a deposition velocity correlation for water slurries

7/18/2019 A Deposition Velocity Correlation for Water Slurries

http://slidepdf.com/reader/full/a-deposition-velocity-correlation-for-water-slurries 1/3

__--

NOTES

A

Deposition Velocity Correlation for

Water Slurries

R .

G. GILLIES“’ and C. A. SHOOK ’)

‘)Saskatchewan Research Council, 1630 Quebec Avenue, Saskatoon, SK S7K I V7

’’Department

of

Chemical Engineering, University

of

Saskatchewan, Saskatoon,

SK S7N OW

A

correlation

to

predict

the

deposition velocity of aqueous slurries

is

presented. The correlation

employs

the viscosity

and

density of the mixture

of

fluid and ( -7 4 pm) particles, as well as the mass median diameter dso) f the fraction

coarser than 74

pm.

The correlation

is

derived from isothermal flow experiments

using

pipelines

of

diameter between

0.053 and 0.495 m .

On presente une correlation pour predire

la

vitesse de dep8t de suspensions aqueuses. La correlation utilise la visco-

site et la densite

du

melange de fluide et de particles (-74

pm)

ainsi que le d iambre moyen de masse dso) e la frac-

tion plus

grosse que 74 p .

La

correlation est calculee a

partir

d’experiences d’ecoulement isotherme menees avec des

conduites

ayant

entre 0,053 et

0,495

m de diamktre.

Keywords: deposition velocity, aqueous slurries.

o aspect of the flow of settling slurries is more im por-

N

ant than the limit-deposit velocity V, . Below this

limit, a stationary deposit of particles forms on the bottom

of the pipe. For flow with such a deposit, frictional head-

losses begin to rise with decreasing velocity. This makes

stable pipeline operation very difficult for flows driven by

centrifugal pumps and thus V , is the normal lower limit for

pipeline design.

Because of its importance , innumera ble correlations have

been proposed to predict V, Carleton and Cheng (1974)

identified 55correlations and many more have been proposed

since that time. Some of these have a theoretical basis but

their validity is entirely depende nt upon the scope of the data

base which they incorporate. In the present communication

we summarize our experience on the basis of tests conducted

with

a variety of slurries and pipelines over the past

10

years

or so.

The

Data

Source

Determination of V,

in

experimental studies can be

difficult when the particles are fine and dark in color because

small quantities of ultra-fine material can make the liquid

phase opaque. In those cases it is difficult to detect the first

thin layer of particles to form on the bottom of the pipe.

If

the

mixture contains very coarse p articles, de posits are

easier

to

see but a different problem arises. At velocities just

above V , . slowly moving dunes form. These dunes are a

few particles

in

depth and advance by simultaneo us erosion

of their upstream surfaces and deposition downstream of their

crests Since they contain particles which are stationary for

some time, a dune is easily mistaken for a stationary deposit

unless the observation is prolonged. The electrical sensors

described by Ercolani et al. (19 79) help to resolve these

difficulties with visual observations unless contaminants in

the flow foul the electrode surfaces.

The pipeline flow tests which provided the data used here

were all isothermal. This restriction has been found to be

essential for the generation of reprodu cible data. There are

two reasons for this. First, deposition velocity is fairly

strongly dependent upon fluid viscosity for fine particles.

Secofidly temperature changes can produce physical and

chemical changes in the solid particles which result

in

unexpected changes in the viscosity of the mixture formed

by the fluid and the finest particles. Experience has shown

that the viscosity must be measured continuously during

testing if reproducible results are to be obtained.

The data we have used were generated during research

sponsored by a variety of Canadian government agencies.

In addition,

it

incorporates experience gained in testing

several industrial slurries. Three quarters of the data points

were obtained with pipes

0.15

m in diameter and larger.

Background Theory

A

layer force balance model for slurry pipeline flow can

be used to interpret the deposition phenomenon and to justify

use of an empirical correlation. Figure

1

shows an idealiza-

tion of the flow of a “settling” slurry before a deposit is

present. There are tw o constant composition regions and the

upper layer contains only particles whose immersed weight

is borne by fluid lift forces. T he density of this mixture deter-

mines the gradient of hydrostatic pressure.

The total concentration

in

the lower layer, C2, is known

to be a function of the mean in-situ concentration

C,

and the

ratio of the mean flow velocity V to the terminal velocity

v , of the mass median particle (Gillies et al, 1990).

v

is

computed for settling in a hypothetical mixture consisting

of

the liquid and the finest (- 74 pm) particles. The differ-

ence

C 2

-

C , )

represents particles which are not supported

by fluid lift forces. T hese particles experience a buoyant force

which depends upon the density

of

the mixture of fluid and

turbulently suspen ded particles. T he particles which are not

suspended generate an interparticle stress which increases

with depth according to the relationship

do,

I dy =

p , - p2) C2 -

Ci .

. . . .

. . 1)

where

p2

is the density of the mixture of fluid and sus-

pended solids in the lower layer. The interparticle stress

a,

is zero at the interface between layers 1 and 2.

Stress

u,

contributes a velocity

-

ndependent frictional

resistance to flow, which increases

as

p 2 increases. From

pressure drop measurements we know (Gillies et al, 1990)

that p depends upon the factors which determine

C2 f

we

now consider flow with a deposit (Figure

2) ,

Equation

I )

THE

CANADIAN JOURNAL OF CHEMICAL ENGINEERING. VOL UME 69, OCTOBER, 1991

1225



Velocity

Concentrat ion

4

0

a -

0

u

0

-

3-

:

0 2 -

> -

-

1 -

Figure 1 dealized velocity and concentration distributions in

a

slurry

before deposition.

Velaclty Concentrat ion

Figure 2 dealized velocity and concentration distributions in

a

slurry

after a stationary deposit forms.

applies within the middle laye r. Within the stationary deposit,

the buoyant force is produced only by the fluid. In terms

of the concentration

C3

in the deposit, the stress gradient

in

this region is:

. . . . . . . . . . . . . . . . . . . . .

a,,ldy

=

p ,

- P L ) g C, 2)

The stress gradient in the stationary layer (Equation 2))

is usually much greater than that in the flowing mixture above

it. Deposition can therefore be consid ered to reflect the ina-

bility of a flowing mixture to support all the available solid

particles in the pipe cross-section.

At the interface between moving solids (laye r

2)

and sta-

tionary solids (layer 3 )

the normal stress obtained from

integrating Equation (1) is in equilibrium with the shea r stress

derived from the flow, 7 2 3 . The coefficient of proportion-

ality between the stresses is tan

+,

where

+

is the angle of

internal friction of the solid particles.

7 2 3

= 0 5 P s

- p 2 ) g D(cos

6 3 -

cos

62)

. . . . . . . . . . . . . . . . . . . . . . . . . .

C,

C , )

tan +

(3)

where p2 and

P3

define the boundaries of layer 2 (Figure 2).

If the stress

7 2 3

is expressed

in

terms of a friction factorh3

for the flow as a whole

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 =

f 2 3 v2p, /2 4)

we have an expression for V 2 n the limiting case where the

deposit is infinitesimal and V becomes V,:

vc2 = P , - P 2 ) 1 cos P 2 ) C2 C,)

. . . . . . . . . . . . . . . . . . . . . . . . . .

D tan +Vz3

m

5 )

E uation (5) shows that to a first approxim ation

V,

varies

the ratio (Vlv,) so that the relationship between V, and D is

rather com plex. The interfacial friction fa ct or h3 is unknown

but presumably varies with the ratio

of

particle diameter to

pipe diameter, and possibly flow Reynolds number.

as

9D. However C2, p 2 and the ratio CIIC, depend upon

5

/

+ 2 0 s

/

/

/

/

/ / / ’

’

/

- 20

0 1 ,

I T I 1 8

, I 8 1

t , ” I I r f 1 I I

Vc Exper imen ta l

Figure

3

Comparison

of

measured deposition velocities with

those predicted by the correlation.

1 2 3 4

Although it has a m echanistic origin, Equation 5 ) s based

upon several sim plifying assumptions and contains a number

of unknown parameters. In the absence of information con-

cerning tan and , we propose a correlation as an alter-

native for pipeline design. Th e dimensionless deposit velocity

FL has been used by many workers since Durand (1953).

. . . . . . . . . . . . . . . . . .

L = V,/ [ 2 g D ~ , ~p f ) / p f ]”‘

6)

In the correlation proposed here,

p f

is the density of the

hypothetical mixture comprising the carrier fluid and the

-74 pm particles in the slurry as a whole. The latter are

considered to determine the density and viscosity of an

“equivalent liquid”. The density

p f

can be calculated and

the viscosity

pf

can be measured.

Because we now associate the -74 pm solids with the

fluid, the total in situ concentration is denoted as C, and the

symbol

C,

is used for only the

+74

pm solids.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

,

=

f +

c,

(7

so that

p f = [psCf

+

l - C , ) p , ] / [ l - C , +

Cfl 8 )

FL depends upon the drag coefficient of

the

particles

settling in an equivalent fluid of dens ity

p f

and viscosity py

Experimental drag coefficients can be determined by mea-

suring single particle terminal falling velocities. These can

be correlated

in

terms of the Archimedes number

for particles of a particular shape

The correlation for Ft is:

FL =

exp [0 165

-

0.073 C 12.5

K 2 ]

. . . . . (9)

where

. . . . . . . . . . . . . . . . . . . . . . . .= [ K i

- 0.14]*

(10)

I

1226

THE CANADIAN JOURNAL

OF

CHEMICAL ENGINEERING, VOLUME

69,

OCTOBER,

1991

a deposition velocity correlation for water slurries

Documents