CFD-Driven Valve Shape Optimization for Performance Improvement of a Micro Cross-Flow Turbine

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Article CFD-Driven Valve Shape Optimization for Performance Improvement of a Micro Cross-Flow Turbine

Endashaw Tesfaye Woldemariam 1, Hirpa G. Lemu 1,* and G. Gary Wang 2

1 Department of Mechanical and Structural Engineering and Materials Science, University of Stavanger, 4036 Stavanger, Norway; [email protected] 2 School of Mechatronics System Engineering, Simon Fraser University, Surrey, BC V5A 1S6, Canada; [email protected] * Correspondence: [email protected]

Received: 13 December 2017; Accepted: 16 January 2018; Published: 19 January 2018

Abstract: Turbines are critical parts in hydropower facilities, and the cross-flow turbine is one of the widely applied turbine designs in small- and micro-hydro facilities. Cross-flow turbines are relatively simple, flexible and less expensive, compared to other conventional hydro-turbines. However, the power generation efficiency of cross-flow turbines is not yet well optimized compared to conventional hydro-turbines. In this article, a Computational Fluid Dynamics (CFD)-driven design optimization approach is applied to one of the critical parts of the turbine, the valve. The valve controls the fluid flow, as well as determines the velocity and pressure magnitudes of the fluid jet leaving the nozzle region in the turbine. The Non-Uniform Rational B-Spline (NURBS) function is employed to generate construction points for the valve profile curve. Control points from the function that are highly sensitive to the output power are selected as optimization parameters, leading to the generation of construction points. Metamodel-assisted and metaheuristic optimization tools are used in the optimization. Optimized turbine designs from both optimization methods outperformed the original design with regard to performance of the turbine. Moreover, the metamodel-assisted optimization approach reduced the computational cost, compared to its counterpart.

Keywords: CFD-driven optimization; NURBS function; micro-hydropower; cross-ﬂow turbine; metamodel-assisted optimization; turbine performance

1. Introduction Due to the dynamically increasing population, competitive market economy and modernization, the global demand for energy is increasing dramatically. Projections from various studies indicate a large increase in energy consumption in the coming decades, especially in developing countries [1]. In the last two decades, the increased consumption of renewable energy sources has shown a surge (Figure1). In fact, the studies project that this trend will also continue in the future. Among others, hydropower is the major source of renewable energy. As of 2015, hydropower sources constitute around 61% of the total global renewable energy share. Of these, micro- and small-hydropower constitute around 4.5% and 7% (Figure2), respectively. Hydropower facilities with an installed capacity between 100 kW and 10 MW are categorized under small-hydropower, while those below 100 kW are categorized under micro-hydropower [2,3]. Hydropower is one of the least expensive forms of renewable energy [2]. Moreover, despite its long history in global civilization and the increasing energy demand, more than half of the global hydropower potential still remains unexploited [4,5].

Energies 2018, 11, 248; doi:10.3390/en11010248 www.mdpi.com/journal/energies Energies 2018, 11, 248 2 of 18 EnergiesEnergies 2018 2018, ,11 11, ,248 248 22 of of 18 18

FigureFigure 1. 1. World World energy energyenergy consumption consumptionconsumption history history and and projection projection by by energy energy sources sources [[1]. [1].1].

FigureFigure 2. 2. Global Global share share of of renewable renewable energy energy (%) (%) [6]. [6]. Figure 2. Global share of renewable energy (%) [6]. InIn thethe existingexisting globalglobal hydrohydro potential,potential, small-small- andand micro-hydropowermicro-hydropower constituteconstitute significantsignificant portions,portions,In the and and existing they they play globalplay a a significant hydrosignificant potential, role role in in small- exploiting exploiting and micro-hydropower the the remaining remaining potential, potential, constitute particularly particularly significant in in portions, remote remote areasandareas they in in developing developing play a significant and and less less role developed developed in exploiting countries. countries. the remaining These These hydropower hydropower potential, sources particularlysources play play inan an remoteimportant important areas role role in indevelopingin off-gridoff-grid rural-area andrural-area less developed electrificationelectrification countries. withwith These insignificantinsignificant hydropower impactimpact sources onon the playthe surroundingsurrounding an important ecosystem roleecosystem in off-grid [7].[7]. Moreover,rural-areaMoreover, electrification ofof allall off-gridoff-grid with technologies,technologies, insignificant they impactthey conscons ontitutetitute the surrounding thethe leastleast expensive ecosystemexpensive [formform7]. Moreover, ofof electricityelectricity of all generationoff-gridgeneration technologies, [6,8]. [6,8]. they constitute the least expensive form of electricity generation [6,8]. InInIn order order toto bestbest exploitexploit thethe existingexisting small-small- andand micro-hydromicro-hydro potential,potential, withwith thethe ultimate ultimate goalgoal ofof meetingmeeting the the growinggrowing energyenergy demand,demand, deployingdeploying efficie efficientefficientnt equipmentequipment to to thethe hydrhydropowerhydropoweropower facilitiesfacilities is is required.required.required. The The hydro-turbine hydro-turbine is is one one of ofof the thethe most mostmost critical critical parts parts in in hydropower hydropower facilities, facilities, and and among among otherother turbine turbine designs, designs, the the cross-flow cross-flowcross-flow turbine turbine is is one one of of the the most most widely-applied widely-applied designs designs in in small- small- and and micro-hydropowermicro-hydropower facilities facilities aroundaround thethe globe,globe, particularlyparticularly forfor off-gridoff-grid run-of-the-riverrun-of-the-river run-of-the-river applications.applications. applications. TheThe cross-flowcross-flow cross-flow hydro-turbinehydro-turbine hydro-turbine isis isrelativelyrelatively relatively simple,simple, simple, flexibleflexible flexible andand lessandless expensive,expensive, less expensive, comparedcompared compared toto thethe conventionaltoconventional the conventional hydro-turbines.hydro-turbines. hydro-turbines. ContraryContrary Contrary toto thethe to thedevelopmentsdevelopments developments inin invariousvarious various computercomputer computer and and experiment-basedexperiment-based optimization optimizationoptimization approaches aapproachespproaches within withinwithin the lastthethe few lastlast decades, fewfew decades,decades, the power thethe generation powerpower generationgeneration efficiency efficiencyefficiency of of cross-flow cross-flow turbines turbines is is not not yet yet well well optimized, optimized, and and insignificant insignificant research research attention attention is is given given

Energies 2018, 11, 248 3 of 18 of cross-flow turbines is not yet well optimized, and insignificant research attention is given to this area. Obviously, the design methodology used to develop the turbine has a significant impact on the performance. Such a design methodology should account for not only the efficient conversion of fluid flow to mechanical energy, but also define, among others, the exact geometry of the blades, the rotational speed of the turbine and the upstream velocity that should be sustained under severe cyclic loading [9]. In this regard, some simulation-based design optimization approaches are proposed in the literature. Signagra et al. [10] investigated the reasons for the reduction of turbine efficiency and proposed a design methodology based on Computational Fluid Dynamics (CFD) simulation. The study reported by Soenoko [11] reviewed cross-flow turbine developments and suggested, among others, optimization of the nozzle construction to improve the performance of the turbine. Sammartano et al. [12] carried out computer-based tests utilizing numerical CFD and hydrodynamic analysis to obtain an optimum design aimed to come up with a theoretical framework for the turbine sequential design. However, the study was limited only to turbine configurations without a guide valve and mainly on parameters embedded in the rotor design. Anagnostopoulos and Papantonis [13], on the other hand, conducted size optimization of small hydropower plants using a then newly-developed evaluation algorithm. However, their focus was on the overall plant performance using basic plant size parameters to optimize project cost. Apart from that, there is considerable inconsistency, as regards the optimum efficiency of cross-flow turbines, in the analysis study results of various reports, both theoretical and experimental [12,14–17]. This paper has followed a new approach, in which a CFD-driven metamodel-assisted and metaheuristic design optimization tools are employed to optimize the shape of the valve profile of a T15-300 micro-cross-flow turbine design. This is one of the T-series turbine designs widely applied around the globe. The profile of the valve is generated using the NURBS (Non-Uniform Rational B-Spline) function. The main objective of the research is to improve the performance of the turbine design at the optimum valve angle. Moreover, the study aims to promote the simulation-based optimization approach for further applications on similar turbines. The paper is organized as follows. The next section discusses in detail the turbine type employed in the research. Section3 then discusses the NURBS function utilized to represent the valve profile. The methodologies followed in the paper are discussed in Section4, and the numerical modeling of a case study is presented in Section5. Thereafter, correlation studies between important hydraulic parameters in the nozzle and the entire turbine model are discussed in Section6. Section7 presents the sensitivity analyses’ results of the optimization parameters, while the results of the research work are discussed in Section8. Finally, conclusions drawn from the study and recommendations are presented in Section9.

2. Cross-Flow Turbine Cross-flow hydro-turbines, also known as Michell-Banki’s turbines, have been in development for decades since they were first patented by Donat-Banki in Budapest in the 1920s [16]. The benefits of the turbine, in addition to those mentioned in Section1, are that most of cross-flow turbine parts can be manufactured with hands-on technologies [13]. Moreover, they have better power generation capability at part load conditions and favorable run-of-the-river application. Figure3 shows the efficiency curves of various turbine designs per percentage rated flow. Despite the benefits, the figure shows that, the optimum efficiency of cross-flow turbines is lower compared to the other turbines. In recent cross-flow turbine design developments, nozzles, along with guide vanes or valves, are being utilized in order to control the flow and improve the turbine’s power-generating capacity [19,20]. Since the turbine is categorized as an impulse turbine, the valve is the critical part. Along with other design parameters [12], the valve design determines the magnitude of some of the important hydraulic parameters that in turn determine the power-generating capacity of the turbine. Among others, the T-series cross-flow turbine designs are the most widely-applied turbine designs that contain guide valves in their nozzles (see Figure4). The T15-300 is one of the latest T-series models designed Energies 2018, 11, 248 4 of 18

Energies 2018, 11, 248 4 of 18

and has been supplied by a well-known Switzerland/Indonesia-based company called ENTEC ag, St. (Gallen, Switzerland) [21]. For this study, we used the single-compartment design of the T15-300 turbine model, which has a width of 68 mm. As can be understood from Figure4a, the ﬂuid ﬂow starts fromEnergies the 2018 inlet,, 11, 248 passes through the nozzle and then crosses the rotor before leaving through the outlet.4 of 18

Figure 3. Efficiency curve of various turbine designs per percentage of rated flow [18].

In recent cross-flow turbine design developments, nozzles, along with guide vanes or valves, are being utilized in order to control the flow and improve the turbine’s power-generating capacity [19,20]. Since the turbine is categorized as an impulse turbine, the valve is the critical part. Along with other design parameters [12], the valve design determines the magnitude of some of the important hydraulic parameters that in turn determine the power-generating capacity of the turbine. Among others, the T-series cross-flow turbine designs are the most widely-applied turbine designs that contain guide valves in their nozzles (see Figure 4). The T15-300 is one of the latest T-series models designed and has been supplied by a well-known Switzerland/Indonesia-based company called ENTEC ag, St. (Gallen, Switzerland) [21]. For this study, we used the single-compartment design of the T15-300 turbine model, which has a width of 68 mm. As can be understood from Figure 4a, the fluid flow starts from the inlet, passes through the nozzle and then crosses the rotor before Figure 3. Efficiency curve of various turbine designs per percentage of rated flow [18]. leaving throughFigure the 3. outlet.Efﬁciency curve of various turbine designs per percentage of rated ﬂow [18]. In recent cross-flow turbine design developments, nozzles, along with guide vanes or valves, are being utilized in order to control the flow and improve the turbine’s power-generating capacity [19,20]. Since the turbine is categorized as an impulse turbine, the valve is the critical part. Along with other design parameters [12], the valve design determines the magnitude of some of the important hydraulic parameters that in turn determine the power-generating capacity of the turbine. Among others, the T-series cross-flow turbine designs are the most widely-applied turbine designs that contain guide valves in their nozzles (see Figure 4). The T15-300 is one of the latest T-series models designed and has been supplied by a well-known Switzerland/Indonesia-based company called ENTEC ag, St. (Gallen, Switzerland) [21]. For this study, we used the single-compartment design of the T15-300 turbine model, which has a width of 68 mm. As can be understood from Figure 4a, the fluid flow starts from the inlet, passes through the nozzle and then crosses the rotor before leaving through the outlet.

FigureFigure 4. 4. T15-300T15-300 cross-flow cross-flow turbine turbine design: design: (a ()a )3D 3D geometry, geometry, (b (b) )front front view view with with detailed detailed blade blade geometry. geometry.

According to experimental investigations and numerical analysis results reported by Costa According to experimental investigations and numerical analysis results reported by Pereira et al. [17], cross-flow turbines with a guide valve inside the nozzle have better performance Costa Pereira et al. [17], cross-flow turbines with a guide valve inside the nozzle have better as compared to those without a guide valve. Various numerical studies also concluded that the nozzle performance as compared to those without a guide valve. Various numerical studies also concluded region and guide valves highly impact the fluid characteristics inside the turbine, which thus that the nozzle region and guide valves highly impact the fluid characteristics inside the turbine, which determines the performance of the turbine [11,22,23]. Fluid jet velocity components (Figure 5) at thus determines the performance of the turbine [11,22,23]. Fluid jet velocity components (Figure5) at different locations of the cross-flow turbine rotor are some of the important parameters in the power conversion of the impulse type cross-flow turbines. In impulse turbines, the theoretical power conversion computation is calculated using Euler’s turbomachinery equation. The angles α1–α4 and β1–β4 (Figure5), the angles that the actual velocity and relative velocities make with the horizontal,

Figure 4. T15-300 cross-flow turbine design: (a) 3D geometry, (b) front view with detailed blade geometry.

According to experimental investigations and numerical analysis results reported by Costa Pereira et al. [17], cross-flow turbines with a guide valve inside the nozzle have better performance as compared to those without a guide valve. Various numerical studies also concluded that the nozzle region and guide valves highly impact the fluid characteristics inside the turbine, which thus determines the performance of the turbine [11,22,23]. Fluid jet velocity components (Figure 5) at

Energies 2018, 11, 248 5 of 18 different locations of the cross-flow turbine rotor are some of the important parameters in the power

Energiesconversion2018, 11of, 248the impulse type cross-flow turbines. In impulse turbines, the theoretical power5 of 18 conversion computation is calculated using Euler’s turbomachinery equation. The angles α1–α4 and β1–β4 (Figure 5), the angles that the actual velocity and relative velocities make with the horizontal, respectively, playplay importantimportant rolesroles inin transferringtransferring hydraulichydraulic andand mechanical powers. The guide valve shape and rotor blades’ proﬁleprofile determinedetermine thethe magnitudesmagnitudes ofof thethe transferredtransferred power.power.

Figure 5. Cross-flow turbine rotor: (a) fluid jet trajectory at different locations and (b) velocity Figure 5. Cross-flow turbine rotor: (a) fluid jet trajectory at different locations and (b) velocity triangles triangles at Locations 1–4. w1–4 refer to relative velocities at Points 1–4; u1 and u2 refer to peripheral at Locations 1–4. w1–4 refer to relative velocities at Points 1–4; u1 and u2 refer to peripheral velocities velocities tangent to the outer and inner diameters respectively; v1–4 refer to actual fluid velocities at tangent to the outer and inner diameters respectively; v1–4 refer to actual fluid velocities at Points 1–4. Points 1–4. 3. NURBS Function 3. NURBS Function Most Computer-Aided Design (CAD) tools utilize basis functions that are based on B-spline functions. Most Computer-Aided Design (CAD) tools utilize basis functions that are based on B-spline Among others, the NURBS function is one of the state-of-the-art design basis functions applied in the functions. Among others, the NURBS function is one of the state-of-the-art design basis functions latest CAD tools. NURBS can be used to describe arbitrarily-shaped curves, surfaces or bodies, have a applied in the latest CAD tools. NURBS can be used to describe arbitrarily-shaped curves, surfaces high level of continuity and allow refinement ability on the CAD geometries [24]. or bodies, have a high level of continuity and allow refinement ability on the CAD geometries [24]. The basis function is based on a parameterized recursive function, which begins from a piecewise The basis function is based on a parameterized recursive function, which begins from a constant value at a polynomial degree of p = 0. For a single dimensional problem, for example, piecewise constant value at a polynomial degree of =0. For a single dimensional problem, for the function is formulated as: example, the function is formulated as: ( 1 ξ ≤ ξ < ξ N (ξ) = i i+1 (1) i,0 10 otherwise≤ () = (1) , 0 For polynomial functions of a higher degrees, p ≥ 1, the function can be recursively obtained usingFor Equation polynomial (2), functions of a higher degrees, 1, the function can be recursively obtained using Equation (2), ξ − ξi ξi+p+1 − ξ Ni,p(ξ) = − Ni,p−1(ξ) + −Ni+1,p−1(ξ) (2) ,() = ξi+p − ξi,() +ξi+p+1 − ξi+1 ,() (2) − − where: Ni,p is the basis function of the polynomial degree p with knot index i = 1, 2, ... , n. ξ determines where: , is the basis function of the polynomial degree with knot index = 1,2, … , . knot values from a given knot vector Ξ = ξ1, ξ2, ξ3,..., ξn+p+1 . The B-Spline basis curve function, determines knot values from a given knot vector Ξ=,,,…,. The B-Spline basis curve Cb(ξ), is given by Equation (3). function, (), is given by Equation (3). n Cb(ξ) = ∑ Ni,p(ξ)Bi (3) i=1 () =,() (3) where Bi = {b1, b2, b3,..., bn} are the corresponding control points. NURBS differs from B-spline in such a way that it introduces a weighting value for the corresponding wherecontrol points = to, enable,,…, local are control the corresponding of the design variables. control points. The NURBS curve equation is given as,

n n Ni,p(ξ)wi C (ξ) = B = R B (4) n ∑ ( n ( ) ) i ∑ i,p i i=1 ∑i=1 Ni,p ξ wi i=1 Energies 2018, 11, 248 6 of 18

Ni,p(ξ)wi Ri,p = n (5) (∑i=1 Ni,p(ξ)wi) where Ri,p is the NURBS basis function and wi are the given weighted values for the control points.

4. Methodology and Approaches In this study, a new optimization approach has been employed to optimize the design of the valve profile in the nozzle of the cross-flow turbine under consideration. The optimization approach, illustrated in Figure6, interconnects an optimization tool, a curve function and a modeling and analysis tool. In this approach, two optimization methods have been utilized. The first method, a Multi-Objective Metamodel-Assisted Optimization Method (MO-MMAO), uses the Optimization Assisted System Integrated Software (OASIS) optimization tool. The optimization tool, OASIS [25], has been developed for the optimization of computationally-expensive implicit problems. It was originated from the Product Design and Optimization Laboratory (PDOL) at Simon Fraser University, Canada, and developed and marketed by Empower Operation Corp (V1.3, Surrey, BC, Canada). OASIS incorporates various optimization algorithms, which are assisted by various metamodels, machine learning, statistical analysis and other tools. The optimization tool follows a direct sampling strategy, where the metamodel adaptively updates itself during the optimization process [26]. For this study, the Multi-Objective Global Optimization (MOGO) algorithm of the tool has been used. The second optimization tool is the well-known metaheuristic optimization tool, Genetic Algorithm (GA). It is a population-based optimization tool motivated by the evolutionary concept of survival of the fittest [27]. Energies 2018, 11, 248 7 of 18

Figure 6. Optimization approach flowchart: inter-connection of OASIS, MATLAB and ANSYS Figure 6. Optimization approach ﬂowchart: inter-connection of OASIS, MATLAB and ANSYS Workbench tools. Workbench tools.

The other important concept in the optimization approach is to separately optimize the nozzle regionIn (Figure the general 8b). We approach, have followed which this employs concept either for three of the important two optimization reasons: tools, a MATLAB (MathWorks, Inc., Natick, MA, USA) script is used to generate construction points from a NURBS (i).function Since with the theturbine construction is assumed points as the being impulse used toturbine, generate this the part proﬁle of the of turbine the valves plays of thea critical cross-ﬂow role in the entire power generation; (ii). It enables us to easily identify and control the important hydraulic parameters at the nozzle that are important in the entire power transfer and are mainly affected by the shape of the guide valve; (iii). It reduces the computational cost in the optimization process.

Figure 7. NURBS curve generated from 12 control points.

Energies 2018, 11, 248 7 of 18

Energies 2018, 11, 248 7 of 18 turbine in the modeling and analysis tool, i.e., ANSYS Workbench (ANSYS, Inc., V17.1, Canonsburg, PA, USA) [28]. The MATLAB script uses a total of twelve (12) control points (see Figure7) to generate a second-order polynomial NURBS curve equation [29]. Of the 12 control points, x or y-coordinates of only ﬁve selected control points are selected as optimization parameters: the y-coordinates of Control

Points 3, 4 and 11 and the x-coordinates of Control Points 6 and 7 are the parameters chosen. These Figure 6. Optimization approach flowchart: inter-connection of OASIS, MATLAB and ANSYS parameters are represented by P (where i = 1–5). The remaining control points maintain their original Workbench tools. i design values as they are assumed to be less important in the optimization, but critical for the overall valve operation.The other Forimportant instance, concept Control in the Points optimization 1, 2, 8 and ap 9proach are at is critical to separately locations optimize of the the valve nozzle and are usedregion for proper (Figure sealing 8b). We whenhave followed the valve this is concept in a closing for three position important (Figure reasons:8a), while Point 5 maintains the(i). gap betweenSince the turbine the valve is assumed profile and as the the impulse shaft outer turbine, diameter, this part which of the is turbine used toplays maneuver a critical the role valve. A total ofin the fifty entire (50) power construction generation; points is generated from the NURBS curve equation. The x- and y-axis(ii). components It enables us of to theeasily 50 identify construction and control points the are important mapped hydraulic to 100 design parameters parameters at the nozzle in the that ANSYS Workbenchare important CAD modeling in the tool.entire The power construction transfer and point are parameters mainly affected are represented by the shape by ofC i the(where i = 1–100).guide Based valve; on the sensitivity to the objectives (see Section7), the five optimization parameters are given(iii). It differentreduces the weighting computational values cost in in the the NURBS optimization function. process.

Figure 7. NURBS curve generated from 12 control points. Energies 2018, 11, 248 Figure 7. NURBS curve generated from 12 control points. 8 of 18

FigureFigure 8. The8. The nozzle nozzle region region and and boundaries boundaries of T15-300 of T15-300 cross-ﬂow cross-flow turbine turbine model: model: (a) the (a) full the cross-ﬂow full turbinecross-flow model turbine and ( bmodel) separate and (b nozzle) separate region nozzle with region both with the inletboth andthe inlet outlet and boundaries. outlet boundaries.

It is important to note that the change in the trends of the output parameters from the separate nozzleThe otherdesign important are initially concept correlated in the with optimization the change approach in the istrends to separately of the important optimize output the nozzle regionparameters (Figure of8b). the We full have turbine followed design thisdue conceptto the change for three of valve important angle (with reasons: respect to the turbine’s performance), which is discussed in Section 6. After the optimization process, a comparison of the optimized valve designs with the original design is carried out using the full turbine model. The multi-objective optimization problem has two objectives and two constraint functions. The two objectives of the optimization, based on the correlation studies are (1) minimizing the

x-component (maximizing the magnitude) of the area-weighted average velocity () of the nozzle outlet fluid and (2) minimizing the drag force (Drag) on the valve profile. The general formulation of the problem is given in Equation (6). Both objective functions are obtained after the CFD computation of the turbulent numerical model using the implicit analysis tool, ANSYS Fluent tool:

Minimize = () and = (); (=1,2,…5)

Constraints () ≤0,( − ≤0,and − ≤0) (6)

∈[,]

where and represent the x-component of the area-weighted average velocity at the nozzle outlet and drag force functions; where both are functions of the optimization parameters (); () represents the constraint functions respectively; and and are the lower and upper bounds of the optimization parameters, respectively. The lower and upper bounds of the parameters are obtained from the design configuration of the original turbine design so that they should not create any irregular valve profile and unacceptable nozzle region; see Table 1. Therefore, the parameters are bounded inside the nozzle region. The same principle is followed when the constraints on the design variables are determined from the turbine configuration.

Table 1. Design points, lower and upper bound values of the optimization parameters.

Name of points (m) (m) (m) (m) (m) Design point 0.15160 0.16092 0.24500 0.25000 0.13700 Lower bound 0.14014 0.14100 0.22000 0.22000 0.11972 Upper bound 0.17000 0.17000 0.25000 0.26000 0.14565

Energies 2018, 11, 248 8 of 18

(i). Since the turbine is assumed as the impulse turbine, this part of the turbine plays a critical role in the entire power generation; (ii). It enables us to easily identify and control the important hydraulic parameters at the nozzle that are important in the entire power transfer and are mainly affected by the shape of the guide valve; (iii). It reduces the computational cost in the optimization process.

It is important to note that the change in the trends of the output parameters from the separate nozzle design are initially correlated with the change in the trends of the important output parameters of the full turbine design due to the change of valve angle (with respect to the turbine’s performance), which is discussed in Section6. After the optimization process, a comparison of the optimized valve designs with the original design is carried out using the full turbine model. The multi-objective optimization problem has two objectives and two constraint functions. The two objectives of the optimization, based on the correlation studies are (1) minimizing the x-component (maximizing the magnitude) of the area-weighted average velocity (Vx) of the nozzle outlet ﬂuid and (2) minimizing the drag force (Drag) on the valve proﬁle. The general formulation of the problem is given in Equation (6). Both objective functions are obtained after the CFD computation of the turbulent numerical model using the implicit analysis tool, ANSYS Fluent tool:

Minimize Vx = f (Pi) and Dr = f (Pi); (i = 1, 2, . . . 5)

Constraints C(Pi) ≤ 0, (P1 − P2 ≤ 0, and P5 − P1 ≤ 0) (6)

Pi ∈ [PiL, PiU] where Vx and Dr represent the x-component of the area-weighted average velocity at the nozzle outlet and drag force functions; where both are functions of the optimization parameters (Pi); C(Pi) represents the constraint functions respectively; and PiL and PiU are the lower and upper bounds of the optimization parameters, respectively. The lower and upper bounds of the parameters are obtained from the design configuration of the original turbine design so that they should not create any irregular valve profile and unacceptable nozzle region; see Table1. Therefore, the parameters are bounded inside the nozzle region. The same principle is followed when the constraints on the design variables are determined from the turbine configuration.

Table 1. Design points, lower and upper bound values of the optimization parameters.

Name of Points P1 (m) P2 (m) P3 (m) P4 (m) P5 (m) Design point 0.15160 0.16092 0.24500 0.25000 0.13700 Lower bound 0.14014 0.14100 0.22000 0.22000 0.11972 Upper bound 0.17000 0.17000 0.25000 0.26000 0.14565

5. Numerical Modeling Based on our experience of comparative studies of the applications of different numerical models on similar problems and various other numerical studies, the standard K-ε numerical model [30,31] is found to best represent the fluid flow inside the cross-flow turbine. According to studies and user guides of simulation tools, this model is robust, economical and provides reasonable accuracy not only for the current problem, but also for a wide range of problems. Therefore, in this paper, we have employed the standard K-ε model for both the time-dependent (transient) and steady state analyses in both the separate nozzle and entire turbine models. In the general transport equation for the standard K-ε model, the turbulence kinetic energy and the rate of dissipation, k and ε, respectively, are obtained from Equations (7) and (8) [30]. A scalable wall function is selected for the near-wall treatment. Energies 2018, 11, 248 9 of 18

" # ∂ ∂ ∂ µt ∂k (ρk) + (ρkVi) = µ + + Gk + Gb − ρε − YM + Sk (7) ∂t ∂xi ∂xj σk ∂xj

" # 2 ∂ ∂ ∂ µt ∂ε ε ε (ρε) + (ρεVi) = µ + + C1ε (Gk + C3εGb) − C2ερ + Sε (8) ∂t ∂xi ∂xj σε ∂xj k k where ρ, µ and Vi are density, viscosity and velocity vectors of the flowing fluid, respectively. σk and σε are the turbulent Prandtl numbers for k and ε, respectively; whose default values in the tool are used for our analyses. Gk and Gb are the generation turbulence kinetic energy due to the mean velocity gradients and buoyancy, respectively; however, the generation buoyancy is negligible for this problem. YM represents the contribution of the fluctuating dilation in compressible turbulence, which is negligible in our analyses. Sk and Sε are the source terms for kinetic energy and dissipation, which ∂ are not applicable for this problem. C1ε, C2ε and C3ε are given constant values; ∂t is the gradient vector with respect to time; ∂ and ∂ are the spatial gradient vectors at the i and j coordinates. ∂xi ∂xj The turbulent viscosity term, µt, is computed using Equation (9), where the term Cµ is a constant value. All constants are listed in Table2 with their corresponding valves.

k2 µ = ρC (9) t µ ε In order to identify an allowable and efficient head for the analyses, an efficiency curve study from steady numerical analyses on the full turbine models is carried out at various flow rate ratios computed from the head ranges from 5–22.2 m. The mass flow rate at each head over the mass flow rate at the maximum head is calculated to find each flow rate ratio. Based on the result, the average allowable head of 12.5 m was maintained for all analyses, as it is where the model provides the maximum computed hydraulic efficiency. In addition, the valve angle position is set at its maximum opening position where the maximum numerical power output is obtained. Moreover, for all simulations on the full turbine model, the rotor speed is maintained at 360 rpm.

Table 2. Given and standard values for constant variables.

Numerical Model Constants Values ρ (kg/m3) 998.2 µ (kg/ms) 0.001003 Cµ (-) 0.09 σk (-) 1 σε (-) 1.3 C1ε (-) 1.44 C2ε (-) 1.92

5.1. Meshing Qualities and Convergence After a number of trials and convergence tests, valid mesh sizes of the full turbine model and the separate nozzle model are chosen; see Figure9. The meshing details and characteristics of the selected meshes of the models (circled in blue in the figures) are shown in Table3. The total area-weighted average pressure values at the outlets of the models were the convergence test parameters in the test; whereas, built-in mesh quality measures in the meshing tool of ANSYS Fluent, such as element quality, skewness and orthogonal qualities of the meshes, are checked while maintaining the meshing quality. Moreover, due to the configuration of the problem and the existence of interfacing bodies, the unstructured meshing method is selected, and local meshing techniques are used to refine meshes around the small-sized rotor blades’ edges and surfaces. The mesh qualities are then controlled to fall within the recommended range of values of the mesh quality measures. Energies 2018, 11, 248 10 of 18

such as element quality, skewness and orthogonal qualities of the meshes, are checked while maintaining the meshing quality. Moreover, due to the configuration of the problem and the existence of interfacing bodies, the unstructured meshing method is selected, and local meshing techniques are used to refine meshes around the small-sized rotor blades’ edges and surfaces. The mesh qualities are then controlled to fall within the recommended range of values of the mesh Energies 2018, 11, 248 10 of 18 quality measures.

FigureFigure 9.9. Model mesh validation:validation: (a)) fullfull turbineturbine model;model; ((b)) separateseparate nozzlenozzle model.model.

Table 3.3. Meshing size details of thethe fullfull turbineturbine geometrygeometry andand separateseparate nozzlenozzle geometry.geometry. Housing Elem. Rotor Elem. Total Elem. Mesh Min. Mesh Max. Max. Tet Geometries Housing Rotor Elem. Total Elem. Mesh Min. Mesh Max. Max. Tet Geometries Size Size Size Size Face size size Elem. Size Size Size Size Face Size Size Full turbine Full turbine 57,662 3,234,243 3,291,905 0.0005 0.0065 0.015 geometry 57,662 3,234,243 3,291,905 0.0005 0.0065 0.015 Separategeometry nozzle 60,270 - 60,270 0.00035 0.0045 0.015 Separategeometry nozzle 60,270Elem. stands for - mesh element 60,270 and Tet stands 0.00035 for tetrahedral. 0.0045 0.015 geometry 5.2. Solution Methods Elem. stands for mesh element and Tet stands for tetrahedral. To solve the transport equations for the standard K-ε numerical model in CFD tool, the SIMPLE 5.2. Solution Methods algorithm scheme has been employed to compute the pressure-velocity coupling, and least squares cell-basedTo solve method the transport has been equations used to compute for the standard the gradientK-ε numerical in the spatial model discretization. in CFD tool, Considering the SIMPLE algorithmthe complex scheme geometry has been of the employed rotor and to the compute interface the between pressure-velocity the housing coupling, and rotor and domains, least squares the cell-basedsecond order method method has has been been used employed to compute to compute the gradient the pressure in the spatial and the discretization. second order Considering upwind to thecompute complex the momentum, geometry of turbulent the rotor kinetic and the energy interface and turbulence between the dissipation housing rate and equations rotor domains, in the thespatial second discretization. order method In both has beenthe full employed turbine to and compute separate the nozzle pressure models, and the all secondwalls are order given upwind a no- toslip compute condition, the momentum,and the outlet turbulent surfaces kinetic are energygiven andpressure-outlets turbulence dissipationwith zero rategauge equations pressure in theboundary spatial conditions. discretization. In both the full turbine and separate nozzle models, all walls are given a no-slipAfter condition, solving the and transport the outlet equations, surfaces the are area given-weighted pressure-outlets average values, with zero which gauge are the pressure valid boundaryvelocity magnitudes conditions. of the numerical fluid flow, are considered to compute the actual velocities at the outletAfter surfaces solving theof both transport geometries. equations, In addition, the area-weighted the x-components average of values,the combined which arepressure the valid and velocityviscous forces magnitudes are used of to the determine numerical the fluid drag flow, forc aree on considered the guide valve to compute surface. the Pressure actual velocitiesand viscous at theforces outlet ( surfaces and of, bothrespectively) geometries. are Incomputed addition, based the x-components on their computed of the combined and user-determined pressure and viscousreference forces values; are for used instance, to determine the pressure the drag force force is computed on the guide as Equation valve surface. (10). On Pressure the other and hand, viscous the forcestotal moment (Fp and Floadv, respectively) vector, moment are computed coefficients, based output on their hydraulic computed power and and user-determined numerical efficiencies reference values;are computed for instance, using Equation the pressures (11)–(14), force is respectively. computed as Equation (10). On the other hand, the total moment load vector, moment coefficients, output hydraulic power and numerical efficiencies are computed using Equations (11)–(14), respectively. =(−) (10) n = − Fp =∑ P +Pre fAnˆ (11)(10) i=1

→ → → → → M = r × Fp + r × Fv (11) M Cm = (12) 1 2 2 ρrVr Ar Lr Energies 2018, 11, 248 11 of 18

Energies 2018, 11, 248 = 11 of 18 1 (12) 2

Po==.M.ω (13)(13) M..ω η== (14)(14) Qin((Pin−−Pout))//ρ → wherewhere Pand andP re fare are the the computed computed and and user-determined user-determined reference reference pressures pressures (Pa), (Pa), respectively; respectively;r and → and are the moment arm and moment vectors. is the unit-less moment coefficient. , , M are the moment arm and moment vectors. Cm is the unit-less moment coefﬁcient. ρr, Vr, Ar and Lr 3 2 areand the reference are thedensity reference (in kg/mdensity3), velocity(in kg/m (m/s),), velocity area (m (m/s),2) and area length (m (m)) and values, length respectively, (m) values, set respectively, set in the analysis tool. , and are the magnitude values of moment (Joules), in the analysis tool. M, Qin and ω are the magnitude values of moment (Joules), mass ﬂow rate (kg/s) andmass rotational flow rate speed (kg/s) ofand the rotational rotor (rad/s), speed respectively. of the rotor (rad/s), respectively.

6.6. CorrelationCorrelation StudyStudy betweenbetween thethe SeparateSeparate NozzleNozzle andand FullFull TurbineTurbine Models Models ToTo identifyidentify importantimportant outputoutput parametersparameters fromfrom thethe separateseparate nozzlesnozzles thatthat havehave aa directdirect impactimpact onon thethe performanceperformance of the turbine and and to to understand understand their their trend trend with with the the change change of of the the valve valve design, design, a acorrelation correlation study study between between the the separate separate nozzle nozzle and and fu fullll model model was carried out. While conducting thisthis study,study, severalseveral numericalnumerical simulationssimulations werewere runrun onon bothboth thethe separateseparate nozzlenozzle andand fullfull turbineturbine modelsmodels atat differentdifferent valvevalve positions,positions, andand theirtheir resultsresults havehave beenbeen analyzed.analyzed. TheThe valvevalve angleangle openingopening positionposition waswas setset toto rangerange fromfrom 50–100%50–100% opening.opening. TheThe other boundaryboundary conditions,conditions, suchsuch asas inlet head, rotational speedspeed ofof thethe rotorrotor forfor thethe fullfull model,model, numericalnumerical modelmodel settingssettings andand thethe geometricgeometric parametersparameters exceptexcept valvevalve position,position, areare setset constantconstant inin allall analyses.analyses. TheThe outputoutput powerpower isis computedcomputed fromfrom thethe fullfull turbineturbine model,model, andand variousvarious outputoutput parametersparameters areare studiedstudied fromfrom thethe nozzlenozzle model.model. TheThe mainmain purposepurpose ofof thisthis correlationcorrelation studystudy isis toto identifyidentify thethe twotwo outputoutput parametersparameters fromfrom thethe nozzlenozzle modelmodel withwith higherhigher impactimpact onon turbineturbine performanceperformance andand setset themthem asas objectiveobjective parameters.parameters. AsAs depicted in Figure Figure 10a,10a, the the results results show show that that the the output output power power increases increases as expected as expected with with the thevalve valve opening opening from from 50–100% 50–100% from from the the closing closing positi position.on. The The simulation simulation results results from the separateseparate nozzlenozzle model,model, FigureFigure 1010b,b, revealreveal thatthat thethe dragdrag forceforce onon thethe valvevalve proﬁleprofile andand thethe xx--componentcomponent ofof thethe area-weightedarea-weighted averageaverage velocityvelocity (X-Velocity)(X-Velocity) fromfrom thethe nozzlenozzle outletoutlet surfacesurface havehave aa visiblevisible correlationcorrelation withwith thethe outputoutput power.power. TheThe velocityvelocity magnitudemagnitude hashas aa directdirect relationrelation toto thethe outputoutput power,power, whilewhile thethe dragdrag forceforce hashas anan inverseinverse relation.relation. Therefore, it isis validvalid toto setset thethe objectivesobjectives ofof thethe optimizationoptimization toto minimizeminimize thethe xx--componentcomponent ofof thethe velocityvelocity atat thethe nozzlenozzle outletoutlet andand minimizeminimize thethe dragdrag forceforce onon thethe valvevalve proﬁle.profile.

FigureFigure 10.10. CorrelationCorrelation studystudy betweenbetween separateseparate nozzlenozzle andand entireentire turbineturbine model:model: ((aa)) OutputOutput powerpower response;response; ((bb)) OutputOutput velocitiesvelocities andand dragdrag forceforce responses.responses.

7.7. SensitivitySensitivity AnalysesAnalyses ToTo assignassign logicallogical weightingweighting valuesvalues toto thethe correspondingcorresponding controlcontrol pointspoints forfor thethe NURBSNURBS curve,curve, thethe signiﬁcancesignificance ofof thethe optimizationoptimization parametersparameters forfor thethe outputoutput objectiveobjective parameters should be studied. Such signiﬁcancy studies (sensitivity analysis) can be achieved by employing the concept of the

Energies 2018, 11, 248 12 of 18

Such significancy studies (sensitivity analysis) can be achieved by employing the concept of the statistical Design Of Experiment (DOE). The factorial DOE method is a widely-applied method in the scientific community. For this study the fractional factorial design method has been applied to

estimate the direct effects of the five optimization parameters ( ) on the responses of the two objective parameters (V and Dr) in the separate nozzle model. Unlike the full factorial experimental Energies 2018, 11, 248 12 of 18 design, the fractional method does not estimate all the interaction effects of the parameters, but can correctly estimate the main effects with fewer runs, compared to the intensive full factorial. statisticalA three-level Design Of five-parameter Experiment (DOE).analysis The with factorial two responses DOE method was employed. is a widely-applied Level 1 is method the lower in thebound; scientific Level community. 2 is the design For point; this study and theLevel fractional 3 is the factorialupper bound. design The method custom has optimal been applied factorial to estimatedesign option the direct with effects random of thesampling five optimization from the Design-Expert parameters (P tooli) on [32] the responsesdictated that of the26 combinations two objective parametersof samples should (V and Drbe )run. in the The separate settings nozzle of the model.other important Unlike the parameters full factorial are experimental carefully controlled design, thebased fractional on the standard method does recommendations not estimate all of the the interaction tool. effects of the parameters, but can correctly estimateAccording the main to effectsthe sensitivity with fewer test runs,results compared (Figure 11), to the the intensive first and fifth full factorial.parameters ( and ) are highlyA three-levelsensitive to five-parameter both objective analysis parameters, with two compared responses to wasthe employed.other three, Level while 1 isthe the second lower bound;parameter Level () 2 is is more the design significant point; to andthe second Level 3 response is the upper than bound.the remaining The custom two. optimal factorial designTherefore, option with the random corresponding sampling control from thepoints Design-Expert for the first tool and [32 fifth] dictated parameters that 26 are combinations given higher of samplesweights shouldin the NURBS be run. curve The settings function, of theso that other more important construction parameters points are will carefully be generated controlled from based their onsurrounding the standard points recommendations on the curve, thus of the generating tool. smooth curves in their vicinity. Note that the total numberAccording of construction to the sensitivity points in test the results NURBS (Figure curv e11 can), the be firstdetermined and fifth in parameters the function’s (P1 and MATLABP5) are highlyscript by sensitive the user to bothdepending objective on parameters,the interest. compared On the other to the hand, other the three, two while least thesensitive second parameters parameter ((P2) and is more ) significantare given only to the fair second weighting response values. than the remaining two.

Figure 11. Sensitivity test results of parameters: (a) sensitivity on the first objective parameter; Figure 11. Sensitivity test results of parameters: (a) sensitivity on the first objective parameter; (b) sensitivity on the second objective parameter. (b) sensitivity on the second objective parameter. 8. Results and Discussion Therefore, the corresponding control points for the first and fifth parameters are given higher weightsThe in MMAO the NURBS optimization curve function, was performed so that more by constructionsetting the OASIS points tool will to be a generated maximum from function their surroundingevaluation of points 100 (based on the on curve, authors’ thus generatingprevious experience smooth curves on similar in their pr vicinity.oblems). Note Apart that from the total the numberdefault convergence of construction tolerance points setti in thengs, NURBS the maximum curve can number be determined of evaluations in the is function’s used as a MATLAB stopping scriptcriterion by for the the user optimization depending process on the interest. in the tool. On In the the other multi-objective hand, the two GA least optimization, sensitive parametersthe number of generations is set to be six and the stall generation limit of five are set as stopping criteria. The (P3 and P4) are given only fair weighting values. population size is chosen to be 20. This combination gives at least 100 function evaluations in the GA 8.optimization Results and process. Discussion The same computational workstation was used for both optimization

The MMAO optimization was performed by setting the OASIS tool to a maximum function evaluation of 100 (based on authors’ previous experience on similar problems). Apart from the default convergence tolerance settings, the maximum number of evaluations is used as a stopping criterion for the optimization process in the tool. In the multi-objective GA optimization, the number of generations is set to be six and the stall generation limit of ﬁve are set as stopping criteria. The population size Energies 2018, 11, 248 13 of 18 Energies 2018, 11, 248 13 of 18 processes: HP Z600 of processor Model Intel(R) Xeon(R) X5650 (HP Inc., Palo Alto, CA, USA) with Energies 2018, 11, 248 13 of 18 twoprocesses: processors HP Z600 each of around processor 2.67 Model GHz Intel(R)capacity Xeon(R). The optimization X5650 (HP Inc., results Palo and Alto, the CA, comparative USA) with analysestwo processors are discussed each around in detail 2.67 in Se GHzctions capacity 8.1 and. 8.2,The respectively. optimization results and the comparative isanalyses chosen are to bediscussed 20. This in combination detail in Sections gives at8.1 least and 1008.2, functionrespectively. evaluations in the GA optimization process.8.1. Optimization The same Results computational workstation was used for both optimization processes: HP Z600 8.1. Optimization Results of processorFigure 12 Model shows Intel(R) the Xeon(R)convergence X5650 plot (HP with Inc., Palothe number Alto, CA, of USA) function with twoevaluations processors for each the aroundMetamodel-AssistedFigure 2.67 GHz12 shows capacity. Optimization the Theconvergence optimization (MMAO), plot i.e., results with OASIS andthe optimization thenumber comparative of prfunctionocess; analyses it canevaluations arebe observed discussed for thatthe in detailtheMetamodel-Assisted optimization in Sections 8.1converged and Optimization 8.2, at respectively. the (M53rdMAO), function i.e., OASISevaluation optimization count. The pr ocess;total timeit can elapsed be observed was that2 h, 50the min optimization and 26 s. Theconverged best Pareto at the point 53rd (with function respect evaluation to the turbine’s count. The performance) total time elapsed from the was Pareto 2 h, 8.1. Optimization Results frontier50 min andpoints, 26 s.shown The best in FigurePareto point13, is (withtaken respectas the bestto the optimum turbine’s point performance) (Table 4) fromfor our the furtherPareto comparativefrontierFigure points, 12 analyses. showsshown thein Figure convergence 13, is taken plot as with the thebest number optimum of point function (Table evaluations 4) for our forfurther the Metamodel-AssistedcomparativeUnlike the analyses. MMAO Optimization optimization, (MMAO), the multi-object i.e., OASISive GA optimization optimization process; carried it out can a be total observed of 121 thatfunction theUnlike optimizationevaluation the MMAO counts, converged optimization, which at led the theto 53rd themulti-object function fact that evaluation iveit took GA optimizationaround count. 3.69 The h.carried totalSeven time out (7) a elapsedPareto total of front was 121 2pointsfunction h, 50 are min evaluation returned and 26 from s.counts, The the bestoptimization which Pareto led to point process the (withfact (Figur that respecte it 13); took similarly, to around the turbine’s the 3.69 best h. performance) ParetoSeven point(7) Pareto (Table from front 4) the is Paretotakenpoints as are frontier the returned best points, optimum from shown the point optimization in Figure for our 13 furtherprocess, is taken comparative(Figur as thee 13); best similarly, analyses. optimum the point best (TablePareto4 )point for our (Table further 4) is comparativetaken as the best analyses. optimum point for our further comparative analyses.

Figure 12. Convergence line plot of the Metamodel-Assisted Optimization (MMAO) FigureoptimizationFigure 12.12.Convergence tool.Convergence line plotline of theplot Metamodel-Assisted of the Metamodel-Assisted Optimization (MMAO)Optimization optimization (MMAO) tool. optimization tool.

Figure 13. Pareto front plot for MMAO (OASIS) optimization and GA multi-objective optimization.optimization. Figure 13. Pareto front plot for MMAO (OASIS) optimization and GA multi-objective optimization.

Energies 2018, 11, 248 14 of 18

Table 4. Parameter values from the optimization results and steady analyses’ results.

Turbine Model P1 (m) P2 (m) P3 (m) P4 (m) P5 (m) Power Output (Watt) Original 0.151600 0.160920 0.24500 0.25000 0.13700 4841.83 GA 0.143077 0.16730 0.23089 0.24575 0.13308 5070.89 MMAO (OASIS) 0.142430 0.15588 0.23567 0.24059 0.12091 5099.85

Unlike the MMAO optimization, the multi-objective GA optimization carried out a total of 121 function evaluation counts, which led to the fact that it took around 3.69 h. Seven (7) Pareto front pointsEnergies are2018 returned, 11, 248 from the optimization process (Figure 13); similarly, the best Pareto point (Table14 of 418) is taken as the best optimum point for our further comparative analyses. Apart from the the differences differences in in computational computational time time and and number number of of the the function function evaluations, evaluations, a differencea difference is isalso also observed observed in inthe the shape shape of the of the optimized optimized valve valve models models (see (see Figure Figure 14). 14However,). However, the shapesthe shapes from from both both methods methods are arenot notas complex as complex as one as one may may expect. expect.

Figure 14. Original and optimize optimizedd shapes of the guide valve generated from the construction points.

Table 4. Parameter values from the optimization results and steady analyses’ results. 8.2. Comparative Analyses and Discussions Turbine Model (m) (m) (m) (m) (m) Power Output (Watt) To see the actual impact of the optimized shapes on the full turbine design, both steady and Original 0.151600 0.160920 0.24500 0.25000 0.13700 4841.83 transient CFD analyses (simulations) have been carried out and the responses collected and compared GA 0.143077 0.16730 0.23089 0.24575 0.13308 5070.89 against the original design. In the transient simulations, the time step size was set at 0.001 and ran MMAO (OASIS) 0.142430 0.15588 0.23567 0.24059 0.12091 5099.85 for 1000 time steps. Given the rotational speed of the rotor, around 5.83 rotations are expected in one second. An output parameter that has a direct relation to the performance of the turbine, which is the 8.2. Comparative Analyses and Discussions surface moment about the rotational axis of the rotor, has been chosen for comparison. The moment is the resultTo see of the effectactual of impact the sum of ofthe both optimized pressure shapes and viscous on the forces full turbine (Equation design, (11)). both Time-dependent steady and transientresponses CFD of the analyses moments (simulations) have been collected have been from thecarried surfaces out ofand the twothe parts:responses collected and compared against the original design. In the transient simulations, the time step size was set at 0.001 (i). From the surfaces of one of the rotors’ blade (1); see Figure4b. and ran for 1000 time steps. Given the rotational speed of the rotor, around 5.83 rotations are expected (ii).in oneFrom second. the An surfaces output of parameter the entire setthat of has rotor a direct blades relation (2). to the performance of the turbine, which is theNote surface that moment in order toabout avoid the some rotational irregularities axis of seenthe rotor, on the has responses been chosen at the beginningfor comparison. of the ﬁrstThe momentcycle, it was is the decided result to of consider the effect the of data the from sum the of second both pressure cycle in bothand casesviscous of theforces comparative (Equation studies. (11)). Time-dependentThe moment responses responses of and the theirmoments peak have values been of thecollected second from rotation the surfaces (cycle) of from thethe two transient parts: (i).analyses From are the examined surfaces of using one comparativeof the rotors’ graphs. blade (1); The see comparative Figure 4b. graph of the moment responses (ii). From the surfaces of the entire set of rotor blades (2). Note that in order to avoid some irregularities seen on the responses at the beginning of the first cycle, it was decided to consider the data from the second cycle in both cases of the comparative studies. The moment responses and their peak values of the second rotation (cycle) from the transient analyses are examined using comparative graphs. The comparative graph of the moment responses from the entire set of rotor blade surfaces, depicted in Figure 15, shows that both the optimized designs return better responses than the original design at most of the time steps, except at the beginning of the cycles. On the other hand, the responses from the MMAO optimized design are seen to outperform the GA-optimized design. Therefore, as can be seen from Table 4, the total power output of the MMAO-optimized design from the time-independent (steady) analyses is higher than that of the other two designs.

Energies 2018, 11, 248 15 of 18 from the entire set of rotor blade surfaces, depicted in Figure 15, shows that both the optimized designs return better responses than the original design at most of the time steps, except at the beginning of the cycles. On the other hand, the responses from the MMAO optimized design are seen to outperform the GA-optimized design. Therefore, as can be seen from Table4, the total power output of the MMAO-optimized design from the time-independent (steady) analyses is higher than that of the other twoEnergiesEnergies designs. 20182018,, 1111,, 248248 1515 ofof 1818

Figure 15. Moment responses from the entire rotor blade surface. FigureFigure 15.15. MomentMoment responsesresponses fromfromthe theentire entirerotor rotor blade blade surface. surface.

UnlikeUnlike thethe responsesresponses fromfrom thethe surfacessurfaces ofof thethe entientirere setset ofof rotorrotor bladeblade surfacessurfaces case,case, thethe plotplot inin Unlike the responses from the surfaces of the entire set of rotor blade surfaces case, the plot in FigureFigure 1616 indicatesindicates thatthat thethe momentmoment responsesresponses collectecollectedd fromfrom thethe singlesingle bladeblade (1)(1) showshow littlelittle variation.variation. Figure 16 indicates that the moment responses collected from the single blade (1) show little variation. However,However, itit isis clearclear andand possiblepossible toto deducededuce thatthat ththee optimizedoptimized designsdesigns stillstill returnreturn betterbetter responsesresponses inin However, it is clear and possible to deduce that the optimized designs still return better responses in magnitudemagnitude atat mostmost timetime stepssteps thanthan thethe originaloriginal design.design. magnitude at most time steps than the original design.

Figure 16. Moment responses from the single blade surfaces (1). FigureFigure 16.16.Moment Moment responsesresponsesfrom fromthe the single single blade blade surfaces surfaces (1). (1).

Moreover,Moreover, thethe steadysteady analyses’analyses’ resultsresults (Table(Table 3)3) estimateestimate aa powerpower improvementimprovement ofof aroundaround 5.33%5.33% Moreover, the steady analyses’ results (Table3) estimate a power improvement of around 5.33% andand 4.73%4.73% fromfrom thethe MMAO-optimizedMMAO-optimized designdesign andand thethe GA-optimizedGA-optimized design,design, respectively.respectively. OnOn thethe and 4.73% from the MMAO-optimized design and the GA-optimized design, respectively. On the otherother hand,hand, fromfrom thethe visualvisual analysisanalysis ofof thethe fluidfluid stream-bandstream-band widthwidth inin thethe firstfirst quarterquarter ofof thethe velocityvelocity other hand, from the visual analysis of the ﬂuid stream-band width in the ﬁrst quarter of the velocity streamlinestreamline fromfrom thethe steadysteady analyses,analyses, FigureFigure 1717 illustratesillustrates thethe volumesvolumes ofof thethe ineffectiveineffective fluidfluid crossingcrossing thethe rotorrotor inin thethe threethree models.models. TheThe optimizedoptimized modemodelsls havehave aa thinnerthinner fluidfluid stream-bandstream-band widthwidth thanthan thatthat ofof thethe originaloriginal model.model.

Energies 2018, 11, 248 16 of 18 streamline from the steady analyses, Figure 17 illustrates the volumes of the ineffective ﬂuid crossing the rotor in the three models. The optimized models have a thinner ﬂuid stream-band width than that Energiesof the original2018, 11, 248 model. 16 of 18

FigureFigure 17. Velocity streamlinesstreamlines from from steady steady analyses: analyses: (a )(a original) original model; model; (b) ( GA-optimizedb) GA-optimized model; model; and and(c) MMAO-optimized (c) MMAO-optimized model. model.

InIn addition, addition, considering considering the the ratio ratio of of the the sum sum of of all all peak peak moment moment responses responses from from the the three three models models fromfrom the the full full rotor rotor surface surface to to the the single single blade blade surfac surface,e, we wehave have seen seen that that an estimated an estimated average average of 8.61 of blades8.61 blades contributes contributes to the to power the power genera generationtion at each at eachcycle cycle of the of rotor. the rotor. Moreover,Moreover, asas oneone can can see see from from both both moment moment response response ﬁgures, figures, the response the response curves curves from all from models all modelscoincide coincide in some in cases, some but cases, show but a show similar a similar trend in tr allend cases, in all whichcases, which veriﬁes verifies that the that numerical the numerical model modelemployed employed is valid is and valid converging. and converging.

9.9. Conclusions Conclusions and and Recommendations Recommendations InIn this this article, article, CFD-driven CFD-driven shape shape optimization optimization is is performed performed on on the the cross-flow cross-ﬂow turbine turbine design. design. TheThe approachapproach providedprovided promising promising results results with with regard regard to improving to improving the performance the performance of the cross-ﬂow of the cross-flowturbine. The turbine. motivation The motivation for the study for isthe the study fact that,is the since fact that, hydropower since hydropower is one of the is one least of expensive the least expensiveand most abundantand most renewable abundant energyrenewable sources energy around sources the globe, around efforts the toglobe, improve efforts performance to improve by performancedesigning optimized by designing critical optimized parts for critical hydropower parts for facilities hydropower are indispensable. facilities are indispensable. TwoTwo different optimization methods have beenbeen employedemployed inin thethe approach.approach. One One is is the the widely-appliedwidely-applied multi-objective multi-objective Genetic Genetic Algorithm Algorithm (G (GA)A) optimization optimization method, method, and and the thesecond second is the is Metamodel-Assistedthe Metamodel-Assisted Optimization Optimization (MMAO) (MMAO) method method from fromthe OASIS the OASIS tool. Optimized tool. Optimized models models from bothfrom optimization both optimization methods methods showed showed an estimated an estimated 4.73% 4.73% and and 5.33% 5.33% improvement improvement on on the the output output power,power, respectively,respectively, compared compared to to the the original original un-optimized un-optimized design design model. model. Apart Apart from that, from the that, MMAO the MMAOoptimization optimization process process converged conver at fewerged at function fewer function evaluations evaluati andons lesser and processing lesser processing time than time its than its counterpart. One of the main reasons is that, unlike the GA method, the MMAO method generates samples towards the optimum region to assist the optimization process to converge with fewer function evaluations. Therefore, we can conclude that the approach we have introduced is effective for such a shape optimization problem to improve the performance of impulse turbines. This approach can be applied to optimize the performance of similar turbines. It is also revealed through the study that the MMAO

Energies 2018, 11, 248 17 of 18 counterpart. One of the main reasons is that, unlike the GA method, the MMAO method generates samples towards the optimum region to assist the optimization process to converge with fewer function evaluations. Therefore, we can conclude that the approach we have introduced is effective for such a shape optimization problem to improve the performance of impulse turbines. This approach can be applied to optimize the performance of similar turbines. It is also revealed through the study that the MMAO method offers a better performance design, and it is more effective than the GA method in identifying the Pareto frontier. The MMAO method has also been shown to be computationally less expensive than GA.

Acknowledgments: The study is funded by the Norwegian Ministry of Education and Research through University of Stavanger (IN-10603). The authors would like to acknowledge both institutions. The authors acknowledge the support from colleagues in the Product Design and Optimization Laboratory at Simon Fraser University, Canada. Author Contributions: The first author is responsible to develop the framework and the models, to run the analysis and write the paper. The second and the third authors provided valuable advices on the used methodology, evaluated reported results and conducted critical proofreading and review of the paper. Conflicts of Interest: The authors declare no potential conflict of interest.

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