Flow Characteristics in Mixers Agitated by Helical Ribbon Blade Impeller

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Engineering Applications of Computational Fluid Mechanics

ISSN: 1994-2060 (Print) 1997-003X (Online) Journal homepage: http://www.tandfonline.com/loi/tcfm20

Flow Characteristics in Mixers Agitated by Helical Ribbon Blade Impeller Yeng-Yung Tsui & Yu-Chang Hu To cite this article: Yeng-Yung Tsui & Yu-Chang Hu (2011) Flow Characteristics in Mixers Agitated by Helical Ribbon Blade Impeller, Engineering Applications of Computational Fluid Mechanics, 5:3, 416-429, DOI: 10.1080/19942060.2011.11015383 To link to this article: http://dx.doi.org/10.1080/19942060.2011.11015383

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Date: 26 March 2016, At: 09:30

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3, pp. 416–429 (2011)

FLOW CHARACTERISTICS IN MIXERS AGITATED BY HELICAL RIBBON BLADE IMPELLER Yeng-Yung Tsui* and Yu-Chang Hu

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Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Chinese Taipei * E-Mail: [email protected] (Corresponding Author)

ABSTRACT: The main concern of this study is to investigate the flow mixing generated by helical ribbon blade impellers and to show that with the help of CFD the performance of the mixing system can be significantly improved by optimizing the geometric configuration of the impeller. To fulfill this objective, a numerical model is developed to solve the Navier-Stokes equations for the flow field. However, difficulties arise due to the rotation of the impeller in the vessel. In order to ease the problem, the velocity field is assumed to be in a quasi-steady state and the multiframe of reference is adopted to tackle the rotation of the impeller. For discretization the fully conservative finite volume method, together with unstructured grid technology, is incorporated. It is shown that the flow in the mixer can be regarded as a flow in an open channel with a wall moving at an angle with respect to the channel. The influences of the blade pitch, the blade width, and the clearance gap between the blade and the surrounding wall are examined. The mechanism to cause these effects is delineated in detail. It is demonstrated that after optimization of the blade geometry, the circulating flow rate induced by the impeller is largely increased, leading to significant reduction in mixing time. In addition, the power demand is reduced. It is also evidenced that by enlarging the clearance, it is difficult for the fluid in this region to be mixed. Keywords: mixing flows, stirred mixers, helical ribbon blade impellers, multiframe of reference, unstructuredgrid methods

system with helical ribbon impellers, mixing proceeds first in the region near the blades and the vessel wall where the fluid is subject to high shear strains. Fluid homogenization is then fulfilled by the axial vortex flow induced by the rotation of the ribbon impeller. It has been shown that this kind of impeller is very effective in mixing high viscous fluids (Gray, 1963). To characterize performance of mixing systems two parameters are usually adopted: the power number and the mixing time. One kind of power number is defined in terms of viscosity  as N *p  P /  N 2 D 3 , where D is the diameter

1. INTRODUCTION The mixing of fluids is a common operation encountered in productions of polymer, food, paint, and greases, to name a few. Poor mixing may result in formation of dead zones, hot spots, and temperature and concentration gradients, which will affect the quality of the final products. The selection of mixing systems depends on operating conditions such as agitation speeds and fluid properties. When the viscosity of the fluid is low, the rotational speed of the agitator can be high enough to produce turbulent flows. Most of these systems involve the use of turbine impellers such as Rushton turbines or pitched blades. For highly viscous liquids, the flow is more likely in the laminar regime because, otherwise, an extremely high demand of power is required. The use of small turbine impellers becomes inefficient as stagnant zones may be formed in the region at far distance from the impeller. To obtain adequate mixing under laminar flow conditions, closeclearance impellers are usually adopted. Impellers such as anchors, gates, or paddle impellers, which produce mainly circumferential flow, perform poorly in mixing because of lack of axial flow to sweep through the entire vessel. In an agitating

of the impeller, N the rotational speed and P the power consumption. A more common definition of the power number is N P  P /  N 3 D 5 . These two dimensionless numbers are related by

N *p  N p Re

(1)

where Re   ND 2 /  is the Reynolds number. For low-Reynolds number flows the power consumption of the agitator is proportional to the square of the rotational speed. As a * consequence, N p is independent of rotational

Received:1 Dec. 2010; Revised:19 Apr. 2011; Accepted:25 Apr. 2011 416

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

Another means is taken from the periodicity of the response curve of a detector placed in the vessel (Ryan et al., 1988; Hayes et al., 1998; Dieulot et al., 2002; Delaplace et al., 2000b; Curran et al., 2000). The dimensionless mixing time Km represents the number of revolutions of the impeller required to complete mixing and the dimensionless circulation time Kc is that for a complete flow loop. Both are independent of the Reynolds number and are functions of the impeller geometry. Although they are closely related, the data for Kc are less scattered than those for Km. The circulation time is directly related to the pumping capability of the impeller. The discharge rate of the impeller can be estimated by

speed and becomes a constant (Käppel, 1979b; Takahashi et al., 1982a).

N *p  c1

(2)

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This relation is valid for Reynolds numbers less than about 40. The power number N *p is a function of the geometric configuration of the mixing system. Various correlations in terms of geometric variables have been proposed in a number of experimental studies (Käppel, 1979b; Takahashi et al., 1980; Takahashi et al., 1982a; Espinosa-Solares et al., 1997; Delaplace et al., 2000a). The mixing time is the time required for the flow to reach a certain level of homogeneity. It was revealed by experiments that the product of the mixing time tm and the rotational speed N, defining a dimensionless mixing time Km, is a constant (Käppel, 1979b; Takahashi et al., 1982b; Delaplace et al., 2000b)

Km  Ntm  c2

Qd 

(5)

where vtot is the total fluid volume of the vessel. Another method to find Qd is to integrate the axial velocity profile obtained from measurements (Carreau et al., 1976; Tanguy et al., 1992). A circulation number KQ can be defined in terms of Qd as

(3)

Various methods, including conductivity techniques (Rieger et al., 1986; Dieulot et al., 2002), thermal techniques (Delaplace et al., 2000b), coloration/decoloration techniques (Carreau et al., 1976; Käppel, 1979a; Ryan et al., 1988), liquid crystal techniques (Takahashi et al., 1982b; Takahashi et al., 1988), and chemical reaction techniques (Hayes et al., 1998), have been adopted to measure the mixing time. However, the determination of the degree of the mixing homogeneity depends on the techniques used. This results in considerable scatter in the constant, which makes the comparison of mixing performance difficult. In addition to the mixing time, circulation time is also often employed as a criterion for evaluation of the mixing performance. It is the time for a fluid element to complete a vertical circulating loop in the vessel during the mixing process. It is generally recognized that the circulation time is proportional to the mixing time. Therefore, the dimensionless circulation Kc, defined as the product of the circulation time tc by the rotational speed, is also a constant for a specific mixing system (Takahashi et al., 1989; Delaplace et al., 2000b).

Kc  Ntc  c3

vtot tc

KQ 

Qd ND 3

(6)

The circulation number is affected by the geometry of the mixing system and the fluid properties (Curran et al., 2000; Carreau et al., 1976). The geometrical configuration of a mixer plays a significant role in determination of the mixing performance. The design of a mixing system was mainly based on correlations obtained from experiments. These empirical correlations are usually applicable in limited ranges and their use in scaling up may be questionable. The development of computational fluid dynamics (CFD) provides an alternative tool to fulfill this purpose. Finite element methods were adopted by Tanguy and coworkers to study the mixers with different kinds of impellers as described in the following. A helical ribbon screw impeller, i.e. a helical ribbon blade in the outer region and a screw blade attached to the impeller shaft, was under examination by Tanguy et al. (1992). The entire volume of fluid in the tank is assumed to be mounted on a rotational frame in the simulation. The flow field can then be regarded as steady. Good agreements with experiments were reported in terms of the circulation time and torque. This method was also used to model the mixing flow of second-order fluids stirred by a helical ribbon

(4)

A direct way to find the circulation time is to follow the trajectory of a suspended particle (Takahashi et al., 1989; Guérin et al., 1984). 417

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rate near the tip of the impeller. The proportionality constant associated with this linear relation is largely independent of the flow behavior index and dependent on the geometric parameters of the system. The mixing of pseudoplastic fluids by anchor impellers was investigated by Prajapati and Ein-Mozaffari (2009) using the FLUENT. The mixing time and power consumption results indicate that an anchor impeller with four blades is more efficient than that with two blades. The FLUENT was also employed by Rahimi et al. (2010). It was shown that after modifying a helical ribbon impeller by placing two 450 flat blades on the bars connecting the ribbon blade and the shaft, the mixing performance is improved with a negligible increase in power consumption. The requirement of optimization of a design to improve its efficiency becomes essential in recent days due to the increasing cost of energy and the growing concern about the environment. The fast development of CFD makes it a useful tool for this goal. As examples, it was used by Idahosa et al. (2008) to optimize a fan blade and by Wu et al. (2008) and Yedidiah (2008) to improve performances of centrifugal pumps. In such studies, commercial codes are most often utilized. A computational method, based on the fully conservative finite volume method and the unstructured grid technology, had been developed by the group of the authors (Tsui and Pan, 2006). It is efficient and robust in dealing with flows with complex domain geometry. This method was further extended to include multiframe of reference to handle the rotation of impeller in mixing systems (Tsui et al., 2006; Tsui and Hu, 2008). Although attempts had been made to compare different setups of mixing systems in previous studies, comparisons were undertaken only in limited ranges of variation of geometrical parameters. In this study, the geometric variables of a mixing system agitated by a helical ribbon blade impeller are allowed to vary to a large extent. The aim is to look into their effects and to optimize the configuration so that the mixing performance is improved.

blade by Bertrand et al. (1999). In the study conducted by Devals et al. (2008), the flow characteristics in a Maxblend impeller mixer were examined. It focused on the effects of the Reynolds number and the bottom clearance of the impeller on the power consumption, the distribution of shear rates and the overall flow pictures. For dual, coaxial impeller systems the two impellers may rotate at different speeds. To cope with this situation, the above method needs to be extended to include two rotational frames with each impeller located in a different frame. In the mixer considered by Tanguy et al. (1997), a Rushton turbine is placed beneath a helical ribbon impeller. It was shown that the dual impeller outperforms the standard helical ribbon in terms of pumping. Thibault and Tanguy (2002) considered a coaxial mixer with an anchor in the outer region and eight rods along with a pitchedblade turbine in the inner region. The predictions of power consumption agree with experiments closely. A lot of studies simply employed commercial codes as the analytical tool. The finite-element software POLY3D was utilized by de la Villéon et al. (1998) to analyze three different impeller mixers of the helical ribbon type. In this code, the surface is represented by a series of control nodes located on its surface. These nodes are placed inside the elements of the vessel. Impeller speed is imposed on the nodes using constrained optimization techniques. At each time step, the velocity and position of the control nodes are updated (Bertrand et al. 1997). Their simulations lead to the conclusion that the double helical ribbon impeller is more efficient than the single helical ribbon one and that adding a central screw does not enhance mixing efficiency. Rivera et al. (2006) also adopted the POLY3D to investigate a dual impeller mixer consisting of an anchor in the near wall region and a Rushton turbine in the core. The agitator operates in either co-rotating or counter-rotating mode. It was found that the corotating mode is more efficient than the counterrotating mode in terms of energy consumption, pumping rate and mixing time. The finite-volume software FLUENT was employed to analyze the mixing of pseudoplastic fluids with a helical ribbon impeller by both Ihejirika and EinMozaffari (2007) and Shekhar and Jayanti (2003). The multiframe of reference is used to cope with the rotation of the impeller. The numerical results in the former study showed good agreement with experiments and correlations. It was seen from the later study that there exists a linear relationship between the impeller speed and the local shear

2. MATHEMATICAL METHOD A sketch of the top view and side view of the agitating system is shown in Fig. 1. The flow in the vessel is inevitably unsteady and threedimensional. Fully time-dependent computations for 3D flow are very time-consuming. The simulation of the velocity field in the agitating 418

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

interface between the two regions is nearly placed at half the distance between the inner edge of the blade and the shaft. It is noted that if the rotational region is severely restricted to the blade, the large variation of the velocity through the interface may cause numerical instability. The governing equations can then be cast into the following dimensionless form.

ds W

h1

rotational frame

Vj rotational frame

H h

x j

S

pressure p are non-dimensionalized by D, ND, and  N 2 D 2 , respectively. In the equations,

D

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T

Fig.1

(8)

where the coordinates x j , velocities Vj and

stationary frame

C

(7)

  p  1 Vi (V jVi )    ( )  qi x j xi x j Re x j

t

h1

0

V j represents the fluid velocity relative to the

Illustration of mixer configuration.

rotational grid.

Vj  V j  V jg

mixer can become easier by assuming a quasisteady state. This steady state assumption can be regarded as a snapshot in photography. The steady-state predictions by Wechsler et al. (1999) for the flow induced by a pitched-blade impeller showed a close agreement with the fully unsteady calculations, at a cost of only a fraction of the computer time of the latter. In steady-state calculations, the impeller is frozen at a specific position without moving. To make the fluid flow, the volume swept by the impeller is mounted on a rotational frame. The body forces generated by the rotational frame trigger off fluid movement. If the driving momentum of the impeller is large enough, the fluid flow follows the rotation of the impeller closely. Thus, the entire volume of the vessel can be regarded as moving with the impeller and it can be assumed that the whole vessel rotates with the impeller in the simulation (Tanguy et al., 1992; Bertrand et al. 1999; Devals et al., 2008). However, the geometry of the impeller blade is largely varied in our study. As the blade width or the impeller pitch becomes small, the rotational effect will be limited only to the region around the impeller. The assumption of a single rotational frame becomes not appropriate. Therefore, multiple frames are adopted in our calculations. The multiframe of reference has also been adopted by Rahimi et al. (2010) in calculating the flow in helical ribbon impeller mixers. As shown in Fig. 1, the vessel is divided into two parts. The inner part is stationary while a rotational frame is imposed on the outer part where the helical blade impeller is located. The

(9)

Here V j is the absolute velocity and V jg the velocity of the computational grid, defined as

V jg  0 in the stationary frame

(10a)

V jg   jpq p x q in the rotational frame

(10b)

where  jpq is the alternating unit tensor and

 p the angular velocity of the impeller. It is noted that although velocity is assigned to the grid in the rotational region, the mesh in this region is fixed without motion. The source term in the momentum equation represents the body forces induced by the rotation of the impeller and does not appear in the stationary frame. In the rotational frame, it can be expressed as

q j   mnj mVng  2 mnjm (Vn  Vng )

(11)

The first term stands for the centrifugal force and the second term the Coriolis force. As for boundary conditions, the upper boundary is assumed to be a free surface at which the shear stress is zero. The other boundaries are solid walls with no-slip conditions being imposed. Hence, a rotational velocity is assigned at the surface of the impeller shaft. The surrounding wall and part of the bottom wall of the vessel are located on the rotational frame. Rotational velocities in the direction opposite to that of the impeller are 419

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

the face. In the equation  is a weighting value between 0 and 1, corresponding to the first-order and second-order upwind difference schemes, respectively. Similar to the high-order schemes used in structured-grid calculations, this secondorder upwind difference scheme may also result in solution oscillations and cause instability. To solve this problem, a value of 0.9 is assigned to  . The diffusive flux on the right hand side of Eq. (13) is modeled by the following approximation.

specified there so that the no-slip condition is satisfied. A usual means to cope with irregular geometry of the flow field is the use of the finite element method which adopts unstructured grids. However, with this method the system of algebraic equations raised requires large amounts of computer resources to solve these equations. Another way is to incorporate curvilinear coordinates into the finite difference method. For complex geometries the domain can not be covered by one single curvilinear mesh; it must be partitioned into a number of blocks with a curvilinear coordinate system defined in each of the blocks. Solutions are sought in each block and iteration must be performed among these blocks. Special care must be taken at the interface between two neighboring blocks to ensure coupling. Thus, the solution procedure is complicated and requires a lot of computing efforts during iteration. To overcome the above difficulties, the unstructured grid technology is incorporated into the finite volume method in this study. With this method the principle of conservation law is obeyed by using the divergence theorem of Gauss. To discretize the continuity and momentum equations, they are integrated over a control volume to yield

  V   f  sf  0

s 2f   f  s f    (C   P )  PC  s f  s 2f   f  (s f     PC )  PC  s f

(15)

where, see Fig. 2a, the subscripts P and C denote the principal and the neighboring nodes sharing a



common face f, and  PC is a distance vector connecting these two nodes. The face gradient

 f is obtained via interpolation from the gradients at the nodes P and C.

(12)

f

  1  ( V   f  s f ) f   Re  f  s f  qv f f where  represents

each

of

the

(13) velocity

 components V , the subscripts f denote the face  value, s f is the surface vector of a face (see Fig. 2a), v is the volume of the considered cell and the source term q includes both the pressure

(a) stationary domain

gradient and the body forces. The summation is over all the faces surrounding the cell. The term on the left hand side of Eq. (13) represents convective flux through the surface of the control volume. The face value  f needs to be

rotating domain

estimated using neighboring nodal values, which is approximated by the following scheme.



 f  UD   ( )UD  

(14) (b)

where the superscripts UD denote the value evaluated at a node upstream of the face under  consideration and  is the distance vector directed from the upwind node to the centroid of

Fig. 2

420

(a) a control volume with a neighboring cell and (b) a typical grid.

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

diameter ds=0.048D. It rotates at a speed N=30 rpm, corresponding to Reynolds number 5.

It needs to be noted that the difference equations for the nodes just next to the interface between the two reference frames include nodal velocities in the other frame. To correctly account for the momentum flux transported through the interface, the velocities at the other frame must be transformed onto the frame where the considered node is located. After discretization, the momentum equation can be solved to find velocities using prevailing pressure. However, the resulting velocity field does not satisfy the continuity and the pressure field needs to be updated. A pressure equation can be obtained by adjusting the velocities such that the continuity equation is satisfied in each cell. The solution procedure is to solve the momentum equation and the pressure equation in an iterative manner until convergence is reached. Details about the method can be found in Tsui and Pan (2006). With the velocity field obtained, the mixing of two fluids can be calculated by solving the following mass transport equation.

C    1 C  (VjC )  ( ) t x j x j P e x j

3.1

Grid sensitivity and validation tests

A typical grid required in simulation is displayed in Fig. 2b. Grid sensitivity tests have been conducted using meshes with 106080, 204000, 304800, and 405408 cells. The resulting power numbers N *p are 317, 337, 344, and 345 for the different grids, which are in good agreement with the measured value 334 given by Käppel (1979b). The corresponding circulation numbers KQ are 0.0634, 0.0626, 0.0624, and 0.0624. It is clear that the solution reaches grid independence for the two highest resolution meshes. In the following, meshes with about 300000 cells are used in calculations. To further validate the present method, comparison of the power numbers with the experimental data of Käppel (1979b) for two pitch values S=0.5D and 1D and three clearance values C=0.0105D, 0.029D and 0.053D is provided in Table 1. It is obvious that reasonably good agreement is obtained by our calculations.

(16) 3.2

where C is the concentration of one of the fluids, based on mass fraction, being in the range of 0 and 1. The Peclet number Pe is related to the Reynolds number Re by Pe=Re*Sc. Here Sc is the Schmidt number defined as Sc=ν/D, whereνis the kinematic viscosity and D the mass diffusivity.

Flow structure

As illustrated in Fig. 3a, the main feature of the flow in the vessel is a downward flow in the outer region near the wall, which is dragged by the ribbon blade due to the rotation of the impeller. It is followed by an upward stream in the inner region to complete a looping flow. The flow field is complicated by having some minor loops at the inner edge of the blade impeller. The pressure distribution on a cylindrical surface at r=0.45D (corresponding to the surface at the mid-width of the ribbon blade) is shown in Fig. 3b. It can be detected that the pressure at the bottom of the vessel is higher than that at the top and there exists a pressure difference across the channel formed by the impeller blade. The cause of the flow patterns seen in Fig. 3 can be illustrated in a schematic drawing of the flow in an open channel. As shown in Fig. 4, the open channel is formed by the blade of the impeller and the wall of the vessel. The wall on the bottom is the impeller shaft. The open part in the lower region represents the core

3. RESULTS AND DISCUSSION The arrangement of the mixing system has been given in Fig. 1. A basic configuration consists of a helical ribbon impeller of diameter D=330mm and a vessel of diameter T=337mm (T/D=1.0212). The ribbon blade is assumed to have a thickness of 5mm and a width W=0.1D. The height of the impeller is h=1D and that of the vessel is H=1.1D. Therefore, the clearance between the impeller and the surrounding wall of the vessel is C=0.0106D and the off-bottom clearance is h1= 0.05D. The impeller has a pitch S= 0.5D and the shaft has a *

Table 1 Comparison of power number N p with experiments for different pitches and clearances

Predictions Experiments

S/D=0.5

S/D=1.

C/D=0.0105

C/D=0.029

C/D=0.053

344 334

212 242

348 334

240 246

196 208

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

region in the vessel. The clearance between the blade and the wall of the vessel is ignored. For an impeller with a single blade there is only one channel formed and for a double-blade impeller two channels exist. The flow can be regarded as periodic with respect to the channel sides. It is assumed that the impeller remains motionless and the wall of the vessel moves at a velocity Vw. This velocity is decomposed into a component Vc along the channel and a component Vt in the transverse direction. The pitch angle  is defined as the helical angle of the ribbon blade with respect to the horizontal plane. The flow in the channel is driven by the velocity component Vc, which gives rise to a pressure increase from pin at the top of the vessel to pout at the bottom in Fig. 4. Owing to the adverse pressure gradient, reverse flow forms in the open part (i.e. the lower part) of the channel, corresponding to the upward stream in the inner region of the vessel in Fig. 3a. The flow in the transverse direction induced by Vt results in a pressure gradient across the channel with a higher pressure pp on the left side of the channel and a lower pressure ps on the right side. The pressure side of the channel corresponds to the lower side of the blade and the suction side to the upper side of the blade. The pressure difference between the two sides is approximately related to the pressure rise in the channel by the following equation.

(a)

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p 129 127 125 123 121 119 117 115 113 111 109 107 105 103 101 99 97 95 93 91 89 87 85

(b)

Fig. 3

(a) Flow field on a vertical plane and (b) pressure distribution on the cylindrical surface at r=0.45D

( p out  pin ) Az  ( p p  p s ) As cos   Fz

(17)

where Az is the cross-sectional area, As the sidepin

Vw



pp As

Vt

wall area, and Fz the axial component of the frictional force exerted on the channel walls. The relationship is obtained from the force balance for the channel in which the momentum fluxes are ignored. As a result of the pressure difference between the two sides of the blade, secondary vortices directing from the pressure side (the lower side of the blade) to the suction side (the upper side of the blade) are formed at the inner edge of the blade, as observed in Fig. 3a. Fluid leak may be detected at the outer edge if the clearance between the blade and the wall of the vessel is large enough. The opening of the channel at the two edges results in decrease of the pressure difference on the two blade sides as well as the axial pressure gradient.

Az

Vc

ps As

pout Az

Fig. 4

3.3

Illustration of flow field in vessel as flow in open channel.

Effects of impeller pitch

The influence of the impeller pitch on the power consumption is shown in Fig. 5a. The power is

422

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

*

the impeller blade becomes negligible because the transverse velocity component Vt is extremely small. The moving wall mainly drives the fluid to flow along the channel. As a consequence, large amounts of power are required to overcome the high frictional resistance. On the contrary, the flow driven by the wall is mainly in the transverse direction at large pitches, where the blade is in the vertical position at the limit of 900. Therefore, the flow in the vessel rotates with the impeller and the power exerted is simply used to overcome the pressure difference between the two sides of the impeller blade. There is a crossover of the two curves for N *p , f and N *p , p , which is located at

W/D=0.1

NP

exp(Kappel) 400

*

NP N*P,p *

NP,f 200

0

0

2

(a)

4

S/D

about S/D=2.1. The discharge flux induced by the impeller is defined as

0.12

KQ

0.11

W/D=0.1 W/D=0.25

0.09

0.07 0.06 0.05 0.04

Fig. 5

0

1

2

S/D

3

4

Variation of (a) power number and (b) circulation number against impeller pitch.

given by P  2 N , where  is the torque exerted on the impeller and is obtained by taking moments about the central axis for the pressure and the frictional forces over the surface of the impeller. As seen from the figure, the power number N *p , defined as N *p  P /  N 2 D 3 , is large

10 8

at small pitches. It falls quickly when the pitch is increased, followed by a gradual decrease at sufficiently large pitches. The power number approaches 160 as the pitch becomes infinite. The limited experimental data provided by Käppel (1979b) were also shown in the figure for comparison. A similar trend between the present calculations and the measurements can be seen as S/D is enlarged from 0.5 to 1. The power consumed by the impeller can be divided into two parts, corresponding to the actions of the frictional force ( N *p , f ) and the pressure force (N

* p, p

Rv

R0

(18)

where Rs is the radius of the shaft, Rv the radius of the vessel, and R0 the location at which the velocity Vz is zero. The velocity Vz is the mean axial velocity averaged over the circumferential direction at mid-height of the vessel. Fig. 6 shows examples of the variation of the mean axial velocity. The absolute values of the two integrals in the equation must be identical because the mass must be conserved. The variation of the circulation number KQ, defined in terms of Qd as in equation (6), with respect to the impeller pitch is shown in Fig. 5b. The flow rates pumped by the impeller are low at low and high values of pitch. As noted above, although the velocity component

0.08

(b)

R0

Rs

Q d  2  Vz rdr   2  Vz rdr

S/D=1/3 S/D=0.5 S/D=0.9 S/D=2 S/D=4 S/D=

Vz

6 4

8

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0.1

2 0 -2 -4 -6 -8 0

). At low values of pitch, i.e. at small

values of pitch angle  in Fig. 4, the channel is elongated and the transverse width is reduced. The pressure difference between the two sides of

Fig. 6

423

0 .2

r/D

0.4

Distribution of mean axial velocity component at mid-height of the vessel for different pitches.

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

at low values of width, followed by a gradual decrease. As for the part due to the pressure, it increases with the blade width in a monotone manner. The variation of the circulation number illustrated in Fig. 7b reveals that the circulating flow induced by the impeller is strengthened by widening the blade at low values of width. When the width of the blade W becomes greater than 0.26D, the flow rate starts to decline. This is ascribed to the open space in the central region of the vessel being reduced severely and, thus, the returning flow in this open space extends into the channel region. This phenomenon becomes clear in view of the mean axial velocity profile shown in Fig. 8. In the present case, the impeller pitch is set at S/D=0.5. It can be seen in Fig. 7b that when the pitch is increased to S/D=1, the location of maximum KQ is decreased to W/D=0.21.

*

NP 400

300

S/D=0.5 *

NP * NP,p * NP,f

200

100

0

0

0.1

0.2

(a)

W/D

0.3

0.4

0.12

KQ

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0. 1

0.08

25

S/D= 0.5 S/D= 1.0

0.06

Vz

20

W/D=0.1 W/D=0.25 W/D=0.4

0.04

15 0.02

0.4

10

Variation of (a) power number and (b) circulation number against blade width.

5

0

0.1

0.2

(b)

Fig. 7

0.3

W/D

0

Vc along the channel increases with the decreasing pitch angle, the width of the channel is reduced. There is no flow in the limiting case  =0 because the impeller is transformed into a solid cylinder. In the other limit  =900, the channel becomes vertical. The fluid simply rotates with the impeller and Qd also becomes zero. There is a peak KQ at S/D=0.9 for the present configuration. The location of peak KQ is affected by the arrangement of the geometry. For example, when the blade width W is enlarged from 0.1D in the benchmark case to 0.25D, the peak location is shifted to S/D=0.7, as shown in the figure. It can also be seen that the circulation number becomes higher. 3.4

-5 -10

0

Fig. 8

3.5

0.1

0.2

r/D

0 .3

0.4

0.5

Distribution of mean axial velocity component at mid-height of vessel for different blade widths.

Effects of clearance gap

It is shown in Fig. 9a that the power number decreases with the enlargement of the clearance gap, with a large decline rate at small clearances. Comparing with the measurements of Käppel (1979b), the same trend can be identified. The variation of N *p follows that of N *p , f closely. Also

Effects of blade width

Fig. 7a shows that the power number increases with increasing width of the blade. It is the frictional part dominating the power consumption due to the small impeller pitch S/D=0.5 as discussed above. The power number of the frictional part N *p , f increases with the blade width

shown in the figure are the power numbers due to the contributions of the two working blade surfaces and the one facing the clearance gap. Although the area of the latter is much smaller than the two main ones, its contribution to the power number can not be ignored. Especially at

424

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

number. To support this point, the performances of two blade configurations are compared in the following. The first is the benchmark one with pitch S=0.5D and width W=0.1D and the second is an optimized one with S=0.7D and W=0.25D. Initially, the volume of 10% of the vessel height at the top is covered by a fluid with concentration C=1 and the rest is occupied by the other fluid with concentration C=0. The Schmidt number is assumed to be 1.6*107, which corresponds to that of glucose syrup. Calculations are conducted up to 200 revolutions. To measure the degree of mixing, a mixing index is defined by

W /D=0.1, S/D=0.5 exp (Kappel) N*P N*P ,p N*P ,f lower side part of N*P,f upper side part of N*P,f clearance side part of N*P,f

500 *

NP 400

300

200

100

0

0

0.02

0.04

0.06

0.08

 C  C v  C  C v    C  C v 2C (1  C )v i

0.1

(a)

i

i

C/D

i

o i

K 0 .0 8 Q

(19)

tot

i

0 .0 6

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where Ci is the concentration at a computational cell which has a volume v , Cio the initial value,

W/D=0.1, S/D=0.5

0 .0 4 0 .0 2 0

(b)

Fig. 9

0

0.02

0 .0 4

C/D0.0 6

0 .08

C the concentration after fully mixing, vtot the

0.1

total volume of the vessel. The summation is taken over all the vessel cells. The mixing index  stands for deviation of the mixing away from uniform distribution of the two fluids. Its value is one initially and becomes zero when the mixing is complete. It is evident from Fig. 10 that the performance of the optimum configuration is much superior to the benchmark one. The mixing indices at a number of revolution numbers are shown in Table 2. This index is 0.169 after 100 revolutions and 0.107 after 200 revolutions for the benchmark case, comparing with the values 0.082 and 0.024 at the corresponding times for the

Variation of (a) power number and (b) circulation number against clearance.

small clearances, it is higher than those of the two working surfaces. The fast decline in the power number at low clearances is mainly ascribable to the quick decrease of this part because the strain rates generated by the rotating blade are greatly decreased. The part of power number due to the pressure force N *p , p also decreases, but at a slow rate, when the clearance is enlarged. This is resulted from the reduction of the pressure difference between the two blade sides. As seen from Fig. 9b, the circulating flow rate is not much affected by the appearance of clearance. The circulation number slightly falls off for large clearances. It can be understood that the appearance of the clearance gap causes reduction of the pressure difference between the two blade sides. This, in turn, leads to decrease of the axial pressure gradient according to equation (17). As a consequence, the force to drive the fluid to flow upward in the central region of the vessel is reduced. 3.6



1

0.8

benchmark blade W/D=0.1, S/D= 0.5 0.6

optimized blade W/D=0.25, S/D= 0.7 0.4

0.2

Mixing performance

It can be drawn from the above results that the pitch and the width of the blade must be optimized for larger circulating flow rate and lower power consumption. Examination of Figs. 5b and 7b reveals that the pitch in the range 0.7D1D and the blade width in the range 0.2D-0.26D give better performance in terms of the circulation

0

0

50

100

150

200

Nt(no. of revolutions) Fig. 10 Comparing mixing performance of benchmark with optimized configurations.

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

Table 2 Comparison of mixing index revolution numbers

 at four

0.5

C/D=0.01

c

Nt=20 Nt=100 Nt=200

0.4

Benchmark case Optimized case Nt=0

Nt=50

Nt=100

Nt=150

Nt=200

0.266

0.169

0.131

0.107

0.170

0.082

0.044

0.024

Nt=20

N t=1 00

0.2

0.1

C 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

(a) Nt=0

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0.3

N t=2 0

Nt=10 0

(b)

0

0.1

(a)

0. 2

0.6

0.3

r/D 0.4

0.6

C/D=0.1

c

Nt=20 Nt=100 Nt=200

0.5

C 0.2 0.1 9 0.1 8 0.1 7 0.1 6 0.1 5 0.1 4 0.1 3 0.1 2 0.1 1 0.1 0.0 9 0.0 8 0.0 7 0.0 6 0.0 5 0.0 4 0.0 3 0.0 2 0.0 1 0

0.5

0.4

0.3

0.2

Fig. 11 Concentration contours on a vertical plane after 0, 20 and 100 revolutions for (a) benchmark impeller and (b) optimized impeller.

0.1 0

0.1

0.2

(b)

0.3

r/D

0.4

0.5

0.6

Fig. 12 Distribution of mean concentration at midheight of vessel for two different clearances: (a) C=0.01D and (b) C=0.1D.

optimized one. This result of comparison is not surprising in view of the circulation number of 0.114 for the optimum blade and 0.062 for the benchmark blade. As for the power number, the optimum configuration has a value 309, which is lower than the value 348 for the benchmark. Thus, not only the mixing effectiveness is enhanced by optimizing the initial design, but also less power is consumed. The above results are also evidenced in Fig. 11, in which the concentration contours on a vertical plane are shown. It can be seen that the distribution of the concentration becomes fairly uniform after 100 revolutions for the optimum one, but not for the benchmark one. The central region in each half plane is not well mixed because it is the core location of the circulating flow and the flow velocity is low there. Calculations have also been undertaken to examine the effect of clearance on the mixing. Fig. 12 shows the concentration averaged over the circumference at mid-height of the vessel for two gap clearances. It is clear that the fluid remains not to be well mixed near the wall as the clearance is increased from 0.01D to 0.1D, even after 200 revolutions. High concentrations persist in this region for the latter case. The peak concentration there is 0.249 after 100 revolutions

and 0.218 after 200 revolutions while the mean concentration for complete mixing is 0.113. This is the main reason why the clearance needs to be kept small in most industrial mixing systems despite its higher power consumption. It can also easily be identified that the mixing is poor in the central region, as indicated above. 4. CONCLUSIONS The flow in a vessel stirred by a helical ribbon impeller has been investigated by the numerical method. The method is based on the assumption of quasi-steady state for the velocity field and the multiframe of reference is used to deal with the rotation of the impeller. A summary of the main findings is drawn as follows. 1. The flow in the vessel can be interpreted as a flow in an open channel with a moving wall. The channel is formed by the two side surfaces of the ribbon blade and the wall of the vessel which moves at the pitch angle of the blade with respect to the channel. The velocity component along the channel drives 426

Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

because the large velocity gradient prevailing in this region may be under-predicted. In the near future, we will examine its effect on prediction accuracy via comparing uses of both two and three frames. Mixing operations encountered in chemical and food industries frequently involve highly viscous fluids such as polymers, resin and pastes. These fluids usually exhibit non-Newtonian nature due to the dependence of the viscosity on the flow strain rate. Therefore, it is important to investigate the impact of rheological complexities on mixing performances for industrial applications. The extension of the present method to account for the non-Newtonian behavior of such fluids is currently underway.

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the flow downward in the outer region of the vessel, creating a pressure gradient in the axial direction. This adverse pressure gradient forces the fluid to flow upward in the central region of the vessel to form a looping flow. The velocity component in the transverse direction brings about a pressure gradient across the channel, with a higher pressure at the lower side of the blade. As a result, secondary vortices directing from the lower side toward the upper side are formed at the inner edge of the blade. 2. The pitch of the blade has a significant influence on the performance of the mixer. When the pitch decreases, the velocity component of the moving wall to drive the flow along the channel is increased. However, the channel is lengthened and the crosssectional area is decreased, resulting in increase of flow resistance. Therefore, to have a better design the pitch needs to be optimized. For the configurations considered in this study this value falls in the range S=0.7D to 1D.

ACKNOWLEDGEMENTS This work was supported by the National Science Council under the contract number NSC-96-2221E-009-135-MY2. Acknowledgment is due to one of the reviewers who indicates the necessity to separate the clearance gap from the rotating frame and to divide the vessel into three frames.

3. At small values of blade width, the discharge flow rate increases with the width. However, as the width becomes sufficiently large, the reverse flow in the central region of the vessel will extend into the blade channel because the open space is limited. Thus, the flow rate is reduced. A better blade width in the present study is in the range W=0.2D to 0.26D.

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4. The enlargement of the clearance gap results in large reduction of power consumption, but has little effect on the circulating flow rate. However, the fluid in the clearance region is difficult to be mixed with the other fluid if the clearance is large. 5. It is demonstrated that after the width and the pitch of the impeller blade are optimized, the circulating flow rate is largely increased, resulting in significant decrease of mixing time. Besides, the power consumption is also reduced. It is also seen that by enlarging the clearance gap, it is difficult for the fluid in the clearance region to be well mixed. In our calculations, the vessel was separated into two frames of reference (one stationary and one rotating frame) and reasonably good agreement with measurements was obtained in terms of power and circulation numbers. However, the inclusion of the clearance gap in the rotating frame may lead to a certain degree of error 427

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Engineering Applications of Computational Fluid Mechanics Vol. 5, No. 3 (2011)

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