Mathematical modeling of a horizontal axis shrouded wind turbine

Renewable Energy 146 (2020) 856e866

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Renewable Energy journal homepage: www.elsevier.com/locate/renene

Mathematical modeling of a horizontal axis shrouded wind turbine Nemat Keramat Siavash a, G. Najafi a, *, Teymour Tavakkoli Hashjin a, Barat Ghobadian a, Esmail Mahmoodi b a b

Tarbiat Modares University, Tehran, P.O. Box: 14115-111, Iran Shahrood University of Technology, Shahrood, P.O. Box: 3619995161, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 April 2019 Received in revised form 28 June 2019 Accepted 3 July 2019 Available online 9 July 2019

In this work, an accurate mathematical modeling of a shrouded wind turbine is presented. This work claims new equations to predict the power coefficient and speed-up ratio effects. In the proposed model, the duct-based pressure drop is considered and the velocity in the far-wake is not suggested to be the same as the bare wind turbines. The mathematical model revealed that any shroud with an optimum design hardly can provide a CP ¼ 1 for a wind turbine. However, using a properly designed duct, CP ¼ 0:93 can be reached practically. This work also reveals that the speed-up ratio should be 0:6 < g < 1:7; in order to exceed Betz limit and work in the optimum condition. The developed model presents a clear relation between the duct efficiency (hd ), duct exit area ratio (b) and its pressure loss coefficient (Kpd ). Finally, a practical limitation for duct parameters is suggested in order to exceed the Betz limit. Further, a physical assumption is applied to approximate the power coefficient, which indicated that the power coefficient growth never can reach the speed-up ratio scale. © 2019 Published by Elsevier Ltd.

Keywords: Shrouded wind turbine Mathematical modeling Power coefficient Speed-up ratio Augmentation

1. Introduction Limitation of fossil fuels, security of alternative energy sources, and environmental issues, like global warming and air pollution are the reasons to invest in developing sustainable energy sources. Among all renewable energy sources, wind-driven turbines have developed rapidly [1] and it has a significant contribution in supplying electric power of some countries like Denmark. The amount of mounted wind power in Iran is 70 MW while the energy council program for 2025 is 1700 MW [2]. Concerning complex wind patterns and wind farms with low wind speeds in the majority percentage of Iran's wind energy sources and researching new wind power system that produces higher power output adapted to Iran is strongly desired. Horizontal axis wind turbines utilize directed upstream thrust force to generate power. As the blades are rotating, they will be like a disc in the rotor plane that prevents free airflow. As a result, a pressure drop will occur across the rotor disc. The more thrust will result in more pressure drop. But this decline has an unwanted effect, which is airflow velocity decreasing through the rotor plane. However, for a bare wind turbine, the available power is 16/27 of

* Corresponding author. E-mail address: g.najafi@modares.ac.ir (G. Najafi). https://doi.org/10.1016/j.renene.2019.07.022 0960-1481/© 2019 Published by Elsevier Ltd.

whole power that is known as Betz limit [3]. To exceed the Betz limit, it's enough to develop a mechanism to increase airflow through the rotor plane. As wind power generation is proportional to cubed wind speed, a slight increase in approaching wind speed to a wind turbine will increase generated power. One of the ordinary approaches to augment a wind turbine performance is to use a diffuser-nozzle shroud surrounding it. This method of augmentation is used at first by ancient Iranian in vertical windmills to grind grains that some of them are still working in eastern states of Iran. But for modern wind turbines, it was initiated by Igra [4e6], Gilbert [7e9], and Foreman [10,11]. Surveying the shrouded wind turbines continued by several researchers and reported in the references that some of them are referred to in the next section. These researchers efforts were to concentrate wind energy in a diffuser and after that use a controlled boundary layer to prevent pressure loss by flow separation and increment of the mass flow inside the diffuser [12]. Phillips has studied shrouded wind turbines in his doctoral thesis and coordinated this issue [13] and Ohya et al. [14e16] have developed a brimmed diffuser (so-called wind-lens) to create a low-pressure region using strong vortex wake formation behind the shroud; in order to draw more air mass through in it. The object of shrouding wind turbines is to increase the generated power and exceed the Betz limit. The mechanism of

N. Keramat Siavash et al. / Renewable Energy 146 (2020) 856e866

Nomenclature11

Lowercase a a' p r

letters Axial Induction Factor Tangential Induction Factor Pressure Radial Distance from Rotor Axis

Greek letters a Angle of Attack b Duct Exit Area Ratio ¼ VVer g Speed-Up Ratio ¼ VVur h Efficiency of x Duct Pressure Drop Coefficient Uppercase A C P

letters Area of Coefficient of … Generated Power

efficiency increment is to produce a negative pressure zone in the duct exit. This will make a suction so as a result, the air flow rate in the rotor plane will be raised. Developing ducted wind energy traps is still the goal of many research centers, due to its cleanliness spirit. Wind energy can be found at every point of planet earth and people can have their own power plant hybrid with other renewable energy sources like solar power. This will bring an inactive defense due to the distributed generation that can make the households independent of regional/global political and economic vicissitude. Finally, it can be expressed that all these efforts are to reduce the ratio ‘cost of production per generated kWh’ and use low regime wind power and also start the turbine in low wind speeds. In the next section, a brief review of the reported results was prepared that assisted us to improve this work. 2. Background The issue (wind turbine augmentation) was started to be studied and investigated by more than 70 years ago. In the beginning, it was surveyed analytically, meanwhile, the result was such attractive for researchers that they rushed to continue it practically in order to examine the mathematical models validity. With the advent of powerful computers, computational fluid dynamics ran into the issue. 2.1. Beginning The first published document concerning ducted wind turbines was by Lilley and Rainbird in 1956 that had theoretically studied the mechanism [17]. Experimental research was started by Kogan [18] in 1962 up to 1963. But the economic attraction of being cheap and having oil in abundance stopped these efforts until the oil crises in 1973 due to the Yom Kippur War between Arabs and Israel. The embargo forced some oil purchaser countries to invest in developing alternative energy sources. Therefore, Igra in BenGurion University in Israel started to investigate and just immediately Grumman aerospace research center [19] in the USA started valuable experimental efforts to obtain the optimum shape for a suitable duct. These efforts opened a distinct path in front of researchers up to now. Hansen [3], Phillips [13], Van Bussel [20], Rio Vaz [21], Liu [22], and many others have studied this issue

R T V K Q

m r u U lr

Rotor Radius Generated Torque Velocity Total Pressure Loss Coefficient Air Flow Rate Turbine Speed Reduction Factor Air Density Angular Velocity of Air Behind Rotor Angular Velocity of Rotor r Local Tip Speed Ratio ¼ U Vu

Subscripts P T d u r e w p

Power Torque Duct Upstream Rotor Plane Duct Exit Far-Wake Pressure

857

theoretically and some of them have different results that come from initial assumptions and some simplifications. In 1979, a conference in the subject of wind energy innovative systems was conducted; the conclusion showed that although power augmentation is considerable, the cost of the duct makes it unusable [23]. An optimum design of a shrouded wind turbine should be done in a combination of three different methods as Rio Vaz suggested [24,25]. The methods consist of classical blade element theory (BET), vortex theory and CFD, which are developed separately by several researchers. For example, Fletcher [26] and Igra [6] have used the BET method, Wood [27] and Okulov et al. and Bontemps et al. [28,29] studied vortex theory and Abe [30], Aranake [31] and Palmer [32] developed the issue numerically. In the following, it is proposed to categories the literature review in two analytical and CFD sections, which in any part, experimental investigations will be concluded as many of works are a combined study; analyticalpractical and CFD-practical. Nevertheless, there are some effective pure experimental works that may appear in any of these categories. 2.2. Analytical studies In many studies, as same as the current work, the radial velocity gradient is ignored for simplification. But Gilbert in 1976 had studied the speed distribution in the radial direction of the rotor plane. He also studied the position of the rotor inside the duct and tried to achieve an optimum geometry for the shroud [8,9]. Before Gilbert, Igra tried to examine different shapes for shrouds. He also used flaps and slots that have made an augmentation coefficient of 2.4 compared to a bare wind turbine [6]. He has built a 3-m diameter rotor that produces 750 W meanwhile using a shroud (with duct exit area ratio of 1.6) in wind speed of 5 m/s. After these struggles, some researchers suggested an innovative layout for the shroud; a multi-slot duct. the new arrangement was able to expand the negative-pressure zone in the duct exit [12,13]. In 1981, Fletcher developed a theoretical method to model the flow pattern around the shroud considering the wake rotation and blade Reynolds number effects. He states that diffuser efficiency and its exit pressure coefficient controls the shroud impression on turbine performance [26]. Erik calculated the power coefficient as a function of mass conservation, energy augmentation and extracted

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power coefficients [33]. Over time, manufacturing technology improvements made plenty of work easy and cheap. Consequently, some activities that were stopped or has a very low growth rate, like shrouded wind turbines accelerated to grow again. Commercialized manufacturing process made researchers eager to work more seriously and plenty of publications came up. Hansen et al., In 2000 employed a simple 1-D mathematical model to analyze a ducted wind turbine. They indicated that the output power can exceed the Betz-limit with a ratio depending on mass flow rate increment, but the augmentation can never reach the relative speed-up [3]. This issue will be further discussed in the next section (see Fig. 13). In 2007, Van Bussel used the continuity equation, which was based on Betz's theory to model a shrouded wind turbine. He declares that maximizing the power coefficient depends on mass flow increment, which is proportional to the duct exit area ratio and the desired augmentation will be provided whenever the back pressure at duct exit exceeds a distinct limit. Van Bussel indicated that the power augmentation through an experiment will be less than what has been calculated; due to flow separation in the shroud wall, which is not covered in developed analytical model [20]. A unified analytical model is also developed by Werle and Presz, which is different from other suggested models. They used a force vector against the flow direction, which was proportional to rotor resisting torque. Werle's model has some breaks; for example, when the disc loading coefficient reaches 2, the relation cannot support the theory [34]. In 2014, Bontempo and Manna employed a semi-analytical method to model the exerted actuator disc as a bare/shrouded wind turbine. The method was nonlinear, so it was able to cover some nonlinear patterns of flow, like the wake rotation and flow velocity non-uniformity along blade radii. The developed model has the capability to compare bare wind turbine and ducted ones. Validation of the method was done by CFD, which shows a remarkable agreement [28]. Later they developed an axial momentum theory through a semi-analytical method to study the effects of a duct thrust on the performance of shrouded wind turbines. The investigation claims that in comparison to other linearized methods, this method fully accounts the wake rotation and rotor-duct interactions [35]. In 2014, Rio Vaz et al. developed a 1-D mathematical analysis, based on BEM, for diffuser augmented wind turbines. For the first time, Glauert's correction was applied at shrouded wind turbines by Rio Vaz. The application was done at high thrusts to avoid high values of axial induction [21]. However, the model has some limitation such as the speed-up ratio estimation. Speed-up ratio comes from CFD investigations for the model. The analysis was based on a hypothesis Vw ¼ ð1  2aÞVu , which is not used in this paper; the reason for not using this assumption will be discussed in the next section. In order to extend the model, Rio Vaz and D. Wood presented an innovative aerodynamic optimization for a ducted wind turbine based on BET. In the aforementioned study, they also accepted and applied the assumption of velocity equality in farwake for both bare and ducted wind turbine. Blade chord and twist angle for a shrouded wind turbine has been optimized for the first time, which were functions of duct speed-up ratio. The model has some limitation such as calculation of speed-up ratio and blade tip-loss correction formulation. The maximum augmentation by the model was estimated to be 1.35 [25]. Besides the limitations of the model, it is commercially cheap, accurate and fast to design a shrouded wind turbine considering some experimental inputs like tip-loss correction and speed-up ratio.

1

All units are in ISO system.

Hjort and Larsen mathematically modeled a multi-element diffuser and validated the results by CFD. They showed that a multi-element diffuser can exceed the Betz limit by 50% and using a passive stall control, noise propagation can be reduced making the turbine silent [12]. 2.3. Computational fluid dynamics 2.3.1. Simple diffuser ducts Vortec 7 is an augmented wind turbine that was developed during the 1970s and early 1980s by a research team in New Zealand. The turbine was optimized using wind tunnel data, which was based on Grumman works [7,8]. Vortec 7 uses a multi-slot diffuser to prevent separation and make the suction more powerful [36,37]. Phillips et al., in 1998 have written a CFD code in PHONICS to optimize Vortec 7. Their study was done to minimize the shroud manufacturing cost as much possible and maximize the output power. They optimized several parts of the shroud as nacelle size and shape, slot sizes, and duct expansion area ratio [38]. The study cited in the previous section by Hansen was accompanied by a CFD analysis to verify the results; using the NaviereStokes solver. The work indicates that for modeling a rotor of shrouded wind turbines, the actuator disc theory is sufficient [3]. Hansen has used an airfoil (NACA 0015) profile for the duct crosssection. Bet and Grassmann also used an airfoil profile for a twopiece duct with augmentation ratio 2 [39]. Airfoil profiles for big and controllable ducts do not seem to be financially appropriate. A controllable duct can alter the duct shrouding angle. For example, Siavash et al. have developed an innovative layout that the diffuser part of the duct is in two pieces, which can open the duct wall up to 180 [40]. Kesby et al. [41] developed a combined CFD/BEM method to determine diffuser augmented wind turbine performance. The method combines numerically predicted flow patterns through the diffuser and the flow behavior modified by blade element theory to predict the generated power and diffuser augmentation ratio proposed under a variety of thrust, swirl, and free-stream conditions. Aranake et al. investigated the flow fields around the shrouded wind turbines to find the separation regions and velocity profiles at any point of the duct. The investigation was performed in a full 3-D CFD simulation, where the results show 1.9 times power augmentation. The study suggests using a high-lift airfoil for the duct crosssection. Power augmentation was 3.39 for Selig S1223 when it is 1.9 for NACA0006 at the wind speed of 5 m/s in comparison to a bare wind turbine [31]. Interactions between the turbine wake and the duct boundary layer are also studied by Aranake. In order to design a rotor for shrouded wind turbines, Hjort and Larsen advanced a swirled actuator disc CFD code and computed power efficiency versus thrust coefficient for low and high values of TSR, for different nacelle, and different span-wise rotor thrust loading. Their study consisted of a bare wind turbine, a ducted and a multi-element ducted configuration. The investigation revealed that in order to avoid stall-induced power drops, the rotor should be designed for low operating TSRs; as it is for a high capacity augmentation ducted wind turbine [42]. In 2017, Aranake and Duraisamy advanced an axisymmetric Reynolds averaged NaviereStokes model using an actuator disc to wind flow pattern around a shrouded wind turbine analysis. The method yields an accurate blade design technique to maximize power generation. Results showed 2.64 and 1.43 times power augmentation in the rotor plane and duct exit respectively. Structural problems are claimed for large turbines and the suggested solution is to use smaller scales [41]. In 2016, Kosasih and Hudin investigated a micro wind turbine using a shroud to determine its performance. On different levels of

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turbulence (2e29% generated by turbulence grids), they showed that the performance decreases with raising turbulence intensity. However, the ducted wind turbine can produce twice power compared to a bare one; independent from turbulence intensity and TSR [43].

new analytical approach is introduced to calculate the maximum power coefficient for a ducted wind turbine.

2.3.2. Brimmed diffuser ducts In the previous section, the survey held on diffuser augmented wind turbines that uses a straight diffuser or multi-element shroud that may have a tiny nozzle. However, in 2002, an innovative geometry was unveiled by Inoue [44] in Japan. Inoue suggested a flanged diffuser for the first time that has a considerably high performance. This type of ducts has been studied by Ohya [14,30] and others numerically and later compared with bare turbines and ducted ones. However, developing a comprehensive CFD study is still needed to reach optimum performance. Abe and Ohya in a numerical-experimental study declared that the disc loading coefficient of a flanged duct is considerably lower than bare ones [30]. A brimmed-ducted wind turbine can reach 1.6e2.4 augmentation ratio [45]. This amount of amplification is due to the extended low-pressure region behind the duct which draws more airflow through the diffuser. Ohya and Karasudani [14] also reported a 2.5-time increment in power extraction from windharvesting turbines using a flanged duct. It should be noted that flanged diffuser rotor rotational speed and dynamic strain is higher than flange-free ducts [46]. As flanged shrouds have more dynamic stresses due to flow tensions, a fatigue-problem study should be minded as proposed by Abe and Chaker [47,48]. Hu and Wang developed a self-adaptive flange (that work with centrifugal forces) for a shrouded wind turbine to reduce the wind loads acting on the flanged diffuser at high wind velocities. The suggested mechanism can contain the advantages of the brimmed diffuser up to critical wind speed and when the speed exceeds the limit, it can act as a classical duct to reduce wind loads acting on the duct and rotor. The study claims that 60% of external forces on the duct is due to the brim that they reduced to 34.5% by the self-adaptive flange [49]. Multi-rotor shrouded wind turbine is a recently developed mechanism that Ohya is trying to upgrade. He has placed two and three rotors in a single ducted structure in a horizontal row. Differing axial distance and number of turbines in wind tunnel investigation, 12% increment is claimed in 3-in-line arrangements. € ltenbott and Ohya et al. studied Aerodynamic interactions of Go these new systems. Experimental investigations shows a 9% and 5% power amplification for 2 and 3 rotor mechanisms respectively [50]. The study contains comparing one, two and three rotors for bare, conventional ducted and flanged ducted wind turbines. As seen, a lot of investigations were held by several researchers in each of three available methods (theoretically, numerically and practically). In this work, some of the analytical investigation deficiencies are elicited. As reviewed, several mathematical studies have been done, which model the duct aerodynamic manner and predict its output [21,23,25]. In the following, an accurate mathematical analysis, which has two different features than other works is developed; one is the velocity reduction ratio in the far-wake, which is not considered equal to bare turbines (m s1  2as 1= 3) and the second is inserting a pressure drop factor in Bernoulli equations due to shroud wastes. The share of absorbed energy by the duct and rotor is discussed, which can be used for an optimization process in rotor-duct design.

Mathematical modeling of a shrouded wind turbine is performed using a combination of three different equations. The main structure is based on energy balance, which gets assistance from classical wind turbine aerodynamics relations to develop a new journey to reach the goal.

3.1. Analytical modeling

3.1.1. Energy balance Let's imagine the ducted wind turbine (DWT), as shown in Fig. 1, as an actuator disc through a cylindrical control volume (C.V). For the C.V, total pressure balance can be written as:

1 1 2 pu þ rVu2 ¼ pþ r þ rVr 2 2

(1)

1 2 1 2 p r þ rVr ¼ pe þ rVe þ Dpd 2 2

(2)

1 1 2 pe þ rVe2 ¼ pu þ rVw 2 2

(3)

 Where pþ r and pr are static pressure in front and behind of the rotor plane respectively and Dpd is pressure drop by the duct. For steady state incompressible flow, the energy balance equation is:


Pextraced by rotor ¼ Q

 1 2 1 2 rVu  rVw  Q Dpd 2 2

For an easy calculation process, imagining some constants are needed. Before expressing these dimensionless phrases, it is worthy to discuss the relationship between upstream and far-wake velocities. Many well-known authors like Rio Vaz [21] and Phillips [13] had assumed that:

Vw ¼ ð1  2aÞVu

(5)

Which simply can be proved for a bare wind turbine; Where a ¼ 1  VVur . However, there is no analytical solution that shows the velocity in far-wake for ducted turbines is as same as the bare one. Hence, in this work, the relationship is defined as Eq. (6), that is a general form of it.

3. Materials and methods Regarding the background review, developing a much accurate analytical model is intransitive to discuss shrouded wind turbines, which this work has taken a step toward in this way. In this part, a

(4)

Fig. 1. Actuator disc control volume for a ducted wind turbine.

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m¼

Vw Vu

(6)

Which m is a function of g (velocity speed-up ratio in rotor plane), b (duct inlet area to its outlet cross-section) and hd (duct efficiency). Assuming, air is incompressible, and its density is frigid, the constants are listed:

a ¼

u 2U

(7)

g¼

Vr Vu

(8)

b¼

Ve Vr

(9)

0

It should be noticed that the g is the speed-up ratio for a full ducted wind turbine (not an empty duct). The sign is shown by ε in some well-known works like Rio Vaz [21]. Rio Vaz uses g as the speed-up ratio for an empty duct and makes a relation between g and ðε ¼ gð1  aÞÞ. However, in this work, the axial induction factor is not accepted as it is in the bare wind turbines. To find m, let's rearrange Eq. (3) [13],:

pe  pu ¼

 1  2 r Vw  Ve2 2

(10)

pe  pu ¼ 1 rV 2 u 2

!  g2 b

2

pe  pu 1 rV 2 u 2

(11)

p   hd ¼  e 1 r V2  V2 r e 2

(18)

Eq. (16) can be used for a bare wind turbine considering g ¼ 1  a, b ¼ 1 and hd ¼ 0.

3.1.2. Rotor as a moving disc In a moving disc the extracted power is:

  P ¼ FV ¼ Dpr Ar Vr ¼ Dpr pR2 ðgVu Þ

(19)

DPr is pressure difference between front and back of disc that can be calculated using Eq. (1):  1  2  2 pþ r  pr ¼ ðpu  pe Þ þ r Vu  Ve  Dpd 2

(20)

Assume:

1 2

Dpd ¼ rxVu2

(21)

 1  2 r Ve  Vw2 2

(22)

So, the pressure difference will be:

1  2



1 2

1 2

(12)



 (23)

As a result, Eq. (18) will complete as:

(13)

P¼

  1 rpR2 gVu3 1  m2  x 2

(24)

Appling Eq. (15) equal to Eq. (22) will result:



The duct efficiency is [21]:

p r

h   i 2 2 CP ¼ g 1  Cpe  g2 b  g2 1  b ð1  hd Þ

 2 2 2 2 2 Dpr ¼ pþ r  pr ¼ r Vu  Vw  rxVu ¼ rVu 1  m  x

Therefore, 2 Vw 2 ¼ m2 ¼ Cpe þ g2 b Vu2

CP can be rewritten as:

ðpu  pe Þ ¼

Left side of Eq. (10) is duct exit pressure coefficient [13]:

Cpe ¼

(17)

Rewrite as:

Dividing both sides in 12 rVu2 , equation will result [26]: 2 Vw Vu2

h    i P 2 ¼ g 1  m2  g2 1  b ð1  hd Þ 3 2 rAr Vu

CP ¼ 1



x ¼ g2 1  b2 ð1  hd Þ

(25)

(14)

From Eq. (2) and Eq. (13), Dpd can be calculated as:

   1  Dpd ¼ r Vr2  Ve2 þ p r  pe 2  1    1  1  2 ¼ r Vr2  Ve2  r Vr2  Ve2 hd ¼ r Vr  Ve2 ð1  hd Þ 2 2 2 (15) Now Eq. (4) can be completed as:

    1 2 1 2 1 P¼Q rVu  rVw  r Vr2  Ve2 ð1  hd Þ 2 2 2 h    i 1 2  Vr2  Ve2 ð1  hd Þ ¼ rVr Ar Vu2  Vw 2 h    i 1 ¼ rAr Vu3 g 1  m2  g2 1  b2 ð1  hd Þ 2

T¼

ð

ðR



0

ur2 ðrVr dAÞ ¼ 4rpUgVu a r3 dr

(26)

0




Power coefficient will be:

3.1.3. 3-1-3- Rotor as a rotating disc Exactly behind the rotor, the wake has an angular velocity of u. The torque equilibrium equation on the rotating disc requirement is:

So, the power is:

ðR 0 P ¼ T U ¼ 4rpU2 gVu a r 3 dr (16)

(27)

0

Calculated power by Eq. (22) and Eq. (25) should be equal as it is accepted in bare wind turbine analysis [51], thus:

N. Keramat Siavash et al. / Renewable Energy 146 (2020) 856e866

ðR   0 1 2 2 3 2 rpR gVu 1  m  x ¼ 4rpU gVu a r3 dr 2

(28)

0 0

Solving Eq. (26) results an innovative equation for a distribution along rotor radius. 0

a ¼

1  m2  x

4. Results and discussions The main result of this work is analytical relations that are discussed in this section and interpreted using graphical figures.

When a duct receives more airflow mass, it means that the capture area is bigger than the rotor area. Fig. 2 illustrates the capture area that its mass enters the rotor. Physically estimation of CP needs two features to be determined; total air flow power  1 3 (12 rAr Vu3 ) and extractable power (16 27 2 rAc Vu ). But practically, just x percent of capture area's available energy can get. So the CP will be: The x can be specified as total kinetic energy in rotor plane on kinetic energy in the capture area:

Er Vr2 Ar Ar ¼ ¼ g2 Ec Vu2 Ac Ac

(30)

Therefore, the physical estimation of total maximum power coefficient (the absorbed power by the rotor and the duct) will be:

16 CP ¼ g2 27

(31)

From Eq. (16), the maximum CP Will be reached, whenever hd ¼ 1. So using Eq. (16) and Eq. (30) results:



g 1m

2



But speed-up ratios optimum point need some more mathematical investigations. For m ¼ 1=3, which is the optimum amount in bare wind turbines, the speed-up ratio will be equal to 32. Physically it can be expressed that optimum m for a shrouded wind turbine should be smaller than 1/3; since the duct absorbs wind energy as the rotor do. However, in an ideal ducted wind turbine, considering m ¼ 1=3 will result in g ¼ 32. As a result, the maximum theoretical power coefficient and augmentation ratio will be 43 and 94 respectively.

(29)

2

4lr

3.2. Duct physical consideration

x¼

861

16 y g2 27

(32)

As a result, a distinct relations can be expressed between g and m for an ideal duct as:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16 g 27

my 1 

(33)

From Eq. (32), the speed-up ratio never can exceed

Fig. 2. Capture area in control volume.

27 16

, (g < 27 16).

4.1. Let rewrite Eq. (4) as

 1  2 r Vu  Vw2 ¼ Dpd þ Dpr 2

(34)

Now considering the pressure drops in the rotor and shroud as defined by Palmer et al. [32]:

Dpd ¼ rKpd Vr2

1 2

(35)

1 2

(36)

Dpr ¼ rKpr Vr2 Putting these two equations into Eq. (35) will result:

Kpr þ Kpd ¼

1  m2

g2

(37)

Using Eq. (20), Eq. (23) and Eq. (34) result in Kpd , which is a function of duct geometry.

  2 Kpd ¼ 1  b ð1  hd Þ

(38)

To predict a real duct performance, the running equation parameters range of variation should be limited to practicability and mathematical reasons. According to researchers results, b and hd Variation ranges are [Ref.13 Table 3.5]:

0:3 < b < 0:6 0:6 < hd < 0:9 This range is logical and can be caught experimentally. The aforementioned range is also wider in regards to the last pointed reference (reference 13). Note in this work ‘logic’ means that the parameters are in a math-physically acceptable range, which is

Fig. 3. Kpd variation in a logically domain for different hd and b.

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practical to manufacture. For example, producing a duct with b ¼ 0:1 or hd ¼ 1 is not logical. Fig. 3 was illustrated using Eq. (37) for a wide range of variation for conducted parameters. A smaller range of b makes more suction in duct exit but from a power-losing view, the smaller the b, bigger the Kpd . In other words, according to Fig. 3, Kpd will be reduced by increasing the b in a given hd . Fig. 3 illustrates the contour of Kpd variation in a logical domain for different hd and b. The dashed ellipse region is an acceptable domain of hd and b. The region indicates that a properly designed shrouded wind turbine's hd and b should be around 0.85 and 0.45 respectively. So Kpd of a well-designed duct will be approximately 0.12. Finally, using Eq. (16), Eq. (36), and Eq. (37), CP can be recalculated as:

CP ¼ Kpr g3

(39)

16 . Therefore, considering Eq. (30), the optimum value for Kpr is 27 g Considering Kpd ¼ 0 will result:

  CP ¼ 1  m 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1  m2 ¼ g 1  m2 Kpr

(40)

Assuming m is equal to the optimum value in bare wind turbines (m ¼ 0:333), Eq. (39) will be:

16 CP ¼ 27

sffiffiffiffiffiffiffi 2 8 ¼ g Kpr 9

Fig. 5. CP variation in a logic domain of g for different levels of Kpd and m ¼ 0.33.

growing the Kpr , the curves are going to converge to the Betz limit and Kpd ’s effect is absorbed slowly. Fig. 6 is a 3-D view of Fig. 5 that clearly shows the power coefficient as a function of Kpr and Kpd . Considering some practical points of view, these two figures show that the desired range of Kpd and Kpr are: Considering the mentioned range, the amount of ‘Kpr þ Kpd ’ for a logic range of m and g is presented in Fig. 7. The figure reveals that for a known Kpd , the speed-up ratio ðgÞ could be estimated. As mentioned, From Fig. 3, it can be estimated Kpd ¼ 0:12 for a properly designed shroud. And as we are not able to calculate an

(41)

Kpr , which indicates the rotor pressure coefficient, is a function of Kpd , g, and m. Fig. 4 illustrates, g, m, and Kpd effect on Kpr . This figure shows that minimizing m raises Kpr and also, Kpd alteration makes the same trend. For a distinct Kpd and m, the more accelerating wind flow in the rotor plane, the more Kpr decrease. It should be mentioned that in the whole of this paper, the authors intentionally have drawn the intended curves in purple color. Regarding Eq. (40), for Kpr < 2, the power coefficient will exceed the bare wind turbines. Kpr is a function of g. So Eq. (38) and Eq. (40) have different forms; in Eq. (38), Kpr is numerator and in the Eq. (40) its denominator. Therefore, a quadratic shape is expected for CP against Kpr . Fig. 5 illustrates the power coefficient as a function of Kpr for different levels of Kpd at m ¼ 0:33. The figure indicates that an ideal ducted wind turbine can reach CP ¼ 1:33 in m ¼ 0:33. From Fig. 5, it can be seen that the smaller Kpd, the more reachable power. But for Kpr, which results in more than one unit augmentation, there is an optimum point for maximum power coefficient. This figure is generated using Eq. (41) for a range of m, g, and Kpd . Using this figure, it can be expressed that: ‘a successful duct will be in the upper hand of Betz-limit line’. Also, from this figure, it can be seen that in a logic range of all parameters, by

Fig. 6. Power coefficient as a function of pressure loss in rotor and duct.

Fig. 4. Kpr as a function of speed-up ratio in different levels of Kpr and m.

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Fig. 7. Total pressure loss coefficient as a function of g in different level of m.

Fig. 8. CP variation in a logically domain of g for different levels of m and.Kpd ¼ 0:12

optimum amount for m in ducted wind turbines, so the power coefficient may be higher or lower than 1.33. Therefore, Fig. 8 is drawn to depict m effect on CP at Kpd ¼ 0:12. The figure illustrates that maximum CP for a ducted wind turbine is 0.93 in m ¼ 0:33. Rio Vaz and Wood predicted the maximum CP equal to 0.948 at hd ¼ 1 and g ¼ 1:6 (the ‘g ¼ 1:6’ is for a rotor-less duct) [25]. If any turbine can absorb all of wind flow energy, CP will rise to 1.11, which is impossible. Figs. 5 and 6 also emphasize that a successfully designed ducted wind turbine should work in Kpr < 0:5. Speed-up ratio (g) is one of the important parameters that controls the power augmentation ratio. Figs. 9 and 10 are prepared to picture CP as a function of g in different levels of Kpd and m. In Fig. 9, the line ‘A’ is generated considering the physical

Fig. 9. CP variation by g for different levels of Kps at.m ¼ 0:33

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Fig. 10. CP variation by g for different levels of m in Kpd ¼ 0:12.

assumption in Eq. (30). Later four figures, all points that Kpd should be lower than 0.3 in order to exceed the Betz limit. In Fig. 10, line ‘A’ shows a wind turbine CP that absorbs the wind flows energy entirely using a duct with Kpd ¼ 0:12. In this unreachable condition, maximum CP is 1.11. Line ‘B’ shows the maximum CP for a properly designed shrouded wind turbine. Line ‘D’ and ‘E’ are CP Variation against g for a lossless duct in m ¼ 0 and 0:33 respectively. The line ‘F’ connects the maximum points to each other. This curve can be used in a controlling procedure of a controllable duct. For example, Hu and Wang have suggested a self-adaptive flange for a shrouded wind turbine. The mechanism contains an oscillating brim that can control g and acting drag force on the duct structure [49]. Figs. 9 and 10, play up that g has an optimum point, which depends on Kpd and m. However, in Figs. 9 and 10 the optimum point of g is illustrated to generating maximum CP . Rio Vaz et al. have determined the g as 0:7  g  1[21]. Furthermore, Hoopen has reported the same domain experimentally for velocity speedup ratio in rotor plane [52]. Affirming these two reported result and considering Fig. 10, the power coefficient cannot exceed 0.8 practically. Although it should be noted that the consistency between Rio Vaz theory and Hoopen experimental results shows that the speed-up ratio range is not optimum and maybe it's possible to increase it further. Therefore, regarding the last four figures and the detailed literature review in the previous part, we can express that in a properly designed ducted wind turbine, it can be expected that: Kpr z0:2 and gz1:6. Fig. 11 repeats Fig. 5 as augmentation ratio against Kpr for different levels of Kpd and m ¼ 0:33. This figure emphasis that augmentation ratio of any ducted wind turbine cannot exceed 2.25. But practically, a properly designed ducted wind turbine can augment the power generation 1.57 times at Kpr ¼ 0:16 using a

Fig. 11. Augmentation ratio for m ¼ 0:33 and different levels of Kpd .

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Fig. 12. Rotor and duct share of absorbed power by a ducted wind turbine.

duct with expansion ratio 0.45 and efficiency equal to 0.85. In this figure, some experimental results from Foremen and Gilbert [19] (for three different ducts with b ¼ 0:36) are shown. Foreman's results show that maximum power augmentation occurs in Kpr z 0:6. The trend of Fig. 11 is quite different from palmers [32] CFD results. This notable difference between current work and Palmer's can be due to his assumption about the capture area. Palmer et al. assume the mass conservation between the capture area and the duct entrance, which many of the output of their works is based on this assumption. As Eq. (28) shows, mass conservation between capture area and duct entrance is not used and instead of that, it has been suggested the ‘x’ parameter, which determines the energy percentage. Palmer et al. have illustrated in Fig. 10 of reference [32] that the maximum augmentation ratio is 2.7. This means that the power coefficient of a ducted wind turbine can reach to 1.6, which needs a very high speed-up (in terms of maximum power coefficient equation by Rio Vaz [21] it should be: g ¼ 2:7). Present work has proved that reaching such high speed-up ratio is impossible. This claim is covered in Figs. 9 and 10. Comparing Eq. (16), Eq. (39) and Eq. (40) can specify the share of absorbed power through the rotor, duct, and whole turbine separately as:

CProtor ¼ g3 Kpr

(42)

CPduct ¼ g3 Kpd

(43)

  CPtotal ¼ g3 Kpr þ Kpd

(44)

Fig. 12 shows the duct and rotor share of power for a practically achievable condition. In considered conditions (m ¼ 0:33, b ¼ 0:45, and hd ¼ 0:85), growing the speed-up factor (g) decreases rotor share of absorbed power. These figures clarify that total absorbed power raises by increasing g, but from a distinct g, which depends on given conditions, the rotor CP starts to decrease. The right image in Fig. 12 illustrates the ‘Cpe ’ in dashed black line. It shows that a successfully designed duct exit pressure coefficient should be next to 0.6. This can help the designer to optimize the duct. For example, if a duct is designing for a wind speed of 8 m/s, the static pressure difference between duct exit and the free stream atmosphere should be around 20 Pa's. Van Bussel Calculated the maximum CP using the momentum theory equal to 0.7 or a bit higher, which occurs in a significant back pressure; Cpe <  1, [20]. As mentioned in the literature review section, the output power can exceed the Betz-limit with a ratio depending on mass flow rate increment, but the augmentation can never reach the relative speed-up ratio. Fig. 13, which is generated using Eqs. (30) and (31), illustrates this claim. It should be noticed that Fig. 13 was generated using a physical assumption (not an accurate mathematical). Fig.14 is a repetition of Fig. 13 for a real (left) and ideal (right) calculated ducted wind turbine comparing to the physically

Fig. 13. Power coefficient and rotor plane speed-up ratio vs turbine Speed reduction factor.

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Fig. 14. CP as a function of m in different level of.g

assumed model. The figure was generated for calculated CP against m for different levels of g. The left image is for b ¼ 0:45 and hd ¼ 0:85. The right image was drawn for an ideal duct that its efficiency is perfect. The black line in each illustration is the physical estimation of an ideal ducted wind turbine that m is a function of g as detailed in part 3e2. The right image clearly shows the consistency between physical and analytical estimation. 5. Conclusion In this work, the authors tried to develop a mathematical model for shrouded wind turbines, considering some corrections to previous works and also determining a logic domain to governing parameters. The share of absorbed power in duct and rotor is defined and the maximum power coefficient for a variety of conditions are illustrated. The model revealed that the speed-up ratio has an optimum value that depends on absorbed energy by the rotor and duct. In a well-made duct, the best speed-up ratio is around 1.6. This model also indicates that in an actual machine, the power coefficient cannot reach 1. But a properly designed ducted wind turbine (bz0:45 and hd z0:85) can reach CP ¼ 0:93. For reaching such power coefficient, the static pressure difference between duct exit and the atmosphere should be around 0.6. Exceeding the Betz limit requires a duct pressure loss coefficient to be less than 0.3, which depend on duct efficiency and its exit area ratio. Funding The authors are grateful to the Tarbiat Modares University(TMU) for financial supports given under IG/39705 grant for renewable Energies of Modares research group. Acknowledgment The authors' gratefully thank Professor Yaghoub Fathollahi for his generous financial support as the head of research deputy of Tarbiat Modares University. Also, we thank Mr. Liu Tongjie and Ali Siasarani for their treasured help in some parts of the project. - References [1] G. Najafi, B.J.R. Ghobadian, S.E. Reviews, LLK1694-wind energy resources and

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