**CFD Simulation of Flows in Stirred Tank Reactors Through Prediction of Momentum Source**

Weidong Huang and Kun Li

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51754

## **1. Introduction**

The mixing and agitation of fluid in a stirred tank have raised continuous attention. Starting with Harvey and Greaves, Computational Fluid Dynamics (CFD) has been applied as a power‐ ful tool for investigating the detailed information on the flow in the tank [1-2]. In their work, the impeller boundary condition (IBC) approach has been proposed for the impeller modeling, in which the flow characteristics near the impeller are experimentally measured, and are speci‐ fied as the boundary conditions for the whole flow field computation [1-4]. Because it depends on the experimental data, IBC can hardly predict the flow in the stirred tank and its applicabili‐ ty is inherently limited. To overcome this drawback, the multiple rotating reference frames ap‐ proach (MRF) has been developed, in which the vessel is divided into two parts: the inner zone using a rotating frame and the outer zone associated with a stationary frame, for a steady state simulation. Although it can predict the flow field, the computation result is slightly lack of ac‐ curacy and needs a longer time for convergence [5]. Sliding mesh approach (SM) is another available approach, in which the inner grid is assumed to rotate with the impeller speed, and the full transient simulations are carried out [6-7]. SM approach gives an improved result, but it suffers from the large computational expense [8]. Moving-deforming grid technique was proposed by Perng and Murthy [9], in which the grid throughout the vessel moves with the im‐ peller and deforms. This approach requires a rigorous grid quality and the computational ex‐ penses are even higher than SM [10-11].

Momentum source term approach adds momentum source in the computational cells to rep‐ resent the impeller propelling and the real blades are ignored. In the approach, the genera‐ tions of the vessel configuration and grids are simpler, and the computational time is shorter and the computational accuracy is higher. However, the determination of the momentum source depends on experimental data or empirical coefficients presently [12-13]. The ap‐

© 2013 Huang and Li; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Huang and Li; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

proach proposed by Pericleous and Patel is based on the airfoil aerodynamics and originally aimed at the two-dimensional flow in the stirred vessels. Xu, McGrath [14] and Patwardhan [15] applied this approach to simulate the three-dimensional flow pattern in a tank with the pitched blade turbines. Revstedt et al. [16] modified the approach for the three-dimensional simulation in a Rushton turbine stirred vessel. In their approach, the determination of the momentum source depends on the specified power number. Dhainaut et al. [13] reported a kind of momentum source term approach in which the fluid velocity is linearly proportional to the radius with an empirical coefficient. In our previous study, we proposed to calculate the momentum source term according to the ideal propeller equation [17], which is related to the rotation speed and radius of the blade [18-20], however, the prediction accuracy is just a little better than MRF method.

izontally across the tank, the grid was refined to resolve the large flow gradients (Fig.2). For it is unnecessary to construct the impeller in momentum source term approach, the tank ge‐

CFD Simulation of Flows in Stirred Tank Reactors Through Prediction of Momentum Source

http://dx.doi.org/10.5772/51754

137

**H/T C/T B/T D/T K/D w/D h/D** 1 1/3 1/10 1/3 3/4 1/4 1/5

ometry and mesh generations are simpler than MRF and SM.

**Figure 1.** Geometry of the stirred vessel.

**Table 1.** Dimension scaling of the stirred vessel configuration.

**Figure 2.** Computational mesh for the momentum source term approach simulation.

In order to investigate the computational speed of MRF and SM, they are also applied to simulate the flow field of stirred tank. The total number of computational cells for MRF is 1,652,532 (*r*×*θ*×*z* ≈ 63×204×126) with the impeller blade covered with 88128 cells (*r*×θ×*z* =

Besides, other methods, such as inner-outer approach, snapshot approach and adaptive force field technique, have been developed, but are less applied[1, 21].

In this study, an equation is proposed to calculate the momentum source term after consid‐ ering both impeller propelling force and the radial friction effect between the blades and flu‐ id. The flow field in the Rushton turbine stirred vessel was simulated with the CFD model. The available experimental data near the impeller tip and in the bulk region were applied to validate this approach. Moreover, the comparisons of the computational accuracy and time with MRF and SM were carried out.

## **2. Model and methods**

#### **2.1. Stirred tank and grid generation**

Fig.1 shows the geometry of the investigated standard six-bladed Rushton turbine stirred tank with four equally spaced baffles. The diameter of cylindrical vessel *T* = 0.30 m, the di‐ ameter of rotating shaft *d* = 0.012 m, the hub diameter *b* = 0.020 mm, and the detailed size proportions are listed in Table 1. The working fluid is water with density *ρ* = 998.2 kg m-3 and viscosity *μ* = 1.003×10−3 Pa s. The rotational speed of the impeller *N* = 250 rpm, leading to a tip speed *u tip* = 1.31 m s-1 and Reynolds number Re = 41,467 (Re = *ρND <sup>2</sup> /μ*). Eight axial locations in the bulk region are also shown in Fig.1. They are the same as the experimental study of Murthy and Joshi [22],

It is well known that sufficiently fine grids and lower-dissipation discretization schemes can significantly reduce the numerical errors [23]. And the grid resolution on the blades has an important influence on capturing the details of flow near the impeller [23-24]. Grid inde‐ pendence study has been carried out for momentum source term approach with the stand‐ ard *k-ε* model with different grid resolutions. In this paper, the results of the grid resolution of 1,429,798 hexahedral cells (*r*×*θ*×*z* ≈ 88×120×131) with the impeller blade covered with 96000 cells (*r*×*θ*×*z* = 32×120×25) have been reported for the simulations of standard *k-ε* model and Reynolds stress model (RSM). In the region encompassing the impeller discharge stream, contained within 1.5 blade heights above and below the impeller and extending hor‐ izontally across the tank, the grid was refined to resolve the large flow gradients (Fig.2). For it is unnecessary to construct the impeller in momentum source term approach, the tank ge‐ ometry and mesh generations are simpler than MRF and SM.

**Figure 1.** Geometry of the stirred vessel.

proach proposed by Pericleous and Patel is based on the airfoil aerodynamics and originally aimed at the two-dimensional flow in the stirred vessels. Xu, McGrath [14] and Patwardhan [15] applied this approach to simulate the three-dimensional flow pattern in a tank with the pitched blade turbines. Revstedt et al. [16] modified the approach for the three-dimensional simulation in a Rushton turbine stirred vessel. In their approach, the determination of the momentum source depends on the specified power number. Dhainaut et al. [13] reported a kind of momentum source term approach in which the fluid velocity is linearly proportional to the radius with an empirical coefficient. In our previous study, we proposed to calculate the momentum source term according to the ideal propeller equation [17], which is related to the rotation speed and radius of the blade [18-20], however, the prediction accuracy is just

Besides, other methods, such as inner-outer approach, snapshot approach and adaptive

In this study, an equation is proposed to calculate the momentum source term after consid‐ ering both impeller propelling force and the radial friction effect between the blades and flu‐ id. The flow field in the Rushton turbine stirred vessel was simulated with the CFD model. The available experimental data near the impeller tip and in the bulk region were applied to validate this approach. Moreover, the comparisons of the computational accuracy and time

Fig.1 shows the geometry of the investigated standard six-bladed Rushton turbine stirred tank with four equally spaced baffles. The diameter of cylindrical vessel *T* = 0.30 m, the di‐ ameter of rotating shaft *d* = 0.012 m, the hub diameter *b* = 0.020 mm, and the detailed size proportions are listed in Table 1. The working fluid is water with density *ρ* = 998.2 kg m-3 and viscosity *μ* = 1.003×10−3 Pa s. The rotational speed of the impeller *N* = 250 rpm, leading to a tip speed *u tip* = 1.31 m s-1 and Reynolds number Re = 41,467 (Re = *ρND <sup>2</sup> /μ*). Eight axial locations in the bulk region are also shown in Fig.1. They are the same as the experimental

It is well known that sufficiently fine grids and lower-dissipation discretization schemes can significantly reduce the numerical errors [23]. And the grid resolution on the blades has an important influence on capturing the details of flow near the impeller [23-24]. Grid inde‐ pendence study has been carried out for momentum source term approach with the stand‐ ard *k-ε* model with different grid resolutions. In this paper, the results of the grid resolution of 1,429,798 hexahedral cells (*r*×*θ*×*z* ≈ 88×120×131) with the impeller blade covered with 96000 cells (*r*×*θ*×*z* = 32×120×25) have been reported for the simulations of standard *k-ε* model and Reynolds stress model (RSM). In the region encompassing the impeller discharge stream, contained within 1.5 blade heights above and below the impeller and extending hor‐

force field technique, have been developed, but are less applied[1, 21].

a little better than MRF method.

136 Nuclear Reactor Thermal Hydraulics and Other Applications

with MRF and SM were carried out.

**2.1. Stirred tank and grid generation**

**2. Model and methods**

study of Murthy and Joshi [22],


**Table 1.** Dimension scaling of the stirred vessel configuration.

**Figure 2.** Computational mesh for the momentum source term approach simulation.

In order to investigate the computational speed of MRF and SM, they are also applied to simulate the flow field of stirred tank. The total number of computational cells for MRF is 1,652,532 (*r*×*θ*×*z* ≈ 63×204×126) with the impeller blade covered with 88128 cells (*r*×θ×*z* = 24×204×18), and for SM is 606,876 (*r*×*θ*×*z* ≈ 60×106×90) with the impeller blade covered with 33920 cells (*r*×θ×*z* = 20×106×16).

#### **2.2. Momentum source model and control equations**

Following the assumptions of Euler equation for turbomachinery [25], the momentum source term from the driving of blade is determined as following. For a small area of blade d*S*, assuming that a force exerted by the impeller on the fluid is perpendicular to the blade surface [14], the propelling force from the impeller d*F* is considered to be equal to the prod‐ uct of the mass flow rate across the interaction section d*Q* and the fluid velocity variation along the normal direction of the area element d*S*:

$$\mathbf{d}F = \mathbf{d}Q \cdot \overrightarrow{\boldsymbol{\Lambda}} \,\tag{1}$$

0.2

CFD Simulation of Flows in Stirred Tank Reactors Through Prediction of Momentum Source

(5)

139

http://dx.doi.org/10.5772/51754

5 7 0.0577 Re 5 10 Re 1 10 *f x*

where Re*x* = *ρv <sup>r</sup> x*/*μ* , *x* is distance between cell center and the center of rotation axis. In this paper, we mainly consider the radial friction resistance, ignoring the axial effect for simplicity. Adding the momentum sources of the both direction into the computational

In the previous study, the difference between the ensemble-averaged flow field calculated with the steady-state and the time-dependent approaches was found to be negligible [28-30]. Here, the continuity and momentum equations of motion for three-dimensional incompres‐ sible flow, as well as the standard *k–ε* turbulence equation [31] or Reynolds stress transport

The free surface was treated as a flat and rigid lid, so a slip wall was given to the surface. The disc, hub and shaft of the Rushton turbine are specified as moving walls. A standard wall function was given to the other solid walls, including the bottom surface, the sidewall and baffles. Second order upwind discretization scheme was adopted for pressure interpola‐ tion and the convection term of momentum, turbulent kinetic energy and energy dissipation rate equations, and the discretized equations were solved iteratively by using the SIMPLE

The power number *N <sup>P</sup>* is an important parameter of the stirred tank, which is generally ap‐ plied to validate the CFD predictions [33-34]. One method for calculating the power number is based on the energy balance, in which the power number of the impeller is calculated from the integration of the turbulent energy dissipation rate (*ε*) predicted from the CFD model [35]:

3 5

re

*N D*

In MRF and SM approaches, the power number is usually calculated from the predicted

3 5

In the present study, since the force from blade results in the fluid movement, the impeller power can be calculated from the integration of the momentum source in the impeller re‐

*rdrd dz*

 q

<sup>=</sup> òòò (6)

<sup>=</sup> (7)

2 /2 00 0

r

2 *p*

*NmM <sup>N</sup> N D* p

r

*H T*

p

*P*

in which *m* is the number of the blades, *M* is the torque of each blade.

*N*

equations [32], were solved to calculate single phase flow of the stirred vessel.

*x*

*C* - = × ´ < <´

cells, the real blade is replaced.

algorithm for pressure-velocity coupling.

**2.3. Power number prediction**

torque [36]:

Since the rigid body rotational motion of the blade is considered, it is a reasonable as‐ sumption that the fluid velocity after being acted on is the same as the velocity of the im‐ peller blade, thus *Δu* can be obtained. For the present study on the vertical blade, Eq.(1) can be written as:

$$\mathbf{d}F = \rho \cdot \mathbf{dS} \cdot \mathbf{u} \cdot (\mathbf{u} - \mathbf{v}\_{\theta}) \tag{2}$$

where *u* is linear velocity of the area element on the blade surface d*S*, *v* <sup>θ</sup> is fluid tangential velocity rotating with the impeller before the fluid was propelled by impeller, d*S* is crosssection area of the interface between fluid and the impeller blade. A similar equation has been applied to describe the propelling effect of the ship's impeller [26] which we applied it to calculate the momentum source term [18-19].

Furthermore, the fluid is continuously impelled out from the impeller region, so there is a rela‐ tive motion of the fluid along the radial direction of the blade. In order to consider the friction effect on the fluid movement, the friction resistance equation about the finite flat plate based on the boundary layer theory is introduced to calculate the friction force approximately [27]:

$$\mathbf{d}f = \frac{1}{2}\rho \mathbf{v}\_r^2 \cdot \mathbf{C}\_f \cdot \mathbf{d}S \tag{3}$$

where *f* is friction force in the computational cells; *v <sup>r</sup>* is radial velocity of the fluid; *C <sup>f</sup>* is local resistance coefficient related to Re*x*, which is calculated approximately by:

$$\begin{aligned} C\_f &= 0.664 \cdot \text{Re}\_x^{-0.8} \\ \text{Re}\_x &\ll 5 \times 10^8 \end{aligned} \tag{4}$$

$$\begin{aligned} C\_f &= 0.0577 \cdot \text{Re}\_x^{-0.2} \\ \mathbf{S} \times 10^5 &< \text{Re}\_x < 1 \times 10^7 \end{aligned} \tag{5}$$

where Re*x* = *ρv <sup>r</sup> x*/*μ* , *x* is distance between cell center and the center of rotation axis. In this paper, we mainly consider the radial friction resistance, ignoring the axial effect for simplicity. Adding the momentum sources of the both direction into the computational cells, the real blade is replaced.

In the previous study, the difference between the ensemble-averaged flow field calculated with the steady-state and the time-dependent approaches was found to be negligible [28-30]. Here, the continuity and momentum equations of motion for three-dimensional incompres‐ sible flow, as well as the standard *k–ε* turbulence equation [31] or Reynolds stress transport equations [32], were solved to calculate single phase flow of the stirred vessel.

The free surface was treated as a flat and rigid lid, so a slip wall was given to the surface. The disc, hub and shaft of the Rushton turbine are specified as moving walls. A standard wall function was given to the other solid walls, including the bottom surface, the sidewall and baffles. Second order upwind discretization scheme was adopted for pressure interpola‐ tion and the convection term of momentum, turbulent kinetic energy and energy dissipation rate equations, and the discretized equations were solved iteratively by using the SIMPLE algorithm for pressure-velocity coupling.

#### **2.3. Power number prediction**

24×204×18), and for SM is 606,876 (*r*×*θ*×*z* ≈ 60×106×90) with the impeller blade covered with

Following the assumptions of Euler equation for turbomachinery [25], the momentum source term from the driving of blade is determined as following. For a small area of blade d*S*, assuming that a force exerted by the impeller on the fluid is perpendicular to the blade surface [14], the propelling force from the impeller d*F* is considered to be equal to the prod‐ uct of the mass flow rate across the interaction section d*Q* and the fluid velocity variation

d d *F Qu*®

d d( ) *F Su u v* = × ×× r

Since the rigid body rotational motion of the blade is considered, it is a reasonable as‐ sumption that the fluid velocity after being acted on is the same as the velocity of the im‐ peller blade, thus *Δu* can be obtained. For the present study on the vertical blade, Eq.(1)

where *u* is linear velocity of the area element on the blade surface d*S*, *v* <sup>θ</sup> is fluid tangential velocity rotating with the impeller before the fluid was propelled by impeller, d*S* is crosssection area of the interface between fluid and the impeller blade. A similar equation has been applied to describe the propelling effect of the ship's impeller [26] which we applied it

Furthermore, the fluid is continuously impelled out from the impeller region, so there is a rela‐ tive motion of the fluid along the radial direction of the blade. In order to consider the friction effect on the fluid movement, the friction resistance equation about the finite flat plate based on the boundary layer theory is introduced to calculate the friction force approximately [27]:

> <sup>1</sup> <sup>2</sup> d d <sup>2</sup> *r f f vC S* = ×× r

where *f* is friction force in the computational cells; *v <sup>r</sup>* is radial velocity of the fluid; *C <sup>f</sup>*

Re 5 10 *f x*

*C* - = × <= ´

0.5

5 0.664 Re

resistance coefficient related to Re*x*, which is calculated approximately by:

*x*

q

= ×D (1)

(3)

is local

(4)

(2)

33920 cells (*r*×θ×*z* = 20×106×16).

138 Nuclear Reactor Thermal Hydraulics and Other Applications

can be written as:

**2.2. Momentum source model and control equations**

along the normal direction of the area element d*S*:

to calculate the momentum source term [18-19].

The power number *N <sup>P</sup>* is an important parameter of the stirred tank, which is generally ap‐ plied to validate the CFD predictions [33-34]. One method for calculating the power number is based on the energy balance, in which the power number of the impeller is calculated from the integration of the turbulent energy dissipation rate (*ε*) predicted from the CFD model [35]:

$$N\_p = \frac{\int\_0^H \int\_0^{2\pi} \int\_0^{T/2} \rho \varepsilon r dr d\theta dz}{\rho N^3 D^s} \tag{6}$$

In MRF and SM approaches, the power number is usually calculated from the predicted torque [36]:

$$N\_p = \frac{2\pi NmM}{\rho N^3 D^3} \tag{7}$$

in which *m* is the number of the blades, *M* is the torque of each blade.

In the present study, since the force from blade results in the fluid movement, the impeller power can be calculated from the integration of the momentum source in the impeller re‐ gion [14], which is called integral power based approach, so the power number is calculated in the following manner in our study:

$$N\_p = \frac{\int\_{C - h/2}^{C + h/2} \int\_0^{2\pi} \int\_{D/2 - w}^{D/2} (F\_\theta \cdot 2\pi Nr) r \text{d}r \text{d}\theta \text{d}\varphi}{\rho N^3 D^\circ} \tag{8}$$

**Figure 3.** Comparison of the results predicted by different impeller approaches and experimental data around impel‐ ler tip (a) radial velocity, (b) tangential velocity, (c) axial velocity and (d) turbulence kinetic energy: ○ experimental da‐ ta (Escudie and Line, 2003), experimental data (Wu and Patterson, 1989), standard k-ε momentum source term approach predictions, RSM momentum source term approach predictions, MRF predictions (Deglon and Meyer, 2006)

CFD Simulation of Flows in Stirred Tank Reactors Through Prediction of Momentum Source

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141

Fig.4 shows the radial profiles of the mean axial velocity at various axial levels. Here, *z*<0.10 m is in the lower parts of the tank, while *z*>0.10 m is in the upper parts. It can be noted that curve changes in two parts are just opposite, indicating that axial velocity directions are op‐ posite. The result shows that the predictions of momentum source term approach with standard *k-ε* model and RSM are both in favorable agreement with the experimental data,

Fig.5 depicts the comparison of predicted and experimental radial velocity component. The high speed impeller discharge streams radially. Radial velocity initially increases and then decreases, attaining the maximum at *r/R* = 0.6-0.8 near blade. It can be seen that compared to the experimental data, both the standard *k-ε* model predictions of momentum source term approach and SM exhibit some disparity, particularly at the levels in the upper part. At the top axial levels (*z =* 0.154 m and *z =* 0.244 m), there are the largest differences between the predictions of momentum source term approach and SM, while they have similar accuracy in other positions. However, the results of momentum source term approach and SM pre‐

**3.2. Numerical validation of the flow field in the bulk region**

dicted with RSM are consistent with the experimental data.

similar to the predicted results of SM.

in which *F <sup>θ</sup>* is tangential momentum source term.

## **3. Results and discussion**

## **3.1. Numerical validation of the flow field near the impeller tip**

Fig.3 shows the profiles of the predicted and experimental flow data in the impeller region. Fig.3a gives the comparison of the radial velocity. Escudie and Line [37] summarized the previous experimental works and found due to the differences of the experimental techni‐ que and the stirred-vessel configuration, there existed some inconsistencies among the re‐ ported results, whereas the maximum of radial velocity was in the range of 0.7-0.87*u tip*. Their experiment study gave a maximal radial velocity of 0.80*u tip*, while Wu and Patterson [38] 0.75*u tip*. In present work, momentum source term approach with standard *k-ε* and RSM predicts a maximum radial velocity of 0.79*u tip* and 0.75*u tip*, respectively, which means that momentum source term approach predicts the maximal radial velocity rather well. From fig. 3a, the distribution of radial velocity predictions based on momentum source term approach agrees rather well with the experimental data, which outperforms MRF predictions.

With regard to tangential velocity, the maxima from Wu and Patterson [38], and Escudie and Line [37] are 0.66*u tip* and 0.72*u tip* respectively. For momentum source term approach with standard *k-ε* and RSM, both gave a maximum of 0.63*u tip*. From Fig. 3b, it can be found that the calculated results of momentum source term approach are in good agreement with the two sets of experimental data, better than MRF predictions.

Fig.3c shows the comparison of axial velocity. The measured results from Wu and Patterson [38], Escudie and Line [37] are almost the same. The momentum source term approach re‐ sults match the measured data well in most regions, but predicting the change from the maximum to minimum with several deviations. Compared to standard *k-ε* model, RSM pre‐ dictions of momentum source term approach make some improvements, while MRF predic‐ tions are not provided.

In fig.3d, it can be observed that there are two maxima of different magnitudes in the pro‐ files of turbulent kinetic energy, thus the curve is not symmetrical. RSM simulations of mo‐ mentum source term approach are in accordance with those of standard *k-ε* model. Momentum source term approach and MRF both successfully predict the variations be‐ tween two maxima, but momentum source term approach exhibits a better prediction of the turbulent kinetic energy than MRF.

CFD Simulation of Flows in Stirred Tank Reactors Through Prediction of Momentum Source http://dx.doi.org/10.5772/51754 141

**Figure 3.** Comparison of the results predicted by different impeller approaches and experimental data around impel‐ ler tip (a) radial velocity, (b) tangential velocity, (c) axial velocity and (d) turbulence kinetic energy: ○ experimental da‐ ta (Escudie and Line, 2003), experimental data (Wu and Patterson, 1989), standard k-ε momentum source term approach predictions, RSM momentum source term approach predictions, MRF predictions (Deglon and Meyer, 2006)

#### **3.2. Numerical validation of the flow field in the bulk region**

gion [14], which is called integral power based approach, so the power number is calculated

3 5 ( 2 )dd d *Ch D*

*N D*

Fig.3 shows the profiles of the predicted and experimental flow data in the impeller region. Fig.3a gives the comparison of the radial velocity. Escudie and Line [37] summarized the previous experimental works and found due to the differences of the experimental techni‐ que and the stirred-vessel configuration, there existed some inconsistencies among the re‐ ported results, whereas the maximum of radial velocity was in the range of 0.7-0.87*u tip*. Their experiment study gave a maximal radial velocity of 0.80*u tip*, while Wu and Patterson [38] 0.75*u tip*. In present work, momentum source term approach with standard *k-ε* and RSM predicts a maximum radial velocity of 0.79*u tip* and 0.75*u tip*, respectively, which means that momentum source term approach predicts the maximal radial velocity rather well. From fig. 3a, the distribution of radial velocity predictions based on momentum source term approach

agrees rather well with the experimental data, which outperforms MRF predictions.

With regard to tangential velocity, the maxima from Wu and Patterson [38], and Escudie and Line [37] are 0.66*u tip* and 0.72*u tip* respectively. For momentum source term approach with standard *k-ε* and RSM, both gave a maximum of 0.63*u tip*. From Fig. 3b, it can be found that the calculated results of momentum source term approach are in good agreement with

Fig.3c shows the comparison of axial velocity. The measured results from Wu and Patterson [38], Escudie and Line [37] are almost the same. The momentum source term approach re‐ sults match the measured data well in most regions, but predicting the change from the maximum to minimum with several deviations. Compared to standard *k-ε* model, RSM pre‐ dictions of momentum source term approach make some improvements, while MRF predic‐

In fig.3d, it can be observed that there are two maxima of different magnitudes in the pro‐ files of turbulent kinetic energy, thus the curve is not symmetrical. RSM simulations of mo‐ mentum source term approach are in accordance with those of standard *k-ε* model. Momentum source term approach and MRF both successfully predict the variations be‐ tween two maxima, but momentum source term approach exhibits a better prediction of the

r

q p

*F Nr r r z*

 q


/2 2 /2 /2 0 /2

p

+

**3.1. Numerical validation of the flow field near the impeller tip**

the two sets of experimental data, better than MRF predictions.

*Ch D w*

in the following manner in our study:

140 Nuclear Reactor Thermal Hydraulics and Other Applications

**3. Results and discussion**

tions are not provided.

turbulent kinetic energy than MRF.

*p*

in which *F <sup>θ</sup>* is tangential momentum source term.

*N*

Fig.4 shows the radial profiles of the mean axial velocity at various axial levels. Here, *z*<0.10 m is in the lower parts of the tank, while *z*>0.10 m is in the upper parts. It can be noted that curve changes in two parts are just opposite, indicating that axial velocity directions are op‐ posite. The result shows that the predictions of momentum source term approach with standard *k-ε* model and RSM are both in favorable agreement with the experimental data, similar to the predicted results of SM.

Fig.5 depicts the comparison of predicted and experimental radial velocity component. The high speed impeller discharge streams radially. Radial velocity initially increases and then decreases, attaining the maximum at *r/R* = 0.6-0.8 near blade. It can be seen that compared to the experimental data, both the standard *k-ε* model predictions of momentum source term approach and SM exhibit some disparity, particularly at the levels in the upper part. At the top axial levels (*z =* 0.154 m and *z =* 0.244 m), there are the largest differences between the predictions of momentum source term approach and SM, while they have similar accuracy in other positions. However, the results of momentum source term approach and SM pre‐ dicted with RSM are consistent with the experimental data.

**Figure 4.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean axial velocity in the bulk region: experimental data [39] (Murthy and Joshi, 2008) standard k-ε momen‐ tum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM predictions (Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

**Figure 5.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean radial velocity in the bulk region: experimental data (Murthy and Joshi, 2008) standard k-ε momentum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM predictions

CFD Simulation of Flows in Stirred Tank Reactors Through Prediction of Momentum Source

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143

(Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

**Figure 5.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean radial velocity in the bulk region: experimental data (Murthy and Joshi, 2008) standard k-ε momentum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM predictions (Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

**Figure 4.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean axial velocity in the bulk region: experimental data [39] (Murthy and Joshi, 2008) standard k-ε momen‐ tum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM

predictions (Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

142 Nuclear Reactor Thermal Hydraulics and Other Applications

**Figure 6.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean tangential velocity in the bulk region: experimental data (Murthy and Joshi, 2008) standard k-ε momen‐ tum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM predictions (Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

**Figure 7.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean turbulent kinetic energy in the bulk region: experimental data (Murthy and Joshi, 2008) standard k-ε momentum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM

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Fig.7 shows the numerical comparison of turbulent kinetic energy in various positions. At *z* = 0.082m, it can be noticed that there are two maxima of turbulent kinetic energy, similar to the case of tangential velocity (Fig.6). They are generated by the interactions between fluid and blades a, fluid and wall, respectively. Overall, the CFD results which predicted by standard *k-ε* model are not satisfying, significantly underpredicting the turbulent kinetic en‐ ergy at almost all positions. However, the results of momentum source term approach and SM are similar. Although RSM predictions of momentum source term approach and SM have reduced the errors, both still underestimate turbulent kinetic energy, and their results

predictions (Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

Fig.6 illustrates the radial profiles of the tangential velocity component. Yianneskis et al [40] pointed out that the baffles reduce the vessel cross-section, which results in higher values for tangential velocity and a reduced pressure, thus generating reverse flows. It may be the reason that negative velocities exist at the levels of *z* = 0.01 m and *z* = 0.044 m. It can be seen that the computational results of momentum source term approach and SM with standard *kε* model are similar, and both of them predict the reverse flows. They agree rather well with the experimental data at the lower levels, whereas deviate at the upper levels. For RSM sim‐ ulations, the prediction accuracy of two modeling methods have been improved, and their results are in good agreement with the experimental results.

CFD Simulation of Flows in Stirred Tank Reactors Through Prediction of Momentum Source http://dx.doi.org/10.5772/51754 145

**Figure 7.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean turbulent kinetic energy in the bulk region: experimental data (Murthy and Joshi, 2008) standard k-ε momentum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM predictions (Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

**Figure 6.** Comparison of the results predicted by different impeller approaches and experimental data of the dimen‐ sionless mean tangential velocity in the bulk region: experimental data (Murthy and Joshi, 2008) standard k-ε momen‐ tum source term approach predictions, RSM momentum source term approach predictions, standard k-ε SM

Fig.6 illustrates the radial profiles of the tangential velocity component. Yianneskis et al [40] pointed out that the baffles reduce the vessel cross-section, which results in higher values for tangential velocity and a reduced pressure, thus generating reverse flows. It may be the reason that negative velocities exist at the levels of *z* = 0.01 m and *z* = 0.044 m. It can be seen that the computational results of momentum source term approach and SM with standard *kε* model are similar, and both of them predict the reverse flows. They agree rather well with the experimental data at the lower levels, whereas deviate at the upper levels. For RSM sim‐ ulations, the prediction accuracy of two modeling methods have been improved, and their

predictions (Murthy and Joshi, 2008), RSM SM predictions (Murthy and Joshi, 2008).

144 Nuclear Reactor Thermal Hydraulics and Other Applications

results are in good agreement with the experimental results.

Fig.7 shows the numerical comparison of turbulent kinetic energy in various positions. At *z* = 0.082m, it can be noticed that there are two maxima of turbulent kinetic energy, similar to the case of tangential velocity (Fig.6). They are generated by the interactions between fluid and blades a, fluid and wall, respectively. Overall, the CFD results which predicted by standard *k-ε* model are not satisfying, significantly underpredicting the turbulent kinetic en‐ ergy at almost all positions. However, the results of momentum source term approach and SM are similar. Although RSM predictions of momentum source term approach and SM have reduced the errors, both still underestimate turbulent kinetic energy, and their results are similar. Due to unsteady and complex nature of flow characteristics in the impeller re‐ gion and the continuous conversion of kinetic energy [37], the *k-ε* model and RSM both fail to capture the transfer details of kinetic energy, so the momentum source term approach and SM model underpredict the turbulent kinetic energy with similar deviations when the *k-ε* and RSM turbulent model are applied.

again, repeating above-mentioned process which generates two circulation loops of different directions in the upper and lower part of the tank, respectively. It can be observed that the cir‐ culation loop ranges above the *z*/*T* value of 0.9, which is consistent with the simulations by Ng et al. [24] (1998) (*Re* = 40,000) and Yapici et al. [41] (2008) (*Re* = 60,236). Reynolds numbers of previous and present studies can ensure that the flow in a stirred vessel is fully developed tur‐

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Fig.9 shows the flow field of stirred tank near the impeller tip. It can be noticed that velocity distributions are not symmetrical about the impeller centre plate (*z* = 0.1 m), but shift up‐ wards slightly. This result is in agreement with the previous experimental works, that the impeller is not symmetrically located, the top of the tank is a free surface, and the hub is present on the top side of the impeller [38, 42-43]. In this study, this detail has been success‐ fully captured in the velocity field near impeller tip, further indicating that momentum

Table 2 shows the comparison of predicted power numbers with the experimental value. Here, the experimental power numbers reported by Rushton et al. [44-45], Murthy and Joshi [22] are applied in the reference. The quantities obtained by SM model through integral *ε* based approach are considerably lower than the reported experimental results. Similar re‐ sults have been observed in earlier studies [29, 46-47]. These deviations are usually attribut‐ ed to underestimation of the turbulent quantities associated to the *k–ε* model [46, 48]. The power number predictions of integral power based approach in the momentum source term approach with standard *k–ε* model and RSM are 5.72 and 5.64, respectively, both between

Fig.10 shows a log-log plot of the experimentally obtained and predicted power numbers for nine kinds of Reynolds numbers which cover laminar and turbulent flow regimes. It can be ob‐ served that at lower Reynolds numbers the power numbers predicted by momentum source

term approach are in good agreement with the experimental data of Rushton et al. [44].

the experimental data 5.1 and 6.07, so they are better than those of SM.

bulence, so the circulation pattern predictions can be considered properly.

source term approach prediction is in good agreement with experimental results.

**Figure 9.** Velocity vectors in the middle plane near impeller tip.

**3.4. Comparison of power numbers**

From the comparisons above, it can be found that predictions of momentum source term ap‐ proach with RSM turbulent model are in good agreement with the experimental data as well as SM model, although both SM and our model have some deviations in prediction of the upper part for the radial and tangential velocity. The models combined with *k-ε* turbulent model have less prediction accuracy, Murthy and Joshi [22] attributed these numerical er‐ rors to the overestimation of the eddy viscosity of the *k-ε* model. It is suggested that the standard *k−ε* model performs well when the flow is unidirectional that is with less swirl and weak recirculation [11, 22].

#### **3.3. Velocity vectors**

Fig.8 shows velocity vectors of the vertical plane in the middle of two baffles. It can be seen that after exerting on the momentum source, fluid velocity in the impeller region is very high, and flow gradients are large.

**Figure 8.** Velocity vectors of stirred tank in the middle plane.

As the high speed fluid jets outward, initially almost not affected by the surrounding fluid, the velocity contours are dense. The flow impinges on the tank wall, splits up into two parts and changes the direction. The split water flows at last return to impeller region and accelerated again, repeating above-mentioned process which generates two circulation loops of different directions in the upper and lower part of the tank, respectively. It can be observed that the cir‐ culation loop ranges above the *z*/*T* value of 0.9, which is consistent with the simulations by Ng et al. [24] (1998) (*Re* = 40,000) and Yapici et al. [41] (2008) (*Re* = 60,236). Reynolds numbers of previous and present studies can ensure that the flow in a stirred vessel is fully developed tur‐ bulence, so the circulation pattern predictions can be considered properly.

Fig.9 shows the flow field of stirred tank near the impeller tip. It can be noticed that velocity distributions are not symmetrical about the impeller centre plate (*z* = 0.1 m), but shift up‐ wards slightly. This result is in agreement with the previous experimental works, that the impeller is not symmetrically located, the top of the tank is a free surface, and the hub is present on the top side of the impeller [38, 42-43]. In this study, this detail has been success‐ fully captured in the velocity field near impeller tip, further indicating that momentum source term approach prediction is in good agreement with experimental results.

**Figure 9.** Velocity vectors in the middle plane near impeller tip.

#### **3.4. Comparison of power numbers**

are similar. Due to unsteady and complex nature of flow characteristics in the impeller re‐ gion and the continuous conversion of kinetic energy [37], the *k-ε* model and RSM both fail to capture the transfer details of kinetic energy, so the momentum source term approach and SM model underpredict the turbulent kinetic energy with similar deviations when the

From the comparisons above, it can be found that predictions of momentum source term ap‐ proach with RSM turbulent model are in good agreement with the experimental data as well as SM model, although both SM and our model have some deviations in prediction of the upper part for the radial and tangential velocity. The models combined with *k-ε* turbulent model have less prediction accuracy, Murthy and Joshi [22] attributed these numerical er‐ rors to the overestimation of the eddy viscosity of the *k-ε* model. It is suggested that the standard *k−ε* model performs well when the flow is unidirectional that is with less swirl and

Fig.8 shows velocity vectors of the vertical plane in the middle of two baffles. It can be seen that after exerting on the momentum source, fluid velocity in the impeller region is very

As the high speed fluid jets outward, initially almost not affected by the surrounding fluid, the velocity contours are dense. The flow impinges on the tank wall, splits up into two parts and changes the direction. The split water flows at last return to impeller region and accelerated

*k-ε* and RSM turbulent model are applied.

146 Nuclear Reactor Thermal Hydraulics and Other Applications

weak recirculation [11, 22].

high, and flow gradients are large.

**Figure 8.** Velocity vectors of stirred tank in the middle plane.

**3.3. Velocity vectors**

Table 2 shows the comparison of predicted power numbers with the experimental value. Here, the experimental power numbers reported by Rushton et al. [44-45], Murthy and Joshi [22] are applied in the reference. The quantities obtained by SM model through integral *ε* based approach are considerably lower than the reported experimental results. Similar re‐ sults have been observed in earlier studies [29, 46-47]. These deviations are usually attribut‐ ed to underestimation of the turbulent quantities associated to the *k–ε* model [46, 48]. The power number predictions of integral power based approach in the momentum source term approach with standard *k–ε* model and RSM are 5.72 and 5.64, respectively, both between the experimental data 5.1 and 6.07, so they are better than those of SM.

Fig.10 shows a log-log plot of the experimentally obtained and predicted power numbers for nine kinds of Reynolds numbers which cover laminar and turbulent flow regimes. It can be ob‐ served that at lower Reynolds numbers the power numbers predicted by momentum source term approach are in good agreement with the experimental data of Rushton et al. [44].


**4. Conclusions**

(Wang et al., 2002) as following:

proach is the least of the three.

gy of China for their help in computing.

and Kun Li

\*Address all correspondence to: huangwd@ustc.edu.cn

**Acknowledgements**

**Author details**

Weidong Huang\*

nology of China, P. R. China

An equation to predict the momentum source term is proposed without the help of experimen‐ tal data in the paper. So the momentum source term approach for CFD prediction of the impel‐ ler propelling action has been developed as a tool with predictive capacity. The prediction results of the approach have been compared with the experimental data, MRF and SM model predictions in the literatures. The following conclusions can be drawn from the present work:

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149

**1.** For the plate blade of the Rushton turbine stirred vessel, the tangential momentum

in which *u* is the linear velocity of the area element on the blade surface d*S*; *v* θ is the flu‐ id tangential velocity rotating with the impeller before the fluid was propelled by impel‐ ler; d*S* is cross-section area of the interface between fluid and the impeller blade. The radial friction force of the impeller blade is proposed to be calculated approximately

In which *f* is the friction force in the computational cells, *v <sup>r</sup>* is the radial velocity of the fluid. **2.** The numerical comparisons of flow field show that the momentum source term ap‐ proach predictions are in good agreement with the experimental data. It has been also found that the prediction accuracy of momentum source term approach is better than MRF and similar to SM, whereas the computational time of momentum source term ap‐

This work is supported by the Ministry of Science and Technology of the People's Republic of China (grants No. 2006BAC19B02). The authors would like to thank professor Taohong Ye for helpful discussion, the supercomputing center of University of Science and Technolo‐

Department of Geochemistry and Environmental Science, University of Science and Tech‐

source added by blade is proposed to be calculated by:

**Table 2.** Power number predictions for different approaches at *Re*=41,467.

**Figure 10.** Comparison of experimental and predicted power numbers.

#### **3.5. Computational speed**

The simulations were carried out on a 100 node AMD64 cluster, each node including 2 fourcore processors with 2.26 GHz clock speed and 2 GB memory. 80 processors were applied in all the computations. Present study compared the computational speeds of momentum source term approach, MRF and SM, and the required expenses are shown in Table 3. It can be seen that momentum source term approach and MRF using steady state simulations need less computational requirements than SM, and the computational time of momentum source term approach is the least.


**Table 3.** Comparison of the computational time required by different approaches, h.

## **4. Conclusions**

**Methods** *N* **<sup>P</sup> Error/%** Experimental value (Rushton et al., 1950) 6.07 / Experimental value (Murthy and Joshi, 2008) 5.1 / SM torque based approach (Murthy and Joshi, 2008) 4.9 12.26 SM integral ε based approach (Murthy and Joshi, 2008) 3.9 30.17 MRF torque based approach (Deglon and Meyer, 2006) 5.40 3.31 Standard k-ε integral power based approach 5.72 2.42 RSM integral power based approach 5.64 0.98

**Table 2.** Power number predictions for different approaches at *Re*=41,467.

148 Nuclear Reactor Thermal Hydraulics and Other Applications

**Figure 10.** Comparison of experimental and predicted power numbers.

The simulations were carried out on a 100 node AMD64 cluster, each node including 2 fourcore processors with 2.26 GHz clock speed and 2 GB memory. 80 processors were applied in all the computations. Present study compared the computational speeds of momentum source term approach, MRF and SM, and the required expenses are shown in Table 3. It can be seen that momentum source term approach and MRF using steady state simulations need less computational requirements than SM, and the computational time of momentum source

**Approach Momentum source term MRF SM** Standard k-ε 48 72 340 RSM 60 86 410

**Table 3.** Comparison of the computational time required by different approaches, h.

**3.5. Computational speed**

term approach is the least.

An equation to predict the momentum source term is proposed without the help of experimen‐ tal data in the paper. So the momentum source term approach for CFD prediction of the impel‐ ler propelling action has been developed as a tool with predictive capacity. The prediction results of the approach have been compared with the experimental data, MRF and SM model predictions in the literatures. The following conclusions can be drawn from the present work:

**1.** For the plate blade of the Rushton turbine stirred vessel, the tangential momentum source added by blade is proposed to be calculated by:

in which *u* is the linear velocity of the area element on the blade surface d*S*; *v* θ is the flu‐ id tangential velocity rotating with the impeller before the fluid was propelled by impel‐ ler; d*S* is cross-section area of the interface between fluid and the impeller blade. The radial friction force of the impeller blade is proposed to be calculated approximately (Wang et al., 2002) as following:

In which *f* is the friction force in the computational cells, *v <sup>r</sup>* is the radial velocity of the fluid.

**2.** The numerical comparisons of flow field show that the momentum source term ap‐ proach predictions are in good agreement with the experimental data. It has been also found that the prediction accuracy of momentum source term approach is better than MRF and similar to SM, whereas the computational time of momentum source term ap‐ proach is the least of the three.

## **Acknowledgements**

This work is supported by the Ministry of Science and Technology of the People's Republic of China (grants No. 2006BAC19B02). The authors would like to thank professor Taohong Ye for helpful discussion, the supercomputing center of University of Science and Technolo‐ gy of China for their help in computing.

## **Author details**

Weidong Huang\* and Kun Li

\*Address all correspondence to: huangwd@ustc.edu.cn

Department of Geochemistry and Environmental Science, University of Science and Tech‐ nology of China, P. R. China

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**Chapter 6**

**Hydrodynamic and Heat Transfer Simulation of**

The nuclear energy is suffering from the lack of public acceptance everywhere mainly due to the issues relating to reactor safety, economy and nuclear waste. The Fluidized Bed Nuclear Reactor (FBNR) concept has addressed these issues and tried to resolve such problems. The FBNR is small, modular and simple in design contributing to the economy of the reactor. It has inherent safety and passive cooling characteristics. Its spent fuel being small spherical elements may not be considered nuclear waste, and can be directly used as a source of radiation for applications in industry and agriculture resulting in reduced environmental impact [1].

With the increase of computational power, numerical simulation becomes an additional tool to predict the fluid dynamics and the heat transfer mechanism in multiphase flow. A numerical hydrodynamic and heat transfer model has been developed to simulate the gas fluidized bed. All of CFD, in one form or another, is based on the fundamental governing equations of fluid dynamics (continuity, momentum and energy equations). These equa‐ tions speak physics. They are mathematical statements of three fundamental physical principals upon which all fluid dynamics is based: mass is conserved, Newton's second law

This chapter aims to study a mathematical modeling and numerical simulation of the hydro‐ dynamics and heat transfer processes in a two-dimensional gas fluidized bed with a vertical uniform gas velocity at the inlet. The velocity, volume fraction, temperature distribution for gas phase and particle phase are calculated. Also, gas pressure and a prediction of the average

Such a simulation technique allows performance evaluation for different bed input parame‐ ters, and can evolve into a tool for optimized design of fluidized beds for different industrial use.

> © 2013 Abd El Kawi Ali; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Abd El Kawi Ali; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Fluidized Bed Using CFD**

Osama Sayed Abd El Kawi Ali

http://dx.doi.org/10.5772/52072

and energy is conserved [2].

heat transfer coefficient are also studied.

**1. Introduction**

Additional information is available at the end of the chapter
