Exact solutions for two phase vertical pipe flow

]MECHANICS RESEARCH CO~nMUNICATIONS Vol. 19 (I) ,7-13,1992 0093-8413/92 88.00+ .00 Copyri6h~(c) 1991

Prm~edIn the USA PeP~Inon Press plc

EXACT SOLUTIONS FOR TWO PHASE V E R T I C A L PIPE FLOW M a n o h a r Gadiraju, I'2 John Peddieson, Jr. 2 and Sastry S. M u n u k u t l a 3 1Bristol Babcock, Inc., Watertown, C o n n e c t i c u t 06795, 2Department of M e c h a n i c a l Engineering and 3Center for Electric Power, T e n n e s s e e T e c h n o l o g i c a l University, Cookeville, T e n n e s s e e 38505

(Received 18 April 1991; accepted for print 16 July 1991)

Introduction The p u r p o s e of this note is to report exact solutions for fully d e v e l o p e d steady laminar flow of a p a r t i c l e / f l u i d s u s p e n s i o n in a vertical circular pipe. The analysis is based on a typical set of g o v e r n i n g equations representative of those employed by such investigators as Green and Homsy [i], Tsuo and G i d a s p o w [2], U n g a r i s h [3], [4], and Ganser and Drew [5]. Some special cases of the r e s u l t s are d i s c u s s e d to illustrate the influence of v a r i o u s terms a p p e a r i n g in the model.

G o v e r n i n q Equations The p r e s e n t w o r k is based on the following c o n t i n u u m two p h a s e flow model.

The equations

at~+v. ((l-~)v) -0, at~+v- (~vp) -0 represent

balances

respectively. the

of mass

for the

(i) fluid and p a r t i c u l a t e

In (1) ~ is the gradient operator,

particulate

v e l o c i t y vector,

phase

volume

fraction,

9

is

phases

t is time, # is the

fluid

phase

and 9p is the p a r t i c u l a t e phase v e l o c i t y vector.

The e q u a t i o n s p(l-~)(@tV+V'VV ) -V'~+pg-f,

pp~(@tVp+Vp'VVp)-V'~p+ppg+f

(2)

r e p r e s e n t b a l a n c e s of linear m o m e n t u m for the fluid and p a r t i c u l a t e p h a s e s respectively.

In (2) p is the fluid bulk density, pp is the

p a r t i c u l a t e bulk density, ~ is the fluid phase stress tensor, gp is the p a r t i c u l a t e phase stress tensor, f is the interphase force per unit volume,

and ~ is the gravitational 7

force per unit mass.

In

8

M. GADIRAJU, J. PEDDIESON and S.S. MUNUKUTLA

writing

(1-4) it was assumed that both p and pp are constants.

The

equations

~ - (1-4) ((-p+X(4)~" ~)~+~(4)(vv+vv--~) ) o - 4 ( ( - ( P + q ( ¢ ) ) + k p ( 4 ) ~ ' ~ p ) ~ + ~ p ( 4 ) (VVp+VVp T)) =p

(3)

T- pp4(V-Vp)/T (4) +pT4 represent tensor,

constitutive

assumptions

the p a r t i c u l a t e

force respectively.

phase

for

stress

the

fluid

tensor,

and

phase the

stress

interphase

In (3) p is the indeterminate fluid pressure,

q is the d i f f e r e n c e between the p a r t i c u l a t e phase pressure and the fluid phase pressure, T is the interphase r e l a x a t i o n time, ~ is the unit tensor, and a superposed T indicates the transpose of a second order

tensor.

coefficients

Also

k,

1,=~/p

(and

~, and

kp,

and

Vp=~p/pp

~p

dynamic

are

viscosity

are c o r r e s p o n d i n g kinematic

v i s c o s i t y coefficients).

Pipe Flow Solutions Consider

steady,

fully

circular

pipe

radius

of

developed,

laminar

a subjected

to

p r e s s u r e c h a r a c t e r i z e d by a gradient p6. v o l u m e fraction 4 is constant. balances

(I)

are

identically

flow

in

a vertically

a

vertical

decreasing

Further, assume that the

Under these circumstances the mass satisfied

and

(2)

and

(3)

can

be

c o m b i n e d to yield (rv')'/r+K~(Vp-V) -4(~-i), ~(rvp')'/r+~(v-vp) -4(~-E) In

(4) v

and Vp are

(both n o r m a l i z e d the

radial

the

axial

fluid

velocities

by the characteristic velocity Vc=6a2/4~),

coordinate

(normalized

d i f f e r e n t i a t i o n with respect to r.

~-a2/(pT),

and particulate

(4)

~-ep/e,

by

a),

and

a

prime

r is

denotes

Also

(-p/pp, {-pp4/(p(1-¢)), @-g/6

(5)

The parameters e and K are o f t e n c a l l e d the i n v e r s e Stokes number and

the p a r t i c l e

imposed on

loading respectively.

The

boundary

conditions

(4) in the present w o r k were

V ' ( 0 ) - 0 , Vp'(0) - 0 , V(1) - 0 , Vp(1) - 0 or vp'(1) - 0

(6)

Equations (6a,b) are t h e symmetry c o n d i t i o n s a t the pipe a x i s and (6c) is the usual no slip condition for the fluid phase at the pipe

TWO-PHASE VERTICAL PIPE FLOW

wall.

The

particulate two

corresponding phase

idealized

condition (6d)).

wall

boundary

is poorly understood

conditions

were

9

condition

at this time.

considered.

These

for

the

Therefore,

are

(first of (6d)) and a perfect slip condition

a no

slip

(second of

It is expected that the actual behavior would probably be

somewhere between these two extremes. The

boundary

value

problem

consisting

of

(4)

and

(6)

has

the

solution v - ((I+~K-~(I+K)) (l-r2)+4~K(~-~-~(~-l)) (i -I0 (~r)/I0(~ ) ) / (~ (I+~K)) ) / (I +~K) (7) Vp - ((I+~K-~(I+K)) (l-r 2) -4 (~-~-#(~-I)) (i - I 0 ( ~ r ) / 1 0 ( ~ ) ) / ( ~ ( l + ~ ) ) ) / ( l +~K) for no particulate

slip and

V-(I+EK-#(I+K) ) (l-r2+2~KI0(~) (i -10(~r )/I0(~ ) ) / (~Ii(~) ) ) / (I+~K) Vp

-

((I+EK-#(I+4)) (i -r2+210 (~) (~K +I 0 (~r)/I 0 (~)) / (~I 1 (~)) )

(8)

-4 (~-~-~ (~-i))/~) / (I+~) for

perfect

particulate

slip.

In

(7)

and

(8)

I 0 and

11

are

m o d i f i e d Bessel functions of the first kind of order zero and order one respectively

and

~ . (~(K +i/~)) i/2 It

should

be

(9)

pointed

out

that,

unlike

currently available for two phase flow,

most

exact

solutions

(7) and (8) are not limited

to small values of the volume fraction 4Discussion of Results The solutions reported above have several interesting features. few of these will be singled out for discussion In

order

to

model

successfully, transition sufficiently volume

many

complex

two

phase

A

in this section. flow

situations

theories will eventually be needed which allow for a

of

the

particle

phase

from

fluid

like

behavior

at

low volume fractions to solid like behavior at higher

fractions.

Endowing the particle phase with viscosity

(as

indicated by the inclusion of the terms multiplied by ~

and ~p in

(3b))

kp and ~p

is a step in this direction.

If the coefficients

i0

M. GADIRAJU, J. PEDDIESON and S.S. MUNUKUTLA

are allowed

increase rapidly with ~ the behavior of the particle

phase

approach

will

fractions.

that

of

a

rigid

body

for

higher

volume

The solutions reported in the previous section provide

a concrete example of this phenomenon. For

~<
the

solutions

given

in

the

previous

section

can

be

simplified to

V - (I+(K-@(I+K)) (1 - r 2)

(10)

Vp - (I+(K-@(I+K) ) (i -r 2) -4 (@-() (l-exp(-~0 (l-r)) )/~ for no particulate

slip and

V " (I+EK-@(I+K)) ( 1 - r 2)

(11)

Vp - (I+EK-@(I+K)) (i -r2+2 (~/e) i/2exp (-~0 (l-r)) ) -4 (@-~)/~ for perfect particulate slip.

In (i0) and (ii) terms of 0(~) have

been neglected and ~0 -

(~/~) 1/2

(12)

Except for the exponential terms appearing in (10b) and (llb), and

(ii)

yields

are the

exponential

identical. solution

The

for

omission

an

of

inviscid

the

exponential

particle

(i0) terms

phase.

The

terms represent a boundary layer effect and show that

the particle phase boundary conditions have influence only in the immediate vicinity

of the wall for ~<
This

is illustrated

in

Figures 1 and 2 which are based on the exact solutions

(7) and (8).

The

an

curves

labeled

particle phase. layer

effect

particulate

~=0

represent

the

results

for

inviscid

It can be seen that the magnitude of the boundary

is larger

for no particulate

slip as indicated by

(I0) and

slip than

for perfect

(ii).

For ~>>i the exact solutions can be simplified to v - 4 (i-@) (l-I0 ( ~ r ) / I 0 (~.)) / (~K) (13) Vp- 0 for no particulate

slip and

V - 2 (I+(K-@(I+K)) Io(~.) ( i - I 0 ( ~ r )/I0(~m ) ) / ((~K) 1/211(~m) ) Vp-2((I+(K-@(I+K))I 0 ( ~ . ) / ( ( ~ K ) l / 2 I 1 ( ~ ) ) for perfect

particulate

have been neglected and

slip.

In

(13)

(14)

-2(l-@)/(eK)) and

(14)

terms

of O(i/~)

TWO-PHASE VERTICAL PIPE FLOW

Ii

~® - (eK) 112 Equations packed

(15)

(13) represent flow of a fluid through a stationary rigid

bed.

Equations

(14)

sliding rigid packed bed.

represent

flow of a fluid

through

a

Sliding packed beds can occur in stand

pipes as discussed by Chen, Rangachari,

and Jackson

[6].

The limiting solutions discussed above illustrate the transition of the particle phase from inviscid fluid like behavior immediate vicinity of the pipe wall) like behavior for ~>>i.

(except in the

for ~<
Since ~ depends on the volume fraction ~,

the proper choice of the function ~(#) can make ~<>i correspond to ~m-~<
(~m being the volume

fraction

Thus the inclusion of particle

phase viscosity provides a mechanism for transition of the particle phase from fluid like to solid like behavior as mentioned earlier. In general motion

is

a two phase pipe vertical

configurations). buoyancy.

(due to One

flow will settling

exception

be parallel of

to this

the

only when

particles

is the

case

in of

the

other

neutral

Thus equating ~=i and #=0 in (7) and (8) corresponds to

horizontal flow of a neutrally buoyant suspension.

An interesting

aspect of this special case is the particulate slip velocity which takes the form V p - V - -4 ( ~ - 1 ) ( l - I 0 ( ~ r ) / I 0 ( ~ ) ) / (~ (I+~K))

(16)

for no particulate slip and Vp-V - 2 ( -2 (~-1) I (e (I+~K)) + (I+K) I 0 ( ~ r ) / ( ~ I 1 ( ~ ) ) ) for perfect

particulate

parametric combinations

slip.

It

(17)

can be seen t h a t t h e r e a r e many

(all situations for which ~
for which the slip velocity is predicted to be positive,

that is:

the particle phase is predicted to lead the fluid phase.

This is

because of the inclusion of the fluid pressure term in (3b) is

necessary

phases

are

to

model

subjected

buoyancy). to

the

fluid

In the

present

pressure

(which

problem

gradient.

If

both the

viscous resistance of the particle phase is sufficiently small the particle

phase will

lead the fluid phase.

should be investigated experimentally.

This

is a point

that

12

M. GADIRAJU, J. PEDDIESON and S.S. MUNKUTLA

Conclusion Exact s o l u t i o n s

for steady,

of a p a r t i c l e / f l u i d The

solutions

phase of

were

flow m o d e l

the

exact

exact

were

of

boundary phase

presented

suspension

behavior

a

good

exhibited

some

fraction

limiting

results

illustrate

the

two

forms

based

on

the

nature

of

the

r e s u l t s will be of help

to check c o m p u t e r

represents

volume

routines

problems.

test

case

The in

by the r e s u l t s

for n u m e r i c a l problem

view

of

the

for low p a r t i c l e

viscosities.

and

Massoudi,

Lahey

distributions

and

interphase volume

[9],

Sinclair

fraction

[ii]

those

not p o s s i b l e

force.

This

exist

and

constitutive

Jackson

predict

is

sufficient

in

solutions

fraction which

It is

in c l o s e d

form.

(see

illustrated on

are

work.

produce

based

Lee,

Johnson,

a lift c o n t r i b u t i o n

but not n e c e s s a r y

layer e f f e c t s

theories

and

distributions

to

of the

volume

in the p r e s e n t

included

form

[8], Wang,

[i0],

nonuniform

velocity

reported

c i t e d above

distribution

also

elaborate

simplicity

to e x p r e s s these s o l u t i o n s

e x p e c t e d t h a t the b o u n d a r y would

of the c l o s e d

[7], D r e w and L a h e y

corresponding than

M o s t of the p a p e r s

More

by Soo

Rajagopal

complicated

apparently

(3).

employed

and

that the e x i s t e n c e

h e r e i n is due to the r e l a t i v e

equations

as t h o s e

Jones,

to r e m e m b e r

presented

constitutive

work

Both

numerical to

l a m i n a r flow

pipe w e r e presented.

finite

type.

some

particle/fluid

layer

solutions

more

and

desiring

herein

It is i m p o r t a n t

such

on a t y p i c a l

It is h o p e d that the p r e s e n t

to i n v e s t i g a t o r s solution

based

vertical,

in a c i r c u l a r

of the c o n t i n u u m

solutions

discussed

suspension

solutions

predictions.

fully developed,

a

to the

nonuniform

[I0]).

It is

by the p r e s e n t more

elaborate

formulations. References [i] [2] [3] [4] [5] [6] [7] [8]

D. G r e e n and G.M. Homsy, Int. J. M u l t i p h a s e Flow 13, 443 (1987) Y.P. T s u o and D. Gidaspow, A I C h E J. 36, 885 (1990) M. Ungarish, Int. J. M u l t i p h a s e F l o w 14, 729 (1988) M. Ungarish, Phys. Fluids A 2, 160 (1990) G.H. G a n s e r and D.A. Drew, Int. J. M u l t i p h a s e F l o w 16, 447 (1990) Y.M. Chen, S. Rangachari, and R. Jackson, Ind. Engr. Chem. Fundam. 23, 354 (1984) S.L. Soo, Appl. Sci. Res. 21, 68 (1969) D.A. D r e w and R.T. Lahey, J. Fluid Mech. 117, 91 (1982)

TWO-PHASE VERTICAL PIPE FLOW

13

[9]

S.K. Wang, S.J. Lee, O.C. Jones, and R.T. Lahey, Int. J. Multiphase Flow 13, 327 (1987) [i0] J.L. Sinclair and R. Jackson, AIChE J. 35, 1473 (1989) [ii] G. Johnson, M. Massoudi, and K.R. Rajagopal, Int. J. Engng. Sci. 29, 649 (1991) Acknowledqement This work was supported by Dayton Power, Duke Power, Pennsylvania Power and Light, Southern Company Services, and Virginia Power.

6B=O

P=O

.

i o-

T

t-w

io

|

~6 *,t.~

% /

tB =0.01

~,

?-

~=0.01

!

,e.

'T" s 0.0

d.2

d.4

d.s

d.B

r FIG.

1.0

1.0

0.0

r 1

Velocity Profiles for No Particulate Slip

FIG. 2

Velocity Profiles for Perfect Particulate Slip