Ind. Eng. Chem. Res. 2001, 40, 4731-4740
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Hydrodynamics and Scale-Up of Horizontal Stirred Reactors G. J. S. van der Gulik,* J. G. Wijers, and J. T. F. Keurentjes Process Development Group, Eindhoven University of Technology, Department of Chemical Engineering and Chemistry, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
In this study, the hydrodynamics in horizontal stirred-tank reactors are investigated. The flow state, agitation power, and macromixing time have been determined experimentally. Two flow states, i.e., “slosh” and “ring”, can be distinguished, with transition between the two states that shows hysteresis. The agitation power was determined by measuring the temperature increase upon mixing. The power number appears to be comparable to power numbers in unbaffled vertical vessels. A variation in fill ratio indicates that agitation energy dissipates uniformly throughout the reactor under laminar conditions. Under turbulent conditions, however, most energy is dissipated at the vessel wall. Using pulse-response measurements, macromixing times have been determined. The mixing times correlate with momentum input and liquid volume, thus indicating different hydrodynamics at large and small scales. A combination of mixing times and agitation power shows that, at small scale and intermediate fill ratios, the mixing is most energy-efficient. Introduction Horizontal stirred-tank reactors are widely used in industry. A commercial example is the unbaffled Drais reactor that can be used for multiple purposes such as powder mixing in catalyst preparation, liquid mixing and kneading during CMC production, and polycondensation processes.1,2 In all applications, the reactor is only partially filled. A schematic representation of the Drais reactor is given in Figure 1. Typical for the unbaffled cylindrical reactor is its horizontal position and the heavily designed impeller. The impeller makes possible the application of high mixing power, which is needed to achieve sufficient mixing in viscous fluid processes. The reactor is characterized by its length L, diameter D, and clearance c, which is the distance between the blades and the reactor wall. Because the clearance is small, the blades perform a scraping action that keeps the reactor wall free of sticking material. The clearance also provides a region with high shear rates and good heat exchange with the cooled walls. The blades have a pumping action toward the reactor center, providing an easy way to discharge the reactor through the open hatch as represented in Figure 1. Despite the wide application of Drais reactors in liquid mixing, little is known about the hydrodynamics, mixing performance, and scale-up of such reactors. A literature survey shows that the literature on hydrodynamics in horizontal vessels is rather limited, in contrast with that on vertical stirred vessels. Some literature exists on the hydrodynamics under turbulent conditions. Ganz3 describes power measurements in horizontal stirred gas absorbers. Ando et al.4 measured power input upon stirring in partially filled horizontal vessels in relation to flow behavior. In the vessel, two flow states could be distinguished. The “slosh” state, is obtained at low stirring speed. The liquid is then pushed upward by the impeller and sprayed, which is ideal for use in gas absorption processes.3,5-8 The “ring” state, * Author to whom correspondence may be addressed. E-mail address:
[email protected].
Figure 1. Drais reactor, given in front, top, and side views, with length L, diameter D, blade angles R and β, blade width w, blade height h, and a hatch. The arrows indicate the moving direction during operation.
which is obtained at high stirrer speeds, results in a cylindrical liquid layer on the inside wall. Ando et al.9 also studied turbulent mixing in a horizontal vessel with baffles and multiple impellers. Macromixing times were measured, and a model was proposed for scale-up purposes. It has been established that the dimensionless macromixing time Ntm is proportional to L/D. Because literature on mixing in horizontal vessels under laminar conditions is virtually absent, we have to rely on mixing studies in vertical unbaffled tanks, for which many studies are available. The usual purpose of these studies has been to find geometries that provide good mixing performance. Data on mixing vary from author to author as a result of differences in geometry, definitions, experimental techniques, and fluid proper-
10.1021/ie010328g CCC: $20.00 © 2001 American Chemical Society Published on Web 10/04/2001
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ties. However, a general consensus emerges concerning impeller designs. Judging from power input and mixing time experiments,9,11,12 it can be concluded that the flow pattern of a good laminar mixer should include (1) axial flow, (2) all stream lines passing through the impeller region, (3) no closed stream lines occurring outside the impeller region, and (4) frequent disruption of fluid along the wall. Figure 1 shows that conditions 2 and 4 will probably be achieved in the Drais reactor because the impeller blades pass through the entire reactor volume. In a preliminary study,13 we investigated the flow patterns in a Drais reactor using planar laser induced fluorescence and found that axial flow in the reactor is rather effective. Also, closed stream lines can be present, acting as toroidal vortices. We have studied the macromixing in the Drais reactor filled with low- and high-viscosity liquids with the aim of obtaining a better understanding of the mixing during a polycondensation process in which the viscosity strongly increases as a result of the formation of polyaramid molecules.2,13 To obtain a polymer product with the required qualities in terms of MW (molecular weight) and MWD (molecular weight distribution), it is important to expose the polymerizing liquid to high shear rates.14 The high shear rates increase reaction rates through molecular orientation and rotational diffusion of the rods. This has been shown in an experimental study by Agarwal and Khakhar,14,15 using two reactors in series in which the first reactor is a vertical reactor with a high-speed stirrer ensuring good overall mixing. The second reactor provides Couette-flow hydrodynamics, with nearly homogeneous high shear flow over the entire reactor but with little overall mixing. The authors were able to improve product quality considerably by increasing the shear rate in the second reactor. The Drais reactor investigated here combines the important features of both reactors:1,2 a large impeller provides overall mixing while at the same time high shear rates occur in the small clearance between impeller and vessel wall. As the clearance of the Drais reactor is small, its volume is small compared to the total liquid volume: in completely filled reactors, this volume ratio is 2 × 10-4. Therefore, macromixing in the reactor has to be optimized so that all liquid in the bulk will pass the high-shear region in the clearance frequently. A second reason that emphasizes the importance of short macromixing times is that the polycondensation is performed in a semibatch manner: one reactant is fed to the other and has to be mixed quickly throughout the reactor to prevent the occurrence of a premature termination reaction.2,13 Judging from the available literature, it is unclear what the macromixing time will be and how it will evolve in scale-up. Therefore, we conducted an experimental study at different scales in which we established macromixing times by means of pulse-response measurements. The applied agitation power was also measured to link the mixing performance with power consumption. All measurements were performed in the ring state, as the polycondensation process is performed at high stirrer frequencies, thus forcing the liquid into the ring state. As it is known that application of the ring state is required for a high MW to be obtained,2 we determined the impeller frequency at which the ring state forms or disappears during operation. Also, flow behavior was studied as a function
of vessel and stirrer geometry, impeller frequency, and fluid viscosity. Scale-up Theory For scale-up of the polycondensation process, it is important to know how the mixing time evolves upon reaction. From the literature, it is well-known that the mixing time will be a function of the process conditions and reactor configuration
tm ) f(F, µ, N, Di, g, geometrical dimensions of the system) (s) (1) Using dimensional analysis and omitting the Froude number (Fr), the functional relationship can be rearranged to
Ntm ) f(Re, geometrical dimensions as ratios) (2) Ntm is the dimensionless mixing time and is expected to be independent of the Reynolds numbers (Re) under both turbulent and laminar conditions.16 Under intermediate conditions, Ntm will be a power law in Re. In our specific case, Fr can be omitted as Fr is only important under conditions at which the transition between slosh and ring states occurs. We are only interested in macromixing times in the ring state, as the polycondensation process is performed in this state. To maintain constant tm during scale-up, a constant impeller frequency N is required according to eq 2. This will also result in a constant average shear rate γ˘ a, which is related to N by17
γ˘ a ) -
dv ) k1N dy
(s-1)
(3)
with k1 close to unity.18 The highest shear rate, γ˘ max, occurs in the clearance and can be estimated using
γ˘ max ) -
πNDi dv πND - 0 |c ≈ ) ∝ k2N (s-1) dy c D/2 - Di/2 (4)
Thus, γ˘ max is constant when N is kept constant, provided that the clearance is kept in a constant ratio with the vessel diameter. Keeping N constant during scale-up, however, is strongly reflected in the power requirement. Under turbulent conditions, the required power is given by
P ) FNPN3Di5 (W)
(5)
and the average energy dissipation rate per unit of mass is
NPfgN3Di2 P ) j ) (m2 s-3) FVl x
(6)
where F is the liquid density (kg m-3); NP is the power number; N is the agitation rate (s-1); Di is the impeller diameter (m); x is the fill ratio; Vl is the liquid volume (m3); and fg is the geometrical factor, which is 1.05. For geometrically similar systems, the power number NP can be rewritten as a functional relationship of
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4733 Table 1. Geometric Data Regarding the Four Reactors small-scale models parameter reactor length L (m) reactor diameter D (m) reactor volume Vr (L) impeller diameter Di blade width w blade height h blade thickness shaft diameter β (degrees) shaft cylindrical wall side walls
reactor11
reactor15
0.198
0.27 0.18 6.7 D(29/30) D/2 D/12 D/90 D/6 135
5.0
180
large-scale model reactor20
reactor11-60
0.36
0.66 0.6 185 D(29/30) D/2 D/12 D/54 D/10 180 stainless steel Perspex Perspex
8.9
120 stainless steel glass stainless steel
dimensionless groups. Under turbulent conditions, only Re is relevant for the Drais reactor; thus
NP ) k3Rea
(7)
Under laminar conditions, the following relation applies
P ) k4µN2Di3 (J s-1)
(8) Figure 2. Schematic representation of the experimental setup for the temperature measurements.
and consequently
j )
P ) k5νN2 (J kg-1 s-1) FVl
(9)
with k4 and k5 as constants and ν the kinematic viscosity. For these equations, it is assumed that NP depends on Re-1 only, which is a good approximation when Re < 100. From eqs 5 and 8, it can be seen that, for constant N, P increases with Di5 and Di3, respectively, which are both highly impracticable. Therefore, on larger scales, a lower specific power input has to be applied which is usually obtained by reducing the impeller speed. This results in an increase in tm, which is undesirable in the polycondensation process. With this experimental study, we identify the limitations that can be faced during scale-up.
mixture of the two was used as the working fluid. The power input to the liquid by stirring was measured by determining the temperature increase of the liquid with time. The experimental setup is depicted in Figure 2. Two Pt100 elements, denoted T1 and T2, measured the temperature in the reactors during agitation. There was no difference observed between T1 and T2 under turbulent conditions. Under laminar conditions, the difference never exceeded 0.2 °C. Pt100 element T3 measured the ambient temperature, Ta, during the experiments. All temperatures were measured with an accuracy of 0.1 °C. The power input followed from the energy balance over the mixing vessel, as given in eq 10
FCpVl Experimental Section The Drais Reactor. To investigate the mixing process at different scales, four scale models of the Drais reactor were available. Typical geometrical data are given in Table 1. The small models are named reactor11, reactor15, and reactor20, where the numbers 11, 15, and 20 refer to the L/D ratios. The large-scale model is reactor11-60, which refers to the L/D ratio and the diameter. Reactor11 and reactor11-60 are geometrically similar. Reactor15 and reactor20 differ from reactor11 in angle β between blades (as shown in Figure 1) and in length. A detailed description has been given previously.13 Transition in Flow State. The flow state was determined for different reactor geometries and fluid viscosities by observing the impeller frequency at which a fluid ring was formed or collapsed. The formation was complete when the inner gas/liquid surface was flat. Both transitions were clearly visible. Power Measurements. Power measurements were performed in reactor11, reactor20, and reactor11-60. Depending on the desired value of Re, tap water, glycerin (purity > 99.9%, Heybroek, Amsterdam), or a
dT ) P - h(T - Ta) (J s-1) dt
(10)
where F is the liquid density (kg m-3), Cp is the heat capacity (J kg-1 K-1), Vl is the liquid volume (m3), T is the liquid temperature (K), Ta is the ambient temperature (K), P is the dissipated stirring energy (J s-1), and h is the overall heat transfer coefficient (J K-1 s-1). The term on the left-hand side of eq 10 represents the accumulation term, P represents the dissipated stirring energy, and the last term represents the lumped losses to the environment. Assuming density, heat capacity, overall heat transfer coefficient and ambient temperature to each be constant and temperatureindependent, differential eq 10 can be solved by considering the initial condition T(0) ) T0, resulting in
( ) [ ( )]
T(t) ) Ta + (T0 - Ta) exp
-ht + FCpVl
P -ht 1 - exp h FCpVl
(K) (11)
Pulse-Response Measurements. The experimental setup for the pulse-response experiments is depicted
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Figure 3. Schematic representation of the experimental setup for the pulse-response measurements.
Figure 5. Flow state transition, expressed in Fr as function of the fill ratio for reactor11-60, reactor11, and reactor20.
of concentration, σ2, around the equilibrium value, as defined by
σ2(t) ) (1 - Cn(t))2
Figure 4. Typical response curve with the corresponding variance for D ) 0.6 m, L ) 0.66 m, x ) 100%, and N ) 4.8 Hz.
in Figure 3. It was decided to perform measurements outside the vessel because the clearance was too small for a probe to be placed inside. Therefore, the reactor contents were circulated through a spectrophotometer (A) placed in an external loop. The liquid was withdrawn from an outlet, placed at the same height as the clearance and fed back at the top in the reactor center. The flow rate in the loop was monitored by flow meter F. The measurement started when a small amount of a concentrated aqueous methylene blue solution was injected as a pulse, δ(t), in the reactor center. The concentration was measured by the spectrophotometer, which was connected to a PC for automatic monitoring. For reactor11-60, the volume of the piping was 450 mL, and the flow rate was 50 mL s-1. Because the liquid volume was between 72 and 180 L, higher-order mixing effects can be ignored. The time needed for the tracer to leave the reactor and reach the spectrophotometer was 0.9 s. This delay time was measured by injecting a small amount of tracer at the reactor outlet. Macromixing times were corrected for this delay. For the smaller reactors, the volume of the piping was 250 mL, the flow rate was 25 mL s-1, the total liquid volume varied between 2 and 10 L, and the delay was 2.1 s. The total amount of injected solution was 3 mL. Using a highspeed camera as described previously,13 the injection time was determined to be 0.20 ( 0.03 s. This time was negligible compared to the mixing time. An example of a tracer concentration response curve with time is given in Figure 4. The plotted response on the left Y axis was normalized using
Cn(t) )
C(t) - C(0) C∞ - C(0)
(12)
On the right Y axis in Figure 4 is plotted the variance
(13)
The macromixing time was defined to be the time at which the variance was below 10-3. This resulted in tm ) 12.3 s for the experiment in Figure 4. Every presented macromixing time (shown in Figures 12-14) is the average of at least three measurements. Table 2 provides the impeller frequencies used in the pulse-response experiments. Under turbulent conditions, the chosen frequency ensured that the liquid was in the ring state. Consequently, the values of Fr at large and small scale were similar. The values of Re were not similar, which was less important as it followed from literature that Ntm was independent of Re at the high applied values of Re.19 Using glycerin in reactor11-60 provides Re values up to 1730, which does not really justify the laminar classification. However, for convenience, we have grouped all measurements using glycerin. Results and Discussion Flow State in Water. For the reactor filled with water, the Fr values at which fluid rings form or collapse are plotted in Figure 5 against the fill ratio for reactor11, reactor20, and reactor11-60. The solid symbols mark the Fr value at which the fluid ring forms with increasing impeller speed, and the open symbols mark the Fr value at which the ring collapses with decreasing impeller speed. From Figure 5, it follows that, at all fill ratios, hysteresis occurs between the values of Fr for ring formation and collapse. The values of Fr at which the transition occurs coincide for the ring collapse; however, they do not coincide for ring formation. At low as well as high fill ratios, flow changes occur at higher values of Fr than is the case at intermediate fill ratios. This implies that, at intermediate fill ratios, weaker inertial forces, i.e., less power, are required to form a fluid ring. From Figure 5, the value of Fr can be determined at which a fluid ring is present. For the large-scale reactor, ring formation occurs around Fr ) 1, which can practically be used as a criterion for ring formation. This information is important, as the fluid in the pulseresponse measurements and power measurements has to be in the ring state for the polycondensation process.2
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4735 Table 2. Range of Impeller Frequency and the Corresponding Shear Rate γ3 , Tip Speed, and Reynolds and Froude Numbers as Applied in the Pulse-Response Measurements kinematic
dynamic
reactor
D (m)
regime
N (Hz)
γ˘ (s-1)
Vtip (m s-1)
Re
Fr
reactor11, reactor15, and reactor20 reactor11-60
0.18
laminar turbulent laminar turbulent
3.0-11.2 5.5-9.1 0.6-4.8 1.8-4.8
560-2100 1040-1715 110-900 340-900
1.7-6.3 3.1-5.1 1.1-9.0 3.4-9.0
100-360 (1.77-2.95) × 105 210-1730 (6.48-1.73) × 105
0.17-2.3 0.55-1.5 0.02-1.4 0.20-1.4
0.6
Figure 6. Photograph of glycerin in reactor20 at a fill ratio of 40% at 8 Hz.
Figure 8. Power number as a function of Re with the fill ratio and L/D ratio as parameters.
Figure 7. Temperature rise versus elapsed time for the small(D ) 0.18 m) and large- (D ) 0.6 m) scale model, filled with glycerin or water.
Because of the hysteresis effect, however, it is possible to maintain the ring state at stirrer frequencies lower than the frequency needed for ring formation. Flow State in Glycerin. In reactors partially filled with glycerin, no fluid ring is formed at high stirrer speeds. For illustration, an image of the flow state is recorded with a high-speed camera and is presented in Figure 6. Viscous forces are too high to obtain a fluid ring as in water. The gas phase is finely dispersed in the liquid, providing a milky fluid with small gas bubbles. Figure 6 also shows the presence of large holes in the fluid in the wakes behind the blades. The holes do not completely extend to the cylindrical wall, indicating that the wall is entirely covered by a liquid film. This liquid film probably is important, as this is the region with the highest shear rate. In the liquid film, small gas bubbles are present that are transported tangentially by the impeller with a speed lower than the impeller speed. This effect has been made visible by use of a stroboscope. Power Measurements. In Figure 7, the temperature rise is plotted as a function of time for five different situations. The time for measurement ranged from 5 min in reactor11 with glycerin to over 2 h in reactor1160 with water. By fitting the temperature profiles with eq 11, the applied agitation power P was obtained. The overall heat transfer coefficient h also follows from the fitting procedure. However, as the interpretation of these results appeared not to be very straightforward, these data were not included.
Figure 9. Power as a function of Re with viscosity and diameter as parameters.
In Figure 8, the power number NP, as defined by P/FN3D,5 is given as a function of Re. The solid line represents measurements in a completely filled reactor11 and reactor11-60, which are geometrically similar but differ only in size. The measurements are in close agreement with data for a propeller mixer in an unbaffled vertical vessel as reported by Rushton et al.20 NP becomes independent of Re at high values of Re, i.e., under turbulent conditions. Accordingly, the power a of Re number in eq 7 approaches 0. The applied power is plotted against Re in Figure 9. From the slope of the lines at high values of Re, it follows that the applied power is proportional to the cubic root in impeller speed, which is in accordance with eq 5. Under laminar conditions, NP becomes inversely proportional to Re (the exponent a in eq 7 approaches -1). From Figure 9, it follows that the applied power at low values of Re is proportional to the square root of the impeller speed, which is in accordance with eq 8. The power number NP is given as a function of Fr in Figure 10. Figure 10 shows that NP is virtually independent of Fr, apart from the case in which the reactor
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is limited, which can be explained by the small clearance volume. Using multivariable analysis, the relationship in eq 14 between NP and the varied parameters can be obtained for laminar conditions as
x0.92 L 0.89 NP ) 93 0.85 D Re
()
(14)
and for turbulent conditions as
NP ) 0.15x0.36
Figure 10. Power number as a function of Fr with fill ratio and fluid type as parameters.
is completely filled with glycerin. For a horizontal reactor, Ando et al.4 also found that NP is independent of Fr. In Figure 11A-D, the power P and dissipated energy per unit mass j are given as functions of the fill ratio for water and glycerin. With glycerin, the applied power was around 5 times higher than it was with water. From Figure 11A and 11B, it follows that, for water and glycerin, the applied power increases with increasing fill ratio. For water, the increase is less than proportional, as follows from Figure 11C in which j decreases with increasing fill ratio. In water, most of the energy is dissipated at the impeller tips and at the cylindrical wall. At low fill ratios, this region represents a relatively larger volume than at high fill ratios. Therefore, j will be higher at low fill ratios. Figure 11D shows that, in glycerin, j is independent of the fill ratio, suggesting that energy is dissipated more uniformly throughout the fluid than in water. This also follows from Figure 11F, in which j is plotted against Re. The values for j at all fill ratios coincide and follow a power law of 2.1. As j is homogeneous, the shear rate is also homogeneous. Consequently, the contribution of the high shear rate in the clearance to j
(DL)
0.44
(15)
The standard errors of the exponents are given in Table 3. For laminar conditions, NP proves to be nearly linear with liquid volume as the exponent for the fill ratio comes close to unity. Exponent a approaches -1, which is in agreement with eq 7. Summarizing, it follows that the value of NP in this case is similar to that of a propeller in an unbaffled tank. In water, j is a function of the fill ratio and is highest near the vessel wall. In glycerin, j is independent of the fill ratio and is homogeneous over the reactor. Macromixing Times in Water at Turbulent Conditions. The dimensionless mixing time Ntm in the large reactor11-60 is depicted in Figure 12A as a function of fill ratio, indicating that, at every fill ratio, Ntm is independent of Re. Also, Ntm increases with increasing fill ratio, indicating that more impeller revolutions are required for a given degree of mixing with increasing liquid volume. From Figure 12B-D, in which Ntm is given for reactor11, reactor15, and reactor20, respectively, it follows that the lowest Ntm value is found at a fill ratio of 0.7, in agreement with previous results.13 This suggests that, at low fill ratios (x < 0.4), the total amount of fluid is too low to provide good overall circulation in the fluid ring, whereas at high fill ratios (x ≈ 1), the poorly mixed zone near the shaft reduces the advantages of the better overall circulation.
Figure 11. Results of the power measurements: (A) power in water against fill ratio, (B) power in glycerin against fill ratio, (C) j in water against fill ratio, (D) j in glycerin against fill ratio, (E) j in water against Re , (F) j in glycerin against Re . Parts A-D have N and L/D as parameters, and Parts E and F have fill ratio and L/D as parameters.
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4737
Figure 12. Dimensionless macromixing time against Re and impeller speed using water for (A) reactor11-60, (B) reactor11, (C) reactor15, and (D) reactor20. Table 3. Standard Errors of the Constants in the Correlations for NP conditions
preexponential factor
exponent in x
exponent in (L/D)
exponent in Re
laminar (Re < 400) turbulent (Re > 105)
93 ( 9.6 0.15 ( 0.0055
0.92 ( 0.066 0.36 ( 0.064
0.89 ( 0.064 0.44 ( 0.072
-0.85 ( 0.02 -
A comparison of Ntm for reactor11, reactor15, and reactor20 shows that Ntm increases with increasing L. Also Ntm proves to be independent of Re, although Ntm increases slightly with increasing Re in reactor20. This increase indicates that the fluid tends toward solid-body rotation at higher stirrer speeds. Solid-body rotation can occur more easily in reactor20 than in reactor11 and reactor15 because of the relatively small effect of the sidewalls. The above observations show that reactor11 and reactor11-60 differ significantly in hydrodynamic behavior despite their geometric similarity. In reactor1160, the shortest mixing time is found at a fill ratio of 0.4, the lowest fill ratio applied, whereas in reactor11, it is found at a fill ratio of 0.7. The Ntm values at the two scales are also different. Apparently, turbulent and convective mixing on both scales are different. An important difference between reactor11 and reactor1160 is the wall surface/liquid volume ratio. This ratio is higher at small scale in reactor11 (22.2 m2 m-3) than at large scale in reactor11-60 (6.66 m2 m-3). As a result, the fluid tends more toward solid-body rotation at large scale and, therefore, shows increased mixing times. The observed trends indicate that the mixing time depends on the scale and the liquid volume in contrast to the conclusion of Harnby,16 who stated that, for geometrically similar systems, Ntm is constant. Fox and Gex,21 Middleton,22 and Mersmann et al.23 have observed that, in vertical vessels, mixing times depend on
liquid volume. Fox and Gex correlate mixing times with liquid volume and momentum input according to
tm ∝
Vl0.5 (N2D4)0.42
(s)
(16)
Application of this approach has the advantage that the fill ratio can easily be implemented. When we use all presented mixing data, the following relationship is obtained
π x D L) ( 4 ) 66 2
tm
0.68
(N2D4)0.42
(s)
(17)
Obviously, this is in good agreement with eq 16. This empirical relationship can be used for scale-up of the Drais reactor for processes under turbulent conditions (Re > 105). Macromixing Times in Glycerin at Laminar Conditions. When the reactor is filled with glycerin at fill ratios lower than 1, the gas phase is finely dispersed in the liquid. In the milky liquid thus obtained, as shown in the photograph in Figure 7, it is impossible to perform spectrophotometric pulse-response measurements. Therefore, experiments have only been performed using completely filled reactors.
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Figure 13. (A) Dimensionless macromixing time against Re and impeller speed using glycerin. (B) tm as a function of j using glycerin.
Figure 14. tm as a function of j for (A) reactor11-60, (B) reactor11, (C) reactor15, and (D) reactor20.
In Figure 13A, Ntm is plotted against Re. It follows that the mixing time in glycerin is approximately 2.5-5 times longer than in water for the small and large reactors. Ntm is hardly dependent on Re on the different scales. Thus, in accordance with the observation of Harnby et al.,16 Ntm is independent of Re for both laminar and turbulent conditions. However, this is only valid for a given scale, as the Ntm values for reactor11 and reactor11-60 differ despite their similar geometries. This emphasizes our previous observation that the hydrodynamics in the two geometrically similar systems are significantly different. Fox and Gex21 also provided a correlation for mixing times under laminar conditions
tm ∝
Vl0.5 (N2D4)1.25
(s)
(18)
Using the same approach, we obtain
π D L) ( 4 ) 530 2
tm
0.98
(N2D4)0.44
(s)
(19)
Table 4. Standard Errors of the Constants in the Correlations for tm conditions
preexponential factor
exponent in π/4 D2L
exponent in N2D4
laminar (Re < 1730) turbulent (Re > 105)
519 ( 36.4 66 ( 3.1
0.98 ( 0.038 0.68 ( 0.022
-0.44 ( 0.014 -0.45 ( 0.022
The standard errors are given in Table 4. The differences between eqs 18 and 19 can be a result of different geometries. Also, the results of Fox and Gex might be biased, as their experiments were performed in one single vessel in which only the liquid height was varied. Macromixing Efficiencies. The term Ntm is an efficiency parameter that represents the number of revolutions required to obtain the desired mixing at time tm. According to this term, the highest mixing efficiency is found in reactor11 at a fill ratio of 0.7. Less than 10 impeller revolutions are required to obtain the required mixing, which is 5 times more efficient than in reactor11-60 or reactor20. In Figures 13B and 14A-D, the mixing times are plotted against the average energy dissipation j for laminar and turbulent conditions, respectively. j is calculated from eq 6, from which NP is calculated using the exponents in Table 3. From these figures, it can be
Ind. Eng. Chem. Res., Vol. 40, No. 22, 2001 4739
concluded that, even though j is the same, on average, tm in reactor11-60 is 5 times longer than in reactor11, although the length and diameter are only 3 times greater. In reactor11 at a fill ratio of 0.7, 1 m2 s-3 is required to obtain mixing times of 1 s, whereas in reactor20, the same j value provides a mixing time of 9 s. Therefore, an important conclusion is that scale-up of the Drais reactor based on energy consumption only will lead to longer mixing times. Concluding Remarks In this study, we have examined the hydrodynamics in the Drais reactor at different scales. It has been concluded that the specific power consumption as a function of Reynolds number is the same on small and large scales. Over a wide range of Re values, the power number NP is comparable to the value for vertical unbaffled reactors with propellers. However, it has been shown that scale-up with constant power consumption will lead to longer mixing times at larger scale. This is reflected in Ntm, which depends linearly on the fill ratio at large scale and with the square root at small scale.13 An important conclusion is that the hydrodynamics at different scales are different, making scale-up of the Drais reactor a difficult task. For scale-up purposes, however, the macromixing times have been correlated, using an approach as suggested by Fox and Gex21 that includes liquid volume and flux of momentum. Especially for turbulent conditions, the agreement in the obtained correlation is remarkable. Two flow states occur in this reactor, i.e., a slosh state at low values of Fr and a ring state at high values of Fr. The transition between the two states as a result of changing impeller speed shows hysteresis. Using the criterion Fr ) 1 in scale-up, one can determine the number of revolutions required to obtain the ring state. The ring state applies only for low-viscosity liquids. Using high-viscosity liquids, we observe a liquid phase with a dispersed gas phase. Nevertheless, a liquid film still exists near the cylindrical wall. The existence of this thin liquid film is important because the highest shear rates are obtained in this film, which is crucial for obtaining products with a high molecular weights in polymerization reactions.14 Mixing times under laminar conditions are approximately only 2.5 times higher than under turbulent conditions. In a previous paper, we showed that mixing under laminar conditions is very efficient because of the chaotic nature. Striations were made visible using planar laser-induced fluorescence and shown to stretch, fold, and reorient throughout the whole reactor. The deformation process of the striations appears to occur homogeneously, which corresponds to the observed homogeneous energy dissipation described in this paper. Because of this homogeneous dissipation, it can be concluded that the Drais reactor is a very efficient mixer, especially under laminar conditions. Acknowledgment The authors thank J. Surquin of Acordis Research (Arnhem, The Netherlands) for discussions and financial support. Also, the Stan Ackermans Institute (Eindhoven, The Netherlands) is acknowledged for financial support. Notation a ) constant b ) constant
c ) clearance, m C ) arbitrary concentration Cn ) normalized concentration C∞ ) final concentration Cp ) heat capacity, J kg-1 K-1 D ) vessel diameter, m Di ) impeller diameter, m fg ) geometrical factor Fr ) Froude number g ) acceleration of gravity, m s-2 h ) overall heat transfer coefficient, J K-1 s-1 h ) blade height, m k1-k5 ) constants L ) length of the vessel, m NP ) power number N ) agitation rate, s-1 P ) dissipated stirring energy, J s-1 Re ) Reynolds number t ) time, s tm ) macromixing time, s T ) liquid temperature, K Ta ) ambient temperature, K T0 ) temperature at t)0, K Vl ) liquid volume, m3 Vr ) reactor volume, m3 vtip ) impeller speed, m s-1 w ) blade width, m x ) fill ratio y ) distance, m Greek Symbols R ) blade angle, degrees β ) mutual blade angle, degrees γ˘ ) shear rate, s-1 j ) average power input per unit of mass, m2 m-3 µ ) dynamic liquid viscosity, kg m-1 s-1 ν ) kinematic viscosity, m2 s-1 F ) liquid density, kg m-3 σ2 ) variance
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(14) Agarwal, U. S.; Khakhar, D. V. Enhancement of polymerization rates for rigid rodlike molecules by shearing. Nature 1992, 360, 53. (15) Agarwal, U. S.; Khakhar, D. V. Shear flow induced orientation development during homogeneous solution polymerization of rigid rodlike molecules. Macromolecules 1993, 26, 3960. (16) Harnby, N., Edwards, M. F., Nienow, A. W., Eds. Mixing in the Process Industries, 2nd ed.; Butterworth-Heinemann Ltd.: London, 1992. (17) Metzner, A. B.; Otto, R. E. Agitation of Non-Newtonian Fluids. AIChE J. 1957, 3 (1), 3. (18) Thoenes, D. Chemical Reactor Development; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994. (19) Coulson, J. M.; Richardson, J. F.; Backhurst, J. R.; Harker, J. H. Coulson and Richardson’s Chemical Engineering 5th ed.; Butterworth-Heinemann Ltd.: Oxford, U.K., 1996; Vol. 1, Fluid flow, heat transfer and mass transfer.
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Received for review April 11, 2001 Revised manuscript received August 8, 2001 Accepted August 27, 2001 IE010328G
