Fluid Dynamics of OilâWaterâSand Systems - ACS Publications

5 Fluid Dynamics of Oil-Water-Sand Systems Downloaded by UNIV OF ARIZONA on December 7, 2012 | http://pubs.acs.org Publication Date: May 5, 1992 | doi: 10.1021/ba-1992-0231.ch005

Hisham A. Nasr-El-Din Petroleum Recovery Institute, 3512 33rd Street N.W., Calgary, Alberta, Canada T2L 2A6

Heavy-oil-in-water emulsions have different rheological behaviors for different emulsion qualities. With low oil volume fractions, these emulsions behave as Newtonian fluids. However, with oil volume fractions ≥0.5, they often behave as shear-thinning fluids. The friction loss and power requirements for pipeline transportation of heavy-oil emulsions depends on their rheological behavior. Various formulas for predicting the friction loss of the flow of heavy-oil-inwater emulsions in smooth pipes are discussed. Fine sand particles, which are usually produced with heavy oil, change the friction loss of the flow of these emulsions in pipelines. The effect of the fine particles depends on the solids concentration profile in the pipe. Various methods of measuring in situ solids concentration in pipelines are reviewed, including sampling and electrical conductivity probes.

As

W O R L D RESERVES O F C O N V E N T I O N A L C R U D E O I L c o n t i n u e to d e c l i n e ,

heavy o i l a n d b i t u m e n are b e c o m i n g i n c r e a s i n g l y i m p o r t a n t sources o f e n ergy. I n general, heavy c r u d e oils a n d b i t u m e n have viscosity ranges f r o m a f e w h u n d r e d to several t h o u s a n d centipoises. B e c a u s e o f t h e i r h i g h v i s c o s i ties, it is n o t feasible to transport t h e m i n c o n v e n t i o n a l p i p e l i n e s w i t h o u t r e d u c i n g t h e i r viscosities. T h r e e methods w e r e i n t r o d u c e d to r e d u c e t h e viscosity o f heavy oils a n d e n a b l e t h e m to b e t r a n s p o r t e d i n c o n v e n t i o n a l pipelines: heating the o i l during transportation, adding a diluent, and e m u l s i f y i n g t h e heavy o i l i n water. T h e first t w o m e t h o d s are expensive at 1991 p r i c e s . H o w e v e r , the e m u l s i f i c a t i o n m e t h o d has a p o t e n t i a l a p p l i c a t i o n w h e n e v e r an a m p l e water s u p p l y is available. T r a n s p o r t o f viscous c r u d e oils as c o n c e n t r a t e d o i l - i n - w a t e r e m u l s i o n s 0065-2393/92/0231-0171 $12.75/0 © 1992 American Chemical Society

In Emulsions; Schramm, L.; Advances in Chemistry; American Chemical Society: Washington, DC, 1992.

Downloaded by UNIV OF ARIZONA on December 7, 2012 | http://pubs.acs.org Publication Date: May 5, 1992 | doi: 10.1021/ba-1992-0231.ch005

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EMULSIONS IN THE PETROLEUM INDUSTRY

has b e e n d e m o n s t r a t e d o n a large scale i n a n I n d o n e s i a n p i p e l i n e a n d i n C a l i f o r n i a (J). A major disadvantage o f this m o d e o f transport is that it r e q u i r e s d e w a t e r i n g o f the e m u l s i o n after transport. C o n s e q u e n t l y , the o i l i n - w a t e r e m u l s i o n s m u s t b e p r e p a r e d w i t h the highest possible o i l v o l u m e fraction. T o stabilize e m u l s i o n s , a surfactant, w h i c h increases the r e p u l s i v e force b e t w e e n o i l d r o p l e t s , is u s e d . N o n i o n i c surfactants are the p r e f e r r e d type because they are effective i n b r i n e s , are g e n e r a l l y c h e a p e r , a n d o f t e n f o r m less viscous e m u l s i o n s t h a n d o i o n i c surfactants. I n a d d i t i o n , t h e i r e m u l s i o n s are easier to break, a n d they d o not i n t r o d u c e i n o r g a n i c residues that m i g h t l e a d to r e f i n e r y p r o b l e m s . T h e y are c h e m i c a l l y stable at o i l r e s e r v o i r t e m peratures a n d are n o n c o r r o s i v e a n d n o n t o x i c . T h e surfactant type a n d c o n c e n t r a t i o n r e q u i r e d for a p a r t i c u l a r s i t u a t i o n c a n b e d e t e r m i n e d b y c o n d u c t i n g l a b o r a t o r y tests. A t y p i c a l c o n c e n t r a t i o n o f 0.1 l b o f surfactant p e r b a r r e l o f o i l is u s e d f o r e m u l s i o n s c o n t a i n i n g about 5 0 - 7 0 % o i l (2). T h e p r e s e n c e o f n a t u r a l o r g a n i c acids i n some c r u d e oils, especially asphaltic c r u d e oils, m a y e l i m i n a t e the n e e d for expensive surfactants. T h e s e acids react w i t h s t r o n g alkali (usually N a O H ) to f o r m p e t r o l e u m soaps. T h e s e soaps d i f f u s e i n t o the o i l - w a t e r interface, decrease i n t e r f a c i a l t e n s i o n , and f o r m e m u l s i o n s . M a n y researchers have u s e d d i l u t e alkali solutions (—0.1 w t % N a O H ) to f o r m stable o i l - i n - w a t e r e m u l s i o n s c o n t a i n i n g u p to 7 5 % o i l (I, 3). A n o t h e r aspect o f the t r a n s p o r t a t i o n o f h e a v y - o i l - i n - w a t e r e m u l s i o n s , e s p e c i a l l y f o r short-distance p i p e l i n e s , is the presence o f sand particles. F i n e sand particles are u s u a l l y p r o d u c e d w i t h heavy oils. T h e presence o f these particles w i l l change the flow resistance a n d p u m p i n g r e q u i r e m e n t s for h e a v y - o i l - i n - w a t e r e m u l s i o n s . F i r s t , the d y n a m i c viscosity o f an e m u l s i o n w i l l change i n the p r e s e n c e o f fine particles (4). S e c o n d , sand particles, because o f t h e i r h i g h e r density, w i l l flow i n a d i s t i n c t layer at the b o t t o m o f the p i p e (5). T h e objectives o f this c h a p t e r are (1) to give a b r i e f r e v i e w o f various f o r m u l a s to p r e d i c t f r i c t i o n losses f o r flow o f o i l - i n - w a t e r e m u l s i o n s i n s m o o t h p i p e s , a n d (2) to discuss various m e t h o d s that measure i n s i t u solids concentration i n pipelines.

Predicting the Pressure Drop for Flow of Emulsions in Pipelines A large b o d y o f l i t e r a t u r e is available o n e s t i m a t i n g f r i c t i o n loss for l a m i n a r and t u r b u l e n t flow o f N e w t o n i a n a n d n o n - N e w t o n i a n fluids i n s m o o t h p i p e s . F o r l a m i n a r flow past s o l i d b o u n d a r i e s , surface roughness has no effect (at least f o r c e r t a i n degrees o f roughness) o n the f r i c t i o n pressure d r o p o f e i t h e r N e w t o n i a n o r n o n - N e w t o n i a n fluids. I n t u r b u l e n t flow, h o w e v e r , the nature


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Fluid Dynamics of Oil-Water-Sand

173

Systems

o f t h e flow is i n t i m a t e l y associated w i t h t h e surface r o u g h n e s s . S i g n i f i c a n t increases i n f r i c t i o n loss i n t u r b u l e n t flow o v e r r o u g h surfaces have b e e n r e p o r t e d 6). E x t e n s i v e studies (6) have b e e n c o n d u c t e d to u n d e r s t a n d t h e effect o f p i p e roughness o n f r i c t i o n loss i n t u r b u l e n t flow o f N e w t o n i a n fluids i n r o u g h p i p e s . T h e p h e n o m e n o n o f t u r b u l e n t flow w i t h n o n - N e w t o n i a n fluids i n r o u g h p i p e s , h o w e v e r , has r e c e i v e d v e r y l i t t l e a t t e n t i o n (7).

Flow of Newtonian Fluids in Smooth Pipes.

Estimates of fric


t i o n losses i n l a m i n a r flow Re = £ A ^ < 2 1 0 0

[where R e is t h e R e y n o l d s n u m b e r , p is t h e fluid d e n s i t y (kg/m ), U is t h e 3

f

h

b u l k (average) fluid v e l o c i t y (m/s), D is t h e p i p e i n n e r d i a m e t e r (m), a n d μ is (

fluid viscosity (Pa*s)] o f N e w t o n i a n fluids i n s m o o t h p i p e s c a n b e o b t a i n e d from the Hagen-Poiseuille equation (θ): (1)

/ = 16/Re T h e F a n n i n g f r i c t i o n factor (f) is d e f i n e d as

w h e r e T is t h e shear stress at t h e w a l l o f t h e p i p e (Pa). T h e f r i c t i o n factor w

c a n b e also expressed i n t e r m s o f the pressure g r a d i e n t a l o n g t h e p i p e ( Δ ρ / L , w h e r e ρ is pressure a n d L is p i p e l e n g t h ) . F o r steady flow, a force balance yields

= TTDLT

~D àp 2

W

(3a)

or ^

(3b)

=ψ

4L

w h e r e D is the d i a m e t e r a n d L is l e n g t h o f the p i p e . S u b s t i t u t i n g e q u a t i o n 3 b in equation 2 yields

/-

D

2 Vt P(

L


(4)

174


T h e shear rate at the w a l l (^ ) f o r l a m i n a r flow i n a p i p e c a n b e c a l c u w

l a t e d as f o l l o w s : du

7w =

-— dr

(

ι

(5)

(r)


w h e r e u(r) is the l o c a l fluid v e l o c i t y a n d r is the r a d i a l p o s i t i o n . T h e v e l o c i t y p r o f i l e for f u l l y d e v e l o p e d steady flow o f a N e w t o n i a n fluid flowing u n d e r l a m i n a r c o n d i t i o n s i n a p i p e is u(r)

=

(D/2) m45r 2

2

Ap

(6)

L

4μ

{

Combining equations 1, 4, 5, and 6 gives the shear rate at the wall of the pipe: y

«22k

(7)

M a n y u s e f u l correlations have b e e n p u b l i s h e d to d e t e r m i n e the f r i c t i o n factor f o r f u l l y d e v e l o p e d t u r b u l e n t flow o f N e w t o n i a n fluids i n s m o o t h p i p e s . O n e o f the earliest correlations was g i v e n b y B l a s i u s (8) as f o l l o w s : /= 0.079/Re°

(8)

E q u a t i o n 8 is v a l i d f o r 3 0 0 0 < R e < 100,000. A n o t h e r c o m m o n l y u s e d c o r r e l a t i o n was g i v e n b y D r e w et a l . (9): / = 0.0014 + 0 . 1 2 5 ( R e ) ~ °

32

(9)

E q u a t i o n 9 is v a l i d f o r 3 0 0 0 < R e < 3,000,000.

Flow of Power Law Fluids in Smooth Pipes. O i l - i n - w a t e r e m u l s i o n s h a v i n g o i l v o l u m e fractions greater t h a n 0.5 are o f t e n n o n - N e w t o n i a n s h e a r - t h i n n i n g fluids ( 3 , 1 0 - 1 3 ) . F o r s u c h fluids, the shear stress (τ) a n d the shear rate ( 7 ) c a n be r e l a t e d b y the p o w e r l a w m o d e l : r = ky

n

(10)

F o r a N e w t o n i a n fluid, the p o w e r l a w i n d e x η = 1, a n d k is the fluid viscosity. A l s o , f o r s h e a r - t h i n n i n g (pseudoplastic) fluids, η < 1. T h e f r i c t i o n losses f o r the flow o f n o n - N e w t o n i a n p s e u d o p l a s t i c fluids u n d e r l a m i n a r flow c o n d i t i o n s c a n be d e t e r m i n e d b y u s i n g the m e t h o d suggested b y M e t z n e r a n d R e e d (14) as f o l l o w s :


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Systems

/= 16/Re'

175 (11)

R e ' , the M e t z n e r - R e e d m o d i f i e d R e y n o l d s n u m b e r , is d e f i n e d as Re' =

k'(8)"'-

(12)


w h e r e n ' a n d V are the M e t z n e r - R e e d m o d i f i e d p o w e r l a w constants for p i p e flow. T h e s e constants are r e l a t e d to the p o w e r l a w constants o b t a i n e d w i t h a v i s c o m e t e r as f o l l o w s : η - η k' = k

(13a) l + 3n 4n

(13b)

T h e shear rate at the w a l l f o r the flow o f p o w e r l a w fluids i n s m o o t h p i p e s u n d e r l a m i n a r c o n d i t i o n s c a n b e c a l c u l a t e d as follows (15): _SU

h

7

w

D

l + 3n 4n

(14)

F o r η = 1 ( N e w t o n i a n fluids), e q u a t i o n 14 reduces to e q u a t i o n 7. D o d g e a n d M e t z n e r (16) p r e s e n t e d an extensive t h e o r e t i c a l a n d e x p e r i m e n t a l study o n the t u r b u l e n t flow o f n o n - N e w t o n i a n fluids i n s m o o t h p i p e s . T h e y e x t e n d e d v o n Kârmân's (17) w o r k o n t u r b u l e n t flow f r i c t i o n factors to i n c l u d e the p o w e r l a w n o n - N e w t o n i a n fluids. T h e f o l l o w i n g i m p l i c i t expres sion f o r the f r i c t i o n factor was d e r i v e d i n terms o f the M e t z n e r - R e e d m o d i fied R e y n o l d s n u m b e r a n d the p o w e r l a w i n d e x :

^--IjIogiRe'/^)-^

(15)

D o d g e a n d M e t z n e r (16) o b t a i n e d excellent a g r e e m e n t b e t w e e n c a l c u l a t e d ( w i t h e q u a t i o n 15) a n d e x p e r i m e n t a l f r i c t i o n factors over values o f n ' f r o m 0.36 to 1 a n d R e ' f r o m 2900 to 36,000. T h e flow o f o i l - i n - w a t e r e m u l s i o n s i n p i p e l i n e s was e x a m i n e d b y various researchers b o t h i n l a m i n a r a n d t u r b u l e n t flow regimes ( 3 , 1 0 , 1 8 - 2 0 ) . T h e s e studies s h o w e d that pressure d r o p p r e d i c t i o n s based o n equations 11 a n d 15 are i n some cases h i g h e r t h a n the e x p e r i m e n t a l measurements i n b o t h l a m i n a r a n d t u r b u l e n t flow r e g i m e s . I n the l a m i n a r flow r e g i m e , this d i f f e r e n c e was e x p l a i n e d b y W y s l o u z i l et a l . (3) a n d G i l l i e s a n d S h o o k (5) i n terms o f d r o p l e t m i g r a t i o n away f r o m the p i p e w a l l as a result o f h i g h shear rates. H o w e v e r , i n the t u r b u l e n t flow r e g i m e , the viscoelastic p r o p e r t i e s o f o i l - i n w a t e r e m u l s i o n s m a y be the cause f o r this d i f f e r e n c e (10).


176


Measuring the Solids Concentration in Pipelines I n p r a c t i c e , testing a n e m u l s i o n f o r purposes o f p i p e l i n e design r e q u i r e s a sample to b e r e m o v e d f r o m a c o n t a i n e r o r a p i p e l i n e . A l t h o u g h the testing is o f t e n s t r a i g h t f o r w a r d , s a m p l i n g is not, especially w h e n an e m u l s i o n contains sand. B e c a u s e the c o n c e n t r a t i o n a n d p a r t i c l e size d i s t r i b u t i o n o f the dis


p e r s e d phase are so i m p o r t a n t , the rest o f this r e v i e w w i l l d e a l w i t h this aspect.

Sampling Methods. A n u m b e r o f m e t h o d s have b e e n u s e d to m e a sure solids c o n c e n t r a t i o n i n p i p e l i n e s . R e v i e w s o f these m e t h o d s are g i v e n b y K a o a n d K a z a n s k i j (21) a n d r e c e n t l y b y B a k e r a n d H e m p (22). I n general, the p r i n c i p l e o f any o f these m e t h o d s is to find a specific p r o p e r t y that is significantly d i f f e r e n t for the t w o phases, for e x a m p l e , e l e c t r i c a l c o n d u c t i v ity, d i e l e c t r i c constant, density, refractive index, o r a b s o r p t i o n o f electro m a g n e t i c r a d i a t i o n . Solids c o n c e n t r a t i o n c a n b e d e t e r m i n e d b y m e a s u r i n g this p r o p e r t y f o r the m i x t u r e , t h e n u s i n g a c a l i b r a t i o n c u r v e . A n y o f these m e t h o d s w i l l give inaccurate measurements i f the values o f the specific p r o p e r t y o f the t w o phases a p p r o a c h one a n o t h e r o r i f the solids c o n c e n t r a t i o n is v e r y l o w . S a m p l i n g is w i d e l y u s e d i n i n d u s t r y to measure i n situ solids c o n c e n t r a t i o n , c o m p o s i t i o n , a n d size d i s t r i b u t i o n f r o m fluid-solid systems (23). It is p r o b a b l y the o n l y r e l i a b l e m e t h o d f o r use at l o w solids c o n c e n t r a t i o n . It is also u s e d to calibrate a n d evaluate n e w l y d e v e l o p e d m e t h o d s o f m e a s u r i n g solids c o n c e n t r a t i o n (24). A n u m b e r o f m e t h o d s o f s a m p l i n g d i f f e r p r i m a r i l y i n the g e o m e t r y o f the s a m p l i n g d e v i c e . F i g u r e 1 shows schematic diagrams o f the most c o m m o n l y u s e d s a m p l i n g m e t h o d s . Serious errors i n m e a s u r i n g solids c o n c e n t r a t i o n arise as a result o f i m p r o p e r s a m p l i n g . T h e effectiveness o f s a m p l i n g devices is u s u a l l y ex p r e s s e d as the ratio o f the m e a s u r e d solids c o n c e n t r a t i o n , C , to the u p s t r e a m l o c a l solids c o n c e n t r a t i o n , C . T h e c o n c e n t r a t i o n ratio (C/C ) is also k n o w n as the a s p i r a t i o n coefficient (25), separation coefficient (26), o r s a m p l i n g e f f i c i e n c y (27). T h r e e m a i n factors c a n cause the s a m p l i n g e f f i c i e n c y o f a s a m p l i n g d e v i c e to deviate f r o m u n i t y (i.e., i d e a l s a m p l i n g ) : 0

0

1. p a r t i c l e i n e r t i a 2. p a r t i c l e b o u n c i n g 3. flow structure a h e a d o f the s a m p l e r

I n the f o l l o w i n g sections the effect o f these parameters o n the p e r f o r m a n c e o f various s a m p l i n g devices w i l l be discussed.



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A

Β

C

D

177

Figure 1. Sampling devices: A, L-shaped probe; B, straight probes; C , side-wall sampling; and D, side-wall sampling with a projection. (Reproduced with permission from reference 23. Copyright 1989 Gulf Publishing Company.)

Particle Inertia. P a r t i c l e i n e r t i a is a major source o f s a m p l i n g errors w h e n the densities o f the two phases are significantly d i f f e r e n t . B e c a u s e p a r t i c l e i n e r t i a is d i f f e r e n t f r o m that o f an e q u i v a l e n t v o l u m e o f fluid, p a r t i c l e m o t i o n does not f o l l o w the d i s t o r t e d fluid streamlines. C o n s e q u e n t l y , sample solids c o n c e n t r a t i o n a n d c o m p o s i t i o n w i l l be significantly d i f f e r e n t f r o m those i n the p i p e . S a m p l i n g errors d u e to i n e r t i a d e p e n d o n • h o w the s a m p l i n g device disturbs the flow field • h o w the particles r e s p o n d to this d i s t u r b a n c e



178


T h i n L - s h a p e d p r o b e s are c o m m o n l y u s e d to measure solids c o n c e n t r a t i o n p r o f i l e i n s l u r r y p i p e l i n e s (28-33). H o w e v e r , serious s a m p l i n g errors arise as a result o f p a r t i c l e i n e r t i a . T o illustrate the effect o f p a r t i c l e i n e r t i a o n the p e r f o r m a n c e o f L - s h a p e d p r o b e s , c o n s i d e r the fluid streamlines ahead (upstream) o f a s a m p l i n g p r o b e l o c a t e d at the c e n t e r o f a p i p e , as s h o w n i n F i g u r e 2. T h e p r o b e has z e r o thickness, a n d its axis c o i n c i d e s w i t h that o f the p i p e . T h e fluid ahead o f the s a m p l e r contains particles o f d i f f e r e n t sizes a n d densities. F i g u r e 2 A shows the fluid streamlines f o r s a m p l i n g w i t h a v e l o c i t y e q u a l to the u p s t r e a m l o c a l v e l o c i t y (isokinetic s a m p l i n g ) . O f course, the p r o b e does not d i s t u r b the flow field a h e a d o f the s a m p l e r , a n d conse q u e n t l y , sample solids c o n c e n t r a t i o n a n d c o m p o s i t i o n e q u a l those u p s t r e a m o f the p r o b e . S a m p l i n g w i t h a v e l o c i t y d i f f e r e n t f r o m the u p s t r e a m l o c a l v e l o c i t y

u—

*: U=Uq

u

0

Isokinetic Sampling

Figure 2. Isokinetic and anisokinetic sampling. (Reproduced with permission from reference 23. Copyright 1989 Gulf Publishing Company.)


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179

Systems

(anisokinetic sampling) w i l l distort the fluid streamlines ahead o f the sam p l e r . T h e d i s t o r t i o n o f the fluid streamlines d e p e n d s o n the ratio o f the s a m p l i n g v e l o c i t y , 17, to the u p s t r e a m l o c a l v e l o c i t y , U . I f the v e l o c i t y ratio (U/U ) is less t h a n u n i t y , the fluid streamlines w i l l d i v e r g e away f r o m the p r o b e , as s h o w n i n F i g u r e 2 B . Particles o f l o w i n e r t i a w i l l f o l l o w the fluid streamlines, whereas those o f h i g h i n e r t i a w i l l m o v e i n straight lines l i k e b u l l e t s . A s a result, the sample o b t a i n e d has a h i g h e r solids c o n c e n t r a t i o n , w i t h m o r e coarse a n d dense p a r t i c l e s , t h a n i n the p i p e . 0


0

A n opposite t r e n d occurs i f the v e l o c i t y ratio is h i g h e r t h a n u n i t y . I n this case the fluid streamlines c o n v e r g e i n t o the p r o b e ( F i g u r e 2 C ) , b u t the particles w i l l r e s p o n d a c c o r d i n g to t h e i r i n e r t i a ; particles o f l o w i n e r t i a w i l l f o l l o w the streamlines i n t o the p r o b e , whereas those o f h i g h i n e r t i a w i l l miss the p r o b e . O n e ends u p w i t h a s a m p l e w i t h a l o w e r solids c o n c e n t r a t i o n , w i t h m o r e fine a n d l i g h t particles, t h a n i n the p i p e . T h e p r e c e d i n g d i s c u s s i o n shows that the s a m p l i n g e f f i c i e n c y f o r t h i n Ls h a p e d p r o b e s is a f u n c t i o n o f t w o p a r a m e t e r s : the d e v i a t i o n f r o m the i s o k i n e t i c c o n d i t i o n s a n d the response o f the particles to the d e f l e c t i o n o f the fluid streamlines u p s t r e a m o f the s a m p l e r . T h e d e v i a t i o n f r o m the i s o k i n e t i c c o n d i t i o n s is a f u n c t i o n o f the v e l o c i t y ratio (U/U ), whereas the p a r t i c l e response is a f u n c t i o n o f the ratio o f p a r t i c l e i n e r t i a to fluid d r a g . T h i s ratio i n a d i m e n s i o n l e s s f o r m is k n o w n as the p a r t i c l e i n e r t i a p a r a m e t e r , the Stokes n u m b e r , o r the B a r t h n u m b e r (K), d e f i n e d as: 0

K=

(16)

** °

P

d

U

w h e r e p is the solids density, d is the m e a n p a r t i c l e d i a m e t e r , a n d R is the s a m p l e r radius. T h e effect o f p a r t i c l e i n e r t i a o n s a m p l i n g e f f i c i e n c y f o r t h i n L - s h a p e d p r o b e s has b e e n s t u d i e d extensively i n fluid-solid systems o f l o w solids c o n c e n t r a t i o n . R e v i e w s o n the p e r f o r m a n c e o f t h i n L - s h a p e d p r o b e s to s a m p l e f r o m g a s - s o l i d systems w e r e g i v e n b y F u c h s (27), a n d r e c e n t l y b y Stevens (34). U n l i k e g a s - s o l i d systems, f e w investigations have b e e n c o n d u c t e d o n s a m p l i n g f r o m l i q u i d - s o l i d systems. R u s h t o n a n d H i l l e s t a d (28) m e a s u r e d solids c o n c e n t r a t i o n profiles i n v e r t i c a l a n d h o r i z o n t a l s l u r r y p i p e lines b y u s i n g d i f f e r e n t s a m p l i n g t e c h n i q u e s . F o r L - s h a p e d p r o b e s , t h e y f o u n d a l i n e a r r e l a t i o n b e t w e e n the inverse o f the s a m p l i n g v e l o c i t y (l/U) and the c o n c e n t r a t i o n ratio ( C / C ) , w h e r e C is the average solids c o n c e n t r a t i o n o v e r the p i p e cross s e c t i o n . T h e slope o f the l i n e was f o u n d to be a f u n c t i o n o f the settling p r o p e r t i e s o f the solids. N a s r - E l - D i n et a l . (33) e x a m i n e d b o t h t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y the p e r f o r m a n c e o f Ls h a p e d p r o b e s w h e n u s e d to s a m p l e f r o m s l u r r y p i p e l i n e s . F i g u r e s 3 - 5 s h o w good agreement between their m o d e l and their experimental measurements for s a n d particles h a v i n g a m e a n p a r t i c l e size, d , o f 0.19 m m (fine sand), 0.45 m m ( m e d i u m sand), a n d 0.91 m m (coarse sand), respectively. s

s

sm

b

b

50



Figure 4. Predicted and observed sampling efficiencies for the medium sand. (Reproduced with permission from reference 33. Copyright 1984.)



A n o t h e r w a y to collect a sample f r o m a p i p e l i n e o r a c o n t a i n e r is b y w i t h d r a w i n g the sample f r o m an o p e n i n g i n the w a l l (see F i g u r e 1 C ) . T h i s m e t h o d o f s a m p l i n g , k n o w n as s i d e - w a l l s a m p l i n g , is w i d e l y u s e d i n i n d u s t r y , not o n l y f o r s l u r r y p i p e l i n e s (28), b u t also f o r m i x i n g vessels (35-37) a n d s l u r r y heat exchangers (38). T h e advantage o f this t e c h n i q u e is its s i m p l i c i t y o f o p e r a t i o n , because it uses a s m a l l aperture i n the w a l l o f the p i p e a n d does not d i s t u r b the flow w i t h a p r o b e . O n the o t h e r h a n d , the m a i n disadvantage is that the s a m p l i n g efficiency is a strong f u n c t i o n o f p a r t i c l e i n e r t i a a n d the solids d i s t r i b u t i o n u p s t r e a m o f the sampler. R u s h t o n (35) was the first to d r a w attention to the errors associated w i t h w a l l s a m p l i n g . S h a r m a a n d D a s (37) m e n t i o n e d that the m e c h a n i s m o f p a r t i cle c o l l e c t i o n u s i n g an o p e n i n g flush w i t h the w a l l is d i f f e r e n t f r o m the c o n c e p t o f i s o k i n e t i c s a m p l i n g . M o u j a e s (38) u s e d w a l l s a m p l i n g to m e a s u r e solids c o n c e n t r a t i o n i n u p w a r d v e r t i c a l s l u r r y flows. H e f o u n d the sample c o n c e n t r a t i o n to be consistently l o w e r t h a n the true values i n the p i p e , especially w i t h the coarse s a n d particles. T o r r e s t a n d Savage (39) s t u d i e d c o l l e c t i o n o f particles i n s m a l l b r a n c h e s . T h e s a m p l i n g transport efficiency, E, d e f i n e d as the ratio o f the solids flow rate i n the b r a n c h to that i n the m a i n p i p e , was f o u n d to be a f u n c t i o n o f p a r t i c l e settling v e l o c i t y (V ) a n d the u p s t r e a m b u l k v e l o c i t y (U ) as f o l l o w s : t


h

182

EMULSIONS I N T H E P E T R O L E U M INDUSTRY

40(V + t / ) - 5 8 . 4 t

Ε = 158.7xÇ>

b

(17)

l-125(V +C/ ) (

b

w h e r e Q is t h e b r a n c h flow rate (m /s) a n d ( V + U ) is i n meters p e r s e c o n d . T h i s c o r r e l a t i o n is v a l i d f o r t h e range o f 0.04 < ( V + U ) < 0.4 m/s. N a s r - E l - D i n a n d c o - w o r k e r s (40, 41) s t u d i e d w a l l s a m p l i n g f r o m a n 3

t

h


t

h

u p w a r d v e r t i c a l s l u r r y flow. T h e y f o u n d that this type o f s a m p l i n g caused serious errors i n m e a s u r i n g solids c o n c e n t r a t i o n a n d p a r t i c l e size d i s t r i b u t i o n . F i g u r e s 6 - 8 s h o w that t h e s a m p l i n g e f f i c i e n c y f o r s i d e - w a l l s a m p l i n g f r o m a v e r t i c a l p i p e l i n e is always less t h a n u n i t y a n d is d e p e n d e n t o n p a r t i c l e size, u p s t r e a m solids c o n c e n t r a t i o n , a n d s a m p l e r d i a m e t e r , respectively. F i g u r e s 9 a n d 10 s h o w that t h e sample m e a n p a r t i c l e d i a m e t e r u s i n g s i d e w a l l s a m p l i n g is s m a l l e r t h a n that i n t h e p i p e , e s p e c i a l l y at l o w s a m p l i n g v e l o c i t y ratios. T h e results d i s c u s s e d so f a r i n d i c a t e that t h e s a m p l i n g e f f i c i e n c y o f a s i d e - w a l l s a m p l e r f r o m a v e r t i c a l p i p e l i n e is always less t h a n u n i t y . O n e w a y to increase s a m p l e solids c o n c e n t r a t i o n is b y u s i n g a s i d e - w a l l s a m p l e r w i t h a p r o j e c t i o n (see F i g u r e I D ) . N a s r - E l - D i n et a l . (40) e x a m i n e d t h e p e r f o r m a n c e o f s u c h s a m p l i n g devices. T h e y f o u n d that t h e p r o j e c t i o n i n c r e a s e d the s a m p l e solids c o n c e n t r a t i o n . H o w e v e r , t h e v a r i a t i o n o f t h e s a m p l i n g e f f i c i e n c y w i t h t h e v e l o c i t y ratio was d i f f e r e n t f r o m that o b t a i n e d w i t h a


5.

NASR-EL-DIN


1.0

r—

1

Fluid Dynamics of Oil-Water-Sand 1

1

1

U/U

1

183

Systems 1 '

1

1

1

b

Figure 7. Effect of solids concentration on the sampling efficiency of side-wall sampling. (Reproduced with permission from reference 40. Copyright 1985.)



184


SIZE IN MILLIMETERS

Figure 9. Effect of sampling velocity ratio on the sample particle size distribution using a side-wall sampler. (Reproduced with permission from reference 40. Copyright 1985.)

s i d e - w a l l s a m p l e r w i t h o u t a p r o j e c t i o n . T h i s d i f f e r e n c e o c c u r s because t h e p r o j e c t i o n changes t h e flow p a t t e r n a h e a d o f the s a m p l e r . U n l i k e w a l l s a m p l i n g f r o m v e r t i c a l s l u r r y flows, t h e s a m p l i n g efficiency o f a s i d e - w a l l s a m p l e r f r o m a h o r i z o n t a l s l u r r y flow m a y e x c e e d u n i t y i n s o m e cases. N a s r - E l - D i n et a l . (42, 43) s h o w e d that t h e s a m p l i n g efficiency f o r

Figure 10. Effect of sampler diameter on the sample particle size distribution using a side-wall sampler. (Reproduced with permission from reference 40. Copyright 1985.)


5.

NASR-EL-DIN


185

Systems


w a l l s a m p l i n g f r o m a h o r i z o n t a l s l u r r y p i p e l i n e is a s t r o n g f u n c t i o n o f t h e s a m p l e r o r i e n t a t i o n (upwards, sideways, a n d d o w n w a r d s ) . S a m p l i n g ef ficiencies greater t h a n u n i t y w e r e o b s e r v e d o n l y w i t h t h e d o w n w a r d s o r i entation. Particle Bouncing. A s e c o n d source o f s a m p l i n g errors occurs as a result o f p a r t i c l e b o u n c i n g effects. A t y p i c a l e x a m p l e o f this effect is s a m p l i n g particles o f h i g h i n e r t i a u s i n g t h i c k (blunt) L - s h a p e d p r o b e s . I n this case, particles m a y h i t t h e p r o b e w a l l , lose some o f t h e i r i n e r t i a , a n d e n t e r the p r o b e . C o n s e q u e n t l y , t h e s a m p l e solids c o n c e n t r a t i o n is h i g h e r t h a n t h e u p s t r e a m solids c o n c e n t r a t i o n , e v e n w h e n t h e s a m p l i n g v e l o c i t y equals t h e u p s t r e a m l o c a l v e l o c i t y , that i s , w h e n U/U = 1. 0

T h e effect o f p a r t i c l e b o u n c i n g o n t h e s a m p l i n g e f f i c i e n c y o f t h i c k L s h a p e d probes was first n o t e d i n g a s - s o l i d systems b y W h i t e l y a n d R e e d (44). T h e y f o u n d that s a m p l i n g e f f i c i e n c y f o r t h i c k L - s h a p e d p r o b e s was h i g h e r t h a n u n i t y at U/U = 1. T o estimate t h e s a m p l i n g e f f i c i e n c y d u e to p a r t i c l e b o u n c i n g at t h e i s o k i n e t i c v e l o c i t y , B e l y a e v a n d L e v i n (25) a n d Y o s h i d a et a l . (45) p r o p o s e d a n e m p i r i c a l e q u a t i o n . T h i s e q u a t i o n c a n b e w r i t t e n i n a s l i g h t l y d i f f e r e n t f o r m as 0

C/C

0

=1 + Β ( 2 Γ + Γ )

(18)

2

w h e r e Γ is t h e p r o b e relative w a l l thickness a n d Β is t h e f r a c t i o n o f particles that h i t t h e n o z z l e edge a n d e n t e r t h e p r o b e . T o establish t h e p e r f o r m a n c e o f b l u n t p r o b e s w h e n u s e d to s a m p l e f r o m l i q u i d - s o l i d systems, a set o f L - s h a p e d p r o b e s o f d i f f e r e n t thicknesses was tested. F i g u r e 11, f r o m N a s r - E l - D i n a n d S h o o k (46), shows t h e effect o f the p r o b e relative w a l l thickness o n t h e s a m p l i n g e f f i c i e n c y f o r t h e m e d i u m s a n d at solids c o n c e n t r a t i o n o f 10%. T h e s a m p l i n g e f f i c i e n c y at U/U = 1 is h i g h e r than unity, an observation f o u n d for sampling sand particles using thick p r o b e s . A s t h e relative w a l l thickness is i n c r e a s e d , C / C at U/U = 1 increases. A l s o , to o b t a i n t h e c o r r e c t c o n c e n t r a t i o n u s i n g these p r o b e s , samples s h o u l d b e t a k e n at a v e l o c i t y greater t h a n t h e i s o k i n e t i c one. T h i s v e l o c i t y was f o u n d to b e a f u n c t i o n o f t h e solids c o n c e n t r a t i o n , t h e p a r t i c l e i n e r t i a p a r a m e t e r , and t h e p r o b e relative w a l l thickness. T h e increase i n t h e sample solids c o n c e n t r a t i o n at i s o k i n e t i c c o n d i t i o n s was m u c h less t h a n t h e c o r r e s p o n d i n g values o b t a i n e d f r o m e q u a t i o n 18 w i t h Β = 0.5. 0

0

0

F i g u r e 12 shows t h e s a m p l i n g e f f i c i e n c y versus t h e v e l o c i t y ratio f o r Ls h a p e d p r o b e s h a v i n g a t i p angle (Θ) o f 18° a n d p r o b e relative w a l l thicknesses o f 0.4, 0.8, a n d 1.2. T h e fine s a n d at 6 . 3 % discharge c o n c e n t r a t i o n a n d 2.63-m/s b u l k v e l o c i t y was u s e d i n these e x p e r i m e n t s . A t this angle, the increase o f C / C at U/U = 1 is e l i m i n a t e d . T h e s e results s e e m to c o n f i r m the explanation g i v e n p r e v i o u s l y about t h e b o u n c i n g effect a n d agree w i t h the t r e n d p r e v i o u s l y o b t a i n e d b y W h i t e l y a n d R e e d (44) i n g a s - s o l i d sys tems. 0

0



186 -,

1

j-

K= 11.30, R e = 9 4 5 . 4 0

C = 0.1 , 0


90°

s

ο γ

=0.4

û

ο Τ

=0.8

•

» Τ = 1.2

0

u/u

0

n

Figure 11. Effect of the probe relative wall thickness on the sampling efficiency. (Reproduced with permission from reference 46. Copyright 1985.)

I

1

I

I

I

ι

I

I

FINE SAND C

1.6

ο

-

ο ο

0=

•

•

-

1.4

= 6 . 3 % , Ub = 2.63 m/s

0

•

18°

ο

Τ - 0.4

•

Τ-0.8

•

Τ •

1.2 _

ο

ο

a» 1.2

ο

-

•

1.0

-

•

'4 "ί" ο • • ,

.

° °

π"

D

°

0.8

0.0

I

Ι

0.2

0.4

I 0.6

I 0.8

1.0 U/U

ο

V

•ο

rp •

I

i

I

I

'

1.2

1.4

1.6

1.8

2.0

0

Figure 12. Sampling efficiencies for probes having a tip angle of 18° and various probe relative wall thicknesses. (Reproduced with permission from reference 46. Copyright 1985.)


5.

NASR-EL-DIN


187

Systems

F i g u r e 13, f r o m N a s r - E l - D i n et a l . (47), shows the effect o f the p r o b e relative w a l l thickness o n the s a m p l i n g e f f i c i e n c y f o r p o l y s t y r e n e particles o f 0. 3 - m m m e a n d i a m e t e r . Samples w e r e taken f r o m the c e n t e r o f the p i p e at a m e a n solids c o n c e n t r a t i o n o f 3 7 % a n d a b u l k v e l o c i t y o f 3.4 m/s, w i t h p r o b e s o f relative w a l l thicknesses o f 0.05, 0.5, 0.8, a n d 1.2. U n l i k e the results o b t a i n e d w i t h the sand particles, s h o w n i n F i g u r e 11, the effect o f the s a m p l i n g v e l o c i t y o n the s a m p l i n g efficiency is not significant. T h i s result is reasonable because the polystyrene particles have a density o f 1050 k g / m , w h i c h is v e r y close to water. T h i s finding i m p l i e s that these particles c a n f o l l o w the fluid streamlines, a n d c o n s e q u e n t l y the s a m p l i n g efficiency f o r these particles is v e r y close to u n i t y , no m a t t e r w h a t the s a m p l i n g v e l o c i t y . Downloaded by UNIV OF ARIZONA on December 7, 2012 | http://pubs.acs.org Publication Date: May 5, 1992 | doi: 10.1021/ba-1992-0231.ch005

3

F i g u r e 13 illustrates that the s a m p l i n g e f f i c i e n c y appears to be i n d e p e n d e n t o f the p r o b e relative w a l l thickness at s a m p l i n g velocities e q u a l to a n d h i g h e r t h a n the u p s t r e a m l o c a l v e l o c i t y . T h i s o b s e r v a t i o n contrasts w i t h the results o b t a i n e d w i t h the sand particles s h o w n i n F i g u r e 11. T h i s d i f f e r e n c e c a n be e x p l a i n e d as f o l l o w s : I n the p r e s e n c e o f a b l u n t p r o b e , the fluid streamlines deflect ahead o f the p r o b e n o z z l e e v e n at a v e l o c i t y ratio U/U = 0

1, a n d the d e f l e c t i o n increases as the p r o b e relative w a l l thickness is i n creased. Particles o f h i g h i n e r t i a , such as coarse sand p a r t i c l e s , are not significantly affected b y fluid d e f l e c t i o n , a n d strike the n o z z l e w a l l . S o m e o f the particles b o u n c e i n t o the p r o b e a p e r t u r e a n d t h e r e b y cause h i g h e r s a m p l i n g concentrations. Particles o f l o w i n e r t i a , s u c h as the polystyrene particles, f o l l o w the fluid streamlines to a greater extent a n d s h o u l d not strike the n o z z l e w a l l as f r e q u e n t l y . T o a c c o u n t for p a r t i c l e r e b o u n d a n d i n e r t i a effects s i m u l t a n e o u s l y , a m o d i f i c a t i o n was i n t r o d u c e d b y N a s r - E l -

*

1

1

t

1

f

Τ

0

d · 0 3 mm S«I05

m

i-

I

-

*

4

-

1

POLYSTTREME

• 0 05 • 0 5 • Ο θ • 12 *

1

·

€

t

*

#

»

•

I

V

Ο

a

I

II

i

A

\

l

14

1$

II

Figure 13. Effects of sampling velocity and probe relative wall thickness on the sampling efficiency for polystyrene particles. (Reproduced with permission from reference 47. Copyright 1986.)


188



D i n a n d S h o o k (46). F i g u r e 14 c o m p a r e s the c a l c u l a t e d s a m p l i n g e f f i c i e n c y f o r a t h i c k p r o b e h a v i n g a p r o b e relative w a l l thickness o f 0.8, c o n s i d e r i n g the i n e r t i a l effect alone a n d w i t h the p a r t i c l e b o u n c i n g effect, w i t h the e x p e r i m e n t a l m e a s u r e m e n t s . C l e a r l y , the a g r e e m e n t is m u c h b e t t e r w h e n b o t h effects are c o n s i d e r e d . A s e c o n d example o f the p a r t i c l e b o u n c i n g effect is s a m p l i n g u s i n g straight p r o b e s . A l t h o u g h L - s h a p e d t h i c k p r o b e s are m o r e p r a c t i c a l t h a n t h i n p r o b e s , they w i l l r e q u i r e a r e l a t i v e l y large aperture i n the w a l l o f the p i p e . Straight p r o b e s are robust, s i m p l e to construct, r e q u i r e a m i n i m u m size o f a p e r t u r e i n the w a l l o f the p i p e , a n d c a n b e w i t h d r a w n after s a m p l i n g . T h e p e r f o r m a n c e o f t w o d i f f e r e n t straight p r o b e s , a s i d e - p o r t p r o b e a n d a 45° p r o b e (see F i g u r e I B ) f o r m e a s u r i n g solids c o n c e n t r a t i o n o f l i q u i d - s o l i d systems was e x a m i n e d (46). F i g u r e 15 shows C / C versus U/U f o r the t h i n w a l l e d L - s h a p e d a n d the c i r c u l a r - p o r t p r o b e s . F o r the c i r c u l a r - p o r t p r o b e , the s a m p l i n g e f f i c i e n c y is h i g h e r t h a n u n i t y at the i s o k i n e t i c v e l o c i t y . T h u s , to get the c o r r e c t c o n c e n t r a t i o n , the v e l o c i t y ratio w o u l d have to be greater than unity. 0

0

T h e increase i n the s a m p l e c o n c e n t r a t i o n at the i s o k i n e t i c c o n d i t i o n s resembles that o f t h i c k L - s h a p e d p r o b e s . Particles r e b o u n d i n g f r o m the p r o b e w a l l p r o b a b l y e n t e r the p r o b e a n d thus cause h i g h e r concentrations at s a m p l i n g velocities e q u a l to a n d greater t h a n i s o k i n e t i c .

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Figure 14. Predicted and observed sampling efficiency for a thick probe having a probe relative wall thickness of 0.8. (Reproduced with permission from reference 46. Copyright 1985.)


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Figure 15. Sampling efficiency for circular and thin-walled L-shaped probes. (Reproduced with permission from reference 46. Copyright 1985.)

F i g u r e 16 shows a c o m p a r i s o n b e t w e e n t h e 4 5 ° a n d t h e t h i n - w a l l e d p r o b e s . T h e 45° p r o b e gives h i g h e r s a m p l i n g concentrations at the i s o k i n e t i c v e l o c i t y a n d above, a n d t h e d i f f e r e n c e increases as t h e s a m p l i n g v e l o c i t y is i n c r e a s e d . F o r s u b i s o k i n e t i c velocities, this p r o b e gives l o w e r s a m p l i n g ef ficiencies than the thin L-shaped probe a n d side-port probe. This lower s a m p l i n g efficiency results f r o m t h e d i f f e r e n c e i n t h e flow field ahead o f the s a m p l i n g p o i n t . T h e p r o j e c t e d area f o r e a c h p r o b e was u s e d i n c a l c u l a t i n g the s a m p l i n g v e l o c i t y . F i g u r e s 15 a n d 16 s h o w that t h e scatter f o r straight p r o b e s is greater t h a n that o b t a i n e d u s i n g t h e t h i n - w a l l e d L - s h a p e d p r o b e s . The Flow Structure Ahead of the Sampler. A t h i r d source o f s a m p l i n g errors is n o t d i r e c t l y r e l a t e d t o the g e o m e t r y o f the s a m p l i n g d e v i c e , b u t t o t h e flow structure ahead o f the s a m p l e r . O b v i o u s l y , i f t h e flow field ahead o f the s a m p l e r is strongly t h r e e - d i m e n s i o n a l , i t w i l l b e v e r y d i f f i c u l t to o b t a i n a representative sample. T o illustrate this p o i n t , c o n s i d e r t h e flow field d o w n s t r e a m o f a 90° e l b o w . W h e n e v e r a fluid flows a l o n g a c u r v e d p i p e , a pressure gradient must o c c u r across t h e p i p e t o balance t h e c e n t r i f u g a l f o r c e . T h e pressure is greatest at t h e w a l l f a r t h e r f r o m t h e c e n t e r o f c u r v a ture (pressure w a l l ) , a n d lowest at t h e nearer w a l l (suction w a l l ) . B e c a u s e o f i n e r t i a , t h e fluid i n t h e core moves across t h e p i p e f r o m t h e s u c t i o n w a l l



t o w a r d the pressure w a l l a n d returns to the i n n e r edge a l o n g the w a l l , as s h o w n i n F i g u r e 17. A p a i r o f s y m m e t r i c a l , c o u n t e r - r o t a t i n g vortices is f o r m e d as a result o f the fluid i n e r t i a . T h i s secondary flow is s u p e r i m p o s e d o n the m a i n stream, so the resultant flow consists o f h e l i c a l m o t i o n o n e a c h side o f the p l a n e o f the b e n d passing t h r o u g h the axis o f the p i p e . T h e strength o f the secondary flow d e p e n d s , a m o n g o t h e r factors, o n the flow R e y n o l d s n u m b e r a n d the c u r v a t u r e o f the e l b o w . F l o w i n c u r v e d p i p e s has b e e n s t u d i e d extensively b o t h e x p e r i m e n t a l l y a n d theoretically. A r e c e n t r e v i e w o n this w o r k was g i v e n b y Ito (48). T h i s type o f flow affects s a m p l i n g i n two ways: F i r s t , because o f the h e l i c a l m o t i o n , it is v e r y d i f f i c u l t to a l i g n the p r o b e w i t h the fluid v e l o c i t y v e c t o r . C o n s e q u e n t l y , a n d because o f the i n e r t i a l effect, sample c o n c e n t r a t i o n w i l l be always less t h a n the u p s t r e a m c o n c e n t r a t i o n (49). S e c o n d , the i n e r t i a l effects o n the e l b o w p l a n e a n d the c e n t r i f u g a l f o r c e o n a p l a n e p e r p e n d i c u l a r to that o f the e l b o w w i l l p r o d u c e a n o n u n i f o r m solids d i s t r i b u t i o n d o w n s t r e a m o f the e l b o w . A f e w studies c o n s i d e r e d the solids d i s t r i b u t i o n d o w n s t r e a m o f e l b o w s . A y u k a w a (50) a n d T o d a et a l . (51 ) o b s e r v e d an a c c u m u l a t i o n o f coarse p a r t i cles at the o u t e r w a l l o f v e r t i c a l b e n d s . T o d a et a l . (52) n o t e d some changes i n the solids d i s t r i b u t i o n d o w n s t r e a m o f 90° b e n d s . H o w e v e r , no c o n c e n t r a tion measurements were taken.



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191

T o o v e r c o m e the effect o f 90° bends, various m e t h o d s have b e e n sug gested. S t a i r m a n d (53) r e c o m m e n d e d u s i n g a flow straightener i n a r e d u c e d p i p e section to get a u n i f o r m c o n c e n t r a t i o n p r o f i l e . D a v i s (54) suggested that the secondary flow g e n e r a t e d b y bends c o u l d b e e l i m i n a t e d b y u s i n g straightening vanes. F u c h s (27) r e c o m m e n d e d t a k i n g m e a s u r e m e n t s five p i p e diameters d o w n s t r e a m o f the e l b o w . N a s r - E l - D i n a n d S h o o k (55) s t u d i e d the effect o f a 90° e l b o w o n solids d i s t r i b u t i o n d o w n s t r e a m o f a v e r t i c a l e l b o w . T h e y tested s a n d - w a t e r slurries o f various solids concentrations a n d p a r t i c l e sizes. T h e s l u r r y flows w e r e t u r b u l e n t , a n d the p a r t i c l e Stokes n u m b e r based o n the p i p e d i a m e t e r a n d b u l k v e l o c i t y v a r i e d f r o m 0.5 to 3. T h e solids d i s t r i b u t i o n d o w n s t r e a m o f a v e r t i c a l e l b o w was f o u n d to be a f u n c t i o n o f the radius o f c u r v a t u r e o f the e l b o w , solids c o n c e n t r a t i o n , a n d p a r t i c l e size. F i g u r e 18 shows the solids c o n c e n t r a t i o n p r o f i l e 22 p i p e diameters d o w n s t r e a m o f a short-radius e l b o w . T h e c o n c e n t r a t i o n p r o f i l e is s y m m e t r i cal, a n d a m i n i m u m solids c o n c e n t r a t i o n appears at the c e n t e r o f the p i p e . A l s o , the solids c o n c e n t r a t i o n g r a d u a l l y increases t o w a r d the p i p e w a l l . T h i s v a r i a t i o n i n c o n c e n t r a t i o n across the p i p e is e v i d e n t l y a c o n s e q u e n c e o f the c e n t r i f u g i n g a c t i o n o f the secondary flow that is g e n e r a t e d b y the b e n d u p s t r e a m . F i g u r e 18 also shows that the c o n c e n t r a t i o n profiles are c o n c e n t r a t i o n d e p e n d e n t , a n d as the solids c o n c e n t r a t i o n is i n c r e a s e d , the p r o files b e c o m e flatter. O t h e r results (55) s h o w e d that these profiles are also f u n c t i o n s o f the p a r t i c l e size a n d the radius o f curvature o f the e l b o w . T o u n d e r s t a n d a n d f o l l o w the c o n c e n t r a t i o n variations, m e a s u r e m e n t s w e r e o b t a i n e d just d o w n s t r e a m o f the e l b o w (1.5 p i p e d i a m e t e r s ) . F i g u r e 19 shows the effect o f the inverse o f the s a m p l i n g v e l o c i t y o n the c o n c e n t r a t i o n



192


0.8

0.4

0.0

PRESSURE WALL

0.4

0.8

2r/D

SUCTION WALL

Figure 18. Concentration profiles 22 pipe diameters downstream of a shortradius elbow. (Reproduced with permission from reference 55. Copyright 1987.)

-j

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1

j

ι

ι

I

I

ι

0.4

0.6

ι

1 0.8

ι

ι

ρ

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l/U

Figure 19. Effect of the inverse of the sampling velocity on C/C for sand particles at two positions 1.5 pipe diameters downstream of a short-radius elbow. (Reproduced with permission from reference 55. Copyright 1987.) v


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ratio C / C f o r t h e m e d i u m sand at t w o positions e q u i d i s t a n t f r o m the c e n t e r o f t h e p i p e ( C is the discharge solids c o n c e n t r a t i o n ) . T h e figure shows that the l i n e a r r e l a t i o n b e t w e e n C / C a n d l/U h o l d s , e v e n i n t h e r e g i o n o f s t r o n g secondary flow. v

v

v


A n attempt was m a d e to m e a s u r e the p a r t i c l e l o c a l v e l o c i t y at 1.5 p i p e d i a m e t e r s d o w n s t r e a m o f the e l b o w b y u s i n g a p a r t i c l e v e l o c i t y p r o b e (56). H o w e v e r , the t e c h n i q u e f a i l e d , p r e s u m a b l y because t h e s t r o n g secondary flow p r e v e n t e d t h e v e l o c i t y p r o b e f r o m b e i n g a l i g n e d w i t h t h e v e l o c i t y vector. F o r this reason, velocities o b t a i n e d at 2 2 p i p e d i a m e t e r s d o w n s t r e a m o f the e l b o w h a d to b e u s e d to estimate the c o n c e n t r a t i o n s at this l e v e l (1.5 p i p e d i a m e t e r s ) . F i g u r e 2 0 shows t h e e s t i m a t e d solids c o n c e n t r a t i o n n o r m a l i z e d b y t h e discharge c o n c e n t r a t i o n (C(/C ) f o r fine a n d m e d i u m s a n d p a r t i cles 1.5 p i p e d i a m e t e r s d o w n s t r e a m o f the e l b o w . M o s t o f the relative concentrations are l o w e r t h a n u n i t y , a n d c o n s e q u e n t l y t h e m e a n c o n c e n t r a t i o n b a s e d o n these m e a s u r e m e n t s w o u l d b e l o w e r t h a n t h e t r u e v a l u e . S i m i l a r findings w e r e o b t a i n e d b y Sansone (57) i n g a s - s o l i d systems d o w n stream o f a 90° e l b o w . T h i s p h e n o m e n o n occurs because t h e v e l o c i t y v e c t o r and t h e p r o b e axis are n o t c o l i n e a r , so that t h e c o n c e n t r a t i o n results are o n l y o f qualitative v a l u e . v

F i g u r e 20 also shows that t h e solids are m o r e c o n c e n t r a t e d near the p i p e w a l l , a n d a m i n i m u m solids c o n c e n t r a t i o n appears at t h e c e n t e r o f t h e p i p e . T h e l o c a t i o n o f the m a x i m u m solids c o n c e n t r a t i o n d e p e n d s o n the p a r t i c l e

Figure 20. Effect of particle size on solids concentration profile downstream of a short-radius elbow. (Reproduced with permission from reference 55. Copyright 1987.)


194


size. F o r the m e d i u m sand, p r o b a b l y because o f the h i g h e r i n e r t i a i n the e l b o w p l a n e , the particles are relatively c o n c e n t r a t e d at the pressure w a l l . F o r the fine sand, the secondary flow seems to p l a y an i m p o r t a n t r o l e , a n d the m a x i m u m c o n c e n t r a t i o n occurs at the s u c t i o n w a l l .


T o establish a u n i f o r m c o n c e n t r a t i o n p r o f i l e d o w n s t r e a m o f a 90° e l b o w , straightening vanes 10 c m l o n g w e r e i n s e r t e d just d o w n s t r e a m o f the shortradius e l b o w . F i g u r e 21 shows the effect o f these vanes o n the c o n c e n t r a t i o n p r o f i l e . A l t h o u g h the c o n c e n t r a t i o n b e c o m e s flatter, a distinct m i n i m u m at the c e n t e r o f the p i p e s t i l l exists. T h e s e results i m p l y that the solids are already d i s t r i b u t e d at the exit o f the e l b o w , a n d the vanes m e r e l y increase the rate o f d i f f u s i o n o f the p a r t i c l e s . T h e c o n c e n t r a t i o n profiles discussed so far w e r e o b t a i n e d i n a v e r t i c a l p i p e l i n e d o w n s t r e a m o f an e l b o w w i t h a h o r i z o n t a l a p p r o a c h . C o l w e l l a n d S h o o k (58) e x a m i n e d c o n c e n t r a t i o n profiles i n a h o r i z o n t a l s l u r r y p i p e l i n e d o w n s t r e a m o f a 90° e l b o w . A c c o r d i n g to t h e i r results, a l e n g t h o f at least 50 p i p e diameters d o w n s t r e a m o f the e l b o w is n e e d e d to o b t a i n f u l l y d e v e l o p e d c o n c e n t r a t i o n profiles.

Conductivity Methods.

The electrical conductivity of a mixture of

two o r m o r e phases is an i m p o r t a n t p r o p e r t y o f the m i x t u r e . M a n y details

Figure 21. Effect of straightening vanes on the solids concentration profile 22 pipe diameters downstream of a short-radius elbow. (Reproduced with permission from reference 55. Copyright 1987.)


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r e g a r d i n g the mixture's structure c a n be i n f e r r e d f r o m its e l e c t r i c a l c o n ductivity. A c c o r d i n g to the nature o f the d i s p e r s e d phase i n the m i x t u r e , uses o f e l e c t r i c a l c o n d u c t i v i t y c a n b e d i v i d e d i n t o t w o major g r o u p s . I n the first g r o u p , the d i s p e r s e d phase (the s o l i d particles i n s l u r r y systems o r the o i l droplets i n o i l - i n - w a t e r emulsions) consists o f loose particles d i s p e r s e d i n a c o n t i n u o u s phase (matrix). T h e particles have a d e f i n e d shape a n d size d i s t r i b u t i o n , b u t the c o n c e n t r a t i o n o f the d i s p e r s e d phase is less t h a n the c o r r e s p o n d i n g m a x i m u m p a c k i n g c o n c e n t r a t i o n . I n this g r o u p , e l e c t r i c a l c o n d u c t i v i t y is u s e d to measure the c o n c e n t r a t i o n a n d the p a r t i c l e size d i s t r i b u t i o n o f the d i s p e r s e d phase w i t h i n the system. T y p i c a l examples o f s u c h systems are m e a s u r i n g c o n c e n t r a t i o n o f the d i s p e r s e d phase i n m i x i n g tanks (59, 60), i n p i p e l i n e s (24, 61, 62), a n d i n three-phase fluidized beds (63). I n the s e c o n d g r o u p , the solid-phase c o n c e n t r a t i o n is h i g h , a n d solids particles are e i t h e r loose b u t i n contact, o r c o n s o l i d a t e d . I n this case, the s o l i d phase is the matrix, w h i l e the l i q u i d phase is the d i s p e r s e d phase. I n this g r o u p , e l e c t r i c a l c o n d u c t i v i t y is u s e d to measure the effective p o r o s i t y o f the p o r o u s m e d i u m (64, 65). A l s o , i f t w o i m m i s c i b l e fluids, f o r example, o i l and water, are p r e s e n t i n a p o r o u s m e d i u m , the e l e c t r i c a l c o n d u c t i v i t y c a n b e e m p l o y e d to measure the relative saturations o f the t w o fluids a n d to give an i n d i c a t i o n o f the w e t t a b i l i t y o f the p o r o u s m e d i u m (66, 67). T h e eleetric c o n d u c t i v i t y m e t h o d s are w i d e l y u s e d i n b o t h categories because they are s i m p l e to operate a n d give q u i c k response, accurate results, and a c o n t i n u o u s r e a d i n g ; that is, they c a n b e u s e d as a m e a s u r i n g e l e m e n t i n any c o n t r o l l o o p . Besides these examples, k n o w i n g the r e l a t i o n s h i p b e t w e e n the m i x t u r e effective c o n d u c t i v i t y a n d t h e p o r o s i t y o r the c o n c e n t r a t i o n o f the d i s p e r s e d phase is i m p o r t a n t . S u c h a r e l a t i o n can b e u s e d to p r e d i c t o t h e r transport coefficients s u c h as the d i f f u s i o n coefficient, d i e l e c t r i c constant, a n d ther m a l c o n d u c t i v i t y . O f course, s u c h relations are u s e f u l i n m a n y p r a c t i c a l applications. A l t h o u g h e s t i m a t i n g the e l e c t r i c a l c o n d u c t i v i t y o f a m i x t u r e o f t w o o r m o r e phases looks s i m p l e a n d s t r a i g h t f o r w a r d , it is a v e r y c o m p l i c a t e d p r o b l e m , b o t h t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y . T h i s c o m p l e x i t y explains the huge v o l u m e o f w o r k d e v o t e d to s o l v i n g this p r o b l e m since the p i o n e e r i n g w o r k o f M a x w e l l (68) a n d L o r d R a y l e i g h (69). A l t h o u g h this section deals w i t h the c o n d u c t i v i t y o f l i q u i d - s o l i d systems, s i m i l a r treatment c a n be u s e d i n o t h e r s i m i l a r two-phase systems s u c h as g a s - l i q u i d dispersions (70, 71), o i l - i n - w a t e r e m u l s i o n s (72), a n d foams (73). Definition of the Mixture Effective Conductivity. If a l i q u i d - s o l i d m i x t u r e is p l a c e d b e t w e e n t w o electrodes o f d i f f e r e n t p o t e n t i a l , the r e s u l t i n g p o t e n t i a l d i f f e r e n c e w i l l cause a c u r r e n t to flow f r o m the e l e c t r o d e o f h i g h e r


196


p o t e n t i a l to t h e e l e c t r o d e o f l o w e r p o t e n t i a l . T h e c u r r e n t a n d p o t e n t i a l g r a d i e n t are r e l a t e d b y t h e f o l l o w i n g d i f f u s i o n - t y p e e q u a t i o n : Ι =

Κ?Φ

(19)

w h e r e I a n d φ are t h e v o l u m e - a v e r a g e values o f the c u r r e n t a n d t h e p o t e n t i a l , respectively. T h e p r o p o r t i o n a l i t y constant K is t h e effective c o n d u c t i v ity o f t h e m i x t u r e . F o r a h o m o g e n e o u s a n d i s o t r o p i c m i x t u r e , X is a scalar quantity, whereas f o r a h o m o g e n e o u s a n d a n i s o t r o p i c m i x t u r e , X is a s y m m e t r i c a l s e c o n d - o r d e r tensor. T h i s fact explains w h y most o f t h e p r e v i o u s t h e o r e t i c a l a n d e x p e r i m e n t a l studies w e r e d e v o t e d to r a n d o m mixtures o f m o n o s i z e d spheres. O f course, these mixtures are h o m o g e n e o u s a n d i s o t r o p i c . T h e r e f o r e , i t is relatively easier to measure and/or to d e t e r m i n e the electrical conductivity of the mixture. m

m


m

I n the f o l l o w i n g sections, various m e t h o d s to d e t e r m i n e t h e effective c o n d u c t i v i t y o f t w o phases w i l l b e discussed, especially f o r h o m o g e n e o u s and i s o t r o p i c m i x t u r e s . Mathematical Description of Effective Conductivity. I n very gen e r a l t e r m s , t h e e l e c t r i c a l c o n d u c t i v i t y o f a m i x t u r e is a f u n c t i o n o f the e l e c t r i c a l c o n d u c t i v i t y o f its constituents, t h e i r relative amounts, a n d t h e i r d i s t r i b u t i o n w i t h i n the system. M o d e l s a n d expressions to p r e d i c t t h e m i x t u r e effective c o n d u c t i v i t y c a n b e d i v i d e d a c c o r d i n g to t h e degree o f c o m p l e x i t y o f t h e m i x t u r e i n t o t w o major categories. I n t h e first category, t h e m i x t u r e consists o f particles o f definite shape (e.g., spheres, s p h e r o i d s , a n d ellipsoids) at l o w solids c o n c e n t r a t i o n . F o r these m i x t u r e s , d e s c r i b i n g t h e b o u n d a r y c o n d i t i o n s is s t r a i g h t f o r w a r d . A l s o , the effect o f t h e s u r r o u n d i n g particles c a n b e n e g l e c t e d . F o r this category, rigorous solutions are available f o r particles o f s i m p l e g e o m e t r i c a l shapes. A rigorous s o l u t i o n i n this case means s o l v i n g L a p l a c e ' s e q u a t i o n f o r t h e p o t e n t i a l a n d u s i n g a p p r o p r i a t e boundary conditions. T h e s e c o n d category i n c l u d e s mixtures o f h i g h concentrations. U n l i k e d i l u t e m i x t u r e s , p a r t i c l e - p a r t i c l e interactions cannot b e n e g l e c t e d . A l s o , it is v e r y d i f f i c u l t t o d e s c r i b e t h e b o u n d a r y c o n d i t i o n s . B e c a u s e the p r o b l e m is basically a b o u n d a r y value p r o b l e m , n o rigorous solutions are available f o r c o n c e n t r a t e d m i x t u r e s , except f o r o r d e r e d arrays. T o o v e r c o m e these p r o b l e m s , various approaches have b e e n c o n s i d e r e d . I n this r e v i e w , t h e f o l l o w i n g cases w i l l b e d i s c u s s e d : • a p p r o x i m a t e solutions b a s e d o n M a x w e l l ' s t h e o r y • empirical formulas Effective Conductivity of Dilute Mixtures. T h e simplest, best d e f i n e d case, is a cluster o f s p h e r i c a l particles d i s p e r s e d i n a l i q u i d a n d l o c a t e d i n a


5.

NASR-EL-DIN


Systems

197

u n i f o r m e l e c t r i c a l field. I f the particles have the same c o n d u c t i v i t y as the l i q u i d , the p o t e n t i a l a r o u n d the particles w i l l not be d i s t o r t e d , a n d the m i x t u r e c o n d u c t i v i t y is e q u a l to that o f the l i q u i d . I f the particles have a l o w e r c o n d u c t i v i t y , the streamlines w i l l diverge away f r o m the p a r t i c l e s , a n d the m i x t u r e c o n d u c t i v i t y w i l l b e l o w e r t h a n that o f the l i q u i d . I f the particles have a h i g h e r c o n d u c t i v i t y , the streamlines w i l l converge i n t o the p a r t i c l e ,


a n d the m i x t u r e c o n d u c t i v i t y w i l l b e h i g h e r t h a n that o f the l i q u i d . M a x w e l l (68) c a l c u l a t e d the p o t e n t i a l d i s t r i b u t i o n f o r a single s p h e r i c a l p a r t i c l e i m m e r s e d i n a c o n d u c t i n g m e d i u m a n d subjected t o a u n i f o r m e l e c t r i c a l field: H e s o l v e d L a p l a c e ' s e q u a t i o n w i t h i n the t w o regions subject to c o n t i n u i t y o f p o t e n t i a l , a n d c o n t i n u i t y o f the n o r m a l c o m p o n e n t o f the c u r r e n t density, at the surface o f the p a r t i c l e . M a x w e l l t h e n e x t e n d e d his single-sphere s o l u t i o n to d i l u t e mixtures a n d o b t a i n e d the f o l l o w i n g expres sion f o r X : r a

X

- X

m

2X +X -2(X -X )C f

f

s

L 2X +X f

s

f

s

+ (X -X )C f

(20)

s

w h e r e X a n d X are the e l e c t r i c a l c o n d u c t i v i t i e s o f the l i q u i d a n d s o l i d phases, respectively; a n d C is the v o l u m e t r i c c o n c e n t r a t i o n o f the d i s p e r s e d phase. T h e assumptions u s e d to d e r i v e e q u a t i o n 20 are v e r y i m p o r t a n t : f

s

• T h e particles are s p h e r i c a l , o f u n i f o r m size, a n d have the same electrical conductivity. • T h e e l e c t r i c a l field a r o u n d any p a r t i c l e o r d r o p l e t is not af f e c t e d b y the p r e s e n c e o f o t h e r particles; that is, p a r t i c l e d i a m e t e r is m u c h s m a l l e r t h a n the distance b e t w e e n the particles. O b v i o u s l y this c o n d i t i o n c a n b e m e t o n l y f o r v e r y d i l u t e m i x tures. • T h e effect o f surface c o n d u c t a n c e is n e g l i g i b l e . • T h e m i x t u r e is h o m o g e n e o u s a n d i s o t r o p i c . E q u a t i o n 20 indicates that • M i x t u r e c o n d u c t i v i t y does not f o l l o w the a d d i t i v i t y r u l e , w h i c h is s o m e t i m e s u s e d as a s i m p l i f y i n g a s s u m p t i o n . T h i s r e l a t i o n is not l i n e a r , except at e x t r e m e l y l o w c o n c e n t r a t i o n s . • E q u a t i o n 20 satisfies the f o l l o w i n g t h r e e l i m i t i n g c o n d i t i o n s : 1. A s C -> 0, X 2. A s C - » 1, X

m

-» X .

m

-> X .

3. A s X —» X , X s

f

f

s

m

-> X , for a l l concentrations. f


198


T h e s e c o n d c o n d i t i o n c a n b e o b t a i n e d o n l y w i t h mixtures hav i n g a n i n f i n i t e l y w i d e d i s t r i b u t i o n . F o r m o n o s i z e d particles, C c a n n o t b e greater t h a n t h e m a x i m u m p a c k i n g c o n c e n t r a t i o n ( C ) . F u r t h e r m o r e , t h e t h i r d c o n d i t i o n c a n b e u s e d to m e a sure t h e solids c o n d u c t i v i t y b y u s i n g solutions o f k n o w n c o n ductivities. M


T h e m i x t u r e c o n d u c t i v i t y is i n d e p e n d e n t o f p a r t i c l e size f o r m o n o s i z e d spheres. T h i s c o n d i t i o n is o b s e r v e d t o b e t r u e i n p r a c t i c e p r o v i d e d that t h e p a r t i c l e size is m u c h s m a l l e r t h a n the s p a c i n g b e t w e e n t h e t w o sensor electrodes. F o r a m i x t u r e o f n o n c o n d u c t i n g spheres (X = 0) i n a c o n d u c t i n g l i q u i d , e q u a t i o n 20 reduces t o s

• E q u a t i o n 20 is n o t s y m m e t r i c a l w i t h respect to X a n d X ; that is, o n e has to k n o w w h i c h phase is t h e c o n t i n u o u s phase ( m a trix) a n d w h i c h phase is the d i s p e r s e d . f

s

Effective Conductivity of Concentrated Mixtures. S o far, w e have c o n s i d e r e d d i l u t e mixtures o f r a n d o m spheres (68). T h i s case has d e f i n e d b o u n d a r i e s a n d c o n s e q u e n t l y , e q u a t i o n 19 has a rigorous s o l u t i o n . U n f o r t u nately, a rigorous s o l u t i o n is n o t possible f o r r a n d o m c o n c e n t r a t e d suspen sions f o r w h i c h i t is v e r y d i f f i c u l t t o d e s c r i b e t h e b o u n d a r i e s . B e c a u s e o f this d i f f i c u l t y , i t was necessary to i n t r o d u c e m o r e s i m p l i f y i n g assumptions. I n this s e c t i o n , t h e most i m p o r t a n t approaches are r e v i e w e d . T h e first a p p r o a c h to estimate f o r c o n c e n t r a t e d suspensions was i n t r o d u c e d b y B r u g g e m a n (74). H e c o n s i d e r e d t h e e l e c t r i c a l c o n d u c t i v i t y o f s p h e r i c a l particles o f a r a n d o m size d i s t r i b u t i o n . B a s i c a l l y , his d e r i v a t i o n is an extension o f M a x w e l l ' s theory. A c c o r d i n g to B r u g g e m a n , a suspension o f h i g h solids c o n c e n t r a t i o n is f o r m e d b y c o n t i n u o u s l y a d d i n g the particles ( d i s p e r s e d phase) to the l i q u i d (matrix). T h e a d d i t i o n process starts w i t h the smallest particles; t h e n , i n e a c h step larger particles are a d d e d . A t a n y step, the suspension o f s m a l l e r particles is t r e a t e d as a c o n t i n u u m w i t h a c o n d u c t i v i t y that c a n b e c a l c u l a t e d f r o m M a x w e l l ' s e q u a t i o n . T h e c o n d u c t i v i t y o f t h e s u s p e n s i o n (after a d d i n g larger particles), c a n b e d e t e r m i n e d b y a p p l y i n g M a x w e l l ' s e q u a t i o n o n c e m o r e . T h i s process is r e p e a t e d t o t h e desired concentration. R e g a r d i n g B r u g g e m a n ' s assumptions, t w o points are i m p o r t a n t : 1. A t e a c h step, t h e suspension o f s m a l l e r particles is n o t a continuum. 2. T h e suspension m u s t have a n i n f i n i t e range o f p a r t i c l e sizes. T h i s s i t u a t i o n is s e l d o m e n c o u n t e r e d i n p r a c t i c e .


5.

NASR-EL-DIN


Systems

199

U s i n g these assumptions a n d a p p l y i n g M a x w e l l ' s e q u a t i o n , B r u g g e m a n d e r i v e d the f o l l o w i n g i m p l i c i t e q u a t i o n for X : m

x

- x ) ( x / x ) - · = ( i " C ) ( X f - λ.)

m

s

m

f

0

33

(22)

F o r n o n c o n d u c t i n g solids i n a c o n d u c t i n g l i q u i d , e q u a t i o n 22 gives


X

= X (l-C)

m

f

L 5

(23)

D e L a R u e a n d T o b i a s (75) m e a s u r e d the c o n d u c t i v i t i e s o f r a n d o m suspensions o f spheres, c y l i n d e r s , a n d sand particles i n aqueous solutions o f z i n c b r o m i d e o f a p p r o x i m a t e l y the same densities as the particles. T h e y f o u n d the suspension c o n d u c t i v i t y c o u l d be c a l c u l a t e d f r o m the f o l l o w i n g expression: X

m

= X (l-C)* f

(24)

w h e r e χ —1.5 for a solids c o n c e n t r a t i o n i n the range 0 . 4 5 - 0 . 7 5 . T h i s e q u a t i o n is s i m i l a r t o that o f B r u g g e m a n (74). E q u a t i o n 24 is u s u a l l y w r i t t e n b y p e t r o l e u m engineers i n terms o f the f o r m a t i o n factor (F), w h e r e F is the r e c i p r o c a l o f Xn/Xf. B e g o v i c h a n d W a t s o n (63) f o u n d e x p e r i m e n t a l l y that the m i x t u r e c o n d u c t i v i t y i n a l i q u i d - s o l i d fluidized b e d is p r o p o r t i o n a l to the l i q u i d h o l d u p . T h e i r e q u a t i o n c a n b e w r i t t e n as X

m

= X (l-C)

(25)

f

S t i l l another e m p i r i c a l expression was g i v e n b y M a c h o n et a l . (60): X

m

= X (l-aC)

(26)

f

w h e r e α is a constant to b e d e t e r m i n e d e x p e r i m e n t a l l y . M a c h o n et a l . f o u n d this constant b y m e a s u r i n g the c o n d u c t i v i t y o f a b e d o f n o n m o v i n g p a r t i c l e s . T h e b e d solids c o n c e n t r a t i o n was i n the range 0 . 6 - 0 . 6 5 . E q u a t i o n 26 is l i n e a r , a n d a c c o r d i n g to this e q u a t i o n , X = X f o r C = 0; this result is s i m i l a r to e q u a t i o n 20. H o w e v e r , at C = 1, e q u a t i o n 26 does not agree w i t h M a x w e l l ' s p r e d i c t i o n unless a = 1. T h i s observation a n d the fact that it has n o t h e o r e t i c a l j u s t i f i c a t i o n suggest that e q u a t i o n 26 s h o u l d b e u s e d w i t h c a u tion. m

f

A c o m p a r i s o n o f these expressions is g i v e n i n T a b l e I. T h i s table shows the increase i n the m i x t u r e resistance d u e to the p r e s e n c e o f n o n c o n d u c t i n g particles o r d r o p l e t s , R - R d i v i d e d b y the fluid resistance as a f u n c t i o n o f the dispersed-phase c o n c e n t r a t i o n . R a n d R\ are the m i x t u r e a n d fluid resistances, respectively. M a x w e l l ' s (68) a n d B r u g g e m a n ' s (74) relations give m

h

m


200

E M U L S I O N S I N T H E P E T R O L E U M INDUSTRY

Table I. Comparison Between Various Expressions for (R -Ri)/Ri


m

C(%)

Ref 63

Ref. 68

Ref 74

10 20 30 40 50 60 70 80 90 100

0.111 0.25 0.429 0.667 1.0 1.5 2.33 4.0 9.0 infinite

0.167 0.375 0.643 1.0 1.5 2.05 3.5 6.0 13.5 infinite

0.171 0.398 0.708 1.52 1.829 2.953 5.086 10.18 30.623 infinite

v e r y s i m i l a r results at l o w solids c o n c e n t r a t i o n s . H o w e v e r , at h i g h e r solids c o n c e n t r a t i o n s , B r u g g e m a n ' s r e l a t i o n gives h i g h e r values. B e g o v i c h a n d W a t s o n ' s (63) r e l a t i o n p r e d i c t s l o w e r values f o r a l l c o n c e n t r a t i o n s , a n d the d e v i a t i o n f r o m the o t h e r t w o relations increases as the c o n c e n t r a t i o n is increased. Conductivity Probe for Local Solids Concentration Measurements. O n t h e basis o f t h e p r e c e d i n g d i s c u s s i o n , l o c a l solids c o n c e n t r a t i o n can b e d e t e r m i n e d b y m e a s u r i n g the m i x t u r e c o n d u c t i v i t y , t h e n u s i n g a c a l i b r a t i o n c u r v e , f o r e x a m p l e , e q u a t i o n 20. H o w e v e r , u s i n g this m e t h o d to measure solids c o n c e n t r a t i o n o r dispersed-phase c o n c e n t r a t i o n is n o t a n easy task. I n the f o l l o w i n g sections, t h e d e v e l o p m e n t a n e w c o n d u c t i v i t y p r o b e w i l l b e s u m m a r i z e d (24). A l s o , various p r o b l e m s e n c o u n t e r e d w i t h conductivity methods w i l l be discussed. Description. T h e p r o b e , s h o w n i n F i g u r e 22, has a n L - s h a p e d c o n f i g u r a t i o n . It is c o n s t r u c t e d f r o m 3/16-in. stainless-steel t u b i n g . T o m i n i m i z e the effect o f flow disturbances, t h e p r o b e terminates w i t h a c o n i c a l stainless steel t i p , a n d t h e a p p r o a c h l e n g t h to the sensor electrodes is 10 p r o b e d i a m e t e r s . T h e t w o field electrodes are flush w i t h the surface o f the t u b i n g a n d c o m p l e t e l y i n s u l a t e d f r o m e a c h other. T h e field e l e c t r o d e o f larger area is g r o u n d e d to t h e p i p e l i n e . T h e field e l e c t r o d e c i r c u i t consists o f a f u n c t i o n generator, a ballast resistance, a n d a n a m m e t e r . T h e t w o sensor electrodes are also flush w i t h t h e surface o f t h e t u b i n g , 1 m m apart, a n d are l o c a t e d b e t w e e n the field electrodes. T h e sensor electrodes are c o n s t r u c t e d f r o m 28-gauge p l a t i n u m . T h e y are also c o m p l e t e l y i n s u l a t e d f r o m each o t h e r a n d f r o m t h e field electrodes. T h e sensor electrodes are c o n n e c t e d to a v o l t m e ter f r o m w h i c h a time-average r e a d i n g c a n b e o b t a i n e d . T h e p r o b e has t w o u n i q u e features: 1. T h e field electrodes are m o u n t e d o n t h e p r o b e i t s e l f a n d n o t o n t h e p i p e w a l l , as c o m m o n l y u s e d (59). T h i s feature is v e r y


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•ml—


©

DETAIL A

DETAIL A FIELD

Figure 22. Conductivity probe for local solids concentration measurement. (Reproduced with permission from reference 24. Copyright 1987.)

i m p o r t a n t because i t eliminates t h e n e e d f o r h i g h e r voltages f o r m e a s u r e m e n t s i n large p i p e s . It also allows t h e study o f the solids d i s t r i b u t i o n w i t h i n t h e p i p e . 2. T h e p o t e n t i a l is sensed f o r a s m a l l r e g i o n ( 1 - m m diameter) i n the a p p l i e d field. T h i s means that resistivity a n d solids c o n c e n t r a t i o n c a n b e m e a s u r e d over a s m a l l v o l u m e i n space. Operation. T h e o p e r a t i o n o f t h e p r o b e relies o n t h e v a r i a t i o n o f t h e s l u r r y resistivity as t h e solids c o n c e n t r a t i o n changes. T o u n d e r s t a n d t h e probe's p r i n c i p l e , assume t h e p r o b e is s u r r o u n d e d b y a c o n d u c t i n g l i q u i d s u c h as tap water; t h e n i f a p o t e n t i a l is a p p l i e d across t h e field electrodes ( o f the o r d e r o f 5 V ) , a s m a l l c u r r e n t flows f r o m o n e field electrode to another. T h e value o f this c u r r e n t , f o r a fixed p r o b e g e o m e t r y a n d a p p l i e d signal, d e p e n d s o n t h e total resistance o f t h e m e d i u m s u r r o u n d i n g t h e field e l e c trodes. I f n o n c o n d u c t i n g particles (e.g., sand particles) are a d d e d to this fluid, t h e n t h e resistivity o f t h e m i x t u r e w i l l increase. A s t h e solids c o n c e n t r a t i o n is i n c r e a s e d , t h e m i x t u r e resistivity increases, a n d c o n s e q u e n t l y the field c i r c u i t c u r r e n t d i m i n i s h e s .


202



O n e w a y o f m e a s u r i n g the solids c o n c e n t r a t i o n , s i m i l a r to that u s e d b y p r e v i o u s w o r k e r s , is to relate the field c i r c u i t c u r r e n t change to the solids c o n c e n t r a t i o n . T h i s m e t h o d has a serious disadvantage because the field c i r c u i t c u r r e n t d e p e n d s o n b o t h the s l u r r y resistivity a n d the p o l a r i z a t i o n resistance d e v e l o p e d o n the surfaces o f the field electrodes. T h i s p o l a r i z a t i o n resistance is v e l o c i t y d e p e n d e n t . T h i s m e t h o d y i e l d s c a l i b r a t i o n curves that are f u n c t i o n s o f v e l o c i t y (59). T o a v o i d this p r o b l e m , the total c u r r e n t was not u s e d to m e a s u r e the solids c o n c e n t r a t i o n . Instead the voltage was m e a s u r e d across the two sensor electrodes l o c a t e d b e t w e e n the field electrodes, as s h o w n i n F i g u r e 22. B e c a u s e the i m p e d a n c e o f the sensor c i r c u i t is v i r t u a l l y i n f i n i t e , p r a c t i c a l l y no c u r r e n t flows i n t o the sensors. C o n s e q u e n t l y , n o p o l a r i z a t i o n occurs o n t h e i r surfaces. T h u s , the c a l i b r a t i o n c u r v e o b t a i n e d s h o u l d be i n d e p e n d e n t o f v e l o c i t y , as s h o w n i n T a b l e II. T o m i n i m i z e the effect o f p o l a r i z a t i o n o n the surfaces o f the field electrodes a n d to facilitate a c o n s t a n t - c u r r e n t o p e r a t i o n , a ballast resistance c a n b e u s e d i n the field e l e c t r o d e c i r c u i t (24). A l s o , to e l i m i n a t e fluid elec trolysis, a square wave o f 1 k H z a n d 5 - V a m p l i t u d e c a n b e e m p l o y e d . Calibration. V a r i o u s m e t h o d s w e r e u s e d to calibrate the p r o b e . T h e s e studies w e r e c o n d u c t e d to find the r e l a t i o n b e t w e e n voltage (e) a n d solids c o n c e n t r a t i o n a n d to c o m p a r e the e x p e r i m e n t a l m e a s u r e m e n t s w i t h the p r e d i c t i o n s o f the M a x w e l l a n d B r u g g e m a n relationships. It was also neces sary to establish an efficient p r o b e c a l i b r a t i o n p r o c e d u r e .

Table II. Normalized Sensor Voltages as a Function of Position Measured over a Velocity Range of 0 to 4 m/s Position (Ύ) 0.1 1.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

e(0,Y)/ e(0,0.5)

Standard Deviation

1.03 1.02 1.0 0.99 0.99 0.99 0.99 1.0 1.0 1.0 0.99 0.99 0.99 0.99 1.0 1.02 1.04

0.009 0.005 0.005 0.007 0.005 0.006 0.005 0.005 0 0.007 0.007 0.007 0.007 0.008 0.007 0.007 0.01


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T h e first test was c o n d u c t e d w i t h the c o n d u c t i v i t y p r o b e m o u n t e d i n t h e p i p e l i n e , as s h o w n i n F i g u r e 2 3 . P o l y s t y r e n e particles o f 0 . 3 - m m m e a n d i a m e t e r w e r e u s e d i n these s l u r r i e s . T h e s e particles w e r e c h o s e n because o f t h e i r t e n d e n c y t o give a u n i f o r m c o n c e n t r a t i o n p r o f i l e across t h e p i p e . C o n c e n t r a t i o n s w e r e o b t a i n e d b y i s o k i n e t i c s a m p l i n g at t h e c e n t e r o f t h e p i p e over a t e m p e r a t u r e range o f 8 t o 25 °C. F i g u r e 24 shows t h e results; t h e


effect o f t e m p e r a t u r e i n tap w a t e r is s h o w n f o r c o m p a r i s o n . A t a fixed t e m p e r a t u r e , i n c r e a s i n g solids c o n c e n t r a t i o n causes t h e sensor voltage to increase. T h i s result is reasonable because p o l y s t y r e n e particles are n o n c o n d u c t i n g a n d t h e i r p r e s e n c e increases s l u r r y resistivity. T h e curves o b t a i n e d at various c o n c e n t r a t i o n s are almost p a r a l l e l . T h i s o b s e r v a t i o n means that t h e rate o f change o f voltage w i t h respect t o t e m p e r a t u r e is i n d e p e n d e n t o f t h e solids c o n c e n t r a t i o n . B y c r o s s - p l o t t i n g t h e results s h o w n i n F i g u r e 24, a set o f c a l i b r a t i o n curves c a n b e p r e p a r e d w i t h t e m p e r a t u r e as a p a r a m e t e r . W h e n s u c h curves w e r e p r e p a r e d , they i n d i c a t e d that t h e value o f e at C = 0, o b t a i n e d b y extrapolation, was l o w e r t h a n the c o r r e s p o n d i n g v a l u e o b t a i n e d f o r tap w a t e r at t h e same t e m p e r a t u r e . A r e v i e w o f t h e p r o c e d u r e o f this e x p e r i m e n t i n d i c a t e d that t h e o n l y p o s s i b l e reason f o r this d i f f e r e n c e was t h e fact that a s m a l l a m o u n t o f a w e t t i n g agent (an a n i o n i c surfactant) was a d d e d w i t h t h e solids t o increase t h e w e t t a b i l i t y o f t h e p o l y s t y r e n e particles. T o c h e c k this effect, t h e l o o p was o p e r a t e d w i t h tap water, a n d m e a s u r e ments w e r e t a k e n at various surfactant concentrations ( C ) . F i g u r e 2 5 shows the results o b t a i n e d . A t a g i v e n t e m p e r a t u r e , as t h e surfactant c o n c e n t r a t i o n is i n c r e a s e d , t h e voltage decreases. T h i s decrease is reasonable t h e surfacd

Figure 23. Test loop for conductivity probe. (Reproduced with permission from reference 24. Copyright 1987.)



204


J

ι

ι

ι

12

8

I

16

I

I

20

I

I

24

Τ (°C) Figure 24. Effect of temperature on sensor potential. (Reproduced with permis sion from reference 24. Copyright 1987.)

Τ

I 12

I

I 14

I

I

I

t 16

,

I

ι

ι

ι

I

I 18

I

L 20

Τ (°C) Figure 25. Effect of surfactant (anionic) concentration (Cd, wt%) on sensor potential. (Reproduced with permission from reference 24. Copyright 1987.)


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tant contains s o d i u m salts o f organic acids. Its p r e s e n c e w o u l d decrease t h e fluid resistivity a n d c o n s e q u e n t l y t h e sensor voltage. T h e s e results s h o w 1. A c a l i b r a t i o n c u r v e b a s e d o n these m e a s u r e m e n t s is n o t ac ceptable because o f p o o r c o n t r o l o n t h e a m o u n t o f surfactant. 2. T h e voltage m e a s u r e d is strongly affected b y any s m a l l change


in the chemical composition o f the conducting liquid. T h e results s h o w n i n F i g u r e s 24 a n d 2 5 i n d i c a t e that u s i n g e as a measure f o r solids c o n c e n t r a t i o n is n o t a p p r o p r i a t e because i t is strongly dependent o n temperature, chemical composition, and position. Therefore, the f o l l o w i n g f u n c t i o n was u s e d i n c a l i b r a t i n g t h e p r o b e : e(C,Y)-e(0,Y) e(0,Y) T h i s f u n c t i o n w i l l c o r r e c t t h e sensor voltage at a n y c o n c e n t r a t i o n ( C ) f o r t e m p e r a t u r e , c h e m i c a l c o m p o s i t i o n , a n d p o s i t i o n (Y) i n t h e p i p e . S e d i m e n t a t i o n was t h e s e c o n d m e t h o d tested to calibrate t h e p r o b e . P o l y s t y r e n e particles o f 0 . 3 - m m m e a n d i a m e t e r w e r e again u s e d , w i t h o u t t h e w e t t i n g agent. T h e s e particles w e r e c h o s e n because o f t h e i r v e r y l o w s e t t l i n g v e l o c i t i e s , w h i c h a l l o w e d sufficient t i m e f o r voltage readings. Tests w e r e d o n e i n a 5 - c m a c r y l i c p i p e w i t h b o t h tap w a t e r a n d a g l y c o l s o l u t i o n o f t h e same density as t h e particles. F i g u r e 2 6 shows t h e voltage m e a s u r e m e n t s , expressed i n terms o f t h e f u n c t i o n j u s t d e f i n e d , as a f u n c t i o n o f solids c o n c e n t r a t i o n i n t h e t w o fluids. T h e figure indicates g o o d agreement b e t w e e n t h e e x p e r i m e n t a l m e a s u r e ments a n d M a x w e l l ' s r e l a t i o n . A l t h o u g h M a x w e l l ' s r e l a t i o n was s u p p o s e d t o be v a l i d f o r l o w c o n c e n t r a t i o n s , i t actually agrees v e r y w e l l w i t h a l l t h e e x p e r i m e n t a l results. T h i s o b s e r v a t i o n agrees w i t h p r e v i o u s w o r k (76-78). F u r t h e r m o r e , c h a n g i n g t h e s o l u t i o n c o n d u c t i v i t y h a d n o effect o n t h e m e a surements. T h i s result demonstrates that b y u s i n g t h e sensor o u t p u t f u n c t i o n , t h e effects o f a l l variables o n t h e p r o b e o u t p u t except C s h o u l d b e isolated. T h e s e c o n d step was t o examine t h e effect o f p a r t i c l e size o n t h e c a l i b r a t i o n c u r v e . T h i s step was n o t possible b y s e d i m e n t a t i o n , because coarser particles have h i g h e r settling v e l o c i t i e s . T h e r e f o r e , a l i q u i d - s o l i d fluidized b e d was u s e d . A fluidization c o l u m n was c o n s t r u c t e d w i t h a 5 - c m a c r y l i c p i p e . W e i g h e d quantities o f solids w e r e u s e d , a n d solids c o n c e n t r a t i o n was v a r i e d b y c h a n g i n g t h e l i q u i d flow rate. M e a s u r e m e n t s f o r these e x p e r i ments i n c l u d e d voltage, b e d h e i g h t , a n d t e m p e r a t u r e . T o a l l o w a p r e c i s e d e t e r m i n a t i o n o f c o n c e n t r a t i o n f r o m b e d h e i g h t , n a r r o w sizes o f particles were used.



F i g u r e 2 7 shows t h e e x p e r i m e n t a l data o b t a i n e d f o r a sand f r a c t i o n o f 0 . 6 - m m m e a n d i a m e t e r a n d n a r r o w size d i s t r i b u t i o n . M e a s u r e m e n t s i n this e x p e r i m e n t w e r e taken at t h e p i p e c e n t e r . M a x w e l l ' s a n d B r u g g e m a n ' s r e l a tions are also s h o w n f o r c o m p a r i s o n . G o o d agreement b e t w e e n t h e e x p e r i m e n t a l measurements a n d b o t h relations is o b s e r v e d at l o w concentrations. H o w e v e r , at h i g h solids concentrations, t h e e x p e r i m e n t a l data e x c e e d M a x w e l l ' s p r e d i c t i o n s , b u t show g o o d a g r e e m e n t w i t h B r u g g e m a n ' s e q u a t i o n . F i g u r e s 26 a n d 2 7 show that p a r t i c l e shape has an effect o n the results. G o o d a g r e e m e n t b e t w e e n t h e e x p e r i m e n t a l results f o r s p h e r i c a l particles ( 0 . 3 - m m polystyrene particles) a n d M a x w e l l ' s e q u a t i o n is o b s e r v e d . F o r i r r e g u l a r sand particles, M a x w e l l ' s e q u a t i o n u n d e r p r e d i c t s the e x p e r i m e n t a l results, especially at h i g h concentrations. T h e scatter i n the e x p e r i m e n t a l data is h i g h e r f o r sand particles than f o r polystyrene particles. T h e scatter increases f o r sand particles as c o n c e n t r a t i o n is i n c r e a s e d . T h i s observation i m p l i e s that i t is d i f f i c u l t to r e p r o d u c e t h e same p a r t i c l e p a c k i n g for i r r e g u l a r sand particles. F i g u r e 28 shows measurements f o r the same sand f r a c t i o n at d i m e n s i o n less r a d i a l positions (H = 2r/D) o f 0.0, 0.7, a n d 0.8, w h e r e r is t h e radial p o s i t i o n m e a s u r e d f r o m t h e p i p e center. T h e effect o f p o s i t i o n o n [e(C,R) e(0,R)]/e(Q,R) is significant. Results o b t a i n e d at the o t h e r positions s h o w the same d e v i a t i o n f r o m M a x w e l l ' s r e l a t i o n at h i g h e r concentrations. B e c a u s e measurements refer to a s m a l l v o l u m e i n space, the effect o f p a r t i c l e size is o f interest. T h i s effect was e x a m i n e d t h r o u g h t w o sets o f



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C(%) Figure 27. Probe calibration in fluidization tests. Maxwell's and Bruggeman's relations are shown for comparison. (Reproduced with permission from reference 24. Copyright 1987.)

Figure 28. Effect of probe position on the calibration curve for sand particles. (Reproduced with permission from reference 24. Copyright 1987.)


208


e x p e r i m e n t s . I n t h e first test, sand fractions o f 0.6- a n d 1 . 5 - m m m e a n d i a m e t e r w e r e u s e d . T h e s e sand fractions w e r e o b t a i n e d f r o m t h e m e d i u m a n d coarse sands (see T a b l e I I I ) , respectively. I n t h e s e c o n d , glass beads o f 1.5-, 2.8-, a n d 5 . 5 - m m m e a n d i a m e t e r a n d s p h e r i c a l shape w e r e u s e d .


F i g u r e 2 9 shows t h e results o b t a i n e d f o r the sand particles. T h e coarser particles s h o w t h e same d e v i a t i o n at h i g h c o n c e n t r a t i o n f r o m M a x w e l l ' s e q u a t i o n as o b s e r v e d p r e v i o u s l y for t h e 0 . 6 - m m sand. T h e figure also shows that m e a s u r e m e n t s f o r t h e coarser particles are s l i g h t l y l o w e r t h a n those o f the 0 . 6 - m m sand. T h e d i f f e r e n c e is generally s m a l l a n d c o u l d p r o b a b l y b e n e g l e c t e d , especially i f the p r o b e w e r e u s e d to measure t h e c o n c e n t r a t i o n o f particles o f a w i d e size d i s t r i b u t i o n . F i g u r e 30 shows t h e results o b t a i n e d for the glass beads. F o r particles o f 1 . 5 - m m m e a n d i a m e t e r , g o o d a g r e e m e n t w i t h M a x w e l l ' s e q u a t i o n was f o u n d u p to 3 0 % solids c o n c e n t r a t i o n . F o r h i g h e r concentrations, t h e e x p e r i m e n t a l data are significantly l o w e r t h a n M a x w e l l ' s e q u a t i o n . F o r particles o f 2 . 8 - m m d i a m e t e r , t h e e x p e r i m e n t a l results are slightly l o w e r t h a n M a x w e l l ' s p r e d i c tions u p to 1 5 % , t h e n start to deviate significantly. F o r 5 . 5 - m m d i a m e t e r particles, t h e e x p e r i m e n t a l results are significantly l o w e r than M a x w e l l ' s f o r all concentrations. F i g u r e s 2 9 a n d 3 0 s h o w that p a r t i c l e size has n o significant effect o n t h e p r o b e c a l i b r a t i o n curve f o r particles o f d i a m e t e r c o m p a r a b l e to t h e sensor e l e c t r o d e s p a c i n g a n d s m a l l e r . F o r coarser particles, p a r t i c l e size has a n effect o n t h e c a l i b r a t i o n c u r v e . A s p a r t i c l e size is i n c r e a s e d , t h e sensor voltage decreases, a n d i t b e c o m e s l o w e r t h a n M a x w e l l ' s p r e d i c t i o n s . T h e s e results i n d i c a t e that t h e r e is a l i m i t a t i o n o n t h e use o f a p r o b e w i t h a fixed g e o m e t r y f o r m e a s u r i n g solids c o n c e n t r a t i o n . T h i s l i m i t a t i o n w o u l d have to be c o n s i d e r e d w h e n selecting sensor e l e c t r o d e spacings. T h e effect is p r o b a b l y d u e to p a c k i n g , because as t h e p a r t i c l e d i a m e t e r is i n c r e a s e d , t h e m e a n c o n c e n t r a t i o n at t h e p r o b e surface falls. T h e next step was to test t h e p r o b e p e r f o r m a n c e i n the p i p e l i n e i n c o m p a r i s o n w i t h a c c e p t e d m e t h o d s f o r m e a s u r i n g solids concentrations: isokinetic sampling and 7-ray absorption methods.

Table III. Particle Properties Particles Glass beads Glass beads Glass beads Polystyrene Polystyrene Fine sand M e d i u m sand Coarse sand

Mean Diameter (mm)

Density (g/cm )

Shape

1.5 2.8 5.5 0.3 1.4 0.19 0.45 0.9

3.0 2.5 2.3 1.05 1.06 2.65 2.65 2.65

spherical spherical spherical spherical irregular irregular irregular irregular

3



τ

ι

τ

ι

ι

C

ι

I

'

ι

Γ

(%)

Figure 30. Effect of particle size on the calibration curve for glass beads. (Reproduced with permission from reference 24. Copyright 1987.)


210



F i g u r e 31 shows the resistivity m e a s u r e d b y the c o n d u c t i v i t y p r o b e a n d the l o c a l c o n c e n t r a t i o n m e a s u r e d b y the i s o k i n e t i c s a m p l i n g m e t h o d . M e a surements w e r e m a d e f o r slurries o f p o l y s t y r e n e p a r t i c l e s o f 1 . 4 - m m m e a n d i a m e t e r at the p i p e c e n t e r a n d at r a d i a l positions o f R = 0.8. G o o d agree m e n t w i t h c a l i b r a t i o n results is seen i n these tests. B e c a u s e the particles u s e d i n these e x p e r i m e n t s w e r e large, n o samples c o u l d be w i t h d r a w n f r o m the c e n t e r o f the p i p e at c o n c e n t r a t i o n s h i g h e r t h a n 3 5 % . A l s o , n o voltage m e a s u r e m e n t s c o u l d b e t a k e n closer to the p i p e w a l l because particles t e n d e d to b e t r a p p e d b e t w e e n the p r o b e a n d the w a l l . F i g u r e 32 shows a t y p i c a l sensor voltage p r o f i l e i n the v e r t i c a l p l a n e . S a n d o f 0 . 4 5 - m m m e a n d i a m e t e r a n d 1 0 % solids c o n c e n t r a t i o n was u s e d i n this e x p e r i m e n t , at a b u l k v e l o c i t y o f 2 m/s a n d a t e m p e r a t u r e o f 22 °C. T h i s p r o f i l e shows that the voltage is h i g h at the b o t t o m o f the p i p e w h e r e most o f the solids are e x p e c t e d to b e at these o p e r a t i n g c o n d i t i o n s . T h e voltage decreases as Y increases, that is, as the solids c o n c e n t r a t i o n decreases. F i g u r e 32 also shows the v a r i a t i o n o f the ballast resistance w i t h p o s i t i o n . T h e change o f resistance across the p i p e is m u c h less t h a n that o f the sensor voltage. T h i s d i f f e r e n c e demonstrates the response o f the sensor electrodes to a s m a l l e r spatial r e g i o n t h a n the field electrodes.

C

(%)

Figure 31. Pipe flow calibration curve using isokinetic sampling. (Reproduced with permission from reference 24. Copyright 1987.)


5.

NASR-EL-DIN

Fluid Dynamics of Oil-Water-Sand RB(C.Y) 10

ι η I.U

4.2

1

—

·

α • α

0.8

s

=

c

v

=

l

1

2.0 21

e

•

• •

m/s C

•

α

J


0.4

- r -

• —

η 100

.

·

•

•

0.2

ι

•

α

-

I

•

0.45mm 10%

Ub= α

4.6

|

PARTICLES

cf

α

0.6 -

Ι

SAND

(kn) 4.4

Ι

211

Systems

α

·

-

·

η

•

•

ο ο ο

·

I l 120

1 140

e(C,Y)

I

I

160

I

I

!

ΙθΟ

(mV)

Figure 32. Sensor potential and ballast resistance variations using sand parti cles. (Reproduced with permission from reference 24. Copyright 1987.)

T h e profiles s h o w n i n F i g u r e 32 are t y p i c a l f o r s a n d - w a t e r flows, w h e r e because o f gravity, m o r e particles are f o u n d near the b o t t o m o f the p i p e . Solids concentrations w e r e o b t a i n e d f r o m m e a s u r i n g sensor voltages s u c h as those s h o w n i n F i g u r e 32 a n d f r o m the c a l i b r a t i o n c u r v e . T h i s c a l i b r a t i o n c u r v e was o b t a i n e d b y u s i n g a least-squares fit o f the e x p e r i m e n t a l data o f F i g u r e 27. A s e c o n d test f o r the c o n d u c t i v i t y p r o b e was c o n d u c t e d w i t h the 7 - r a y m e t h o d . T w o scans w e r e c o n d u c t e d s i m u l t a n e o u s l y o n sand slurries, w i t h the c o n d u c t i v i t y p r o b e a n d w i t h the 7-ray m e t h o d . T h e 7 - r a y values w e r e obtained with a collimated beam of 1-mm diameter. F i g u r e 33 shows the c o n c e n t r a t i o n p r o f i l e f o r sand particles o f 0 . 1 9 - m m m e a n d i a m e t e r a n d a b u l k v e l o c i t y o f 2 m/s. G o o d a g r e e m e n t b e t w e e n the 7ray m e t h o d a n d p r o b e measurements was o b t a i n e d . T h e figure shows some scatter at the t o p o f the p i p e w h e r e the solids c o n c e n t r a t i o n is v e r y l o w a n d b o t h m e t h o d s are subject to e r r o r . T h e s e results are i n d i r e c t e v i d e n c e o f a constant c o n c e n t r a t i o n across the p i p e u n d e r the c o n d i t i o n s o f these m e a surements. F i g u r e 34 shows another c o m p a r i s o n f o r the same sand, b u t at a b u l k v e l o c i t y o f 3.5 m/s. A g a i n , g o o d agreement was o b t a i n e d . T h i s result is i m p o r t a n t because it c o n f i r m s the results s h o w n i n T a b l e I I i n that the c a l i b r a t i o n c u r v e is i n d e p e n d e n t o f v e l o c i t y .



212 1.0 Γ

0.8 ho

SAND PARTICLES d = 0.18 mm U = 2 . 0 m/s

·

b

0.6

•·

• Ύ- RAY • PROBE


0.4

0.2

20

15

30

25

C (%) Figure 33. Concentration profiles obtained with γ-ray and the conductivity probe (low velocity). (Reproduced with permission from reference 24. Copy right 1987.)

1.0

SAND PARTICLES d = 0.18 mm Ub= 3.5 m/s

0.8

0.6

•· • ·

• /-RAY • PROBE

0.4

0.2

20

15

C

25

30

(%)

Figure 34. Concentration profiles obtained with γ-ray and the conductivity probe (high velocity). (Reproduced with permission from reference 24. Copy right 1987.)


5.

NASR-EL-DIN


Systems

213

Concluding Remarks


H e a v y - o i l - i n - w a t e r e m u l s i o n s h a v i n g o i l v o l u m e fractions greater t h a n 0.5 are o f t e n n o n - N e w t o n i a n s h e a r - t h i n n i n g fluids. T h e f r i c t i o n loss f o r the flow o f s u c h fluids i n s m o o t h p i p e l i n e s c a n be p r e d i c t e d b y u s i n g the p r o c e d u r e s d e v e l o p e d b y M e t z n e r a n d R e e d (14) a n d D o d g e a n d M e t z n e r (16). A n o t h e r aspect o f the transportation o f h e a v y - o i l - i n - w a t e r e m u l s i o n s , especially for short-distance p i p e l i n e s , is the p r e s e n c e o f sand particles. I n situ solids c o n c e n t r a t i o n a n d e m u l s i o n q u a l i t y can be m e a s u r e d w i t h various s a m p l i n g devices. H o w e v e r , serious errors i n m e a s u r i n g b o t h p a r a m e t e r s arise f r o m i m p r o p e r s a m p l i n g . C o n d u c t i v i t y p r o b e s can b e e m p l o y e d to measure i n situ solids c o n c e n t r a t i o n o r o i l - i n - w a t e r q u a l i t y w i t h M a x w e l l ' s e q u a t i o n . H o w e v e r , the effects o f t e m p e r a t u r e a n d i o n i c strength o n the c o n d u c t i v i t y o f the c o n t i n u ous phase s h o u l d be isolated b e f o r e p r o p e r m e a s u r e m e n t s can b e t a k e n .

List of Symbols Β c C C

b

C C C

d

0

M

f r a c t i o n o f particles that e n t e r a p r o b e l o c a l solids c o n c e n t r a t i o n , v o l u m e f r a c t i o n solids c o n c e n t r a t i o n i n the s a m p l e r , v o l u m e f r a c t i o n average solids c o n c e n t r a t i o n over the p i p e cross section, v o l u m e frac tion surfactant c o n c e n t r a t i o n , w t % l o c a l solids c o n c e n t r a t i o n u p s t r e a m o f the s a m p l e r , v o l u m e f r a c t i o n particle m a x i m u m packing concentration, volume fraction

C d D D e

discharge solids c o n c e n t r a t i o n , v o l u m e f r a c t i o n mean particle diameter, m pipe inner diameter, m sampler diameter, m voltage, V

Ε / F I

s a m p l i n g transport efficiency F a n n i n g f r i c t i o n factor f o r m a t i o n factor volume-average current

k V Κ

P e r l a w constant, P a - s M e t z n e r - R e e d m o d i f i e d p o w e r l a w constant, P a - s p a r t i c l e i n e r t i a p a r a m e t e r b a s e d o n p r o b e radius (also c a l l e d Stokes number or Barth number) pipe length, m power law index

v

s

sm

L η

o w

n

n


214


n'

M e t z n e r - R e e d m o d i f i e d p o w e r law index

ρ Q r R

pressure, P a b r a n c h flow rate, m /s radial position measured from the center o f the pipe, m d i m e n s i o n l e s s radial p o s i t i o n , 2r/D 3

fluid

R\

m i x t u r e resistance, o h m s a m p l e r radius, m Reynolds number, Reynolds number, ρ ϋ /1/μ M e t z n e r - R e e d modified Reynolds number

t Τ u(r) U

p r o b e w a l l thickness, m p r o b e relative w a l l thickness, l o c a l velocity, m/s s a m p l i n g velocity, m/s

U U V y Y

b u l k velocity, m/s u p s t r e a m l o c a l velocity, m/s p a r t i c l e settling velocity, m/s vertical position measured from the bottom o f the pipe, m d i m e n s i o n l e s s v e r t i c a l p o s i t i o n , y/D

m

s m

0


resistance, o h m

R K Re Re Re'

ί

b

0

t

(

ί

t/R

sm

Greek 7 7 θ X X X μ p p τ T

shear rate, s' shear rate at t h e p i p e w a l l , s" p r o b e t i p angle fluid c o n d u c t i v i t y , S 1

W

f

s

m

(

f

s

W

φ

1

solids c o n d u c t i v i t y , S mixture conductivity, S fluid viscosity, Pa-s fluid density, k g / m solids density, kg/m shear stress, P a shear stress at t h e p i p e w a l l , P a 3

3

volume-average potential, V

Acknowledgment T h e assistance o f B . F r a s e r i n t y p i n g this m a n u s c r i p t is greatly a p p r e c i a t e d .


5.

NASR-EL-DIN


Systems

215


References 1. Plegue, T. H.; Frank, S. G.; Fruman, D . H.; Zakin, J. L . Soc. Pet. Eng. Prod. Eng. 1989, 3, 181. 2. Marsden, S. S.; Raghavan, R. J. Inst. Petrol. 1973, 59, 273. 3. Wyslouzil, Β. E.; Kessick, Μ. Α.; Masliyah, J. H . Can. J. Chem. Eng. 1987, 65, 353. 4. P a l ,R.;Masliyah, J. H . Can. J. Chem. Eng. 1990, 68, 25. 5. Gillies, R. G.; Shook, C . A . Presented at the Quarterly Meeting of the Canadian Heavy Oil Association, Calgary, Alberta, Canada, November 14, 1989. 6. Gerhart, P . M.; Gross, R. J. Fundamentals of Fluid Mechanics; Addison-Wesley Reading,MA,1985. 7. Shah, S. N . SPE Prod. Eng. 1990, 4, 151. 8. Blasius, H . Forsch. Ingenieurwes. 1913, 131. 9. Drew, Τ. Β.; Koo, E. C.; McAdams, W . H. Trans. Am. Inst. Chem. Eng. 1932, 28, 56. 10. Zakin, J. L.; Pinaire, R.; Borgmeyer, M. E . Trans. ASME 1979, 101, 100. 11. Mao, M . L.; Marsden, S. S. J. Can. Petrol Technol. 1977, 16, 54-59. 12. Flock, D . L.; Steinborn, R. Presented at the 33rd Annual Petroleum Society Technological Meeting, Calgary, Alberta, Canada, June, 1982; CIM paper 8233-60. 13. Pilehvari, Α.; Saadevandi, B.; Halvaci, M.; Clark, P . E . Proceedings of the 3rd ASME International Symposium on Liquid-Solid Flows; American Society of Mechanical Engineers: Chicago, IL, 1988. 14. Metzner, A . B.; Reed, J. C . AIChE J. 1955, 1, 434. 15. The Flow of Complex Mixtures in Pipes; Govier, G .W.;Aziz, K., Eds.; V a n Nostrand Reinhold: New York, 1972. 16. Dodge, D . W.; Metzner, A . B. AIChE J. 1959, 5, 189. 17. von Kármán, T. National Advisory Committee for Aeronautics Technical Memo, No. 611, 1931, Washington, DC. 18. Pal, R.; Bhattacharya, S. N.; Rhodes, E . Can. J. Chem. Eng. 1986, 64, 3. 19. Pal, R. Chem. Eng. J. 1990, 43, 53. 20. P a l ,R.;Rhodes, E . Presented at the 36th Annual Technology Meeting of the Canadian Institute of M i n i n g and Metallurgy, Edmonton, Alberta, Canada, June 1985; CIM paper 85-36-12. 21. Kao, D . T.; Kazanskij, I. Proceedings of the 4th International Technical Conference on Slurry Transportation; Slurry Transport Association (now Coal and Slurry Technology Association): Washington, DC, 1979,p102. 22. Baker, R. C.; H e m p , J. Fluid Eng. Ser. (British Hydrodynamics Research Asso ciation) 1981, 8, 3. 23. Nasr-El-Din, H . In Encyclopedia of Environmental Control Technology; Cheremisinoff, N . P., Ed.; Gulf Publishing Company: Houston, TX, 1989; Vol. 3, pp 389-422. 24. Nasr-El-Din H.; Shook, C . Α.; Colwell, J. Int. J. Multiphase Flow 1987, 13, 365. 25. Belyaev, S. P.; Levin, L . M . J. Aerosol Sci. 1972, 3, 127. 26. Rehakova, M.; Novosad, Z . Coll. Czech. Chem. Commun. 1971, 36, 3004. 27. Fuchs, N . A . J. Atmos. Environ. 1975, 9, 698. 28. Rushton, J. H . ; Hillestad, J. Y. Paper N o . 52-64, American Petroleum Institute, Washington, DC, 1964,p517. 29. Karabelas, A . J. AIChE J. 1977, 23, 426.


216 30.


31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

61. 62. 63. 64. 65. 66. 67. 68.


Hayashi, H.; Sampei, T.; Oda, S.; Ohtomo, S. Proceedings of the 7th International Conference on Hydraulic Transport of Solids in Pipes; British Hydrody namics Research Association: Cranfield, United Kingdom, 1980; Paper D 2 , p 149. Iinoya, K . Kagaku Kogaku Ronbunshu 1970, 34, 69. Akers, R. J.; Stenhouse, J. I. T. Proc. Inst. Mech. Eng. 1976, 45. N a s r - E l - D i n , H.; Shook, C . Α.; Esmail, M . N . Can. J. Chem. Eng. 1984, 62, 179. Stevens, D . C . J. Aerosol Sci. 1986, 17, 729. Rushton, J. H . AIChE Symp. Ser. No. 10; American Institute of Chemical Engi neers: N e w York, 1965, p 3. Barresi, Α.; Baldi, G . Chem. Eng. Sci. 1987, 42, 2949. Sharma, R. N.; Das, H . C . L . Coll. Czech. Chem. Commun. 1980, 45, 3293. Moujaes, S. F . Can. J. Chem. Eng. 1984, 62, 62. Torrest, R. S.; Savage, R. W . Can. J. Chem. Eng. 1975, 53, 699. N a s r - E l - D i n , H . ; Shook, C . Α.; Esmail, M . N . Can. J. Chem. Eng. 1985, 63, 746. N a s r - E l - D i n , H.; Shook, Int. J. Multiphase Flow 1986, 12, 427. N a s r - E l - D i n , H.; Afacan, Α.; Masliyah, J. H . Chem. Eng. Commun. 1989a, 82, 203. N a s r - E l - D i n , H.; Afacan, Α.; Masliyah, J. H . Int. J. Multiphase Flow 1989b, 15, 659. Whitely, A . B.; Reed, L . E . J. Inst. Fuel 1959, 32, 316. Yoshida, H ; Yamashita, K . ; Masuda, H . ; Iinoya, K. J. Chem. Eng. Jpn. 1978, 11, 48. N a s r - E l - D i n , H . ; Shook, C . A . J. Pipelines 1985, 5, 113. N a s r - E l - D i n , H.; Shook, C . Α.; Colwell, J. Hydrotransport 1986, 10, 191. Ito, H . JSME Int. J. 1987, 30, 543. Lundgren, D . Α.; Durham, M . D.; Mason, K. W . Am. Ind. Hyg. Assoc. 1978, 39, 640. Ayukawa, K . Bull. JSME 1969, 12, 1388. Toda, M.; Ishikawa, T.; Sait, S.; Maeda, S. J. Chem. Eng. Jpn. 1973, 6, 140. Toda, M.; Komori, N.; Sait, S.; Maeda, S. J. Chem. Eng. Jpn. 1972, 5, 4. Stairmand, C . J. Trans. Inst. Chem. Eng. 1 9 5 1 , 29, 15. Davies, R. E . Int. J. Air Water Pollut. 1964, 8, 177. N a s r - E l - D i n , H.; Shook, C . A . J. Pipelines 1987, 6, 239. Brown, N . P.; Shook, C . Α.; Peters, J.; Eyre, D . Can. J. Chem. Eng. 1983, 61, 597. Sansone, E . Am. Ind. Hyg. Assoc. 1969, 30, 487. Colwell, J. M.; Shook, C . A . Can. J. Chem. Eng. 1988, 66, 714. Lee, K. T.; Beck, M. S.; M c K e o w n , K . J. Meas. Control 1974, 7, 341. Machon, V.; Fort, I.; Skrivanek, J. Proceedings of the 4th European Conference on Mixing; B H R A F l u i d Engineering Centre: Cranfield, United Kingdom, 1982; p 289. Ong, K . H.; Beck, M . S. Meas. Control 1975, 8, 453. Pal, R.; Rhodes, E . Proceedings of the 3rd Multi-Phase Flow and Heat Transfer Symposium; Clean Energy Research Institute: Coral Gables, FL, 1983. Begovich, J. M.; Watson, J. S. AIChE J. 1978, 24, 351. Perez-Rosales, C . J. Pet. Technol. 1976, 28, 819. Perez-Rosales, C . Soc. Pet. Eng. J. 1982, 22, 531. Sweeney, S. Α.; Jennings, Η. Y., Jr. J. Phys. Chem. 1960, 64, 551. Keller, G . V . Oil Gas J. 1953, 62. A Treatise on Electricity and Magnetism, 3rd ed.; Maxwell, J. C . , E d . ; Dover: New York, 1954; V o l . 1, Article 314.



217

5.

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69. 70. 71. 72. 73. 74. 75. 76. 77. 78.

L o r d Rayleigh Phil. Mag. 1892, 34, 481. Pearce, C . A . R. Br. J. Appl. Phys. 1955, 6, 113. Clark, N . O. J. Phys. Chem. 1945, 49, 93. Lorentz, H . A . Ann. Phys. Chem. 1880, 9, 641. Lorenz, L. ibid. 1880, 11, 70. Bruggeman, D . A . G . Ann. Physik. 1935, 24, 636. D e L a Rue, R. M.; Tobias, C . W . J. Electrochem. Soc. 1959, 106, 827. Merilo, M.; Dechene, R. L.; Cichowlas, W . M. J. Heat Transfer 1977, 99, 330. Neale, G . H.; Nader, W . K . AIChE J. 1973, 19, 112. Turner, J. C . R. Chem. Eng. Sci. 1976, 31, 487.


Systems

R E C E I V E D for review December 18, 1990. A C C E P T E D revised manuscript May 23, 1991.


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