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## Brazilian Journal of Chemical Engineering

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*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.17 n.4-7 São Paulo Dec. 2000

#### http://dx.doi.org/10.1590/S0104-66322000000400057

**CHARACTERIZING THE FLOW OF STIRRED VESSELS WITH ANCHOR TYPE IMPELLERS**

**S.M.C.Peixoto, J.R.Nunhez* ^{ }and C.G.Duarte**

Departamento de Processos Químicos, Faculdade de Engenharia Química,

Universidade Estadual de Campinas, C.P. 6066,

13083-970, Campinas - SP, BRASIL

E-mail: simone@feq.unicamp.br

E-mail: nunhez@feq.unicamp.br

*(Received:November 7, 1999 ; Accepted: May 18, 2000)*

Abstract -Despite its importance in chemical industries, there are few works which studies anchor type impellers and only a fraction of the works investigate these systems under a computational approach. The great majority refers to turbine impellers, specially Rushton turbines, under turbulent flow. Anchor impellers are used specially for highly viscous flow, typical of polymer reactions. The viscosity is normally in the range 1000-10000 cp. Since this range of viscosity describe highly viscous flows, the reactions for anchor agitated systems are normally carried out under laminar flow. This work presents a detailed computational fluid dynamics (CFD) approach to study the behaviour of stirred vessels using anchor impellers. The axial plane of the tank, which is being modelled, is divided into small control volumes, which collectively is referred to as the mesh, or grid. In each of these cells the momentum balance, energy and mass conservation, which describes the model, are rewritten algebraically using the finite volumes method to relate such variables as velocity, pressure and temperature to values in neighbouring cells. The equations are then solved numerically, and the results yield the flow corresponding to the model. Since the geometry of a vessel with anchor impellers strictly calls for a three dimensional method, an approximation is made to account for the effect of the blades (Kuncewics, 1992). The main objective of this work is to give a detailed description of the flow generated by this axial impeller with a view to indicate ways in which the design and operation of these systems can be improved.

Keywords:anchor impeller, stirred vessels, CFD, finite volumes method.

**INTRODUCTION**

The primary flow generated by anchor impellers using a two dimensional grid have been studied by some investigators (Kaminoyama et al., 1994a; Kaminoyama et al., 1994b). The simulated section for the primary flow refers to a cross sectional plane at the tip of the impeller blade since at this position the effect of the bottom of the vessel can be neglected in a preliminary investigation (Rubart and Bohme, 1991). However, despite being important to know how primary flow of reactors agitated by anchor impellers behaves, it is important to acknowledge that anchor impellers are very much used** **for viscous fluids in heating or cooling processes and especially to avoid the stagnation of the products at the vessel walls, since the blades of the stirrer work as a scraper. Heat transfer in these systems is important and it is dominated by the secondary flow, which is the flow generated by the action of the inertial forces due to the movement of the blades. It is therefore necessary to gain more insight about the secondary flow of these vessels to determine ways in which these systems can be improved.

Many industries employ anchor impellers. Most polymer solutions are pseudo-plastic fluids, and many are reacted in vessels agitated by these impellers. Thus, the knowledge of how flow is generated in these systems have become increasingly important especially because of the development of the synthetic plastics industries. The knowledge of the flow patterns of the secondary flow behaves is of great interest.

The design of vessels today assume uniform temperature and perfect mixing which are clearly not the case, especially for moderate and highly exothermic reactions. Also, the design of vessels are based on experimental works and empirical correlation that often are not suitable for many systems and also can only give** **a global picture as far as performance is concerned. If one intends to improve the design of these systems in terms of how the shape of the vessel, the number of blades, the mode of reaction can be improved, as well as how can dead zones be eliminated or minimised, and also how energy consumption can be minimised, it is necessary to know a detailed picture of the flow, heat transfer and shear stresses in order to enable improvements in these processes. The high demand today for industries to comply to safety and environmental regulations as well as the need to ensure high quality of the products calls for well thought and planned design and rigid process operation. Even though experimental works have improved a lot recently, they unfortunately have not been able to address to all the needs listed above. Improvements in these areas today call for the use of computational studies. The computational fluid dynamics (CFD) have been used in the last two decades to devise solutions and gain insight for the flow inside these systems and the CFD together with experimental validation have been able to improve the design of many reactor systems.

The great majority of computational works for vessels with anchor stirrers presents computational studies for the primary flow. The grid is normally very coarse and many simplifications are imposed to the model (Kaminoyama et al., 1994a, Kaminoyama et al., 1990b, Kaminoyama et al., 1990a, Kaminoyama et al., 1993). A very important detail in the design that can not be simplified for the secondary flow of anchor impellers is that the bottom of the vessel should be modelled curved. The simplification of flat bottom can be assumed in simulations for the majority of impellers since they are normally located at two thirds of the vessel height and, therefore, far from the bottom. However, this is not the case for anchor systems, since the distance between the blades and vessel walls is very small. This is due to the fact that the blades act as scrapers to the viscous solutions. Thus, it is paramount that simulations be able to capture the real vessel shape in order to accurately describe the discharge flow in these systems which occur between blades and wall.

The main objective of this work is to give a detailed picture of the secondary flow of stirred vessels with anchor impellers and indicate how this knowledge can help to improve anchor reactor design and operation.

**MODELING AND SIMULATION**

The model described here calculates the flow and heat transfer for a single phase flow. The three components of the velocities are calculated on a two dimensional grid. Reaction is taken into account by a source term that generates heat inside the vessel. This aspect can be further improved at a later stage by a more appropriate reaction model.

The governing equations for the axi-symmetric model under an Eulerian frame of reference is given below in cylindrical co-ordinates (Bird et al., 1960):

Mass conservation,

(1) |

Momentum balance,

- radial direction,

(2) |

- angular direction,

(3) |

- axial direction,

(4) |

Energy conservation,

(5) |

The properties and other important parameters of the fluid are given in Tables 1 and 2:

Figure 1 shows the geometry being modelled. Due to symmetry, only a half section needs to be modelled. Figure 2 shows the mesh of 5492 control volumes used to generate the numerical experiments.

**Boundary Conditions**

**Free Surface**

At** **the liquid free surface there is no shear stress, which is acceptable for Reynolds numbers below 300 (Edward and Wilkinson, 1972). Therefore a flat surface is assumed and axial velocity is null.

**Bottom and Walls of the Vessel**

There** **is no slip, so the velocity is null.

**Impeller Blades**

The presence of the two blades of the anchor impeller strictly calls for a time dependant, three dimensional method. However, as a first approximation, it will be considered a two dimensional model calculating for the three components of the velocity. An axi-symmetric model calculating for the three components of the velocity can only be fully justified for disk type impellers. In order to enable an averaging of the effects of the blades, the approach of Kuncewicz (1992), which takes into account the number of impeller blades, is used. He assumes that the blades can be approximated by a momentum which acts equally on the whole swept volume of the blades and he uses a coefficient varying between 0 and 1, to account for the blades effect and is dependent on the number of blades. It can be thought of as a drag coefficient which accounts for the fact that the blades do not act on the whole tank, but only at two positions which change with time as the anchor stirrer rotates. This approach have been used by many authors and it is able to provide a good representation for the flow patterns inside the tank, and is enough for this investigation [Nunhez and McGreavy (1995, 1994), Foumeny et al., (1993), Ohta et al., (1985)].

**Bottom and Walls of the Vessel**

For jacketed arrangements it is assumed that there is enough cooling liquid inside the jacket to maintain temperature constant at 283.15 K (10 °C).

At the walls and vessel bottom the boundary condition is:

(6) |

and the free surface is:

(7) |

The set of non linear equations obtained by the model are solved by the finite volumes method and the results of the model are performed using the CFX-4.2 package by AEA technology which has been successfully used for many reactor flow problems.

**Numerical Method**

The finite volumes method splits the domain of the problem in several small blocks, called volumes, where an approximation for the equations of mass, energy and momentum conservation is made. All the individual terms of the equations are approximated by the method in the cells, therefore the quality of the solution will depend heavily on the size of the volumes inside the mesh. The commercial software CFX-4.2 uses the co-localized mesh, where all the variables being investigated are calculated on a single point of the volume (Peric et al., 1988). The body fit method is also used to accommodate the contours of the geometry. The SIMPLEC method (Van Doormaal and Raithby, 1984) is used to avoid spurious pressure oscillations. The sparse non-linear system for the velocities is solved by the Algebraic method and the Preconditioned Conjugate Gradient method was used to calculate for the pressure. The mass residual was set for an error e < 10^{-10} kg/s, which is small enough to guarantee all variables converged.

**RESULTS AND DISCUSSION**

In order to demonstrate how CFD can help to understand the flow in these systems, it will be investigated the flow of concentrated orange juice. During its processing, orange juice is first filtered and centrifuged and it contains about 88% of water and 12% of soluble particles which are responsible for flavour, taste, colour and other properties of the product. The juice is concentrated at high vacuum and low temperature in several stages until a concentrated juice containing 35% of water is obtained. To preserve the properties of the juice, it stays only a short time inside the evaporator and, after leaving it, the concentrated juice is cooled from 313.15K (40°C) to 293.15K (20°C) and sent to tanks where juice from several batches are homogenised at 263.15K (-10°C), before being prepared under rigid quality control. The characteristics of the product are:

In order to ensure the results are independent on mesh size, a mesh independent study was carried out and the approach used by Foumeny et al. (1993), Nunhez and McGreavy (1995,1994) and Peixoto (1998) was adopted. Figures 3 and 4 show a comparison of the radial and axial velocities in two different sections of the reactor for the two mesh densities. The results are very similar. The more pronounced variations for the velocities between both meshes were found in Figure 4 for the variation of the radial velocity as a function of **z** for the radial position r=0.28m, which is halfway between shaft and wall. The finer mesh used 13080 control volumes and the elapsed time for the run was 63 hours and 36 minutes on a 512 MB, ULTRA 5 of SUN systems with a 273 MHz processor. It is a very expensive computational time, specially when compared with the coarser mesh that took 13 hours and 06 minutes to run. Therefore the mesh of 5492 control volumes was used for the experimental runs.

Figure 5 shows the velocity vector plot for the rotational speed of 136 rpm, which, for the tank diameter used in this work, is considered low speed for industrial applications. It can be noticed the formation of a single recirculation zone centered near the curve separating bottom and vessel walls, a little bit above the curve of the anchor blade. The flow is axial, as expected. As it can be seen, fluid is poorly mixed for this rotational speed. The velocities near the free surface of the liquid are very low, indicating poor mixing. This suggests the velocity of the stirrer is not enough for this system. Figure 6 shows the same plot for the rotational speed of 317 rpm which defines a medium industrial speed for the same geometry. It is readily noticed that the mixing inside the vessel is superior to the lower speed and gives a better fluid circulation. The region of low velocity near the free surface is practically eliminated. However, power consumption is much higher so a trade off has to be made in order to decide which mode of operation should be chosen. As expected, heat transfer is superior for the higher speed. Figure 7 shows the temperature contour plots corresponding to the rotational speed of 136 rpm and viscosity of 45 and 60 kg.m^{-1}.s^{-1}. It is assumed there is enough cooling liquid to maintain the jacket temperature fixed at 10°C. As expected, the temperature contours follow the secondary flow and the temperature peak is located at the center of the recirculation zones. This happens because the flow is laminar and under this flow condition the conduction phenomena can be significant. Although being expected, it is interesting to notice that the differences in temperature are more pronounced for the viscosity of 60 kg.m^{-1}s^{-1}. This happens because there is more resistance to flow for the higher viscosity, which is reflected at the temperature field. Table 3 compares the temperature distributions for the two rotational speeds.

It is interesting to note that the difference between maximum and average temperature for the rotational speed of 136 rpm is considerable for the viscosity of 60 kg.m^{-1}s^{-1}. On the other hand these differences are minimal for the rotational speed of 317 rpm, which indicates fluid is well mixed and temperature can be considered uniform. In this case, change in the mode of operation and result in improvements of the process.

**CONCLUDING REMARKS**

The pseudo three-dimensional model presented in this work gives a good representation for the flow and temperature fields for anchor impellers and helps to determine design features which improve the flow inside tank reactors stirred by anchor impellers. Results show that moderate agitation in industrial applications gives good fluid circulation and a trade off between the benefits of higher fluid circulation and power consumption has to bedecided for the process being investigated.

Further work analysing blade height, distance to the wall and other geometrical factors will also determine other ways in which design can be improved in these systems.

**ACKNOWLEDGEMENTS**

The authors would like to thank FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and FAEP (Fundação de Apoio ao Ensino e à Pesquisa - UNICAMP) for the grants received for this project.

**NOMENCLATURE**

C | impeller height [m] |

Cp | specific heat capacity at constant pressure [J.kg^{-1}.K^{-1}] |

D | impeller diameter [m] |

De | impeller diameter [m] |

h_{fs} | heat transfer coefficient at the free-surface of the liquid [W.m^{-2}.K^{-1}] |

h_{w} | heat transfer coefficient at the wall [W.m^{-2}.K^{-1}] |

k_{l} | thermal conductivity of the liquid [W.m^{-1}.K^{-1}] |

n | power number which describes the non-Newtonian attributes |

p | pressure [N.m^{-2}] |

q_{fs} | heat flux at the free-surface [W.m^{-2}] |

q_{j} | heat flux at the walls and bottom of the vessel [W.m^{-2}] |

r | radial direction [m] |

s | impeller width [m] |

T | tank diameter [m] |

T | temperature [K] |

T_{fs} | temperature at the free-surface [K] |

T_{w} | temperature at the walls and bottom of the vessel[K] |

u_{z} | axial velocity [m.s^{-1}] |

u_{r} | radial velocity [m.s^{-1}] |

uq | angular velocity [m.s^{-1}] |

z | axial direction [m] |

Z | liquid height [m] |

*Greek Symbols*

DH | heat source [W.m^{-3}] |

m | Newtonian viscosity [kg.m^{-1}.s^{-1}] |

r | density [kg.m^{-3}] |

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*To whom correspondence should be addressed