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CFD study of flow dynamics in a blade free planetary mixer (BFPM) – A qualitative flow study
chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 100–115
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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
CFD study of flow dynamics in a blade free planetary mixer (BFPM) – A qualitative flow study Nacera Chergui a,∗ , Mohamed Lateb b , Étienne Lacroix a , Louis Dufresne a a
Thermo-Fluid for Transport (TFT) Laboratory, Mechanical Engineering Department, École de Technologie Supérieure (ÉTS) 1100 Notre-Dame West, Montréal, Québec H3C 1K3, Canada b New Jersey Institute of Technology (NJIT), Center for Natural Resources Development and Protection 323 Martin Luther King Jr. Blvd., Newark, NJ 07102, United States
a r t i c l e
i n f o
a b s t r a c t
Article history:
A numerical study is conducted to analyse the flow dynamics in a blade free planetary mixer
Received 19 March 2015
(BFPM). The flow is simulated in the revolution reference frame. The analysis is focused on
Received in revised form 25 May
the departure from solid body rotation to a more complex 3D flow structure. Numerical
2015
calculations are realised using second order finite volume discretisation. The parameters
Accepted 31 May 2015
investigated are the particle trajectories, velocity contours and streamlines. The results
Available online 16 June 2015
show that the flow is generally swirling with the presence of singular points on different
Keywords:
and rotation axes are dependent on the revolution rate. In addition, the dynamics of the flow
Blade free planetary mixer (BFPM)
is subject to significant changes at certain revolution rates.
specific planes. The vortical structures are found to be in spiral forms and their sizes, shapes,
Visualisation technique
© 2015 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers.
Rotating cylinder Vortical structures Qualitative analysis Flow dynamic
1.
Introduction
Blade free planetary mixers (BFPMs) are mainly constituted of a cylindrical container which generally has a flat bottom with no obstacles nor agitators. The container is subject to two rotational movements: one is a rotation around its geometric axis, the second is a revolution around an axis located at a given distance (R0 ) as shown in Fig. 1. The cylinder is usually inclined at an angle ˛ to improve mixing. The combination of the double rotation is similar to that of the Earth around itself and around the sun, thereby the name of “Planetary”. The BFPM technique is known to be a convenient technology since the 1970s for synthesising and mixing fluid phase products (Massing et al., 2008). The mixing action is provided through two movements (rotation and revolution). These two movements combined together improve the mixing without an external intervention. The BFPMs are usually used in industry (pharmaceutical,
∗
biotechnology, biochemical, etc.) to mix, synthesise and aerate high viscosity products (colloids, resins, pastes, creams, etc.) with a minimum degree of contamination (Massing et al., 2008; Raza et al., 2012). However, some deficiencies remain in the design and the application of this technique and some mixing problems were noticed. The products are sometimes of poor quality for certain operating and/or designing conditions (e.g. Chen et al., 2007; Niwa and Hashimoto, 2008; Massing et al., 2008; Hirsch et al., 2009; Raza et al., 2012). Specifically, the link between the operating conditions and the quality of the product could not be determined. For instance, some authors proceed by using trial and error methods to correct the mixing issue. However, the kinematic of the fluid, which is strongly linked to the dynamics of the mixture (Ottino, 1989), is itself a function of operating and designing conditions. In other words, to understand the mixing problem, the dynamics of flow in BFPMs must first be understood. To the best of authors’
Corresponding author. Tel.: +1 514 396 8800x7838. E-mail address: [email protected] (N. Chergui). http://dx.doi.org/10.1016/j.cherd.2015.05.045 0263-8762/© 2015 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers.
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Fig. 1 – Double rotating systems: (a) BFPM with R0 0 and (b) precessing cylinder with R0 ≈ 0.
knowledge no study has been published on the flow dynamics in BFPMs. This is probably due to confidentiality reasons and/or to competitiveness issues. Nevertheless, it should be mentioned that flow studies subject to double rotation, in similar systems (precessing cavities), were conducted and concern the behaviour of fluids in the core of the earth and the satellite reservoirs (e.g. Vanyo et al., 1995; Noir, 2000; Lagrange et al., 2008; Mouhali, 2009). Note that the difference between these two systems lies in the fact that, for the precessing cavity system, the revolution axis intercepts the rotation axis within the cylinder volume, while for the BFPM system the intersection occurs outside of the cylinder volume, as illustrated in Fig. 1. The objective of this study is to characterise qualitatively the flow dynamics within the BFPM. More specifically by considering the departure from solid body rotation. The reminder of this article is organised as follows. Section 2 summarises the methodology used in this work such as the physical model, fundamental equations and the detailed parameters of the numerical model. The results are addressed and developed in Section 3. The section concentrates on the main findings of the visualisation technique such as the particle trajectories, velocity iso-contours and the topology of the flow. Finally, a conclusion is presented in Section 4.
2. Mathematical model and numerical methodology Computational fluid dynamic (CFD) calculations are used for this study. The fluid is assumed to be Newtonian and homogeneous, and the rotating flow in the container is laminar and incompressible.
2.1.
Choice of the departure from solid body rotation
The flow and mixture in agitated reactors are maintained by the presence of the agitating tool which is in simple rotation, while in the BFPM, they are maintained by a balance of forces resulting from the double rotation. For BFPM in single rotation, the equilibrium is established between the centrifugal acceleration (−˝ × ˝ × RBM ) and the pressure gradient (−(1/) ∇ p). The flow is circular and in this case the mixture is not performed. However if the BFPM, previously in rotation, is animated by another movement of revolution, an other acceleration fields such as the Coriolis acceleration (−2˝ × V BM ) is generated. These fields due to the combination of the rotation and revolution movements break the solid rotation and create axial flows. The flow becomes then three-dimensional and the mixing is promoted.
2.2.
Physical model
The mixing process in the BFPM is defined by the dimensionless physical and geometric parameters shown in Fig. 2 and detailed below: - k = R0 /R: is the ratio of the revolution radius on the cylinder radius; - F = Hf /R: is the ratio of the height of the fluid on the cylinder radius; - ˛: is the inclination angle of the container relative to the revolution arm; - n = ω/˝: is the ratio of rotation speed on the revolution speed named the revolution rate; - Fr = ˝2 R0 /g: is the Froude number; - Reω = ωR2 /: is the rotation Reynolds number; - and are the fluid density (kg/m3 ) and dynamic viscosity (kg/m s), respectively.
2.3.
Selection of axes and fundamental equations
Three reference frames are distinguished for the BFPM: the inertial frame (A), the revolution one (B) and the rotational one (C) as depicted in Fig. 2. The study of flow dynamics in double rotation coordinate system is based on the choice of the reference frame. Some seem more appropriate numerically while others are more appropriate physically. Many researchers like Noir (2000), Mouhali (2009) and Lacroix (2010) have experimentally observed that the flow is periodic in the rotational coordinate system (C) and is permanent in the revolution one (B). Although the mixing occurs in reference frame (C), the selection of (B) seems to be more appropriate for the numerical calculations because of the steady state of the flow. The Navier–Stokes equations from the point of view of the revolution reference frame (B) are written.
∂VBM 1 + VBM .∇VBM = − ∇p + ∇ 2 VBM − 2 × VBM − × × RBM ∂t (1)
where RBM indicates the position vector of the point M in reference frame (B) as indicated in Fig. 2, V BM is the flow velocity and ˝ the revolution speed of the reference frame (B) in (A). Two new forces appear in the coordinate system (B): the Coriolis force (2˝ × V BM ) and the centrifugal force (˝ × ˝ × RBM ). The centrifugal force pushes the fluid outwards in the radial direction and the Coriolis force tends to push the fluid perpendicular to the direction of its motion.
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Fig. 2 – Geometric parameters of a BFPM. The mass conservation equation for an incompressible flow is ∇ · VBM = 0
2.4.
(2)
Assumptions and boundary conditions
To study a flow that starts with a solid rotation, the cylinder speed must be constant for all calculations, therefore the rotation Reynolds number will be kept constant and equal to 125 (the optimal value obtained experimentally by Lacroix (2010) for a better mixing). The revolution speed, in turn, is variable and induces a variable revolution rate n. It is to note that the flow regime is always laminar in this range of data and the flow admits a steady-state solution. The negative sign of n, further used in this work, indicates the counter-rotation. This choice is justified by the results of Mio et al. (2004) and Lacroix (2010) who found that the grinding and mixing are more effective in counter-rotation. Within the parametric range studied, the inertial forces are greater than the gravitational forces, consequently the Froude number, which is equal to 74, is much larger than 1 and the weight of the fluid is therefore neglected. Since the thermal effects of the flow are neglected, the energy conservation equation is not taken into account in this study. Consequently the physicochemical properties are assumed to be constant. Table 1 summarises the physical and geometric parameters of the BFPM studied and further justified in the validation section (Section 2.6). The boundary conditions used in this work are detailed below and the surfaces of concern are shown in Fig. 3. - The free surface is assumed to be rigid (flat wall) and parallel to the bottom with slip conditions. From the experimental results of Lacroix (2010), it is observed that for certain operating conditions of mixing, the free surface is slightly deformed relatively to the fluid height. In addition, the friction forces between the air and the free surface of the fluid can be neglected.
Table 1 – Design and operating parameters of the BFPM. Design and operating parameters
k
F
˛
Reω
n
Values
4
1.5
45
125
Variable
- The cylinder is inclined by an angle ˛ with respect to the revolution axis. - The walls and bottom of the cylinder are rotating around the geometric axis of the cylinder. - The free surface, the walls and the bottom are subject to a revolution around another axis located at a distance R0 from the revolution axis as shown in Fig. 2. The different planes used for the presentation of the results are presented in Fig. 3. The meridional planes (xz) and (yz) are parallel and perpendicular to the revolution arm, respectively. The horizontal plane (xy), in turn, is perpendicular to the rotation axis of the cylinder.
2.5.
Numerical method and discretisation schemes
Numerical calculations are conducted using the commercial code Fluent which is based on the second order finite volume method. Several discretisation schemes were tested. The most stable scheme which converges quickly is found to be the third order upwind scheme. Consequently, the latter is used for the solution of the momentum equations and the PRESTO algorithm which, in turn, is used for the calculation of the pressure gradients. Note that the equations of velocity and pressure are simultaneously calculated.
2.6.
Solution validation
The numerical model performed in this work has been already validated by Lacroix (2010) against experimental measurements of an existing platform BFPM. According to the author, the specified values of the geometric parameters, indicated previously in Table 1 were designed by a manufacturer for the purpose of experimental investigations. The flow topology at the free surface, the dissipated power and the mixing time were then measured. In the qualitative validation – which concerns the comparison of the flow topology at the free surface – Lacroix (2010) found that the tested numerical model reproduced the experimental results with good agreement. While, the quantitative validation – in which the dissipated power and the mixing time (as functions of the Reynolds number) were compared between the numerical and experimental results – has shown a relative error of about 12.6% for the dissipated power and 13.6% for the mixing time. Several types of meshing are tested, however the quasistructured hexahedral meshing is selected to minimise error
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Fig. 3 – Boundary conditions and geometric representation of the meridional planes (xz and yz) and the horizontal planes (xy). and noise in the calculation of gradients and the Laplacian (Chergui and Dufresne, 2011). The simulations are performed in parallel on a Cluster using 8 central processing units (CPU). Each processor (Xeon E5405) is a dual core 2.0GHz with 16GB of random access memory (RAM). The numerical solution is considered as converged when the residues of the pressure and the three components of the velocity are less or equal to 10−6 . In order to ensure the independence of the numerical solution with respect to the grid meshing, a grid convergence study is performed. For that, the grid convergence index (GCI) is evaluated, according to work of Celik et al. (2008), on three components (u, v, w) of the flow velocity V BM . Three grids are tested: fine, medium and coarse with 1.3 × 103 , 0.6 × 103 and 0.3 × 103 cells, respectively. Two cases are considered: n = −13.33 where the flow becomes gradually three-dimensional and n = −4 where the flow is more complex. Although various velocity components were examined in the volume of BFPM, only the results obtained for the component velocity u – on the plane (xz) at the horizontal line z/R = 0.4 and on the plane (yz) at the vertical line y/R = 0.4 – are presented in Figs. 4 and 5. It is to note that the other horizontal and vertical line results are similar to those presented. The velocities and the distances are normalised with respect to
the linear rotation speed (ω · R) and the radius of the cylinder (R), respectively. Figs. 4 and 5 show that the curves of the normalised velocities U are substantially superposed for the three grids tested. The GCI calculated for the medium grid, regarding the fine grid, shows that all the points obtained are of the order of 10−1 to 10−4 per cent (%) for the two planes (xz) and (yz). In addition, the noise induced by the computation of the second derivatives is reduced by 40% when using the medium grid compared to the coarse grid (Chergui and Dufresne, 2011). While, between the fine and the medium grids, the noise is reduced by 5%. Consequently, the medium grid is selected for the rest of the study.
Fig. 4 – Normalised absolute velocity component, along (ox) direction, relative to the linear rotation speed (ω · R) for the three grids tested. View in the plane (xz) at the horizontal line z/R = 0.4.
Fig. 5 – Normalised absolute velocity component, along (ox) direction, relative to the linear rotation speed (ω · R) for the three meshes tested. View in the plane (yz) at the vertical line y/R = 0.4.
3.
Results
Computational fluid dynamics (CFD) visualisation techniques are used in this work to analyse the flow in the BFPM. The considered quantities are: the particle trajectories (Alliet-Gaubert et al., 2006), velocity contours (Devals et al., 2008; Fontaine et al., 2013; Hidalgo-Millán et al., 2012; Solano et al., 2012) and the streamlines (Bartels et al., 2002; Chong et al., 1990; Rivera et al., 2009). The solid rotation departure starts with a high revolution rate (n = −50) and a rotation Reynolds number (Reω )
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equal to 125. The results are discussed in terms of absolute value of n. In this section, some figures present the container in vertical position for visualisation purposes only.
3.1.
Particle trajectories
The particle trajectories are used to analyse the three dimensional structure of the flow in the BFPM. They are calculated by integration of velocity field in the revolution reference frame (B). Fig. 6 shows the particle trajectories that initially are in the meridional plane (xz) shown in Fig. 3. For the revolution rate n = −50, that indicates the start of uniform rotation, the fluid moves in circular trajectories parallel to the bottom. At this level of revolution, the flow is almost 2D and very similar to the solid rotation. The axis of these trajectories is slightly inclined towards the revolution axis with respect to the geometric axis of the cylinder. When the revolution rate decreases (i.e. n varies from −13.33 to −1), the flow in the BFPM becomes more complex and several vortical structures are observed. Indeed, the fluid particles leave the two-dimensional plane parallel to the bottom and move mainly by following concentric vortexes that give rise to a more complex 3D flow. However, no separate recirculation zones are observed (i.e. separated vortical structures). These structures have spiral forms and their sizes and shapes dependent on the revolution rate. Thus, for values of n between −50 and −4, spirals with cylindrical forms are observed as can be seen in Fig. 7. For lowest ratios (n equal to −2 and −1), the spirals become more complicated and their forms tend towards conical shapes. The spiral diameters increase with the decrease of n. For lowest revolution speeds, the particle trajectories are generally flattened at the vicinity of the wall as shown in Figs. 6 and 8. In this respect, spiral flows are numerically observed by Mouhali (2009) for a cylinder in orthogonal precession. This means that the precession axis is perpendicular to the rotation axis. The author has investigated a range of revolution rates n included between −6.67 and −16.67, i.e. a precession rate ( = 1/n) varying from −0.06 to −0.15, with a rotation Reynolds number Reω equal to 1000. The shape and size of the observed structures are found to be a function of the precession rate. It is to note that similar vortexes are also found in agitated tanks for a simple rotation movement and with an eccentric turbine or a turbine induced by the bottom of the tank (Torré et al., 2007; Galletti et al., 2009; Woziwodzki, 2014). The results obtained with n = −6 and −2 are particularly interesting because some characteristics of the flow are highlighted and particular behaviours are noticed within the flow structure. Note that these particular behaviours are not observed for the other n values. For n = −6 and at the vicinity of the mid-height plane of the cylinder (see particles in green and black colours in Figs. 9 and 10) a slight symmetry is observed in the trajectories of particles initially located on the plane (xz) particularly far away from the main rotation axis as indicated in Fig. 9a. However, the behaviour of the particles initially located on the plane (yz) is different and more complex: the spiral sizes are different and no symmetry is observed in the flow as can be seen in Fig. 9b. The same observation is found for n = −2 (see Fig. 10) with a reversed tendency. This means that the complex behaviour is observed in the plane (xz) (Fig. 10a) and the slight symmetry of the particle trajectories is noticed in the plane (yz) (Fig. 10b).
In spite of that the rotation and the revolution are in counter-rotation, all the particles rotate in the same direction as the cylinder (i.e. here in the counterclockwise direction). According to the trajectories analysis of Figs. 6–10, two main flows emerge from the flow structure configuration. First, there is a primary flow which is governed by the rotation of the cylinder. The particles located in the vicinity of the wall move following spiral trajectories with large diameters (see Figs. 6 and 8), and their movement is substantially the same for n = −50 to n = −4. These particles follow the same motion as the ones adjacent to the wall which defining the “primary rotation”. There is also a secondary flow which is governed both by the rotation and revolution movements of the cylinder. This time, the particles initially located in the vicinity of the rotation axis define and describe better this “secondary rotation” (Figs. 6 and 7). Note that new appellations, designated by (i) “primary spin axis” (PSA) for the primary rotation axis of the particle movement and (ii) “secondary spin axis” (SSA) for the secondary flow axis, are proposed in this work. From these results of the analysis of the particle trajectories three axes can be distinguished. Two of them are geometrically known: the primary spin axis (PSA) and the revolution axis. The third axis – which represents the secondary spin axis (SSA) – depends on the flow behaviour. This SSA is the main axis of pumping as illustrated by Supplementary material in Appendix A. Furthermore, the SSA changes direction with the variation of revolution rate. More specifically, the SSA follows a diagonal direction change when the revolution rate n decreases (i.e. when the revolution speed increases) as shown by Supplementary material available in Appendix A. However, the geometric determination of the SSA can help to identify the direction of pumping according to n value. To better identify the main vortical structures in the BFPM and to properly show the SSA various techniques were tested. These techniques include the minimum pressure (Jeong and Hussain, 1995; Cucitore et al., 1999), maximum vorticity (Kida and Hideaki, 1998; Kolár, 2007; Hidalgo-Millán et al., 2012), Q criterion (Carmer et al., 2008) and 2 criterion (Jeong and Hussain, 1995; Cala et al., 2006; Ducci and Yianneskis, 2007; Carmer et al., 2008; Roy et al., 2010; Koched et al., 2011). However, some of them are not always reliable to identify the presence, the number and the size of the vortexes for some types of flow (Jeong and Hussain, 1995; Kida and Hideaki, 1998; Cucitore et al., 1999; Chakraborty et al., 2005). It is noted that none of these criteria have described adequately the vortical structures in the BFPM and their axes of rotation. The BFPM system is accelerated by the two movements (rotation and revolution) that induce a background vorticity which is, in turn, strongly present in the confined flow of the cylinder. A geometric method is proposed to locate the SSA (see details in Appendix B). The method is based on the determination of the centroid of the particle trajectory curves in a given plane. This method allows to give a qualitative insight about the SSA. The results obtained are shown in Fig. 11. According to that figure all the axes meet together at z/R = 0.5. For the revolution rate n = −50, the SSA coincides with the PSA. Note that at this speed (n = −50), the fluid is in solid rotation and the particles flow in uniform circular paths. Nevertheless, when the value of n decreases the particles move along a cylindrical spiral from the bottom to the free surface of the cylinder (see Fig. 7 for cases n = −13.33 to −4). The SSA in this case starts to lose its linearity and becomes increasingly helical. For n = −2 and −1, the flow becomes more complicated, the spirals take
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Fig. 6 – Particle trajectories located in the plane (xz) for different values of n with Reω = 125.
conical forms (see Fig. 7 for cases n = −2 to −1), and the secondary rotation starts from the vicinity of the bottom of the cylinder and ends before reaching the free surface. The SSA changes direction and the trajectories of its points draw a cone when the revolution speed varies (Fig. 11). The top of the SSA moves following a circular form (Fig. 12a) while the other end keeps its position at the bottom centre. These results are in good agreement with those obtained by Vanyo et al. (1995) and Noir (2000) for the presence of a secondary spin
axis and the change of its direction.1 The authors conducted their experiments on a fluid in a precessing ellipsoid. Vanyo
1 The observations – concerning the presence and the direction change of the secondary spin axis (SSA) – noticed by Vanyo et al. (1995) and Noir (2000) in their works are not shown in this work. For more details, the reader is advised to refer to the authors’ works.
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Fig. 7 – Particle trajectories located initially on the same straight line (D) and in the vicinity of the rotation axis in the plane (xz) for different values of n with Reω = 125.
et al. (1995) have carried out their experiment with revolution rate varying from n = −2000 to −17.4 ( = −0.0005 to −0.0575) with a Reω = 7700 while the work of Noir (2000) is conducted for n = −300 to −50 ( = −0.0033 to −0.02) with a Reω = 1969. They considered that for the investigated range of n, the flow is not a solid body rotation. In addition, the authors could not identify the rotation axis of the fluid beyond a certain value of the revolution rate (n = −17.4 for Vanyo et al. (1995) and n = −50 for
Noir (2000)) using their visualisation technique because of the nonlinearity of the axis.
3.2.
Velocity iso-contours
Since the cases of n = −2 and n = −6 have shown a particular flow dynamics (as discussed in Section 3.1), an analysis of velocity iso-contours of these two revolution rates are
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Fig. 8 – Particle trajectories located initially in the vicinity of wall of cylindrical surface (C) for different values of n with Reω = 125.
performed with respect to case with n = −50 in order to highlight the flow symmetry observed previously with the particle trajectories. The results presented below are those of the components of the velocity relative to the revolution ( V BM ) on both meridional planes (xz) and (yz) clearly shown in Fig. 3. In the case of a rotation of the fluid as a rigid body, in the plane (xz), the components u and w are equal to zero and the component v is a linear function of R and constant along the rotation axis.
In the plane (yz), v and w are equal to zero while u is a linear function of R and constant along the axis of rotation. During the start of the solid rotation – i.e. the BFPM is in double rotation with a high n value (n = −50 which means a very small revolution rate) – the behaviour of the fluid is very similar to a fluid subjected to a single rotation, the flow is almost symmetrical and two-dimensional. Far from the bottom, the iso-contours are almost straight and parallel to the
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Fig. 9 – Trajectories of 4 particles located initially in (a) the plane (xz) at x/R = 0.4 and (b) the plane (yz) at y/R = 0.4 for n = −6 with Reω = 125. (For interpretation of the references to colour in the text, the reader is referred to the web version of the article.)
Fig. 10 – Trajectories of 4 particles located initially in (a) the plane (xz) at x/R = 0.4 and (b) the plane (yz) at y/R = 0.4 for n = −2 with Reω = 125. (For interpretation of the references to colour in the text, the reader is referred to the web version of the article.) rotation axis for the component v in the plane (xz) (Fig. 13b1) and the component u in the plane (yz) (Fig. 14a1). These two velocity components vary linearly with the radius R. However, near the bottom they have a slightly curved shape. In addition, the magnitude of the components u and w in the plane (xz) and v and w in the plane (yz) is very low. Moreover, a perfect symmetry of the component w with the rotation axis is noticed (see Figs. 13c1 and 14c1). However, the iso-contours of the components u in the plane (xz) (Fig. 13a1) and v in the
plane (yz) (Fig. 14a1) are not symmetrical because of the slip condition at the free surface and the adhesion condition at the bottom of the cylinder. The velocity components that have low value, in case of low revolution rate n = −50, become more significant when the revolution increases (small values of n). This observation confirms the three-dimensional behaviour of the flow. In case of n = −6, the symmetry is totally broken for the components u and w in the plane (yz) (see Fig. 14a2 and c2) and slightly broken
Fig. 11 – Secondary spin axis (SSA) geometrically identified for different values of n with Reω = 125.
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Fig. 12 – Displacement of the secondary spin axis (SSA) with respect to the primary spin axis (PSA) and the revolution axis.
for u and w in the plane (xz) (see Fig. 13a2 and c2). However for n = −2, an opposite observation is made: the symmetry is completely lost in the plane (xz) (see Fig. 13a3 and c3) and slightly broken in the plane (yz) (see Fig. 14a3 and c3).
3.3.
Flow topology
The results of the velocity iso-contours presented in the previous section for n = −6 and −2 confirm the particularity of
Fig. 13 – Contours of the three velocity components relative to the revolution frame in the meridional plane (xz) parallel to the revolution arm. The velocity is normalised to the linear rotation speed (ω . R) and the coordinates x, y and z are normalised with respect to the radius R: (ai ) the velocity component (u) along the ox axis, (bi ) the velocity component (v) along the oy axis and (ci ) the velocity component (w) along the oz axis. The various cases are conducted for: (1) n = −50, (2) n = −6 and (3) n = −2 with Reω = 125.
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Fig. 14 – Contours of the three velocity components relative to the revolution frame in the meridional plane (yz) perpendicular to the revolution arm. The velocity is normalised to the linear rotation speed (ω · R) and the coordinates x, y and z are normalised with respect to the radius R: (ai ) the velocity component (u) along the ox axis, (bi ) the velocity component (v) along the oy axis and (ci ) the velocity component (w) along the oz axis. The various cases are conducted for: (1) n = −50, (2) n = −6 and (3) n = −2 with Reω = 125. flow for these two values of n. To better investigate this particularity, a two-dimensional visualisation technique using streamlines is used. Furthermore, the study is extended to other revolution rates to identify the rate for which the flow topology is subject to significant changes. For incompressible flows, the stream function is obtained from the integration of the velocity field following the equation VBM = ∇ ×
(3)
For instance, in the (xy) plane the stream function obtained such that
z
=
z
is
udy +
vdx
(4)
Fig. 15 shows the results of streamlines in horizontal planes (xy) perpendicular to the rotation axis at different heights within the cylinder. The ratio n varies and Reynolds number Reω is maintained constant and equal to 125. The case
of n = −50 is a typical uniform rotation, the streamlines are circular with a stagnation point at the centre of the plane, which means that the flow is overall 2D. When n decreases, the lines become increasingly spiral in the plane (xy) with a velocity component perpendicular to the plane which allows the evacuation of the fluid along the cylinder axis. The location of spiral centre changes as one moves away from the bottom of the container. Therefore this axial component constrains the fluid to move vertically along a spiral and along a non-linear axis, thereby confirming the results obtained previously with the particle trajectories (Section 3.1). On the other hand, the position of this centre is constantly changing at each plane when the revolution speed increases (n value decreases). The trajectory drawn by these centre positions follows the SSA trajectory. Figs. 16 and 17 show the flow topology in the planes (xz) and (yz), respectively, for different values of n that are −50, −13.33, −10, −8, −6, −4, −2 and −1. These figures show that the particular characteristics of the flow, already mentioned and highlighted in previous sections, appear in the plane (yz)
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Fig. 15 – Flow topology in the BFPM for different horizontal planes (xy) perpendicular to the rotation axis for various n with Reω = 125.
for a range values of n between −8 and −4, while for the plane (xz) these flow particular characteristics start from n = −2.5. The revolution rate values less than −8 are the most critical for the flow. This flow singularity occurs only in one of the two meridional planes, i.e. the flow returns to its original topology in one plane when the change begins in the other plane. For a rigid body rotation, the velocity component values that are
tangent to the planes (xz) and (yz) are equal to zero. Therefore the streamlines are not represented on these planes. However for n = −50, circular streamlines are obtained on the two planes (xz) and (yz). Despite the two-dimensional configuration of the flow, axial components along x and z directions exist within the flow but their values remain insignificant. Figs. 16 and 17 show a flow configuration more complex and the existence
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Fig. 16 – Flow topology in the BFPM for the vertical plane (xz) parallel to the revolution arm for various n with Reω = 125. of several critical points (Tobak and Peake, 1982) on the plane (xz) when |n| ≤ 2.5 and on the plane (yz) when the n value is included between −8 and −5. As shown in Fig. 16 for the plane (xz), the centre starts moving towards the left side at n = −4 and becomes a focus
at n = −2.5. The latter divides into several other focuses when the revolution rate decreases and then saddles and separation zones appear. In addition, the flow is drained away horizontally in form of spirals with different sizes. On the plane (yz), as indicated in Fig. 17, when n decreases a focus takes origin
Table 2 – Summary of the flow characteristics at the two planes (xz) and (yz). Plane (yz)
Plane (xz)
Position of the plane
Perpendicular to the revolution arm
Parallel to the revolution arm
Effects of n
– Flow singularities start at n = −8 – Flow starts returning to its initial state at n = −4
– Flow singularities start at n = −4 – Flow singularities remain existing for n ≤ −4
Flow characteristics
– Separation and recirculation zones exist within the flow – Attachment and detachment of spiral phenomena occur often at walls
– Separation and recirculation zones exist within the flow – Spirals are separated from the wall
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Fig. 17 – Flow topology in the BFPM for the vertical plane (yz) perpendicular to the revolution arm for various n with Reω = 125.
above the bottom then detaches and moves towards the top and pushes the already existing centre. The detachment of the fluid is indicated by the appearance of half-saddle forms on the bottom wall and by the direction of the flow in this zone. The displaced centre is divided into several spiral structures (for n = −6.5, −6, −5.84 and −5) to finally establish itself on the lateral wall (fact observed for two focuses). For the insignificant revolution rates (cases with n = −4 to −1, results not reported here for sake of brevity), the focus that detached before expands itself progressively to occupy all the plane (yz) and to return finally to its initial state (case of n = −50). Furthermore, the attachment and detachment of the flow on the solid walls are mainly observed in the plane (yz) in opposite to the plane (xz) for the studied values of n. Table 2 summarises the characteristics of the flow in the two meridional planes (xz) and (yz).
4.
Conclusion
In this work, the characteristics of a flow within blade free planetary mixer (BFPM) are investigated by means of computational fluid dynamics simulations. The numerical simulations were carried out using second order finite volumes as implemented in the commercial code Fluent. A qualitative analysis of the flow dynamic was performed using visualisation techniques with various revolution rates and a constant rotation Reynolds number Reω = 125. In this respect, the parameters investigated are the particle trajectories, velocity iso-contours and streamlines. The main findings of the study can be summarised as follows. • Vortical structures in spiral forms are observed for various revolution rates and a constant rotation Reynolds number.
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Fig. B-1 – General view of (a) the 3D spiral and (b) the curve of the spiral of a given part (hj ) projected onto the plane (xy)j indicating the centroid (Mj ) location. The flow behaviour is divided into two parts: a primary rotation in the vicinity of the wall and a second rotation in the vicinity of the geometric axis of the BFPM. A total of three rotation axes are identified within the BFPM flow structure: (i) a primary spin axis (PSA), (ii) a secondary spin axis (SSA) and (iii) a revolution axis. The criteria for identifying vortex structures are tested to clearly locate the SSA, however none of them has adequately represented the spirals observed in the cylinder and/or determined their rotation axes. To this purpose, a geometric method based on the calculation of the centroid curves was used. The results are very conclusive and the SSA is determined. The direction of the SSA changes with the revolution rate n. • Some results of this qualitative analysis are in good agreement with those found for other doubly rotating flows by Vanyo et al. (1995), Noir (2000) and Mouhali (2009). Although the range of the studied parameters is not identical. • The analysis of the particle trajectories, velocity isocontours and stream functions showed that (i) there is a symmetry in the flow for some revolution rates and (ii) there are critical revolution rates for which the flow dynamics is subject to major changes. These changes cannot occur at the same time on both meridional planes (xz and yz) studied. Further studies are required to give a more proper explanation on the mechanisms at the origin of these significant flow changes.
Acknowledgement Financial support for this study from the Natural Sciences and Engineering Research Council of Canada (http://www.nserc-crsng.gc.ca/index eng.asp NSERC) is gratefully acknowledged.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.cherd.2015.05.045.
Appendix B. Secondary spin axis (SSA) determination method To approximately evaluate the secondary spin axis (SSA), a geometric method based on the determination of the centroid
of the particle trajectory curves in a plane is proposed. The procedure is as follows. - Particles located initially in the vicinity of the rotation axis are those that best describe the secondary rotation (see Fig. 7). Therefore a straight line (D) of particles is chosen in the vicinity of the cylinder rotation axis. - The 3D spiral of height h is divided into several spiral parts of small heights ( hj ) (see Fig. B-1a). Then, each part of the resulting spiral segments is projected onto a plane (xy)j thus forming a curve such as that depicted in Fig. B-1b. - The coordinates of the centroid (Mj ) of the curve obtained in the plane (xy)j are calculated using the two following equations (Beer and Johnston, 2000). x¯ j =
x L i
(B-1)
y L i
(B-2)
L
and y¯ j =
L
- Finally, the set of the centroids (Mj ) resulting from the different parts of the spiral forms the secondary spin axis (SSA).
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Lagrange, R., Eloy, C., Nadal, F., Meunier, P., 2008. Instability of a fluid inside a precessing cylinder. Phys. Fluids (1994–present) 20 (8), 081701. Massing, U., Cicko, S., Ziroli, V., 2008. Dual asymmetric centrifugation (DAC) – a new technique for liposome preparation. J. Control. Release 125 (1), 16–24. Mio, H., Kano, J., Saito, F., Kaneko, K., 2004. Optimum revolution and rotational directions and their speeds in planetary ball milling. Int. J. Miner. Process. 74 (Suppl.), S85–S92. Mouhali, W., 2009. Study of a Precessing Flow. Transition to Turbulence and Application to Dynamo Effect. Université Paris-Diderot – Paris VII, France (in French) (Ph.D. thesis). Niwa, T., Hashimoto, N., 2008. Novel technology to prepare oral formulations for preclinical safety studies. Int. J. Pharm. 350 (1–2), 70–78. Noir, J., 2000. Fluid Flow in Precessing Cavity: Numerical and Experimental Approaches. Université Joseph Fourier – Grenoble I, France (in French) (Ph.D. thesis). Ottino, J.M., 1989. The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge University Press. Raza, M.A., Westwood, A.V.K., Sterling, C., 2012. Effect of processing technique on the transport and mechanical properties of graphite nanoplatelet/rubbery epoxy composites for thermal interface applications. Mater. Chem. Phys. 132 (1), 63–73. Rivera, C.A., Heniche, M., Takenaka, K., Tanguy, P.A., 2009. Finite element modeling of the laminar and transition flow of the superblend dual shaft coaxial mixer on parallel computers. Chem. Eng. Sci. 64 (21), 4442–4456. Roy, S., Acharya, S., Cloeter, M.D., 2010. Flow structure and the effect of macro-instabilities in a pitched-blade stirred tank. Chem. Eng. Sci. 65 (10), 3009–3024. Solano, J.P., Herrero, R., Espin, S., Phan, A.N., Harvey, A.P., 2012. Numerical study of the flow pattern and heat transfer enhancement in oscillatory baffled reactors with helical coil inserts. Chem. Eng. Res. Des. 90 (6), 732–742. Tobak, M., Peake, D.J., 1982. Topology of three-dimensional separated flows. Annu. Rev. Fluid Mech. 14, 61–85. Torré, J.P., Fletcher, D.F., Lasuye, T., Xuereb, C., 2007. Single and multiphase CFD approaches for modelling partially baffled stirred vessels: comparison of experimental data with numerical predictions. Chem. Eng. Sci. 62 (22), 6246–6262. Vanyo, J., Wilde, P., Cardin, P., Olson, P., 1995. Experiments on precessing flows in the Earth’s liquid core. Geophys. J. Int. 121 (1), 136–142. Woziwodzki, S., 2014. Mixing of viscous Newtonian fluids in a vessel equipped with steady and unsteady rotating dual-turbine impellers. Chem. Eng. Res. Des. 92 (3), 435–446.
Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
CFD study of flow dynamics in a blade free planetary mixer (BFPM) – A qualitative flow study Nacera Chergui a,∗ , Mohamed Lateb b , Étienne Lacroix a , Louis Dufresne a a
Thermo-Fluid for Transport (TFT) Laboratory, Mechanical Engineering Department, École de Technologie Supérieure (ÉTS) 1100 Notre-Dame West, Montréal, Québec H3C 1K3, Canada b New Jersey Institute of Technology (NJIT), Center for Natural Resources Development and Protection 323 Martin Luther King Jr. Blvd., Newark, NJ 07102, United States
a r t i c l e
i n f o
a b s t r a c t
Article history:
A numerical study is conducted to analyse the flow dynamics in a blade free planetary mixer
Received 19 March 2015
(BFPM). The flow is simulated in the revolution reference frame. The analysis is focused on
Received in revised form 25 May
the departure from solid body rotation to a more complex 3D flow structure. Numerical
2015
calculations are realised using second order finite volume discretisation. The parameters
Accepted 31 May 2015
investigated are the particle trajectories, velocity contours and streamlines. The results
Available online 16 June 2015
show that the flow is generally swirling with the presence of singular points on different
Keywords:
and rotation axes are dependent on the revolution rate. In addition, the dynamics of the flow
Blade free planetary mixer (BFPM)
is subject to significant changes at certain revolution rates.
specific planes. The vortical structures are found to be in spiral forms and their sizes, shapes,
Visualisation technique
© 2015 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers.
Rotating cylinder Vortical structures Qualitative analysis Flow dynamic
1.
Introduction
Blade free planetary mixers (BFPMs) are mainly constituted of a cylindrical container which generally has a flat bottom with no obstacles nor agitators. The container is subject to two rotational movements: one is a rotation around its geometric axis, the second is a revolution around an axis located at a given distance (R0 ) as shown in Fig. 1. The cylinder is usually inclined at an angle ˛ to improve mixing. The combination of the double rotation is similar to that of the Earth around itself and around the sun, thereby the name of “Planetary”. The BFPM technique is known to be a convenient technology since the 1970s for synthesising and mixing fluid phase products (Massing et al., 2008). The mixing action is provided through two movements (rotation and revolution). These two movements combined together improve the mixing without an external intervention. The BFPMs are usually used in industry (pharmaceutical,
∗
biotechnology, biochemical, etc.) to mix, synthesise and aerate high viscosity products (colloids, resins, pastes, creams, etc.) with a minimum degree of contamination (Massing et al., 2008; Raza et al., 2012). However, some deficiencies remain in the design and the application of this technique and some mixing problems were noticed. The products are sometimes of poor quality for certain operating and/or designing conditions (e.g. Chen et al., 2007; Niwa and Hashimoto, 2008; Massing et al., 2008; Hirsch et al., 2009; Raza et al., 2012). Specifically, the link between the operating conditions and the quality of the product could not be determined. For instance, some authors proceed by using trial and error methods to correct the mixing issue. However, the kinematic of the fluid, which is strongly linked to the dynamics of the mixture (Ottino, 1989), is itself a function of operating and designing conditions. In other words, to understand the mixing problem, the dynamics of flow in BFPMs must first be understood. To the best of authors’
Corresponding author. Tel.: +1 514 396 8800x7838. E-mail address: [email protected] (N. Chergui). http://dx.doi.org/10.1016/j.cherd.2015.05.045 0263-8762/© 2015 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers.
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Fig. 1 – Double rotating systems: (a) BFPM with R0 0 and (b) precessing cylinder with R0 ≈ 0.
knowledge no study has been published on the flow dynamics in BFPMs. This is probably due to confidentiality reasons and/or to competitiveness issues. Nevertheless, it should be mentioned that flow studies subject to double rotation, in similar systems (precessing cavities), were conducted and concern the behaviour of fluids in the core of the earth and the satellite reservoirs (e.g. Vanyo et al., 1995; Noir, 2000; Lagrange et al., 2008; Mouhali, 2009). Note that the difference between these two systems lies in the fact that, for the precessing cavity system, the revolution axis intercepts the rotation axis within the cylinder volume, while for the BFPM system the intersection occurs outside of the cylinder volume, as illustrated in Fig. 1. The objective of this study is to characterise qualitatively the flow dynamics within the BFPM. More specifically by considering the departure from solid body rotation. The reminder of this article is organised as follows. Section 2 summarises the methodology used in this work such as the physical model, fundamental equations and the detailed parameters of the numerical model. The results are addressed and developed in Section 3. The section concentrates on the main findings of the visualisation technique such as the particle trajectories, velocity iso-contours and the topology of the flow. Finally, a conclusion is presented in Section 4.
2. Mathematical model and numerical methodology Computational fluid dynamic (CFD) calculations are used for this study. The fluid is assumed to be Newtonian and homogeneous, and the rotating flow in the container is laminar and incompressible.
2.1.
Choice of the departure from solid body rotation
The flow and mixture in agitated reactors are maintained by the presence of the agitating tool which is in simple rotation, while in the BFPM, they are maintained by a balance of forces resulting from the double rotation. For BFPM in single rotation, the equilibrium is established between the centrifugal acceleration (−˝ × ˝ × RBM ) and the pressure gradient (−(1/) ∇ p). The flow is circular and in this case the mixture is not performed. However if the BFPM, previously in rotation, is animated by another movement of revolution, an other acceleration fields such as the Coriolis acceleration (−2˝ × V BM ) is generated. These fields due to the combination of the rotation and revolution movements break the solid rotation and create axial flows. The flow becomes then three-dimensional and the mixing is promoted.
2.2.
Physical model
The mixing process in the BFPM is defined by the dimensionless physical and geometric parameters shown in Fig. 2 and detailed below: - k = R0 /R: is the ratio of the revolution radius on the cylinder radius; - F = Hf /R: is the ratio of the height of the fluid on the cylinder radius; - ˛: is the inclination angle of the container relative to the revolution arm; - n = ω/˝: is the ratio of rotation speed on the revolution speed named the revolution rate; - Fr = ˝2 R0 /g: is the Froude number; - Reω = ωR2 /: is the rotation Reynolds number; - and are the fluid density (kg/m3 ) and dynamic viscosity (kg/m s), respectively.
2.3.
Selection of axes and fundamental equations
Three reference frames are distinguished for the BFPM: the inertial frame (A), the revolution one (B) and the rotational one (C) as depicted in Fig. 2. The study of flow dynamics in double rotation coordinate system is based on the choice of the reference frame. Some seem more appropriate numerically while others are more appropriate physically. Many researchers like Noir (2000), Mouhali (2009) and Lacroix (2010) have experimentally observed that the flow is periodic in the rotational coordinate system (C) and is permanent in the revolution one (B). Although the mixing occurs in reference frame (C), the selection of (B) seems to be more appropriate for the numerical calculations because of the steady state of the flow. The Navier–Stokes equations from the point of view of the revolution reference frame (B) are written.
∂VBM 1 + VBM .∇VBM = − ∇p + ∇ 2 VBM − 2 × VBM − × × RBM ∂t (1)
where RBM indicates the position vector of the point M in reference frame (B) as indicated in Fig. 2, V BM is the flow velocity and ˝ the revolution speed of the reference frame (B) in (A). Two new forces appear in the coordinate system (B): the Coriolis force (2˝ × V BM ) and the centrifugal force (˝ × ˝ × RBM ). The centrifugal force pushes the fluid outwards in the radial direction and the Coriolis force tends to push the fluid perpendicular to the direction of its motion.
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Fig. 2 – Geometric parameters of a BFPM. The mass conservation equation for an incompressible flow is ∇ · VBM = 0
2.4.
(2)
Assumptions and boundary conditions
To study a flow that starts with a solid rotation, the cylinder speed must be constant for all calculations, therefore the rotation Reynolds number will be kept constant and equal to 125 (the optimal value obtained experimentally by Lacroix (2010) for a better mixing). The revolution speed, in turn, is variable and induces a variable revolution rate n. It is to note that the flow regime is always laminar in this range of data and the flow admits a steady-state solution. The negative sign of n, further used in this work, indicates the counter-rotation. This choice is justified by the results of Mio et al. (2004) and Lacroix (2010) who found that the grinding and mixing are more effective in counter-rotation. Within the parametric range studied, the inertial forces are greater than the gravitational forces, consequently the Froude number, which is equal to 74, is much larger than 1 and the weight of the fluid is therefore neglected. Since the thermal effects of the flow are neglected, the energy conservation equation is not taken into account in this study. Consequently the physicochemical properties are assumed to be constant. Table 1 summarises the physical and geometric parameters of the BFPM studied and further justified in the validation section (Section 2.6). The boundary conditions used in this work are detailed below and the surfaces of concern are shown in Fig. 3. - The free surface is assumed to be rigid (flat wall) and parallel to the bottom with slip conditions. From the experimental results of Lacroix (2010), it is observed that for certain operating conditions of mixing, the free surface is slightly deformed relatively to the fluid height. In addition, the friction forces between the air and the free surface of the fluid can be neglected.
Table 1 – Design and operating parameters of the BFPM. Design and operating parameters
k
F
˛
Reω
n
Values
4
1.5
45
125
Variable
- The cylinder is inclined by an angle ˛ with respect to the revolution axis. - The walls and bottom of the cylinder are rotating around the geometric axis of the cylinder. - The free surface, the walls and the bottom are subject to a revolution around another axis located at a distance R0 from the revolution axis as shown in Fig. 2. The different planes used for the presentation of the results are presented in Fig. 3. The meridional planes (xz) and (yz) are parallel and perpendicular to the revolution arm, respectively. The horizontal plane (xy), in turn, is perpendicular to the rotation axis of the cylinder.
2.5.
Numerical method and discretisation schemes
Numerical calculations are conducted using the commercial code Fluent which is based on the second order finite volume method. Several discretisation schemes were tested. The most stable scheme which converges quickly is found to be the third order upwind scheme. Consequently, the latter is used for the solution of the momentum equations and the PRESTO algorithm which, in turn, is used for the calculation of the pressure gradients. Note that the equations of velocity and pressure are simultaneously calculated.
2.6.
Solution validation
The numerical model performed in this work has been already validated by Lacroix (2010) against experimental measurements of an existing platform BFPM. According to the author, the specified values of the geometric parameters, indicated previously in Table 1 were designed by a manufacturer for the purpose of experimental investigations. The flow topology at the free surface, the dissipated power and the mixing time were then measured. In the qualitative validation – which concerns the comparison of the flow topology at the free surface – Lacroix (2010) found that the tested numerical model reproduced the experimental results with good agreement. While, the quantitative validation – in which the dissipated power and the mixing time (as functions of the Reynolds number) were compared between the numerical and experimental results – has shown a relative error of about 12.6% for the dissipated power and 13.6% for the mixing time. Several types of meshing are tested, however the quasistructured hexahedral meshing is selected to minimise error
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Fig. 3 – Boundary conditions and geometric representation of the meridional planes (xz and yz) and the horizontal planes (xy). and noise in the calculation of gradients and the Laplacian (Chergui and Dufresne, 2011). The simulations are performed in parallel on a Cluster using 8 central processing units (CPU). Each processor (Xeon E5405) is a dual core 2.0GHz with 16GB of random access memory (RAM). The numerical solution is considered as converged when the residues of the pressure and the three components of the velocity are less or equal to 10−6 . In order to ensure the independence of the numerical solution with respect to the grid meshing, a grid convergence study is performed. For that, the grid convergence index (GCI) is evaluated, according to work of Celik et al. (2008), on three components (u, v, w) of the flow velocity V BM . Three grids are tested: fine, medium and coarse with 1.3 × 103 , 0.6 × 103 and 0.3 × 103 cells, respectively. Two cases are considered: n = −13.33 where the flow becomes gradually three-dimensional and n = −4 where the flow is more complex. Although various velocity components were examined in the volume of BFPM, only the results obtained for the component velocity u – on the plane (xz) at the horizontal line z/R = 0.4 and on the plane (yz) at the vertical line y/R = 0.4 – are presented in Figs. 4 and 5. It is to note that the other horizontal and vertical line results are similar to those presented. The velocities and the distances are normalised with respect to
the linear rotation speed (ω · R) and the radius of the cylinder (R), respectively. Figs. 4 and 5 show that the curves of the normalised velocities U are substantially superposed for the three grids tested. The GCI calculated for the medium grid, regarding the fine grid, shows that all the points obtained are of the order of 10−1 to 10−4 per cent (%) for the two planes (xz) and (yz). In addition, the noise induced by the computation of the second derivatives is reduced by 40% when using the medium grid compared to the coarse grid (Chergui and Dufresne, 2011). While, between the fine and the medium grids, the noise is reduced by 5%. Consequently, the medium grid is selected for the rest of the study.
Fig. 4 – Normalised absolute velocity component, along (ox) direction, relative to the linear rotation speed (ω · R) for the three grids tested. View in the plane (xz) at the horizontal line z/R = 0.4.
Fig. 5 – Normalised absolute velocity component, along (ox) direction, relative to the linear rotation speed (ω · R) for the three meshes tested. View in the plane (yz) at the vertical line y/R = 0.4.
3.
Results
Computational fluid dynamics (CFD) visualisation techniques are used in this work to analyse the flow in the BFPM. The considered quantities are: the particle trajectories (Alliet-Gaubert et al., 2006), velocity contours (Devals et al., 2008; Fontaine et al., 2013; Hidalgo-Millán et al., 2012; Solano et al., 2012) and the streamlines (Bartels et al., 2002; Chong et al., 1990; Rivera et al., 2009). The solid rotation departure starts with a high revolution rate (n = −50) and a rotation Reynolds number (Reω )
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equal to 125. The results are discussed in terms of absolute value of n. In this section, some figures present the container in vertical position for visualisation purposes only.
3.1.
Particle trajectories
The particle trajectories are used to analyse the three dimensional structure of the flow in the BFPM. They are calculated by integration of velocity field in the revolution reference frame (B). Fig. 6 shows the particle trajectories that initially are in the meridional plane (xz) shown in Fig. 3. For the revolution rate n = −50, that indicates the start of uniform rotation, the fluid moves in circular trajectories parallel to the bottom. At this level of revolution, the flow is almost 2D and very similar to the solid rotation. The axis of these trajectories is slightly inclined towards the revolution axis with respect to the geometric axis of the cylinder. When the revolution rate decreases (i.e. n varies from −13.33 to −1), the flow in the BFPM becomes more complex and several vortical structures are observed. Indeed, the fluid particles leave the two-dimensional plane parallel to the bottom and move mainly by following concentric vortexes that give rise to a more complex 3D flow. However, no separate recirculation zones are observed (i.e. separated vortical structures). These structures have spiral forms and their sizes and shapes dependent on the revolution rate. Thus, for values of n between −50 and −4, spirals with cylindrical forms are observed as can be seen in Fig. 7. For lowest ratios (n equal to −2 and −1), the spirals become more complicated and their forms tend towards conical shapes. The spiral diameters increase with the decrease of n. For lowest revolution speeds, the particle trajectories are generally flattened at the vicinity of the wall as shown in Figs. 6 and 8. In this respect, spiral flows are numerically observed by Mouhali (2009) for a cylinder in orthogonal precession. This means that the precession axis is perpendicular to the rotation axis. The author has investigated a range of revolution rates n included between −6.67 and −16.67, i.e. a precession rate ( = 1/n) varying from −0.06 to −0.15, with a rotation Reynolds number Reω equal to 1000. The shape and size of the observed structures are found to be a function of the precession rate. It is to note that similar vortexes are also found in agitated tanks for a simple rotation movement and with an eccentric turbine or a turbine induced by the bottom of the tank (Torré et al., 2007; Galletti et al., 2009; Woziwodzki, 2014). The results obtained with n = −6 and −2 are particularly interesting because some characteristics of the flow are highlighted and particular behaviours are noticed within the flow structure. Note that these particular behaviours are not observed for the other n values. For n = −6 and at the vicinity of the mid-height plane of the cylinder (see particles in green and black colours in Figs. 9 and 10) a slight symmetry is observed in the trajectories of particles initially located on the plane (xz) particularly far away from the main rotation axis as indicated in Fig. 9a. However, the behaviour of the particles initially located on the plane (yz) is different and more complex: the spiral sizes are different and no symmetry is observed in the flow as can be seen in Fig. 9b. The same observation is found for n = −2 (see Fig. 10) with a reversed tendency. This means that the complex behaviour is observed in the plane (xz) (Fig. 10a) and the slight symmetry of the particle trajectories is noticed in the plane (yz) (Fig. 10b).
In spite of that the rotation and the revolution are in counter-rotation, all the particles rotate in the same direction as the cylinder (i.e. here in the counterclockwise direction). According to the trajectories analysis of Figs. 6–10, two main flows emerge from the flow structure configuration. First, there is a primary flow which is governed by the rotation of the cylinder. The particles located in the vicinity of the wall move following spiral trajectories with large diameters (see Figs. 6 and 8), and their movement is substantially the same for n = −50 to n = −4. These particles follow the same motion as the ones adjacent to the wall which defining the “primary rotation”. There is also a secondary flow which is governed both by the rotation and revolution movements of the cylinder. This time, the particles initially located in the vicinity of the rotation axis define and describe better this “secondary rotation” (Figs. 6 and 7). Note that new appellations, designated by (i) “primary spin axis” (PSA) for the primary rotation axis of the particle movement and (ii) “secondary spin axis” (SSA) for the secondary flow axis, are proposed in this work. From these results of the analysis of the particle trajectories three axes can be distinguished. Two of them are geometrically known: the primary spin axis (PSA) and the revolution axis. The third axis – which represents the secondary spin axis (SSA) – depends on the flow behaviour. This SSA is the main axis of pumping as illustrated by Supplementary material in Appendix A. Furthermore, the SSA changes direction with the variation of revolution rate. More specifically, the SSA follows a diagonal direction change when the revolution rate n decreases (i.e. when the revolution speed increases) as shown by Supplementary material available in Appendix A. However, the geometric determination of the SSA can help to identify the direction of pumping according to n value. To better identify the main vortical structures in the BFPM and to properly show the SSA various techniques were tested. These techniques include the minimum pressure (Jeong and Hussain, 1995; Cucitore et al., 1999), maximum vorticity (Kida and Hideaki, 1998; Kolár, 2007; Hidalgo-Millán et al., 2012), Q criterion (Carmer et al., 2008) and 2 criterion (Jeong and Hussain, 1995; Cala et al., 2006; Ducci and Yianneskis, 2007; Carmer et al., 2008; Roy et al., 2010; Koched et al., 2011). However, some of them are not always reliable to identify the presence, the number and the size of the vortexes for some types of flow (Jeong and Hussain, 1995; Kida and Hideaki, 1998; Cucitore et al., 1999; Chakraborty et al., 2005). It is noted that none of these criteria have described adequately the vortical structures in the BFPM and their axes of rotation. The BFPM system is accelerated by the two movements (rotation and revolution) that induce a background vorticity which is, in turn, strongly present in the confined flow of the cylinder. A geometric method is proposed to locate the SSA (see details in Appendix B). The method is based on the determination of the centroid of the particle trajectory curves in a given plane. This method allows to give a qualitative insight about the SSA. The results obtained are shown in Fig. 11. According to that figure all the axes meet together at z/R = 0.5. For the revolution rate n = −50, the SSA coincides with the PSA. Note that at this speed (n = −50), the fluid is in solid rotation and the particles flow in uniform circular paths. Nevertheless, when the value of n decreases the particles move along a cylindrical spiral from the bottom to the free surface of the cylinder (see Fig. 7 for cases n = −13.33 to −4). The SSA in this case starts to lose its linearity and becomes increasingly helical. For n = −2 and −1, the flow becomes more complicated, the spirals take
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Fig. 6 – Particle trajectories located in the plane (xz) for different values of n with Reω = 125.
conical forms (see Fig. 7 for cases n = −2 to −1), and the secondary rotation starts from the vicinity of the bottom of the cylinder and ends before reaching the free surface. The SSA changes direction and the trajectories of its points draw a cone when the revolution speed varies (Fig. 11). The top of the SSA moves following a circular form (Fig. 12a) while the other end keeps its position at the bottom centre. These results are in good agreement with those obtained by Vanyo et al. (1995) and Noir (2000) for the presence of a secondary spin
axis and the change of its direction.1 The authors conducted their experiments on a fluid in a precessing ellipsoid. Vanyo
1 The observations – concerning the presence and the direction change of the secondary spin axis (SSA) – noticed by Vanyo et al. (1995) and Noir (2000) in their works are not shown in this work. For more details, the reader is advised to refer to the authors’ works.
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Fig. 7 – Particle trajectories located initially on the same straight line (D) and in the vicinity of the rotation axis in the plane (xz) for different values of n with Reω = 125.
et al. (1995) have carried out their experiment with revolution rate varying from n = −2000 to −17.4 ( = −0.0005 to −0.0575) with a Reω = 7700 while the work of Noir (2000) is conducted for n = −300 to −50 ( = −0.0033 to −0.02) with a Reω = 1969. They considered that for the investigated range of n, the flow is not a solid body rotation. In addition, the authors could not identify the rotation axis of the fluid beyond a certain value of the revolution rate (n = −17.4 for Vanyo et al. (1995) and n = −50 for
Noir (2000)) using their visualisation technique because of the nonlinearity of the axis.
3.2.
Velocity iso-contours
Since the cases of n = −2 and n = −6 have shown a particular flow dynamics (as discussed in Section 3.1), an analysis of velocity iso-contours of these two revolution rates are
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Fig. 8 – Particle trajectories located initially in the vicinity of wall of cylindrical surface (C) for different values of n with Reω = 125.
performed with respect to case with n = −50 in order to highlight the flow symmetry observed previously with the particle trajectories. The results presented below are those of the components of the velocity relative to the revolution ( V BM ) on both meridional planes (xz) and (yz) clearly shown in Fig. 3. In the case of a rotation of the fluid as a rigid body, in the plane (xz), the components u and w are equal to zero and the component v is a linear function of R and constant along the rotation axis.
In the plane (yz), v and w are equal to zero while u is a linear function of R and constant along the axis of rotation. During the start of the solid rotation – i.e. the BFPM is in double rotation with a high n value (n = −50 which means a very small revolution rate) – the behaviour of the fluid is very similar to a fluid subjected to a single rotation, the flow is almost symmetrical and two-dimensional. Far from the bottom, the iso-contours are almost straight and parallel to the
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Fig. 9 – Trajectories of 4 particles located initially in (a) the plane (xz) at x/R = 0.4 and (b) the plane (yz) at y/R = 0.4 for n = −6 with Reω = 125. (For interpretation of the references to colour in the text, the reader is referred to the web version of the article.)
Fig. 10 – Trajectories of 4 particles located initially in (a) the plane (xz) at x/R = 0.4 and (b) the plane (yz) at y/R = 0.4 for n = −2 with Reω = 125. (For interpretation of the references to colour in the text, the reader is referred to the web version of the article.) rotation axis for the component v in the plane (xz) (Fig. 13b1) and the component u in the plane (yz) (Fig. 14a1). These two velocity components vary linearly with the radius R. However, near the bottom they have a slightly curved shape. In addition, the magnitude of the components u and w in the plane (xz) and v and w in the plane (yz) is very low. Moreover, a perfect symmetry of the component w with the rotation axis is noticed (see Figs. 13c1 and 14c1). However, the iso-contours of the components u in the plane (xz) (Fig. 13a1) and v in the
plane (yz) (Fig. 14a1) are not symmetrical because of the slip condition at the free surface and the adhesion condition at the bottom of the cylinder. The velocity components that have low value, in case of low revolution rate n = −50, become more significant when the revolution increases (small values of n). This observation confirms the three-dimensional behaviour of the flow. In case of n = −6, the symmetry is totally broken for the components u and w in the plane (yz) (see Fig. 14a2 and c2) and slightly broken
Fig. 11 – Secondary spin axis (SSA) geometrically identified for different values of n with Reω = 125.
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Fig. 12 – Displacement of the secondary spin axis (SSA) with respect to the primary spin axis (PSA) and the revolution axis.
for u and w in the plane (xz) (see Fig. 13a2 and c2). However for n = −2, an opposite observation is made: the symmetry is completely lost in the plane (xz) (see Fig. 13a3 and c3) and slightly broken in the plane (yz) (see Fig. 14a3 and c3).
3.3.
Flow topology
The results of the velocity iso-contours presented in the previous section for n = −6 and −2 confirm the particularity of
Fig. 13 – Contours of the three velocity components relative to the revolution frame in the meridional plane (xz) parallel to the revolution arm. The velocity is normalised to the linear rotation speed (ω . R) and the coordinates x, y and z are normalised with respect to the radius R: (ai ) the velocity component (u) along the ox axis, (bi ) the velocity component (v) along the oy axis and (ci ) the velocity component (w) along the oz axis. The various cases are conducted for: (1) n = −50, (2) n = −6 and (3) n = −2 with Reω = 125.
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Fig. 14 – Contours of the three velocity components relative to the revolution frame in the meridional plane (yz) perpendicular to the revolution arm. The velocity is normalised to the linear rotation speed (ω · R) and the coordinates x, y and z are normalised with respect to the radius R: (ai ) the velocity component (u) along the ox axis, (bi ) the velocity component (v) along the oy axis and (ci ) the velocity component (w) along the oz axis. The various cases are conducted for: (1) n = −50, (2) n = −6 and (3) n = −2 with Reω = 125. flow for these two values of n. To better investigate this particularity, a two-dimensional visualisation technique using streamlines is used. Furthermore, the study is extended to other revolution rates to identify the rate for which the flow topology is subject to significant changes. For incompressible flows, the stream function is obtained from the integration of the velocity field following the equation VBM = ∇ ×
(3)
For instance, in the (xy) plane the stream function obtained such that
z
=
z
is
udy +
vdx
(4)
Fig. 15 shows the results of streamlines in horizontal planes (xy) perpendicular to the rotation axis at different heights within the cylinder. The ratio n varies and Reynolds number Reω is maintained constant and equal to 125. The case
of n = −50 is a typical uniform rotation, the streamlines are circular with a stagnation point at the centre of the plane, which means that the flow is overall 2D. When n decreases, the lines become increasingly spiral in the plane (xy) with a velocity component perpendicular to the plane which allows the evacuation of the fluid along the cylinder axis. The location of spiral centre changes as one moves away from the bottom of the container. Therefore this axial component constrains the fluid to move vertically along a spiral and along a non-linear axis, thereby confirming the results obtained previously with the particle trajectories (Section 3.1). On the other hand, the position of this centre is constantly changing at each plane when the revolution speed increases (n value decreases). The trajectory drawn by these centre positions follows the SSA trajectory. Figs. 16 and 17 show the flow topology in the planes (xz) and (yz), respectively, for different values of n that are −50, −13.33, −10, −8, −6, −4, −2 and −1. These figures show that the particular characteristics of the flow, already mentioned and highlighted in previous sections, appear in the plane (yz)
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Fig. 15 – Flow topology in the BFPM for different horizontal planes (xy) perpendicular to the rotation axis for various n with Reω = 125.
for a range values of n between −8 and −4, while for the plane (xz) these flow particular characteristics start from n = −2.5. The revolution rate values less than −8 are the most critical for the flow. This flow singularity occurs only in one of the two meridional planes, i.e. the flow returns to its original topology in one plane when the change begins in the other plane. For a rigid body rotation, the velocity component values that are
tangent to the planes (xz) and (yz) are equal to zero. Therefore the streamlines are not represented on these planes. However for n = −50, circular streamlines are obtained on the two planes (xz) and (yz). Despite the two-dimensional configuration of the flow, axial components along x and z directions exist within the flow but their values remain insignificant. Figs. 16 and 17 show a flow configuration more complex and the existence
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Fig. 16 – Flow topology in the BFPM for the vertical plane (xz) parallel to the revolution arm for various n with Reω = 125. of several critical points (Tobak and Peake, 1982) on the plane (xz) when |n| ≤ 2.5 and on the plane (yz) when the n value is included between −8 and −5. As shown in Fig. 16 for the plane (xz), the centre starts moving towards the left side at n = −4 and becomes a focus
at n = −2.5. The latter divides into several other focuses when the revolution rate decreases and then saddles and separation zones appear. In addition, the flow is drained away horizontally in form of spirals with different sizes. On the plane (yz), as indicated in Fig. 17, when n decreases a focus takes origin
Table 2 – Summary of the flow characteristics at the two planes (xz) and (yz). Plane (yz)
Plane (xz)
Position of the plane
Perpendicular to the revolution arm
Parallel to the revolution arm
Effects of n
– Flow singularities start at n = −8 – Flow starts returning to its initial state at n = −4
– Flow singularities start at n = −4 – Flow singularities remain existing for n ≤ −4
Flow characteristics
– Separation and recirculation zones exist within the flow – Attachment and detachment of spiral phenomena occur often at walls
– Separation and recirculation zones exist within the flow – Spirals are separated from the wall
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Fig. 17 – Flow topology in the BFPM for the vertical plane (yz) perpendicular to the revolution arm for various n with Reω = 125.
above the bottom then detaches and moves towards the top and pushes the already existing centre. The detachment of the fluid is indicated by the appearance of half-saddle forms on the bottom wall and by the direction of the flow in this zone. The displaced centre is divided into several spiral structures (for n = −6.5, −6, −5.84 and −5) to finally establish itself on the lateral wall (fact observed for two focuses). For the insignificant revolution rates (cases with n = −4 to −1, results not reported here for sake of brevity), the focus that detached before expands itself progressively to occupy all the plane (yz) and to return finally to its initial state (case of n = −50). Furthermore, the attachment and detachment of the flow on the solid walls are mainly observed in the plane (yz) in opposite to the plane (xz) for the studied values of n. Table 2 summarises the characteristics of the flow in the two meridional planes (xz) and (yz).
4.
Conclusion
In this work, the characteristics of a flow within blade free planetary mixer (BFPM) are investigated by means of computational fluid dynamics simulations. The numerical simulations were carried out using second order finite volumes as implemented in the commercial code Fluent. A qualitative analysis of the flow dynamic was performed using visualisation techniques with various revolution rates and a constant rotation Reynolds number Reω = 125. In this respect, the parameters investigated are the particle trajectories, velocity iso-contours and streamlines. The main findings of the study can be summarised as follows. • Vortical structures in spiral forms are observed for various revolution rates and a constant rotation Reynolds number.
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Fig. B-1 – General view of (a) the 3D spiral and (b) the curve of the spiral of a given part (hj ) projected onto the plane (xy)j indicating the centroid (Mj ) location. The flow behaviour is divided into two parts: a primary rotation in the vicinity of the wall and a second rotation in the vicinity of the geometric axis of the BFPM. A total of three rotation axes are identified within the BFPM flow structure: (i) a primary spin axis (PSA), (ii) a secondary spin axis (SSA) and (iii) a revolution axis. The criteria for identifying vortex structures are tested to clearly locate the SSA, however none of them has adequately represented the spirals observed in the cylinder and/or determined their rotation axes. To this purpose, a geometric method based on the calculation of the centroid curves was used. The results are very conclusive and the SSA is determined. The direction of the SSA changes with the revolution rate n. • Some results of this qualitative analysis are in good agreement with those found for other doubly rotating flows by Vanyo et al. (1995), Noir (2000) and Mouhali (2009). Although the range of the studied parameters is not identical. • The analysis of the particle trajectories, velocity isocontours and stream functions showed that (i) there is a symmetry in the flow for some revolution rates and (ii) there are critical revolution rates for which the flow dynamics is subject to major changes. These changes cannot occur at the same time on both meridional planes (xz and yz) studied. Further studies are required to give a more proper explanation on the mechanisms at the origin of these significant flow changes.
Acknowledgement Financial support for this study from the Natural Sciences and Engineering Research Council of Canada (http://www.nserc-crsng.gc.ca/index eng.asp NSERC) is gratefully acknowledged.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.cherd.2015.05.045.
Appendix B. Secondary spin axis (SSA) determination method To approximately evaluate the secondary spin axis (SSA), a geometric method based on the determination of the centroid
of the particle trajectory curves in a plane is proposed. The procedure is as follows. - Particles located initially in the vicinity of the rotation axis are those that best describe the secondary rotation (see Fig. 7). Therefore a straight line (D) of particles is chosen in the vicinity of the cylinder rotation axis. - The 3D spiral of height h is divided into several spiral parts of small heights ( hj ) (see Fig. B-1a). Then, each part of the resulting spiral segments is projected onto a plane (xy)j thus forming a curve such as that depicted in Fig. B-1b. - The coordinates of the centroid (Mj ) of the curve obtained in the plane (xy)j are calculated using the two following equations (Beer and Johnston, 2000). x¯ j =
x L i
(B-1)
y L i
(B-2)
L
and y¯ j =
L
- Finally, the set of the centroids (Mj ) resulting from the different parts of the spiral forms the secondary spin axis (SSA).
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