204 Chem. Eng. Technol. 10 (1987) 204-215

Gas/Liquid Interfacial Area per Unit Volume and Volumetric Mass Transfer Coefficient in Stirred Slurry Reactors

Michael Schmitz, Artur Steiff and Paul-Michael Weinspach*

In three-phase systems, where the liquid constitutes the continuous phase, solid is the catalyst and gas represents the dispersed phase, there are decisive criteria which have to be observed in reactor design. These are e.g. the interfacial area per unit volume between gas and liquid, the volumetric mass transfer coefficient and the mass transfer coefficient. The basic aim of the present work was therefore the investigation of these parameters in relation to the main influenc- ing parameters. Process parameters stirrer speed and superficial gas velocity were varied as well as the physical properties such as liquid viscosity, solids concentration, particle diameter and a geometrical parameter, i.e. reactor diameter. The sulphite method was employed for the determi- nation of these values. The test results confirmed the known relationships of power consumption and superficial gas velocity. An increase in the liquid viscosity leads to a decrease in all the tested values. In most cases, suspended solid particles lead to a lowering of the test values. The influence of the tank diameter on the plots of the test values against specific power consumption turned out to be invariant so that a scale-up of geometrically similar systems can be carried out at constant power consumption, superficial gas velocity and liquid viscosity.

1 Introduction

The investigation and optimization of multiphase systems is at present a major activity in process engineering. Although sever- al dimensioning results for single- and two-phase reactors are available, such data material for three-phase reactors is still lacking.

Apart from the scientific aim of determining the complex rela- tionships and interactions in such systems and of describing these processes by physical models, from the industrial point of view, it is just as important to solve the problems of scale-up from a laboratory or pilot plant to an industrial size.

The mentioned three-phase reactors are - fixed bed reactors and - slurry reactors.

The term fixed bed reactor embraces all reactors, where both gas and liquid flow in co- or countercurrent through stationary solid filling material which can be either reactive or inert. In slurry reactors, i.e. three-phase bubble column, three-phase fluidized-bed reactor and stirred slurry reactor, the solid is sus- pended either by mechanical or by fluid-induced agitation. An extensive comparison of these reactor types, together with their process engineering advantages or disadvantages, was given by Shah [l] and Kiirten [2].

The stirred three-phase reactor is employed in industry for the catalytic hydrogenation of unsaturated fats and fatty acids and

* Dr.-Ing. M. Schmitz (present address: Fa. Enka AG, Werk Oberbruch, Boos Fremery StraRe, D-5138 Heinsberg), Dr.-Ing. A. Steiff and Prof. Dr.-Ing. P.-M. Weinspach, Lehrstuhl f i r Therrnische Verfahrenstech- nik der Abteilung Chemietechnik der Universitit Dortmund, Postfach 500 500, D-4600 Dortmund 50.

for the reaction between HCl and CH3OH in the presence of ZnCl2 catalyst.

A common feature of all these processes is the gas component which passes through the gadliquid interface. If mass transfer is a diffusion-controlled process, the magnitude of the interfa- cial area per unit volume exerts a decisive influence. The main objective of this work was to provide a contribution to the dimensioning of stirred slurry reactors, based on the influence of power consumption on interfacial area per unit volume. In addition, the volumetric mass transfer coefficient and the mass transfer coefficient were to be determined at identical fluid dy- namic conditions.

2 Measurement Method

Since conventional physical test methods for the determination of interfacial area per unit volume (photography, conductivity, photoelectric probe method) are not suitable for three-phase systems and would lead to incorrect results (reasons are dis- cussed in [3]), a chemical method, i.e. the sulphite method, was chosen for the determination of the test values.

The overall reaction equation is:

Na2S0, + - 1 O2 - k2 Na2S0, 2 co2+

Absorption of oxygen in cobalt-catalyzed aqueous sodium sul- phite solutions is a method frequently used both in laboratory and in industrial plants to determine the interfacial area per unit volume and the volumetric mass transfer coefficient.

For the basic principles of the sulphite method and derivations of the equations as well as an extensive explanation of the calcu- lation procedure, the reader is referred to [3,4].

0 VCH Verlagsgesellschaft mbH, D-6940 Weinheim, 1987 0930-7516/87/0306-0204 $02.50/0

Chem. Eng. Technol. 10 (1987) 204-215 205

3 Material Systems

The employed mass transfer system was an aerated, aqueous so- dium sulphite solution with suspended inert solid particles. The test conditions were set as follows'):

pH: 8, temperature: 20 "C,

pressure: atmospheric pressure,

sulphite concentration: 0.4 kmol/m3 5 5

catalyst concentration in determination of interfa-

0.8 kmo1/m3,

4 x 1 0 - ~ kmoi/m3 5 cC:+ 5 cial area: 8 x kmol/m3,

catalyst concentration in determination of volu- metric mass transfer coefficient k,a, cC:+ = 5 x kmo1/m3.

The physical properties used for the calculations are described in [3]. CMC (carboxymethyl cellulose) was added to pure sul- phite solution in order to increase liquid viscosity. The calculat- ed liquid viscosities, according to the concept of Metzner and Otto [ 5 ] , were between 1.45 and 140 mPa s, depending on the added CMC-proportions.

1) List of symbols at the end of the paper.

Inert glass beads were used as the solid. Their density was be- tween 2460 and 2490 kg/m3; particle fractions with average di- ameters of 0.088 and 0.32 mm were used.

4 Experimental Set-up

Fig. 1 shows a diagram of the equipment. The central part is a stirred slurry reactor ( I ) , in which the liquid represents the continuous phase, while the gas and solid are the dispersed phases. Two agitated vessels made of glass (internal diameters 0.2 and 0.45 m) are used, with geometries corresponding to those usually mentioned in the literature. A stainless steel ves- sel, 1.5 m in diameter, was used as well. The ratio of non- aerated liquid height to reactor diameter is unity. Four, sym- metrically arranged baffles, utilized as cooling elements, have diameters of 0.1 D.

Sparger rings were employed as gas distributors. Their arrange- ment allowed the same hole gas velocities at equal superficial gas velocities. The turbine impellers have diameters of 1/3 D.

5 Results

5.1 Influence of Impeller Rotational Speed and Superficial Gas Velocity

The influence of stirring intensity and superficial gas velocity was to be described first for the two- and three-phase systems

I Pressure air 1 I

8 (Coollngwoferr I t

I 1

1 Reactor 2 Torque pick-up . . .

3 Direct current motor 4 Tacho-alternator 5 Pulse generator 6 Pressure regulator 7 Flowmeter 8 Adjusting valve 9 Pressure meter

10 Air filter 11 pH measurement 12 Gas analyzer for oxygen 13 sampling of liquid

Fig. 1. Experimental plant.

206 Chem. Eng. Technol. 10 (1987) 204 - 215

with and without added CMC. Fig. 2 shows the interfacial area per unit voIume as a function of stirrer speed for three different superficial gas velocities. In the two-phase system, the interfa- cial area is independent of stirring speed in the range of low speeds, i.e. from 0 to 200 l/min, because the stirrer is flooded.

As shown in Fig. 2 , the influence of superficial gas velocity dominates in the flooded range. The line, drawn according to a correlation of Wiedmann [6], describes the demarcation to the non-flooded range. To the right of this line, there is a strong linear increase in the interfacial area per unit volume with in- creasing stirring speed, produced by the setting-in dispersion ef- fect of the agitator and the corresponding dispersion of gas bub- bles. In this range, an increase of superficial gas velocity leads only to a slight increase in the interfacial area.

For the described three-phase data ($s = 0.3; d, = 0.32 mm), the steeper slope of the regression curve leads to the conclusion that the interfacial area becomes more dependent on stirrer speed. In the range of low stirrer speeds (for the used tank n < 1200 1 /min) where the solid is completely suspended accord- ing to the 1-s criterion, the solid reduces the interfacial area. This is caused by the non-homogeneous suspension of the solid, or rather the significant concentration profile over the reactor height, as experimentally determined by Bohnet [7]. The solid is mainly restricted to the bottom part of the reactor where it restrains the formation of a bubble layer in the lower part of the reactor and impairs the dispersion effect of the stirrer. At high rotational speeds (n > 1200 l/min), whereby the solid attains a nearly homogeneous suspension state, the interfacial areas of

the three-phase system tend to higher values than those of the two-phase system. This can be explained by higher kinetic ener- gy of the particles at high stirrer speeds; the particles deform an ascending gas bubble and thus produce an enlargement of the interfacial area. The statement, that a non-homogeneous solid dispersion adversely affects the interfacial area, is supported by experience, namely, in three-phase systems, in contrast to com- parable two-phase systems, the interfacial areas are smallest at high rotational speeds and the highest set superficial gas veloci- ty of U G ~ = 4.6 cm/s.

The relationship between the interfacial area and stirring speed (see Fig. 2 ) allows no further conclusions as to whether the addi- tion of inert solid particles, in order to increase the interfacial area, is of energetic advantage or not. For an energetic assess- ment, a plot of interfacial area versus power consumption is more suitable (see Fig. 3). This plot also permits a comparison of different reactor types. In lower power consumption ranges ( P t o t / V ~ < 1 kW/m3), the curves for different superficial gas velocities coincide. Within this range, it is of no importance whether the dissipated power is provided by stirring or by aera- tion. In higher power consumption ranges (Ptot/VL > 1 kW/ m3), a fanning of the curves is obtained for the two-phase system. At constant power consumption, higher superficial gas velocities lead to significantly larger interfacial areas.

Therefore, a description of interfacial area per unit volume, with Ptot/ VL as the only parameter, e.g. as presented by Midoux [8], cannot be recommended. At constant power consumption, the values for the three-phase system are, as a rule, lower than those for the two-phase system. Only at the lowest superficial gas velocity of U G ~ = 0.34 cm/s, larger interfacial areas are produced by the employed particles in the three-phase system, at a high power consumption.

The results of Fig. 3 are presented in Fig. 4 for the volumetric mass transfer coefficient. The shapes of the curves in the two

L , I I l l I I I l l I 1 I l l I I 1 rn -

lo3

10’

2 lo-’ 10’ kW/m3 lo’ 3 1 o2 103 I/min G

m n - 9 0 1 I VL - Fig. 2. Interfacial area per unit volume as a function of stirrer speed and superficial gas velocity, for stirred two- and three-phase systems ($CMC = 0).

Fig. 3. Interfacial area per unit volume as a function of total power consump- tion per unit volume in stirred two- and three-phase systems ($cMc = 0).

Chem. Eng. Technol. 10 (1987) 204-215 207

Fig. 4. Volumetric mass transfer coefficient as a function of total power con- sumption per unit volume in stirred two- and three-phase systems ($CMC =

0).

diagrams are almost identical, which leads to the conclusion that the kL-value varies only slightly.

The fanning of the curves in the two-phase system shifts towards lower power consumptions, which can be attributed to the stronger influence of the superficial gas velocity on the ~ L U L -

value or to a relationship between the kL-value and the superfi- cial gas velocity.

As shown by the curve for U G ~ = 0.34 cm/s, the kLaL-value is more reduced by the suspended solid than the interfacial area per unit volume. The mass transfer coefficient kL is obviously influenced by the suspended solid.

The way in which the total power consumption and superficial gas velocity influence the interfacial area in more viscous sodi- um sulphite solutions (vL = 25 to 60 mPa s) is shown graphical- ly in Fig. 5. For comparison, the results for pure sulphite solu- tion are presented as well. Apart from a few tests at high super- ficial gas velocities (uGo = 4.6 c d s ) and, simultaneously, a low stirring intensity, the interfacial areas of the highly viscous sulphite solution are lower than those of the pure sulphite system. The reasons for this are explained later. The gentle curve gradient of the highly viscous system indicates a reduced influence of stirring intensity at increasing liquid viscosity. This effect is particularly distinctive for the high superficial gas ve- locity of uGo = 4.6 cm/s. As shown by the relatively large horizontal distance between the curves, the influence of superfi- cial gas velocity in the highly viscous system is more significant than that in a system of low viscosity.

Corresponding to Fig. 5 , the volumetric mass transfer coeffi- cient is plotted for the total power consumption in Fig. 6 . Very striking is the significantly higher percentage decrease of the k,a,-values when viscosity increases. This decrease is more pronounced than that of interfacial area, and is caused by a simultaneous influence of viscosity increase on the kL-and a,-values.

Fig. 5. Interfacial area per unit volume as a function of total power con- sumption per unit volume in stirred two-phase systems ( ~ C M C = 0 and 0.5%).

The measurements for the results shown in Figs 5 and 6 were also carried out in three-phase systems. These results are sum- marized in Figs 7 and 8. In analogy to the two-phase system, increased liquid viscosity leads to an intensified effect of the su- perficial gas velocity on a,-and k,a,-values. Furthermore, vis- cosity increase causes a stronger reduction of the kLaL-value, in comparison to the aL-value.

Finally, the influence of the total power consumption and super- ficial gas velocity on the mass transfer coefficient k, should be discussed. Figs 9 and 10 present the mass transfer coefficient as a function of the total power consumption per unit volume for the two superficial gas velocities. Liquid viscosity and solids

Fig . 6 . Volumetric mass transfer coefficient as a function of total power consumption per unit volume in stirred two-phase systems ($CMC = 0 and 0.5%).

208 Chem. Eng. Technol. 10 (1987) 204-215

t aL

5

1 m -

lo3

lo2

1 0’ 10-1 100 10’ kWlm3 102

ptot 1 VL - Fig. 7. Interfacial area per unit volume as a function of power consumption per unit volume in stirred threephase systems ($CMC = 0 and 0.5%).

L 10-l 10’ kW/m3 10’ L

h o t I VL - Fig. 8. Volumetric mass transfer coefficient as a function of power con- sumption per unit volume in stirred three-phase systems ($CMC = 0 and 0.5%).

fraction were varied as well. Since the mass transfer coefficient can be obtained by division of the k,a,-value by the a,-value, i.e. from two different test runs, the k,-value obviously shows a wider scatter range than the other test values.

All mass transfer coefficient measurements can be approximat- ed by horizontal lines; no influence of power consumption on k,-value is present. Similar results were obtained by Calder- bank [9, 101, Reith [ l l ] , Juvekar [12] and Matheron [13]. Op- posite effects may possibly be responsible for the fact that the k,-value is independent of power consumption:

1. On account of enhanced dispersion effect, an increasing power consumption leads to smaller bubble diameters. Since

lo5

m - S

t k,

D.2Ocm. dp=0 .32mm, P s = 2 L 9 O 2

10-1 10‘ 3 100 kW/m3

PIo, I V L - Fig. 9. Mass transfer coefficient as a function of total power consumption (uc0 = 4.6 cm/s).

1 o5

m - S

t kL

10”

P t O l VL - Fig. 10. Mass transfer Coefficient as a function of total power consumption (uc0 = 1.9 c ~ s ) .

small bubbles behave as rigid spheres, the liquid film, sur- rounding the bubble, increases in thickness which reduces the k,-value.

2. At the same time, liquid turbulence increases with increasing power consumption. A decreasing liquid film thickness and an increasing k,-value are the result.

The two effects neutralize each other in the investigated aqueous sodium sulphite solution, with or without added CMC.

5.2 Effect of Liquid Viscosity

The measurements under variation of liquid viscosity were car- ried out with CMC-concentrations of $CMC = 0 to 0.75 mass-% and, in exceptional cases, $CMC = 1 mass-%.

Fig. 11 shows the interfacial area per unit volume, for different stirring speeds, as a function of representative liquid viscosity.

Chem. Eng. Technol. I0 (1987) 204-215

2 100 10' 102 mPas lo3

'I, - Fig. 11. Interfacial area as a function of representative liquid viscosity.

For all the tested stirrer speeds, an increasing liquid viscosity leads to decreasing interfacial areas. In agreement with Hattori [14], up to 20 mPa s, the decrease is relatively small but it be- comes larger at higher viscosities. For the correlation of results, this implies that the relationship between the interfacial area and viscosity cannot be described by a simple power law.

Fig. 12 shows the interfacial area per unit volume as a function of the total power consumption for the two-phase system at a superficial gas velocity of L J G ~ = 1.9 cmls. Addition of CMC leads to smaller interfacial areas as a result of increasing vis- cosity. Several theories were developed in literature in order to explain this phenomenon. For example, Yagi [15] points out that, at high viscosities, a disk-shaped gas pocket forms around the stirrer, which restrains the radial pumping capacity and for- mation of small bubbles. The formation of stable gas trails be- hind the blades of the stirrer is regarded by Hocker [16] as responsible for the altered bubble spectrum with a larger mean

1 m -

I lo3 O L

102

f kL

209

1 0 0

1 - S

10"

I

10' 1 o-z

1 0.' 100 kW/rn3

Ptd 1 VL - Fig. 13. Volumetric mass transfer coefficient as a function of total power consumption, with viscosity as parameter (two-phase system, D = 20 cm).

diameter. Own observations are restricted to a 2 cm thick layer on the reactor wall, since the sulphite solution, containing CMC, was rather turbid. At high viscosities, very small bubbles were observed, which remain in the liquid for quite a while and also individual large bubbles which possibly form by detach- ment from the gas trails behind stirrer blades or from the disk- shaped gas pockets. The small bubbles depleted of oxygen, due to their long residence time, do not contribute to mass transfer and thus to the active interfacial area. Therefore, the mean bub- ble size is displaced to higher values.

Fig. 13 shows the measured kLaL-values at the same operating conditions against the total power consumption. The kLaL- values of the 0.75 % CMC-solution could not be included in the diagram, on account of a much too slow decrease in sulphite concentration, which could not be analyzed by the available

I o - ~

m - S

1 0"

I 1 0-' 10" kW/m3 1 0'

Ptot / V L - h o t I V l - Fig. 12. Interfacial area per unit volume as a function of total power con- sumption, with viscosity as parameter (two-phase system, D = 20 cm).

Fig. 14. Mass transfer coefficient as a function of the total power consump- tion, with viscosity as parameter (two-phase system, D = 20 cm).

210 Chem. Eng. Technol. I0 (1987) 204-215

L

1 m -

t Io3 OL

102

lo’ I 1 o1 10’ kW/m3 1 0‘

PI, /VL - Fig. 15. Interfacial area per unit volume as a function of total power con- sumption, at different viscosities (three-phase system, D = 45 cm).

measuring techniques. In comparison to Fig. 12, the curve slopes are nearly the same which underlines the independence of the kL-value of power consumption. It is quite obvious, that the lowering of the kLaL-value with increasing viscosity is more pronounced than in the case of interfacial area per unit volume. For Pt,t/ VL = 4 kW/m3, the interfacial area is reduced by 55 % , if addition of CMC is increased from 0 to 0.5%; on the other hand, the kLaL-value is reduced by 75 % . This indicates that the kLaL-value is on the one hand influenced by interfacial area, for reasons which were already mentioned, and, on the other hand, by the mass transfer coefficient.

The effect of viscosity increase on the mass transfer coefficient is shown in Fig. 14. The decrease of the kL-value with increas- ing viscosity can be explained by reduced turbulence in the li- quid which results in a thicker laminar liquid film around the gas bubbles. Lower mass transfer coefficients are obtained as a result.

Figs 15 and 16 represent the influence of liquid viscosity on in- terfacial area and volumetric mass transfer coefficient in stirred three-phase systems. As in the two-phase system, an increase of liquid viscosity leads to decreasing test values.

5.3 Influence of Particle Concentration and Particle Diameter

The measurements should provide answers to the following questions:

1. From what particle concentration will the tested parameters

2. In what way are the values influenced? be significantly affected?

In some test series, the particle concentration was varied be- tween 0 and 30 mass-%. Fig. 17 presents one of these test series,

lo[

1 s -

t k~ OL

lo-’

10’ lo” 10’ k W / m 3 10’

Ptot I V L - Fig. 16. Volumetric mass transfer coefficient as a function of total power consumption, at different viscosities (three-phase system, D = 45 cm).

throughout the whole concentration range. This was to be ex- pected since the stirrer speed and the superficial gas velocity were adjusted to match the conditions at the point of intersection between the two- and three-phase curves in Fig. 2. Assessment of the interactions between gas bubbles and solid particles was based on the equation derived by Lee [7]. For a collision be- tween a gas bubble and a particle, Lee obtained a critical Weber number using a balance of forces, under the consideration of in- ertial force of the particles and the interfacial force:

= 3 . Ps d* U L Wec,it = U

If the Weber number is higher than 3, the particle is capable of dividing or deforming the bubble, so that a larger interfacial area is generated.

The disadvantage of using this equation lies in the fact that the relative velocity between gas bubble and particle uIel in stirred

t D.2Ocm vu =L.2rnls , n = 1 2 0 0 l l m l n

*CMC = 0 p, T 2190kglm’

3 0 5 10 15 20 25 30 % LO

JIS - Fig. 17. Interfacial area per unit volume as a function of solids mass fraction - .

in absence of CMC. There was no change in the interfacial area ( ~ M C = 0; dp = 0.32 m, D = 20 cm).

Chem. Eng. Technol. 10 (1987) 204-215 21 1

0 5 10 15 20 25 30 % LO 50

JI, - Fig. 18. Interfacial area per unit volume as a function of solids mass fraction ($CMC = 0.5%; d, = 0.32 mm; D = 20 cm).

reactors is unknown and cannot be estimated by approximate methods. In bubble columns, as a first approximation, Urel can be set equal to the bubble rise velocity. Since, compared to the two-phase system, there is no change in the interfacial area in Fig. 17, i.e. the Weber number is around 3, urel can be calculat- ed by using Eq. (2). It leads to a value of urel = 0.48 m/s.

A sulphite solution with 0.5% CMC was also used at the same stirring speed and superficial gas velocity (see Fig. 18). Up to a solids fraction of $s = lo%, the interfacial area per unit volume remains constant; between 10 and 30%, there is a linear reduction. The interfacial area is reduced by 30% of the initial value. It is possible that the viscosity increase, which reduces liquid turbulence as well as the settling velocity of the particles, decreases the relative velocity between bubbles and particles since both are more strongly affected by the superimposed li- quid flow. Collisions between particles and bubbles do not lead to a deformation or division of the bubbles and, therefore, no enlargement of interfacial area is produced.

Fig. 19 shows the measured interfacial areas under comparable operating conditions but with particles of diameter dp = 0.088 mm added to the pure sulphite solution. Up to 5 % solids content makes no difference to the interfacial area. The increase of the solids fraction to 30% produces a linear increase in the interfa- cial area by 70%. On account of particle diameter reduced to d, = 0.088 mm, small particles require a markedly higher ve- locity relative to that of the bubbles, in order to exceed the criti- cal Weber number Wecrit = 3, according to the theory of Lee [7 ] . Using Eq. (2), a value of ure1 = 0.93 m/s is calculated for relative velocity. With the present state of knowledge there is no explanation as yet as to why small (dp = 0.088 mm) and not large (dp = 0.32 mm) particles lead to an increase of interfacial area at the set operating conditions of n = 1200 l/min and UG,, = 1.9 cm/s.

The solids concentration was varied in the reactor with diameter D = 0.45 m, at a constant tip velocity of uu = 4.2 m/s (see

2

1 rn -

t 103 aL

L 0 5 10 15 20 25 30 % LO

$5 - Fig. 19. Interfacial area per unit volume as a function of solids fraction (&-Mc = 0; dp = 0.088 mm; D = 20 cm).

Fig. 20). As in the case of Fig. 18, the increasing solids concen- tration produces a decrease of interfacial area on using the high- ly viscous solution.

On the one hand, the last four diagrams have shown that sus- pended solids influence the interfacial area only from a concen- tration of $s = 5 to 10% upwards and, on the other hand, it is very difficult to predict whether the interfacial area will be enlarged or reduced. The interfacial areas plotted for the differ- ent particle diameters in Fig. 21 as a function of stirring speed, can be transferred to Fig. 22, provided that certain assumptions are made.

This is described in the following for large particles (d, = 0.32 mm), based on the assumptions:

1. The interfacial area remains constant up to a solids fraction

2. The interfacial area varies linearly with solids concentration. of $s = 7.5%.

0 5 10 15 20 25 30 % LO

JIS - Fig. 20. Interfacial area per unit volume as a function of solids fraction ($CMC = 0.5%; dp = 0.32 mm; D = 45 cm).

212 Chem. Eng. Technol. 10 (1987) 204-215

t OL

n-

Fig. 21. Interfacial area per unit volume as a function of stirring speed foi different particle diameters.

Both assumptions are confirmed by Figs 17 to 20. If a linear relationship exists between interfacial area and stirring speed, only four experiments are required to obtain the graphs shown in Fig. 22, for example, at n = 700 l/min and n = 1800 l/min in the two- and three-phase systems. The resultant curves ex- hibit a point of intersection at about 1200 l/min in which there is no influence of solids throughout the whole concentration range. The line n = 1200 l/min in Fig. 22, resulting from this point of intersection, has no inversion point. The horizontal sec- tions of the curves (0 5 GS I 7.5%) in Fig.22 were obtained by linear interpolation between two-phase values while the end points (GS = 30%) of the ascending or descending curve sec- tions were obtained by interpolation between three-phase values of Fig. 21. Combination of the horizontal sections with the end

lo2 0 5 10 15 20 25 % 30

$5 - Fig. 22. Interfacial area per unit volume as a function of solids fraction and stirrer speed.

t Q L

l o L

1 m -

lo3

10' kW/m3 10' '-lo-' PI01 /VL -

Fig. 23. Interfacial area per unit volume as a function of total power con- sumption, for different particle diameters ( U G ~ = 4.6 cm/s).

values for GS = 30% leads to the curves presented in Fig.22 and thus to a rough qualitative estimation of the influence of in- ert particles on interfacial area in the overall stirring speed and particle concentration range.

Subsequent diagrams allow a discussion of the influence of par- ticle diameter at a solids fraction of GS =30%. Fig. 23 shows the interfacial area per unit volume as a function of the total power consumption per unit volume for different particle di- ameters, at a superficial gas velocity of U G ~ = 4.6 cm/s. Both particle fractions reduce the interfacial areas per unit volume practically in the whole power consumption range. This effect is particularly pronounced for large particles (dp = 0.32 mm). Small particles, with dp = 0.088 mm, cause a smaller reduction of the interfacial area. At low power consumptions, at which stirrer speed is 1.3 times higher than the suspension rotational speed, the suspension state of the particles is of some impor- tance. On account of their high settling speed, large particles are not nearly as well suspended as the small ones and they res- trict the formation of a bubble layer in the bottom part of the reactor and considerably reduce the interfacial area.

As shown in Fig. 24, the same particle fractions were also used in a sulphite solution with 0.25 % CMC. A general reduction of the interfacial area can be observed on addition of solids since liquid turbulence is lowered by the viscosity increase. This results in a reduced relative velocity between gas bubbles and particles and in a restricted ability of the particles to deform the gas bubbles.

Finally, the relationship between the mass transfer coefficient and particle diameter should be discussed. In Fig.25, the kL- value for a given superficial gas velocity is plotted against the total power consumption. The kL-value of the two-phase system is compared with the value of the stirred three-phase system, with the addition of equivalent particle fraction to pure sulphite solution (VL = 1.45 mPa s).

Chem.

3

1 m -

lo3

f aL

lo2

3

Eng. Technol. 10 (1987) 204-215

lo-' 100 k W / m 3 10' PI01 / v, -

Fig. 24. Interfacial area per unit volume as a function of total power con- sumption, for different diameters (uc0 = 1.9 cm/s).

lo-3, 1 I 1

k L

I FL(o)= 3.3x10-'

/ 5

10.'

5 lo-' loo k W / m 3 10' 2

p,,, ' VL - Fig. 25. Mass transfer coefficient as a function of total power consumption, for different particle diameters (uG,, = 1.9 cm/s).

The plot confirms the assumption that the mass transfer coeffi- cient is independent of the power consumption per unit volume. The suspended solid leads to a 10% decrease of the mass trans- fer coefficient when using the small particles (dp = 0.088 mm) and to a 20% decrease on using the large particles (dp = 0.32 mm). A possible explanation is provided by the average bubble diameter, which is reduced, especially in the presence of larger particles. Smaller rigid bubbles form a thicker liquid film which, in turn, results in lower kL-values.

5.4 Influence of Reactor Diameter

In mixing technology, scale-up methods exploit the test results obtained in a small-scale system to produce equivalent process results by the application of suitable scale-up rules. Since there are many stirring tasks (homogenization, suspension, disper- sion, heat transfer), it is impossible to solve all the scale-up problems with one scale-up criterion. A complete scale-up of

213

L , I I I l l I I I l l I

I I I / / I

10' 10' kW/m3 10' 2 ptot VL -

Fig. 26. Interfacial area per unit volume as a function of total power con- sumption, for different reactor diameters (uc0 = 1.9 cmis; $CMC = 0).

model tests (Index M) to larger reactors (Index H) requires geo- metrical similarity, the same material system and the same phy- sical properties for the main construction. These three condi- tions are often only approximately or partly fulfilled.

Own results confirmed the condition that for the interfacial area per unit volume as well as for the volumetric mass transfer coefficient, a scale-up should be performed at constant specific power consumption of the stirrer.

This statement was verified for stirred two- and three-phase sys- tems of different viscosities. Figs 26 and 27 illustrate this asser- tion for the system of low viscosity at a superficial gas velocity of uGo = 1.9 cm/s. Further results on scale-up in presence of CMC are presented in [3].

5.5 Correlation of Data

For a statistical analysis of the results, a regression analysis was used to determine a model for chemical engineering processes.

100

1 - S

1 o4

1 d 10-1 10' k W / m 3 10' 2

Ptot ' VL - Fig. 27. Volumetric mass transfer coefficient as a function of total power consumption, for different reactor diameters (UC, = 1.9 c d s ; $CMC = 0).

214 Chem. Eng. Technol. 10 (1987) 204-215

A power law, extended by a viscosity term, was used as the regression model:

Both equations were first used for the two-phase system. Since the relationships between the tested values and liquid viscosity are linear only in parts (see Fig. 1 l ) , a function, describing the whole viscosity range, would be far too imprecise.

Therefore, the viscosity spectrum was divided into two ranges. The low viscosity range covers viscosities between 1.45 and 20 mPa s while the high viscosity range extends to 75 mPa s. The results of the correlation calculation are presented in Table 1 .

The validity range of the correlations comprises the following parameter ranges:

300 w/m3 I P R / v L 5 10000 w/m3,

0.34 x m/s 5 uGo I 4.6 x m/s.

It should be mentioned that the correlations are only valid for geometrically similar plants and for non-coalescing systems.

The results in Table 1 show that the exponents ai in Eqs (3) and (4) are the same for equivalent viscosity niveaus, which con- firms that the mass transfer coefficient k , is independent of the power consumption per unit volume. In absolute terms, the ex- ponent c1 is always higher in the correlations of the k,a,-values than in those of the a,-value, since the influence of viscosity on the k,-and a,-values is incorporated into the k,a,-value.

The correlations of three-phase data are restricted to liquids of viscosities in excess of 5 mPa s since, at these viscosities, the addition of solids always reduces the investigated values. At low viscosities (pure sulphite solution) the tested parameters were reduced and also increased (see Figs 3 and 4), so that a single correlation would be too imprecise. Both the following correla- tion equations, which permit a rough estimation of the solids in-

Table 1. Correlation of Two-phase Values.

Objective A value

a L 2.87 0.76 0.34 -0.18 0.926 1 . 4 5 ~ 1 0 - ’ 5 q L 5 2 0 ~ 1 0 - ~

aL 3.17 0.39 0.48 - 1 . 1 1 0.920 20X10-351)L c 7 5 x l o - ’

kLaL 5 . 9 7 x 1 0 - 4 0.78 0.36 -0.27 0.922 1 . 4 5 ~ 1 0 - ’ 5 q L

5 2 0 ~ l o - ’

kLaL 1.39X10-4 0.32 0.33 -1.51 0.903 2 0 ~ 1 0 - ’ 5 3 , 5 7 5 x

fluence on the tested parameters, are based on the assumption that, below a concentration of $, = 0.075, the influence of the solid is negligibly small:

1.33 ($s - ULZ

1 1.42 ($s - 0.075) .

Ranges of validity:

0.075 I GS I 0.3,

dp = 0.32 mm, p s = 2490 kg/m3,

300 w/m3 5 P R / v L 5 10000 w/rn3,

0.34 x m/s I uGo 5 4.6 x lO-’m/s,

5 x 1 0 - ~ Pa s < vL 5 75 x P a s .

The values aLz and k,aLz for the two-phase system must be cal- culated by using Eqs (3) and (4).

In order to obtain a generally valid correlation which, apart from the solids concentration, also reflects the solids density and particle shape, it would be necessary to conduct a large number of additional experiments. Unfortunately, the time al- lotted to this investigation was insufficient.

Received: May 20, 1986 [CET 171

Symbols used

interfacial area per unit volume concentration of cobalt ions concentration of sulphite ions reactor diameter impeller diameter particle diameter rate constant (2nd order reaction) volumetric mass transfer coefficient stirrer speed total power consumption per unit volume stirrer power consumption per unit volume correlation coefficient superficial gas velocity relative velocity between gas bubbles and particles stirrer tip speed representative liquid viscosity according to [5] liquid viscosity density of solid surface tension mass fraction of carboxymethyl cellulose mass fraction of solid critical Weber number

Chem. Eng. Technol. 10 (1987) 204-215

Indices

D three-phase system Z two-phase system

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215

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