4. Optimization techniques

In this chapter, the geometric parameters related to the internal flow through the crosssectional area of the impeller and volute are selected as design variables to simultaneously optimize the hydraulic efficiency and radial force sources, considering the interaction between the rotating impeller and volute of the single-channel pump. The distribution of internal flow in the cross-sectional area of the impeller and volute can be changed smoothly by adjusting the control points represented by third-order and fourth-order Bezier-curves, respectively, as shown in Figure 5. Therefore, the variations in the y-axes for five control points (CP1, CP2, CP3, CP4, and CP5) of both the impeller and volute are selected as design variables to obtain the most sensitive results for the variation in curve among the control points [17].

The aim of the current optimization problem is to simultaneously improve the hydraulic efficiency (η) and reduce the radial force sources considering the impeller-volute interaction in the single-channel pump. Here, one of the three objective functions, that is, the hydraulic efficiency, is defined as follows.

$$
\eta = \frac{\rho g H \mathcal{Q}}{\tau \omega} \tag{18}
$$

where r, g, H, Q, τ, and ω denote the density, acceleration of gravity, total head, volume flow arte, torque, and angular velocity, respectively.

Figure 5. Definition of the design variables. (a) Impeller part (b) Volute part.

adiabatic and no-slip conditions. The stage average and transient-rotor-stator methods are respectively applied to connect the interface between the rotating impeller and volute domains

The convergence criteria in a steady computation consist of the root-mean-square (RMS) values of the residuals of the governing equations, which are set to less than 10<sup>5</sup> for all equations. The physical time scale was set to 1/ω, where ω is the angular velocity of the impeller. The computations are carried out using an Intel Xeon CPU with a clock speed of 2.70 GHz, and the converged solutions are obtained after 1000 iterations with a computational

The results of the steady RANS analysis are used in the unsteady RANS analysis to obtain the characteristics of the radial force sources in the region of the exit surface of the impeller according to the impeller-volute interaction in the single-channel pump. In an unsteady simulation, the time step and coefficient loop for the time scale control are set to 0.000947 s and three times, respectively. The solutions are obtained after 180 iterations with an unsteady total time duration of 0.170478 s (five revolutions), and the computational time for the unsteady

In this chapter, the geometric parameters related to the internal flow through the crosssectional area of the impeller and volute are selected as design variables to simultaneously optimize the hydraulic efficiency and radial force sources, considering the interaction between the rotating impeller and volute of the single-channel pump. The distribution of internal flow in the cross-sectional area of the impeller and volute can be changed smoothly by adjusting the control points represented by third-order and fourth-order Bezier-curves, respectively, as shown in Figure 5. Therefore, the variations in the y-axes for five control points (CP1, CP2,

in the steady and unsteady analyses.

Figure 4. Computational domain and grids.

188 Wastewater and Water Quality

time of approximately 4 h.

calculation was approximately 8 h.

4. Optimization techniques

Figure 6. Definition of objective functions related to the radial force sources [10].

The other objective functions related to the radial force sources are defined as the sweep area (As) of the radial force during one revolution of impeller and the distance (Ds) of the mass center of the sweep area from the origin, as shown in Figure 6. These functions are defined as follows:

$$A\_s = \frac{1}{2} \sum\_{i=0}^{n-1} \left( x\_i y\_{i+1} - x\_{i+1} y\_i \right) \tag{19}$$

where As is the signed area of the polygon as the sweep area of the radial force during one revolution of impeller. The centroid of a non-self-intersecting closed polygon, defined by n vertices (x0, y0), (x1, y1), …, (xn-1, yn-1), is defined as the point (Cx, Cy) as follows:

$$\mathbf{C}\_{\mathbf{x}} = \frac{1}{6A\_s} \sum\_{i=0}^{n-1} \left( \mathbf{x}\_i + \mathbf{x}\_{i+1} \right) \left( \mathbf{x}\_i y\_{i+1} - \mathbf{x}\_{i+1} y\_i \right) \tag{20}$$

The RSA models are employed to construct the response surfaces based on the objective function values at the 54 design points generated in the design space using LHS. A hybrid multiobjective genetic algorithm (MOGA) is used to obtain the global Pareto-optimal solutions (POSs). The approximate POSs are obtained using a controlled elitist genetic algorithm (a variant of NSGA-II [20] as the MOGA function for three objective functions. The optimization algorithm and functions in the MATLAB OPTIMIZATION TOOLBOX [21] are used to finally generate the global POSs. Figure 7 shows an example of the multiobjective optimization procedure [22]. The detailed optimization procedure can be referred to in the previous litera-

State-of-the-Art Design Technique of a Single-Channel Pump for Wastewater Treatment

http://dx.doi.org/10.5772/intechopen.75171

191

A hybrid MOGA based on the response surface constructed from the RSA model is employed to obtain the global POSs by using a controlled elitist genetic algorithm (a variant of NSGA-II) for three objective functions. Figure 8 shows the three-dimensional POSs based on the three objective functions obtained using a hybrid MOGA combined with the RSA model. Here, the values of all the objective function are normalized according to the corresponding values in the reference design. Three-dimensional POSs are obviously the trade-off among the conflicting objective functions. As a result, a trade-off analysis shows an obvious correlation between the hydraulic efficiency and radial force sources. The arbitrary optimum design (AOD) is randomly extracted near the end of the POSs, which exhibits the best performance in terms of all objective functions, as shown in Figure 8. The AOD has objective function values that are remarkably improved relative to those in the reference design. Consequently, the value of each objective function in the AOD shows improvements of approximately 49%, 80%, and 4% in the sweep area (As) of the radial force during one revolution, the distance (Ds) of the mass center of the sweep, and the hydraulic efficiency (η), respectively, in comparison with the reference design. On the other hand, a relatively large error among the three objective functions is observed, especially for the distance of the mass center of the sweep. Nevertheless, the values

obtained by the numerical analysis are better compared with the reference design.

To understand the optimization results, the trade-off of the POSs in each two-dimensional functional space is shown in Figure 9. As shown in Figure 9(a) and (b), the decrement in the distance of the mass center of the sweep clearly leads to deterioration in the other objective

tures [23, 24].

5. Results of multiobjective optimization

Figure 7. Multiobjective optimization procedure [22].

$$\mathbf{C}\_{y} = \frac{1}{6A\_{s}} \sum\_{i=0}^{n-1} \left( y\_{i} + y\_{i+1} \right) \left( \mathbf{x}\_{i} y\_{i+1} - \mathbf{x}\_{i+1} y\_{i} \right) \tag{21}$$

In these formulas, the vertices are assumed to be numbered in the order of their occurrence along the perimeter of the polygon. Therefore, the distance of the mass center of the sweep area from the origin is finally defined as follows:

$$D\_s = \sqrt{\mathcal{C}\_x^2 + \mathcal{C}\_y^2} \tag{22}$$

The Latin hypercube sampling (LHS) is employed to generate 54 design points that are used as the initial base data for constructing the response surface from five design variables. LHS, as an effective sampling method for designing and analyzing computer experiments (DACE) [18], is a matrix of order i � j, where i is the number of levels to be examined and j is the number of design variables. Each of the j columns of the matrix containing levels 1, 2, …, i is randomly paired. LHS generates random sample points, ensuring that all portions of the design space are represented. Finally, the objective function values at these design points are evaluated by steady and unsteady numerical analyses.

The response surface approximation (RSA) model is applied as a surrogate model to predict the objective function values based on the 54 design points generated in the design space by using LHS. The RSA model, as a methodology of fitting a polynomial function to discrete responses obtained from numerical calculations, represents the association between the design variables and response functions [19]. The construction function for a second-order polynomial RSA can be expressed as follows:

$$f(\mathbf{x}) = \beta\_0 + \sum\_{j=1}^{N} \beta\_j \mathbf{x}\_j + \sum\_{j=1}^{N} \beta\_{jj} \mathbf{x}\_j^2 + \sum\_{i \neq j} \sum\_{i \neq j}^{N} \beta\_{ij} \mathbf{x}\_i \mathbf{x}\_j \tag{23}$$

where β, N, and x represent the regression analysis coefficients, number of design variables, and a set of design variables, respectively, and the number of regression analysis coefficients (β0, βi, etc.) is [(N + 1) � (N + 2)]/2.

State-of-the-Art Design Technique of a Single-Channel Pump for Wastewater Treatment http://dx.doi.org/10.5772/intechopen.75171 191

Figure 7. Multiobjective optimization procedure [22].

The other objective functions related to the radial force sources are defined as the sweep area (As) of the radial force during one revolution of impeller and the distance (Ds) of the mass center of the sweep area from the origin, as shown in Figure 6. These functions are defined as follows:

where As is the signed area of the polygon as the sweep area of the radial force during one revolution of impeller. The centroid of a non-self-intersecting closed polygon, defined by n

yi <sup>þ</sup> yiþ<sup>1</sup>

In these formulas, the vertices are assumed to be numbered in the order of their occurrence along the perimeter of the polygon. Therefore, the distance of the mass center of the sweep area

q

The Latin hypercube sampling (LHS) is employed to generate 54 design points that are used as the initial base data for constructing the response surface from five design variables. LHS, as an effective sampling method for designing and analyzing computer experiments (DACE) [18], is a matrix of order i � j, where i is the number of levels to be examined and j is the number of design variables. Each of the j columns of the matrix containing levels 1, 2, …, i is randomly paired. LHS generates random sample points, ensuring that all portions of the design space are represented. Finally, the objective function values at these design points are

The response surface approximation (RSA) model is applied as a surrogate model to predict the objective function values based on the 54 design points generated in the design space by using LHS. The RSA model, as a methodology of fitting a polynomial function to discrete responses obtained from numerical calculations, represents the association between the design variables and response functions [19]. The construction function for a second-order polynomial

> xj <sup>þ</sup><sup>X</sup> N

> > j¼1

where β, N, and x represent the regression analysis coefficients, number of design variables, and a set of design variables, respectively, and the number of regression analysis coefficients

<sup>β</sup>jjx<sup>2</sup>

<sup>j</sup> <sup>þ</sup>XX N

i6¼j

βijxixj (23)

xiyiþ<sup>1</sup> � xiþ<sup>1</sup>yi

ð Þ xi <sup>þ</sup> xiþ<sup>1</sup> xiyiþ<sup>1</sup> � xiþ<sup>1</sup>yi

� � xiyiþ<sup>1</sup> � xiþ<sup>1</sup>yi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cx <sup>2</sup> <sup>þ</sup> Cy 2

� � (19)

� � (20)

� � (21)

(22)

As <sup>¼</sup> <sup>1</sup> 2 Xn�1 i¼0

vertices (x0, y0), (x1, y1), …, (xn-1, yn-1), is defined as the point (Cx, Cy) as follows:

Xn�1 i¼0

Xn�1 i¼0

Ds ¼

Cx <sup>¼</sup> <sup>1</sup> 6As

Cy <sup>¼</sup> <sup>1</sup> 6As

from the origin is finally defined as follows:

190 Wastewater and Water Quality

evaluated by steady and unsteady numerical analyses.

f xð Þ¼ <sup>β</sup><sup>0</sup> <sup>þ</sup><sup>X</sup>

N

j¼1 βj

RSA can be expressed as follows:

(β0, βi, etc.) is [(N + 1) � (N + 2)]/2.

The RSA models are employed to construct the response surfaces based on the objective function values at the 54 design points generated in the design space using LHS. A hybrid multiobjective genetic algorithm (MOGA) is used to obtain the global Pareto-optimal solutions (POSs). The approximate POSs are obtained using a controlled elitist genetic algorithm (a variant of NSGA-II [20] as the MOGA function for three objective functions. The optimization algorithm and functions in the MATLAB OPTIMIZATION TOOLBOX [21] are used to finally generate the global POSs. Figure 7 shows an example of the multiobjective optimization procedure [22]. The detailed optimization procedure can be referred to in the previous literatures [23, 24].
