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Hydrodynamics of a planetary mixer used for dough process: Influence of impeller speeds ratio on the power dissipated for Newtonian fluids
Journal of Food Engineering 118 (2013) 350–357
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Hydrodynamics of a planetary mixer used for dough process: Influence of impeller speeds ratio on the power dissipated for Newtonian fluids Frederic Auger a,f, Guillaume Delaplace c, Laurent Bouvier c, Andreas Redl e, Christophe André c,d,⇑, Marie-Hélène Morel b a
Syral, Burchtstraat 10, 9300 Aalst, Belgium INRA, UMR 1208 IATE, 2 Place Pierre Viala, 34060 Montpellier Cedex 1, France INRA, UR 638 Processus aux Interfaces et Hygiène des Matériaux, F-59651 Villeneuve d’Ascq, France d UC Lille, HEI, 13 rue de Toul, 59036 Lille Cedex, France e Tereos Syral SAS, ZI et Portuaire BP 32, 67390 Marckolsheim, France f CSM Bakery, Mainzer Straße 152-160, 55411 Bingen, Germany b c
a r t i c l e
i n f o
Article history: Received 3 April 2012 Received in revised form 15 April 2013 Accepted 16 April 2013 Available online 25 April 2013 Keywords: Planetary mixer Hydrodynamics Power consumption Newtonian fluids
a b s t r a c t The objective of this paper is to better characterize the influence of process parameters (impeller revolution speeds) on the performance of a planetary flour-beater mixer (mixer bowl P600 from Brabender) used in dough production. Firstly, we have theoretically described the path followed by the impeller tip into the vessel and the variation of the absolute velocity during its trajectory. This gives us indications during the transient mixing action of the material induced by this mixer. Secondly, we have theoretically and experimentally shown that for Newtonian fluids: (i) The power dissipated by this mixer is strongly dependent on the impeller speed ratios. (ii) It is possible to obtain for this planetary mixer a unique master power curve, gathering on the same characteristic the influence of the dual impeller speeds on power consumption. This requires the introduction of a characteristic velocity, known as the maximal impeller tip velocity, into power and Reynolds numbers. The constant Kp of the mixer bowl P600, determined as the product of the modified Reynolds and power numbers, was found to be equal to 48.6. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Planetary mixers combining dual impeller revolution speeds are commonly used in the gluten network development but have rarely been investigated in depth as classical mixing systems (mixing equipments equipped in which the agitator is vertically and centrally mounted in the tank and performs a single revolutionary motion around the vertical axis). Mixing wheat flour with water results in an even distribution of the ingredients and the development of a continuous network of gluten proteins within the dough (Sandstedt et al., 1954). According to Tipples and Kilborn (1975), mixing has three distinct functions in dough development: (i) distribution and homogenization of dough ingredients, (ii) hydration of flour particles, and (iii) energy input to develop the homogeneous protein structure. Several types of planetary mixers and off-centered double agitators are ⇑ Corresponding author at: UC Lille, HEI, 13 rue de Toul, 59036 Lille Cedex, France. Tel.: +33 3 28 38 48 58; fax: +33 3 28 38 48 04. E-mail address: [email protected] (C. André). 0260-8774/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfoodeng.2013.04.012
widely used in flour dough mixing processes. It is well known (Frazier et al., 1975; Jongen et al., 2003; Kilborn and Tipples, 1972; Lee et al., 2001; Oliver and Allen, 1992; Peighambardoust et al., 2006; Schluentz et al., 2000; Wilson et al., 1997; Zounis and Quail, 1997) that several process parameters strongly influence the gluten network and dough development: the design of mixing equipment, which mainly determines the type of dough deformation, the material properties and the mixing variables, such as mixing speed. Among the mixing variables, we recently showed that the instantaneous power delivered to a dilute flour–water dough is a key parameter to control the process of dough development (Auger et al., 2008). This work was done using a particular planetary mixer, which combines two parallel revolutionary motions around a vertical axis: the mixer bowl P600 from Brabender that we used to mix flour doughs. Therefore, characterizing the power consumption of this particular mixer and, especially, quantifying its dependence on mixing speed is an essential step towards optimizing wheat dough development. Characterizing the power consumption of a stirred tank requires the knowledge of the Newtonian power curve of the mixing
F. Auger et al. / Journal of Food Engineering 118 (2013) 350–357
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Nomenclature d dG dR C H HL HT Kp l N NG Np NpM NR
agitator diameter (m) diameter of the gyrational motion diameter of the rotational motion (m) bottom clearance between the tank and the agitator (m) height of the blade (m) height of liquid in the vessel (m) height of the impeller tip of the hook (m) proportionality constant of the power number defined by Eq. (12) (–) altitude of the point M in the reference frame R (m) rotational speed of the drive shaft for classical mixing system (rev s1) rotational speed of the gyrational motion (rev s1) power number (–) modified power number for the mixer bowl P600 (–) rotational speed of the rotational motion (rev s1)
system, which is the plot of the power number versus the Reynolds number. With classical mixing equipment, in which the agitator is vertically and centrally mounted in the tank and performs a single revolutionary motion around the vertical axis, the determination of such dimensionless numbers is actually well established from mixing torque/speed measurements obtained with known Newtonian fluids. Unfortunately, it is not yet the case for planetary mixers. Indeed, such a characteristic power curve does not exist, to our knowledge, for planetary mixers used in dough mixing in general, and not for the mixer bowl P600 in particular. This state of knowledge may be explained partially by the fact that planetary mixers are more recent mixing equipments, and that analysis of their mixing characteristics have appeared only recently in the literature. However, this mixing equipment is taking a big sweep in the processing industry, fulfilling the needs of consumers for complex products with specific functionalities. Pioneer works have been conducted on planetary mixers by Tanguy et al. (1996), but scientific studies devoted to the performances of this mixing equipment are still scarce in the literature (Delaplace et al., 2004, 2005, 2007, 2011, 2012; Jongen, 2000; Tanguy et al., 1996, 1999). These experimental and numerical works have clearly highlighted the difficulties of comparing the mixing performances of planetary mixers with those of well-established conventional mixers. This is mainly due to the fact that dual revolutionary motions of the agitator, characteristic of a planetary mixer, induce an increase in the number of the relevant process parameters required to perform the dimensional analysis. To solve this difficulty, Delaplace et al. (2005, 2007) modified the dimensional analysis established for classical mixers and adapted it for non-conventional mixers. This was done for a particular planetary mixer: the TRIAXEÒ system, which combines two perpendicular revolutionary motions around a horizontal axis. These authors proposed modified Reynolds, power and mixing time numbers, which involved the maximum impeller tip speed as the characteristic velocity and a dimension perpendicular to the vertical axis of revolution as the characteristic length. These authors showed experimentally that such modified dimensionless numbers allowed us to obtain a unique power and mixing characteristics of the TRIAXEÒ system, regardless of a variation in speed ratio. Moreover, the modified dimensionless numbers proposed by these authors were consistent with the definition of classical Reynolds, power and mixing time numbers if the impeller was forced to perform only one motion around the vertical axis of the tank, as in the case for a classical mixing system.
Ns P Re ReM RG RR t uimpellertip uch
a b
l C h
q
rotational speed of the drive shaft for planetary mixer (rev s1) power consumption of the mixer (W) Reynolds number (–) modified Reynolds number for the mixer bowl P600 (–) radius of the gyrational motion (–) radius of the rotational motion (–) time (s) instantaneous impeller tip speed of the hook (m s1) characteristic velocity of the impeller (m s1) gyrational angle (Rad) rotational angle (Rad) apparent dynamic viscosity (Pa s) torque acting on the drive shaft (N m) temperature (°C) fluid density (kg m3)
To our knowledge, such an approach has not been applied for other planetary mixers, and not for the planetary mixer bowl P600. In this paper, we propose to use and develop the methodology proposed by Delaplace et al. (2005), on the particular mixer bowl P600 from Brabender. This mixing system was used in a previous paper to mix flour doughs (Auger et al., 2008). However, we can note that the revolutionary motion performed by the mixer bowl P600 is quite different from the one performed by the TRIAXEÒ system studied by Delaplace et al. (2005), since revolution axes of the hook shaped planetary mixer are parallel, whereas they are perpendicular in the case of the TRIAXEÒ system. Moreover, the geometry of the agitator is also strongly different. Consequently, the aim of this paper is: (i) To propose a characteristic velocity for this planetary dough, such as the pi-space describing the power consumption of the hook shaped planetary mixer, is described only by a modified power and Reynolds numbers. In particular, attention will be paid to the ways of obtaining the characteristic velocity for this planetary dough mixer. (ii) To ascertain the reliability of the modified dimensionless numbers proposed from experimental measurements of the power consumption obtained with the mixer bowl P600 by mixing several Newtonian fluids. 2. Materials and methods 2.1. Mixing equipment The mixing equipment used was a planetary mixer bowl P600 (Brabender OHG, Germany) equipped with a helicoidal dough hook (Kenwood stirring insert, Brabender OHG, Germany) and thermostated with a water double jacket (see Fig. 1). This mixer is usually used in non-food area for testing the properties of plastic powders, like their liquid absorption or their plasticizer absorption rate but we recently used it for mixing flour–water doughs. The mixer was coupled to a Plastograph labstation (Brabender OHG, Germany) which allowed continuous torque, speed and temperature recording. The experimental temperature (25 °C or 40 °C) varied according to the tested fluid (Table 1). The hook displacement is characterized by two parallel revolutionary motions, gyration and rotation, around the vertical central axis. Gyration is the revolution of the vertical axis of the hook
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The mixing equipment was driven by the unique variable speed motor of the lab-station, which induced rotations of the drive shaft (N) ranging from 0 to 150 rpm. The reduction gearbox ratio for the gyration (NG/N) was equal to 1.5, whereas the reduction gearbox ratio for the rotation (NR/N) was equal to 5 (Fig. 2). The lab-station was equipped with a torquemeter in the range of 0–10 N m, directly connected to the drive shaft. The total power consumption of the planetary mixer bowl P600 was obtained from torque and speed measurements carried out directly on the motor drive shaft, upstream from the reduction gearbox. Experimental measurements were carried out with 1 kg of several Newtonian fluids. This corresponds to a liquid height HL of ca. 5.5 cm (Fig. 2). The diameter of the vessel measured at the surface of the fluid HL was 18.5 cm. 2.2. Mixed fluids
Fig. 1. Picture of the planetary mixer bowl P600 investigated, in connection to the lab-station unit.
Table 1 Rheological and physical properties of the Newtonian fluids. Fluids
Polybutene Polybutene Polybutene Polybutene N15000
oil S8000 oil N4000 oil S8000 oil
Density q (kg m3)
Viscosity l (Pa s)
Temperature of the agitated liquid h (°C)
874.0 875.6 881.7 888.9
5.94 9.13 21.13 40.69
40 25 25 25
Four Newtonian fluids were mixed in the planetary mixer bowl P600 at drive shaft speeds (N) ranging from 20 to 120 rpm for power consumption measurements. The four Newtonian fluids (Polybutene oils, Poulten Selfe and Lee Ltd., UK) covered a wide range of viscosity (6 < l < 41 Pa s). Details of the different fluids tested are given in Table 1. 2.3. Viscous properties of mixed fluids The viscous properties of the fluids were determined using a strain-controlled rheometer (ARES, TA Instruments, USA) equipped with a cone-and-plate geometry (50 mm diameter, 0.04 rad angle, 0.0457 mm gap). Viscosity measurements were performed at 25 °C or 40 °C, as during mixing experiments (Table 1). 3. Theory 3.1. Dimensional analysis for a conventional agitated vessel Newtonian power curves of conventional agitated vessels, in which impellers are vertically and centrally mounted in the tank and achieve only a single revolutionary motion around the vertical axis (Fig. 3), are now well established when mixing highly viscous Newtonian fluids. For a given conventional mixing system, only two dimensionless numbers interfere: the Reynolds number Re and the power number Np, defined as follows:
Re ¼
q N d2 l
ð1Þ
Fig. 2. Main dimensions of the mixer bowl P600 equipped with a helicoidal dough hook.
around the center of the mixing bowl, whereas rotation is the revolution of the hook on its own vertical axis (Fig. 2). NG and NR respectively refer to the gyrational and the rotational impeller speeds. These double motions, turning in opposite directions, allowed the hook to periodically come in contact with the entire volume of the vessel. In this work, the mixing tool was a helicoidal dough hook with a diameter dR = 9 cm fixed inside the rotating central orbit with a diameter dG = 6 cm (Fig. 2). The vessel used was a cylinder with a conical bottom and the height of the impeller tip HT was 4 cm. The diameter of the vessel at the impeller tip height HT was 18 cm.
Fig. 3. Classical geometric parameters and notations used for a mixing vessel equipped with an impeller vertically and centrally mounted inside the tank.
F. Auger et al. / Journal of Food Engineering 118 (2013) 350–357
Np ¼
P
q N3 d5
¼
2p C q N 2 d5
ð2Þ
where q is the fluid density (in kg m3), N is the rotational speed (in rev s1), d is the agitator diameter (in m), l is the dynamic viscosity (in Pa s), P is the power (in W) and C is the mixing torque (in N m). In such a conventional mixer, the characteristic length and the characteristic velocity of the impeller are, respectively, the agitator diameter d and uch (=N d). Note that uch is proportional to the maximum linear velocity encountered in the vessel (=p N d) and corresponds to the maximum impeller tip speed divided by p. 3.2. Dimensional analysis for the hook shaped planetary mixer For a planetary mixer, the dimensional analysis is more complex due to the occurrence of two revolutionary motions. In such a mixer, the characteristic length of the mixing system is somewhat more complex to define and the maximum linear velocity encountered in the vessel does not depend any more on a single revolutionary speed (N) but on the two revolutionary speeds NG and NR. Therefore, the classical Reynolds and power numbers should be modified adequately to take into account the complexity of the combined motion followed by the agitator. In Fig. 4, we propose a simplified sketch of the hook shaped planetary mixer to take into account its particular geometry. Assuming that the acceleration due to gravity does not influence the mixing process of highly viscous fluids, the list of relevant dimensional parameters influencing power consumption, when mixing Newtonian fluids with the planetary mixer bowl P600, is:
f1 ðNR ; N G ; dR ; dG ; q; l; P; H; HL ; CÞ ¼ 0
ð3Þ
where NR and NG are respectively the rotational and the gyrational impeller speeds, dR and dG are respectively the diameter of the rotational and the gyrational motions, q is the fluid density, l is the fluid dynamic viscosity and P is the power of the mixer. The position of the agitator is given here by the bottom clearance position of the agitator in vessel C, instead of HT (Fig. 3), which is another way to refer to the mixer installation inside the bowl. HL and H respectively refer to the liquid height and the height of the blade. A closer look at Eq. (3) facilitates a reduction in the number of physical quantities in the list of relevant parameters. Indeed, it is
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possible to introduce as a characteristic velocity uch, a value which is proportional to the maximum impeller tip speed (discussed later) and consequently to reduce the list of relevant physical variables by one parameter:
f2 ðuch ; dG ; q; l; P; dR ; H; HL ; CÞ ¼ 0
ð4Þ
This method which allows us to reduce the number of dimensionless variables is called variable fusion in Szirtes’s book (1997). In this way, the number of variables appearing in the relevant list is reduced and consequently the number of associated dimensionless ratios is reduced. For a fixed condition of installation and a given planetary system (dR, H, HL, C are constant), this 5-parametric dimensional space leads to a power characteristic consisting of only two pi-numbers:
f3 ðReM ; NpM Þ ¼ 0
ð5Þ
with
ReM ¼
NpM ¼
q uch dG l P
q u3ch d2G
ð6Þ
¼
2p N C q u3ch d2G
ð7Þ
where q is the fluid density (in kg m3), uch is the characteristic velocity (in m s1), dG is the diameter of the gyrational motion (in m), l is the dynamic viscosity (in Pa s), P is the power (in W), C is the mixing torque (in N m) and Ns is the mixing speed of the drive shaft (in rev s1). To sum up, the dimensional analysis of the power consumption of this planetary mixer, in a fixed installation condition, leads to a relationship between only two pi-numbers (see Eq. (5)) if a characteristic velocity, proportional to the maximum impeller tip speed, is introduced in the parametric dimensional space. The two pinumbers proposed in Eqs. (6) and (7) correspond to the modified Reynolds and power numbers. In the following section, the way to compute the characteristic velocity for the mixer bowl P600 will be detailed. Then, the reliability of the two pi-numbers proposed will be ascertained, using power consumption measurements. 3.3. Determination of the characteristic velocity for the planetary mixer bowl P600 For the planetary mixer bowl P600, which combines the dual motions reported in Appendix A, the instantaneous position of a point M located at the impeller tip in a fixed reference frame R ! ! ! (0; X ; Y ; Z ) is given by:
RG cosð2pNG tÞ þ RR cosð2pðNG NR Þ tÞ ! ðOMÞR ¼ RG sinð2pNG tÞ þ RR sinð2pðN G NR Þ tÞ l
ð8Þ
where NR and NG are respectively the rotational and the gyrational impeller speeds (in rev s1), RR and RG are respectively the radius of the rotational and the gyrational motions (in m), t represents the time (in s) and l represents the altitude of the point M (in m) in the reference frame R. For the planetary mixer bowl P600, the magnitude of the instantaneous impeller tip speed in an inert reference frame is defined as follows (see Appendix B): ! k V ak ¼ uimpellertip ðtÞ ¼2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðRG NG Þ2 þ R2R ðN R N G Þ2 2 RG RR N G ðN R N G Þ cosð2 p N R tÞ Fig. 4. Sketch and symbols used for the planetary mixer bowl P600.
ð9Þ
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For a classical mixing system, the characteristic velocity, which interferes in Reynolds and power numbers, is N d. Unit of N d is m/s. This characteristic velocity, N d corresponds to the maximum linear speed (=p N d) encountered in the vessel divided by p. By analogy with the dimensional analysis used for classical mixing systems, we propose a characteristic velocity for the planetary mixer bowl P600 defined as follows:
uch ¼
uimpellertip maxðtÞ
p
ð10Þ
The derivation of Eq. (9) allows us to obtain the instants for which the magnitude of instantaneous impeller tip speed reaches its extremes (see Appendix C), and thus to compute the maximum value for instantaneous impeller tip speed (uimpellertip max(t)). Finally, using Eq. (10), it can be shown that uch can be defined as follows (see Appendix C):
uch ¼ N G dG þ ðNR NG Þ dR
ð11Þ
compare, on a same chart, the power consumption performances of this mixing equipment to those of other classical mixing systems. It was made possible by introducing the characteristic speed uch in the list of the relevant physical parameters. Otherwise, the set of dimensionless numbers proposed for the mixer bowl P600 mixer would have been enlarged, by the gyrational and rotational speeds of the impeller, in comparison to that of a classical mixing system. This point has been already discussed elsewhere (Delaplace et al., 2005, 2007). From further analysis of Fig. 5, it also appears that the relationship between NpM and ReM is linear in the bi-log plot on the overall range of ReM values obtained, which did not exceeded a value of 10. This trend, which is well known for classical mixing systems when power measurements are carried out under laminar regime, suggests that all the mixing experiments were performed in the laminar regime. Moreover, it can be observed from Fig. 5 that the product NpM ReM is constant, as reported for classical mixing systems when mixing highly viscous fluids. The value of the product is Kp = 48.6.
4. Results and discussion
K p ¼ NpM ReM ¼ 48:6
4.1. Newtonian power curve of the planetary mixer bowl P600
To our knowledge, this is the first time that such a constant is given for a planetary mixer used for dough mixing. Combining Eqs. (6), (7), (11), and (12) leads to:
Power consumption measurements obtained for the planetary mixer bowl P600, when mixing various Newtonian fluids at different mixing speeds Ns, are shown in Fig. 5. Results are presented in terms of the modified power and Reynolds numbers, such as suggested in combining Eqs. (6), (7), and (11). Fig. 5 clearly shows that the proposed pi-numbers space allows us to obtain a unique master curve, which is independent of the mixing speed Ns and of the viscosity of the Newtonian fluids. This result proves to a certain extent the reliability of using the maximum impeller tip speed uch as the characteristic velocity to reduce the set of dimensionless numbers. It can also be noticed that the set of dimensionless numbers proposed above has never been ascertained for a planetary mixer combining dual parallel revolutionary motions around vertical axes, like the mixer bowl P600. Indeed, the previous work dealing with the power curve of a planetary mixer with Newtonian fluids (Delaplace et al., 2007) refers to a planetary mixer combining dual revolutionary motions around perpendicular axes and not around dual parallel axes as in the case in this paper. In this sense, experimental measurements carried out with the planetary mixer bowl P600 extend the validity of the previous work done by Delaplace et al. (2005), who had given ways to analyze power consumption for planetary mixers. It can also be noticed that the pi-numbers space proposed for the mixer bowl P600 has other advantages since it allows us to
Fig. 5. Newtonian power curve obtained for the mixer bowl P600 by mixing Newtonian polybutene oils (see Table 1). Log–log plot.
P ¼ 48:6lðNG dG þ ðNR NG Þ dR Þ2 dG
ð12Þ
ð13Þ
Eq. (13) shows clearly how the power consumption varies with the viscosity of the Newtonian fluid used and with the dual revolutionary motion of the mixer bowl P600. Eq. (13) has been proven to be valid when ReM 6 10. 4.2. Path of the impeller tip inside the mixer bowl P600 From Eq. (8), it is possible to plot the trajectory of the point M located at the impeller tip into the bowl (see Appendix A). Fig. 6 is an example of the complete path performed by the impeller tip inside the vessel for Ns = 100 rpm. The large circle appearing in Fig. 6, with a diameter of 18 cm, corresponds to the cross section of the vessel with a horizontal plane located at the impeller tip height (HT = 4 cm). We have chosen to draw the instantaneous position of the impeller tip over one rotation period, over one gyration period and over its complete path inside the vessel (Fig. 6). It can be noticed that the impeller tip comes in contact with the entire volume of the vessel, except for: (i) An area close to the vessel wall, where a gap of 1.5 cm is observed. (ii) An area located in the middle of the bowl, around the central axis. This dead zone is contained in a circle with a diameter of 3 cm. We might suggest that such dead zones exist to avoid contacts between the hook and the vessel wall, in order not to damage the mixing device (see Fig. 2). Moreover, we can observe that the complete path of the impeller tip requires three gyration periods. Consequently, due to the reduction gearbox ratio of the motions (NR/ NG = 5/1.5 and NG/NS = 1.5), the complete path of the impeller tip requires 10 rotation periods and two revolutions of the drive shaft. Plotting the path of the impeller tip into the vessel also shows that the mixer bowl P600 induces a transient mixing action of the material. Indeed, the material located at the initial position of the impeller tip in Fig. 6 was deformed only when the hook came back to this local zone, i.e. when the drive shaft performed exactly two revolutions. So, in this particular mixer, the material locally suffered a succession of mixing stresses and relaxations.
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Fig. 6. Path of the impeller tip inside the vessel, over one complete path (thin continuous line), over one gyration period (large continuous line) and over one rotation period (circle symbols). The size of circle symbols increases as the absolute velocity of the impeller tip speed increases. Seen from of the top of the vessel, in the reference frame R defined in Appendix A. Units are in m.
Fig. 7. Example of evolution of the instantaneous impeller tip speed with time over one gyration period, at N = 100 rpm (black line) and N = 80 rpm (gray line).
4.3. Variation of the absolute velocity of the impeller tip inside the mixer bowl P600 The evolution of the instantaneous impeller tip speed with time is illustrated in Fig. 7, for two mixing speeds (Ns = 80 rpm and Ns = 100 rpm) and over one gyration period, which required 0.5 s at Ns = 80 rpm and 0.4 s at Ns = 100 rpm. For this planetary mixer, the instantaneous impeller tip speed uimpellertip (t) in an inert reference frame is not constant with time. We can observe successions of acceleration and deceleration phases, revealing another transient nature of the mixing action inside the vessel of this mixer. Increasing the drive shaft mixing speed (Ns) increases the maxima and minima impeller tip speed values but also increases the amplitude of oscillation between these two extreme speed values (uimpellertip max uimpellertip min). From Eq. (9), it can be established that:
uimpellertip max ¼ pðNG dG þ ðNR NG Þ dR Þ ¼ pNsð1:5dG þ 3:5dR Þ
ð14Þ
uimpellertip min ¼ pððNR NG Þ dR N G dG Þ ¼ pNsð3:5dR 1:5dG Þ
ð15Þ
uimpellertip max uimpellertip min ¼ 2pðN G dG Þ ¼ 2pNsð1:5dG Þ ¼ uimpellertip max½3dG =ð3:5dR þ 1:5dG Þ
ð16Þ
For the mixer bowl P600, we found that this amplitude is equal to 44.4% of the maximum impeller tip speed (see Eq. (16)), irrespective of the mixing speed (Ns). This result reveals that the transient nature of the mixing increases with Ns inside the vessel of the mixer. We visualized the variation of the impeller tip speed inside the
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F. Auger et al. / Journal of Food Engineering 118 (2013) 350–357
vessel in Fig. 6, by increasing the size of circle symbols according to the corresponding impeller tip speed. In Fig. 6, the smallest circles correspond to the lowest impeller tip speed values, whereas the largest circles correspond to the highest impeller tip speed values. From this figure, we clearly note that maximal impeller tip speeds are encountered near the center of the vessel, whereas minimal impeller tip speeds are encountered near the vessel wall.
Appendix A Considering M to be a point located at the impeller tip, the instantaneous position of M in the fixed reference frame R ! ! ! (0; X ; Y ; Z ) is given by:
RG
R
5. Conclusion
O
X
Y Modified Reynolds and power numbers, obtained from a dimensional analysis, are proposed in this work to be able to predict the power dissipated by the mixer bowl P600 (from Brabender) from the knowledge of impeller revolution speeds and viscosity of the product. The dimensionless numbers proposed involve the maximum impeller tip speed uch as the characteristic velocity of the impeller, and the diameter of the gyrational motion dG as its characteristic length. Experimental measurements performed by mixing several Newtonian fluids ascertain that such pi-space allows us to obtain a unique power characteristic of the mixing system, regardless of the mixing speed used, as in the case for classical mixing systems. Under laminar regime, occurring when ReM 6 10, the constant Kp of this mixer is equal to 48.6. This work makes it possible to better understand the influence of the mixing speeds, but also that of the viscosity of a Newtonian fluid, on the power consumption of this mixer. Quantifying how these process parameters affect the power consumption was the main purpose of this work. However, the results suggested by this study are much more profound. This study validates that the characteristic velocity is a good indicator to sum up the hydrodynamic induced by the dual revolutionary motion of the planetary mixer whatever the revolutionary axes orientation. This set of tools is an essential step towards optimizing wheat dough development by tuning the desired impeller speed ratios in order to control power dissipated by planetary mixer equipment and by establishing the link between the dynamic of gluten network development and instantaneous power supplied by the mixing equipment. Moreover, this result is not only restricted to monophasic fluids. Indeed, recently Delaplace et al. (2012) and André et al. (2012) have shown that this characteristic velocity governs both the gas retention when whipping a foaming solution aerated by the surface with a planetary mixer, and also mixing times of a powder mixture with a planetary mixer. Note that the knowledge of this power characteristic is also interesting since it allows us to monitor the apparent viscosity of the product using on line torque measurements during the dough processing. This work is also interesting since it points out clearly that the hydrodynamic of a planetary mixer is described only by a modified Reynolds numbers containing a characteristic velocity (the maximum of the absolute impeller tip speed) Finally, this work allows us to better characterize the mechanical action of this mixer and reveals the transient nature of its mixing action. Firstly, the material locally suffers a succession of mixing stresses and relaxations. Secondly, the velocity of the impeller tip is heterogeneous within the vessel and the hook follows a succession of acceleration and deceleration phases. Thirdly, the amplitude between the minimal and maximal impeller tip speed increases with the mixing speed.
H Z l
RR O’ . M
! ! ! ! ðOMÞR ¼ OH þ HO 0 þ O0 M Seen from above, we have:
RG cosð2p NG tÞ 0 ! ! ! ðOHÞR ¼ RG sinð2p NG tÞ ðHO0 ÞR ¼ 0 ðO0 MÞR 1 0 RR cosð2p ðNG NR Þ tÞ ¼ RR sinð2p ðNG NR Þ tÞ 0 Therefore,
RG cosð2p NG tÞ þ RR cosð2p ðNG NR Þ tÞ ! ðOMÞR ¼ RG sinð2p NG tÞ þ RR sinð2p ðNG NR Þ tÞ 0
Acknowledgments
Appendix B
The authors thank Sara Quinger for the correction of the English spelling or grammatical and Bruno Dumont and Jean-Pierre Godart for the checking of the mathematical description of the movement.
Considering M to be a point located at the impeller tip, the instantaneous velocity of M in the fixed reference frame R ! ! ! (0; X ; Y ; Z ) is given by:
F. Auger et al. / Journal of Food Engineering 118 (2013) 350–357
2p RG NG sinð2p N G tÞ 2p RR ðN G NR Þ sinð2p ðNG N R Þ tÞ !! dOM ¼ 2p RG NG cosð2p N G tÞ þ 2p RR ðN G NR Þ cosð2p ðN G NR Þ tÞ dt R 0
Hence, the magnitude of instantaneous velocity of M is given by: ! k V ak ¼ uimpellertip ðtÞ ¼2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðRG N G Þ2 þ R2R ðN R NG Þ2 2 RG RR NG ðN R NG Þ cosð2 p N R tÞ
Appendix C It can be shown that maximum values of the function t#uimpellertip ðtÞ is obtained when cos(2p NR t) is equal to (1). In this case, uimpellertipmax(t) = 2p (NG RG + (NR NG) RR) u maxðtÞ Consequently, uch ¼ impellertip ¼ N G dG þ ðN R N G Þ dR . p References André, C., Demeyre, J.F., Gatumel, C., Berthiaux, H., Delaplace, G., 2012. Dimensional analysis of a planetary mixer for homogenizing of free flowing powders: mixing time and power consumption. Chemical Engineering Journal 198–199, 371– 378. Auger, F., Morel, M.H., Lefebvre, J., Dewilde, M., Redl, A., 2008. A parametric and microstructural study of the formation of gluten network in mixed flour–water batter. Journal of Cereal Science 2, 349–358. Delaplace, G., Bouvier, L., Moreau, A., Guerin, R., Leuliet, J.C., 2004. Determination of mixing time by colourimetric diagnosis – application to a new mixing system. Experiments in Fluids 36, 437–443. Delaplace, G., Guerin, R., Leuliet, J.C., 2005. Dimensional analysis for planetary mixer: modified power and Reynolds numbers. Journal of American Institute of Chemical Engineers 51, 1–7. Delaplace, G., Thakur, R.K., Bouvier, L., André, C., Torrez, C., 2007. Dimensional analysis for planetary mixer: mixing time and Reynolds numbers. Chemical Engineering Science 62, 1442–1447.
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Delaplace, G., Bouvier, L., Moreau, A., André, C., 2011. An arrangement of ideal reactors as a way to model homogenizing processes with a planetary mixer. AIChE Journal 57, 1678–1783. Delaplace, G., Coppenolle, P., Cheio, J., Ducept, F., 2012. Influence of whip speed ratios on the inclusion of air into a bakery foam produced with a planetary mixer device. Journal of Food Engineering 108, 532–540. Frazier, P.J., Daniels, N.W.R., Russel-Eggitt, P.W., 1975. Rheology and the continuous breadmaking process. Cereal Chemistry 52, 106–130. Jongen, T.R.G., 2000. Characterization of batch mixers using numerical flow simulations. Journal of American Institute of Chemical Engineers 46, 2140– 2150. Jongen, T.R.G., Bruschke, M.V., Dekker, J.G., 2003. Analysis of dough kneaders using numerical flow simulations. Cereal Chemistry 80, 383–389. Kilborn, R.H., Tipples, K.H., 1972. Factors affecting mechanical dough development. I. Effect of mixing intensity and work input. Cereal Chemistry 49, 34–47. Lee, L., Ng, P.K.W., Whallon, J.H., Steffe, J.F., 2001. Relationship between rheological properties and microstructural characteristics of nondeveloped, partially developed, and developed doughs. Cereal Chemistry 78, 447–452. Oliver, J.R., Allen, H.M., 1992. The prediction of bread baking performance using the farinograph and extensograph. Journal of Cereal Science 15, 79–89. Peighambardoust, S.H., Van der Goot, A.J., Van Vliet, T., Hamer, R.J., Boom, R.M., 2006. Microstructure formation and rheological behaviour of dough under simple shear flow. Journal of Cereal Science 43, 183–197. Sandstedt, R.M., Schaumburg, L., Fleming, J., 1954. The microscopic structure of bread and dough. Cereal Chemistry 31, 3–49. Schluentz, E.J., Steffe, J.F., Ng, P.K.W., 2000. Rheology and microstructure of wheat dough developed with controlled deformation. Journal of Texture Studies 31, 41–54. Szirtes, T., 1997. Applied Dimensional Analysis and Modeling, Ed. McGraw-Hill, ISBN: 13:978 0070628113. Tanguy, P.A., Bertrand, F., Labrie, R., Brito-de la Fuente, E., 1996. Numerical modelling of the mixing of viscoplastic slurries in a twin blade planetary mixer. Chemical Engineering Research and Design 74, 499–504. Tanguy, P.A., Thibault, F., Dubois, C., Aït-Kadi, A., 1999. Mixing hydrodynamics in a double planetary mixer. Chemical Engineering Research and Design 77, 318– 323. Tipples, K.H., Kilborn, R.H., 1975. ‘‘Unmixing’’ – the disorientation of developed bread doughs by slow mixing speed. Cereal Chemistry 52, 248–262. Wilson, A.J., Wooding, A.R., Morgenstern, M.P., 1997. Comparison of work input requirement on laboratory-scale and industrial-scale mechanical dough development mixers. Cereal Chemistry 74, 715–721. Zounis, S., Quail, K.J., 1997. Predicting test bakery requirements from laboratory mixing tests. Journal of Cereal Science 25, 185–196.
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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng
Hydrodynamics of a planetary mixer used for dough process: Influence of impeller speeds ratio on the power dissipated for Newtonian fluids Frederic Auger a,f, Guillaume Delaplace c, Laurent Bouvier c, Andreas Redl e, Christophe André c,d,⇑, Marie-Hélène Morel b a
Syral, Burchtstraat 10, 9300 Aalst, Belgium INRA, UMR 1208 IATE, 2 Place Pierre Viala, 34060 Montpellier Cedex 1, France INRA, UR 638 Processus aux Interfaces et Hygiène des Matériaux, F-59651 Villeneuve d’Ascq, France d UC Lille, HEI, 13 rue de Toul, 59036 Lille Cedex, France e Tereos Syral SAS, ZI et Portuaire BP 32, 67390 Marckolsheim, France f CSM Bakery, Mainzer Straße 152-160, 55411 Bingen, Germany b c
a r t i c l e
i n f o
Article history: Received 3 April 2012 Received in revised form 15 April 2013 Accepted 16 April 2013 Available online 25 April 2013 Keywords: Planetary mixer Hydrodynamics Power consumption Newtonian fluids
a b s t r a c t The objective of this paper is to better characterize the influence of process parameters (impeller revolution speeds) on the performance of a planetary flour-beater mixer (mixer bowl P600 from Brabender) used in dough production. Firstly, we have theoretically described the path followed by the impeller tip into the vessel and the variation of the absolute velocity during its trajectory. This gives us indications during the transient mixing action of the material induced by this mixer. Secondly, we have theoretically and experimentally shown that for Newtonian fluids: (i) The power dissipated by this mixer is strongly dependent on the impeller speed ratios. (ii) It is possible to obtain for this planetary mixer a unique master power curve, gathering on the same characteristic the influence of the dual impeller speeds on power consumption. This requires the introduction of a characteristic velocity, known as the maximal impeller tip velocity, into power and Reynolds numbers. The constant Kp of the mixer bowl P600, determined as the product of the modified Reynolds and power numbers, was found to be equal to 48.6. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Planetary mixers combining dual impeller revolution speeds are commonly used in the gluten network development but have rarely been investigated in depth as classical mixing systems (mixing equipments equipped in which the agitator is vertically and centrally mounted in the tank and performs a single revolutionary motion around the vertical axis). Mixing wheat flour with water results in an even distribution of the ingredients and the development of a continuous network of gluten proteins within the dough (Sandstedt et al., 1954). According to Tipples and Kilborn (1975), mixing has three distinct functions in dough development: (i) distribution and homogenization of dough ingredients, (ii) hydration of flour particles, and (iii) energy input to develop the homogeneous protein structure. Several types of planetary mixers and off-centered double agitators are ⇑ Corresponding author at: UC Lille, HEI, 13 rue de Toul, 59036 Lille Cedex, France. Tel.: +33 3 28 38 48 58; fax: +33 3 28 38 48 04. E-mail address: [email protected] (C. André). 0260-8774/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfoodeng.2013.04.012
widely used in flour dough mixing processes. It is well known (Frazier et al., 1975; Jongen et al., 2003; Kilborn and Tipples, 1972; Lee et al., 2001; Oliver and Allen, 1992; Peighambardoust et al., 2006; Schluentz et al., 2000; Wilson et al., 1997; Zounis and Quail, 1997) that several process parameters strongly influence the gluten network and dough development: the design of mixing equipment, which mainly determines the type of dough deformation, the material properties and the mixing variables, such as mixing speed. Among the mixing variables, we recently showed that the instantaneous power delivered to a dilute flour–water dough is a key parameter to control the process of dough development (Auger et al., 2008). This work was done using a particular planetary mixer, which combines two parallel revolutionary motions around a vertical axis: the mixer bowl P600 from Brabender that we used to mix flour doughs. Therefore, characterizing the power consumption of this particular mixer and, especially, quantifying its dependence on mixing speed is an essential step towards optimizing wheat dough development. Characterizing the power consumption of a stirred tank requires the knowledge of the Newtonian power curve of the mixing
F. Auger et al. / Journal of Food Engineering 118 (2013) 350–357
351
Nomenclature d dG dR C H HL HT Kp l N NG Np NpM NR
agitator diameter (m) diameter of the gyrational motion diameter of the rotational motion (m) bottom clearance between the tank and the agitator (m) height of the blade (m) height of liquid in the vessel (m) height of the impeller tip of the hook (m) proportionality constant of the power number defined by Eq. (12) (–) altitude of the point M in the reference frame R (m) rotational speed of the drive shaft for classical mixing system (rev s1) rotational speed of the gyrational motion (rev s1) power number (–) modified power number for the mixer bowl P600 (–) rotational speed of the rotational motion (rev s1)
system, which is the plot of the power number versus the Reynolds number. With classical mixing equipment, in which the agitator is vertically and centrally mounted in the tank and performs a single revolutionary motion around the vertical axis, the determination of such dimensionless numbers is actually well established from mixing torque/speed measurements obtained with known Newtonian fluids. Unfortunately, it is not yet the case for planetary mixers. Indeed, such a characteristic power curve does not exist, to our knowledge, for planetary mixers used in dough mixing in general, and not for the mixer bowl P600 in particular. This state of knowledge may be explained partially by the fact that planetary mixers are more recent mixing equipments, and that analysis of their mixing characteristics have appeared only recently in the literature. However, this mixing equipment is taking a big sweep in the processing industry, fulfilling the needs of consumers for complex products with specific functionalities. Pioneer works have been conducted on planetary mixers by Tanguy et al. (1996), but scientific studies devoted to the performances of this mixing equipment are still scarce in the literature (Delaplace et al., 2004, 2005, 2007, 2011, 2012; Jongen, 2000; Tanguy et al., 1996, 1999). These experimental and numerical works have clearly highlighted the difficulties of comparing the mixing performances of planetary mixers with those of well-established conventional mixers. This is mainly due to the fact that dual revolutionary motions of the agitator, characteristic of a planetary mixer, induce an increase in the number of the relevant process parameters required to perform the dimensional analysis. To solve this difficulty, Delaplace et al. (2005, 2007) modified the dimensional analysis established for classical mixers and adapted it for non-conventional mixers. This was done for a particular planetary mixer: the TRIAXEÒ system, which combines two perpendicular revolutionary motions around a horizontal axis. These authors proposed modified Reynolds, power and mixing time numbers, which involved the maximum impeller tip speed as the characteristic velocity and a dimension perpendicular to the vertical axis of revolution as the characteristic length. These authors showed experimentally that such modified dimensionless numbers allowed us to obtain a unique power and mixing characteristics of the TRIAXEÒ system, regardless of a variation in speed ratio. Moreover, the modified dimensionless numbers proposed by these authors were consistent with the definition of classical Reynolds, power and mixing time numbers if the impeller was forced to perform only one motion around the vertical axis of the tank, as in the case for a classical mixing system.
Ns P Re ReM RG RR t uimpellertip uch
a b
l C h
q
rotational speed of the drive shaft for planetary mixer (rev s1) power consumption of the mixer (W) Reynolds number (–) modified Reynolds number for the mixer bowl P600 (–) radius of the gyrational motion (–) radius of the rotational motion (–) time (s) instantaneous impeller tip speed of the hook (m s1) characteristic velocity of the impeller (m s1) gyrational angle (Rad) rotational angle (Rad) apparent dynamic viscosity (Pa s) torque acting on the drive shaft (N m) temperature (°C) fluid density (kg m3)
To our knowledge, such an approach has not been applied for other planetary mixers, and not for the planetary mixer bowl P600. In this paper, we propose to use and develop the methodology proposed by Delaplace et al. (2005), on the particular mixer bowl P600 from Brabender. This mixing system was used in a previous paper to mix flour doughs (Auger et al., 2008). However, we can note that the revolutionary motion performed by the mixer bowl P600 is quite different from the one performed by the TRIAXEÒ system studied by Delaplace et al. (2005), since revolution axes of the hook shaped planetary mixer are parallel, whereas they are perpendicular in the case of the TRIAXEÒ system. Moreover, the geometry of the agitator is also strongly different. Consequently, the aim of this paper is: (i) To propose a characteristic velocity for this planetary dough, such as the pi-space describing the power consumption of the hook shaped planetary mixer, is described only by a modified power and Reynolds numbers. In particular, attention will be paid to the ways of obtaining the characteristic velocity for this planetary dough mixer. (ii) To ascertain the reliability of the modified dimensionless numbers proposed from experimental measurements of the power consumption obtained with the mixer bowl P600 by mixing several Newtonian fluids. 2. Materials and methods 2.1. Mixing equipment The mixing equipment used was a planetary mixer bowl P600 (Brabender OHG, Germany) equipped with a helicoidal dough hook (Kenwood stirring insert, Brabender OHG, Germany) and thermostated with a water double jacket (see Fig. 1). This mixer is usually used in non-food area for testing the properties of plastic powders, like their liquid absorption or their plasticizer absorption rate but we recently used it for mixing flour–water doughs. The mixer was coupled to a Plastograph labstation (Brabender OHG, Germany) which allowed continuous torque, speed and temperature recording. The experimental temperature (25 °C or 40 °C) varied according to the tested fluid (Table 1). The hook displacement is characterized by two parallel revolutionary motions, gyration and rotation, around the vertical central axis. Gyration is the revolution of the vertical axis of the hook
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The mixing equipment was driven by the unique variable speed motor of the lab-station, which induced rotations of the drive shaft (N) ranging from 0 to 150 rpm. The reduction gearbox ratio for the gyration (NG/N) was equal to 1.5, whereas the reduction gearbox ratio for the rotation (NR/N) was equal to 5 (Fig. 2). The lab-station was equipped with a torquemeter in the range of 0–10 N m, directly connected to the drive shaft. The total power consumption of the planetary mixer bowl P600 was obtained from torque and speed measurements carried out directly on the motor drive shaft, upstream from the reduction gearbox. Experimental measurements were carried out with 1 kg of several Newtonian fluids. This corresponds to a liquid height HL of ca. 5.5 cm (Fig. 2). The diameter of the vessel measured at the surface of the fluid HL was 18.5 cm. 2.2. Mixed fluids
Fig. 1. Picture of the planetary mixer bowl P600 investigated, in connection to the lab-station unit.
Table 1 Rheological and physical properties of the Newtonian fluids. Fluids
Polybutene Polybutene Polybutene Polybutene N15000
oil S8000 oil N4000 oil S8000 oil
Density q (kg m3)
Viscosity l (Pa s)
Temperature of the agitated liquid h (°C)
874.0 875.6 881.7 888.9
5.94 9.13 21.13 40.69
40 25 25 25
Four Newtonian fluids were mixed in the planetary mixer bowl P600 at drive shaft speeds (N) ranging from 20 to 120 rpm for power consumption measurements. The four Newtonian fluids (Polybutene oils, Poulten Selfe and Lee Ltd., UK) covered a wide range of viscosity (6 < l < 41 Pa s). Details of the different fluids tested are given in Table 1. 2.3. Viscous properties of mixed fluids The viscous properties of the fluids were determined using a strain-controlled rheometer (ARES, TA Instruments, USA) equipped with a cone-and-plate geometry (50 mm diameter, 0.04 rad angle, 0.0457 mm gap). Viscosity measurements were performed at 25 °C or 40 °C, as during mixing experiments (Table 1). 3. Theory 3.1. Dimensional analysis for a conventional agitated vessel Newtonian power curves of conventional agitated vessels, in which impellers are vertically and centrally mounted in the tank and achieve only a single revolutionary motion around the vertical axis (Fig. 3), are now well established when mixing highly viscous Newtonian fluids. For a given conventional mixing system, only two dimensionless numbers interfere: the Reynolds number Re and the power number Np, defined as follows:
Re ¼
q N d2 l
ð1Þ
Fig. 2. Main dimensions of the mixer bowl P600 equipped with a helicoidal dough hook.
around the center of the mixing bowl, whereas rotation is the revolution of the hook on its own vertical axis (Fig. 2). NG and NR respectively refer to the gyrational and the rotational impeller speeds. These double motions, turning in opposite directions, allowed the hook to periodically come in contact with the entire volume of the vessel. In this work, the mixing tool was a helicoidal dough hook with a diameter dR = 9 cm fixed inside the rotating central orbit with a diameter dG = 6 cm (Fig. 2). The vessel used was a cylinder with a conical bottom and the height of the impeller tip HT was 4 cm. The diameter of the vessel at the impeller tip height HT was 18 cm.
Fig. 3. Classical geometric parameters and notations used for a mixing vessel equipped with an impeller vertically and centrally mounted inside the tank.
F. Auger et al. / Journal of Food Engineering 118 (2013) 350–357
Np ¼
P
q N3 d5
¼
2p C q N 2 d5
ð2Þ
where q is the fluid density (in kg m3), N is the rotational speed (in rev s1), d is the agitator diameter (in m), l is the dynamic viscosity (in Pa s), P is the power (in W) and C is the mixing torque (in N m). In such a conventional mixer, the characteristic length and the characteristic velocity of the impeller are, respectively, the agitator diameter d and uch (=N d). Note that uch is proportional to the maximum linear velocity encountered in the vessel (=p N d) and corresponds to the maximum impeller tip speed divided by p. 3.2. Dimensional analysis for the hook shaped planetary mixer For a planetary mixer, the dimensional analysis is more complex due to the occurrence of two revolutionary motions. In such a mixer, the characteristic length of the mixing system is somewhat more complex to define and the maximum linear velocity encountered in the vessel does not depend any more on a single revolutionary speed (N) but on the two revolutionary speeds NG and NR. Therefore, the classical Reynolds and power numbers should be modified adequately to take into account the complexity of the combined motion followed by the agitator. In Fig. 4, we propose a simplified sketch of the hook shaped planetary mixer to take into account its particular geometry. Assuming that the acceleration due to gravity does not influence the mixing process of highly viscous fluids, the list of relevant dimensional parameters influencing power consumption, when mixing Newtonian fluids with the planetary mixer bowl P600, is:
f1 ðNR ; N G ; dR ; dG ; q; l; P; H; HL ; CÞ ¼ 0
ð3Þ
where NR and NG are respectively the rotational and the gyrational impeller speeds, dR and dG are respectively the diameter of the rotational and the gyrational motions, q is the fluid density, l is the fluid dynamic viscosity and P is the power of the mixer. The position of the agitator is given here by the bottom clearance position of the agitator in vessel C, instead of HT (Fig. 3), which is another way to refer to the mixer installation inside the bowl. HL and H respectively refer to the liquid height and the height of the blade. A closer look at Eq. (3) facilitates a reduction in the number of physical quantities in the list of relevant parameters. Indeed, it is
353
possible to introduce as a characteristic velocity uch, a value which is proportional to the maximum impeller tip speed (discussed later) and consequently to reduce the list of relevant physical variables by one parameter:
f2 ðuch ; dG ; q; l; P; dR ; H; HL ; CÞ ¼ 0
ð4Þ
This method which allows us to reduce the number of dimensionless variables is called variable fusion in Szirtes’s book (1997). In this way, the number of variables appearing in the relevant list is reduced and consequently the number of associated dimensionless ratios is reduced. For a fixed condition of installation and a given planetary system (dR, H, HL, C are constant), this 5-parametric dimensional space leads to a power characteristic consisting of only two pi-numbers:
f3 ðReM ; NpM Þ ¼ 0
ð5Þ
with
ReM ¼
NpM ¼
q uch dG l P
q u3ch d2G
ð6Þ
¼
2p N C q u3ch d2G
ð7Þ
where q is the fluid density (in kg m3), uch is the characteristic velocity (in m s1), dG is the diameter of the gyrational motion (in m), l is the dynamic viscosity (in Pa s), P is the power (in W), C is the mixing torque (in N m) and Ns is the mixing speed of the drive shaft (in rev s1). To sum up, the dimensional analysis of the power consumption of this planetary mixer, in a fixed installation condition, leads to a relationship between only two pi-numbers (see Eq. (5)) if a characteristic velocity, proportional to the maximum impeller tip speed, is introduced in the parametric dimensional space. The two pinumbers proposed in Eqs. (6) and (7) correspond to the modified Reynolds and power numbers. In the following section, the way to compute the characteristic velocity for the mixer bowl P600 will be detailed. Then, the reliability of the two pi-numbers proposed will be ascertained, using power consumption measurements. 3.3. Determination of the characteristic velocity for the planetary mixer bowl P600 For the planetary mixer bowl P600, which combines the dual motions reported in Appendix A, the instantaneous position of a point M located at the impeller tip in a fixed reference frame R ! ! ! (0; X ; Y ; Z ) is given by:
RG cosð2pNG tÞ þ RR cosð2pðNG NR Þ tÞ ! ðOMÞR ¼ RG sinð2pNG tÞ þ RR sinð2pðN G NR Þ tÞ l
ð8Þ
where NR and NG are respectively the rotational and the gyrational impeller speeds (in rev s1), RR and RG are respectively the radius of the rotational and the gyrational motions (in m), t represents the time (in s) and l represents the altitude of the point M (in m) in the reference frame R. For the planetary mixer bowl P600, the magnitude of the instantaneous impeller tip speed in an inert reference frame is defined as follows (see Appendix B): ! k V ak ¼ uimpellertip ðtÞ ¼2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðRG NG Þ2 þ R2R ðN R N G Þ2 2 RG RR N G ðN R N G Þ cosð2 p N R tÞ Fig. 4. Sketch and symbols used for the planetary mixer bowl P600.
ð9Þ
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For a classical mixing system, the characteristic velocity, which interferes in Reynolds and power numbers, is N d. Unit of N d is m/s. This characteristic velocity, N d corresponds to the maximum linear speed (=p N d) encountered in the vessel divided by p. By analogy with the dimensional analysis used for classical mixing systems, we propose a characteristic velocity for the planetary mixer bowl P600 defined as follows:
uch ¼
uimpellertip maxðtÞ
p
ð10Þ
The derivation of Eq. (9) allows us to obtain the instants for which the magnitude of instantaneous impeller tip speed reaches its extremes (see Appendix C), and thus to compute the maximum value for instantaneous impeller tip speed (uimpellertip max(t)). Finally, using Eq. (10), it can be shown that uch can be defined as follows (see Appendix C):
uch ¼ N G dG þ ðNR NG Þ dR
ð11Þ
compare, on a same chart, the power consumption performances of this mixing equipment to those of other classical mixing systems. It was made possible by introducing the characteristic speed uch in the list of the relevant physical parameters. Otherwise, the set of dimensionless numbers proposed for the mixer bowl P600 mixer would have been enlarged, by the gyrational and rotational speeds of the impeller, in comparison to that of a classical mixing system. This point has been already discussed elsewhere (Delaplace et al., 2005, 2007). From further analysis of Fig. 5, it also appears that the relationship between NpM and ReM is linear in the bi-log plot on the overall range of ReM values obtained, which did not exceeded a value of 10. This trend, which is well known for classical mixing systems when power measurements are carried out under laminar regime, suggests that all the mixing experiments were performed in the laminar regime. Moreover, it can be observed from Fig. 5 that the product NpM ReM is constant, as reported for classical mixing systems when mixing highly viscous fluids. The value of the product is Kp = 48.6.
4. Results and discussion
K p ¼ NpM ReM ¼ 48:6
4.1. Newtonian power curve of the planetary mixer bowl P600
To our knowledge, this is the first time that such a constant is given for a planetary mixer used for dough mixing. Combining Eqs. (6), (7), (11), and (12) leads to:
Power consumption measurements obtained for the planetary mixer bowl P600, when mixing various Newtonian fluids at different mixing speeds Ns, are shown in Fig. 5. Results are presented in terms of the modified power and Reynolds numbers, such as suggested in combining Eqs. (6), (7), and (11). Fig. 5 clearly shows that the proposed pi-numbers space allows us to obtain a unique master curve, which is independent of the mixing speed Ns and of the viscosity of the Newtonian fluids. This result proves to a certain extent the reliability of using the maximum impeller tip speed uch as the characteristic velocity to reduce the set of dimensionless numbers. It can also be noticed that the set of dimensionless numbers proposed above has never been ascertained for a planetary mixer combining dual parallel revolutionary motions around vertical axes, like the mixer bowl P600. Indeed, the previous work dealing with the power curve of a planetary mixer with Newtonian fluids (Delaplace et al., 2007) refers to a planetary mixer combining dual revolutionary motions around perpendicular axes and not around dual parallel axes as in the case in this paper. In this sense, experimental measurements carried out with the planetary mixer bowl P600 extend the validity of the previous work done by Delaplace et al. (2005), who had given ways to analyze power consumption for planetary mixers. It can also be noticed that the pi-numbers space proposed for the mixer bowl P600 has other advantages since it allows us to
Fig. 5. Newtonian power curve obtained for the mixer bowl P600 by mixing Newtonian polybutene oils (see Table 1). Log–log plot.
P ¼ 48:6lðNG dG þ ðNR NG Þ dR Þ2 dG
ð12Þ
ð13Þ
Eq. (13) shows clearly how the power consumption varies with the viscosity of the Newtonian fluid used and with the dual revolutionary motion of the mixer bowl P600. Eq. (13) has been proven to be valid when ReM 6 10. 4.2. Path of the impeller tip inside the mixer bowl P600 From Eq. (8), it is possible to plot the trajectory of the point M located at the impeller tip into the bowl (see Appendix A). Fig. 6 is an example of the complete path performed by the impeller tip inside the vessel for Ns = 100 rpm. The large circle appearing in Fig. 6, with a diameter of 18 cm, corresponds to the cross section of the vessel with a horizontal plane located at the impeller tip height (HT = 4 cm). We have chosen to draw the instantaneous position of the impeller tip over one rotation period, over one gyration period and over its complete path inside the vessel (Fig. 6). It can be noticed that the impeller tip comes in contact with the entire volume of the vessel, except for: (i) An area close to the vessel wall, where a gap of 1.5 cm is observed. (ii) An area located in the middle of the bowl, around the central axis. This dead zone is contained in a circle with a diameter of 3 cm. We might suggest that such dead zones exist to avoid contacts between the hook and the vessel wall, in order not to damage the mixing device (see Fig. 2). Moreover, we can observe that the complete path of the impeller tip requires three gyration periods. Consequently, due to the reduction gearbox ratio of the motions (NR/ NG = 5/1.5 and NG/NS = 1.5), the complete path of the impeller tip requires 10 rotation periods and two revolutions of the drive shaft. Plotting the path of the impeller tip into the vessel also shows that the mixer bowl P600 induces a transient mixing action of the material. Indeed, the material located at the initial position of the impeller tip in Fig. 6 was deformed only when the hook came back to this local zone, i.e. when the drive shaft performed exactly two revolutions. So, in this particular mixer, the material locally suffered a succession of mixing stresses and relaxations.
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Fig. 6. Path of the impeller tip inside the vessel, over one complete path (thin continuous line), over one gyration period (large continuous line) and over one rotation period (circle symbols). The size of circle symbols increases as the absolute velocity of the impeller tip speed increases. Seen from of the top of the vessel, in the reference frame R defined in Appendix A. Units are in m.
Fig. 7. Example of evolution of the instantaneous impeller tip speed with time over one gyration period, at N = 100 rpm (black line) and N = 80 rpm (gray line).
4.3. Variation of the absolute velocity of the impeller tip inside the mixer bowl P600 The evolution of the instantaneous impeller tip speed with time is illustrated in Fig. 7, for two mixing speeds (Ns = 80 rpm and Ns = 100 rpm) and over one gyration period, which required 0.5 s at Ns = 80 rpm and 0.4 s at Ns = 100 rpm. For this planetary mixer, the instantaneous impeller tip speed uimpellertip (t) in an inert reference frame is not constant with time. We can observe successions of acceleration and deceleration phases, revealing another transient nature of the mixing action inside the vessel of this mixer. Increasing the drive shaft mixing speed (Ns) increases the maxima and minima impeller tip speed values but also increases the amplitude of oscillation between these two extreme speed values (uimpellertip max uimpellertip min). From Eq. (9), it can be established that:
uimpellertip max ¼ pðNG dG þ ðNR NG Þ dR Þ ¼ pNsð1:5dG þ 3:5dR Þ
ð14Þ
uimpellertip min ¼ pððNR NG Þ dR N G dG Þ ¼ pNsð3:5dR 1:5dG Þ
ð15Þ
uimpellertip max uimpellertip min ¼ 2pðN G dG Þ ¼ 2pNsð1:5dG Þ ¼ uimpellertip max½3dG =ð3:5dR þ 1:5dG Þ
ð16Þ
For the mixer bowl P600, we found that this amplitude is equal to 44.4% of the maximum impeller tip speed (see Eq. (16)), irrespective of the mixing speed (Ns). This result reveals that the transient nature of the mixing increases with Ns inside the vessel of the mixer. We visualized the variation of the impeller tip speed inside the
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vessel in Fig. 6, by increasing the size of circle symbols according to the corresponding impeller tip speed. In Fig. 6, the smallest circles correspond to the lowest impeller tip speed values, whereas the largest circles correspond to the highest impeller tip speed values. From this figure, we clearly note that maximal impeller tip speeds are encountered near the center of the vessel, whereas minimal impeller tip speeds are encountered near the vessel wall.
Appendix A Considering M to be a point located at the impeller tip, the instantaneous position of M in the fixed reference frame R ! ! ! (0; X ; Y ; Z ) is given by:
RG
R
5. Conclusion
O
X
Y Modified Reynolds and power numbers, obtained from a dimensional analysis, are proposed in this work to be able to predict the power dissipated by the mixer bowl P600 (from Brabender) from the knowledge of impeller revolution speeds and viscosity of the product. The dimensionless numbers proposed involve the maximum impeller tip speed uch as the characteristic velocity of the impeller, and the diameter of the gyrational motion dG as its characteristic length. Experimental measurements performed by mixing several Newtonian fluids ascertain that such pi-space allows us to obtain a unique power characteristic of the mixing system, regardless of the mixing speed used, as in the case for classical mixing systems. Under laminar regime, occurring when ReM 6 10, the constant Kp of this mixer is equal to 48.6. This work makes it possible to better understand the influence of the mixing speeds, but also that of the viscosity of a Newtonian fluid, on the power consumption of this mixer. Quantifying how these process parameters affect the power consumption was the main purpose of this work. However, the results suggested by this study are much more profound. This study validates that the characteristic velocity is a good indicator to sum up the hydrodynamic induced by the dual revolutionary motion of the planetary mixer whatever the revolutionary axes orientation. This set of tools is an essential step towards optimizing wheat dough development by tuning the desired impeller speed ratios in order to control power dissipated by planetary mixer equipment and by establishing the link between the dynamic of gluten network development and instantaneous power supplied by the mixing equipment. Moreover, this result is not only restricted to monophasic fluids. Indeed, recently Delaplace et al. (2012) and André et al. (2012) have shown that this characteristic velocity governs both the gas retention when whipping a foaming solution aerated by the surface with a planetary mixer, and also mixing times of a powder mixture with a planetary mixer. Note that the knowledge of this power characteristic is also interesting since it allows us to monitor the apparent viscosity of the product using on line torque measurements during the dough processing. This work is also interesting since it points out clearly that the hydrodynamic of a planetary mixer is described only by a modified Reynolds numbers containing a characteristic velocity (the maximum of the absolute impeller tip speed) Finally, this work allows us to better characterize the mechanical action of this mixer and reveals the transient nature of its mixing action. Firstly, the material locally suffers a succession of mixing stresses and relaxations. Secondly, the velocity of the impeller tip is heterogeneous within the vessel and the hook follows a succession of acceleration and deceleration phases. Thirdly, the amplitude between the minimal and maximal impeller tip speed increases with the mixing speed.
H Z l
RR O’ . M
! ! ! ! ðOMÞR ¼ OH þ HO 0 þ O0 M Seen from above, we have:
RG cosð2p NG tÞ 0 ! ! ! ðOHÞR ¼ RG sinð2p NG tÞ ðHO0 ÞR ¼ 0 ðO0 MÞR 1 0 RR cosð2p ðNG NR Þ tÞ ¼ RR sinð2p ðNG NR Þ tÞ 0 Therefore,
RG cosð2p NG tÞ þ RR cosð2p ðNG NR Þ tÞ ! ðOMÞR ¼ RG sinð2p NG tÞ þ RR sinð2p ðNG NR Þ tÞ 0
Acknowledgments
Appendix B
The authors thank Sara Quinger for the correction of the English spelling or grammatical and Bruno Dumont and Jean-Pierre Godart for the checking of the mathematical description of the movement.
Considering M to be a point located at the impeller tip, the instantaneous velocity of M in the fixed reference frame R ! ! ! (0; X ; Y ; Z ) is given by:
F. Auger et al. / Journal of Food Engineering 118 (2013) 350–357
2p RG NG sinð2p N G tÞ 2p RR ðN G NR Þ sinð2p ðNG N R Þ tÞ !! dOM ¼ 2p RG NG cosð2p N G tÞ þ 2p RR ðN G NR Þ cosð2p ðN G NR Þ tÞ dt R 0
Hence, the magnitude of instantaneous velocity of M is given by: ! k V ak ¼ uimpellertip ðtÞ ¼2p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðRG N G Þ2 þ R2R ðN R NG Þ2 2 RG RR NG ðN R NG Þ cosð2 p N R tÞ
Appendix C It can be shown that maximum values of the function t#uimpellertip ðtÞ is obtained when cos(2p NR t) is equal to (1). In this case, uimpellertipmax(t) = 2p (NG RG + (NR NG) RR) u maxðtÞ Consequently, uch ¼ impellertip ¼ N G dG þ ðN R N G Þ dR . p References André, C., Demeyre, J.F., Gatumel, C., Berthiaux, H., Delaplace, G., 2012. Dimensional analysis of a planetary mixer for homogenizing of free flowing powders: mixing time and power consumption. Chemical Engineering Journal 198–199, 371– 378. Auger, F., Morel, M.H., Lefebvre, J., Dewilde, M., Redl, A., 2008. A parametric and microstructural study of the formation of gluten network in mixed flour–water batter. Journal of Cereal Science 2, 349–358. Delaplace, G., Bouvier, L., Moreau, A., Guerin, R., Leuliet, J.C., 2004. Determination of mixing time by colourimetric diagnosis – application to a new mixing system. Experiments in Fluids 36, 437–443. Delaplace, G., Guerin, R., Leuliet, J.C., 2005. Dimensional analysis for planetary mixer: modified power and Reynolds numbers. Journal of American Institute of Chemical Engineers 51, 1–7. Delaplace, G., Thakur, R.K., Bouvier, L., André, C., Torrez, C., 2007. Dimensional analysis for planetary mixer: mixing time and Reynolds numbers. Chemical Engineering Science 62, 1442–1447.
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