Transient behaviour of particles in flow bubble column slurry reactors

Computersand Chemical Printed in Great Britain

Engineerit~g Vol. 10, No. 2, pp. 1 IS- 118, 1986

TRANSIENT

0098-1354/86 Pergamon

$3.00 + .OO Journals Ltd

BEHAVIOUR OF PARTICLES IN FLOW BUBBLE COLUMN SLURRY REACTORS

M. CHIDAMBARAM Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Bombay 400 016, India (Received

1984; revision received 8 February 1985; received for publication 21 May 19851

7 September

Abatraet-An analytical solution is derived for the unsteady-state sedimentation-diffusion model for the transient concentration of particles in flow bubble column slurry reactors. The time required to reach the steady-state distribution of the particles is evaluated from this solution. The effect of backmixing of the particles and the effect of particle diameter, particle density and liquid velocity on the transient behaviour of the particles are evaluated. Numerical evaluation shows that in cocurrent reactors the response becomes faster with increase. in particle backmixing, whereas in counter-current reactors the response becomes faster with decrease in particle backmixing. For identical operating conditions, the transient response of the concentration of the particles in counter-current reactors is compared with that in cocurrent reactors. The response is faster in counter-current reactors. INTRODUCl’ION

af

Bubble column slurry reactors (BCSRS) find extensive applications in chemical and petrochemical industries and in pollution control. The mixing condition of the particles is the most important variable that affects the performance of the BCSRs[l,2]. The concentration distribution of the particles is described adequately by the sedimentation-diffusion mode1[3-51. Cova[3] has given an approximate solution for the unsteady-state sedimentation-diffusion model. In the present work, exact analytical solution of the unsteady-state sedimentation model is derived for the transient concentration of the particles for a step change in feed solid concentration for both cocurrent and counter-current BCSRs. From this solution, the time required to reach the steady-state distribution of the particles is determined. The effect of backmixing of the particles and the effect of various process variables on the transient behaviour of the particles are evaluated. For identical operating conditions, the cocurrent and counter-current reactors are compared for the transient behaviour of the particles. MODEL

EQUATIONS

f = 0,

8 = U,tlL,

Up = UL - Us, = (U, + U,),

af iaZo

= - (Pe,f,

- Pepfo)

Pep = UpLID, f=

,

,

wlWf

for cocurrent

,

(44 .

for counter-current

(4b)

when the slurry and gas streams enter at the bottom of the reactor and to be in counter-current mode when the slurry enters at the top and the gas at the bottom of the reactor. The Z-direction is taken from the slurry entering point. That is, Z = 0 at the bottom of the reactor for current and 0 at the top for the countercurrent mode. Accordingly, the particle velocity in the reactor (U,) is considered along the Z-direction. The value of Up is positive whenever settling velocity of the particle (U,) is less than the liquid velocity (U,) in the reactor and negative otherwise [cf. eqn (4a)]. For the counter-current mode, Up is always positive along the Z-direction because Up = U, + Us. Taking the Laplace transformation of eqns (l)-(4) with step input in the feed solid concentration gives

1 d2F --_-_ Pep dZ2

(1)

’

(41

It should be noted that in writing the eqns cl)(4), the BCSR is considered to be in cocurrent mode

In BCSRs, the mixing characteristics of the particles for the unsteady-state condition are given by the sedimentation-diffusion mode1[3-51. Here the longitudinal movement of the suspended solid particles is considered to be caused by longitudinal dispersion and settling of the solid particles and by the flow of the liquid. The governing dimensionless equations are[3,5,6]

af

,

ate=0

Pe pL = ULLiDp,

AND ANALYTICAL

aZ

,

- Pep)

where

SOLUTION

1 a*f af _=---Pep aZ* ae

= - f,(Pe,

iaZ,

dF -

=-

dF dZ (Pe,/s

sF=O

-

,

Pe,FJ

,

- Pe,)

.

dz0

dF

,

dz,=115

F, (Pe,

(6)

116

M.

CHIDAMBARAM

NUMERICAL

Solution of eqn (5) is given by F = [PI cosh(mZ) + JIz sinh(

e”lZ ,

(8)

.

(9)

where m = 0.5Pep [1 + (4s/PeJ]“.s

P I and /I2 are obtained by use of the boundary conditions [eqns (6) and (7)] and are given by P, = Pep‘ d, 1 Mmd,

+ a,d,)l

P2 = - PepLddz/ [s(md, + aIdI)]

,

(10) .

EVALUATION

Table 1 gives the parameter values used for the present numerical evaluation. The dispersion coefficient Dp is calculated from the standard correlation of Kato et al. [S]: P,dr -

= (, +lTF:o.a)

. (17)

(1 + 0.009Re Fr-O.*)

DP

The settling velocity of the particles is estimated from the correlation of Kato et al. [5]: ,

us = 1.33u, (vo/uJ0.r5

(11)

(18)

where the particle terminal velocity (U,) is given by Stokes’ law[5]:

Now F = [d, cosh(mZ)-

d2 sinh(mZ)]

PepL e”lZ/ [s(md*+ uld,)]

.

U, =

(12)

(pp

-

pddh

/

(18

PL)

(19)

.

Inverting eqn (12) by the method of residues gives j7Z0)

= (exp[-0,0-Z)]

- PePLexPbA andatZ

(13)

+ u2 sinh[a,(l-Z)]jexp[-(1-Z)u,]

1

x exp[_(q2+ 025p& B,Pe ] 2qJqr c4(l-zkrl + u3 sin w-aI,lJ P ’ ' , -1 (43+ 0.25PeU ((qf + a,) cosq, + a@, sinq,)

= 1, 2?$ exp [-Cd + 0.25 Pe$) B/Pep] ?-r (4: + 0.25 Pea) {(qt + a,) cosq, + u5q, sinq,]

f(ldY) = 1 - U6 z

where q, are the roots of the equation QW, = q, PepL/(q$ + 0.25PeZp- 0.5PepPepL). (15)

The value of 8, is defined as the time necessary for the concentration of the solid particles at the exit of the reactor to reach 99% of the steady-state value. From eqn (14), we get

u6

z

1-1

(qf

+

The liquid velocity (U,) in the reactor is calculated from u, = V‘ /(I - e)

=

0.01

.

(16)

Figure 1 shows t, vs dp behaviour for cocurrent reactors with liquid velocity and particle density as

Table 1. Parameters used for numerical evaluation Reactor diameter, d + Reactor length, L: !Juperficial velocity of the gas, vq: Supeficial velocity of the liquid, vL: Density of the particle, pp: Viscosity of the liquid, p: Density of the liquid: Particle diameter, dp: Gas holdup, 8:

,

where l is the gas holdup. The volume of solid particles is assumed to be negligible when compared to the volume of the liquid. The particle velocity in the reactor (Up) is calculated from eqn (4a) or (4b).

2qf eXP[- (4’ + 0.25P&) 8, /Pep] 0.25~4) ((s” + a,) COST, + a3qq, sinq,]

Since the equation is implicit in 8,, the NewtonRaphson algorithm is used to get the value of 6,.

(14)

’

20 cm 2OOcm 7.5 cm/w 2.5 cmlsec, 4.0 anlsec 3.35 glcm 0.31 centi Poise 0.85 g/cm’ 10,20,30,40,50,60,70,80 0.25

pm

Behaviour of particles

in flow bubble column slurry reactors

117

2401

200f

2

160i

1

OOJ 0

I

I

20 Particle

I

60

I

60

diameter,d,-,

60 pm

diameter

key

kocurrent

0

20

LO

Particle

PP’ g/cm’

Fig. 3. The time required diameter (counter-current

6.70 3.35 3.35

as in Table

SOJ 120.‘-:::

versus particle

2.5 2.5 4.0

Other conditions

100

reactors):

cm/set

vL,

1 2 3

1

-

Fig. 1. The time required to attain steady-state

3

_

1.

parameters. The value of t, increases as particle density and diameter increase and liquid velocity decreases. This is due to decrease in the particle velocity in the reactor U, (because of increase in U, and U,) and dispersion coefficient. Figure 2 shows t, versus particle dispersion coefficient (D,) behaviour. As D, increases t, decreases. That is, the completely backmixed particle gives faster response. Figure 3 shows tyi versus particle diameter behaviour for counter-current reactors with liquid velocity and particle density as parameters. The value of t, decreases as particle density, particle diameter and liquid velocity increase. This is due to increase in particle velocity and decrease in dispersion coefficient. Figure 4 shows t, versus dispersion coefficient (0,). The value of t,, increases as Dp increases. That is, the response becomes faster with decrease in the mixing of the particles. It is important to note that the behaviour of the particles in counter-current reactors is significantly different from that in cocurrent reactors.

diameter,

60

)O

60 dp pm

+

to attain steady-state versus particle reactors). Key: same as in Fig. 1.

Comparing the values of t, in Fig. 2 and in Figs. 1 and 3, we find that the response of the concentration of the particles in counter-current is faster than that in cocurrent reactors. CONCLUSION An analytical solution is derived for the unsteadystate sedimentation-diffusion model for the transient concentration of particles in flow bubble column slurry reactors. From the solution, the time required to reach the steady-state distribution of the particles is evaluated. The effect of particle diameter, particle density, liquid velocity and effect of backmixing of the particles on the transient behaviour are evaluated. In cocurrent reactors the response is faster as particle diameter and particle density decrease and liquid velocity increases, whereas in counter-current reactors, the response is faster as particle diameter, particle density and liquid velocity increase. In cocurrent reactors the response is faster with increase in backmixing of the particles, whereas in counter-current reactors, it is faster with decrease in backmixing. For identical operating conditions the response in counter-current reactors is faster than that in cocurrent reactors.

l-200-

t t aoo-

i

i

C.

2

400-

lQ0

P

____________--------_ 0

110

0

I

I

200

100 Dp

( ml*/*.c

60

1

I

600

600

I

1000

-

Fig. 2. The time required to attain steady-state versus axial dispersion coefficient of the particle: solid line, cocurrent; dashed line, counter-current; dp = 60 pm, vL = 2.5 cm/ sec. Other conditions: as in Table 1.

Fig. 4. The time required to attain steady-state versus axial dispersion coefficient of the particle (counter-current reactors); d, = 60 pm, v, = 2.5 cmlscc. Other conditions as in Table 1.

hf. CHIDAMBARAM NOMENCLATURE OSPe, PepLIar Pe PL -aI 0.25Pe5 - Pep‘ (0.5Pe,

W

Wf

+ 1)

2 + Pe, PepL exp(a,) m co&(m) + a3 sinh(m) m sinh(m) + a) cash(m) diameter of the particles, cm reactor diameter, cm axial dispersion coefficient of the particles, cm* set’ diamensionless concentration of the particles, WI w, Laplace transformed f Froude number, ~,/(981d,)~~~ Length of the reactor, cm (1 < 4s/PeJ”.5 aI UPL lDP, Peclet number of the particles U, L/Dp roots of &fn (15) dp U, pP/p, particles Reynolds number Laplace variable time, set liquid velocity in the reactor, cm set-r net velocity of the particle in the reactor, cm SK-’ particle setting velocity, cm see-’ particle terminal velocity, cm se-’ superficial velocity of the gas, cm so-’ superficial velocity of the liquid, cm seC’

e 0,

&bscrims 0 .: ss

particles concentration in the reactor, g cm-’ particles concentration in the feed slurry, g crnm3 height from the slurry entering point, cm z/L, normalized height from the slurry entering point UP t/L, dimensionless time dimensionless time required to reach the 99% of the steady-state distribution of the particles. particle density, gm cm -a defined by eqns (10) and (111, respectively fractional gas holdup viscosity of liquid, gm cm-’ set-r

atZ=O atZ=l at feed point steady-state REFERENCES

1. V. M. H. Govindarao, Chem Engng J. 9, 229 (1975). 2. V. M. H. Govindarao Kt M. Chidamharam, Chem. Engng J. 27, 29 (1983). 3. D. R. Cova, Ind. Engng Chem. &UC Des Dev. St 20 (1966). 4. K. Imafirku, T. V. Wang, K. Koide & H. Ku&a, J. Chem. Engng Jpn 1, 153 (1968). 5. Y. Kate, A. Nishiwaki, T. Fukuda & S. Tanaka, J. Chem. Engng Jpn 5, 112 (1972). 6. M. Chidambaram, Ph.D. thesis, Indian Institute of !kiexe, Bangalore, India (1983).