A novel design of nozzle-diffuser to enhance performance of INVELOX wind turbine

Journal Pre-proof A novel design of nozzle-diffuser to enhance performance of INVELOX wind turbine

S. Rasoul Hosseini, Davoud Domiri Ganji PII:

S0360-5442(20)30189-4

DOI:

https://doi.org/10.1016/j.energy.2020.117082

Reference:

EGY 117082

To appear in:

Energy

Received Date:

26 August 2019

Accepted Date:

31 January 2020

Please cite this article as: S. Rasoul Hosseini, Davoud Domiri Ganji, A novel design of nozzlediffuser to enhance performance of INVELOX wind turbine, Energy (2020), https://doi.org/10.1016/j. energy.2020.117082

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A novel design of nozzle-diffuser to enhance performance of INVELOX wind turbine S. Rasoul Hosseini, Davoud Domiri Ganji  Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran

Abstract In this study, the performance of Invelox Wind Turbine, under the effect of geometric changes in the nozzle-diffuser section, is investigated by Finite Volume Method. To follow a more realistic approach, the wind speed equation is applied to the inlet. The effects of ratio of the length to nozzle cross-sectional area, diffuser length, diffusion angle, various nozzle design standards, also the angle and height of the added flange were described by the pressure and velocity distribution contours. The results showed that when the ratio of the nozzle length to throat diameter rises up to an optimum value, the channel flow rate increases and after that it begins to decrease. Then, the RSM optimization method was employed to optimize the diffuser, and optimal values of diffuser length to throat diameter and diffuser opening angle to nozzle opening angle ratios were obtained. Also, to apply the ideal nozzle the system was geometrically optimized to meet the Invelox system requirements. After that, the power rise due to adding the outlet flange was studied, and an appropriate geometric ratio was achieved. Finally, the system performance, when the turbine was modeled as a uniformly loaded actuator disc was investigated to find the most efficient turbine. Keywords: Wind energy, INVELOX wind turbine, Actuator disc, RSM optimization, Nozzle-diffuser.

Nomenclature Symbols

𝑢𝑒

Free stream wind velocity

𝐶2

Pressure jump

𝑢1

Wind turbine inlet wind velocity

𝐶𝑡

Load factor

𝑢2

Wind turbine exit wind velocity

𝐶𝑝 ∗

Input power coefficient

𝐶𝑝𝑏

base pressure coefficient

𝜏𝑤

𝐷 𝐷∗ 𝐸

Throat diameter Nozzle inlet diameter Nozzle-diffuser length

𝜌 𝜇 α

Greek symbols wall shear stress air density molecular viscosity material permeability

 Corresponding author: Assistant professor, Faculty of Mechanical Engineering, Babol Noshirvani University of Technology Email: [email protected] (Davoud Domiri Ganji)

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𝐻 𝐾 𝐿

Flange height Acceleration ratio Nozzel length

𝑛

diffuser length to diffuser inlet diameter Wind shear coefficient

𝑃1

Static pressure before turbine

𝑃2

Static pressure after turbine Static pressure at the channel output Static pressure of free flow Venturi velocity

𝐿∗

𝑃𝑏 𝑃𝑒 𝑢 𝑢𝑏

𝜇𝑡 𝜆 𝛾 𝜃 ∆𝑛

turbulent viscosity nozzle length to throat diameter opening angle of the diffuser to opening angle of the nozzle angle of the flange thickness of the porous layer

Abbreviations AD

Actuator disc

IWT

Invelox Wind Turbine

RSM N-D

Response surface methodology Nozzle diffuser

Wind speed in diffuser exit

1. Introduction Suitable energy resources, after human force, are the most important factor effective to the economy of industrialized countries, since energy is a key factor to sustain economic development, social welfare, improvement of the quality of life and community security. On the other hand, the limitations of fossil fuels are manifested, and also the security of alternative energy resources is a major issue. After the solar energy, wind energy is the second greatest source of energy and with the annual growth rate of 30% is the fastest growing renewable energy resource in the world [1]. Despite all advances in the field of wind energy, there are great challenges to the operation of traditional wind turbines, such as installation, operating at low wind speeds or frost situation, fins fatigue, repair and maintenance of the generator at high altitudes. Such challenges have put wind farms in the way of extinction [2]. In addition to the aerodynamic performance of turbines, since the output power of a turbine is proportional to the cubic wind speed, the wind flow plays a crucial role in the economization of this industry [3]. This means even a small increment in wind speed causes a significant increase in the power output .Hence, many researchers have tried to increase the wind speed in the turbine rotors. In ducted turbines, turbine rotors are surrounded by a chamber to direct the airflow into the system to increase the wind speed in the rotors, showing interesting results in the economic viability of the wind energy industry [4-7]. The first modern duct wind turbine was reported by Igra [8]. In this report, the positive effect of adding a duct wind turbine in the shape of a diffuser is studied. He argued that the presence of a turbine in a channel, as well as the flow separation at the end of the diffuser, reduces the channel efficiency. There are several numerical studies taking into consideration the effect of adding flange and various profiles of the nozzle (inlet) and diffuser [9-11]. Abe et al. [12] studied wind turbines with flange diffuser. They showed that the presence of a flange significantly reduces the load factor in numerical simulations.

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They also indicated that the opening angle of the diffuser has a major impact on the flange performance, because it greatly affects the flow separation in the channel. Liu and Yoshida [13] examined the physical phenomena associated with a turbine in a diffuser-shaped channel by simulating the turbine with the actuator disk method. In another research, Aranake et al. [14] studied the channel with the internal cross-section in the shape of Airfoil S1223. They tried to improve channel efficiency by increasing the flow rate through it. The selection of airfoil S-1223 is attributable to its high lift force because high lift force results in flow rate increment in the channel. They also achieved the power extraction up to 90% beyond the Betz limit. Recently, there has been a renewed interest in the optimization of ducted wind turbines, and many related numerical and experimental investigations can be found in the state-of-the-art literature. Kannan et al. [15] tried to optimize the design of a diffuser-augmented wind turbine. They studied the effect of wind velocity on different shapes of flanged diffusers to obtain a suitable diffuser for wind turbines. Aranake, Duraisam et al. [16] analyzed shrouded wind turbines using RANS solver method and designing fins for used turbines in the channel, simultaneously. These two methods were combined and optimized to obtain optimum turbine efficiency. There was no flange in the channel and the actuator disk model was employed. By using a 3D body-fitted RANS solver, the optimal solution was achieved confirming the precision of the design. Using an experimental model, Tang et al. [17] studied geometrical parameters of the airfoil shaped ducted wind turbine to evaluate the aerodynamic performance of the channel. In this model, it is assumed that the rotor is simulated by the actuator disk model. The effects of the tip clearance, angle of attack and screen position along the airfoil chord are investigated through a Design of Experiments (DoE)-based approach. The results revealed that when the angle of attack and tip clearance increase, the DWT performance improves. Increment in the angle of attack results in the reduction of back pressure coefficient. The pressure distribution shows that the rise in the tip clearance creates a favorable negative pressure behind the, therefore, an increased velocity would be induced through the rotor. However, up to now, using turbine channel is shown to be economical only for small-scale applications, and when the turbine power reaches over 500kw, it will not be economical anymore, considering the size of the turbine. As a result, the wind energy industry remained the same as traditional turbines which were set on the top of towers. A new concept called Invelox is introduced by Alaei and Andreopoulos [18] which significantly increases the turbine output power by performing initial tests. This system is designed in such a way to direct the wind from high altitudes to the ground and, and by passing it through a nozzlediffuser and increasing the wind speed, the wind turbine efficiency rises. Alaei and Andreopoulos [19] later investigated the Invelox system experimentally and without turbine, and their results showed that the ratio of the velocity at the throat to the free stream velocity increased in the range of 50-110%. Also, in another experimental research [20], they placed a turbine in the Invelox system and studied its effects on the system and the optimal number of turbines, and concluded that by placing three turbines in the system its efficiency will be 2.2 times the system efficiency with one turbine. Anbarsooz et al. [21] analyzed the important geometric parameters of the Invelox system, including the venturi diameter and the channel entrance height. The results indicated a

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significant impact of venturi diameter on the system performance. The results also revealed that one of the main problems with the initial system design is the inlet section that sends out the major amount of the inlet wind from the other-side inlet and does not direct it to the nozzle-diffuser section. In another recent research [23], they came up with an innovative curtain design to improve the Invelox system aerodynamically. Their results indicated that using this new design results in 25% increment of the average wind velocity in the venturi. In addition, a novel structure consisting of a two-storey Invelox turbine is recently proposed to enhance system efficiency. It was observed that, when this structure was used, a rise of about 44% in the output power was seen, revealing that when a storey is added to this device, it is possible to reach a higher performance without raising the maintenance cost. In another numerical study, Gohar et al. [23], by altering the design of a conventional Invelox system, studied a system with multiple wind turbines and comparatively analyzed the effects. Their results indicated that for the configuration of multiple wind turbines, the generated power employing multistage wind turbine is higher than the conventional ones. In the present study, ANSYS Fluent software is used to simulate the process of air flowing through an Invelox channel and to geometrically enhance the nozzle-diffuser section. First, in order to validate and achieve an accurate solution, the system was modeled numerically; then, the nozzle section was studied parametrically. To optimize the diffuser section, length and angle parameters were investigated by using response surface optimization method and computational fluid dynamics, and suitable results were obtained. Also, other design standards were studied. At this stage, using the historical data optimization method led to the optimal nozzle design. At the end of the geometric change process, the addition of the end ring to the channel and its opening angle were investigated. In the end, the actuator disk model was studied to obtain the most suitable wind turbine for the system to attain the characteristics of the ideal turbine applicable to the system.

2. Description of model and the CFD simulations 2.1. INVELOX geometry Figs. 1 and 2 show the dimensions and design of the Invelox system. This system presented by Alaei and Andreopoulos [19] is designed to entrain wind from all directions. Four fins are embedded in the inlet of the Invelox tower, contributing to further improvement in the performance of the inlet when capturing the free stream flow. Also, the intaken wind is delivered through a pipe carrier to the nozzle-diffuser. The system was modeled using the commercially available packages ANSYS16.2.

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Fig. 1. Problem geometry and computational domain.

Fig. 2. Geometrical dimensions of the Invelox channel.

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2.2. CFD simulations 2.2.1. Reynolds-averaged Navier-Stokes equations The governing equation for the conservation of mass in a compressible fluid flow can be written as: ∂ (u ) = 0 ∂xi i

(1)

where ui is the velocity component in each principal direction and ρ is the fluid density. In addition, the momentum conservation equation in each principal direction can be written as: ρ

∂ ∂P ∂τij (uiuj) = ― + ∂xj ∂xi ∂xj

(2)

τij is the Reynolds stress tensor which is defined as:

(

τij = μ

∂ui ∂xj

+

∂uj

)

2 ∂ui ― μ δij ∂xi 3 ∂xi

(3)

where δij is the Kronecker delta, which is unity when i and j are equal and is zero otherwise. According to turbulent flow characteristics, the field variables ui and p must be expressed as the sum of mean and fluctuating parts as: ui = Ui + u′i

𝑝 = 𝑃 + 𝑝′

(4)

with the bar denoting the time average. By inserting these definitions into Eqs. (1) and (2), the Eqs. (5) and (6) are obtained as follows: ∂ (U ) = 0 ∂xi i

(5)

∂ ∂P ∂(τij ― ρu′iu′j) ( ) UU =― ρ + ∂xj i j ∂xi ∂xj

(6)

In Eq. (6) the term u′iu′j is representative of Reynolds stress terms that has to be modeled via an appropriate turbulence model [24].

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2.2.2. Turbulence model Selecting an appropriate model for CFD turbulence simulation is very important because not every model can predict exactly separation phenomenon, and separation from nozzle-diffuser surface greatly affects the channel performance. For turbulence description, the k ω based Shear Stress Transport (SST) model with automatic wall functions, developed by Menter et al. [25], was employed. k represents the turbulence kinetic energy and ω represents the turbulence specific dissipation rate. The SST model combines the k ω and k ― ε models by applying a blending function which changes the model from k ω to k ― ε when the distance from the wall rises. Hence, this model uses good accuracy of k ― ω model near the wall, and the good convergence rate of k ― ε high-Reynolds model. This two-equation model is the most appropriate RANS-based turbulence model for predicting flow separation [25]. The two-equation SST turbulence model accounting for the effect of turbulence can be presented in the following form: ∂ ∂ ∂ ∂k (ρk) + (ρkuj) = P ― βρωk + [(μ + σkηt) ] ∂t ∂xj ∂xj ∂xj

(7)

ρσw2 ∂k ∂ω ∂ ∂ λ ∂ ∂w (μ + σwηt) (ρω) + (ρωuj) = P ― βρω2 + + 2(1 ― F1) ω ∂xj ∂xj ∂t ∂xj νt ∂xj ∂xj

[

{

∂ui P = min τij ,20βρωk ∂xj τij = ηt

(

∂ui ∂xj

+

∂uj ∂xi

―

]

}

(8)

(9)

)

2∂uk 2 δij ― ρkδij 3 ∂xk 3

(10)

The turbulent eddy viscosity is calculated from Eq. (11): ηt =

ρa1k

(11)

max (a1ω,ΩF2)

Each of the constants is a blend of an inner (1) and outer (2) constant, blended via blend functions F1 and F2. The expressions used for F1 and F2 are as Eqs. (12) and (13), respectively.

{{

(

)

k 500ν 4ρσw2k F1 = tan h min [max , ] , βωy y2ω CDkwy2

}} 4

(12)

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{{

(

2 k 500ν F2 = tan h max , βωy y2ω

)} } 4

(13)

y is the distance to the nearest wall and CDkω is calculated using Eq. 14: 1 ∂k ∂ω ―10 CDkw = man[2ρσω2 ,10 ] ω∂xi ∂xi

(14)

F1 equals zero, away from the surface (k ε model), and switches to one inside the boundary layer (k ω model). The solution of the governing equations is obtained using the Ansys-Fluent 16.2 commercial software with the Pressure-Based Segregated Algorithm. The pressure-velocity decoupling is done using the SIMPLE algorithm and the second-order upwind scheme is used for discretization of the pressure and momentum equations.

2.2.3 Actuator Disc model To model the Actuator Disc (AD) accurately, the experimental data of the pressure drop across the AD for various free stream velocities were used. A semi-empirical correlation is employed to model the effect of AD by adding a source term to the momentum equation (Eq. (2)). The following relation is obtained by using a second-order polynomial as a trend-line: ∆𝑝 = 𝜁𝑢2𝑖 ― 𝜂𝑢𝑖

(15)

where ∆𝑝 is the uniform pressure drop across the AD, and 𝜁 and 𝜂 are two arbitrary constants. The momentum source term includes the Darcy coefficient representing the viscous loss and the Forchheimer coefficient representing the inertial loss that are first and second term on the righthand side of Eq. (16), respectively, originated from the orifice equation [26]. Considering a simple homogeneous porous medium, the source term would be achieved as: 𝜇 1 𝑆𝑖 = ―( 𝑢𝑖 + 𝐶2 𝜌𝑢2𝑖 ) 𝛼 2

(16)

here α is permeability and C2 is pressure jump coefficient in the source term Si for the i ― th momentum equation. The relationship between the momentum source term and pressure drop would be as: ∆𝑝 = ― 𝑆𝑖 ∆𝑛

(17)

where ∆n is the porous medium thickness. Arbitrary constants of Eq. (15) are then obtained by:

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1

𝜁 = 𝐶22𝜌∆𝑛 𝜇

―𝜂 = 𝛼∆𝑛

(18) (19)

2.2.4 Fluidic parameters of the channel The load factor or 𝐶𝑡 represents the pressure drop of the diffuser; it means that the effect of replacing rotor into the channel and the resulted pressure drop is replaced by a volume force. It is stated as: Ct =

P1 ― P2 1 2 ρu 2 2

(20)

The ratio of the airflow velocity at the inlet to the free stream velocity is acceleration coefficient (K) and is defined as: K=

u1

(21)

ue

Another parameter that affects the performance of a turbine embedded in a flow channel is the input power factor Cp ∗ which is proportional to the load factor and it is stated as: C p ∗ = C tK 3

(22)

Base pressure coefficient Cpb represents the ratio of the duct pressure drop to the dynamic pressure of the free air stream: C pb =

Pb ― Pe 1 2 ρu 2 e

(23)

The pressure recovery coefficient 𝐶𝑝𝑑is another affecting parameter that shows the ratio of the turbine pressure drop to the dynamic pressure inside the turbine: 𝐶𝑝𝑑 =

𝑃𝑏 ― 𝑃2 1 𝜌𝑢 2 2 2

(24)

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2.3. Optimization The process of design optimization generally includes three main elements: design variables or parameters, an objective function, and constraints. Such process initiates by introducing input factors that affect the system of interest, which is defined by considering the design variables. Then, system outputs are generated by going through a design process for specified design variables. The objective function is achieved by analyzing the relationship between design variables and system outputs, which can be optimized based on the requirements. At the same time, both equality and inequality constraint functions and the bounding upper and lower limits of design variables can be defined and applied to the optimal solution. To perform the optimization process, the Response Surface Methodology (RSM) of the Design Expert software is employed.

2.3.1 Response surface methodology Response surface methodology [27-29] includes a group of mathematical and statistical techniques for empirical model building. The RSM is used to optimize a response (dependent variable) that is affected by other independent variables and to develop the relationship between them. This relationship can be described in a general form of [30]: y = F(x1,x2,…,xn𝜈) + ε

(25)

Here, y is the response variable, 𝑥𝑖 represents design variables, and 𝜀 is the total error, which is generally considered having a normal distribution with zero mean. 𝐹 is the response surface model considered as a second-order polynomial, written for n𝜈 design variables as: 𝑦 = 𝑐0 +

∑𝑐 𝑥 + ∑ 𝑖 𝑖

𝑖

𝑐𝑖𝑗𝑥𝑖𝑥𝑗 , 𝑝 = 1,…, 𝑛𝑠

(26)

1 ≤ 𝑖 ≤ 𝑗 ≤ n𝜈

where 𝑦 is the RSM dependent variable, 𝑐0, 𝑐𝑖, and 𝑐𝑖𝑗 are regression coefficients, and 𝑛𝑠 represents the number of observations. Eqs. (25) and (26) can be further stated in a matrix form, and the leastsquares method is normally used to estimate the regression coefficients. (27)

𝑦 = 𝑋𝑐 𝑐 = (𝑋𝑇𝑋)

―1 𝑇

𝑋 𝑦

(28)

where 𝑦, 𝑐 and 𝑛𝑟𝑐 are representative of the response vector, regression coefficients vector, and the number of regression coefficients, respectively; also, X is a 𝑛𝑠 × 𝑛𝑟𝑐 matrix. In the present study, the Central Composite Design (CCD), which is widely used in the literature, is used as shown in Fig. 3. CCD was first introduced by Box and Wilson [31] and for two design variables, it includes four factorial points, four axial points, and one or more central points. Generally, DOE uses the principle of replication to enhance the reliability of the experimental results by carrying out some

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tests. However, the principle of replication is not applied here since DOE is performed with a CFD problem without any experiment. Hence, only one numerical analysis was performed at the central point in the CCD. The curvilinear regression model, as expressed in Eq. (26), is used for detecting variations of dependent variable with those of independent ones. As it is observed in Fig. 3, for two design variables in CCD, each independent variable has observation results at five different levels (-α, -1, 0, +1, +α) allowing estimation of a second-order response surface.

Fig. 3. Central composite design for two design variables at two levels.

2.4. Mesh and boundary conditions In order to reach an accurate solution, it is essential to use appropriate boundary conditions and gridding. ANSYS 16.2 software package is employed for grid generation. To increase the simulation accuracy and observe the effect of flow separation, fine gridding is used in the nozzlediffuser part and in the vicinity of the channel walls. In addition, the boundary layer mesh (prism elements) is used on the inner surfaces of the channel, as shown in Fig. 4.

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a)

b)

Fig. 4. Mesh distributions: a) outside the computational domain, and b) inside and adjacent to the walls.

To verify the solution convergence, a mesh sensitivity test was conducted for the validation case. The grid independency check is carried out using the inlet velocity of 6.71 m/s. For this purpose, different meshing models have been applied, details of which are given in Table 1. Table 1 Study of mesh independency.

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Number of the cells

Y+

𝑢 𝑢∞

Error

(a)

613427

2-200

1.42

9.75%

(b)

1214957

1-80

1.48

5.8%

(c)

1835021

1-5

1.61

2%

(d)

2454878

1-5

1.61

2%

According to the suitable accuracy of this grid and considering the computational cost, mesh model (c) was used to perform simulations. All boundary conditions applied to the solution domain can be seen in Fig. 1. In this simulation, the velocity-inlet boundary condition is used for the inlet of the system. To make the solution configurations closer to the real condition, the real wind speed equation is used at the inlet. In a normal condition, the wind speed increases with distancing from the ground. Generally, the wind speed profile is obtained by the following equation [32-33]: ∗

𝑢0 = 𝑢0(

𝐻0 ∗

𝑛

𝐻0)

(29)

𝑢0: Known velocity at the altitude of 𝐻0 𝑢0 ∗ : Unknown velocity at the altitude of 𝐻0 ∗ 𝑛: Wind shear coefficient, which is chosen according to the environmental conditions of the research (the type of grand cover) [20], and is considered 𝑛 = 0.15 here. This equation is applied to the inlet boundary condition, using a user-defined function (UFD) in Fluent. Also, 𝑢0 in Eq. 29 is determined in a way that the known velocity at the altitude of 𝐻0 equals the average free-stream velocity obtained by the experimental data [20]. Hence, the wind speed equation applied to the inlet of the solution domain is equal to the wind speed of the test environment in [20]. Pressure outlet boundary condition is used for the outlets to minimize geometrical disturbances in the flow and reach a fully developed state in the outflow. No-slip boundary condition is used on the channel walls, as well as the ground surface. Due to the symmetric nature of the problem and to reduce the computational costs, symmetry boundary condition is utilized. The convergence criterion for the solution is considered 10 ―5.

3. Validation The computational fluid dynamics analysis of Alaei and Andreopoulos [19] has been carried out again to validate the solution methods and to further improve the accuracy of the validation process, the experimental process of the same study is validated. For numerical solution, the inlet

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velocity was considered to be constant, in ―𝑍 direction and equal to 6.71. The results are compared in Table 2. As can be seen, computational errors are less than 2%. Table 2 Comparison of the results of the present study with numerical results. Flow direction

Free stream (m/s)

Alaei and Andreopoulos [19]

-z

Present study

-z

Venturi velocity (𝑚 𝑠)

Speed ratio (SR) error

Average

Max

Average

Max

6.71

10.6

12.1

1.58

1.8

6.71

10.81

12.4

1.61

1.85

2%

In the experiment of Alaei and Andreopoulos [19], the air flow rate was determined by 23 different data from a sensor at the altitude of 8 ft above the surface of the Invelox and a sensor at the throat. The average free stream velocity obtained 8 ft above the Invelox is inserted to the equation of wind stream in the environment (Eq. 29) to achieve the equation of free stream velocity in the experimental environment. This equation is then used as the inlet boundary condition. Since wind flows in any direction to the system, the inlet equation is applied in three directions of ―𝑍 and + 𝑍 , ―𝑋, the results of which are shown in Table 3. Table 3 Comparison of the results of the present study with experimental results. Flow direction

Present study

Venturi velocity (m/s)

Speed ratio (SR)

Average

Max

Average

Max

+z

5.653

6.514

1.58

1.82

-z

6.159

7.305

1.72

2.04

-x

5.87

6.863

1.64

1.92

Average

5.894

6.927

1.646

1.92

Alaei and Andreopoulos [19] Error

1.77 -7 %

+8.85%

Considering that the speedometer sensors used in the experiment measure the velocity of one point of the throat section and the sensor location is not known, the results of the present study are obtained by averaging the velocity on the cross-section of the throat. Regarding the fact that the throat velocity is minimum on the centerline and becomes maximum adjacent to the wall, and the numerical results obtained the average and maximum velocity on the surface with -7% and +8.85% errors, respectively without having the exact location of the sensor, it can be concluded that the results are predicted within the range and with an acceptable accuracy.

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In view of the validation results, it is also found that if the flow is in ―𝑍 direction, the mean velocity is 9.4% higher than that of + 𝑍 direction and 4.8% higher than that of X direction. As a result, the system can be placed in a direction that the flow is dominant in the ―𝑍 direction.

4. Results and discussion 4.1. Parametric analysis of nozzle To initiate the process of applying geometrical changes, the nozzle section is analyzed parametrically. The procedure of applying changes can be observed in Fig. 5.

Fig. 5. Geometrical variation of the nozzle section.

Fig. 6 shows the effect of the nozzle length on the flow behavior inside the nozzle-diffuser section of the Invelox system; the velocity rate (the ratio of the velocity at the throat to the velocity of the free stream) and pressure coefficient graphs are presented for different ratios of the nozzle length L

to throat diameter (λ = D = 1, 1.16, 1.33, 1.5, 1.66 and 1.83). From Fig. 6 (a) it can be concluded that the maximum velocity at the nozzle-diffuser throat increases by increasing the nozzle length to throat diameter ratio up to 1.33, by exceeding which the velocity begins to decrease. The axial pressure coefficient distribution is plotted in Fig. 6 (b) for the same values of λ. In this figure, it can be seen that the pressure at the nozzle-diffuser throat decreases up to λ = 1.33.

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(a)

(b)

Fig. 6. Distributions of (a) wind velocity and (b) axial pressure coefficient in the entire section of the nozzle-diffuser in terms of nozzle changes.

Fig. 7 depicts the streamlines and corresponding pressure contours inside the INVELOX for the best and worst condition of the parametric nozzle analysis. As shown in Fig. 7 (b), whenλ = 1.83 , the fluid flow on the upper wall of the diffuser separates and a large vortex appears, resulting in a significant drop in the flow velocity through the channel. The effect of this phenomenon can be observed in the pressure contour inside the diffuser section. Thus, by choosing λ = 1.33 and applying corresponding changes, the diffuser section is analyzed. (a)

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(b)

Fig. 7. Streamline and pressure coefficient contours in the nozzle-diffuser in (a) the best and (b) the worst condition of the analysis.

4.2. Geometric analysis of the diffuser In order to analyze the diffuser section in terms of two effective parameters, including 𝐿 ∗ (ratio of the diffuser length to throat diameter) and γ (ratio of the opening angle of the diffuser to that of the nozzle), the Response Surface Method was used in Design Express software. γ varies between 0.57 and 1.04 and 𝐿 ∗ ranges from 1.16 to 2.16, as shown in Fig. 8.

Fig. 8. Geometrical variation of the diffuser section.

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These two parameters are considered as input variables to the model and maximum velocity at the throat is the desired output. Doing the optimization process leads into a mathematical equation relating the influential parameters and the throat velocity: 2 𝑢 = ―16.265 + 8.57 × L ∗ + 38.654 × γ ― 4.31 × 𝐿 ∗ × 𝛾 ― 1.3248 × 𝐿 ∗ ― 20.639 × 𝛾2 (30)

The 3D graph of the equation (30) is displayed in Fig. 9.

Fig. 9. 3D surfaces of the speed rate of nozzle-diffuser throat based on the diffuser length and diffusion angle.

Doing the optimization process resulted in the geometry with 𝐿 ∗ = 1.87 (the diffuser length of 3.42 m or 11.23 ft) and γ = 0.81 (the diffuser opening angle of 11.33 degrees), as the optimal geometric characteristics of the diffuser section. Distributions of the average wind velocity and the average pressure coefficient in the whole section of the nozzle-diffuser are shown in Fig. 10. The results indicate a 6% growth in the ratio of the flow velocity in the nozzle-diffuser section to the velocity of the free stream, which is achieved by the optimization process. By comparing the obtained results in this section, it can be concluded that the length of the diffuser part has a direct relationship with the velocity at the throat, and when it increases, higher efficiencies can be achieved. By excessive rise or decline of the length or opening angle of the diffuser, the channel flow separates, resulting in a significant drop in the flow velocity.

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Fig. 10. Mean distributions of axial wind velocity and axial pressure coefficient in the nozzle-diffuser section considering diffuser changes.

As illustrated in Fig. 11 (b), by optimizing the diffuser, the low-pressure zone at the diffuser outlet expands which leads to the increment of the flow suction to the inside of the Invelox system; this results in velocity increase that can be observed in the velocity contour presented in Fig. 11 (a). (a)

(b)

Fig. 11. velocity magnitude and pressure coefficient contours in the nozzle-diffuser after optimizing the diffuser.

4.3. Optimization of the ideal nozzle To this section, nozzles and diffusers are considered to be in the form of a cone when analyzing the nozzle-diffuser cross-section. However, performance enhancement is achieved only when the

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system is aerodynamically fitted to the airflow. Hence, regarding the design standards of the nozzle, a new nozzle with better aerodynamic characteristics is designed. An ideal nozzle contains smooth and parallel lines (the flow, either at the inlet or the outlet, is not turbulent or not separated due to a sudden change in the velocity direction) and the velocity field should be uniform at the nozzle outlet, meaning that the outlet velocity should be in the axial direction. Such conditions should also be met next to the nozzle walls. Among studied nozzle design standards, as illustrated in Fig. 12, the ideal nozzle (Fig. 12 (b)) is obtained to be the one with the best performance; therefore, it is chosen to be optimized for further simulations.

Fig. 12. Different types of nozzle design standards.

The key factor to design this type of nozzle is finding the coordinates of the inflection point, which is the main and determining parameter in the design of the nozzle shape. In this research, in order to find the best coordinates for the inflection point, the historical data optimization method in the response surface section of the Design Expert Software is used. By performing the optimization process, the mathematical correlation between the throat velocity and the ideal nozzle shape is expressed as: 3 u = ―0.51 + 2.03X + 13.87Y ― 1.88XY + 2.26X2 + 16.9Y2 ― 3.1X2Y ― 0.706X3 ― 41.77Y(31 )

X and Y are the inflection point coordinates from the nozzle inlet, whose ranges of changes are well displayed in Fig. 13 to better understand the coordination. Variations of the throat velocity contour by the inflection point coordinates are shown in Fig. 13.

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Fig. 13. The contour of interaction between inputs and outputs of the model in the nozzle optimization process.

At this stage, by analyzing the input data and optimization code, the point with coordinates of (2.16,0.055) was selected as the optimized location. Considering this point as the inflection point, the shape of the nozzle-diffuser would change into as shown in Fig. 14.

Fig. 14. Final shape of nozzle-diffuser.

Distributions of mean wind velocity and pressure coefficient across the nozzle-diffuser section before and after the optimization of the nozzle design are shown in Fig. 15. The results showed

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that when the ideal nozzle is used, a growth of 6.6% in the ratio of the flow velocity in the nozzlediffuser section to the free stream velocity is achieved.

Fig. 15. Mean distributions of axial wind velocity and axial pressure coefficient in the nozzle-diffuser section of the ideal nozzle.

In addition, the streamlines corresponding to the geometrical changes made to the nozzle section of the Invelox system is depicted in Fig. 16. It can be seen that by applying the ideal nozzle and optimizing the geometry, when the system characteristics were considered, our expectations were met, including the uniform streamlines, delayed flow separation, and increased flow velocity at the nozzle-diffuser section.

Fig. 16. Streamlines of the system with the ideal nozzle.

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4.4. Flange effect 4.4.1. Flange height Here, the effect of adding a flange to the output of the system is studied. The effective parameter H examined at this stage is D ratio (H is flange height and D is throat diameter ). Therefore, flanges H with D ratios of 0.083, 0.125, 0.133, 0.15, 0.166, 0.208, 0.25, and 0.33 were studied. As depicted in Fig. 17, by increasing this ratio from 0 to 0.15, local pressure drop increases and as a result, suction increases. However, in geometries with ratios more than 0.15, the reduction of mass flow rate initiates.

Fig. 17. The effect of flange height on the velocity ratio and pressure coefficient.

Figs. 18 and 20 show the overview of the pressure field and streamline around the flange H diaphragm, when D=0.15. As is shown, adding flange causes flow separation on its wall, which creates a low-pressure zone in the outlet of the system. This low-pressure zone in the outlet of the system gives rise to a more intense suction inside the channel resulting in the velocity rise at the throat.

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Fig. 18. Streamlines of the system with flange diffuser.

Fig. 19. Pressure coefficient contour in the Invelox system with flange diffuser.

4.4.2. Flange angle To investigate the influence of end flange angle (𝜃), angles of 2.5, 5, 7.5, 10, and 15 degrees were considered. As it is observed in Fig. 20, the change in the flange angle slightly affects the amount

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of flow passing through the channel. However, the flange with the angle of 5° shows the best performance and improves the velocity ratio by about 2%.

Fig. 20. The effect of flange angle on the velocity ratio and pressure coefficient.

4.5. Load factor effect In order to show the channel's performance when a turbine is embedded inside of it, the effect of the turbine rotor can be replaced by a load factor. The reason is that both of the wind turbine rotor and the load factor have a common function of creating a pressure drop in their downstream [34].

4.5.1. Validation of the load factor (actuator disk) In order to apply the load factor to simulate turbine performance conditions, three parameters of 𝛼 the material permeability, ∆𝑛 the thickness of the layer, and C2 the jump factor are required. These parameters were obtained using the experimental data from the study [20] and relationships in Eq. (15) to (19), as presented in Table 4. Table 4 Model inputs to generate the load factor. Special parameters

Achieved values

material permeability

2.82E-06 𝑚2

thickness of the layer

0.005 m

jump factor

4.946

The results obtained by the numerical simulation of the Invelox system in the presence of the load factor and parameters shown in Table 5 are as follows:

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Table 5 Comparison of the results of the present study in the presence of the load factor with experimental results. ratio of the velocity at the throat to the free stream velocity (experimental study[20])

ratio of the velocity at the throat to the free stream velocity (present study)

velocity at the throat (m/s) (present study)

1.64

1.67

5.77

Given that the average free flow velocity at the inlet of the Invelox system was 3.4 m/s, to compare the results with an experimental study, the experimental samples were chosen with the inlet velocity between 3 to 4 m/s. The reason for that is, regarding the experimental data, as the free flow rate rises, the velocity ratio grows. Validation results show the similar effects of the parameters of Table 4 on the turbine performance in the experimental study.

4.5.2. Load factor effect In order to select an appropriate wind turbine with the most efficient performance, the effects of different load factors on the performance of the Invelox system were investigated. By comparing Figs. 21 and 22 representing the pressure and velocity contours for the load factor of 0.28 (a) and without load factor (b), it is observed that the pressure drop and velocity distributions are significantly affected by the load factor in the channel.

(a)

(b)

Fig. 21. The contours of the pressure coefficients, (a) 𝐶𝑡 = 0.28 and (b) 𝐶𝑡 =0

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(a)

(b)

Fig. 22. Velocity magnitude contour in N-D region of the duct, (a) 𝐶𝑡 and (b) 𝐶𝑡=0 = 0.28

The average velocity rate and the average pressure coefficient for the entire nozzle-diffuser section are displayed in Fig. 23. As it is shown, raising the load factor intensifies the pressure drop. Also, the non-uniformity of the pressure coefficient profile shows the effect of the load factor. However, the average velocity rate remains without breakpoints, due to the limitation of the continuity equation.

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(a)

(b)

Fig. 23. Comparison of on-axis distributions at different load factors: (a) wind velocity and (b) pressure coefficient.

Considering the relationship between the acceleration factor (the ratio of inlet air velocity to the free stream velocity) and the load factor in Fig. 24, it is clear that by increasing the load factor, the acceleration factor decreases until it reaches 𝐶𝑡 = 0.3. For this value of the load factor, the acceleration factor is still greater than one. In other words, the designed channel is still effective and able to increase the air velocity inside. As 𝐶𝑡 increases to be more than 0.3, the channel will not be effective in velocity increment, and the acceleration factor will be less than one.

Fig. 24. Invelox performance: the acceleration factor.

Fig. 25 shows the relationship between the load factor and the input power. (𝐶𝑝∗ is the load factor multiplied by the cubic acceleration factor). Evidently, in the range of 𝐶𝑡 < 0.11, the diffuser does

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not function properly. Hence, the best 𝐶𝑡 is estimated to be close to 0.11. The base pressure coefficient 𝐶𝑝𝑏, which is defined as the ratio of the difference between the pressure at the diffuser outlet and the ambient pressure to dynamic pressure of the free stream, is plotted versus the load factor in Fig. 26. From this figure, it is found that the rise in the load factor increases the base pressure coefficient. Following that, the base pressure coefficient reaches zero, meaning that the pressure in the diffuser output is equal to the ambient pressure; it eliminates the main function of the diffuser and also the channel effect on increasing the velocity of the passing air.

Fig. 25. Invelox performance: the input-power coefficient.

Fig. 26. Invelox performance: the base-pressure coefficient.

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Conclusion In this study, the performance of the Invelox system was investigated by analyzing the geometry of the nozzle-diffuser section. Some geometrical changes were applied, including changes in the nozzle length, diffuser length, diffuser opening angle, and also replacing conical nozzle by the ideal one and adding the exit flange. Then, efforts were made to investigate the height and opening angle and their effects on flow characteristics and channel's ability to increase the flow velocity. The main findings of the first part of the simulation can be summarized as follows:    

Increasing the nozzle and diffuser length or the diffuser angle, as long as the separation in the channel flow is avoided, increases the channel efficiency. Replacing the conical nozzle by the ideal nozzle significantly affects the flow rate and flow stream through the channel. Addition of the exit flange substantially increases the flow velocity through the channel, however, changing its angle has a slight impact on the flow rate increment. Each factor causing the flow separation in the channel reduces the channel flow, while the separation and local vortices outside of the diffuser results in the increment of the system efficiency due to the increase in pressure drop and flow rate.

In the second part of this study, the turbine conditions were applied by using Actuator Disc method to find a suitable turbine with the highest performance in the system. The results showed that by increasing Ct, due to increased pressure drop, the velocity ratio decreases at the throat, but the input power rises and reaches a constant value; also, the base pressure approaches zero. When the input power increases, it means that the turbine performance in the system improves, however, when the base pressure approaches zero, the channel performance deteriorates. The most suitable type of turbine for the system is selected by taking into account both base pressure and input power characteristics, simultaneously.

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Declarations of interest: none

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Highlights     

Employing three dimensional FVM to analyze the Invelox system Finding optimum values for nozzle and diffuser length to throat diameter ratios Optimizing the system to apply the ideal nozzle considering system requirements Adding an outlet flange to raise the output power Using actuator disk model to obtain the most suitable wind turbine for the system