Mixing phenomena in a continuous flow stirred tank reactor

Chemical

Engmeering

Scmce,

1972, Vol. 27, pp. 1783-1795.

Pergamon

Press.

Printed in Great Britain

Mixing phenomena in a continuous flow stirred tank reactor A. BURGHARDT and L. LIPOWSKA Polish Academy of Sciences, Research Centre of Chemical Engineering and Equipment Design, Gliwice, ul.G.Walow, Poland (Received

3 December

197 1)

Abstract -The aim of the work was to determine the range of parameters in which, for water and other liquids with higher viscosities, the phenomenon of ideal mixing of the liquid occurs in a CFST reactor. The conditions have been determined in which the liquid is ideally mixed by the inlet flow energy only, as well as mixing intensity required when the mechanical stirrer is used. For non-ideal mixing the Peclet number has been correlated as a function of the remaining parameters characterizing mixing intensity. 1. INTRODUCTION

of CFST reactors is as a rule based on the assumption of the validity of the ideally mixed tank reactor model. However, in the existing literature there areno datadeterminingexactly the range of parameters in which the actual mixing corresponds to the model. Former research on mixing phenomena in single continuous flow tank reactors has been done using mainly water solutions (7 = 1 cP) [l-5] and over a limited range of remaining parameters like flow rate or mean residence time. This was probably the cause that the authors did not obtain correlations of more general character. The influence of viscosity on the flow pattern in the reactor has been investigated only in two works [4,6]. From the above mentioned experimental works it can be seen that for water the phenomenon of ideal mixing by the inlet flow only, without stirring, can occur over a fairly wide range of flow rates, when the optimal geometrical dimensions of the reactor are preserved [3,4]. But in case of non-typical tanks, as in the works [ 1,23, and when the densities of solutions fed to the tank differ considerably [51, or for liquids of higher viscosity than that of water[4,61, the flow cannot be considered ideally mixed. In such cases the use of a mechanical stirrer is required to obtain ideal mixing. On the basis of research data published to this day the following aims have been set forth in this work: THE DESIGN

To examine in a systematic way the range of parameters within which for water and other liquids with higher viscosities the phenomenon of ideal mixing takes place, without using a mechanical stirrer, that is by the energy of inlet flow only, and to determine the limits of parameters outside which the mixing is non-ideal. To determine the conditions necessary for obtaining ideal mixing of the liquid, in the case when the inlet flow energy is not sufficient and mechanical stirring has to be used. To examine what model fits best the actual phenomena occurring in the non-ideal mixing region.

2. ANALYSIS

ENERGY

OF IDEAL OF INLET

MIXING BY THE FLOW ONLY

Quantitative analysis of the ideal mixing process in the reactor without use of mechanical stirrer has led to the conclusion that mixing is caused by the kinetic energy transported to the tank with the inlet flow. The inlet flow transmits its energy to the liquid in the tank and causes generation of convective streams and turbulent eddies, which give rise to the mixing of the contents of the tank. The energy of the stream is in the ultimate effect dissipated by viscous friction. From a purely theoretical point of view the velocity field in the tank produced by the inlet flow can be described by the Navier-Stokes equation:

1783

A. BURGHARDT

i=

y~=~VpW,

and L. LIPOWSKA

The integral in the above equation is defined

(1) by the dimensionless velocity field wp(.r~, x$,

1,2,3.

In the above equation pressure forces are omitted because in comparison with inertial and viscous forces they are negligible, and the inlet flow energy is dissipatedmainly by viscous forces. In order to transform Eq. (1) into a dimensionless form, the following quantities have been chosen as characteristic ones: mean flow velocity of liquid in the tank W and the tank diameter D. Then the following dimensionless variables can be defined:

x$)

and therefore is a function of the Reynolds number only, so: Ed =

~)W’DQ

(Ret),

and, after some transformations:

Ed= &&et) cy3yD2, or, finally: Ed = f (Ret) W3yD2,

wi”=!%w’

xi*=E;’

X

fGD

t*

w in dimensionless

and Eq. (1) takes form:

j&yz=fUW.

Lwh=I~w;” Dt"

Re,

*

(2)

Therefore, according to the law of dynamic similarity, dimensionless velocity fields in a tank are identical when the Reynolds numbers are the same, and, of course, the tanks are geometrically similar. The energy dissipated in the tank due to viscous friction can be calculated from the relation: (3) where c&,is Rayleigh’s dissipation function given by the relation:

Upon introducing into Eq. (3):

form:

the dimensionless

Ed = r]W2D j- 4; dV” v*

The relations obtained show, that the energy dissipated in the tank due to friction is a function of Reynolds number only. Hence it is necessary to determine some limiting value of Reynolds number, Ret* based on the tank diameter. For Re, > Ret* the velocity field produced by the inlet flow only conforms to the ideal flow model. The model is defined by the residence time distribution function. The range of the dimensionless modulus Ret variability in the experimental part of this work has resulted from the examined ranges of viscosity, density and flow rates required for obtaining appropriate mean residence times, as also from the designed tank diameter. Obviously, apart from the Reynolds number, the phenomenon is also effected by the values of geometrical modules, which in the above given rough analysis of the differential equations have been omitted (the identical linear characteristic dimension has been assumed for all directions).

variables

(4)

(5)

3. DESCRIPTION METHOD

OF THE EXPERIMENTAL AND EQUIPMENT

In the experiments the internal age distribution function Z(0) was measured by using the method of step-wise change of concentration on

1784

Continuous flow stirred tank reactor

inlet from the value of C,- to Co+ > Co-. Potassium chloride was used as a tracer. Its concentration in the outlet stream was measured by the conductometer. The values of the function Z(e) were calculated from the formula ii I(@

=

II

co+ - C(t)

co+-co- -

The relative density difference of the solutions used, caused by the content of KCl, was not greater than b = 1.7 x 10-S. Y For the experiments the vessel with dimensions H = D = 170 mm was used. The tank was provided with four baffles having width D/12. Inlet and outlet tubes had dia. d = 6.6 mm and were situated on the opposite sides of the vessel near its bottom and on the level of the liquid face accordingly. Six-blade turbine impeller was used with flat blades, with dimensions complying to those recommended by Rushton [7]. The ratio of the impeller dia. to the tank dia. was O-31. All measurements of the residence time distribution function were carried out at liquid temperatures equal to 20 * 0.1°C and with gemoetrical dimensions of the system kept constant. The schematic diagram of the experimental equipment is shown in Fig. 1. 4. DISCUSSION

ON THE

1 I

1

1 c,

14

I

I

I

12u

I

Fig. 1. Diagram of the experimental apparatus, (1) tank reactor, (2) stirrer, (3) batlIe, (4) water jacket, (5) insulation, (6) conductometric vessel, (7) flexible rubber hose, (8) variable speed electric motor, (9) vee-belt transmission, (10) disc with scratched line, (11) thermostat, (12) tank, (13) three-way glass valve, (14) valve, (15) conductormeter, (16) autotransformer, (17) stabilizer. +- circulation of solution with concentration C t, --+----- circulation of solution with concentration C,-, ======:= electric wires.

The value of Reynolds number, based on the tank dia. varied during the experiments within the following limits:

EXPERIMENTS

Ret = 218*4+4*7.

The first part of the experiments was aimed at determining the region in which the liquid is in the state of ideal mixing caused by the energy of the inlet flow only. These experiments were carried-out using a vessel without baffles and impeller, on water and water solutions of glycerine with viscosities of 4.2, 6.2, 8-O and 11 cP. A total of 52 measurements of the residence time distribution function has been carried out in the following range of variables: I’* = 2.28 + 105 l/h r= 1.80+87.40min y = 1000 f 1152 kg/m3.

c,’

The values of the parameters from the experiments are listed in Table 1. The second part of the experimental work comprised the measurements necessary for determining the conditions required for obtaining ideal mixing with the use of the mechanical stirrer, for cases when the inlet flow energy does not suffice. For the experiments water solutions of glycerine were used with viscosities of 11, 2 1 and 43 cP. For each solution four series of experiments have been carried-out differing from each other with values of the mean residence time which

1785

A. BURGHARDT

and L. LIPOWSKA

Table 1. Measured parameters of the residence time distribution for water and glycerine solutions with viscosities 4.2,6.2,8-O and 11-OcP. without mechanical mixing No.of experiment 1

I’* (l/h) 2

7 (min) 3

Ret 4

7) = l*OcP 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

10.5 105 102.5 85 80 80 75 75 70 65 60 60 55 55 50 50 45 40 35 30 30 25 25 19.7 19.7 14.3 14.3 12 11 10 9 9

No.of experiment 1

Description of flow 5

218.4 218.4 213.2 176.8 166.4 166.4 156.0 156.0 145.6 135.2 124.8 124.8 114.4 114.4 104,o 104-o 93.6 83.2 72.8 62.4 62.4 52.0 52-O 41.0 41.0 29.6 29.6 24.9 22.9 20.8 18.7 18.7

ideally mixed flow ,I I! I, I, ,, I, I,

3.45-45.3 l/h 1152-1200 kg/m3 lo-300 revlmin O-21-8-17 26-389-l.

Description of flow 5

Ret 4

n = 1000 kg/m3

33 34 35 36

8 7 6 5.5

24.26 28.97 33.30 36.65

16.6 14.6 12.5 11.4

ideally mixed flow small perturbations ideally mixed flow n

37 38 39

4.5 3.4 2.3

43.87 5740 8740

9.4 7.1 4.7

non-ideal flow I, ,,

q = 4.2 CP

y=

1110kg/m3

I,

,,

I! I,

I,

40 41

26.70 24.45

8.08 8.77

14.7 13.5

ideally mixed flow n

42 43

22.20 20.25

9.77 1064

12.2 11.1

non-ideal flow ,,

,,

n = 6.2 CP

II

y = 1130 kg/m3

n * I

I,

I,

44 45

40.35 35.70

5.32 6.01

15.4 13.7

ideally mixed flow 0

46 47

29.70 24.75

7.16 8.62

11.4 9.5

non-ideal flow ,,

I,

7) = 8.0 CP

I,

,, I,

,,

I,

y = 1144 kg/m3

48 49

54.00 4590

4.12 4-71

16.1 13.7

ideally mixed flow D

50 51

4290 41.40

5.00 5.14

12.8 12.2

non-ideal flow

n

I

7)= 1lcP

I

y = 1152 kg/m3

I

52

I

has fallen in within the ranges of 5-7, 12-15, 22-30 and 50-60 min. In each of twelve series with r) = const and T = cons& the number of the impeller revolutions was increased from zero to the value for which the effect was ideal mixing. The remaining parameters in this part, consisting of 71 experimental points, were varied in the following ranges: V* = y= II = Ret = Re, =

7 (mitt) 3

7)= l.OcP

y = 1000 kg/m3 1.80 1.85 1.93 2.33 244 2.49 266 266 2.88 3.17 3.23 3.31 3.73 3-74 399 4.02 4.48 4,94 5.87 6.65 684 8.15 8.15 9.96 10-02 13.75 13.89 15.95 17.54 19.32 21.70 22.37

I’* (l:h)

45.3

4.79

9.9

The data for these experiments Table 2. 5. RESULTS

non-ideal flow

are listed in

OF THE EXPERIMENTS THEIR EVALUATION

AND

Experiments in which the process of ideal mixing without use of the mechanical stirrer was investigated verified the prediction resulting from the theoretical analysis that in this case the value of Ret is decisive. Independently from the liquid viscosity the phenomenon investigated agreed well with the ideal mixing model for the values of Ret a 13.5. The graphs of the function Z(0) obtained in this region are shown in Fig. 2 for few experimental points.

1786

Continuous flow stirred tank reactor Table 2. Experimental parameters of residence time distribution for glycerine solutions 11*0,21*0 and 43.0 CP with mechanical mixing. No.of experiment 1

I’* (l/h)

7 (min)

Ret

n (revlmin)

2

3

4

5

Re,

ti

Pe

Description of flow

6

7

8

9

1)= 1lcP

y = 1152 kg/m3

53 54 55 56 57 58 59

37.50 3540 14.85 15.45 15.15 1560 690

5.82 6.00 1444 13.77 14.16 13.56 31.17

8.17 7.71 3.24 3.36 3.30 340 1.50

60 61 62 63 64 65 66 67 68 69 70 71

7.80 7.65 7.50 7.35 7.50 360 360 3.45 3.50 3.45 3.45 3.45

27.33 27.88 28.16 29.10 28.76 60.58 59.41 63.65 60.43 63.30 60.70 63.91

1.70 166 1.63 160 1.63 0.784 o-784 0.753 0.760 0.753 0.753 0.753

10 20 30

48.6 97.2 145.8 -

0.180 0.263 -

940 5.45 -

10 20 20 30 40

48.6 97.2 97.2 145.8 194.4 -

0.190 0.290 0.420 -

8.70 4.50 1.83 -

10 10 20 20 30 40

48.6 48.6 97.2 97.2 145.8 194.4

0.269 0.246 0.341 0.316 0.519 -

5.20 6.05 3.18 3.75 0.63

-

-

7)=21cP 41.10 40.50 42.75 39.45

5.74 5.87 5.56 5.98

4.80 4.73 5.00 4.61

76 77 78 79 80

40.65 19.25 19.20 19.20 19.20

5.89 12.36 12.38 12.38 12.38

4.75 2.25 2.24 2.24 2.24

81 82 83 84 85

19.05 10.23 9.80 960 9.95

12.47 2144 24.42 24.75 24.06

86 87 88 89 90

9.62 444 4.42 4.34 4.46

91

4.35

-

-

irregular ideally mixed flow plug-flow dispersion flow I, ideally mixed flow plug-flow with dead space dispersion flow n I, dispersion flow ideally mixed flow plug-flow dispersion flow ,, n I, I, ideally mixed flow

1180kg/m3

_ 12 25 25

_ 31.3 65.2 65.2

_ 0.197 0.241 0.254

_ 8.3 6.23 5.75

50 12 25 50

130.5 31.3 65.2 130.5

0.144 0.303 0447

12.5 4.13 1.45

ideally mixed flow plug-flow dispersion flow I, I,

2.23 1.20 1.15 1.12 1.16

60 12 25 50

156.6 31.3 65.2 130.5

0.191 0.331 0.452

8.68 3.40 140

ideally mixed flow plug-flow dispersion flow ” c

24.70 53.92 53.76 54.74 53.27

1.13 0.520 0.516 0.508 0.520

70 12 25 50

182.7 31.3 65.2 130.5

0.166 0.298 0.477

10.55 4.27 1.08

ideally mixed flow plug flow dispersion flow ,I n

5464

0.508

70

182.7

-

-

71=43cP 29.10 29.04 28.65 29.25 28.50 28.20

-

y=

72 73 74 75

92 93 94 95 96 97

-

48.6 -

-

-

10

7.45 7.82 8.15 7.81 7.51 7.70

-

plug-flow dispersion flow n n

ideally mixed flow

y = 1200 kg/m3 1.69 1.68 166 1.70 1.65 164

20 50 100 100 200

_ 25.9 64.9 129.7 129.7 259.4

1787

0.239 0.347 -

6.32 3.08 -

plug-flow dispersion flow non-active dead space dispersion flow non-active dead space ideally mixed flow

A. BURGHARDT

and L. LIPOWSKA

Table 2(Contd.) No.of experiment 1

Y* (l/h) 2

7 (min)

Ret

n (rev/min)

3

4

5

Re,

~9

Pe

Description of flow

6

7

8

9

l7=43cP

y = 1200 kg/m3

98 99 100 101 102 103 104

13.50 13.95 13.54 12.75 13.50 16.80 13-95

1560 15.10 16.12 16.52 1560 12.54 15.10

0.784 0.811 0.788 0.741 0.784 0.974 0.811

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

13.65 7.35 7.55 7-15 7.00 7.50 740 7.35 6.35 7.25 7.50 4.01 3.72 3.81 3.78 3.81 3.80 3-67 360

15.40 30.28 28.49 29.45 30.69 28.64 29.66 29.17 32.15 29.05 29.56 52.14 58.63 55.67 56.75 56.30 56.05 58.45 59.25

0.792 0.427 0.438 0.415 0.407 0.435 0.43 1 0.427 0.380 0.419 O-435 0.233 0.216 o-221 0.219 0.221 0.220 0.213 0.209

-

-

-

-

25 25 50 80 100 150

32.4 32.4 64.9 103.8 129.7 194.6

0.310 0.306 0.3% 0.479 0.510 -

200 -

259.4 -

-

25 50 50 80 80 100 160 200 300 -

32.4 64.9 64.9 103.8 103.8 129.7 207.5 259.4 389.1

0.171 0.355 0.360 0.406 0.428 0446 0.518 -

10.1 2.93 2.82 2.03 1.70 1.47 0.65 -

25 50 50 80 100

32.4 64.9 64.9 103.8 129.7 259.4 389.1

0.258 0.358 0.354 0.406 0.424 -

5-63 2.87 2.93 2.03 1.75 -

3.93 4.03 2.19 1.07 0.73 -

-

plug-flow dispersion flow 0 ,, ,, non-active dead space non-active dead space ideally mixed flow plug-flow dispersion flow ,,

I,

n n

non-active dead space ideally mixed flow plug-flow dispersion flow R n I I, non-active dead space ideally mixed flow

For Ret < 13.5 the experimental curves departed from the curve of Z(0) = e-e, what is illustrated in Fig. 3. On the basis of the obtained results the value of Ret has been determined, which is the limit of the ideal mixing region of the liquid by the inlet flow, as

For twelve experimental points with mechanical stirring for which the mixing was ideal (see Table 2) the mixing intensity represented as:

Ret* = 13.5.

has been correlated as a function of the ratio RetIRe:. By the least square method the following equation has been obtained.

(6)

Some examples of the residence time distribution functions, obtained in the second part of the experiments (Table 2) are shown in Figs. 4-6. Results of these experiments allowed to determine the conditions for ideal mixing with the use of a mechanical stirrer. They also allowed for analysis of the flow model in the region of nonideal mixing.

Re, = 89.3 ln$$.

t

(7)

The above is valid for Ret 6 Ref. In Fig. 7 there is shown the curve described by Eq. (7), together with the experimental points.

1788

Continuous flow stirred tank reactor

Fig. 2. Diagram of the Z(0) function in the region of ideally mixed flow without mechanical mixing.

Naof

exp.

fle,

006 004

/

02

04

06

08

I IO

12

14

16

I*

2.0

2.2

24

Fig. 3. Diagrams of the functions Z(0) in the region Ret < 13.5 for experiments without mechanical mixing.

e

e

Fig. 4. Residence time distribution functions Z(0) for parameters q = 11 cP, Q-= 60 min.

Fig. 5. Residence time distribution functions Z(e) for parameters 7) = 21 cP, 7 I 54 min.

1789 Cl3

Vol. 27, No. 10-C

A. BURGHARDT

and L. LIPOWSKA

t 400t* 360360340320300260-

e

Fig. 6. Residence time distribution functions f(0) for parameters 7) = 43 cP, 7 = 15min.

The equation constitutes the limit of the ideally mixed region, defined by two dimensionless numbers: Ret and Re,. For Re, 3 Re: the liquid is ideally mixed only by the inlet flow energy, therefore without mechanical mixing. On the other hand, for Ret < Re:, the following condition must be fulfilled to obtain the same effect Re, 2 89.3 In%.

6040200

0.1 02 0.3 04

0.5 0.6 07

0.6 09

%,/RI,*

t

For cases when Ret < Re: and Re, < 89.3 In Re?lRet (the region below the limiting line) the phenomena occurring in the reactor cannot be described by the ideally mixed model. The curves of the residence time distribution function obtained in that region fitted well to the longitudinal dispersion model functions. Consequently, evaluation of the results had been reduced to determining such values of the Peclet number which would fit best the individual curves of Z(e). To this effect the relation between the variance of the age distribution function and the value of the Peclet number, has been used. The variance of the experimental curves was determined from the formula:

Fig. 7. Correlating line Re, = 89.3 In (ReF/Re,) and experimental points, corresponding to the ideally mixed flow with mechanical mixing.

by graphical integration of the graphs Z(e) = f(0”) and Z(0) = cp(B) within the limits O-6.25 and O-2.5 accordingly. The integration limits resulted from the conditions of experiment. The relation v2 = f (Pe) was determined by integrating graphically, within the above limits, the theoretical curves of the longitudinal dispersion model. The graph of this relation was used for reading the values of the Peclet number corresponding to the variances of the experimental curves. The value of cr2 and Pe obtained by this method for individual curves Z(0) are collected with other parameters in Table 2. For experimental curves obtained in experiments without mechanical mixing, the values of

1790

Continuous flow stirred tank reactor

single reactor, and, on the other hand, to the imperfect method of Peclet number determination, for establishing the model parameter on basis of two moments of the age distribution function curve, which always causes some errors. More accurate methods of comparing whole experimental curves with the theoretical ones and fitting such Peclet number which would minimize the integral of squares of distances of both curves in the range 0-m require the use of very arduous numerical calculations because of the fact that for the longitudinal dispersion model the age distribution function is given by an infinite series. Analysing the physical meaning of the obtained correlating equation it should be stated that the rise of Re, form zero to the value of Re, = 89.3 In RePIRe, corresponds to the change of the Peclet number within the limits of Pe =

Peclet number have not been determined because the curves lay very near to the plug-flow function curve. Therefore the values of the Peclet number for these curves were very high and they would have been determined with very poor accuracy. On the basis of the obtained results, the Peclet number has been correlated as a function of the remaining parameters describing the phenomenon of mixing, grouped in dimensionless numbers Ret and Re,. By the method of least squares the following relationship has been obtained: 89.3 ln$$ Re,

t ’

(9)

valid for Ret < Ret*

03+-o.

and

Re, < 89.3 ln$.

As to the influence of Ret on the value of Pe, the equation obtained shows that maximum occurs when the ratio:

t

In Fig. 8 the curve Peexp = f (Peealc) is shown. The mean deviation of the values of PeexP from Pecaro calculated from the Eq. (9) is 39.91 per cent. Scatter of the experimental points is rather considerable. This may be ascribed to the stochastic character of the phenomenon of mixing in a

Ret 89’3 In Ret _ 2 Re, =.

Below this value, i.e. near the state of the ideal mixing the rise of Ret causes the drop of Pe, and in the region of 89.3 ln$$ Re,

pe

CDIC.

Fig. 8. The relationship Peelp = f( Pecale) .

‘>2

the influence of Ret is opposite. That kind of influence is caused probably by the fact that for the description of the complicated phenomenon, a one-parameter model has been used. Presumably, far from the state of ideal mixing massexchanging dead spaces exist. In connection with this the three-parameter model of Villermaux and Van Swaaij [S] should have been used. Then, at the constant value of Re,, the rise of Ret would have caused probably the drop of the dead space fraction or the rise of the number of mass transfer units. 1791

A. BURGHARDT

and L. LIPOWSKA

On assumption, that in the active space the conditions of mixing undergo only slight changes, i.e. the Peclet number is nearly constant, with rise of Ret the curves of the age distribution function approach the curve for a = 1 or N = m. This is illustrated in Fig. 9 which has been made on the basis of the work [8].

l-l

I.0

09

08

a=070 0=0@5

e

O-7

Q 7

0=100

Fig. 10. The effect of Ret on experimental Re, = const = 65.

curves I(0) at

05

an unquestionable way this very complex phenomenon. However, it attaches to the given conditions (Re, and Ret) the age distribution curve, which differs only slightly from the actual one.

in

04 I

03t

\\

02 t 01

0

I

I

02

04

I

06

I

08

I

I

IO

12

I.4

16

I8

!2

8 Fig. 9. The effect of a dead space fraction on the function I (0) for a three-parameter model.

Only after disappearance of the dead spaces the mixing intensity in whole volume of the reactor would rise, and the value of Pe would decrease. But, as the one-parameter model is used, the changes of the age distribution curves, resulting from the increase of the parameter a, i.e. the active volume fraction, are connected with the Peclet number increase, till the disappearance of the dead spaces, and next only with its decrease. In Fig. 10 is shown the influence of Ret, at Re, = const = 65, on the experimental curves of Z(f3) and corresponding values of Pe. The obtained Eq. (9) describes therefore correctly the changes of the residence time distribution with the increase of Ret. Adoption of the simplified one-parameter model in place of the three-parameter one does not explain physically

6. ANALYSIS OF THE OBTAINED AND COMPARISON WITH OTHER WORKS

RESULTS AUTHORS’

The relationships (6) and (7) obtained in this work define the region of parameters (grouped in the dimensionless numbers Ret and Rem) in which the reactor used in the experiments the phenomenon of the ideal mixing of liquid occurred. Comparison of the relationships with other authors’ results have been performed by checking whether their values of Ret and Re, for the ideal mixing fall within the range defined by the Eqs. (6) and (7). The authors of the work[4] have got in their experiments the ideal mixing for water without use of the mechanical mixing, for mean residence times of G-~= 7.65 min and 72 = 37.54 min. Considering the dimensions of their vessel, to the above times the following values of Re, can be ascribed Ret, = 111 and Reta = 22.3. They fall well within the region in which the inlet flow energy suffices for ideal mixing (Ret 3 13.5). Similarly for a corn syrup with viscosity 10 cP,

1792

Continuous flow stirred tank reactor

impeller revolutions n = 225 revlmin and r = 28.1 min the authors got ideal mixing. In this case the values of dimensionless numbers were: Re, = 3.22 and Re, = 522. These values fulfill the condition Re, > 89.3 ln$$

t

at

Ret < Ret*.

Snider and Corrigan[3] had performed the experiments with water solutions and got ideal mixing irrespective of the flow rate and other parameters. But except the vessel diameter they gave no data allowing to compare their results. From the value of the diameter D = 248 mm and physical parameters of water one may conclude that to attain Ret -C 135 the authors would have used flow rates V* < 9.5 l/h and mean residence times r > 60 min. Because such great residence times are rarely found in experiments it may be only believed that the authors had carried out their experiments in the range of Ret > Ret and in connection with this they always obtained the ideal mixing. Cholette and Cloutier [ 1,2] investigated the processes of mixing of water at Ret > 100. Nevertheless, in their experiments without the mechanical stirrer they did not obtain the state of ideal mixing but only non-ideal flow with dead spaces and a by-pass. Their results however, are hard to compare with the present work because the tank used by them had different dimensions and inlet tube situation. The inlet in the works [ 11 and [2] was placed over half of the height of the tank which undoubtedly caused the formation of dead spaces in the lower part of the tank, especially as the density of the water fed into the reactor was lower than the density of NaCl solution with which their tank was filled. The authors of the work[5], according to the direction of the step-wise temperature change on the inlet, obtained either an ideally mixed flow or the plug flow with superimposed backmixing (Pe = 78.4). In their experiments the value of Ret was

greater than that of Ret, but the relative difference of the densities of solutions exceeded ten times the values of Ay/y used in this work and that probably was the cause of the phenomenon described above. The results of the present work for the range of non-ideal mixing have been compared with those of the works[4] and [6]. Corrigan and his co-workers[4] carried out the experiment without mechanical stirring at density 10 cP, 7 = 28 min, and Ret = 3.22. Similarly as in the present work for experiments without mechanical stirring at Ret < Ret* the authors got plug-flow. Zaloudik[6] carried out his experiments for the range of viscosities considerably higher than used in the present work. He used liquids of viscosities n = 400-9500 cP. Hence, the maximum value of Ret in his experiments was O-13 and was lower than the minimum value obtained in this work which was Ret = 0.21. The author observed the flow not corresponding to the longitudinal dispersion model, but to the one with the dead spaces in parallel with ideally mixed region, and with minimum part of the plug-flow. The differences between the flow models obtained in the work [6] and in the present experiments can result from the use of totally different viscosities. In connection with this one may come to the following conclusions with regard to the flow model which would describe in a possibly general way the phenomena of mixing in the tank reactor in a wide range of viscosities. It seems probable that such model is just the longitudinal dispersion model with active dead spaces [8]. It is worthwhile emphasizing the fact that for the range of viscosities investigated in the present work it was possible to approximate the actual flow by the longitudinal dispersion model, which, as results from the mathematical analysis of the model described in he work [S] proves that the fraction of dead spaces in the reactor was very small (a + l), or that the dead spaces exchanged mass with the active space very well (N --j 00). In some experiments with the highest

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A. BURGHARDT

and L. LIPOWSKA

value of viscosity (‘I = 43 cP) it was observed that small dead spaces not exchanging mass were present. On the other hand Zaloudik finds total lack of mass transfer between the dead spaces and the active space, and in his experiments the volume fraction of the dead spaces is large. So it may be concluded that in the proposed model the fraction of dead spaces rises with the viscosity increase, and the mass transfer intensity decreases to zero. Thereby the model assumed in this work and the model of Zaloudik are approximations of the presented general model. Obtaining a correlation which would define the parameters of the general model in such wide range of viscosities seems rather improbable. This would demand very large number of experimental data and much tedious and time-consuming work of fitting the experimental curves to the theoretical ones. The authors of the work[8] state that for finding the parameters N and a by the method of least squares with the assumed value of Pe it took them on average about 20 hr of computer time per curve. Moreover, as in some cases the explicit determination of these parameters had been impossible, there was needed further optimisation of the results obtained by the former method. Generally, it has been ascertained that the experimental data of other authors, obtained for reactors of similar dimensions and design, correspond to the ranges of ideal and non-ideal flow delimited in the present work. The obtained relationships (6), (7) and (9) are valid only for the geometrical dimensions of tank, impeller and inlet tube used in this work. With the purpose of attaining more general correlations further research is being done, concerning primarily the influence of the geometrical dimensionless numbers on the region of the ideally mixed flow in the tank reactor.

co-

C(t) D d d, Ed H N n t V I/* W

Greek symbols

density, kg/m3 .T=- : V* (j+

dimensionless

volume

time

CT2 variance

of a residence tribution function

time dis-

Dimensionless

numbers

Re,~E+

impeller Reynolds number

Re,=m

Reynolds number for tank Jef

fraction of active reactor

mean residence time, min

7) dynamic viscosity, cP, kg/mh

NOTATION a

initial concentration of a tracer in in a reactor, mole/l. concentration of a tracer fed into a reactor after a step-wise change, mole/l. concentration of a tracer at an outlet from a reactor at the moment I, mole/l. tank dia., mm inlet tube dia., mm impeller dia., mm energy dissipated in a tank by viscuous friction, kgm2/sec3 liquid head in a tank, mm number of mass-transfer units number of impeller revolutions, revlmin time, min reactor volume, 1. volumetric flow rate, l/hr linear velocity of liquid in a tank, mlsec

in a 1794

Boundary value number for tank

of

modified Peclet number.

Reynolds

Continuous flow stirred tank reactor REFERENCES [ll CHOLETTE A. and CLOUTIER L., Can. .I. Chem. Engng 1959 37 105. [21 CLOUTIER L. and CHOLETTE A., Can. J. Chem. Engng 1969 46. [31 SNIDER D. and CORRIGAN T. E.:A.I.Ch.E.JI 1968 19 813. [41 CORRIGAN T. E., BETSCHEL E. and AXLINE A., Chem. Engng Sci. 1967 22 1535. [51 AKITA S., SHIBATA S., NISHIMURA Y. and MATSUBARA M.,Chem. Engng Sci.1969 24433. [61 ZALOUDIK P., Bit. Chem. Engng 1969 14 657. [71 RUSHTON J. H., Chem. Engng Progr. 1950 46 395. 181 VILLERMAUX J. and VAN SWAAIJ W. P. M., Chem. Engng Sci. 1969 24 1097. Resume-Cette etude cherche a determiner la gamme de parametres pour lesquels le phtnomtne de melange parfait du liquide se produit dans un reacteur agitateur a Ccoulement continu, dans le cas de l’eau et de produits ayant des viscosites superieures. L&s auteurs determinent d’une part les conditions dans lesquelles le liquide est parfaitement melange uniquement par l’energie du courant a l’entree, et d’autre part l’intensite necessaire du m&urge lorsqu’un agitateur mtcanique est utilise. Dans le cas dun melange imparfait, le nombre de Peclet est calcule comme Btant une fonction des parametres restants, qui caracterisent l’intensite du melange. Zusammenfassung- Das Ziel dieser Arbeit bestand darin den Bereich der Parameter festzustellen, in welchem fiir Wasser und andere Fliissigkeiten mit hijheren Viskositaten die Erscheinung idealer Mischung der Fliissigkeit in einem CFST Reaktor auftritt. Die Bedingungen wurden bestimmt unter welchen die Fliissigkeit durch die Zufluss-Stromungsenergie allein ideal gemischt wird, sowie die erforderliche Mischenergie wenn ein mechanischer Riihrer verwendet wird. Fur nichtideale Mischung wurde die Pecletsche Zahl als eine Funktion der die Mischintensitat charakterisierenden iibrigen Parameter in Korrelation gebracht.

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