a study and modelling of liquid-phase mixing in a flotation...

International Journal of Mineral Processing, 26 (1989) 1-16 1 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

A Study and Modelling of Liquid-Phase Mixing in a Flotation Column

P. MAVROS, N.K. LAZARIDIS and K.A. MATIS

Laboratory of General and Inorganic Chemical Technology, Department of Chemistry, University of Thessaloniki, GR-54006 Thessaloniki (Greece)

(Received March 9, 1988; accepted after revision July 4, 1988)

ABSTRACT

Mavros, P., Lazaridis, N.K. and Matis, KA., 1989. A study and modelling of liquid-phase mixing in a flotation column. Int. J. Miner. Process., 26: 1-16.

The mixing process of the liquid phase in a gas-liquid flotation column and the effect of the flowrates on it have been investigated. The increase of the gas and liquid flowrates had opposite effects: the former enhanced the mixing process, whereas the latter caused the behaviour of the column to approach plug flow.

A mixed zones-in-series model is proposed for interpreting the experimental results, which in- volves two parameters: the number of zones and the ratio of backwards to net liquid flow. In this way, the mixing was related to the backflow and the increase in mixing was due to an intense internal recirculation. The model was able to predict accurately the column behaviour in the experimental range investigated. Its major advantage lies in its structured description of the column contents.

INTRODUCTION

Bubble columns, commonly used as reactors and/or absorbers in the chemical industry, are finding lately new applications in the mineral processing and the effluent treatment plants as flotation columns. In the flotation process, as it is known, a liquid-solid slurry is fed into an aerated vessel; the solid particles collide with the bubbles and, if adhesion occurs, are eventually transported to the top of the vessel - given that no detachment occurs. There, a froth rich in these particles is generally formed and removed, whereas the carrying liquid and any remaining solids {e.g. gangue) are removed, usually from the bottom of the vessel.

Compared to the more widely used mechanically agitated cells, flotation columns may offer some advantages: the longer contact times achieved between gas and the solid-liquid slurry, the lack of mechanical agitation which may achieve savings in terms of energy required (Ek, 1986; Hyde and Stojsic, 1987)

0301-7516/89/$03.50 © 1989 Elsevier Science Publishers B.V.

and reduction of the maintainance costs, the countercurrent flow pattern of- fering better at tachment probability, etc. The latter would also be advanta- geous, as compared to the concurrent mode of operation, due to a highly efficient "filtration" effect and the provision of a quiescent exit - which is most important in effluent treatment, where a total removal of fine particulate matter is required. In particular it is also hoped that it may be more efficient in the treatment of fines (Somasundaran, 1986).

The design of a flotation column requires both experimental data about the physical system to be treated in it and a sound theoretical model describing the flow, mixing and mass transfer phenomena occurring in the column. Con- siderable work has already been done related to the chemistry and operation of flotation columns (see for example, Matis, 1980, 1982, Dobby and Finch, 1985, 1986; Stalidis et al., 1988).

The present work investigates the mixing in a flotation column through the determination of the residence time distribution (RTD) of the liquid phase, since it is known that the efficiency of the flotation process is largely a function of the hydrodynamic characteristics of the system. This is most important for the flotation process, since the state of mixing affects the overall process.

The interpretation of the experimental results is done through a model describing the mixing process in the column; the simplest models for this are either the plug flow model or the completely stirred tank reactors (CSTR) in the series model.

In the first case, mixing is regarded as a deviation from ideal plug flow and a single parameter, the axial dispersion coefficient Da, characterises the degree of mixing in the column. The model is described by an unsteady-state material balance equation written for the tracer:

OC uL OC _ 02C Ot ÷ 1 - ~ ~ x = D a ' o x '2 (1)

where C is the concentration of some liquid species, UL is the superficial liquid velocity, t is the time, x is the distance along the column and e is the gas holdup, due to the gas flow in the column (see also the Notation). The value of Da is determined by comparison of the experimentally obtained RTD curve with the model prediction. Numerous studies have been conducted on various parameters such as the viscosity, the surface tension, the presence of electrolytes, the geometrical features of the column, etc. and excellent reviews exist: see for example Joshi (1980), Shah et al. (1982) and Pandit and Joshi (1983). The model has even been extended so as to obtain both the axial and radial dispersion coefficients. D, and Dr (see Deckwer and Schumpe, 1987).

On the other hand, the column may be regarded as either one or a series of CSTRs, i.e. well-mixed zones. In this case, the RTD curve is given by the fol- lowing well known equation:

E = N" (N'O)N-1 e -N° (2) ( N - I ) ]

where E is the dimensionless RTD, N is the number of well-mixed zones and 0 the dimensionless time: 0= t /r for a mean residence time z= V/QL, where V is the volume of the liquid in the column. Since this model allows only for integer values of N, it has been generalised by Buffham and Gibilaro (1968) so that N can take real values as well:

N.(N.O) N-I E - e-N° (3)

F(N)

where F(N) is the gamma function.

EXPERIMENTAL DETAILS

The experimental apparatus presented in Fig. 1 consisted off a column and its flow accessories, a conductivity cell and corresponding meter, and a recorder.

The column was acrylic-made, approximately 1.00 m high and with an internal diameter of 8.0 cm. It was operated in a countercurrent fashion. The gas-free liquid height was kept constant at 86 cm, corresponding to a liquid volume of 4395 cm 3.

Tap water was fed at the top of the column, at a liquid depth of 4 cm and withdrawn from the bottom of the column, assisted by a peristaltic pump. In

Vl A! -'~ ~

- ~CM CR

Fig. 1. Schematic diagram of the flotation column set-up: A =flotation column; B = g ~ spm~ger; (?=overhead constant level holding tank; CC--conduetivity cell; CM=eonduetivity meter; CR = chart reeorder; D = thermostatic bath; P = peristaltic pump; R = rot~metem; V= valves.

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order to assure an even and constant- temperature liquid, this was fed first through a thermostatic bath and then into an overhead constant-level holding tank.

The gas was sparged at the bot tom of the column through a sintered-glass disk (Schott, G4), with pores of 10-16 #m (mean diameter). Both gas and liquid flowrates were measured with calibrated flowmeters.

A flow-through conductivity-metering cell, with a constant k equal to 1.05 cm - 1 was connected to the liquid exit. The connecting tube was kept as short as possible, so as to avoid any effects of the tube volume on the RTD measure- ment. The cell was connected to a conductivity meter (Metrohm) and the signal from it was recorded on a strip-chart recorder (Hewlett Packard).

A carefully measured amount of solution of 4 M KC1 was injected into the liquid stream entering the flotation column, about 1 cm away from the entering point and the recorder was simultaneously started. The time required for the injection was very short, as compared with the mean residence time, so that the injected tracer could be considered as a perfect pulse. The conductivity of the outgoing liquid was recorded for a time period equivalent to approximately three mean residence times and from the obtained trace of conductivity versus time, the residence time distribution was computed.

R E S U L T S A N D D I S C U S S I O N

Experiments were performed in the flotation column for various values of the gas and liquid flowrates (see Table 1). The values of the liquid flowrate were chosen so as to obtain a mean residence time, r, ranging from a few min- utes to the 15-20 min typical of the flotation cells. The gas flowrates, on the other hand, were chosen so as to obtain fine (and possibly non-coalescing) bubbles, as usually existing in flotation vessels.

The RTD curve obtained for the lowest liquid flowrate QL = 200 cm a min - 1,

T A B L E I

E x p e r i m e n t a l va lues of l iquid a n d gas f lowrates

U * 1 * 2 ' Q~, Q~ l, u(; r *~ ( cm :~ r a i n - J ) ( cm :~ m i n - ~ ) ( cm m i n - ~ ) ( c m m i n - 1 ) ( m i n )

0 0 200 100 4.0 2.0 22 400 300 8,0 6.0 11 600 500 11.9 9.9 7.3 850 824 16.9 16.4 5.2

*lu~, = QL/S, where S: c ross - sec t iona l a rea o f c o l u m n = 50.3 cm 2.

*'~u,:; = Q c , / S . *:~r= V/QI, V = 4 3 9 5 c m 3.

5

in the absence of gas, showed that the tracer moved through the flotation column in a more or less plug flow fashion marked by some spreading of the tracer pulse, as shown by curve a in Fig. 2 (where each curve is the average of three replications).

When the gas was introduced, however, the mixing effect was immediately obvious: the turbulence caused by the interaction between the two countercurrent flows mixed the contents of the column and its behaviour soon approached that of a well-mixed vessel.

The increase of gas flowrate affected the mixing process up to a certain limit; for example, at a liquid flowrate of 200 cm 3 min - 1 the effect of the gas flowrate was noticeable only for Qc= 100 and 300 cm a min-1; at the latter, the mixing was complete and the introduction of gas at a higher rate had no additional effect.

The same pat tern of behaviour was also displayed at the higher liquid flowrates: Figs. 3, 4 and 5 show the RTD for QL = 400, 600 and 850 cm 3 min-1, respectively.

From these figures it is interesting to note that the behaviour of the column, as seen by its RTD, depended also upon the relative magnitude of the liquid and gas flowrates. For instance, at a liquid flowrate of 400 cm ~ min -1, the "maximum mixing" observed for QG----300 cm 3 min-1 started deviating from the 1-CSTR RTD curve (Fig. 3 ), whereas this deviation was most obvious for the highest liquid flowrate of 850 cm a min-1 (Fig. 5 ).

Thus, it is obvious that the simple, one-parameter models could not account for the complex flow features usually encountered in a bubble column, due

1210 t_ !i I L r 200 = ]

T x...~. .,ja

02 I i " ~--:.~.~.:~ ,_. \ " " ~ , ~ . . ~ , ._

0 05 1.0 1.5 2.0 2.5 e [-}

Fig. 2. Experimental RTD (E) vs dimensionless time (0) for QL=200 cm 3 min -1. QG: a=0 cm 3 min-1; b= 100 cm a rain-l; c=300 cm a rain-l; . . . . . 1 CSTR.

11! I ,~-~'--., i ~ i F i : ,~00 cm>, n~'r '

1 o k ~ ,. . . . . .

ti,%, ",,

'o.~UJl ~ ) . , ',

o~.1tI I " ' . . " . 4 .

Ilrl 0 2

0 0 5 10 1 5 2 0 2 5

e [ ]

Fig. 3. E x p e r i m e n t a l R T D for QL = 400 c m '~ m i n 1. QG: a = 0 c m 3 m i n - ~; b = 100 c m 3 r a i n - ~; c = 300 c m 3 m i n - 1; . . . . 1 C S T R .

f " ' "~ ' - ' - - -" '~, QL = 600 cma/min 1 0 ,,"~ ' ~, J I

~x ~ - - , .J" II I '

0 8 ~'t

o2w, ' I " ' - " ~ " L~ I .... - - " ~ ~ _ ~ 0 I I I " ' ,--, ..._._,._.__._,_.~'

0 5 10 15 2 0 2 O [ ]

Fig. 4. E x p e r i m e n t a l R ' [ 'D for QL = 600 cm :~ r a i n - ~. QG: a = 0 cm ~ r a i n - ~; b = t 0 0 cm ~ r a i n - ~; c = 300 c m 3 m i n - 1 ; . . . . . 1 C S T R .

mainly to the interaction of the gas and liquid phases as demonstrated by Figs. 2-5. The introduction of the gas at the bottom of the column establishes a circulation pattern in it, with the liquid entrained upwards by the gas in the central core of the column and flowing downwards near the column walls (Freedman and Davidson, 1969; Molerus and Kurtin, 1986 ). This flow pattern results in a radial non-uniformity of liquid velocity (B.P. Pablov, 1965, and H. Yoshitome, 1967, cited in Ohki and Inoue, 1970; Hills, 1974) and gas-holdup

7

12

t '~'~\ [ QL 850 cm3lmin ] 10 i t , f ,.. =

' \ ] \ ,,

LU ii' XC...~.. " , b \ a J

o.,. "-.',-.':... ",

0 05 10 15 20 2.5 g [ - ]

Fig . 5. E x p e r i m e n t a l R T D f o r QL ---- 8 5 0 c m ~ m i n - 1 QG: a = 0 c m 3 m i n - ~; b = 1 0 0 c m 3 m i n - ~; c = 3 0 0

c m 3 m i n - ~ ; . . . . . 1 C S T R .

(Hills, 1974; Steinemann and Buchholz, 1984; Guy et al., 1986 ), which depends upon the geometrical factors, the relative flowrates of gas and liquid etc. In the case of flotation columns, the capture and removal of solid particles or particulate matter by the rising bubbles imposes a gradient of particle concentration along the column, which also needs accounting for.

Therefore, it seemed more realistic to visualise the flotation column rather as a network of connected well-mixed zones, with provision for flow of the phases involved from one zone to all adjacent ones. This approach has already been followed for agitated vessels (Mann et al., 1981) and bubble columns (Todt et al., 1977; JiliSn)/et al., 1979; Pandit and Joshi, 1983; Deckwer and Schumpe, 1987).

In this context, the simplest possible column configuration comprises a series of well-mixed zones (Fig. 6). Flow is allowed from the ith zone to both the ( i+ 1 )th and the ( i - 1 )th zones. The material balance for the tracer in the ith zone is:

Vi "~-~=Qi-l.i°Ci-1 + Qi+l,i'Ci+l - (Qu+l + Qi.i-1)"Ci (4)

where the "i , i+ 1" denotes flow from the ith to the ( i+ 1 )th zone and Vi is the zone volume. There are essentially two parameters in this model: the first one is the number of zones that is used to represent the column flow structure, while the amount of flow flowing backwards, Qi,i- 1, is the second parameter of the model.

In the case of the bank of flotation cells {Frew, 1984), it has been possible to determine this backflow for each individual zone-to-zone connection. In the

&, 01:~

! li I

. . . . . . . . . . . . . . . I

i OL(!.x) I '~ t ~0:-

i i i I . _ _ _ j

OL QG

Fig. 6. Model of the N-zones-in-series for the representation of the flotation column.

f lotat ion column, however, no such geometrical separation exists and, therefore, a universal value may be considered as characterising the backflow. If 2 is defined as the ratio of backflow, Qi.i- 1, to the net liquid flow into the column, QL, as follows:

it_Qi,i-1 (5 ) QL

then, the previous material balance equation may be rewrit ten as:

dCi Vi "-~-= (1+ it ).QL.Ci_ l + )..QL.Ci+ l - (1+ 2.)[)QL.C~ (6)

for i = 2 . . . N - 1, whereas for the first and last zone it becomes:

V1 " ~ t l -~- QL.Cin +LQL.C2 - (I+It) .QL.C, (7)

and:

dCN VN " - ~ = ( 1 + It).QL.CN-, -- ( 1 + it).Qi.CN (8)

with initial condit ions at t ime t = 0:

CI(O)=N.C o andCi ( 0 ) = 0 , i=2 . . .N

where Co is the initial concentra t ion of the tracer in the whole column and Cin is the concentrat ion of the tracer of the incoming liquid flow.

The backflow parameter , thus, is an indication of the degree of mixing in the column: low values of it denote relatively little mixing, whereas large values of

9

4 mean that there is intense mixing of the fluid contents of the various zones, seen here as a backflow of liquid against the main liquid flow.

This model has been used in the past for the interpretation of experimental data from stagewise operations in chemical engineering (Miyauchi and Ver- meulen, 1963) and has been compared with the axial dispersion model (Roe- mer and Durbin, 1967). Recently, however, it has also been considered for modelling the flotation process for the case of communicating banks of agitated flotation cells (Lynch et al., 1981; Frew, 1984).

The eqs. 6-8 may be rewritten in dimensionless and compact form as:

1 d - - ' - - C * = [A]. C* +B (9) N d0 -

_C* being an array of dimensionless concentrations:

Ci* =Ci/Co, i= l...N (10)

B_ is an array of constants:

B1 =Cin, Bi =0, i=2...N (11)

and [A] is a matrix N , N of coeficients:

al. 1 =aN.N = -- (1+4) (12) ai,i+l =4 i = l . . . N - 1 (13) ai,i_l = (1+4) i=2...N (14) ai.i = - (1+24) i=2 . . .N-1 (15)

it is assumed that all zones have the same volume ( Vi= V/N) . Eq. 9 constitutes a set of ordinary differential equations, which may be solved

either numerically or through the Laplace transform for various initial conditions. For example, in the case of a pulse stimulus, at time 0= 0:

Cl*(O)=NandCi* ( 0 ) = 0 i=2...N

The curve CN* (0), which is obtained from this solution is the theoretical RTD of the column (the case for N = 2 and N = 3 is illustrated in the Appendix).

The backflow model has two degrees of freedom: the number of zones N and the backflow ratio 4. One of these parameters, therefore, may be set in advance and the other one is computed by matching the theoretical RTD curve to the experimental data: setting the value of N and for a given value of 4, the sum of squares of the deviation of the theoretical from the experimental value is calculated and the minimum of this sum corresponds to the best fit. For example, in the case of QL----200 cm 3 min -1 and QG=100 cm 3 m i n - ' , for N=5, the fit was poor for 4= 1 whereas the best fit corresponded to 4= 10, as shown in Fig. 7.

Each experimental condition, therefore, could be fitted by various pairs of (N,4) values, as can be seen in Fig. 8. For all but the lowest liquid flowrate, the

i 2

, Oi ° ,. q-

I 0 i " : ' l'JF] f',:r, :

08 " ' / -..

-~ 0 5 ~ ~ % "' ....

h'l I ""<:~4:-..

0 0 5 10 15 2 0 2 5

o [ ]

Fig. 7, Fitt ing of the theoretical to the experimental RTD curve ( F-l, experimental point).

backflow ratio ~t was lying between 1 and 16 at most, denoting a more or less intense recirculation of the liquid contents of the column. As the number of zones N increased, the backflow ratio increased too, reflecting the need for more backwards flow, in order to represent the mixing of the various zones in the column. It is noted that for the lowest liquid flowrate of 200 cm :~ min-; , the mixing caused by the introduction of gas was very intense and the behaviour of the column approached soon that of a single CSTR. An attempt to obtain a reasonable fit with this model for QG = 300 c m a min- 1 yielded exceed- ingly large values of ~ ( > 40), even for a large number of zones.

The backflow parameter seems to reach a limiting value, as the number of zones gets closer to 10; this could be correlated to the observation made by Joshi and Sharma (1979) that in bubble columns the recirculating liquid establishes a pattern of loops, each one having a height approximately equal to the column diameter (later, Zehner (1986) used a similar approach to model the behaviour of a bubble column). Therefore, the number of distinct zones in this case could be given by the ratio of liquid height vs column diameter, which in this case has been kept close to 10.

The intensity of mixing in the column was thus represented by various pairs of (N,2) values, which possibly makes interpretation of the experimental results difficult. It has been proposed, however, to combine these two parameters into a single one, the Peclet number, which is traditionally used for describing the mixing process through the axial dispersion coefficient, Da (Roemer and Durbin, 1967; Todt et al., 1977):

2.N P e = l + 2 . ~ (= ) (16)

11

Qo~QL 200 LO0 600 850

100 0 (] A %/ 300 I A V

16 (b) /,_~\ I -~_1~- l - - I

72 I /

10 / A A~A 7 , / v j,

8 / A / V / -< ~ v

2

i i I i i I

s [ - }

12 ( a ] O ~ O ~ o ~ o _ o /

10 O 01~-0

7 8 o ~ n ~D~

0 / 0 A/h V---- V L / 0 / A./" ~ / V " / "

2 ~V~V

i , I l

N[-I

Fig. 8. Determination of pairs of (N,~) fitting the experimental RTD curves. QG: (a) i00 cm 3

min-l; (b) 300 cm 3 min -I.

where L is the height of the liquid in the column. So, each pair of (N,;L) values corresponds to a single value of Pe: high values of Pe denote little mixing, i.e. the liquid flow approaches plug flow, whereas low values of Pe correspond to intense mixing of the column contents.

In this way, the effect of the gas and liquid flowrates on the mixing of the flotation column could be clearly demonstrated as shown in Figs. 9 and 10: the increase of QG obviously increased the turbulence in the column, therefore causing an increase in the degree of mixing, observed here as a decrease of the Pe number. On the other hand, the increase of QL had the opposite effect: the more liquid flowing into the column, the less it was disturbed by the gas flow, hence the degree of mixing decreased, approaching plug flow. This is made obvious in Fig. 11, where effectively the increase of the liquid flowrate is re- flected by the increasing remoteness from the loCSTR curve.

Figs. 8-10 clearly show that it could be possible to obtain an empirical equa-

!5 ', w,, A

\ v - - - v . . . . . ~ . . . . v -

0,5 \ ~ 0 5 A / , ~A ~A - - - - A %(

~ ~ I r t 6'00 0 200 L00 600 800 200 L00 8 0

Q@ [ cm3/min ', O [c:'p3/m[r

Fig. 9. T h e Pe n u m b e r as a f u n c t i o n o f t h e gas f l o w r a t e Q(;. Q~: z& = 4 0 0 c m :~ r a i n - J; T = 6 0 0 c m :~ min- 1; [] = 850 cm s min- 1.

Fig. 10. The Pe number as a function of the liquid flowrate QL. QG: m = 1 0 0 C11[1 s min 1; V =300 am :1 min- ~.

12 I I0 [ "~G= I00 cm3'min

o8 l-t I O ' ',o".,."',.}',.,

o6 ~H~, "".. "..," '..~:-.,

H | I

0 05 1 0 1 5 20 ?5

Fig. 11. Decrease in mixedness caused by the increase of the liquid flowrate QL: a = 200 cm ~ min 1; b=400 cm 3 min- ~; c=600 cm 3 min- 1; d=850 cm 3 min-L

t ion, re lat ing the degree or i n t e n s i t y o f m i x i n g (Pe n u m b e r ) or the backf low ratio as a f u n c t i o n of: the n u m b e r o f z o n e s and the gas and l iquid f lowrates . H o w e v e r , no a t t e m p t was made, at th i s stage, to obta in such an equat ion , s ince it was t h o u g h t that more w o r k is required in order to inves t igate the ef fect o f several o ther p a r a m e t e r s like: (1) the geometr ica l factors ( c o l u m n d iameter and height , pos i t i on o f l iquid feed etc. ); (2) the s ize and type o f the gas dis- tr ibutor (perforated plate , di f fuser or d i sso lved-a ir injector); (3) the in f luence o f the var ious addi t ives ( frothers , co l l ec tors etc. ) added for t h e e n h a n c e m e n t

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of the flotation process; (4) the size, density and concentration of the solids etc., before such a predictive equation could be reliably and safely obtained.

CONCLUSIONS

The degree of mixing in the gas-liquid flotation column, which is of major importance for its performance, is clearly affected by the relative flowrates of the gas and liquid being fed into it, among other parameters. The increase of the gas flowrate has been found to increase the degree of mixing in the column, whereas the increase of the liquid flowrate decreases it.

A relatively simple model was proposed for representing the behaviour of the liquid stream in it. It consists of a series of well-mixed zones, with flow allowed both forward and backwards. The model has two parameters: the number of zones N and the ratio of backflow to net forward liquid flow 2. If an appropriate number of zones is selected, then the behaviour of the column may be predicted for any gas and liquid flowrates, in the range of values for which experiments have been performed.

With this configuration, it should be possible to overlay information about the distribution of values for the solids concentration, the local gas hold-up and bubble size distribution and obtain a full model for predicting the performance of the gas-liquid flotation column.

A C K N O W L E D G E M E N T S

The help of Ms A. Daniilidou and Mr P. Zarbos in the experimental part of this work is gratefully acknowledged.

NOTATION

a : e l emen t of m a t r i x [A] [A] : m a t r i x N , N of coeff ic ients

B : a r ray of c o n s t a n t s [ k m o l c m -3] C : t r ace r c o n c e n t r a t i o n [ k m o l c m -3] C* : d imens ion le s s c o n c e n t r a t i o n D : axial d i spe r s ion coeff ic ient [cm 2 s -1 ] k : conduc t iv i t y cell c o n s t a n t [ c m - , ] L : h e i g h t of l iquid in c o l u m n [cm] N • n u m b e r of zones in model Pe : Pec le t n u m b e r (= ua.L/Da) Q : n e t flow in to co lumn [cm 3 m i n - 1 ] t : t ime [ m i n i u : superf icial veloci ty [cm r a i n - ' ] V : co lumn vo lume [cm 3 ] x : d i s t ance [cm]

I4

/ : g a m m a func t ion : gas ho ld-up

f.~ : d imens ion le s s t ime ). : backflow p a r a m e t e r

: mean res idence t ime [ m i n i Subscripts a : axial G : gas parameter i : i th cell in : cond i t ions a t the co lumn inle t L : l iquid parameter r : radial o : in i t ia l cond i t ion

A P P E N D I X

The ana ly t ica l so lu t ion of eqs. 9 m ay easily be ob t a ined when the n u m b e r of zones- in-ser ies is 2. The unsteady-state mass ba lance equa t ions for b o t h zones are:

v .dC~ 1 dt - - ( I + ) J . Q L . C ~ +),.QL.C,~

V dC~ 2 " - -~ -= (1 +)~) QL.C1 -- ( I+) - ) .QL.C2

with in i t ia l condi t ions : C~ (0) = 2.Co a n d C2 (0) = 0. These may be written in d imens ion less form:

d C F - 2 . ( I+) j .C~*+2A.C2*

dO

dC2* - 2. (1 + ) , ) . C 1 " - 2. ( 1 + i ) .C , ,*

d0

where C~* (0) = 2 and C..,* (0) = 0 and they may be t r a n s f o r m e d now in to t he Laplace d o m a i n as:

s .L(CF) - C ~ * ( 0 ) = - 2 . ( I +~, ) .L(Cf l )+2.LL(C2*)

s.L(C2*) = 2 . ( 1 + I ) . L ( C ~ * ) - 2.(1 + ) d . L ( C 2 * )

where s is the Laplace variable and L(CI*) is the Laplace-transformed C,*. From these, L(C2*) may be calculated:

4 . ( 1 + i ) L(C,F) -

[s + 2.(1 + ) j ]2 -4 . ) , . (1 + 2 )

which is easily inversed in to the time domain to yield the theoretical RTD:

4 . (1+) . ) C.2"_ {4.):. (1 + i ) } ~/2 exp{ - -2 . (1 + i ) . 0 }

× s i n h ({4.), ( l + ) d }L/2.0)

which is a func t ion of d imens ion le s s t ime 0 a n d the backflow ra t io )L. In the case of 3 zones- in-ser ies , t he ana ly t i ca l so lu t ion is still ob ta inab le , a f te r some cons iderab le

calculus. T h e mass ba lance equa t ions are, in th i s case:

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dC1 V1 " - ~ ' = - (1 +~.).QLC1 +~t.QL.C2

dC2 V2 "--~- = (1 +)~).QL.C1 - ( 1 + 2.Jl).QL.C2 + X.QL.C3

dC3 V3 " -~-= (I +•).QL.C2 - (1 + ~-).QL.C3

with initial conditions: C1 (0) = 3.Co and C2 (0) = Ca (0) = 0. These may be non-dimensionalised and written in the Laplace domain:

s.L(CI*)-CI*(O)= -3.(I+,t) .L(CI*)+3A.L(C2*)

s.L(C2*) = 3. (1 +~I).L(CI*) - 3. (1 + 2A).L(C2*) + 3A.L(C3*)

s.L(C3*) = 3. (1 + 2).L(C2") - 3. (1 +~).L(C3*)

and L(C3* ) is readily obtained:

27.(1+)~) 2 L(C3*) =

D

where:

D={s+ 3.( l +,t ) }.{ [s+ 3.( l + ~.) ].[s+ 3.(1+ 2~.) ] - lSJ..( l + ).) }

The inversion of this function is still feasible, although lengthy: the denominator is put in the form:

D = ( s - a ) . ( s - b ) . ( s - c )

where:

a = - 3 . ( l + X ) ,

b=3+9.~./2+{9.~..(S+9A)}l/2/2

c = 3 +9A/2 - {9JL. (8 + 9A) } 1/2/2

from which Ca* is obtained as:

( b - c ).exp(a.9) + ( c - a ).exp(b.9) + ( a - b ).exp(c.9) 63*

( a - b ) . ( b - c ) . ( c - a )

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