Pressure waves in solid—liquid suspensions

-Powder

Technology,

40 (1984)

303 - 307

. 303

Pressure Waves in Solid-Liquid

Suspensions

V. BATI’ARRA Snamprogetti

U

S pA

, Centro di Progettarzone

di Fano,

Fano (PS) (Italy)

BILARDO

Facoltd dr Ingegnerq

Unwersitd dr Roma,

Rome

(Italy)

and C. CERCIGNANI Drpartrmento

di

Matematica,

Polrtecnico

di Mdano,

Mzlan

SUMMARY

The equations governing the unsteady onedimensional matron of a solrd-lrquid mixture m an elastic tube previously derived by the authors are reconsidered m order to take into account the change rn concentration_ Such equations are applied to the problem of pressure surge generation resulting from rapid flow shutdown and a comparrson with the results of other authors is made. In particular, the fact that Wood and Kao’s boundary conditzon produces the correct results is explained The previous findings concerning the long-term behaviour are confirmed.

INTRODUCTION

In the last few years, the pipeline transport of solids m a hquid medium has become an important method of carrying small particles of coal, gravel, and mineral ores over large distances_ The relatively high flow velocities required to maintam these sohd particles m suspension may produce excessive surge pressures under conditions resulting from loss of power to pumps or rapid flow shutdown. The latter problem has been considered by several authors [l - 61. In particular, Wood and Kao [ 1, 21 indicated a new expression to calculate pressure wave velocities of sohdliquid suspensions, whrle Thorley and Hwang [4] criticized that expression on the grounds that it contains the average specific volume rather than average density The present authors [5] used a slmphfied model of the flow, which had the advantage of being amenable to a ciosed-form solution by the 003s591oia4is3.00

(Italy)

Laplace-transform technique. While the mam aim of OUT previous work was to investigate the practical consequences of the existence of two different speeds of wave propagation in a two-phase mixture, we pointed out that the results concerning the pressure surge generation were crucially dependent on the boundary conditions at the valve. In our model, as well as in Wood and Kao’s calculations [l, 21, it was impossible to impose more than one boundary condrtlon, because of the reduced number of variables (the solid particle concentration was assumed to be constant) Wood and Kao’s results were obtained under the assumption of vanishmg volumeaveraged velocity. In our paper, we calculated the solution for a certam number of boundary conditions. We recovered Wood and Kao’s result for pressure when we used their boundary condition, but the results changed when the boundary condition was changed. Bechteler and Vogel [6] approached the problem m a more general way. They did not neglect the change m concentration and accordingly were able to impose two boundry conditions at the valve, as is physically satisfactory. They used the method of characteristics to handle the problem and found that the pressure surge was exactly the one predicted by Wood and Kao by means of then rather ud hoc boundary condition of vanishing volume-averaged veIocity. In this paper, we tone up again the model presented in our previous work and compare it with the model of Bechteler and Vogel. Then we compute the pressure surge without neglecting the change in concentration and find agreement with Bechteler and Vogel as welI as with Wood and Kao, except for the effect of added masses. Finally, we discuss 0 Elsevier Sequoia/Pnnted

ir, The Netherlands

-L

304

the surprising fact that Wood and Kao were able to select the only boundary conditions that produce an exact result from the approximate equations

d(v, - ~1)

The conservation of mass and momentum for the two phases form the system of partral differential equations governing the onedimensional motion of the mixture. The continuity equation for the solid phase may be written as follows:

(1) where pS, c, and u, denote density, volume concentration and velocity of the sohd phase, u the duct cross-section, x the spatial coordmate and f the time. If the first and the last terms in the righthand side of eqn. (1) are assumed to be negligible, the equation simphfres to C,

ap,

aUs

ps

at

ax

& (&P)

= Fd

(6)

BASIC EQUATIONS

c, au k -++_+__~--cs_ at u at

+

dt

(2)

Fd is the drag force (per unit volume) between solid particles and liquid phase, p the pressure and p,’ the so-called ‘added density’ of the sohd partrcles. If the particles are assumed to have a spherical shape with radius (I, the followmg relation may be used:

(7)

P.5

In eqn. (5), we consrdered the added mass effect to depend on the relative acceleration and not on the absolute acceleration. This is a sublect hardly discussed in the hterature and worth a more thorough investigation. For the sake of slmphcity, we omit the added mass term; this will make calculatrons simpler and comparisons with results of other authors easier If, in addition, we assume that in eqn (5) the substantial denvatrves can be replaced by partial derrvatives, we obtain from eqn. (6)

(8) With the same assumptions, rt is possible to denve the following simplified form of momentum equation for the liquid phase:

If u and pB are assumed to depend on time only through the pressure, one obtains

-ac, +c,ay,at

ap +c,-- ah at

ax

=o

(9) (3)

where

cy,2 &A

+t E,

D, E and s are the diameter, Young modulus and thickness of the pipe, respectively, and E, the compressibility of the solid phase. An analogous procedure for the liquid phase gives

-acl +qa,at

From the basic system of eqns. (3), (5), (8) and (9), the following equatron for characteristic speeds can be derived:

aP

at

+c1-

ah =o at

Det

ffscsx

cs

0

--x

CqC~X

0

=I

P

CS

P&J

0

-P

Cl

0

PlClX

= 0 (10)

i_e 2_ EL!-=0

(5)

In order to write the momentum balance for the solid phase, we shall neglect the mteraction of the solid particles with the duct Thus we have

h

PIPS

where (11)

305

This equation is rather cumbersome However, it simplifies considerably the assumptron

to solve under

cup 4 1

and at

CSCI

(12)

= (Plal-

which is satisfied for all cases of practical mterest. In such a case, we can neglectp in eqn. (11) to obtain the following roots: x = *(;

and

X=0

+ ~)1’201-112=

*a,

which agrees with Z2.ef.s 5 and 6 (when the added mass is neglected) Equation 12 implies also that the last terms in the r.h s. of eqns. (3) and (9) may be neglected, as was done in our previous paper, as long as the mam interest is m the calculation of the pressure rise The root A = 0 m a second approximation becomes:

ii =

i(clps :,pl)“2

The physical meaning of the two couples of speeds is clear the first (*co) is the sound velocity and the second represents, m our approximation, the velocity of propagation of mass perturbations_ PRESSURE TION

DUE

RISE

AND

TO RAPID

CHANGE VALVE

IN CONCEN’I’RACLOSURE

The equations discussed in the previous section may serve to handle the interesting case of a sudden flow shutdown. If we handle the closure of the valve as an impulsive phenomenon, it is very easy to compute the initial change, provided that we neglect non-linear effects. We first separate the unknowns in such a way as to decouple the effects of lower order in alp from those of next order. Accordingly, we form the following two systems : a-

aP

at

+c,-

aUs+q- a4 =o ax

ax

1

ac, +A,x au, -P1-

pscl + ~6

a4

1

ax

(14)

P$L) g

pat,

1

-_=_F’

CsCl

In the first system

the speed emerges: (15)

while the second

is ruled by 112

P

al =

(16)

PsCl + PIG

Note that these are (except for the sign) the charactenstic speeds computed before. The two systems have different meanmgs. the first describes the unpulsrve phenomena and can be used to predict the mitial pressure rise caused by an mstantaneous flow shutdown. Accordingly, we can neglect partial derivatives of c, and

cl and

the

inhomogeneous

deriving the following

srmplified

term

F’,

form of eqn

(13).

ap at

au ax

(y-+---o

-au

+ao20L-

at

ap =o ax

with

1

(1’7)

v = c,v, +civi The form of the above system is sufficient to compute the mitial pressure jump. It is clear that rt must be given by the fcrmula for a one-component fluid with a,-, in place of the speed of propagation. In other words, if fluids have velocity v0 before the closure of the valve, then Ap=

?-

ws

aa0

This is the result of Wood and Kao [ 1, 2j. It was also recovered by us [ 51 when we used the boundary condition of zero volumeaveraged velocity At that time, we did not see any justification for this choice and stated

306

that both u, and ur must be zero at the valve. This was, however, mathematically impossible because we had put c, constant from the beginning. The simplified form used here Indicates that, as long as we are interested in the pressure jump, LJ= 0 is the correct boundary conditron_ The second system, eqn. (14), allows a calculation of the concentration jump. Neglecting the terms cr,_5 X ap/at, the tihomogeneous term Fd and the partial derivatives of ps and pl, rt sunphfies top ---1 a,?- c,c,

ar

at

ar -l--=0 ax

ac,

at

P +--co c,c,

a,=

(19)

ac,

1

asr

where r = psvs - pru,. From eqn. (19), It is possible to derive the following espresslon for the concentration jump : Ara, AC-=

-_

C,Cl

P

=

(P,

C(P,Cl

-

Pl)UOCs~l

+

PdPl

“2

(20)

These results agree with those of Ref. 6, escept for the added mass terms, which have been deliberately neglected here. Note, however, that JI, c, and cr are not uniquely defined at the jump: accordmgly, they should be taken to suitable average values between the values before and after the jump. A more rigorous treatment should consider the jump as a shock wave

CObIhIENTS

AND

these changes before. The first was purely a matter of mathematical convenience - we reduced the number of equations and made them linear A second reason for neglecting the changes m c, was that we were interested more in the long-term behaviour than in the mitral jump. The simplified form of the equatrons allowed a detailed study of the solutron for large times and drstances. Thus, we were able to show that a new speed a0 was more important than a, in the long-term behavior. This speed is given by

FURTHER

DEVELOPMENTS

The second goal of the present work is to investigate the long-term behaviour of pressure varration; in fact, espenmental measures showed that the pressure vanation across the wave front originating from a rapid flow shutdown occurs in two stages. After the initial and principal change described by eqn. (lS), a smaller but srgnificant secondary change occurs in a short transition penod; in fact, due to the momentum exchange between the phases, pressure increases until the phases are brought to rest_ The changes in c, were neglected in our previous treatment [5] ; they may be important, however, at least from a conceptual pomt of vrew. There were at least two reasons that led us to neglect

&

(21)

with p = p,c, + plcI, i e. the speed appropriate for a homogeneous mrxture of two flmds. This is the speed advocated by ThorIey and Hwang [ 4]_ Here we give a different derivation of this result, startmg from the more accurate system of eqns. (3), (5), (8) and (9), with

F, = c,c,(q-us)-

PH

(22)

PsPl

where H 1s a quantity depending in general on Iv, - u,l as well as c,, p,, pr. and havmg the drmensron of the inverse of a time r T 1s the characteristic tune for equilibration of the velocities of the two phases. Typically, we espect it to be of the order of 10m4 s. Accordingly, there should be transients on the time scale of r which decay fast if we look at the phenomenon on the scale of seconds. The behavlour after the transrent can be found by expanding in power of r the solution If we let us1

=

us.1

O+ TU,,ll

(23)

we easily find that UO-5

Ul

o=o

(24)

and hence, if we denote the common velocity at lowest order by u, we can rewrite the simplified basic system as follows:

ac P-~Lycap at

-s+~cap at

5

‘at 11x

au +ap=o ‘at ax

Acs_

au=(-J

ax

+yg

aLJ

=

0

(25)

so7 CONCLUDING

and hence

ap

au

CK-+-=o

ax

at

(26)

au

pat+%=0

i

i e. the equations appropnate for a homogeneous fluid. From eqn (26), the final pressure nse Ap_, can be easrly calculated according to the followmg-

APO.= V

%I, a!

- 1 -ac, +((Y,-acy1)at

that c, is not necessarily

ap =o at

le log c, 1 -c,

+ (a, -

Our present treatment of the pressure waves in slurry pipelines improves our treatment m Ref. 5, because we consider changes in concentrations Such changes, although small, are conceptually important because taking them into account indicates that Wood and Kao I:1,23 were correct in choosing the boundary condition of vanishing volumeaveraged velocity at the valve, when the changes in c, are neglected. REFERENCES

We remark, however, constant. In fact

c,~l

REMARKS

ar,)p = constant

The result expressed m eqn. (26) was previously obtained by means of an asymptotic study of the Lsplace-transform solution of a snnplified version of the basic system of equations (with H = constant) and confu-med by nmnencal calculations. The derivation given here, although less complete, applies to the more complete system and does not use the assumption N = constant

D. J Wood and T. Y_ Kao, Proc ASCE, J Eng Alechr Div. Vol 92, No. EM6 (1966) 117 D. J Wood and T. Y Kao, Advances in solidlrquld flow zn popes and its appkatzon, Pergamon Press, Oxford and Ntw York, 1971. pp 87 - 100 T. Y. Kao and D. J Wood, in Proc. Hydrotransport 5, Hanover. Brltlsh Hydromechanuzs Research As.oc , Bedford, 1978, Paper El, pp 1-14. A R D. Thorley and L Y. Hwang. Rot. Hydrotransport 6, Canterbury. British Hydromechanics Research Arson .Bedford. 1979.PaperDS.pp.229242 V. Battarra and C. Cercignam, hleccamco. 17 (1982) 67. W. Bechtelcr and G. Vogel, Proc. Hydrotmnsport 8. Johannesburg, British Hydromechamu Research Asscr , Bedford, 1982, Paper H2, pp 383 398 L M. Milne-Thompson, Theoretical Hydradynamics, McMlllan, New York, 1960 S. L Soo, Fluid dynamrcs of multiphase systems, Blalsdell Pubhshmg Co.. Waltham. MA, 1967.