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Scale-up of the power draw of inline-rotor–stator mixers with high throughput
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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Scale-up of the power draw of inline-rotor–stator mixers with high throughput Björn Schönstedt a, Hans-Joachim Jacob b, Carsten Schilde c,∗, Arno Kwade c a b c
Siegwerk Druckfarben AG & Co. KGaA, Alfred Keller Str. 55, 53721 Siegburg, Germany Ystral GmbH, Wettelbrunner Straße 7, 79282 Ballrechten-Dottingen, Germany Institute for Particle Technology, TU Braunschweig, Volkmaroder Str. 5, 38118 Braunschweig, Germany
a b s t r a c t Inline-rotor–stator mixers are widely used in diverse industrial mixing and dispersing processes. Despite their expanded presence in industrial processes, no general accepted method for the scale-up of this machine type has been established so far. This paper reports of a new developed approach to predict flow rate and power consumption within the scale-up of these machines. Based on an earlier published idea, the operating behavior of these machines is characterized by power curves. The approach is further developed, considering an often used procedure for the scale up of stirring processes with non-Newtonian fluids. Additionally, characteristic flow curves are used within the new approach, which allow the prediction of flow rate and power consumption in the scale-up of these machines. The experimental work was done with inline-rotor–stator mixers whose construction design is close to centrifugal pumps. Thus, very high throughputs are generated. Power curves and characteristic flow curves were recorded with polyglycerol–water mixtures. For the validation of the method, a shear thinning model dispersion with pyrogenic nanoparticles was processed in 4 machines of different sizes. The throughput of the machines ranged from 10 to 90 m3 /h. These experiments showed that the proposed model enables to predict flow rates and net power consumptions with good accuracy. © 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Scale-up; Rotor–stator mixer; Process engineering; Dispersing; Power draw
1.
Introduction
The scope of applications of inline-rotor–stator mixers ranges from the dispersing of pyrogenic nanoparticles over pharmaceutical products to bitumen (Hall et al., 2011). Basically, rotor–stator mixers consist of a dispersing chamber, including concentric mounted toothed rings, where rotating and static rings alternate. Due to the centrifugal forces induced by the rotating rings, a radial flow is created and the fluid in the dispersing chamber is pressed through the radial gaps of the rotating and the static rings. The stressing of the disperse phase is caused by turbulent and shear stresses, which occur due to the rotor–stator interaction of the toothed rings. Generally, it is one of the major issues within the development of new products to scale-up the production processes from laboratory- or pilot-scale to production scale. In a scaleup process it is necessary to appoint characteristic operating
parameters of the respective machines, which can be kept constant in order to ensure constant product quality. Furthermore a method is required that allows the dimensioning of the driving motor of the up-scaled machine. Commonly, for the scale-up of rotor–stator mixers, the tip speed of the rotor is kept constant in order to generate constant stress intensities due to the rotor–stator interaction. For the characterization of the power consumption of inline-rotor–stator mixers, several studies were published in the past few years (Baldyga et al., 2007; Hall et al., 2011; Kowalski et al., 2011; Ozcan-Taskın et al., 2011). However, no model is established by now which is generally accepted and used for the prediction of power draw in the scale-up of rotor–stator devices. The power consumption of common stirred devices is often described with a Reynolds-Diagram. In this diagram, the power consumption of the stirrer is described by the Power number Po [−] that can be plotted as a characteristic
∗
Corresponding author. Tel.: +49 5313919604; fax: +49 531 391 9633. E-mail address: [email protected] (C. Schilde). http://dx.doi.org/10.1016/j.cherd.2014.04.004 0263-8762/© 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Schönstedt, B., et al., Scale-up of the power draw of inline-rotor–stator mixers with high throughput. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.04.004
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function of the Reynolds number Re [−]. The Power number Po is defined as: Po =
P − P0 · n3 · d5
(1)
here, P is the power consumption measured on the driving shaft, P0 is the idle power measured without product in the stirred system, is density of the product suspension, n is the number of stirrer revolutions per seconds and d is the stirrer diameter. The stirrer Reynolds number depends on the fluid viscosity and represents the ratio of inertia to viscosity forces: Re =
· d2 · n
(2)
One of the first studies dealing with the power consumption of continuous operated rotor–stator-mixers was accomplished by Sparks (Sparks, 1996). Sparks carried out experiments with a inline-rotor–stator-mixer in industrial scale. One of the results of this study was that the commonly used Power number Po was inadequate to describe the power consumption of the used test inline-rotor–stator mixer. Since the construction of inline-rotor–stators shows strong analogies to centrifugal pumps, Sparks proposed an alternative Power number, which takes the flow rate Q˙ V into account. The initially proposed form of this equation is: PoRS =
P − P0 f
· QV · ng · dh
(3)
For the determination of the exponents, f, g and h, experimental data were fitted by numerical regression to Eq. (3). The resulting values were f = 0.75; g = 2.17; h = 2.4. Since the values found for f, g, and h were close to 1, 2 and 2, it was assumed that the exponents are integers, and that the found decimals were due to measuring inaccuracies. Thus, the empirical found Power number, PoRS , is a dimensionless characteristic number for the description of the power characteristics of inline-rotor–stator mixers: PoRS =
P − P0 · QV · n2 · d2
(4)
Baldyga (Baldyga et al., 2007) and Kowalski (Kowalski, 2009) introduced an alternative model for the description of the power consumption of inline-rotor–stator mixers. According to their model, the power consumption (P–P0 ) can be described by Eq. (5): P − P0 = Poz · · n3 · d5 + k1 · Q˙ m · n2 · d2
(5)
The term PoZ ··n3 ·d5 can be derived from Eq. (1) and describes the hydraulic power necessary to stir the fluid in the dispersing chamber in the absence of flow. The term on the right hand, k1 · Q˙ m · n2 · d2 , represents the power consumption required for the initiation of the mass flow. Poz and k1 are machine parameters, which have to be determined experimentally. Later works from Hall et al. (2011), Cooke et al. (2011) and Ozcan-Taskın et al. (2011) showed that this model is adequate to describe also the power consumption of rotor–stator mixers from Silverson Machines Ltd. (Chesham, UK) and from Ytron GmbH & Co. KG (Bad Endorf, Germany). Besides the modeling of measured power consumption, Hall et al. (2011) used this model for a scale-up prediction of the power draw of a
Silverson inline mixer with a 68 mm diameter rotor, based on experiments with a 38 mm diameter machine. In this study from Hall et al. the flow rate of normal operating conditions was adjusted without an external pump. At the highest rotor speeds and highest flow rates an external pump was integrated into the test rig. Thus, a slightly positive pressure on the inlet to the Silverson inline mixer could be ensured. The diversity of designs of different inline-rotor–stator mixers which were developed by the manufacturers does not step back behind the diversity of applications. One of the frequently used systems is the Conti TDS from Ystral GmbH (Ballrechten-Dottingen, Germany). This system has as an outstanding feature the ability to induct dry powder into the liquid and to disperse the inducted agglomerates in one step. The outer ring of the rotor–stator geometry consists of blades similar to the blade wheel of a centrifugal pump, generating a considerable flow rate. Because of the high flow rate which is created by the Conti TDS rotor–stator mixer, it is operated without an additional pump in the product line in the predominant number of cases. The intention of the here presented study was to develop and to verify a method which enables the prediction of power consumption of Conti TDS mixers in production scale based on experiments with machines in laboratory- or pilot scale.
2.
Experimental
Four inline-rotor–stator mixers were used for the experimental work. The size of the used machines ranged from small pilot scale (80 mm rotor diameter, flow rate ≈10 m3 /h)) to production scale (225 mm rotor, flow rate ≈90 m3 /h). The used machines were the Conti TDS 1, 2, 3, and 5, all supplied from Ystral GmbH. The arrangement of the test rigs was for all four machines similar and is shown in Fig. 1 together with the rotor stator geometry of the used inline-mixers. The machines were integrated into the test rig with horizontal arranged axis of rotation. A stirred vessel containing the product was located above the rotor–stator mixer in a manner that the containing product flows into the dispersing chamber due to the gravity. The product was conveyed backed into stirred vessel by the pumping effect of the rotor. This arrangement is typical for the operation of the inline-rotor–stator mixers and complies with the advisements of the supplier. The stirred vessels were equipped with a cooling jacket, respectively. Furthermore, each test rig contained a measurement of the electric power, product temperature measurement and a measurement of the flow rate. The main geometric parameters can be found in Fig. 2 and Table 1. The electrical power measurement of alternating current motors via current and voltage can lead to relevant inaccuracies due to low measured value acquisition frequencies as well as an incorrect measurement of the number of revolutions. Low frequencies lead to an erroneous modeling of the sine curve. A missing or insufficient measurement of the number of revolutions leads to a negligence of the
Table 1 – Main geometric parameters of the test rigs. Conti TDS 1 2 3 5
d1 [mm]
d2 [mm]
80 111.8 168 225
59 76.8 124 153
Product pipe diameter [mm] 32 32 65 80
Please cite this article in press as: Schönstedt, B., et al., Scale-up of the power draw of inline-rotor–stator mixers with high throughput. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.04.004
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Fig. 1 – Arrangement of the test rigs and geometry of the used rotor–stator geometry.
Table 3 – Experimental parameters for the evaluation of power consumption with glycerol-water mixtures.
Fig. 2 – Top view of the rotor with the main diameters. motor slip and a wrong calculated energy input. For this reasons, all tested devices were equipped with high-quality AC motors. For the motor control and the measurement of the power input a frequency converter of the company Danfoss was used. This converter is able to realize high frequencies of (4 kHz) and is equipped with a slip compensation in order to consider the correctnumber of revolutions for the power measurement and avoid electrical power losses. For the evaluation of the power characteristics of the rotor–stator mixers, the machines were operated with Newtonian model fluids of defined viscosities. As model fluids,
Machine
Rotor speed [min−1 ]
Glycerol content [%]
Conti TDS 3 Conti TDS 5
1500, 2000, 2500, 3000 1500, 2000, 2500, 3000
0, 30, 60, 70, 75, 80, 83, 86 0, 30, 60, 70, 75, 80, 83, 86
polyglycerol–water mixtures and glycerol–water mixtures with varying water content were used. The viscosity of these fluids is strongly dependent on the temperature and the water content. To avoid any non-conformity due to variations in process temperature or fluid compositions, the viscosity of each mixture was measured at 20, 30 and 40 ◦ C in preliminary tests. With these data a regression curve for the viscosity as a function of temperature was determined for each mixture. Combined with the measured process temperature, these regression curves allowed to assign the respective product viscosity to the measured power consumption. The detailed experimental parameters for operation of the rotor–stator mixers with the Newtonian fluids can be found in Tables 2 and 3 (viscosities and tip speed variations were used for the Reynolds number variation). Furthermore, the machines were operated with a shear thinning dispersion. This dispersion served as model fluid to verify the model under realistic operating conditions. For the shear thinning model dispersion pyrogenic silica nano particles of the type R812S supplied by Evonik Industries GmbH (Essen, Germany) were used as received. These particles were wetted in technical ethanol (purity > 98%, denatured with MEK). The solid content was adjusted to cm = 0.15. For stabilization hexamethyldisilazane was added to the dispersion. The detailed operating conditions of the experiments with the shear thinning model dispersion are listed in Table 4. As mentioned before, the tip speed is kept constant in the scale-up of rotor–stator processes in order to keep the stress intensities constant. In the case of the Conti TDS machine, the rotor–stator interaction does not take place at the outer rotor
Table 2 – Experimental parameters for the evaluation of power consumption with polyglycerol-water mixtures. Machine Conti TDS 1 Conti TDS 2 Conti TDS 3 Conti TDS 5
Rotor speed [min−1 ] 5659, 6600, 7545, 8488, 9431 3450, 4140, 4830, 5520, 6210, 6900 1000, 1500, 2000, 2500, 3000, 3600 1000, 1500, 2000, 2500, 3000, 3600
Polyglycerol content [%] 0, 15, 37, 50 0, 5, 10, 15, 20, 37, 50 0.45, 0.5, 0.55 0.45, 0.5, 0.55
Please cite this article in press as: Schönstedt, B., et al., Scale-up of the power draw of inline-rotor–stator mixers with high throughput. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.04.004
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Table 4 – Experimental parameters for the evaluation of the scale-up with the model dispersion. Machine
Rotor–stator tip speed vs [m/s]
Conti TDS 1 Conti TDS 2 Conti TDS 3 Conti TDS 5
24; 28 24; 28 24 24
diameter, but at the diameter d2 (compare Fig. 2). Therefore, the tangential speed at the diameter d2 , in the following called rotor–stator tangential speed vt,RS , should be kept constant for the scale-up of the Conti TDS.
3.
Results and discussion
3.1. Characterization of flow rate and power consumption
Fig. 4 – Net power consumption of the Conti TDS 2 as function of the flow rate.
The machines were operated with fluids of varying viscosities at different rotor speeds. Due to these variations, different power consumptions and flow rates appeared. Fig. 3 shows exemplarily the measured flow rates of the Conti TDS 5. The general characteristics of the shown flow rate curves are valid for all four examined machines, although the absolute values differ naturally. For moderate rotor speeds, the flow rates increase almost linear with the rotor speeds. For rotor speeds exceeding 1500 min−1 , the slopes of the flow rate curves decrease and only a slight increase of the flow rate can be detected for increasing rotor speeds. For the two lowest viscosities, 1 and 3 mPa s, it even can be detected a maximal flow rate at approximately 2000 min−1 . For these viscosities the flow rates diminishes slightly for higher rotor speeds. The reason for this behavior is presently not sure. It is thinkable that the blades of the rotor head cause cavitation for higher rotor speeds. Due to increasing cavitation effects with rising rotor speeds, the pumping effect is thus decreased, what might lead to the found characteristic flow rates. Another reason could be of course the flow restriction due to the pressure drop at high flow rates. For further examination, the net power consumption of the rotor–stator mixers was measured as function of the flow rate, according to Cooke et al. (2011). The flow rate was adjusted by closing the valve at the product back flow of the test rig (see Fig. 1). Fig. 4 shows measured power data gained by this procedure for the Conti TDS 2. In principal, the power consumption increases with the flow rate. For high flow rates the
increase of power consumption approaches a constant value for both examples shown. The behavior that was found for the Conti TDS machines and is shown in Figs. 3 and 4 differs from that which was found by Cooke et al. (2011). In the study of Cooke, a strongly linear relation between rotor speed and flow rate and between flow rate and net power consumption was detected for Silverson mixers. The operating behavior of the Conti TDS machines seems to differ significantly from that of Silverson and Ytron mixers. Furthermore, the detected nonlinear dependency of the Conti TDS net power consumption from the flow rate is in contradiction with the model of Kowalski and Baldyga. Thus, other approaches are required for the description of the Ystral machines used here. Fig. 5 shows the power number Po (Eq. (1)) and the Power number for rotor–stator-mixers according to Sparks (Eq. (4)) plotted as function of the Reynolds number Re. The data of the Power number Po do not show a unique development. In fact, each set of data that was collected with one polyglycerol–water mixture has its own development and slope. In contrast, the progression of the PoRS values according to Sparks fulfills the expectations to a characteristic power curve. Since the Conti TDS concept is relatively close to the design of centrifugal pumps, it makes sense to take the flow rate into account for the modeling of the power consumption of this rotor–stator geometry.
Fig. 3 – Flow rates of the Conti TDS 5 measured as function of the rotor speed for different viscosities.
Fig. 5 – Power number Po and Power number for rotor–stator mixers PoRS of the Conti TDS 2 plotted as function of the Reynolds number.
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Table 5 – For the power curves determined fit paramters. Machine
Fig. 6 – Power numbers PoRS for all examined Conti TDS machines as function of the Reynolds numbers. A comparison of the results in Fig. 5 with former studies indicates that each geometry concept of rotor–stator dispersing machines has its own characteristic operating behavior. Cooke et al. (2011) gained for the Silverson data a well looking power curve by plotting the “classic” Power number Po versus the Reynolds number. The same is true for the study of Ozcan-Taskın et al. (2011). Note that the PoRS curve in Fig. 5 becomes flat (indicating turbulent regime) for Re > 100,000, what I quite high compared to “normal” power curves. However, the normally used power number Po was originally developed for simple stirrer in large reactors. Since the investigated rotor–stator geometries in this study are much more complex compared to these reactor geometries, a different power curve can be expected. Similar results were observed for the power curves of stirred media mills (Becker et al., 2001). In this devices, the curves depend strongly on the mill geometry and for some geometries a flat power curve was also only observed for Re > 1,000,000. The value PORS is an empirical found Power number, which takes the flow rate into account. In the study of Özcan-Taskin it was shown that the Power number PoRS according to Sparks did not deliver decent results for the characterization of Silverson and the Ytron mixers. However, for the here used rotor–stator geometry of the Conti TDS mixers, the empirical power number PoRS gives reasonable power curves for all investigated machines, as can be seen in Fig. 6. Normally, all the different sized machines should all have a similar power curve. The fact, that the PORS curves do not look all similar may be due to the fact that the dispersing chambers are not all exactly geometrical similar. However, one of the strength of the presented method is that the scale up is possible even when the flow curves are not identical. For the plot of the data points as function of the Reynolds number in a double logarithmic scaled diagram, each graph can be divided into three regions of constant slopes. Analogously to plots with the classical Power number, these regions can be assigned to a laminar, a turbulent and a transition region. In Fig. 6, the progression of the measured data for the regions with constant slope was fitted with exponential functions of the form: PoRS = a · Reb
(6)
The values for the parameters a and b in Eq. (6) were determined with least square method and are given in Table 5.
Region
A
b
Conti TDS 1
Laminar Transition
5349.76 61.89
−0.68 −0.10
Conti TDS 2
Laminar Transition Turbulent
198042.57 128.38 8.70
−1.12 −0.22 0.02
Conti TDS 3
Laminar Transition Turbulent
35100000 44926 54.20
−1.80 −0.89 −0.11
Conti TDS 5
Laminar Transition Turbulent
8656.19 124.72 44.66
−0.68 −0.18 −0.09
Furthermore, the mean deviations between the measured data and the fit function were calculated. They were determined to 4.42% for the Conti TDS 1, 5.45% for the Conti TDS 2, 5.27% for the Conti TDS 3 and 4.32% for the Conti TDS 5. The deviations may be due to the measurement of the power consumption via the electrical power. Kowalski (Kowalski et al., 2011) showed, that the determination via measurement of torque on the shaft is more accurate. The measurement of flow rate by electromagnetic flowmeters may be a further source of impreciseness.
3.2.
Prediction of the power consumption
The ability to model the operating parameters of Conti TDS machines by means of the power number PoRS shall be used in the following to establish a scale-up model for these machines. The power curves presented in Fig. 6 were determined by the use of Newtonian fluids with defined viscosities. However, in real processes Newtonian fluids are rather rarely processed. Especially in the case of dispersions, a shear thinning flow behavior can mostly be observed. The viscosity of shear thinning fluids depends on the applied shear rate and, therefore, no unique viscosity value exists for the calculation of the Reynolds number. For the scale up of stirred systems with shear thinning fluids an often used approach is that of Rieger and Novak (1974), where an apparent viscosity app of the shear thinning fluid is determined. In the first instance of this approach, the power number of the “small scale” stirrer is calculated for the respective operating parameters. With this Power number, the corresponding Reynolds number can be determined from the power curve. By rearranging Eq. (2) to , a so called apparent viscosity app can be calculated with this value of the Reynolds number. With the so determined apparent viscosity, app , the corresponding Reynolds number of the “large scale” stirrer is calculated and the power curve of the “large scale” stirrer gives the corresponding Power number. Thus, by rearranging Eq. (1), the net power consumption can be determined. This approach is frequently used and has been adopted also to other machines than stirred vessels (Becker et al., 2001). Fig. 7 shows the application of this approach to the power curves of the Conti TDS 1 and Conti TDS 3. On the basis of operating parameters of the “small scale” Conti TDS 1, the corresponding Power number PoRS of the “large scale” Conti TDS 3 is determined. Similar to the scale up of stirred devices, Eq. (4) has to be rearranged to the net power consumption: P − P0 = · QV · n2 · d2 · PoRS
(7)
Please cite this article in press as: Schönstedt, B., et al., Scale-up of the power draw of inline-rotor–stator mixers with high throughput. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.04.004
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Fig. 7 – Adoption of the approach from Rieger and Novak for the scale up of Conti TDS rotor–stator mixers. For the calculation of the net power consumption according to Eq. (7), almost all values are known. The Power number PoRs is delivered by the approach from Rieger and Novak. The density of the dispersion and the rotor diameter are usually known and the required rotor speed, n, is determined by the rotor–stator tangential velocity vr,RS that is kept constant within the scale-up process. However, the flow rate that results from the respective operating parameters has still to be identified. The flow rate which is initiated by a certain machine depends on the rotor speed and the product viscosity. The characteristic curves which were gained for the flow rate as function of rotor speed and viscosity (Fig. 3) can hence deliver the necessary data. The resulting flow rate for a certain set of operating parameters can easily be read off. To ensure that the correct viscosity is used, again the apparent viscosity app from the approach of Rieger and Novak can be used. If the calculated apparent viscosity app corresponds not to one of the recorded characteristic curves, the respective value of flow rate can be gained by linear interpolation. In case of a zero flow rate, Eq. (4) is divided by zero and is hence not defined. This means, that the presented model is not intended or able to predict the power consumption with a flow rate of zero or very low flow rates around zero. However, the case of a zero flow rate is no operating condition that is of practical interest for the dimensioning of the driving motors or the design of rotor–stator processes. Neither is it of special importance for the daily operation of rotor–stator mixers. Moreover, the similarity of the terms k1 · Q˙ m · n2 · d2 (right hand of Eq. (5)) and the term P − P0 = · QV · n2 · d2 · PoRS (Eq. (7)) is obvious. In other words, k1 = PoRS . Despite the fact, that the cited publications found k1 to be constant, in this study k1 depends on Re. However, the challenge is to determine k1 (or PoRS ) for given operating conditions. For this determination, the approach described in this study manuscript is currently the only investigated method. In order to verify the presented scale-up model, all 4 machines were operated with a shear thinning model fluid, containing 15 wt.% pyrogenic silica particles (Oswald de Waale: = k· n ; with k = 0.319 and n = 0.564; R2 = 0.985). Based on the data measured on the Conti TDS 1, the power consumption of the “large scale” machines was calculated theoretically according to the presented model. These predictions are compared to measured data. As mentioned initially, for the scale up of Conti TDS machines the rotor–stator tangential velocity vt,RS of the rotors is commonly kept constant. To verify this approach, the model
Fig. 8 – Comparison of the agglomerate sizes that are reached at tangential speeds of 24 and 28 m/s with machines of different sizes. dispersion was processed with two machines until no more decrease of the agglomerate size was detectable. This was done with the Conti TDS 1 and the Conti TDS 2, respectively with a tangential speed of 24 and 28 m/s. The measured agglomerate sizes are plotted in Fig. 8 as function of the consumed specific energy, Em . Though the progression of the agglomerate sizes are not exactly analogously for the different machines, is can be seen, that for the same tangential speeds, similar agglomerate sizes are reached at the end of the dispersing process. Similar product fineness at the end of a dispersion process is equivalent to similar maximum stress intensities acting on the product (if the particle, aggregate or agglomerate strength is constant, e.g. the same product is dispersed) (Schilde et al., 2010, 2013). Furthermore, the specific energy inputs required to reach the final agglomerate sizes are also comparable. Consequently, for the prediction of power consumption, the rotor speeds were adjusted .in a way that the tangential velocity vt,RS was constant at 24 and 28 m/s, respectively. The operation data of the Conti TDS 1, which served for the prediction of the data of the larger machines is listed in Table 6. In Table 7 the calculated and the actual measured data of the “large scale” machines are compared. In principal, the theoretical calculated and the measured data are in good accordance. Nevertheless, the theoretical calculated values for flow rate and power consumption are systematically deviating from the measured ones. In this case, the deviations lay between 0.7% and 11.3% for the predicted power. The reason for these deviations might be found in the use of the apparent viscosity for the assignment of the flow rate. The values found for the apparent viscosities reflect the extreme high shear rates appearing in the dispersing chamber due to rotor–stator interactions. However, the shear rates appearing in the product line can be assumed considerably
Table 6 – Operating parameters of the Conti TDS 1 with the shear thinning model dispersion. Shear velocity [m/s] 28 24
Rotor speed [min−1 ] 9400 8200
Net-power [kw] 9.14 6.72
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Table 7 – Comparison of measured and modeled operation data of “large scale” rotor stator machines processing the shear thinning model fluid. Shear velocity [m/s]
Rotor speed [min−1 ]
Flow rate modeled [m3 /h]
Flow rate measured [m3 /h]
Conti TDS 2 28 24
6900 6210
14.43 13.2
14.07 13.05
Conti TDS 3 24
3600
50.36
47.5
Conti TDS 5 24
3000
72
69
Net power modeled [kw]
Net power measured [kw]
7.84 5.8
Deviation power [%]
7.8 5.4
0.7% 6.9%
19.6
18.0
9.0%
38.7
36.6
5.6%
Table 8 – Measured power consumption and with measured flow rate calculated power consumption of “large scale” rotor stator machines processing the shear thinning model fluid. Shear velocity [m/s]
Rotor speed [min−1 ]
Net power modeled [kW]
Net power measured [kW]
Deviation power [%]
Conti TDS 2 28 24
6900 6210
Conti TDS 3 24
3600
18
18.6
3.5%
Conti TDS 5
3000
36.6
37.1
1.2%
7.8 5.40
lower and the apparent viscosity of the shear thinning fluid accordingly higher. Higher viscosities result in lower flow rates due to increased frictional forces. According to Eq. (7) these systematically overestimated flow rates have a linear impact on the predicted power consumption. Table 8 shows power data which were calculated according to Eq. (7) with the actual measured flow rates. As Power number PoRS , the values gained by the approach of Rieger and Novak were taken as before. The net power consumption that was calculated this way, is considerably closer to the measured data than the before calculated values. The accuracy of the predicted power consumption could thus be enhanced by a more accurate prediction of the occurring flow rates. Therefore it is necessary to draw more sophisticated conclusions from the experiments with the “small scale” machine on the flow rates occurring in the “large scale” machines. The measured and predicted flow rates of all 4 investigated machines are shown in Table 9. The prediction of the flow rates of the “small scale” machine Conti TDS 1 was gained by reading off the values of the respective characteristic curves of the Conti TDS 1 machine. As it was seen before for the “large scale” machines, the actual measured flow rate is slightly higher than it would have been predicted based on the apparent
7.78 5.8
0.2% 7.8%
viscosity app . Additionally, the ratios R of predicted to measured flow rate are listed in Table 9. The ratios R of predicted to measured flow rate range between 1.01 and 1.07. The ratio of predicted to measured power consumption is, within the accuracy of measurement, for all examined machines in the same order of magnitude. This indicates that the shear rates in the product line deviate in the same range from those in the dispersing chamber for all experimental set-ups. Regarding an accurate prediction of the flow rates, these similar ratios R are important since this finding allows concluding from the flow rates of the “small scale” on those of the “large scale” rotor–stator machines. The predicted net power consumption can thus be calculated by correcting the flow rate determined by the apparent viscosity app with the ratio R determined by the small scale experiments: P − P0 =
QV · n2 · d2 · PoRS · R
(8)
The values of the net power consumption predicted according to Eq. (8) are compared in Table 10 with the measured net power consumptions. The values of R are taken from the corresponding rows of the Conti TDS 1 in Table 9. As can be seen,
Table 9 – Measured and modeled flow rates of all 4 investigated machines. Shear velocity [m/s]
Rotor speed [min−1 ]
Flow rate measured
Flow rate predicted
Ratio R of predicted to measured flow rate
Conti TDS 1 28 24
9400 8200
10.56 10.20
10.92 10.96
1.03 1.07
Conti TDS 2 28 24
6900 6210
14.1 13.0
14.43 13.20
1.03 1.01
Conti TDS 3 24
3600
47.5
50.4
1.06
Conti TDS 5 24
3000
69
72
1.04
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Table 10 – Comparison of measured and modeled operation data of “large scale” rotor stator machines processing the shear thinning model fluid. Shear velocity [m/s]
Rotor speed [min−1 ]
Net power modeled [kW]
Net power measured [kW]
Deviation power [%]
Conti TDS 2 28 24
6900 6210
Conti TDS 3 24
3600
18
18.6
3.5%
Conti TDS 5
3000
36.6
36.0
−1.7%
7.5 5.40
the predicted power consumptions are close to the measured data and the deviations do not exceed the mean deviations of the measured data to the fit functions. Due to the proposed scale-up model it is thus possible, to predict the power consumption and the flow rates of inlinerotor–stator mixers of the type Ystral Conti TDS within in the limits of measurement accuracy. However, one has to be aware, that the predicted power depends strongly on the occurring flow rates. Hence for the scale-up of machines it has to be considered, that the product pipes of the later production installation has to be similar to that one, with that the characteristic curves of the respective “large scale” machine were recorded. A significant change in the piping installation results in a change of the pressure drop due to frictional forces and leads to changing flow rates. However, this restriction seems to be unavoidable since also an earlier developed scale-up model for other types of inline-rotor–stators has the occurring flow rate as central parameter for the calculation of the power consumption (Bałdyga et al., 2008; Hall et al., 2011; Kowalski, 2009; Kowalski et al., 2011). The adjustment of the flow rate by an external pump might overcome this problem. However, this solution is, at least for inline-rotor–stator mixers of the type Conti TDS, not in accordance with actual practice. Continuous operated rotor–stator mixers of the type Conti TDS generate a strong flow rate, wherefore a separate pump is usually not integrated into the product line. Furthermore, the piping installations that were used within this work can be regarded as typical installations for the respective machines. These installations are recommended by the manufacturer and are similarly used in the majority of practical applications. A direct comparison of the equation for the calculation of power consumption of centrifugal pumps (Eq. (9)) with Eq. (7) shows an interesting aspect of the Power number PoRS : P − P0 = w · p · QV
(9)
Here, w is the pump efficiency factor and p is the pressure drop. Equalizing both terms of Eqs. (7) and (9) leads to: PoRS · · n2 · d2 · QV = w · p · QV
(10)
The term PoRS ··n2 ·d2 has the dimension [N/m2 ] and can thus be seen as an equivalent of the product of pressure drop and pumping efficiency factor. The characteristic number PoRS seems thus to be suited to describe the ratio of power on shaft to the induced hydraulic power for rotor stator mixers of the type Conti TDS. Compared to the classical power number Po, which was initially developed for disk agitators, it has the advantage that it takes flow rate, which is induced by the rotor, into account. On the other hand, compared to the approach usually used for centrifugal pumps, it has the advantage, that
7.78 5.4
3.2% −0.5%
it has a unique characteristic curve for a given piping and can be used also for non-Newtonian fluids. From this point of view, one might state that by means of this characteristic curve, the product of pump efficiency factor and pressure drop can be calculated for different product viscosities and rotor speeds.
4.
Conclusion
A model was developed that allows the prediction of the net power consumption and the occurring flow rates of inlinerotor–stator mixers of the type Conti TDS. The scale-up model is based on an earlier idea from Sparks to model the power consumption of inline-rotor–stator mixers with a Power number which takes the flow rate into account. This idea of a Power number for inline-rotor–stator mixers is combined with the approach of Rieger and Novak for the scale up of stirring processes with non-Newtonian fluids. An apparent viscosity is determined and used for the determination of a power number of the large scale machine. Furthermore, the occurring flow rate has to be predicted by the use of characteristic flow curves. A critical point in the scale-up approach is that in the rotor–stator-machine and in the pipe different apparent viscosities appear. The flow rates which are predicted based on the apparent viscosity inside the rotor–stator-machine show a systematic deviation to the actual occurring flow rates. A correction factor for the actual occurring flow rates can be gained by experiments with the small scale machine. The proposed model was validated by experiments with machines of different sizes, ranging from small pilot-scale to production scale. For the validation of the new approach, a shear thinning dispersion was processed with all machines. The power consumptions and the flow rates of inline-rotor–stator mixers of the type Ystral Conti TDS were predicted within the limits of measurement accuracy. Thus, the approach presented in this study enables the scale-up of power draw of inline-rotor–stator with high throughputs.
Acknowledgements The authors would like to thank P. Hartmann and C.-D. Laser (TU Braunschweig) as well as K. Ebbinghaus (Ystral GmbH) for the help with the experimental part of the work.
References Bałdyga, J., Makowski, Ł., Orciuch, W., Sauter, C., Schuchmann, H.P., 2008. Agglomerate dispersion in cavitating flows. Chem. Eng. Res. Des. 87, 474–484. Baldyga, J.A., Kowalski, A.J., Cooke, M., Jasinska, M., 2007. Investigations of micromixing in the rotor–stator mixer. Chem. Process Eng. 28, 867–877.
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Becker, M., Ford, W.N., Gutsche, O., 2001. Power draw and scale-up of stirred media mills. In: 3rd European Congress of Chemical Engineering, Chemie Ingenieur Technik, Nürnberg, p. 703. Cooke, M., Rodgers, T.L., Kowalski, A.J., 2011. Power consumption characteristics of an in-line silverson high shear mixer. AIChE J. 58, 1683–1692. Hall, S., Cooke, M., Pacek, A.W., Kowalski, A.J., Rothman, D., 2011. Scaling up of Silverson rotor–stator mixers. Can. J. Chem. Eng. 89, 1040–1050. Kowalski, A.J., 2009. An expression for the power consumption of in-line rotor–stator devices. Chem. Eng. Process. 48, 581–585. Kowalski, A.J., Cooke, M., Hall, S., 2011. Expression for turbulent power draw of an in-line Silverson high shear mixer. Chem. Eng. Sci. 66, 241–249.
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Ozcan-Taskın, G., Kubicki, D., Padron, G., 2011. Power and flow characteristics of three rotor–stator heads. Can. J. Chem. Eng. 89, 1005–1017. Rieger, F., Novak, V., 1974. Power consumption scale-up in agitating non-Newtonian fluids. Chem. Eng. Sci. 29, 2229–2234. Schilde, C., Breitung-Faes, S., Kampen, I., Kwade, A., 2013. Grinding kinetics of nano-sized particles for different electrostatic stabilizing acids in a stirred media mill. Powder Technol. 235, 1008–1016. Schilde, C., Kampen, I., Kwade, A., 2010. Dispersion kinetics of nano-sized particles for different dispersing machines. Chem. Eng. Sci. 65, 3518–3527. Sparks, T., 1996. Fluid Mixing in Rotor Stators, Biotechnology Centre. Cranfield University, Cranfield.
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Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Scale-up of the power draw of inline-rotor–stator mixers with high throughput Björn Schönstedt a, Hans-Joachim Jacob b, Carsten Schilde c,∗, Arno Kwade c a b c
Siegwerk Druckfarben AG & Co. KGaA, Alfred Keller Str. 55, 53721 Siegburg, Germany Ystral GmbH, Wettelbrunner Straße 7, 79282 Ballrechten-Dottingen, Germany Institute for Particle Technology, TU Braunschweig, Volkmaroder Str. 5, 38118 Braunschweig, Germany
a b s t r a c t Inline-rotor–stator mixers are widely used in diverse industrial mixing and dispersing processes. Despite their expanded presence in industrial processes, no general accepted method for the scale-up of this machine type has been established so far. This paper reports of a new developed approach to predict flow rate and power consumption within the scale-up of these machines. Based on an earlier published idea, the operating behavior of these machines is characterized by power curves. The approach is further developed, considering an often used procedure for the scale up of stirring processes with non-Newtonian fluids. Additionally, characteristic flow curves are used within the new approach, which allow the prediction of flow rate and power consumption in the scale-up of these machines. The experimental work was done with inline-rotor–stator mixers whose construction design is close to centrifugal pumps. Thus, very high throughputs are generated. Power curves and characteristic flow curves were recorded with polyglycerol–water mixtures. For the validation of the method, a shear thinning model dispersion with pyrogenic nanoparticles was processed in 4 machines of different sizes. The throughput of the machines ranged from 10 to 90 m3 /h. These experiments showed that the proposed model enables to predict flow rates and net power consumptions with good accuracy. © 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
Keywords: Scale-up; Rotor–stator mixer; Process engineering; Dispersing; Power draw
1.
Introduction
The scope of applications of inline-rotor–stator mixers ranges from the dispersing of pyrogenic nanoparticles over pharmaceutical products to bitumen (Hall et al., 2011). Basically, rotor–stator mixers consist of a dispersing chamber, including concentric mounted toothed rings, where rotating and static rings alternate. Due to the centrifugal forces induced by the rotating rings, a radial flow is created and the fluid in the dispersing chamber is pressed through the radial gaps of the rotating and the static rings. The stressing of the disperse phase is caused by turbulent and shear stresses, which occur due to the rotor–stator interaction of the toothed rings. Generally, it is one of the major issues within the development of new products to scale-up the production processes from laboratory- or pilot-scale to production scale. In a scaleup process it is necessary to appoint characteristic operating
parameters of the respective machines, which can be kept constant in order to ensure constant product quality. Furthermore a method is required that allows the dimensioning of the driving motor of the up-scaled machine. Commonly, for the scale-up of rotor–stator mixers, the tip speed of the rotor is kept constant in order to generate constant stress intensities due to the rotor–stator interaction. For the characterization of the power consumption of inline-rotor–stator mixers, several studies were published in the past few years (Baldyga et al., 2007; Hall et al., 2011; Kowalski et al., 2011; Ozcan-Taskın et al., 2011). However, no model is established by now which is generally accepted and used for the prediction of power draw in the scale-up of rotor–stator devices. The power consumption of common stirred devices is often described with a Reynolds-Diagram. In this diagram, the power consumption of the stirrer is described by the Power number Po [−] that can be plotted as a characteristic
∗
Corresponding author. Tel.: +49 5313919604; fax: +49 531 391 9633. E-mail address: [email protected] (C. Schilde). http://dx.doi.org/10.1016/j.cherd.2014.04.004 0263-8762/© 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
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function of the Reynolds number Re [−]. The Power number Po is defined as: Po =
P − P0 · n3 · d5
(1)
here, P is the power consumption measured on the driving shaft, P0 is the idle power measured without product in the stirred system, is density of the product suspension, n is the number of stirrer revolutions per seconds and d is the stirrer diameter. The stirrer Reynolds number depends on the fluid viscosity and represents the ratio of inertia to viscosity forces: Re =
· d2 · n
(2)
One of the first studies dealing with the power consumption of continuous operated rotor–stator-mixers was accomplished by Sparks (Sparks, 1996). Sparks carried out experiments with a inline-rotor–stator-mixer in industrial scale. One of the results of this study was that the commonly used Power number Po was inadequate to describe the power consumption of the used test inline-rotor–stator mixer. Since the construction of inline-rotor–stators shows strong analogies to centrifugal pumps, Sparks proposed an alternative Power number, which takes the flow rate Q˙ V into account. The initially proposed form of this equation is: PoRS =
P − P0 f
· QV · ng · dh
(3)
For the determination of the exponents, f, g and h, experimental data were fitted by numerical regression to Eq. (3). The resulting values were f = 0.75; g = 2.17; h = 2.4. Since the values found for f, g, and h were close to 1, 2 and 2, it was assumed that the exponents are integers, and that the found decimals were due to measuring inaccuracies. Thus, the empirical found Power number, PoRS , is a dimensionless characteristic number for the description of the power characteristics of inline-rotor–stator mixers: PoRS =
P − P0 · QV · n2 · d2
(4)
Baldyga (Baldyga et al., 2007) and Kowalski (Kowalski, 2009) introduced an alternative model for the description of the power consumption of inline-rotor–stator mixers. According to their model, the power consumption (P–P0 ) can be described by Eq. (5): P − P0 = Poz · · n3 · d5 + k1 · Q˙ m · n2 · d2
(5)
The term PoZ ··n3 ·d5 can be derived from Eq. (1) and describes the hydraulic power necessary to stir the fluid in the dispersing chamber in the absence of flow. The term on the right hand, k1 · Q˙ m · n2 · d2 , represents the power consumption required for the initiation of the mass flow. Poz and k1 are machine parameters, which have to be determined experimentally. Later works from Hall et al. (2011), Cooke et al. (2011) and Ozcan-Taskın et al. (2011) showed that this model is adequate to describe also the power consumption of rotor–stator mixers from Silverson Machines Ltd. (Chesham, UK) and from Ytron GmbH & Co. KG (Bad Endorf, Germany). Besides the modeling of measured power consumption, Hall et al. (2011) used this model for a scale-up prediction of the power draw of a
Silverson inline mixer with a 68 mm diameter rotor, based on experiments with a 38 mm diameter machine. In this study from Hall et al. the flow rate of normal operating conditions was adjusted without an external pump. At the highest rotor speeds and highest flow rates an external pump was integrated into the test rig. Thus, a slightly positive pressure on the inlet to the Silverson inline mixer could be ensured. The diversity of designs of different inline-rotor–stator mixers which were developed by the manufacturers does not step back behind the diversity of applications. One of the frequently used systems is the Conti TDS from Ystral GmbH (Ballrechten-Dottingen, Germany). This system has as an outstanding feature the ability to induct dry powder into the liquid and to disperse the inducted agglomerates in one step. The outer ring of the rotor–stator geometry consists of blades similar to the blade wheel of a centrifugal pump, generating a considerable flow rate. Because of the high flow rate which is created by the Conti TDS rotor–stator mixer, it is operated without an additional pump in the product line in the predominant number of cases. The intention of the here presented study was to develop and to verify a method which enables the prediction of power consumption of Conti TDS mixers in production scale based on experiments with machines in laboratory- or pilot scale.
2.
Experimental
Four inline-rotor–stator mixers were used for the experimental work. The size of the used machines ranged from small pilot scale (80 mm rotor diameter, flow rate ≈10 m3 /h)) to production scale (225 mm rotor, flow rate ≈90 m3 /h). The used machines were the Conti TDS 1, 2, 3, and 5, all supplied from Ystral GmbH. The arrangement of the test rigs was for all four machines similar and is shown in Fig. 1 together with the rotor stator geometry of the used inline-mixers. The machines were integrated into the test rig with horizontal arranged axis of rotation. A stirred vessel containing the product was located above the rotor–stator mixer in a manner that the containing product flows into the dispersing chamber due to the gravity. The product was conveyed backed into stirred vessel by the pumping effect of the rotor. This arrangement is typical for the operation of the inline-rotor–stator mixers and complies with the advisements of the supplier. The stirred vessels were equipped with a cooling jacket, respectively. Furthermore, each test rig contained a measurement of the electric power, product temperature measurement and a measurement of the flow rate. The main geometric parameters can be found in Fig. 2 and Table 1. The electrical power measurement of alternating current motors via current and voltage can lead to relevant inaccuracies due to low measured value acquisition frequencies as well as an incorrect measurement of the number of revolutions. Low frequencies lead to an erroneous modeling of the sine curve. A missing or insufficient measurement of the number of revolutions leads to a negligence of the
Table 1 – Main geometric parameters of the test rigs. Conti TDS 1 2 3 5
d1 [mm]
d2 [mm]
80 111.8 168 225
59 76.8 124 153
Product pipe diameter [mm] 32 32 65 80
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Fig. 1 – Arrangement of the test rigs and geometry of the used rotor–stator geometry.
Table 3 – Experimental parameters for the evaluation of power consumption with glycerol-water mixtures.
Fig. 2 – Top view of the rotor with the main diameters. motor slip and a wrong calculated energy input. For this reasons, all tested devices were equipped with high-quality AC motors. For the motor control and the measurement of the power input a frequency converter of the company Danfoss was used. This converter is able to realize high frequencies of (4 kHz) and is equipped with a slip compensation in order to consider the correctnumber of revolutions for the power measurement and avoid electrical power losses. For the evaluation of the power characteristics of the rotor–stator mixers, the machines were operated with Newtonian model fluids of defined viscosities. As model fluids,
Machine
Rotor speed [min−1 ]
Glycerol content [%]
Conti TDS 3 Conti TDS 5
1500, 2000, 2500, 3000 1500, 2000, 2500, 3000
0, 30, 60, 70, 75, 80, 83, 86 0, 30, 60, 70, 75, 80, 83, 86
polyglycerol–water mixtures and glycerol–water mixtures with varying water content were used. The viscosity of these fluids is strongly dependent on the temperature and the water content. To avoid any non-conformity due to variations in process temperature or fluid compositions, the viscosity of each mixture was measured at 20, 30 and 40 ◦ C in preliminary tests. With these data a regression curve for the viscosity as a function of temperature was determined for each mixture. Combined with the measured process temperature, these regression curves allowed to assign the respective product viscosity to the measured power consumption. The detailed experimental parameters for operation of the rotor–stator mixers with the Newtonian fluids can be found in Tables 2 and 3 (viscosities and tip speed variations were used for the Reynolds number variation). Furthermore, the machines were operated with a shear thinning dispersion. This dispersion served as model fluid to verify the model under realistic operating conditions. For the shear thinning model dispersion pyrogenic silica nano particles of the type R812S supplied by Evonik Industries GmbH (Essen, Germany) were used as received. These particles were wetted in technical ethanol (purity > 98%, denatured with MEK). The solid content was adjusted to cm = 0.15. For stabilization hexamethyldisilazane was added to the dispersion. The detailed operating conditions of the experiments with the shear thinning model dispersion are listed in Table 4. As mentioned before, the tip speed is kept constant in the scale-up of rotor–stator processes in order to keep the stress intensities constant. In the case of the Conti TDS machine, the rotor–stator interaction does not take place at the outer rotor
Table 2 – Experimental parameters for the evaluation of power consumption with polyglycerol-water mixtures. Machine Conti TDS 1 Conti TDS 2 Conti TDS 3 Conti TDS 5
Rotor speed [min−1 ] 5659, 6600, 7545, 8488, 9431 3450, 4140, 4830, 5520, 6210, 6900 1000, 1500, 2000, 2500, 3000, 3600 1000, 1500, 2000, 2500, 3000, 3600
Polyglycerol content [%] 0, 15, 37, 50 0, 5, 10, 15, 20, 37, 50 0.45, 0.5, 0.55 0.45, 0.5, 0.55
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Table 4 – Experimental parameters for the evaluation of the scale-up with the model dispersion. Machine
Rotor–stator tip speed vs [m/s]
Conti TDS 1 Conti TDS 2 Conti TDS 3 Conti TDS 5
24; 28 24; 28 24 24
diameter, but at the diameter d2 (compare Fig. 2). Therefore, the tangential speed at the diameter d2 , in the following called rotor–stator tangential speed vt,RS , should be kept constant for the scale-up of the Conti TDS.
3.
Results and discussion
3.1. Characterization of flow rate and power consumption
Fig. 4 – Net power consumption of the Conti TDS 2 as function of the flow rate.
The machines were operated with fluids of varying viscosities at different rotor speeds. Due to these variations, different power consumptions and flow rates appeared. Fig. 3 shows exemplarily the measured flow rates of the Conti TDS 5. The general characteristics of the shown flow rate curves are valid for all four examined machines, although the absolute values differ naturally. For moderate rotor speeds, the flow rates increase almost linear with the rotor speeds. For rotor speeds exceeding 1500 min−1 , the slopes of the flow rate curves decrease and only a slight increase of the flow rate can be detected for increasing rotor speeds. For the two lowest viscosities, 1 and 3 mPa s, it even can be detected a maximal flow rate at approximately 2000 min−1 . For these viscosities the flow rates diminishes slightly for higher rotor speeds. The reason for this behavior is presently not sure. It is thinkable that the blades of the rotor head cause cavitation for higher rotor speeds. Due to increasing cavitation effects with rising rotor speeds, the pumping effect is thus decreased, what might lead to the found characteristic flow rates. Another reason could be of course the flow restriction due to the pressure drop at high flow rates. For further examination, the net power consumption of the rotor–stator mixers was measured as function of the flow rate, according to Cooke et al. (2011). The flow rate was adjusted by closing the valve at the product back flow of the test rig (see Fig. 1). Fig. 4 shows measured power data gained by this procedure for the Conti TDS 2. In principal, the power consumption increases with the flow rate. For high flow rates the
increase of power consumption approaches a constant value for both examples shown. The behavior that was found for the Conti TDS machines and is shown in Figs. 3 and 4 differs from that which was found by Cooke et al. (2011). In the study of Cooke, a strongly linear relation between rotor speed and flow rate and between flow rate and net power consumption was detected for Silverson mixers. The operating behavior of the Conti TDS machines seems to differ significantly from that of Silverson and Ytron mixers. Furthermore, the detected nonlinear dependency of the Conti TDS net power consumption from the flow rate is in contradiction with the model of Kowalski and Baldyga. Thus, other approaches are required for the description of the Ystral machines used here. Fig. 5 shows the power number Po (Eq. (1)) and the Power number for rotor–stator-mixers according to Sparks (Eq. (4)) plotted as function of the Reynolds number Re. The data of the Power number Po do not show a unique development. In fact, each set of data that was collected with one polyglycerol–water mixture has its own development and slope. In contrast, the progression of the PoRS values according to Sparks fulfills the expectations to a characteristic power curve. Since the Conti TDS concept is relatively close to the design of centrifugal pumps, it makes sense to take the flow rate into account for the modeling of the power consumption of this rotor–stator geometry.
Fig. 3 – Flow rates of the Conti TDS 5 measured as function of the rotor speed for different viscosities.
Fig. 5 – Power number Po and Power number for rotor–stator mixers PoRS of the Conti TDS 2 plotted as function of the Reynolds number.
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Table 5 – For the power curves determined fit paramters. Machine
Fig. 6 – Power numbers PoRS for all examined Conti TDS machines as function of the Reynolds numbers. A comparison of the results in Fig. 5 with former studies indicates that each geometry concept of rotor–stator dispersing machines has its own characteristic operating behavior. Cooke et al. (2011) gained for the Silverson data a well looking power curve by plotting the “classic” Power number Po versus the Reynolds number. The same is true for the study of Ozcan-Taskın et al. (2011). Note that the PoRS curve in Fig. 5 becomes flat (indicating turbulent regime) for Re > 100,000, what I quite high compared to “normal” power curves. However, the normally used power number Po was originally developed for simple stirrer in large reactors. Since the investigated rotor–stator geometries in this study are much more complex compared to these reactor geometries, a different power curve can be expected. Similar results were observed for the power curves of stirred media mills (Becker et al., 2001). In this devices, the curves depend strongly on the mill geometry and for some geometries a flat power curve was also only observed for Re > 1,000,000. The value PORS is an empirical found Power number, which takes the flow rate into account. In the study of Özcan-Taskin it was shown that the Power number PoRS according to Sparks did not deliver decent results for the characterization of Silverson and the Ytron mixers. However, for the here used rotor–stator geometry of the Conti TDS mixers, the empirical power number PoRS gives reasonable power curves for all investigated machines, as can be seen in Fig. 6. Normally, all the different sized machines should all have a similar power curve. The fact, that the PORS curves do not look all similar may be due to the fact that the dispersing chambers are not all exactly geometrical similar. However, one of the strength of the presented method is that the scale up is possible even when the flow curves are not identical. For the plot of the data points as function of the Reynolds number in a double logarithmic scaled diagram, each graph can be divided into three regions of constant slopes. Analogously to plots with the classical Power number, these regions can be assigned to a laminar, a turbulent and a transition region. In Fig. 6, the progression of the measured data for the regions with constant slope was fitted with exponential functions of the form: PoRS = a · Reb
(6)
The values for the parameters a and b in Eq. (6) were determined with least square method and are given in Table 5.
Region
A
b
Conti TDS 1
Laminar Transition
5349.76 61.89
−0.68 −0.10
Conti TDS 2
Laminar Transition Turbulent
198042.57 128.38 8.70
−1.12 −0.22 0.02
Conti TDS 3
Laminar Transition Turbulent
35100000 44926 54.20
−1.80 −0.89 −0.11
Conti TDS 5
Laminar Transition Turbulent
8656.19 124.72 44.66
−0.68 −0.18 −0.09
Furthermore, the mean deviations between the measured data and the fit function were calculated. They were determined to 4.42% for the Conti TDS 1, 5.45% for the Conti TDS 2, 5.27% for the Conti TDS 3 and 4.32% for the Conti TDS 5. The deviations may be due to the measurement of the power consumption via the electrical power. Kowalski (Kowalski et al., 2011) showed, that the determination via measurement of torque on the shaft is more accurate. The measurement of flow rate by electromagnetic flowmeters may be a further source of impreciseness.
3.2.
Prediction of the power consumption
The ability to model the operating parameters of Conti TDS machines by means of the power number PoRS shall be used in the following to establish a scale-up model for these machines. The power curves presented in Fig. 6 were determined by the use of Newtonian fluids with defined viscosities. However, in real processes Newtonian fluids are rather rarely processed. Especially in the case of dispersions, a shear thinning flow behavior can mostly be observed. The viscosity of shear thinning fluids depends on the applied shear rate and, therefore, no unique viscosity value exists for the calculation of the Reynolds number. For the scale up of stirred systems with shear thinning fluids an often used approach is that of Rieger and Novak (1974), where an apparent viscosity app of the shear thinning fluid is determined. In the first instance of this approach, the power number of the “small scale” stirrer is calculated for the respective operating parameters. With this Power number, the corresponding Reynolds number can be determined from the power curve. By rearranging Eq. (2) to , a so called apparent viscosity app can be calculated with this value of the Reynolds number. With the so determined apparent viscosity, app , the corresponding Reynolds number of the “large scale” stirrer is calculated and the power curve of the “large scale” stirrer gives the corresponding Power number. Thus, by rearranging Eq. (1), the net power consumption can be determined. This approach is frequently used and has been adopted also to other machines than stirred vessels (Becker et al., 2001). Fig. 7 shows the application of this approach to the power curves of the Conti TDS 1 and Conti TDS 3. On the basis of operating parameters of the “small scale” Conti TDS 1, the corresponding Power number PoRS of the “large scale” Conti TDS 3 is determined. Similar to the scale up of stirred devices, Eq. (4) has to be rearranged to the net power consumption: P − P0 = · QV · n2 · d2 · PoRS
(7)
Please cite this article in press as: Schönstedt, B., et al., Scale-up of the power draw of inline-rotor–stator mixers with high throughput. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.04.004
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Fig. 7 – Adoption of the approach from Rieger and Novak for the scale up of Conti TDS rotor–stator mixers. For the calculation of the net power consumption according to Eq. (7), almost all values are known. The Power number PoRs is delivered by the approach from Rieger and Novak. The density of the dispersion and the rotor diameter are usually known and the required rotor speed, n, is determined by the rotor–stator tangential velocity vr,RS that is kept constant within the scale-up process. However, the flow rate that results from the respective operating parameters has still to be identified. The flow rate which is initiated by a certain machine depends on the rotor speed and the product viscosity. The characteristic curves which were gained for the flow rate as function of rotor speed and viscosity (Fig. 3) can hence deliver the necessary data. The resulting flow rate for a certain set of operating parameters can easily be read off. To ensure that the correct viscosity is used, again the apparent viscosity app from the approach of Rieger and Novak can be used. If the calculated apparent viscosity app corresponds not to one of the recorded characteristic curves, the respective value of flow rate can be gained by linear interpolation. In case of a zero flow rate, Eq. (4) is divided by zero and is hence not defined. This means, that the presented model is not intended or able to predict the power consumption with a flow rate of zero or very low flow rates around zero. However, the case of a zero flow rate is no operating condition that is of practical interest for the dimensioning of the driving motors or the design of rotor–stator processes. Neither is it of special importance for the daily operation of rotor–stator mixers. Moreover, the similarity of the terms k1 · Q˙ m · n2 · d2 (right hand of Eq. (5)) and the term P − P0 = · QV · n2 · d2 · PoRS (Eq. (7)) is obvious. In other words, k1 = PoRS . Despite the fact, that the cited publications found k1 to be constant, in this study k1 depends on Re. However, the challenge is to determine k1 (or PoRS ) for given operating conditions. For this determination, the approach described in this study manuscript is currently the only investigated method. In order to verify the presented scale-up model, all 4 machines were operated with a shear thinning model fluid, containing 15 wt.% pyrogenic silica particles (Oswald de Waale: = k· n ; with k = 0.319 and n = 0.564; R2 = 0.985). Based on the data measured on the Conti TDS 1, the power consumption of the “large scale” machines was calculated theoretically according to the presented model. These predictions are compared to measured data. As mentioned initially, for the scale up of Conti TDS machines the rotor–stator tangential velocity vt,RS of the rotors is commonly kept constant. To verify this approach, the model
Fig. 8 – Comparison of the agglomerate sizes that are reached at tangential speeds of 24 and 28 m/s with machines of different sizes. dispersion was processed with two machines until no more decrease of the agglomerate size was detectable. This was done with the Conti TDS 1 and the Conti TDS 2, respectively with a tangential speed of 24 and 28 m/s. The measured agglomerate sizes are plotted in Fig. 8 as function of the consumed specific energy, Em . Though the progression of the agglomerate sizes are not exactly analogously for the different machines, is can be seen, that for the same tangential speeds, similar agglomerate sizes are reached at the end of the dispersing process. Similar product fineness at the end of a dispersion process is equivalent to similar maximum stress intensities acting on the product (if the particle, aggregate or agglomerate strength is constant, e.g. the same product is dispersed) (Schilde et al., 2010, 2013). Furthermore, the specific energy inputs required to reach the final agglomerate sizes are also comparable. Consequently, for the prediction of power consumption, the rotor speeds were adjusted .in a way that the tangential velocity vt,RS was constant at 24 and 28 m/s, respectively. The operation data of the Conti TDS 1, which served for the prediction of the data of the larger machines is listed in Table 6. In Table 7 the calculated and the actual measured data of the “large scale” machines are compared. In principal, the theoretical calculated and the measured data are in good accordance. Nevertheless, the theoretical calculated values for flow rate and power consumption are systematically deviating from the measured ones. In this case, the deviations lay between 0.7% and 11.3% for the predicted power. The reason for these deviations might be found in the use of the apparent viscosity for the assignment of the flow rate. The values found for the apparent viscosities reflect the extreme high shear rates appearing in the dispersing chamber due to rotor–stator interactions. However, the shear rates appearing in the product line can be assumed considerably
Table 6 – Operating parameters of the Conti TDS 1 with the shear thinning model dispersion. Shear velocity [m/s] 28 24
Rotor speed [min−1 ] 9400 8200
Net-power [kw] 9.14 6.72
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Table 7 – Comparison of measured and modeled operation data of “large scale” rotor stator machines processing the shear thinning model fluid. Shear velocity [m/s]
Rotor speed [min−1 ]
Flow rate modeled [m3 /h]
Flow rate measured [m3 /h]
Conti TDS 2 28 24
6900 6210
14.43 13.2
14.07 13.05
Conti TDS 3 24
3600
50.36
47.5
Conti TDS 5 24
3000
72
69
Net power modeled [kw]
Net power measured [kw]
7.84 5.8
Deviation power [%]
7.8 5.4
0.7% 6.9%
19.6
18.0
9.0%
38.7
36.6
5.6%
Table 8 – Measured power consumption and with measured flow rate calculated power consumption of “large scale” rotor stator machines processing the shear thinning model fluid. Shear velocity [m/s]
Rotor speed [min−1 ]
Net power modeled [kW]
Net power measured [kW]
Deviation power [%]
Conti TDS 2 28 24
6900 6210
Conti TDS 3 24
3600
18
18.6
3.5%
Conti TDS 5
3000
36.6
37.1
1.2%
7.8 5.40
lower and the apparent viscosity of the shear thinning fluid accordingly higher. Higher viscosities result in lower flow rates due to increased frictional forces. According to Eq. (7) these systematically overestimated flow rates have a linear impact on the predicted power consumption. Table 8 shows power data which were calculated according to Eq. (7) with the actual measured flow rates. As Power number PoRS , the values gained by the approach of Rieger and Novak were taken as before. The net power consumption that was calculated this way, is considerably closer to the measured data than the before calculated values. The accuracy of the predicted power consumption could thus be enhanced by a more accurate prediction of the occurring flow rates. Therefore it is necessary to draw more sophisticated conclusions from the experiments with the “small scale” machine on the flow rates occurring in the “large scale” machines. The measured and predicted flow rates of all 4 investigated machines are shown in Table 9. The prediction of the flow rates of the “small scale” machine Conti TDS 1 was gained by reading off the values of the respective characteristic curves of the Conti TDS 1 machine. As it was seen before for the “large scale” machines, the actual measured flow rate is slightly higher than it would have been predicted based on the apparent
7.78 5.8
0.2% 7.8%
viscosity app . Additionally, the ratios R of predicted to measured flow rate are listed in Table 9. The ratios R of predicted to measured flow rate range between 1.01 and 1.07. The ratio of predicted to measured power consumption is, within the accuracy of measurement, for all examined machines in the same order of magnitude. This indicates that the shear rates in the product line deviate in the same range from those in the dispersing chamber for all experimental set-ups. Regarding an accurate prediction of the flow rates, these similar ratios R are important since this finding allows concluding from the flow rates of the “small scale” on those of the “large scale” rotor–stator machines. The predicted net power consumption can thus be calculated by correcting the flow rate determined by the apparent viscosity app with the ratio R determined by the small scale experiments: P − P0 =
QV · n2 · d2 · PoRS · R
(8)
The values of the net power consumption predicted according to Eq. (8) are compared in Table 10 with the measured net power consumptions. The values of R are taken from the corresponding rows of the Conti TDS 1 in Table 9. As can be seen,
Table 9 – Measured and modeled flow rates of all 4 investigated machines. Shear velocity [m/s]
Rotor speed [min−1 ]
Flow rate measured
Flow rate predicted
Ratio R of predicted to measured flow rate
Conti TDS 1 28 24
9400 8200
10.56 10.20
10.92 10.96
1.03 1.07
Conti TDS 2 28 24
6900 6210
14.1 13.0
14.43 13.20
1.03 1.01
Conti TDS 3 24
3600
47.5
50.4
1.06
Conti TDS 5 24
3000
69
72
1.04
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Table 10 – Comparison of measured and modeled operation data of “large scale” rotor stator machines processing the shear thinning model fluid. Shear velocity [m/s]
Rotor speed [min−1 ]
Net power modeled [kW]
Net power measured [kW]
Deviation power [%]
Conti TDS 2 28 24
6900 6210
Conti TDS 3 24
3600
18
18.6
3.5%
Conti TDS 5
3000
36.6
36.0
−1.7%
7.5 5.40
the predicted power consumptions are close to the measured data and the deviations do not exceed the mean deviations of the measured data to the fit functions. Due to the proposed scale-up model it is thus possible, to predict the power consumption and the flow rates of inlinerotor–stator mixers of the type Ystral Conti TDS within in the limits of measurement accuracy. However, one has to be aware, that the predicted power depends strongly on the occurring flow rates. Hence for the scale-up of machines it has to be considered, that the product pipes of the later production installation has to be similar to that one, with that the characteristic curves of the respective “large scale” machine were recorded. A significant change in the piping installation results in a change of the pressure drop due to frictional forces and leads to changing flow rates. However, this restriction seems to be unavoidable since also an earlier developed scale-up model for other types of inline-rotor–stators has the occurring flow rate as central parameter for the calculation of the power consumption (Bałdyga et al., 2008; Hall et al., 2011; Kowalski, 2009; Kowalski et al., 2011). The adjustment of the flow rate by an external pump might overcome this problem. However, this solution is, at least for inline-rotor–stator mixers of the type Conti TDS, not in accordance with actual practice. Continuous operated rotor–stator mixers of the type Conti TDS generate a strong flow rate, wherefore a separate pump is usually not integrated into the product line. Furthermore, the piping installations that were used within this work can be regarded as typical installations for the respective machines. These installations are recommended by the manufacturer and are similarly used in the majority of practical applications. A direct comparison of the equation for the calculation of power consumption of centrifugal pumps (Eq. (9)) with Eq. (7) shows an interesting aspect of the Power number PoRS : P − P0 = w · p · QV
(9)
Here, w is the pump efficiency factor and p is the pressure drop. Equalizing both terms of Eqs. (7) and (9) leads to: PoRS · · n2 · d2 · QV = w · p · QV
(10)
The term PoRS ··n2 ·d2 has the dimension [N/m2 ] and can thus be seen as an equivalent of the product of pressure drop and pumping efficiency factor. The characteristic number PoRS seems thus to be suited to describe the ratio of power on shaft to the induced hydraulic power for rotor stator mixers of the type Conti TDS. Compared to the classical power number Po, which was initially developed for disk agitators, it has the advantage that it takes flow rate, which is induced by the rotor, into account. On the other hand, compared to the approach usually used for centrifugal pumps, it has the advantage, that
7.78 5.4
3.2% −0.5%
it has a unique characteristic curve for a given piping and can be used also for non-Newtonian fluids. From this point of view, one might state that by means of this characteristic curve, the product of pump efficiency factor and pressure drop can be calculated for different product viscosities and rotor speeds.
4.
Conclusion
A model was developed that allows the prediction of the net power consumption and the occurring flow rates of inlinerotor–stator mixers of the type Conti TDS. The scale-up model is based on an earlier idea from Sparks to model the power consumption of inline-rotor–stator mixers with a Power number which takes the flow rate into account. This idea of a Power number for inline-rotor–stator mixers is combined with the approach of Rieger and Novak for the scale up of stirring processes with non-Newtonian fluids. An apparent viscosity is determined and used for the determination of a power number of the large scale machine. Furthermore, the occurring flow rate has to be predicted by the use of characteristic flow curves. A critical point in the scale-up approach is that in the rotor–stator-machine and in the pipe different apparent viscosities appear. The flow rates which are predicted based on the apparent viscosity inside the rotor–stator-machine show a systematic deviation to the actual occurring flow rates. A correction factor for the actual occurring flow rates can be gained by experiments with the small scale machine. The proposed model was validated by experiments with machines of different sizes, ranging from small pilot-scale to production scale. For the validation of the new approach, a shear thinning dispersion was processed with all machines. The power consumptions and the flow rates of inline-rotor–stator mixers of the type Ystral Conti TDS were predicted within the limits of measurement accuracy. Thus, the approach presented in this study enables the scale-up of power draw of inline-rotor–stator with high throughputs.
Acknowledgements The authors would like to thank P. Hartmann and C.-D. Laser (TU Braunschweig) as well as K. Ebbinghaus (Ystral GmbH) for the help with the experimental part of the work.
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Please cite this article in press as: Schönstedt, B., et al., Scale-up of the power draw of inline-rotor–stator mixers with high throughput. Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.04.004