The operation and modelling of a periodic, countercurrent, solid—liquid reactor

ChemicalEnRineering Science, 1973. Vol. 28. pp. 1233-1248. Pergamon Press. Printed inGreat Britain

The operation and modelling of a periodic, countercurrent, solid-liquid reactor

R. DODD&P P. I. HUDSON,S L. KERSHENBAUM and M. STREAT Department of Chemical Engineering and Chemical Technology, Imperial College

London S.W.l., England

(Firs? received 28June 1972; in reoisedform 2 Augusf 1972)

Abstract- Several models are proposed for the Cloete-Streat stage-wise solid-liquid reactor in which the solids are periodically transferred from stage to stage countercurrent to the net flow of liquid. The behaviour predicted by the models is compared with experimental data on an ion exchange reaction system. The results illustrate a range of operating conditions for which unsteady-state operation (with periodic solid transfer) is superior to continuous steady-state operation.

The various models differ in the treatment of the composition distribution of the ion exchange resin. They include (a) a discretisation of the distribution; (b) use of the continuous distribution function and solution of the resulting hyperbolic partial differential equations; and (c) approximation of the state of the resin by the leading moments of the distribution.

INTRODUCTION

IT HAS already been established, both theoretical: ly and experimentally, that unsteady-state operation of a process can be superior to conventional steady-state operation. Belter and Speaker[ 11 and McWhirter and Lloyd [ 161 have shown experimentally that controlled cycling in liquid-liquid extraction and distillation respectively leads to considerable improvements in efficiency. A review of the work carried out in several fields of chemical engineering was under- taken by Schrodt [2]. Little experimental work has been done on the unsteady-state operation of chemical reactors, but Douglas and Rippin [3], Douglas and Gaitonde [4] and Horn and Lin [5] have shown theoretically that the periodic operation of stirred-tank reactors can be superior to steady-state operation in some cases. Often, however, the improvement is small.

In recent years the ion-exchange industry has been searching for an effective continuous solid-liquid contactor which will overcome some of the disadvantages of conventional fixed-bed operation. Although a packed bed ion-exchange column is relatively simple to operate and control, since there is no movement of the solid particles, only a small fraction of the bed is doing useful work at any instant in time. As the

reaction zone moves through the bed, much of the column is either exhausted or unreacted and serves only to increase the pressure drop for flow. The regeneration step is inefficient in most cases due to unfavourable kinetics and this therefore tends to be wasteful of chemical reagents.

In an attempt to overcome these disadvantages, several continuous solid-liquid contactors have been developed. In the main, these com- prise adaptations of a moving fixed-bed column and the most notable are the Higgins contactor [61 and the Japanese Asahi contactor, a version of which is described by Bouchard[7]. Both these designs have had a measure of commercial success.

The development of fluidised ion exchange contactors was prompted by the application of resins to the recovery of metals from unclarified aqueous leach solutions. The earliest techniques were rather crude, involving the agitation of resins in contact with an ore suspension. Later more sophisticated concepts were considered, culminating in the development of several multistage fluidised bed contactors. Weiss [8], Grimmett and Brown [9], George et al. [IO] and many others have described equipment in pilot- plant and under full-scale development. Cloete

tPresent address: Unilever Research Laboratories, Port Sunlight, Cheshire, England. SPresent address: B.N.F.L., Windscale Works, Seascale, Cumberland, England.

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CBSVol.ZSNo.6-A

R. DODDS, P. I. HUDSON, L. KERSHENBAUM and M. STREAT

and Streat [ 1 l-131 have developed a fluidised bed contactor consisting of a cascade of horizon- tal stages in which resin transfer is achieved between stages by a periodic reversal of flow. A vertical column, operating by the same prin- ciple, is under commercial development in the field of water treatment and has been described by Stevenson [ 141.

The Cloete-Streat contactor is periodic in operation md, bearing in mind the recent work on unsteady-state processes, it may prove to be superior to a truly continuous contactor. Horn [ 151 has done a theoretical study of an analogous system: namely, a periodically operated distillation column in which a fraction of the liquid hold-up on each stage is periodically moved to the stage below. It was shown that a periodic steady-state (pseudo-steady-state) was attained for a periodic or constant input and that the improvement in performance over continuous operation was greater for higher fractional transfer of liquid, reaching a maximum when the total liquid hold-up on each stage is transferred to the stage below.

EXPERIMENTAL SYSTEM

The vertical Cloete-Streat contactor was developed specifically for ion-exchange processes and achieves semi-continuous counter-

current flow of particulate solids and liquid. The contactor described in this paper consists of a series of stages arranged vertically and separated by pairs of specially designed per- forated plates. For most of the time liquid flow is upwards, fluidising the particles on each stage. Periodically, resin particles are moved countercurrent to the net liquid flow. A schematic diagram of the experimental system is shown in Fig. 1.

The contactor was built of standard industrial glassware. Each stage comprised a glass pipe section 5 in. tall by 3 in. dia. containing specially designed distributor plates. The top plate contained twelve 3/16 in. holes on a square pitch giving 4.7 per cent free area. The bottom plate contained five 3/16 in. holes, which were located in the centre of the square pitch arrange- ment of holes in the top plate. The plates were separated by a spacer of sufficient thickness to give an off-set angle between the holes in the top and bottom plate of 12%‘. This angle of repose ensured that there was no drainage of particles under a no-flow condition.

The movement of a magnetic float switch between reed switches in a vertical tube was utilised for periodically reversing the direction of flow. On opening an air vent valve, the head of liquid in the contactor caused resin to be

I Liquid Feed Tank

-‘, “r”’ $npressed

Fig. 1. Experimental set up.

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transferred downwards between adjacent stages, whilst the float moved upwards in the oscillator tube. When the float reached the upper reed switch, a relay operated to close the vent and open the compressed air valve. This forced the magnetic float back to the lower reed switch, which actuated a further relay switch to close off the compressed air. This sequence could be repeated in each cycle to transfer the same volume of resin from stage to stage. The fractional transfer could be varied by adjusting the separation of the reed switches, thus altering the amplitude of the reversal stroke.

The resin feed was pumped continuously into the top stage of the contactor using the technique of “dense-phase” flow[27]. During the reversal of flow, the liquid feed was diverted through a recycle line back to the feed tank. All other controls were achieved by using a programmed cycle timer operating pneumatically actuated diaphragm valves. The column was capable of performing any required number of cycles.

Unless the cycle times are short, most of the cycle is occupied by the forward flow operation during which liquid is fed continuously to the bottom of the column and removed continuously from the top. Particulate solid is fed continuously to the top of the column and removed periodically from the bottom

Just before solid transfer, the volume of solid on all stages should be equal. With the same fractional solid transfer for each stage, equal volumes of solid are transferred between stages, solid from the bottom stage leaving the column. Therefore the volume of solid on every stage except the top stage remains unchanged. The hold-up volume of solid on the top stage is suddenly reduced upon solid transfer, but the continuous feed builds up the volume to the proper value before the next solid transfer. The top stage is therefore not a typical stage because its voidage continually varies.

The reaction studied was a simple ion- exchange (neutralisation) reaction

R-H+ + NaOH + R-Na+ + H,O

where R denotes the resin matrix. The resin

The operation of a solid-liquid reactor

used was a hydrogen-form strong acid cation exchange resin, Zeo Karb 225, 8% DVB-a sulphonated styrene-divinyl benzene copolymer.

During the runs, liquid samples were collected from the feed tank, and from sample points in each stage and in the effluent from the column. Samples were analysed for sodium ion concentration using a “Unicam” SP 70 Atomic Absorption Spectrophotometer. The instrument was calibrated in the range l-5 ppm sodium and samples were diluted to this concentration range if necessary, using distilled water. Resin samples were taken from the resin product tank and analysed for residual hydrogen ion concentration by passing sodium chloride solution through a 5 ml. sample and titrating the effluent against standard alkali. The sodium uptake on the resin was obtained by difference.

The neutralisation reaction is a convenient one to study since for dilute solutions (less than about 0.1 M) the reaction rate is predominantly liquid film controlled. The rate of reaction is thus given by

r = kLa(c-c*).

Helfferich [ 171 has stated that c* = 0 for this reaction provided the resin is not saturated with sodium ions. Therefore

r = kLac for x < 1

r=O for x = 1

where x is the resin conversion. It should be noted that these kinetics (zero-order in resin conversion) imply that the resin can become completely reacted in a finite time.

THEORY

During the reaction period the column is described completely by the voidage, liquid concentration and solid conversion distribution on each stage. Voidage changes are easily accounted for by overall mass balances if necessary. If the contents of each stage are perfectly mixed, changes in liquid concentration are described by an ordinary differential equation

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for each stage. The variation of the solid conversion distribution on each stage is governed by a partial differential equation.

Considering a simple ion-exchange reaction in which A ions are transferred from the liquid to the resin, the equations for any stagej (Fig. 2) are :

further assumed, as a first approximation, that the liquid phase is perfectly mixed on each stage and that the expanded fluidised bed fills the entire stage volume. Similar equations can be derived for any solid-fluid reaction with or without solid transfer.

$+(Cj-,-cj)-~~~~(x,t)r(x,cj)dx 1 3 (1)

m-5 t) 1 afjk 0 at +zr(x,cj) ax

Three approaches have been considered to solve the system equations: a discretisation of the composition distribution, solution of the actual system equations by the method of characteristics, and approximation of the state of the resin by the leading moments of its distribution.

Additional assumptions made in modelling this specific process are

These equations are derived in the Appendix, and are based on the assumption that when fluidised, the particles are perfectly mixed on each stage. This is a reasonable assumption so long as all particles are similar in size and density, so that segregation does not occur. It is

(1) No reaction takes place during the reverse flow segment of the cycle. This is equivalent to the assumption that this operation is instant- aneous and that the time for forward flow is equal to the cycle time.

(2) No liquid is transferred with the solid particles during the reversal of flow.

L,C, S

Stage 5

C c, S-l

St age s-l

cs-2 4

(3) The solid particles are of uniform size and density at all conversion levels.

c, 1 a Stage J

‘j-1 $

A necessary condition for pseudo-steady-state is that the volume of solid transferred from each stage to the stage below must be the same for all stages. If the fraction of solid which is transferred from each stage is to be a meaningful parameter of the system, it must be the same for all the stages. Therefore, the solid volume on all stages must be equal just before the solid transfer.

(l-Ej)Vj=COIlStZiIlt= (l-Es*)V&

c2

Stage 2

c1

8

Stage 1

LJCO 4

j=1,2,...s-1 (3)

where E,* is the voidage of the top stage just before solid transfer. When all the hold-up volumes Vj, V, are equal, then l j = E: and will be denoted as E.

1236

Fig. 2. Schematic representation of the contactor.

Furthermore, the continuous solids feed rate, S, to the top stage must satisfy an overall material balance for the column over the cycle to ensure no build-up or depletion of solid in the column.


Thus: (1 -~~)I’~/ld= ST. (4)

(a) Discrete model As an alternative to solving the partial differen-

tial Eq. (2), the conversion variable x can be discretised by dividing its range, 0- 1, into it equal intervals, with x,+~ = 1. A discrete probability density function is defined, the fraction of solid particles on stagej at conversion xi being denoted by pj(xi) with

n+1

iz &(Xj) = 1; j= 1,2,. . .s. (b) Use of method of characteristics

In order to characterise the movement of the Discretisation of the conversion distribution conversion distribution with time, it is con- led to numerical difficulties which are discussed venient to introduce an additional set of conver- later. Accordingly, Eqs. (1) and (2), a pair of sion variables yi, for i = 1, 2, . . . n + 1, which (generally) non-linear hyperbolic partial differen- vary continuously rather than discretely over a tial equations were solved directly by the method small time step. At a time to, the beginning of any of characteristics. This reduces the problem to time interval, the initial values of yi are taken as the solution of a pair of ordinary differential the corresponding discrete conversions xi. Then equations along two (characteristic) directions during a time interval all resin at any discrete in the x-t plane. Detailed descriptions of the conversion level will obey the differential method of characteristics are given by Courant equation and Hilbert [ 181 and Abbott [ 191.

It is shown in the Appendix that the direction of I-characteristics in the x - t plane is k-1

dt -Fr(yi, Cj); i=1,2,...n+l

where c is the capacity of the solid for reactant ions. Thus, the change of conditions on stage j from time to to time tO+At can be found by integrating the following n + 2 equations simultaneously:

dyi _ 1 ~-;r(yi,cj); i= 1,2,...n+l.

At time to + At each solid fraction pj (xii) has reached a corresponding conversion yi 2 xi (the conversion cannot decrease). A discrete probability function now exists with II + 1 frac- tions at conversion values yi which do not correspond, in general, to the original discrete

conversion levels, xi. These must then be trans- formed back into a new distribution at the original (n + 1) conversion levels. Thus, some material (i.e. that which has not undergone much reaction in this time interval) will remain at the same discrete conversion level, xi, while others will move to some other level xk, depending upon the amount of reaction occurring during that time interval at that conversion level. The movement of the conversion distribution on each stage can be determined in this manner without requiring the direct solution of Eq. (2).

dx &‘O

and in that direction,

2 = :(cj_l -cj) -2Rj (6) J I

where

Rj = Ji fj(x, t)r(x, cj) dr.

The direction of II-characteristics in the x-t plane is

and in that direction one must solve

&f,(X, t) =-*&t-(x, cj). 7 3

(8)

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In the case of a zero-order reaction, the method of characteristics is especially simple to apply. The rate of reaction is described by:

T(X, cj) = kLacj; x < 1

T(X, Cj) = 0; x= 1.

And hence

&[r(x, Cj)] = 0 for 0 S X S 1.

Equation (2), describing the time history of the conversion density function, reduces to:

a-++1,(x at c

c.) af.(x t) = 0 )‘axJ 7 (9)

a form of the Liouville equation from the field of statistical mechanics. Various applications of this equation are discussed by Kurth[20], Hulbert and Katz [2 l] and others.

Using Eqs. (5)-Q) the characteristic directions for stagej are:

dx for I-characteristics: ,z I = 0

dx for II-characteristics: z II _ kLaci c

and the following ordinary differential equations

C= I Character,st,cs

'0 01 02 03 OL 05 06 07 06 09 C0nWSlOn,X

Fig. 3. An example of a characteristics grid for a zero-order reaction. The initial part of the solution for the single-stage

reactor.

can be integrated along these characteristic directions, respectively,

dhk t) - 0


where Fj( l-) is the fraction of the resin on the jth stage which is not completely converted (saturated). A sketch of the x-t plane under these conditions is shown in Fig. 3

(c) Use ofmoments

For practical purposes, a unimodal density function is adequately described by its leading moments. Equations describing the movement in time of the leading moments of a size distribution during polymerization and particle growth have been developed by Katz[21, 221 and others. In systems where the distribution is, in fact, unimodal, this leads to considerable simplification of the system equations. From Eq. (2),

afib, t) 1 a at --1;~[_f_i(x9 t)r(x7Cj)l- (11)

The kth moment of the continuous density func- tionf,(x, t) is

k = 0, 1,2, . . .

since x lies in the range zero to unity. From Eqs. (ll)and(l2)

~=-~I’x~[~(x,t)r(x,cj)]dx. 0

Integrating by parts yields

* = -f [Xkfj(X, t)r(X* Cj)]l 0

+i I1 x"-tfj(X, t)r(X, Cj) dX* 0

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If any solid exists at x = 0 or x = 1, discon- tinuities in the density function occur at this point, and it cannot be unimodal. However, if f( 1, t) = f(0, t) = 0 the equation above reduces to

* = $ I l xy(x, t)r(x, Cj) dx. (13) 0

Now, r(x, cj) can usually be expressed as a poly- nomial in x, and the integrand can be expanded and replaced by moments.

For example, consider the second order rate expression T(X, cj) = A,( 1 -x)” where Aj is in general a function of the liquid concentration c,. No solid reaches complete conversion in a finite time in this case and the distribution is unimodal. Equation (13) becomes

and, using Eq. ( 12)

In this situation a moment truncating procedure such as the one developed by Hulbert and Katz [21] is necessary before these equations can be solved. This is because dpk/dt is dependent on the next higher IIIOIIIent pk+l, etc.

In the case of a zero-order reaction, solid can reach complete conversion in a finite time; but for all time such that f( 1, t) =f(O, t) = 0, (i.e. before any saturation occurs) Eq. (13) applies with

T(X, CJ) = kLacj = A,.

Thus, the system equations

dpi_& F kllk-13 k=1,2,...

(14)


are solved simultaneously for all time that f(1.t) =o.

(d) Special aspects of the models (1) Treatment of solid feed. The top stage

must be treated slightly differently from the others in that the feed is added continuously during the operation. It was convenient com- putationally to add the solid feed discretely at the end of each integration time step at the same average rate as the continuous feed rate. In other words, a volume SAt of solid is added to the top stage after each time step over the cycle. The shorter the time step, the better the approximation to continuous solid addition.

(2) Treatment of solid transfer. It has been assumed that the solid particles are perfectly mixed when fluidised. The particles are of uniform size on entering the column and as it has been assumed that no change in size or density takes place on reaction, they will all settle at the same rate when the forward flow of liquid is stopped. Therefore the conversion distribution of the transferred solid will be that of the fluidised solid at the end of the forward operation.

The new density function on any stage j after solid transfer can be obtained from a simple mass balance with any of the three computational methods. Since no solid is added to the top stage during transfer, the conversion distribution on this stage remains unchanged although the volume of solid decreases.

(3) Finding the pseudo-steady-state. The pseudo-steady-state can be found in two basic ways: (a) Relaxation method: starting with chosen initial conditions, the system equations are integrated forward in time until consecutive cycles are identical. (b) Boundary condition iteration: Horn and Lin [5] discuss this technique of matching boundary conditions of a cyclic process. If the state variables of the system are defined and the dynamic equations for these state variables are linearised, a Newton Raphson iteration scheme can be used. In this process state variables would be the voidage, liquid concentration and solid conversion density function on each stage.

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Each of these methods has disadvantages. The relaxation method may take a large number of cycles to reach pseudo-steady-state. The boundary condition interation would be faster, but only when convergence is assured. Because of these difficulties, a combination of these two methods usually proves the most successful. The relaxation method was chosen for this preliminary work. Although it may be less economical in computation time, there are no convergence problems, and it yields interesting information on the approach to steady-state.

The possibility of a multiplicity of stable or unstable steady-states for one set of inputs to the system must not be overlooked. For this reason initial conditions must be chosen with some care.

At pseudo-steady-state it is useful to calculate the average liquid concentration on each stage over the cycle, (cj,), defined by:

1 Cj, = -

r T 0

q(t) dt (15)

which will also serve as a performance criterion for the column.

(4) Estimation of physical parameters. The resin capacity (i;) was measured experimentally and an approximate value of a (external surface per unit volume) was calculated from the surface mean diameter of the particles, assuming they were spherical.

The mass transfer coefficient kL depends on the physical properties of the system and the liquid flowrate. A generalised correlation for fluidised beds relating kL to the Reynolds and Schmidt numbers and the liquid diffusivity has been suggested by Snowdon and Turner[23] and is discussed by Beek[241. The expression takes the following form:

k,d’ O-81 - = TRe’i2SCl13

D

where E, d’ and D are the voidage, particle diameter and liquid ditfusivity of sodium hydroxide respectively. The range of experimental conditions covered in this work are given in Table 1. The range of kL values predicted by the correlation for these conditions was from 6.3 x low5 to 8.4 X lo+’ m set-l.

RESULTS

The range of operating conditions for the experimental runs are summarised in Table 1. The principal operating parameters were liquid flowrate, voidage per stage, cycle time and the fractional transfer of resin.

Initial experiments were performed in order to establish the attainment of pseudo-steady-state operation. Samples were repeatedly taken at each tray in the column and in the solution outflow. These data are plotted in Fig. 4 for a typical experimental run. There is an initial

Table 1. Range of conditions used experimentally and in simulation

Parameter Experimental range [26] Value in simulation

Liquid feed concentration, c0 (kg.equiv./n+) Liquid feed rate, L (mYsec) Cycle time, T (set) Solid hold-up per tray, (1 - l )V, (n?) Fractional transfer of solid, d Number of stages Mass transfer coefficient,kl (m/set) Specific surface area of particles, (I (m-i) Surface mean diameter of particles (m) Resin capacity, F(kg.equiv./n+) Voidage, E Solid feed rate, S (m3/sec)

0~01-0~10 0.04 3 x 10-5-7 x 10-S 6 x IO+

30-60 30 65 x 10-6-200 x 1O-6 70 x 10-G

0.27-0.83 040 3-4 4

6.3 x IW-8.4 x 1W 6.3 x 1O-5 9250 9250 6.5 x lo+ 6.5 x 1O-4

1.469 1.469 - 0.7 0.7

- 2x 10-s 2 x 10-B

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* I- 800

O=Sample G (centrel l =Sample G (wall1

I

o=Sample H

V=Sample I

100 I I I I I I I I 0 8 16 2L 32 LO I.8 56

Cycle Number

Fig. 4. Concentration from run 13.

transient period after which cyclic steady-state is attained. Each sample point determination was taken at the same instant in the operating cycle. No attempt was made to observe the change of liquid concentration during the cycle.

Typical mass transfer results are reported in Table 2. A comparison of runs 10 and 14 shows the effect of increasing the fractional transfer of resin per cycle. There is a significant increase in performance when expressed as the ratio of solution concentration in the influent and effluent.

The effect of liquid flow-rate and solids hold-up can be seen by comparing run 4 with the others, and the effect of cycle time is evident in run 13. An increase in cycle time tends to increase the solids residence time per cycle and can lead to saturation of particles. This will lower performance, for a fixed flowrate, solid hold-up and fractional transfer (cf. runs 10 and 13).

Fourteen simulation test runs were made for the discrete model using various combinations of conversion division size and time step, and for the method of characteristics using varying numbers of characteristics. The initial conditions chosen were resin at various conversion levels on each stage and liquid concentrations on each stage equal to the feed concentration.

The following observations were made:

(1) A pseudo-steady-state was reached unless excessively large time steps led to instabilities in the integration procedure.

(2) The fourth-order Runge-Kutta integration procedure is of sufficient accuracy for the model for the operating conditions listed above.

(3) For the range of parameters studied, the choice of initial conditions was found to have no effect on the pseudo-steady-state attained.

For the typical conditions listed in Table 1, the concentration profiles for each stage over the cycle are shown in Fig. 5. The conversion distribution function at both the beginning and end of the reaction time is shown in Fig. 6. Little reaction is seen to occur on the bottom stage in this case because of resin saturation. The results were similar for both the discrete method and the method of characteristics, the latter being

Table 2. A comparison of theoretical and experimental results for a four-stage contactor

Run No. L T CO

(mYsec) (set) (kg.equiv./n?)

Predicted Total Resin k Voidages Hold-up colc*r COlC,T

(mlsec) (-) W) By exp’t. From model

4 5.08 x 1O-5 30 0.10 0.83 6.8 x 10-S 0.76 3.80 x lo+ 3.5 4.8 10 6.65 x 10-s 30 0.04 0.61 7.4 x 10-S 0.62 6.14 x 1O-4 10.3 15.4 13 6.62~ 1W 40 0.04 0.59 7.6 x 1O-5 0.62 6.11 x 1O-4 6.7 11.6 14 6.55 x 1O-5 30 0.04 0.43 7.5 x 10-S 066 5.52 x 10-a 6.4 7.7

c,r = exit liquid concentration at the end of a pseudo-steady-state cycle.

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-3 Liquid concentration unbts -kg-equlv m Time units -s Liquid feed concentratjon z 0 OL kg-equlv me3

Fig. 5. Liquid concentration changes at pseudwteady-state.

generally more reliable and requiring less computing time.

From Fig. 5 it is seen that the liquid concentration on all except the top stage falls and passes through a minimum before rising to equal the initial concentration again at the end of the cycle time. For each of these stages, solid of a lower average conversion is added on solid transfer. In this zero-order case, the fraction of solid which has completely reacted is the important factor. On solid transfer this fraction is decreased and the “available” surface area is increased, so that more reaction takes place. Over the cycle, as more solid becomes completely converted, the concentration rises again.

On the top stage, no change in the conversion distribution takes place on solid transfer. The

IO STAGE 3

08

02 Okx 06 06 10

06 FIxI

OL


0 Dlstrlbutlon function before reactlo” 0 Distrlbutlon function after reactaon

Fig. 6. Movement of conversion distribution functions F(x), during a cycle at pseudo-steady-state.

only effect of solid transfer is to increase the voidage and decrease the “available” surface area. Less reaction therefore occurs and the concentration rises. Addition of fresh resin during the cycle causes the concentration to fall again.

The effects of variations in operating conditions It must be emphasised at this point that the

system has been tested for one type of kinetics under a limited range of operating conditions only. The general trends indicated by the results can be accepted with reasonable confidence, however.

The column is represented by a set of initial conditions plus inputs, outputs and controls as follows:

Inputs: solid feed rate, S; liquid feed rate, L;

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inlet solid conversion (zero in these runs); inlet liquid concentration, co.

Outputs: exit solid conversion, asfi (x, t); exit liquid concentration, c,.

Controls: fractional solid transfer, d; voidages, Ej; cycle time, T.

For any feed conditions (inputs), one would like to adjust the controls to minimise the average exit liquid concentration over the cycle, c sm, as defined by Eq. (15). This is equivalent to maximisation of the mean exit solid conversion at pseudo-steady-state.

If the hold-up volumes in all stages are equal, then from Eqs. (3) and (4), the three controls, d, E and T are related by the solids mass balance

ST= (l-•)l’d. (16)

Now the average solid and liquid residence times in a column of s stages, Fs and FL are:

7 s

= solidhold-up =sV(l--E) =sT solid feed rate s d

(17) liquid hold-up SVE

” = liquid feed rate = r ’

For a series of runs, the three control parameters can be varied in three patterns by keeping one of the parameters constant and allowing the

other two to vary so that they satisfy Eq. (16). The most interesting and important case is in

the variation of T and d while keeping E constant. From Eq. (16) it is seen that for constant l :

z= (l--E)V=conSfant d S

and therefore from Eqs. (17), the solid and liquid residence times remain constant while T and d are varied in this manner. The effects of varying T and d on the liquid conversion are shown in Fig. 7. Increasing d and T decreases the exit liquid concentration (increasing the liquid conversion). A small but definite advantages results from increasing the fractional transfer.

As T and d become small, cycle times become very short with small amounts of solid being transferred at the end of each cycle. If d and T are taken sufficiently small, the system could be considered equivalent to a steady-state column with continuous liquid flow and continuous solids overflow. Such a column has been investigated experimentally by Turner and Church

WI. From Fig. 7, the conversion is increased from

0.502 to 0.538, about 7 per cent of the conversion value, by operating the column cyclically with a fractional solid transfer of unity instead of operating it in a “continuous” manner. The

No of stages=2

E J

5 0.530 - :

E 0.520 - o

5 o- -0.019 .z (

Z -m 0 510-

=

2 m

al z

40 L90 I I I I I I I I I 0018 2 0 0.1 02 0.3 OL 05 06 07 08 09 10

FractIonal solld transfer, d

Fig. 7. The effect of varying cycle time and fractional solid transfer on the average exit liquid concentration.

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performance with physically realisable values of d somewhat less than one are shown in Fig. 7 as well. As was shown experimentally, there are other feed conditions for which cycling leads to even more significant improvement over steady- state operation.

Results have also been obtained for the variation of controls while keeping either T or d constant. However, since the solid and liquid residence times are variable under these conditions, the results are difficult to interpret and are not included here.

The experimental results [26] can be compared with the results predicted by the model under the same conditions at pseudo-steady-state. Mass transfer coefficients predicted by the correlation of Snowdon and Tumer[23] are used. From Table 2, the model is seen to predict the same trends as shown by the experimental results. Exact duplication of the results cannot be expected because of the assumption in the model of perfect liquid mixing on each stage.

DISCUSSION

In all of the work that has been described, the basic assumption has been made that the liquid phase is perfectly mixed on each stage. In practice the movement of the liquid phase is complex, lying between the two extremes of perfectly mixed and plug flow. A plug flow model has been solved for one stage using the method of characteristics to solve the partial differential equations in space and time, a mean conversion being used to represent the solid conversion distribution. It was decided that the assumption of perfect mixing would enable more to be discovered about the process in this preliminary work, which is a basis for developing a more exact model. Eventually, describing the liquid flow by axial dispersion or a series of tanks may be necessary for an adequate description of the contactor.

The model with the discretised conversion distribution discussed above is a simple but rather crude hrst attempt at a mathematical description of the system. Using the discrete model, it was difficult to find a combination of

time step and discretised conversion step which led to consistently rapid convergence to a pseudo-steady-state. This was, in fact, related to an inherent numerical difficulty in that method. As the time step is decreased one gets a more accurate approximation to the time derivative, and hence to the true solution. However, as the step size decreases, the updating procedure for the conversion distribution fails. The updating depends on the conversion yi at the end of a time interval being in the vicinity of a discrete level xk different from the conversion xi at the beginning of the time interval. As the time step becomes small xk becomes equal to Xi and the conversion distribution therefore fails to update. This method is generally inferior to the use of the method of characteristics where one can always be certain of increased accuracy upon increasing the number of characteristics (de- creasing the step size).

The sensitivity of the method of characteristics to the number of characteristics was investigated for a typical set of conditions given in Table 1, but with a fractional transfer d = 0.8. The initial conditions were completely unreacted solid and liquid of concentration 0.04 kg-equiv. me3 on each stage.

The liquid concentrations for each stage at the end of a cycle at pseudo-steady-state are given in Table 3, for the characteristics model with 40, 60, 70, 200 and 300 characteristics. The top stage concentration is plotted against n on Fig. 8 and is seen to asymptotically approach a concentration close to 0.00528 kg-equiv.me3 (the true steady-state as calculated by overall mass balance) as n is increased.

Table 3. Concentrations at the end of pseudo-steady-state cycles for increasing numbers of characteristics

n c1 cz c3

40 0.026970 0.016211 0.014940 60 0.024915 0.014764 ow9741 70 0.024793 0~014700 O+lO8830

200 0.024569 0.014571 o%u%40 300 0.024564 0.014568 OW8639

Concentration units - kg-equiv.m-3. n = number of characteristics.

c4

0.014180 O+lO8832 0037320 o+lO5377 0.005285


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0 015

t

co: O.OLkg-eqw W3

0,OlL

K e

0 0055 LO 100 200 300

Number of character1stlcs.n

Fig. 8. Exit liquid concentration at the end of pseudo-steady- state cycle for increasing numbers of characteristics (n)-

multistage model.

The discrete model in this case gives an average exit liquid concentration of OGl6230 kg-equiv.mm3 and a concentration at the end of the cycle of OGl5364 kg-equiv.me3 at pseudo- steady-state. Although agreement between the two methods of computation was quite good in this case, the results for the discrete model were

strongly dependent upon the time interval chosen.

Although only the zero-order case was solved by characteristics, the same technique is easily applied to any kinetics. The calculation procedure will be marginally more complicated because the family of II-characteristics will not all be parallel at any given time and the density function will not be constant along them. However, Eq. (7) shows that the family of I-characteristics will still be parallel to the time axis.

The moments of the composition distribution were computed from the results of the other methods of solution and compared with those obtained by direct integration of the moment equations, as shown in Table 4. As long as no saturated resin is found, the results of the two methods agree quite closely. Clearly, saturation will not occur as long as the rate expression is of the form

r=A(B-x)” wherem 5 1.

Of all the models, the one using moments is certainly the best in situations where it can be used. For practical purposes the exit liquid concentration and the mean and variance of the exit solid conversion distribution are usually sufficient to describe the process. Only a few ordinary differential equations are then required for each stage, resulting in a substantial reduction

Table 4. Testing of moment equations for a zero-order reaction

Time Liquid concentration (set) (kg-equiv.m-3) CL1 PT lJ.2 & 1*3 &

0 0~1OOOOO 0.046667 0.046667 0.002503 0.002503 oGOO147 0@00147 4 0.065328 0.163333 0.162641 0.027003 0.026725 oGO45 17 0.004460 8 0.061545 0.256667 0.257564 0.066203 0.066662 0.017158 0.017337

12 0.061133 0.35OOoO 0.350191 0.122825 0.122956 0.043216 0.043285 16 0.061088 0443373 0442566 0.196869 0.196188 0.087567 0.087112 20 0.061083 0.536667 0.534915 0.288336 0.286457 0.155089 0.153575 24 0.061082 0.630000 0.627260 0.397225 0.393719 0.250661 0.247407 28 0.061082 0.723333 0.719605 0523536 0.518155 0.379161 0.373332 32 0.061082 0.816667 0.811950 0667269 0.659587 0.545468 0.540077

pk = kth moment determined by discrete model. & = kth moment determined by moment equations. /4l= 1.

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in computational time. However, this approach is satisfactory only when the density function is unimodal.

CONCLUSIONS

Three methods have been proposed to describe the vertical Cloete-Streat contactor: the most efficient one would depend on the form of the kinetics of the ion-exchange reaction. The accuracy of the method of characteristics can be confidently increased by using a finer characteristics grid and less computational time is required than for the discrete model for similar accuracy.

The approach using the leading moments to describe the continuous conversion density function can be used only for solid-liquid reaction kinetics for which the solid cannot become completely converted in a finite time. This approach gives a simple set of ordinary differential equations to solve and looks promising for future work, especially in situations in which intraparticle pore diffusion controls the overall kinetics.

A

a

Cjm

c

d

d’ D

NOTATION

proportionality factor in rate expression, kg-equiv.m-3sec-1

external surface area per unit solid volume, m-l

liquid concentration, kg-equiv.mm3 liquid feed concentration, kg-equiv.me3 equilibrium liquid concentration at

solid-liquid interface, kg-equiv.me3 mean liquid concentration leaving stage

j during cycle, kg-equiv.mP3 resin capacity, kg-equiv.me3 fraction of solid hold-up transferred per

cycle particle diameter, m diffusivity of the liquid, m2sec-’

.fk t> F(x)

k kr,

L n

PC%)

dx, c)

R

Re

; SC

t

T

x9 Y V

continuous conversion density function cumulative conversion distribution

function moment number liquid-film mass transfer coefficient,

m set-’ liquid volumetric feed rate, m3sec-’ number of conversion divisions or

characteristics discrete conversion density function at

conversion Xi

reaction rate per unit solid volume, kg- equiv.m-3sec-1

reaction rate per unit volume of a solid mixture of different conversions, kg- equiv.m-3sec-1

particle Reynolds number number of stages in the column solid volumetric feed rate, m3sec-’ Schmidt number for the liquid time, set cycle time, set conversion of solid stage bulk hold-up volume, m3

Greek symbols E voidage (fractional liquid hold-up) per

stage E,* voidage of top stage just before solid

transfer r liquid residence time per stage, set

FL average liquid residence time in the column, set

Ts average solid residence time in the column, set

,..&k kth mOIYM?ntOf density fUnCtiOn

Subscripts i, k discrete values of resin conversion

j the general stagej

REFERENCES

[l] BELTER P. A. and SPEAKERS. M., Ind. Engng Chem. Proc. Des. Dev. 1967 6 36. [2] SCHRODT V. N., Znd. Engng Chem. 1967 59661. [3] DOUGLAS J. M. and RIPPIN D. W. T., Chem. Engng Sci. 1966 21305. WI DOUGLAS J. M. and GAITONDE N. Y., Znd. Engng Chem. Proc. Des. Den 1967 6 265. [S] HORN F. J. M. and LIN C., Znd. Engng Chem. Proc. Des. Deu. 1967 6 21.

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HIGGINS I. R., Ind. Engng Chem. 196153 635. BOUCHARD J., Proc. international Conference on Ton-Exchange in the Process Industries S.C.I. 91 1969. WEISS D. E. and SWINTON E. A., Aust. J. Appl. Sci. 1953 4 3 16. GRIMMETT E. S. and BROWN B. P., Ind. Engng Chem. 1962 54 24.

[61 171 181 191

[lOI t111 [121 [I31

- -

1141 t151 [I61 t171 [ISI [I91 PO1 [211 WI [231

[241

GEORGE D. R., ROSS J. R. and PRATER J. D., Mining Engng 1968 73 1. CLOETE F. L. D. and STREAT M.. Nature 1963 200 1199. CLOETE F. L. D., STREAT M. and MILLER A. I., A.1.Ch.E. Instn Chem. Engng Symp. Ser. 1965 154. BENNETT B. A., CLOETE F. L. D. and STREAT M., Proc. International Conference on lon-Exchange in the Process Industries. S.C.I. 173 1969. STEVENSON D. G., Proc. International Conference on Ion-Exchange in the Process Industries. S.C.I. 114 1969. HORN F. J. M., Ind. Engng Chem. Proc. Des. Deu. 1967 6 30. McWHIRTER J. R. and LLOYD W. A., Chem. Engng Progr. 1963 54 58. HELFFERICH F.,J. phys. Chem. 1965 69 1178. COURANT R. and HILBERT D., Methods in Mathematical Physics, Vol. 2. Interscience, New York 1961. ABBOTT M. B., An Introduction to the Method of Characteristics. Thatnes and Hudson, London 1966. KURTH R. (Ed.), Axiomatics of Classical Statistical Mechanics. Pergamon Press, London 1960. HULBERT H. M. and KATZ S., Chem. Engng Sci. 1964 19 555. KATZ S. and SAIDEL G. M., A.1.Ch.E. Jll967 13 3 19. SNOWDON C. B. and TURNER J. C. R., International Symposium on Fluidisation. Eindhoven, The Netherlands University Press 1967. BEEK W. J., Fluidization (Eds. J. F. DAVIDSON and D. HARRISON), Chapt. 9. Academic Press, London and New York 197 1.

r251 TURNER J. C. R. and CHURCH M. R., Trans. lnstn Chem. Engrs 1963 41283.

t261 HUDSON P. I., MSc. Thesis, University of London 1970. t271 BENNETT B. A., CLOETE F. L. D., MILLER A. I. and STREAT M., Chem. Engr CE412, No. 233, Nov. 1969.

APPENDIX I I I Derivation of Eqs. (1) and (2)

A mass balance on hydroxide ion in the liquid phase gives for any stage j,

where Rj is the rate of reaction per unit volume of solid and i is given by

.

Rj = I,’ fr(x, t)r(x, cj) dx.

y

I I Solving for dc,flt, yields Eq. (l), 1 I

X x+Ax x

2 = ; (cj+ - c,) -y R, (A-‘) tion function and the r.h.s. of (A-3) = d/dt[F(x+ Ax) -F(x)].

Expanding F(x+ Ax) in a Taylor’s series yields for small where 7, = l ,cI#.. Ax

A mass balance‘on ions in the solid phase yields for any conversion x,

(1 --q)V,r(x, c~) =$[(l -e,)Fjxc]

&(x+A+-FW =$$x=$rf(x,tMxl.

Substituting in (A-3) and dividing by Ax gives or

$=$(x, Cj). (A-2)

C

dx f(x+ k t) z

( > I+~ -“KG t)(Z),

-

Equation (2) is derived by considering the movement of the Ax 1 =+x5 t)l.

conversion distribution with time. Considering the fraction of Taking the limit as Ax + 0 the total number of particles having the conversion between x and x + Ax, a mass balance gives

f(x. t)($)Z-f(x+hr, t)(g),+, =$ _/r”.fx~, t) dz.

$(x7 t) +&[f(x, t@] = 0

or

(A-3) But, f(z) = (d/dz)F(z) where F(z) is the cumulative distribu-

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R. DODDS, P. 1. HUDSON, L. KERSHENBAUM and M. STREAT

And substitution of Eq. (A-2) yields (A-5) which is Eq. (2) The characteristic directions are given by in the text,

d.~ QI dr Qz a.m. t) 1 at+;r(x,c)yy-- ‘(” t, - -;f(x t)%(x, c) (A-5)

;T;,=p, a”d Tt,,=p, ‘ax . and along those directions one must solve

Characreristics ofEqs. (1) and (2) Equations (1) and (2) can be written

2 = $ (+ -c,) -?Rj I 1

and

_ =% and df =!!? dc dt I P, dx II p*.

Thus, for this case, the sets of characteristic directions are given by

dx dX 1 ;ii,=O and z,,=zr.

Along the first set, one must integrate

That is, in the form dc 1 =‘(c,_~-c~)-~R~ dr 1 Tj %

p %+Q and along the second set

’ at !%=,

‘ax 1

and 3 =___=- dr,r 1 rax’

rr

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