Viscosity Correlation Whit SG and MW

8/10/2019 Viscosity Correlation Whit SG and MW

1/8

E L S E V I E R

PII: S0016-2361 97)00090-2

uel Vol. 76, No. l 1, pp. 10 49-10 56, 1997

1997 E lsev ier Science L td . A l l r igh ts reserved

Printed in Great Britain

0016-2361/97 17.00+0.00

i s c o s i t y c o r r e l a t i o n w i t h s p e c i f i c

g r a v i t y a n d m o l e c u l a r w e i g h t o f

c r u d e o i l f r a c t i o n s

akeshige W akabayashi

Nat iona l Ins t i tu te fo r Resources and E nv i ronment , 16-30n ogaw a, Tsukuba 305, Japan

Received 7 July 1996; revised 3 M arch 1997)

An a t t empt i s made to co r re la t e the v i scos i ty o f c rude f r ac t ions wi th o the r cha rac te r i s t i c s . The fo l lowing equa t ion

i s de r ived by s t epwise mul t ip le r eg ress ion :

log v = 1 .37 (d log M) 2 + 16 .12(d log M) - 2 _ 9 .06

wh ere v i s the k inem at ic v i sco s i ty (mm 2s -~) o f the c rude f r ac t ion a t 50C , d the spe c i f i c g rav i ty an d M the

mo l e c u l a r we i g h t . T h e c a l c u la t e d v i s c o s i t y h as a n a v e r a g e a b s o lu t e d e v i a t io n o f 1 5 .9 . C o mp a r i s o n b e t we e n t h e

a v e r a g e d e v i a t i o n c a l c u l a te d f r o m t h e e q u a t i o n a n d t h a t f r o m l o c a l c o rr e l a ti o n s f o r c o n n a t e c r u d e f r a c t i o n s s h o ws

tha t th i s equa t ion i s app l i cab le no t on ly to a va r i e ty o f c rude f r ac t ions bu t a l so to those be long ing to a spec i f i c

g r o u p .

1997 Elsevier Science Ltd.

Keywords: cru de oil fract ions; visc os ity; p r e d i c t i o n )

I n f o r m a t i o n o n t h e v i s c o s i t y o f o il f r a c t io n s a n d t h e i r

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

t r a n s p o r t p r o p e r t i e s in p e t r o l e u m p r o c e s s i n g . T h e e f f e c t o f

t e m p e r a t u r e o r c h a r a c t e r i s ti c s o f o i l o n i ts v i s c o s i t y i s

e s p e c i al l y n o t e w o r t h y . T y p i c a l v i s c o s i t y - t e m p e r a t u r e c o r -

r e l a t i o n s a r e :

l o g

v = a + b/ T 1)

l o g [ l o g ( v + 0 . 7 ) ] = a + b l o g T ( 2 )

w h e r e v i s t h e k i n e m a t i c v i s c o s i t y a t t e m p e r a t u r e T , a n d a

a n d b a r e c o n s ta n t s . A n d r a d e d e r i v e d a f o r m u l a f o r t h e

v a r i a t i o n o f v i s c o s i ty w i t h t e m p e r a t u r e , b a s e d o n a th e o r e -

t i c a l a n a l y s i s o f l i q u i d v i s c o s i t y 1. T h e u s e o f k i n e m a t i c

v i s c o s i t y i n h is f o r m u l a r e s u l t s i n t h e s a m e f o r m a s E q u a t i o n

( 1 ), w h i c h h a s b e e n s h o w n t o b e a p p l i c a b l e t o c r u d e f r a c -

t i o n s 2 . E q u a t i o n ( 2 ) i s s t a n d a r d i z e d f o r l i q u id h y d r o c a r b o n s

b y A S T M 3.

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

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

v i s c o s i t y is s t a n d a r d i z e d b y A S T M i n r e l a t i o n t o t h e

m o l e c u l a r w e i g h t 4 a n d h a s b e e n s h o w n t o v a r y i n v e r s e ly

w i t h t h e s p e c i f i c g r a v i t y o f s t e a m c r a c k e r o i l s 5 . S t u d i e s o f t h e

c o n s t a n t s i n v i s c o s i t y - t e m p e r a t u r e c o r r e l a t i o n s c o u l d a l s o

s u g g e s t s u i t a b l e c h a r a c t e r i s t i c s w h i c h a r e r e la t e d t o t h e

v i s c o s i t y .

A m i n a n d c o - w o r k e r s e x p r e s s e d t h e c o n s t a n t a a s a

f u n c t i o n o f t he a v e r a g e b o i l i n g p o i n t T b a n d t h e s p e c i f i c

g r a v i t y d , a n d t h e c o n s t a n t b a s a f u n c t i o n o f T h i n E q u a t i o n

( 1 ), f o r A m e r i c a n 2 a n d S a u d i A r a b i a n 6 c r u d e f r a c t io n s

r e s p e c t i v e l y . T h i s m e a n s t h a t t h e k i n e m a t i c v i s c o s i t y i s

e x p r e s s e d a s a f u n c t i o n o f T b a n d d o f t h e c r u d e f r a c t i o n a t a

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

c h a r a c t e r i s t ic s o f l iq u i d w e r e a l s o r e v e a l e d i n a s t u d y o f t h e

e s t i m a t i o n o f t h e K f a c t o r o f l u b e b a s e o i l s f r o m t h e i r

c h a r a c t e r i s ti c s 7 a n d i n a p p l i c a t i o n o f a c o r r e s p o n d i n g - s t a t e s

m o d e l o f n o n - p o l a r l i q u id s f o r e s t i m a t i o n o f th e v i s c o s i t y o f

8

p e t r o l e u m f r a c t i o n s , a l th o u g h t h e f o r m s o f th e s e r e la t io n s

d i f f e re d . T h e a b o v e S a u d i A r a b i a n c r u d e f r a c t i o n s a l s o h a v e

d i f f e r e n t f o r m s o f r e l at i o n . T h e c o n s t a n t s a a n d b w e r e

r e c e n tl y e x p r e s s e d b y A m i n a n d B e g 9 i n m o d i f ie d f o r m a s

f u n c t i o n s o f m o l e c u l a r w e i g h t M a n d o f T b , M a n d d ,

r e s p e c t i v e l y .

T h e c o n s t a n t s a a n d b i n E q u a t i o n ( 2 ) h a v e b e e n s h o w n t o

b e l i n e a rl y p r o p o r t i o n a l t o l o g M f o r C o l d L a k e b i t u m e n

f r a c t i o n s I . T h e c o n s t a n t a i n E q u a t i o n ( 2 ) h a s a l t e r n a t iv e l y

b e e n e x p r e s s e d 11 a s a f u n c t i o n o f d f o r M i d d l e E a s t e r n c r u d e

o i l s a t a t m o s p h e r i c p r e s s u re , w h e n b = - 3 . 5 . M o h a r a m

et

aI. ~2

d e v e l o p e d t h e f o l l o w i n g e m p i r i c a l c o r r e l a ti o n o n t h e

b a s i s o f E q u a t i o n ( 2 ):

l o g ( l o g v + 3 . 2 4 2 ) = 0 . 0 1 8 3 +

b/T

w h e r e b i s a f u n c t i o n o f T h a n d d .

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

c h a r a c t e r i s ti c s w h i c h i s a p p l i c a b l e t o a v a r i e t y o f p e t r o l e u m

f r a c t io n s , a n d t h e v i s c o s i t y o f p e t r o l e u m o i l a n d i t s fr a c t i o n s

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

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

m e n t i o n e d a b o v e . S u c h i n c o n s i s t e n c y m a y b e d u e to th e fa c t

t h a t a n y r e l a ti o n w a s d e r i v e d f r o m e x p e r i m e n t a l d a t a o n

f r a c t i o n s o r i g i n a t i n g f r o m o n l y o n e b i t u m e n o r o n c o n n a t e

o i l s .

T h e a i m o f th e p r e s en t w o r k w a s t o d e v e l o p a n e w

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

c r u d e f r a c t io n , i n c l u d i n g b i t u m e n , a n d c h a r a c t e r i s ti c s t h a t

c a n b e e a s i l y m e a s u r e d . S i n c e t h e c o n t r ib u t i o n o f a n y

F u e l 1 9 97 V o l u m e 7 6 N u m b e r 11 1 4 9


2/8

Viscosity correlation o f crude oil fractions: T Wakaba yashi

Ta ble 1 Propert ies of heavy crude oi ls

Crude oil Specif ic gra vi ty Mo lecular e ight Conradson arbon H/C atomic ra t i o A ro m ati c i t y Visc os i ty t 50C

(15/4C) residue (wt%) (mm 2s-=)

O i l s a n d b i t u m e n s b

A t h a b a s c a ( G C O S ) 0 . 9 9 7 3 4 6 8 1 1 .7 1 . 46 0 . 3 9 8 5 7

A t h a b a s c a ( A R C ) 1 . 0 15 5 5 1 8 1 3 . 2 1 . 48 0 . 3 9 2 2 8 0

C o l d L a k e - 1 0 . 9 8 0 4 4 6 6 1 2 . 6 1 . 49 0 . 3 9 4 9 8

Col d L a ke - 2 0 . 9944 497 13 .7 1 .45 0 , 39 10 4 9

O r i noc o M e r r y 1 . 0007 549 16 .3 1 . 46 0 . 37 3369

O r i noc o M o r i c ha l 1 . 0087 579 15 .5 1 . 45 0 . 38 8337

T o p p e d c r u d e s

A r a b i a n L i gh t 0 . 9521 463 8 . 23 1 . 57 0 . 31 160 . 2

K u w a i t 0 . 9643 524 10 . 18 1 .55 0 . 32 404 . 6

K ha f j i 0 . 9821 567 13 .73 1 . 50 0 . 34 1344

G a c h S a r a n 0 . 9 5 9 4 5 0 3 9 . 6 0 1 . 54 0 . 3 2 3 5 3 . 7

A g ha J a r i 0 . 9529 478 8 . 22 1 . 54 0 . 31 225 . 6

T a c h i n g 0 . 9 0 8 6 5 6 3 4 . 3 5 1 . 74 0 . 2 0 1 3 9 . 5

S he ng l i 0 . 9446 593 8 . 54 1 . 62 0 . 28 839 . 9

a Ra t i o o f a r oma t i c c a r bon t o t o t a l c a r bon a t oms c a l c u l a t e d a c c or d i ng r e f . [ 14 ]

b G C O S , G r e a t C a n a d i a n O i l S a n d s , c o m m e r c i a l p l a n t, o p e n - p i t m i n i n g ; A R C , A l b e r t a R e s e a r c h C o u n c i l , p i l o t p la n t , o p e n - p i t m i n i n g ; C o l d L a k e - l ,

n s tu

d i s c o n t i n u o u s r e c o v e ry i n p i l o t p l an t ; C o l d L a k e - 2 n s tu c o n t i n u o u s r e c o v e r y i n p i l o t p l a n t

c h a r a c t e r i s t i c t o t h e v i s c o s i t y c a n n o t b e t h e o r e t i c a l l y

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

o f c r u d e f r a c t i o n s , it h a s b e e n d e t e r m i n e d b y m u l t i p l e

regres s ion ana lys i s .

E X P E R I M E N T A L

Materials

B e n z e n e ( e x t r a - p u r e ) w a s u s e d t o r e m o v e w a t e r , s a lt et c .

f r o m c r u d e o i l s a n d b i t u m e n s . T h e e x t r a c t e d o i l s a n d

b i t u m e n s Iw e r e h e a t e d t o 2 0 0 C a t a t m o s p h e r i c p r e s s u r e t o

e v a p o r a t e t h e b e n z e n e . T o p p e d c r u d e s w e r e r e s i d u e s

d i s ti l le d f r o m c o r r e s p o n d i n g c r u d e s a t 0 . 13 3 k P a . T h e

b o i l i n g p o i n t o f t h e t o p p e d c r u d e s w a s > 3 4 0 C at

a t m o s p h e r i c p r e s s u r e .

Measurements

S p e c i f i c g r a v i t i e s w e r e d e t e r m i n e d p y c n o m e t r i c a l l y .

M o l e c u l a r w e i g h t s o f th e o i l s a n d b i t u m e n s a n d t o p p e d

c r u d e s ( h e r e i n a f t e r d e s c r i b e d a s t h e tw o b i t u m e n s ) d i s s o l v e d

i n b e n z e n e w e r e d e t e r m i n e d w i t h a v a p o u r - p h a s e o s m -

o m e t e r a t 3 1 8 K . E l e m e n t a l c o m p o s i t i o n w a s d e t e r m i n e d

w i t h a n i n s t r u m e n t a l a n a l y s e r .

R e l a t i v e v i s c o s i t i e s w e r e c a l c u l a t e d f r o m f l o w r a t e s i n a

c a p i ll a r y v i s c o m e t e r o f C a n n o n - F e n s k e t y p e . Pr o t o n n . m .r .

s p e c t r a o f t h e t w o b i t u m e n s d i s s o l v e d i n c a r b o n t e t r a -

c h l o r i d e ( e x t r a - p u r e ) w e r e m e a s u r e d w i t h a J E O L J M N -

M H - 1 0 0 s p e c t r o m e t e r . C o n r a d s o n c a r b o n r e s i d u e s w e r e

d e t e r m i n e d a c c o r d i n g t o J I S s t a n d a r d K 2 2 7 0 .

R E S U L T S A N D D I S C U S S I O N

P r o p e r t i e s o f t h e t w o b i t u m e n s a r e s h o w n i n Table 1. T h e

a r o m a t i c i t y w a s c a l c u l a t e d f r o m d a t a o n t h e e l e m e n t a r y

a n a l y s i s , t h e p r o t o n d i s t r i b u t io n d e t e r m i n e d f r o m n . m . r , d a t a

o n t h e b a s i s o f C h a m b e r l a i n ' s r u l e 13, a n d t h e m o l e c u l a r

w e i g h t a c c o r d i n g t o t h e p r o c e d u r e o f S p e i g h t 14.

Applica tion of publish ed correlati ons to the two bitumens

A n a t t e m p t w a s m a d e t o c o r r e l a t e t h e v i s c o s i t y a t 5 0 C

w i t h t h e s p e c i fi c g r a v i t y o r th e m o l e c u l a r w e i g h t f o r t h e t w o

0.7 -

5

+

0 6 m

0 . 5 - -

0 . 4 - -

0 . 3 - -

t l I

o

d o

f i l l S ~

,O

o , .

..,O'

O

O

o.2 I I I I

2 . 60 2 . 65 2 . 70 2 . 75 2 . 80

l o g M

Fig ure 1 Double logarithm of viscosity as a function of

logarithm of molecular weight for (0) oil sand bitumens and (O)

topped crudes

b i t u m e n s b y t h e f o l l o w i n g p u b l i s h e d r e l a t i o n s 5 ' j 'f l :

l o g v = a b/d

(3)

log[ log (v + 0 . 7 ) ] = a + b log M

(4)

log[ lo g(v + 0 . 7 ) ] = a bd cd 2 (5)

w h e r e c i s a c o n s ta n t . ( N o t e : h e r e a n d b e l o w , ' l o g ' s i g n if i es

l o g a r i t h m t o b a s e 1 0 .) T h e l o g a r i t h m o f th e v i s c o s i t y i s

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

b o t h t h e t w o b i t u m e n s a t t h e 9 5 % c o n f i d e n c e l e v e l , w i t h

c o m p a r a b l e v a l u e s o f t h e c o r r e l a t i o n c o e f f i c i en t f o r d i f f e r e n t

l eve l s o f s ign i f i cance iS. Ho we ver , i t i s d i f f icu l t to de f ine a

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

g r a v i t y , s i n c e t h e c o r r e l a t i o n c o e f f i c i e n t o f 0 . 8 1 7 i s t o o l o w ,

a n d b e c a u s e t h e r e i s n o c o r r e l a t i o n f o r o i l s a n d b i t u m e n

a l o n e o r t o p p e d c r u d e a l o n e .

1 5 F u el 9 97 V o l u m e 76 N u m b e r


3/8

Viscos i t y co r re la t i on o f c rude o i l frac t i ons : T Wa kabayash i

0.7 --

0.6

~ 0.5

~ 0.4

0.3

0.2 I I I I

0.85 0.90 0.95 1.00 1.05

d

Fig ure 2 Double logarithm of viscosity as a function of specific

gravity for (O) oil sa nd bitumens and () topped crudes

m o r e v i s c o u s i n a c o m p a r i so n b e t w e e n o i l s o f s i m i l a r

m o l e c u l a r w e i g h t .

T h e v i s c o s i t y s e e m s t o b e c o r r e l a t e d m o r e w i t h t h e

spec i f i c g rav i ty than wi th the molecu la r we ight fo r bo th

b i t u m e n s Figure 2), and the cor re l a t ion i s s t a t i s t i ca l ly

s ign i f i can t a t 95% . H ow ever , i t has the fo l low ing problems.

There i s no cor re l a t ion for o il sand b i tumen a lone or t opped

c r u d e a l o n e . T h e c u r v e i n

Figure 2

i s concave upward , bu t

the cor re sponding curve for Middle Eas t e rn c rude o i l i s

convex upward in a lmost t he same range of spec i f i c

grav i ty ~ l. These re su l t s sug ges t t ha t Equa t ion (5) a l so i s

inadequa te to cor re l a t e v i scos i ty w i th spec i f i c g rav i ty for

d i f fe ren t k inds of heavy o i l .

I t fo l lows f rom the abov e tha t t he v i scos i ty is r e l a t ed to some

exten t t o the spec i f ic g rav i ty or t he mo lecu la r we igh t , and a l so

cannot be exp ressed a s a func t ion of spec i f i c g rav i ty a lone or o f

molecu la r we igh t a lone . There fore the v i scos i ty has to be

expressed a s a func t ion of two or mo re cha rac t e ri s t ic s .

Developm ent of new correlations at 50C

Selection o f characteristics.

Only cha rac t e r i s t i c s t ha t

Ta ble 2 Correlation coefficients of relations between characteristics

Molecular weight Specific gravity No. of samples

13

il sand bitumens and topped crudes

Molecular weight 1.0000 - 0.0911

Specific gravity - 0.0911 1.0000

Conradson carbon residue 0.0996 0.9262

Atomic ratio H/C 0.2734 - 0.9394

Aromat icity - 0.3375 0.9458

Vacuum residues

Molecular weight 1.0000 0.5644

Specific gravity 0.5644 1.0000

Conradson carbon residue 0.5209 0.9391

Atomic ratio H/C - 0.5717 - 0.9764

Aromat icity 0.4836 0.9917

Softening point 0.8868 0.5102

Saudi Arabian crude fractions b

Molecular weight 1.0000 0.9918

Specific gravity 0.9918 1.0000

Average boiling point 0.9892 0.9944

UOP K - 0.9131 - 0.9170

10

8 )

20

~See ref.16

bSee ref.9

A p l o t o f d o u b l e l o g a r i t h m o f v i s c o s i t y v e rsu s l o g a r i t h m

o f m o l e c u l a r w e i g h t a p p a r e n t l y su g g e s t s n o r e l a ti o n

b e t w e e n t h e t w o b i t u m e n s

Figure 1).

H o w e v e r , c a r e f u l

e x a m i n a t i o n p r o v i d e s t h e f o l l o w i n g i n f o r m a t i o n . T h e

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

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

shown by the fu l l l i ne . The dashed l i ne g ives a s imi l a r

r e l a t i o n f o r t o p p e d c r u d e s e x c e p t K h a f j i a n d T a i c h i n g ,

which have s imi l a r molecu la r we igh t s bu t d i f fe ren t spec i f i c

grav i t i e s . The dashed l i ne i s be low and a lmost pa ra l l e l t o

the fu l l l i ne . The v i scos i ty i s t hus cor re l a t ed wi th

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

so m e o f t h e t o p p e d c r u d e s a l o n e , b u t n o t f o r b o t h b i tu m e n s

a t 95% s ign i f i cance . The p lo t r evea l s t ha t t he v i scos i ty i s

cor re l a t ed wi th molecu la r we igh t fo r o i l s w i th s imi l a r

prope r t i e s , and tha t o i l s w i th h ighe r spec i f i c g rav i ty a re

a re no t i n t e rcor re l a t ed should be cons ide red a s va r i ab le s

for e s t ima t ing the v i scos i ty o r i n mul t ip l e regress ion

analysis.

Table 2 shows cor re l a t ions of cha rac t e r i s t i c s o f o i ls w i th

the i r molecu la r we igh t o r spec i f i c g rav i ty . The molecu la r

we ight i s r e l a t ed to ne i the r t he spec i f i c g rav i ty nor the o the r

three cha rac t e r is t i c s fo r the tw o b i tumens , whereas the spec i f ic

grav i ty i s cor re l a t ed wi th those th ree cha rac t e r i s ti c s a t 99%

signif icance. The sam e is t rue for v acu um res idues 16. On the

o the r hand , M, d and Tb of Arab ian c rude o i l f rac t ions a re

c lose ly cor re l a t ed wi th each o the r9. Table 2 sugges t s t ha t bo th

molecu la r we igh t and spec i f i c g rav i ty should be necessa ry

and a l so suf f i c i en t fo r e s t ima t ion of v i scos i ty , a s t he o the r

cha rac t e r is t i c s a re cor re l a t ed wi th a t l eas t on e o f t hese two.

Thi s s tudy a t t empt s to express the v i scos i ty a s a

polynomia l o f va r i ab le s to which the two cha rac t e r i s t i c s ,

Fuel 1997 Volume 76 Number 11 1 51


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Viscos i t y co r re la t i on o f c rude o i l f rac t ions : T Wakabayash i

Tab le 3 Properties of crude fractions

Specific grav ity 15/4C) Mo lecular weight Viscosity at 50C mm 2s ~)

Cold L a ke o i l s a nd b i tum e n

W ho le b i tum e n 0 .995 582 3074

Cut 1 0 .879 209 2 .77

Cut 2 0.941 310 27.4

Cut 3 0 .990 667 2664

Cut 4 0 .999 800 24 700

Cut 5 1.072 2500 6.809

X 1010

Sa ud i A ra b ia n c rude f r a c t ions b

Berr i : 260/345 0.8431 230 2.95

345/455 0.8886 323 10.97

Heav y: 260/345 0.8513 232 3.04

345/455 0.9020 331 13.17

Med ium : 260/345 0.8477 228 2.82

345/455 0.9006 326 12.10

Light: 260/345 0.8476 217 2.99

345/455 0.9057 332 13.16

a Se e r ef . [10 ]; the t e m pe ra tu re a t w h ic h the s pe c i f i c g ra v ity w a s m e a s ure d i s unknow n; the v i s c os i ty i s c a lc u la ted f rom pa ra m e te r s

bSee ref. [9]

Tab le 4 F-values in analysis of variance for simple logarithm of viscosity of crude fraction

Inde pe nd e n t va r ia b le S ing le c o r re la t ion F -va lue wi th (d log 114)2 F -va lue wi th (d log

M 2

a n d

1 / d

l o g M )

log M 0.9576 351.06 2977.77

1/log M 0.9152 593.41 2981.83

d 0 .8422 891 .59 2976 .52

1 / d

0.8139 1160.49 2991.56

d log M 0 .9546 4367 .04

l l d

log M) 0 .8731 4571 .33 - -

( log

M ) / d

0.5977 412.08 2984.62

d / ( log M) 0 .5669 395 .92 2977 .92

( log M) 2 0 .9724 347 .78 2974 .79

1/( log M) 2 0.8896 764.16 2988.28

d 2 0 .8560 777 .87 2970 .39

l i d 2 0.7994 1296.82 2995.35

(d log

M 2

0 .9 8 2 6 - - - -

l l d

log

M 2

0 .8290 4394 .64 a

[ ( log M) /d ] 2 0 .6125 421 .03 2988 .39

(d / log M) 2 0 .0725 337 .26 2928 .94

UCa lcu lat ion s toppe d be c a us e va lue o f c o r re la t ion m a t r ix < 0 .001

t ha t i s, l og M and d , a re t rans form ed by mul t ip l i ca t ion and

d i v is i o n . O t h e r m o r e c o m p l e x t y p e s o f v a r i a b l e a r e o m i t t e d

f r o m c o n s i d e r a t i o n , s o a s n o t t o m a k e t h e e x p r e s s i o n t o o

c o m p l e x . T h i s r e su l t s i n v i s c o si t y b e in g e x p r e s s e d b y s o m e

of the fo l low ing 16 va r i ab les : l og M, 1 / logM , d, 1/d,

d l o g M , 1 / d l o g M ) , l og M ) / d , d / l o g M ) a n d t h e i r

squa res . The squa res a re inc luded s ince a quadra t i c

expres s ion i s gene ra l ly prac t i ca l i n a po lynomia l . S imple

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

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

3 ) - 5 ) .

A l a rge r number of s amples wi th d ive rse cha rac te r i s t i c s

inc reases conf idence in the regres s ion ana lys i s , which has

t h e r e f o r e b e e n c a r r i ed o u t f o r 2 7 s a m p l e s o f p e t r o l e u m o i ls

i n c l u d i n g C o l d L a k e b i t u m e n a n d A r a b i a n o i l s , w h o s e

vi scos i ty a t 50C, spec i f ic g rav i ty and m olecu la r we igh t a re

shown in Table 3, bes ides the two b i tumens . Arab ian o i l s

wi th a v i s cos i ty of > 2 .0 mm2s - l a re a t t he l imi t o f

appl i cab i l i t y o f Equ a t ion 2) .

Multiple regression analysis. A p o p u l a r p r o c e d u r e i s

the s t epwise me thod for adequa te s e l ec t ion of va r i ab les

f rom the above 16 for mul t ip l e regres s ion ana lys i s . The

s tepwise me thod i s e s sen t i a l ly a fo rward s e l ec t ion proce -

dure bu t w i th the added provi so tha t a t each s t age the pos -

s ib i li t y o f de le t ing a va r i ab le , a s in back ward e l imina t ion , i s

cons id e red 17 a~. Th e f i rs t co lu mn of Table 4 shows the

s ing le cor re l a t ion of each va r i ab le w i th the s imple loga r i thm

o f t h e v i s co s i ty . T h e v a r i a b le d l o g M ) ~ e n t e rs t h e

v i scos i ty -cha rac te r i s t i c equa t ion a s the f i r s t va r i ab le , a s i t

has the h ighes t cor re l a t ion wi th an F -va lue of 257 , which

e x c e e d s t h e t a b u l a t ed v a l u e o f 7 . 8 2 (FI,24,0.01). T h e s e c o n d

c o l u m n s h o w s F - v a l u e s f o r th e l o g a r it h m o f v i s c o s it y c o r -

re l a t ed wi th each o f the 15 va r i ab les o the r than d log M) z .

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Viscos i t y co r re la t i on o f c rude o i l f rac t i ons : 7 Wakabayash i

T a b l e 5 A n a ly s i s o f v a r i an c e fo r s im p le lo g a r ith m o f v i s c o s i ty o f c ru d e f r a c tio n c o r re l a te d w i th fu n c tio n s o f d lo g M

(1) With (d log M)2 and 1/ d og M), referring to Equation (6)

Adjusted multiple correlation = 0.998581

Sum of squ are s De gre es f freedom Mean square F-value

Regression 109.51 2 54.75 4571.33

Residual 0.29 24 0.01

Total 109.80 26

Independent variable Partia l regression St an da rd rror St an da rd artia l St an da rd rror F-value VIF

coefficient regression coefficient

(d log M) 2 1.601 0.034 1.509 0.033 2154.23 9.69

1/ d log M ) 19.887 1.163 0.556 0.033 292.33 9.69

Const, - 15.883 0.686

(2) W ith (d log M ) 2 and

1/ d

og M) 2, referring to E quation (8)


Sum of squ are s De gre es f freedom Mean square F-value

Reg ressio n 37.49 2 18.74 1547.97

Residual 0.28 23 0.01

Total 37.77 25

Independen t va riable Part ia l regr ess io n S ta nd ard r ror S t an da rd a r tial S t an dar d rror F-value VIF ~


(d log M) 2 1.372 0.084 1.567 0.096 265.02 28.89

1/ d log M) 2 16.119 2.644 0.587 0.096 37.17 28.89

Const. - 9.061 0.970

Variance inflation factor

T a b l e 6 A n a ly s i s o f v a r i a n c e fo r d o u b le lo g a r ith m o f v i s c o s i ty o f c ru d e f r a ct io n c o r rel a t e d w i th

l / d

l o g M)


Sum of squ ares De gre es f freedom Mean square F-value

Regression 2.94 1 2.94 4795.54

Residual 0.02 25 0.00

Total 2.96 26

Independent variable Partial regression St an da rd rror St an da rd artia l St an da rd rror F-value VIF


l/ d

log M) - 5.849 0.084 - 0.997 0.014 4795.54 1.00

Const. 2.658 0.035

T h e c o m b i n a t i o n o f ( d l o g M ) 2 a n d

1 / d

l o g M ) i s c l e a r l y t h e

m o s t s u i t a b l e f o r d e t e r m i n a t i o n o f v i s c o s i t y . N o F - v a l u e i n

t h e t h ir d c o l u m n e x c e e d s t h e g r e a t e s t F - v a l u e i n t h e s e c o n d

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

c o r r e l a t i n g v i s c o s i t y w i t h t h e c h a r a c t e r i s t i c s . S e c t i o n ( 1 ) o f

T a b l e 5

s h o w s t h e a n a l y s i s o f v a r i a n c e f o r t h e c o r r e l a t i o n .

T h e c o r r e l a t i o n c a n b e r e c o g n i z e d t o b e e f f e c t i v e a t 9 9 %

s i g n i fi c a n c e , s i n c e th e F - v a l u e o f e a c h v a r i a b l e e x c e e d s 7 . 8 2

a n d s i n c e t h e v a r i a n c e i n f l a t io n f a ct o r ( V I F ) o f < 1 0 i n d i -

17 b)

c a t e s t h a t t h e r e i s n o m u l t i c o l h n e a r l t y . T h e v i s c o s i t y i s

e x p r e s s e d a s f o l l o w s :

l o g v = 1 . 6 0 ( d l o g M ) 2 + 1 9 . 8 9 / ( d l o g M ) - 1 5 . 88 ( 6 )

H o w e v e r , i t c a n n o t b e c o n c l u d e d t h a t E q u a t i o n ( 6 ) i s t h e

o n l y f o r m o f e x p r e s s i o n f o r th e v i s c o s i t y . T h e F - v a l u e f o r

t h e c o m b i n a t i o n o f t h e t w o v a r i a b l e s in E q u a t i o n ( 6 ) is

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

c o m b i n a t i o n o f ( d l o g M ) 2 a n d ( d lo g M ) o r o f ( d lo g M ) 2

a n d

l / d

l o g M ) 2

T a b l e 4 ) .

A d d i t i o n o r d e l e t i o n o f a f e w

s a m p l e s m a y m a k e i t p o s s i b l e f o r t h e F - v a l u e s f o r t h e tw o

l a t t e r c o m b i n a t i o n s t o e x c e e d t h a t f o r th e c o m b i n a t i o n i n

E q u a t i o n ( 6 ). T h e f o r m o f E q u a t i o n ( 6 ) c a n e a s i l y b e c h a n -

g e d t o e x p r e s s s e v e r a l f u n c t i o n s o f ( d l o g M ) i n c l u d i n g i t s

s q u a r e a n d r e c i p r o c a l .

T h e s a m e a n a l y s i s w a s c a r r i e d o u t f o r t h e d o u b l e

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

T a b l e 6

s h o w s t h e a n a l y s i s

o f v a r i a n c e f o r t h e c o r r e l at i o n . T h e f o r m o f t h e b e s t

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

t h e s i m p l e l o g a r i t h m , a n d i s e x p r e s s e d a s :

l o g [ l o g ( v + 0 . 7 ) ] = - 5 . 8 5 / ( d l o g M ) + 2 . 6 6 ( 7 )

A s t h e F - v a l u e f o r 1 / d l o g M ) i s v e r y fa r f r o m a n y o t h e r F -

v a l u e , t h e f o r m o f E q u a t i o n ( 7 ) c o u l d n o t b e c h a n g e d m u c h .

T h e r e i s h a r d l y a n y d i f f e r e n c e b e t w e e n t h e t w o c o r r e l a -

t i o n s w h i c h a r e e x p r e s s e d a s E q u a t i o n s ( 6 ) a n d ( 7 ) , a s b o t h

h a v e l a r g e F - v a l u e s , v e r y g o o d a d j u s t e d m u l t i p l e c o r r e l a t i o n

a n d a V I F o f < 1 0. T h e d e v i a t i o n s o f t h e c a l c u l a t e d f r o m

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

m e r i t s o f t h e t w o c o r r e l a t i o n s . P r e d i c t i o n f r o m t h e s i m p l e

l o g a r i t h m g i v e s a l o w e r a v e r a g e a b s o l u t e d e v i a t i o n o f

1 6 .0 % , c o m p a r e d w i t h 2 3 . 9 % f o r t h e d o u b l e l o g a r i t h m .

T h e r e f o r e E q u a t i o n ( 6 ) i s s u p e r i o r . T h e a v e r a g e a b s o l u t e

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Viscos i t y co r re la t i on o f c rude o i l f rac t ions : T Wakabayash i

5

3

:

2

J

2 0 2 5

d log M

3 0

Figu re 3 Correlation of logarithm of viscosity with product of

specific gravity and logarithm of molecular weight for crude

fractions: O, oil sand bitumens; , topped crudes; &, Cold Lake

bitumen; 4, Saudi Arabian crude

next s ec t ion . The s t anda rd pa r t i a l r egres s ion coe f f i c i en t

ind ica te s tha t t he cont r ibu t ion of (d log M ) 2 i s g rea te r than

tha t o f 1 / d l og M ) 2 T a b l e 5 ) .

The v i s cos i ty i s we l l expres sed by Equa t ion (8) fo r a

va r i e ty of c rude f rac t ions F i g u r e 3 ) . I s Equa t ion (8) a l so

su i t ab le for p red ic t ing the v i s cos i ty of o i l s be longing to a

spec i f i c g roup? The f rac t ions for which mul t ip l e regres s ion

ana lys i s was ca r r i ed ou t in th is s tudy can b e d iv ided in to the

fo l lowing th ree groups :

(1) O i l s and b i tume ns and topped c rudes wi th bo i l ing po in t

> 3 4 0 C a n d m o l e c u l ar w e i g h t r a n ge - 4 5 0 - 6 0 0 . T h i s

group i s e f fec t ive for t e s t ing whe the r Equa t ion (8) i s

appl i cab le to heavy o i l s o r no t .

(2 ) Cold Lake o i l s and b i tumen and i t s f rac t ions wi th a

m o l e c u l a r w e i g h t r a n g e o f 2 0 9 - 2 5 0 0 . T h i s g r o u p i s

e f fec t ive for t e s t ing whe the r Equa t ion (8) i s app l i cab le

o v e r a w i d e m o l e c u l a r w e i g h t r a n g e o r n o t .

(3 ) F rac t ions of the Middle Eas te rn c rude o i l s w i th bo i l ing

p o i n t > 1 7 4 C a n d m o l e c u l a r w e i g h t r a n g e o f 3 0 0 -

400 , which a re l i gh te r than the o i l s o f g roup (1) .

M u l t i p le r e g r e s s i o n a n a ly s i s g a v e E q u a t i o n s ( 9 ) - ( 1 1 ) t o

expres s the bes t cor re l a t ions for the f i r s t , s econd and th i rd

groups re spec t ive ly :

log v = 1 .10(d log M) 2 - 4 .82 (9)

log v = 3 .32(d log M) 2 + 118 .66 / ( log M) - 64 .49 (10)

log v = 0.5 4(d log M) 2 - 1.69 (11)

T h e f o r m o f e a c h o f t h e se e q u a t i o n s i s d i f fe r e n t f ro m t h a t o f

Equa t ion (8) , which apparen t ly s eems d i f f i cu l t t o apply to

o i l s o f a spec i f i c g roup . Th e av e rage dev ia t ions for the f ir st ,

s econd and th i rd groups a re 20 .7 , 12 .6 and 4 .8 re spec -

t ive ly in pred ic t ion f rom the above th ree equa t ions , and

17 .5 , 26 .5 and 7 .0 in p red ic t ion f rom Equ a t ion (8)

Ta b le 7 ) . I t s eems incomprehens ib le tha t t he dev ia t ion

us ing Equa t ion (9) fo r the f i r s t g roup i s g rea te r than tha t

us ing Equa t ion (8) . Mul t ip l e regres s ion ana lys i s w i th a

smal l number of s amples i s somet imes s e l f -de fea t ing . A

la rge r number of s amples in the f i r s t g roup would y ie ld a

more su i t ab le re l a t ion in the s ame form as Equa t ion (8) . A

la rge dev ia t ion in pred ic t ion for the s econd group us ing

Equa t ion (8) can be obse rved . Thi s i s based on the ex t ra -

ord ina ry e r ror o f 98 .8 for cu t 3 , and exc lus ion of th i s cu t

reduce s the dev ia t ion to 8 .4 .

Ta b le 7

sugges ts that the

v i scos i ty of o i l s be longing to a spec i f i c g roup can be

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

Equa t ion (8) fo r a l l c rude f rac t ions , and tha t t he ca l cu la t ed

va lues a re a lmos t the s ame . I t i s reasonable to co nc lude tha t

Equa t ion (8) i s app l i cab le for p red ic t ing the v i s cos i ty of a

va r i e ty of c rude f rac t ions and a l so tha t o f o i l s be longing to

any spec i f ic g roup . The va r i ab le (d log M) 2 i s a key cont r i -

bu tor in Equa t ion (10) a s we l l a s in Equa t ion (8) , and i s t he

mos t impor tan t fac tor in any of the equa t ions for p red ic t ing

viscos i ty.

The v a r i ab le d log M m ay be s e l ec t ed pre fe ren t i a l ly in the

f i r s t s t ep of the s t epwise me thod in th i s s tudy , a s the

v i scos i ty is cor re l a t ed wi th bo th d and log M. Th ere fo re

the re may be a r i sk of mis s ing a func t ion o f d and/or log M.

Mu l t ip l e regres s ion ana lys i s w i th d log M e xc lud ed in the

f i r s t s t ep was a l so ca r r i ed ou t . However , t he cor re l a t ion

equa t ion thus de r ived was no t supe r ior to Equa t ion (8) e t c .

C o r r e l a t i o n s a t o t h e r t e m p e r a t u r e s

The v i s cos i ty of c rude f rac t ions a t 50C has b een fou nd in

the preced ing s ec t ion to be cor re l a t ed wi th the i r spec i f i c

grav i ty and molecu la r we ight . I s a s imi l a r expres s ion

poss ib le a t o the r t empera tures? M ul t ip l e regres s ion ana lys i s

w a s u s e d t o d e r i v e E q u a t i o n s ( 1 2 ) - ( 1 4 ) a s t h e b e s t

c o r r e l a t i o n s r e s p e c t i v e l y f o r v a c u u m r e s i d u e s w h o s e

vi scos i t i e s a t 140 and 160C have been repor t ed 9J6 and

for pe t ro leu m f rac t ions f ro m v a r ious sources a t 37 .8C19:

log v = 0.3 8(d log M) 2 - 1.28 (12)

log v = 0.34 (d log M) 2 - 1.18 (13)

log v = 0 .83(d log M) 2 - 8 .80(d / log M) 2 - 1 .61 (14)

The independent va r i ab le s e l ec t ed in the cor re l a t ions for

vacuum res idues i s t he s ame as tha t i n Equa t ions (9) and

( 1 1 ). T h e v i s c o s it y c a n b e c o r r e l a te d w i t h ( d l o g M ) 2 a t

h ighe r t empera tures such a s 140 or 160C as we l l a s a t

50C. The co r re l a t ion coe f f i c ien t s fo r vacuum res idues how-

ever a re low er than tha t fo r c rude f rac t ions a t 50C, and the

ave rage dev ia t ions a re 42 a t 140C and 36 a t 160C.

Such l a rge dev ia t ions a r i s e no t f rom ce r t a in spec i f i c re s idues

but equa l ly f rom a l l . The dev ia t ion in pred ic t ing v i s cos i ty

f rom the v i s co s i ty -ch a rac te r i s t i c s cor re l a t ion inc reases w i th

the bo i l ing po in t o f the f rac t ion , a s ment ioned prev ious ly .

Er rors would a r i s e in ca l cu la t ing the v i s cos i t i e s o f heav ie r

f rac t ions f rom the i r cha rac te r i st i c s .

The reac t iv i ty and prope r t i e s o f pe t ro leum o i l f rac t ions

a re somet imes cor re l a t ed wi th cha rac te r i s t i c s such a s

spec i f i c g rav i ty , ave rage bo i l ing po in t and aspha l t ene

conten t . These re l a t ionsh ips would gene ra l ly apply on ly to

o i l f rac tions who se prope r t i e s a re conna te to a ce r t a in ex ten t

or va ry con t inuous ly wi th the cha rac te r i s t i c index . A l igh te r

c rude f rac t ion conta ins a l a rge quant i ty o f component s o f

l o w e r b o i l in g p o i n t w h i c h a r e h o m o l o g o u s . T h e v i s c o si t y o f

such a l i gh te r f rac t ion would be ch ie f ly de te rmined by the

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

represen ta t ive prope r t i e s o f the f rac t ion . A f rac t ion such as

v a c u u m r e s i d u e h a s f e w s u ch h o m o l o g o u s c o m p o n e n t s , a n d

the re fore the v i s cos i ty depends ma in ly on the f lu id i ty of the

res idue i tse l f . In this case , the viscos i ty cannot be wel l

cor re l a t ed so le ly w i th ce r t a in represen ta t ive prope r t i e s .

F u el 9 9 7 V o l u m e 76 N u m b e r 1 5 5


8/8

Viscos i t y co r re la t i on o f c rude o i l f rac t i ons : T Wakabayash i

T h e v a c u u m r e s i d u e s h a v e m o l e c u l a r w e i g h t s e x c e e d i n g

~ 9 0 0 a n d a sp e c i fi c g r a v i t y e x c e e d i n g - 1 . 0 L6. T h e r e f o r e

t h e v i s c o s i t y - c h a r a c t e r i s ti c c o r r e l a t io n w o u l d b e u n su i ta b l e

for appl i ca t ion to a f rac t ion wi th a molecu la r we igh t o f

> 900 . Equa t ion (8) , which i s de r ived f rom regress ion

ana lys i s fo r 26 samples exc lud ing cu t 5 , seems reaso nable a s

a gene ra l i zed v i scos i ty -cha rac t e r i s t i c cor re l a t ion a t 50C in

this study.

The reg ress ion ana lys i s fo r pe t ro l eu m f rac t ions a t 37 .8C

se lec t ed the va r i ab le (d / log M) 2 , which was nev e r se l ec t ed

in the ana lyses a t 50 , 140 and 160C. H ow ever , t he va r i ab le

( d l o g M ) 2 i s a k e y c o n t r i b u t o r t o d e t e r m i n a t i o n o f v i s c o s i ty

for t hese f rac t ions a s we l l a s fo r a l l o the r c rude f rac t ions .

T h e a v e r a g e d e v i a t io n o f 2 1 i s c o m p a r a t i v e l y l a rg e . T h i s

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

the d ive rs i ty o f sources .

C O N C L U S I O N S

Mul t ip l e regress ion ana lys i s w i th 16 independent va r i ab le s

to which the two cha rac t e r i s t i c s d ( spec i f i c g rav i ty ) and

log M (m olecu la r we igh t ) a re t r ansforme d by mul t ip l i ca tion ,

d iv i s ion and/or squa r ing , and wi th the two dependent

va r i ab le s log v and log[ log(v + 0 .7) ] , gave the fo l low ing

vi scos i ty cor re l a t ion for 26 samples of c rude f rac t ions a t

50C:

log v = 1 .37(d log M) 2 + 16 .12 / (d log M) 2 - 9 .06

T h e v i s c o s i ty c a l c u la t e d f r o m t h i s e q u a t i o n h a s a n a v e r a g e

abso lu te dev ia t ion of 15 .9 , and the re fore th i s equa t ion i s

use fu l fo r e s t ima t ing the t ranspor t p rope r t i e s o f c rude o i l .

H o w e v e r , i t s h o u ld b e n o t e d t h a t t h e f o r m o f c o r r e l a t io n

equa t ion va r i e s w i th the number of samples , t he cha rac t e r -

i s t i c s o f t he c rude o i l , t empera ture e t c . The v i scos i t~

fundam enta l ly i s a f fec t ed m ost b y the va r i ab le (d log M) ~,

and a l so b y o the r va r i ab le s such a s log M and (d / log M) ,

accord ing to c i rcumstances .

C o m p a r i so n b e t w e e n t h e a v e r a g e d e v i a t i o n s c a l c u l a t e d

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

e q u a t i o n s f o r c o n n a t e c r u d e f r a c t i o n s h a s sh o w n t h a t t h e

above equa t ion i s app l i cab le no t on ly to a va r i e ty of such

f rac t ions bu t a l so to f rac t ions be longin g to a spec i f i c g roup .

A m o r e g e n e r a l e x p r e s s i o n a p p l i c a b l e a t a n y t e m p e r a t u r e

sh o u l d f o l l o w f r o m a c c u m u l a t i o n o f a l a r g e r q u a n t i t y o f

sys t emat i c expe r imenta l da t a .

R E F E R E N C E S

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