0 3) p oi nt ed o ut t ha t t he

r ed u ce d s ec on d v i ri al c oe ff ic i en t (B2,CPC / RTC) of

m os t f lu id s a t t h e c ri t ic al p o in t i s n e ar ly 0 . 34 . W h e n

t h i s c o n d i t i o n i s a p p l i e d t o E q . ( 4 ) , o n e c a n o b t a i n :

Xb Xac 0:34 5

E q . ( 5 ) w o u l d b e t h e c o n s t r a i n t f o r o b t a i n i n g t h e E o S o f

n a tu r a l g a s m i x tu r es .

Nasrifar and Bolland (2004) i m p r ov e d S o av e 's

f u n c t i o n o n t h e b a s i s o f t h e e q u a l i t y o f t h e s e c o n d v i r i a l

c oe ff ic ie nt f ro m t he R KS E oS a nd t he s qu ar e- we ll

potential. The obtained f u n c t i o n , i n g e n e r a l , i m p r o v e s

t he a cc ur ac y o f t he R KS E oS i n p re di ct in g t he p ur e

c om po ne nt c om pr es si bi li ty f ac to r a nd f ug ac it y a t

s u pe r c ri t i ca l t e mp e ra t u re s . A c cu r a te f u ga c i ty o f p u re

c o m p o u n d s i s p a r t i c u l a r l y i m p o r t a n t , a s s h o w n b y Flter

e t a l . ( 1 9 9 8 ), i n c a l c u l a t i n g t h e f l u i d p h a s e e q u i l i b r i a o f a s y m m et r i c h y d r o c a rb o n m i x t u r es c o n t a i ni n g m e t h a ne .

H ow e ve r, t h e f ug ac it y o f f lu i ds f or a n E oS i s u su al l y

F i g . 1 . P e r c e n t b i a s b e t w e e n e x p e r i m e n t a l ( D a u b e r t a n d D a n n e r , 1 9 9 2 ) a n d p r e d i c t e d v a p o r p r e s s u r e o f s o m e n a t u r a l g a s c o n s t i t u e n t s f r o m t h e t r i p l e

point to the critical point. The predictions are from Eq. (6) and %Bias is defined by: %Bias = (100)(caldexp l) / e x p l.

F i g . 2 . P e r c e n t a b s o l u t e d e v i a t i o n s b e t w e e n t h e I U P A C - r e c o m m e n d e d

v al ue s ( Wa gn er a nd d e R eu ck , 1 99 6) a nd p re di ct ed f ug ac it ie s o f

m e t h a n e a s f u n c t i o n s o f t e m p e r a t u r e a n d p r e s s u r e . T h e p r e d i c t i o n s a r ef r o m E q . ( 6 ) a n d % D e v i s d e f i n e d b y : % D e v = ( 1 0 0 ) | c a l d exp l| / e x p l.

255Kh . Na srifa r, O. B o lla n d / Jo u rn a l o f P etro leu m S cien ce a n d E n g in eerin g 5 1 ( 2 0 0 6 ) 2 5 32 6 6

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f ix ed i nd ir ec tl y b y c or re la ti ng t he E oS t o t he v ap or

pressure of pure compounds along the coexistence

c u r v e . I t i s a l s o w o r t h n o t i n g t h a t a l l t w o - c o n s t a n t E o S s

s i m i l a r i n f o r m t o E q . ( 1 ) h a v e a f i x e d v a l u e o f f u g a c i t y

a t t h e c r it i c al p o i nt . T h e c r it i c al f u ga c i ty c o ef f ic i en t o f

t h e R K S E o S i s e q u a l t o 0 . 6 6 5 7 a n d 0 . 6 4 5 7 f o r t h e P R Eo S (M i ch e ls en a nd M ol l er up , 2 0 04 ) . T he r ep or te d

v al ue f or t he f ug ac it y c oe ff ic ie nt o f m et ha ne a t t he

c ri ti ca l p oi nt (Yo un g lo v e a nd E ly, 1 98 7) i s 0 .6 64 0,

h ow ev er. I f E q. ( 1) i s t o b e u se d t o p re di ct t he t he r-

m od yn am ic p ro pe rt ie s o f m et ha ne ( na tu ra l g as ) i n

s up er cr it ic al r eg io n, t he s ta rt in g p oi nt w hi ch i s t he

c ri t ic al p o in t , s ho ul d a ls o b e p re di c te d a cc ur at e ly. I t

s h o u l d b e n o t e d t h a t t h e a c c u r a c y o f E o S s i n e n g i n e e r i n g

i s b as ed o n t h e a de q ua cy o f t h e c ri t ic al p oi n t t o p re di ct

t h e s u b c r i t i c a l a n d s u p e r c r i t i c a l p r o p e r t i e s . A s s u c h , i t i s

e s s e n t i a l f o r t h e E o S o f n a t u r a l g a s s y s t e m s t o p r e d i c t t h ec r it i ca l f u ga c it y o f m e th a n e ( a s t h e m a jo r c o mp o n en t )

a cc ur at el y. H ow ev er, i t i s w or th s tr es si ng t ha t t he

a c c u r a t e p r e d i c t i o n o f f u g a c i t y o f m e t h a n e b y t h e E o S o f

n a t u r a l g a s s y s t e m i s o n l y a n e c e s s a r y c o n d i t i o n . O n t h e

basis of this premise and Eq. (5) as the constraint, we

c o n c l u d e d t h a t 1 =2 = 1/ffiffiffi

3p

w o u ld m e et t h e se r e qu i re -

m e n t s . I n c o r p o r a t i n g t h e s e v a l u e s i n E q . ( 1 ) , a n e w PVT

r e l a t io n s h i p i s o b t a i n e d:

P RTv b

aCaTr v

b=

ffiffiffi3

p

2 6

with

b 0:079246 RTCPC

7

aC 0:421875 R2 T2C

PC8

A t t h e c r it i c al p o in t , t h e c o mp r es s i bi l i ty f a ct o r, s e co n d

v ir ia l c oe ff ic i en t, a n d f ug ac i ty c o ef fi ci e nt o f E q. ( 6) w er e

f o u n d t o b e 0 . 3 2 9 , 0 . 3 4 2 , a n d 0 . 6 6 4 0 , r e s p e c t i v e ly.

F or t h e f u n c ti o n , t h e m o di f i ed S o a ve ' s function

by N a s ri f a r a n d B o ll a n d ( 2 0 04 ) i s u s e d :

a Tr 1 m 1 ffiffiffiffiffiTr p

2Tr V1

b1

Tr b2

T2r b3

T3r Tr N1

8