**3. Data analysis**

For a given point, the instantaneous velocity components uij in terms of a mean velocity ui and a temporal fluctuating term u<sup>0</sup> ij is given as

$$\mathbf{u}\_{\rm ij} = \overline{\mathbf{u}}\_{\rm i} + \mathbf{u}'\_{\rm ij} \tag{1}$$

where i refers to the velocity component r, z, or θ and j refers to the instant at which the velocity was measured for each acquisition.

The mean velocity is calculated as follows:

$$\overline{\mathbf{u}}\_{\text{i}} = \frac{\mathbf{1}}{\mathbf{N}\mathbf{n}} \sum\_{\text{j=1}}^{\text{Nn}} \mathbf{u}\_{\text{ij}} \tag{2}$$

With Nn refers to the snapshot total number. The average of the temporal fluctuating terms which known as a root mean square (rms) value is given by

$$\mathbf{u}\_{\mathbf{i},\mathbf{r}\mathbf{s}\mathbf{s}} = \left(\overline{\mathbf{u}'^2\mathbf{i}}\right)^{1/2} = \left(\frac{\mathbf{1}}{\mathbf{N}\mathbf{n}}\sum\_{j=1}^{\mathbf{N}\mathbf{n}} \left(\mathbf{u}\_{\mathbf{i}\mathbf{j}}\overline{\mathbf{u}}\_{\mathbf{i}}\right)^2\right)^{1/2} \tag{3}$$

For two-dimensional velocity data, in the Cartesian coordinate, the turbulent kinetic energy is equal to

$$\mathbf{k} = \frac{\mathbf{3}}{4} \left( \overline{\mathbf{u}'}^2 + \overline{\mathbf{v}'}^2 \right) \tag{4}$$

The proper orthogonal decomposition (POD) is obtained by computing the auto-covariance matrix (R). According to Liné et al. [14], the matrix of instantaneous velocity vector data is calculated as

$$\mathbf{M} = \begin{bmatrix} \mathbf{u}\_1(1) & \dots & \mathbf{u}\_{\text{th}}(1) \\ \vdots & & \ddots \\ \vdots & & \ddots \\ \mathbf{u}\_1(n) & \dots & \mathbf{u}\_{\text{th}}(n) \\ \mathbf{v}\_1(1) & \dots & \mathbf{v}\_{\text{th}}(1) \\ \vdots & & \ddots \\ \vdots & & \ddots \\ \mathbf{v}\_1(n) & \dots & \mathbf{v}\_{\text{th}}(n) \end{bmatrix} \tag{5}$$

Nn refers to the number of snapshots, and n refers to total number of the interrogation area of the whole flow field. The auto-covariance matrix, which is associated to the fluctuating velocity components for each snapshot, is calculated as follows:

$$\mathbf{R} = \frac{1}{\text{N } \mathbf{n}} \mathbf{M} \text{ } \mathbf{M}^{\text{\textquotedblleft}r} \tag{6}$$

described at the blade tip. In fact, the velocity has some deviation in the other azimuthal planes by the effect of the propagation of trailing vortices. The jet flow is more intensive for the convex blade turbine. For the concave blade, a maximum velocity is created at the top edge that deviated to the blade tip. Afterward, near the wall of the vessel, the velocity is divided into the upward and downward flow. Then, it returns to the shaft to create the circulation loop. As a matter of fact, the concave configuration produces a larger lowest loop than the other configurations. Hence, the region located below the impeller (at the bottom of the tank) is more turbulent than the flat and the convex configurations. Moreover, the distribution of the turbulent flow at the upper and the downer regions is more similar at the

*The Effects of Curved Blade Turbine on the Hydrodynamic Structure of a Stirred Tank*

*DOI: http://dx.doi.org/10.5772/intechopen.92394*

The flow distribution of the flat and the curved blade is similar to that presented by Driss et al. [15] for a laminar flow. In fact, a radial jet is created, and then an axial flow is obtained by the wall effect. In addition, the flow slows down significantly far away from the impeller. The shape and the position of the recirculation loops and

the trailing vortices are affected significantly by the blade design [15, 16].

The coordination of the center of the highest loops (zh\*, rh\*) and the lowest loops (zb\*, rb\*) and its radial and the axial extension (zb, rb, zh, rh) are presented in

Highest loops zh\* (z/H) 0.78 zh\* (z/H) 0.78 zh\* (z/H) 0.78

Lowest loops zb\* (z/H) 0.32 zb\* (z/H) 0.25 zb\* (z/H) 0.27

**Flat impeller Concave impeller Convex impeller**

rh\* (r/R) 0.76 rh\* (r/R) 0.81 rh\* (r/R) 0.76 zh (z/H) 0.32 zh (z/H) 0.21 zh (z/H) 0.32 rh (r/R) 0.42 rh (r/R) 0.33 rh (r/R) 0.4

rb\* (r/R) 0.72 rb\* (r/R) 0.75 rb\* (r/R) 0.8 zb (z/H) 0.22 zb (z/H) 0.37 zb (z/H) 0.33 rb (r/R) 0.4 rb (r/R) 0.3 rb (r/R) 0.3

convex configuration.

*Velocity field for one-staged system.*

**Table 1**.

**Table 1.**

**167**

*Loop coordination and shape of the one-staged system.*

**Figure 3.**

Eigenvalues are calculated by solving the Fredholm integral eigenvalue problem and Karhunen-Loeve analysis:

$$\iint\_{\Omega} \mathbf{R}(\varkappa, z, \varkappa', z') \overrightarrow{\phi^{K\overleftarrow{\text{li}}}}(\varkappa', z') d\varkappa' dz' = \lambda^{ki} \overrightarrow{\phi^{K\overleftarrow{\text{li}}}}(\varkappa, z) \tag{7}$$

where *Ki* refers to the POD mode and Ω refers to the domain of interest. The eigenfunction is calculated as follows:

$$\phi = \frac{\sum\_{\mathbf{n}^{\text{in}}}^{\text{N}\cdot\text{n}} \mathbf{a}\_{\mathbf{n}^{\text{in}}}^{\text{K}\cdot\text{n}} \mathbf{u}^{\text{in}}}{\left\| \sum\_{\mathbf{n}^{\text{in}}}^{\text{N}\cdot\text{n}} \mathbf{a}\_{\mathbf{n}^{\text{in}}}^{\text{K}\cdot\text{n}} \mathbf{u}^{\text{in}} \right\|} \tag{8}$$

## **4. Experimental results**

#### **4.1 Velocity field**

**Figure 3** shows the velocity field of different types of curved blade turbines for one-staged system. According to these results, two circulation loops were observed, in which the first one is localized in the upper region of the tank near the free surface and the second one is localized in the bottom of the tank. A radial jet is

*The Effects of Curved Blade Turbine on the Hydrodynamic Structure of a Stirred Tank DOI: http://dx.doi.org/10.5772/intechopen.92394*

#### **Figure 3.**

For two-dimensional velocity data, in the Cartesian coordinate, the turbulent

The proper orthogonal decomposition (POD) is obtained by computing the auto-covariance matrix (R). According to Liné et al. [14], the matrix of instanta-

Nn refers to the number of snapshots, and n refers to total number of the interrogation area of the whole flow field. The auto-covariance matrix, which is associated to the fluctuating velocity components for each snapshot, is calculated as

Eigenvalues are calculated by solving the Fredholm integral eigenvalue problem

, *z*<sup>0</sup> ð Þ*dx*<sup>0</sup>

**Figure 3** shows the velocity field of different types of curved blade turbines for one-staged system. According to these results, two circulation loops were observed, in which the first one is localized in the upper region of the tank near the free surface and the second one is localized in the bottom of the tank. A radial jet is

where *Ki* refers to the POD mode and Ω refers to the domain of interest.

(4)

ð5Þ

ð6Þ

ð8Þ

*dz*<sup>0</sup> <sup>¼</sup> *<sup>λ</sup>kiϕKi* �!ð Þ *<sup>x</sup>*, *<sup>z</sup>* (7)

<sup>k</sup> <sup>¼</sup> <sup>3</sup> <sup>4</sup> <sup>u</sup><sup>0</sup> 2 þ v<sup>0</sup> 2 � �

kinetic energy is equal to

follows:

and Karhunen-Loeve analysis:

**4. Experimental results**

**4.1 Velocity field**

**166**

ðð

Ω

R *x*, *z*, *x*<sup>0</sup>

The eigenfunction is calculated as follows:

, *<sup>z</sup>*<sup>0</sup> ð Þ*ϕKi* �! *<sup>x</sup>*<sup>0</sup>

neous velocity vector data is calculated as

*Vortex Dynamics Theories and Applications*

*Velocity field for one-staged system.*

described at the blade tip. In fact, the velocity has some deviation in the other azimuthal planes by the effect of the propagation of trailing vortices. The jet flow is more intensive for the convex blade turbine. For the concave blade, a maximum velocity is created at the top edge that deviated to the blade tip. Afterward, near the wall of the vessel, the velocity is divided into the upward and downward flow. Then, it returns to the shaft to create the circulation loop. As a matter of fact, the concave configuration produces a larger lowest loop than the other configurations. Hence, the region located below the impeller (at the bottom of the tank) is more turbulent than the flat and the convex configurations. Moreover, the distribution of the turbulent flow at the upper and the downer regions is more similar at the convex configuration.

The flow distribution of the flat and the curved blade is similar to that presented by Driss et al. [15] for a laminar flow. In fact, a radial jet is created, and then an axial flow is obtained by the wall effect. In addition, the flow slows down significantly far away from the impeller. The shape and the position of the recirculation loops and the trailing vortices are affected significantly by the blade design [15, 16].

The coordination of the center of the highest loops (zh\*, rh\*) and the lowest loops (zb\*, rb\*) and its radial and the axial extension (zb, rb, zh, rh) are presented in **Table 1**.


#### **Table 1.**

*Loop coordination and shape of the one-staged system.*

**Figure 4.** *Velocity field for staged system.*


end of the turbine. The bulk region of the tank is described with the lowest value. The flat blade turbine generates a larger radial velocity. Hence, the area of the maximum radial velocity is larger than the other configurations. The maximum radial velocity is spread to reach places farther than the blade, which can be explained by the ability of the blade shape to generate training vortices. The development of the radial velocity component for the concave blade configuration is closer to the axial turbine. In fact, the maximum velocity area is localized at the top edge of the blade and spreads to the same direction as the Von Karman vortex street. The highest value of the radial velocity component is generated by the convex blade configuration that is equal to umax = 0.35 Utip, whereas it is equal to umax = 0.15 Utip for the other configurations. However, the maximum value remained closer to the blade tip. Therefore, it can be seen that the convex shape of the blade gives the ability to the turbine to move easily within the water and transmit more velocity while not giving it enough capacity to expand much. For the staged system (**Figure 6**), the maximum value of the radial velocity component is localized between the two blades. This explains the oblique direction of the velocity field at this region. The largest maximum area is defined at the association of the concave and the flat turbines that confirm that the maximum value cannot spread with the convex shape. In addition, the development of the trailing vortices is not as great as the use of the flat and the concave shapes. However, the maximum value of the radial velocity component is found for the second configuration which represents the association of the flat blade at the top and the convex blade at the bottom (umax = 0.31 Utip). The lowest value is found by using the concave blade instead of the convex blade, due to the high strain created by the interaction between the blade and the flow (umax = 0.22 Utip). It can be seen that the blade at the bottom of the tank has the greatest effect on the flow. In fact, the maximum value of the radial velocity is almost similar while we use the flat

*The Effects of Curved Blade Turbine on the Hydrodynamic Structure of a Stirred Tank*

**Figure 7** shows the distribution of the axial velocity component of the curved blade turbine. According to these results, the highest value region is localized at the bottom of the tank close to the blade that represents the suction of the flow of the blade. The second one is localized besides the wall of the tank above the blade at the same direction with the recirculation loops. Then, two lowest value regions were presented. In fact, the largest one is localized close to the free surface, while the

blade turbine at the bottom (umax = 0.27 Utip).

**Figure 5.**

**169**

*Radial velocity for one-staged system.*

*DOI: http://dx.doi.org/10.5772/intechopen.92394*

**Table 2.**

*Loop coordination and shape of the staged system.*

In addition, the maximum velocity is greater for the convex blade turbine and weaker for the concave blade turbine due to the interaction between the blade and the flow that is lower for the convex blade and greater for the concave blade (flat turbine Umax = 0.22 Utip, concave turbine Umax = 0.18 Utip, convex turbine Umax = 0.35 Utip).

For the staged system (**Figure 4**), the flow becomes more turbulent, and the loops can reach the free surface as the bottom of the tank. In addition, an oblique flow is created between the two impellers. The highest velocity is produced in the case of the second configuration (Umax = 40% Utip) (a, Umax = 34.54% Utip; c, Umax = 32,72% Utip; d, Umax = 32,72% Utip). The position and the shape of the recirculation loops of each configuration are presented at **Table 2**.

The combination of inclined blade turbine and flat turbine shows no great change in terms of acceleration and shape of the recirculation loops. This found locks similar to that developed by Bereksi et al. [17] for the combination between the Rushton and the curved blade. In addition, they proved that the gas holdup is better by the combination between the curved and the Rushton turbine than by the combination between two Rushton turbines.

#### **4.2 Radial and axial velocity**

**Figure 5** shows the distribution of the radial velocity component of the curved blade turbine. According to these results, the highest value region is localized at the *The Effects of Curved Blade Turbine on the Hydrodynamic Structure of a Stirred Tank DOI: http://dx.doi.org/10.5772/intechopen.92394*

**Figure 5.** *Radial velocity for one-staged system.*

end of the turbine. The bulk region of the tank is described with the lowest value. The flat blade turbine generates a larger radial velocity. Hence, the area of the maximum radial velocity is larger than the other configurations. The maximum radial velocity is spread to reach places farther than the blade, which can be explained by the ability of the blade shape to generate training vortices. The development of the radial velocity component for the concave blade configuration is closer to the axial turbine. In fact, the maximum velocity area is localized at the top edge of the blade and spreads to the same direction as the Von Karman vortex street. The highest value of the radial velocity component is generated by the convex blade configuration that is equal to umax = 0.35 Utip, whereas it is equal to umax = 0.15 Utip for the other configurations. However, the maximum value remained closer to the blade tip. Therefore, it can be seen that the convex shape of the blade gives the ability to the turbine to move easily within the water and transmit more velocity while not giving it enough capacity to expand much.

For the staged system (**Figure 6**), the maximum value of the radial velocity component is localized between the two blades. This explains the oblique direction of the velocity field at this region. The largest maximum area is defined at the association of the concave and the flat turbines that confirm that the maximum value cannot spread with the convex shape. In addition, the development of the trailing vortices is not as great as the use of the flat and the concave shapes. However, the maximum value of the radial velocity component is found for the second configuration which represents the association of the flat blade at the top and the convex blade at the bottom (umax = 0.31 Utip). The lowest value is found by using the concave blade instead of the convex blade, due to the high strain created by the interaction between the blade and the flow (umax = 0.22 Utip). It can be seen that the blade at the bottom of the tank has the greatest effect on the flow. In fact, the maximum value of the radial velocity is almost similar while we use the flat blade turbine at the bottom (umax = 0.27 Utip).

**Figure 7** shows the distribution of the axial velocity component of the curved blade turbine. According to these results, the highest value region is localized at the bottom of the tank close to the blade that represents the suction of the flow of the blade. The second one is localized besides the wall of the tank above the blade at the same direction with the recirculation loops. Then, two lowest value regions were presented. In fact, the largest one is localized close to the free surface, while the

In addition, the maximum velocity is greater for the convex blade turbine and weaker for the concave blade turbine due to the interaction between the blade and the flow that is lower for the convex blade and greater for the concave blade (flat turbine Umax = 0.22 Utip, concave turbine Umax = 0.18 Utip, convex turbine

**PD8 h, PI8 concave PD8 h, PI8 convex PD8 b, PI8 concave PD8 b, PI8 convex**

rh\* (r/R) 0.7 rh\* (r/R) 0.71 rh\* (r/R) 0.7 rh\* (r/R) 0.65 zh (z/H) 0.29 zh (z/H) 0.23 zh (z/H) 0.26 zh (z/H) 0.26 rh (r/R) 0.46 rh (r/R) 0.44 rh (r/R) 0.44 rh (r/R) 0.46

rb\* (r/R) 0.74 rb\* (r/R) 0.67 rb\* (r/R) 0.75 rb\* (r/R) 0.7 zb (z/H) 0.33 zb (z/H) 0.32 zb (z/H) 0.24 zb (z/H) 0.36 rb (r/R) 0.46 rb (r/R) 0.46 rb (r/R) 0.43 rb (r/R) 0.47

Highest loops zh\* (z/H) 0.78 zh\* (z/H) 0.76 zh\* (z/H) 0.74 zh\* (z/H) 0.82

Lowest loop zb\* (z/H) 0.21 zb\* (z/H) 0.25 zb\* (z/H) 0.19 zb\* (z/H) 0.26

For the staged system (**Figure 4**), the flow becomes more turbulent, and the loops can reach the free surface as the bottom of the tank. In addition, an oblique flow is created between the two impellers. The highest velocity is produced in the case of the second configuration (Umax = 40% Utip) (a, Umax = 34.54% Utip; c, Umax = 32,72% Utip; d, Umax = 32,72% Utip). The position and the shape of the

The combination of inclined blade turbine and flat turbine shows no great change in terms of acceleration and shape of the recirculation loops. This found locks similar to that developed by Bereksi et al. [17] for the combination between the Rushton and the curved blade. In addition, they proved that the gas holdup is better by the combination between the curved and the Rushton turbine than by the

**Figure 5** shows the distribution of the radial velocity component of the curved blade turbine. According to these results, the highest value region is localized at the

recirculation loops of each configuration are presented at **Table 2**.

combination between two Rushton turbines.

*Loop coordination and shape of the staged system.*

**4.2 Radial and axial velocity**

Umax = 0.35 Utip).

**Figure 4.**

**Table 2.**

**168**

*Velocity field for staged system.*

*Vortex Dynamics Theories and Applications*

**Figure 6.** *Radial velocity for staged system.*

**4.3 Rms of velocity fields**

*Axial velocity for staged system.*

**Figure 8.**

**Figure 9.**

**171**

*Rms velocity for the one-staged system.*

**Figure 9** shows the root mean square of the velocity field of the curved blade turbine. The root mean velocity presents the fluctuation of the periodic velocity. According to these results, the highest value region is localized at the same direction of the trailing vortices. In fact, for the convex configuration, the fluctuated velocity is localized close to the blade tip. For the flat blade turbine, the turbulent fluctuation is propagated to the vessel wall. For the concave blade, the fluctuation is localized at the upper and the downer edges of the blade. The maximum variability of the flow

*The Effects of Curved Blade Turbine on the Hydrodynamic Structure of a Stirred Tank*

*DOI: http://dx.doi.org/10.5772/intechopen.92394*

For the staged system (**Figure 10**), the greatest fluctuation is localized at the blade that is placed at the bottom of the tank. These fluctuations can be explained by the suction of the flow from the bottom of the tank. In addition, the maximum values of the turbulent fluctuations are created between the two blades due to the interaction between the blades. The fluctuations generated due to the association of the convex impeller with the flat impeller are narrowed when they are compared to the association of the concave and the flat impeller. This effect reveals that the

occurs due to the trailing vortices following the recirculation loops.

**Figure 7.** *Axial velocity for one-staged system.*

narrowed one is localized besides the wall of the tank at the bottom. The development of the maximum value of the axial velocity component is larger with the convex blade, which confirms that the recirculation loops associated to the convex blade are more important according to the other configurations. According to the velocity field, it can be seen that the recirculation loops associated to the convex blade are larger than the other configurations that are in conjunction with the amelioration of the axial velocity component with the convex blade. The highest maximum axial velocity component is generated by the flat turbine by vmax = 0.22 Utip followed by the concave blade by vmax = 0.16 Utip and by the convex blade by vmax = 0.15 Utip.

For the staged system (**Figure 8**), the maximum value of the axial velocity component is localized at the lowest blade that represents the suction of the flow. The highest maximum value is produced by the second configuration by vmax = 0.4 Utip. For the first configuration, the highest value is equal to vmax = 0.29 Utip. In addition, the maximum value of the axial velocity is almost similar while we use the flat blade turbine at the bottom (vmax = 0.33 Utip).

*The Effects of Curved Blade Turbine on the Hydrodynamic Structure of a Stirred Tank DOI: http://dx.doi.org/10.5772/intechopen.92394*

**Figure 8.** *Axial velocity for staged system.*
