The effects of shape and size on duct-augmented horizontal axis turbine performance

Abstract

This article seeks to contribute to knowledge on duct-augmented turbines by investigating the influence of the key geometric parameters of the duct on the turbine performance: (i) duct expansion angle and length, (ii) position of the duct relative to the rotor and (iii) added geometric features to the duct. A new analytic model is proposed for the duct-augmented turbine and used for the investigation. The proposed analytic model used in this study was developed with existing momentum and blade element analysis methodologies serving as its basis. Using the proposed analytic model, the duct length is found to be more influential on the duct turbine system performance than the duct expansion angle. In addition, the performance can be enhanced by addition of a flange to the duct trailing edge. The study also highlights that the optimum rotor location within a duct is slightly behind the minimum duct area.

Keywords

Duct turbine augmentation geometrical parameters parametrical study mathematical model

Introduction

Duct-augmented horizontal axis turbines are known to be more efficient in capturing moving fluid energy and converting the captured energy to mechanical power than bare turbines (Gaden and Bibeau, 2010; Ohya et al., 2005; Ohya and Karasudani, 2010). The performance enhancement is primarily attributed to the ability of the duct to concentrate the flow and hence increase mass flux through the rotor. The mechanism that increases the flux relates to the outward radial force, a radially inward reaction to the aerodynamic force acting radially inward on the duct. This force deflects the flow radially outward expanding the flow downstream of the turbine (Phillips, 2003). For an incompressible fluid, enlargement of the downstream streamtube has an associated enlargement of the corresponding upstream streamtube. As a consequence of the enlarged upstream zone, the mass flux at the duct inlet is increased. Another factor responsible for the enhanced performance by the duct turbine is the low-pressure region behind the duct that draws more flow into the rotor. This mechanism is explained in several studies such as Ohya et al. (2008; Ohya and Karasudani, 2010) and Mansour and Meskinkhod (2014).

The net fluid force that works to concentrate fluid into a duct is influenced by the flow around the duct and the presence of the rotor within the duct. It is important that duct shape is designed well to gain the benefit. Several duct shapes have been studied (Foreman et al., 1978; Hansen et al., 2000; Hjort and Larsen, 2014; Igra, 1981; Mansour and Meskinkhod, 2014; Matshusima et al., 2006; Ohya et al., 2008; Ohya and Karasudani, 2010; Phillips, 2003). Frustum profile (as shown Figure 1(a)) is the most studied duct shape (Mansour and Meskinkhod, 2014; Matshusima et al., 2006; Ohya et al., 2008; Ohya and Karasudani, 2010) and its simple geometric form avoids complexity in manufacture. With this design, the wake expansion effect is achieved by the enlargement of the duct section area towards the downstream. The next widely studied duct is the duct with airfoil wall section shown in Figure 1(b) (Foreman et al., 1978; Hansen et al., 2000; Hjort and Larsen, 2014; Igra, 1981; Phillips, 2003). Duct airfoil utilises the lifting effect (Hansen et al., 2000) that induces flow acceleration at the leading edge and the inlet of the duct where the rotor is located. Another considered duct shape is the convergent–divergent profile (as shown in Figure 1(c))–Lawn (2003), Phillips (2003) and Khamlaj and Rumpfkeil (2017). The profile incorporates a nozzle at the inlet as flow concentrator and a diffuser section to provide the acceleration effect by the flow expansion downstream. The convergent–divergent duct configuration also enables the turbine operation under bi-directional flow (Belloni et al., 2018).

Figure 1.

Various duct shapes: (a) Furtsum, (b) airfoil and (c) nozzle–diffuser.

In addition to the shape of the duct, the duct size also influences the flow augmentation. A large duct area ratio (i.e. large length to diameter ratio or large duct expansion angle) enhances the flow augmentation (Foreman et al., 1978; Igra, 1981; Jafari and Kosasih, 2014b; Matsushima et al., 2006; Owis et al., 2015). For the frustrum type duct, increased duct length results in increased flow augmentation around the duct entrance (Matsushima et al., 2006). Ducts with high area ratio have been applied (Foreman et al., 1978; Igra, 1981) to maximise the flow augmentation. However, an excessive duct area ratio not only results in high structural loading (Gaden and Bibeau, 2010) but also can have the undesirable consequence of flow separation occurring at some point along the inner duct wall. The flow separation weakens the expansion which, as a consequence, negatively affects the flow augmentation (Jafari and Kosasih, 2014b; Matsushima et al., 2006; Owis et al., 2015).

Modification of the duct wall also affects the power augmentation of the duct turbine. Incorporating a flange on the external duct wall (Figure 2(a)) can enhance the flow augmentation of the duct turbine. The flange generates a large flow circulation at the rear duct region that causes the back pressure to be sub-ambient (Mansour and Meskinkhod, 2014; Ohya et al., 2008; Ohya and Karasudani, 2010; Shives and Crawford, 2011). The sub-ambient pressure induces a suction zone behind drawing more flow into the duct. In addition to the external duct wall modification, internal modification of the duct is considered in a few studies. The addition of a step device at the internal duct wall (Figure 2(b)) can reduce the back-flow at the duct rear region, improving the flow concentration at the entrance (Roshan et al., 2015). Prasad et al. (2020) install vortex generator (VG) at the internal duct wall, adopting the principle in the wing of the aeroplane to assist the turbine to produce more flow circulation behind and hence low-pressure regions formed. The incorporation of slots (Figure 2(c)) is another added feature into the duct wall which aims at suppressing the flow separation to maintain the high augmentation for the duct turbine with the high area ratio (Foreman et al., 1978; Igra, 1981; Phillips, 2003). The slot enables external high energy flow to be injected tangent to the duct wall, enhancing the axial momentum within the boundary layer, suppressing severe adverse pressure gradient and frictional losses in the wall region of the large-angle duct (Foreman et al., 1978).

Figure 2.

Added features on the duct: (a) flange, (b) step and (c) slot.

To analyse and optimise the design parameter on the duct-augmented axis turbine, including the duct geometry, few mathematical models are available. A few of the models modify the thrust and mass flow equations in the actuator disc model to account for the effect of the duct on the performance of the turbine. Vaz et al. (2014) modify the mass flow equation at the rotor plane in the disc model by including a parameter of duct speed-up ratio γ, a ratio of the velocity at the duct entrance to the ambient velocity that is experimentally determined and is characterised by the duct shape and size. Bussel (2007) also includes similar flow augmentation factor in the mass flow equation where the factor is related to the diffuser area ratio and the back negative pressure generation under an empty duct. The thrust equation in these models (Bussel, 2007; Vaz et al., 2014) is similar to the momentum model for the bare turbine. The derivation of the models indicates the power augmentation is simply proportional to the flow augmentation factor by the duct. Another model by Jamieson (2008) uses a different dimensionless parameter of an axial induction factor at a zero loading (empty duct) to include the flow augmentation factor. The zero load induction factor is also characterised by the duct geometry and dimension. In deriving the thrust equation, the model applies a boundary condition at the full blockage rotor; the thrust coefficient is zero to be consistent with the ideal inviscid flow model. However, in reality, in the total blockage of flow through the system (axial induction factor equal 1), the rotor would experience high thrust due to drag associated the wake mixing. The model developed in Jamieson (2008) can be applied to the turbine with no duct case by assuming the induction factor at zero loading to be zero. Some other models such as Foreman et al. (1978), Lawn (2003), Palmer et al. (2017), Phillips (2003) and Siavash et al. (2020) and use the parameters that link the augmentation factor, that is, duct area ratio, duct efficiency and duct exit pressure. These models also account the pressure loss within the duct that is included in the thrust equation.

The existing models for the duct turbine predominantly rely on the use of the augmentation parameters under an unloaded condition (empty duct) where no rotor–duct interaction aspect is included. The presence of the rotor in the duct can influence the flow augmentation, thereby the exclusion of this effect into the models can yield a less accurate calculation. The duct–rotor interaction can add axial momentum on the boundary layer on the internal flow near the duct wall and thereby delay the onset of flow separation with the beneficial consequence of increasing the flow augmentation under loaded condition (Hansen et al., 2000; Oman et al., 1977; Phillips, 2003; Shives and Crawford, 2011). A few studies (Allsop et al., 2017; Palmer et al., 2017; Shives and Crawford, 2011) have accounted for the influence of the duct–rotor interaction by incorporating empirical expressions (obtained from an experiment) that link the rotor loadings and the augmentation parameters into their models. However, the developments of such an empirical model require many experimental or computational data to be generated for specific duct design. Using the empirical model approach, a parametric study of the influence of the various duct parameters on the performance of the duct turbine system would necessitate the use of significant time and material resources to obtain the required data set from experiment. Therefore, the analysis and optimization of the design parameters of the duct turbine using the mathematical models incorporated with the empirical solution are not ideal from a practical standpoint.

From the above review, there is a clear need for a systematic approach and workflow in the design of duct shapes to maximise the power gain of the duct-augmented turbine. However, most of the tools available rely on the experimental data, which is not practical when many duct shape variations are to be investigated. This article seeks to contribute to this need through a parametric study of the influence of the key duct geometric parameters, the duct expansion angle and length, the use of bi-directional duct and flange and the rotor location within the duct, on the performance of the duct-augmented turbine using a newly developed simplified model based on momentum and blade elements methodologies.

Derivation of the model

In this section, the model for the duct turbine is developed. In the developed model, an added force associated with the enhanced augmentation by the duct–rotor interaction as well as the wake mixing is accounted in the thrust equation. The duct–rotor interaction can suppress the flow separation by the mechanism of the enhanced momentum on the boundary layer on the internal flow near the duct wall, thereby increasing the flow augmentation. As a result, the force under loaded condition increases. The model modifies the boundary condition of the thrust coefficient at full blockage condition in the model by Jamieson (2008) which assumes to be zero to follow the ideal inviscid model. In the current model, the thrust coefficient at full blockage condition is experimentally determined. By utilising the experimental data, both the effect of the enhanced augmentation and the wake mixing are captured, through the thrust coefficient, in the model.

The model schematic shown in Figure 3 consists of a single-stream tube with the free-stream region (0), the rotor (inlet) region (1 and 2) and the outlet region (wake) (3) as indicated. The rotor is treated as a porous disc.

Figure 3.

The duct–rotor system.

A modification of the equation of the pressure drop across the rotor in the disc model is conducted, written as

Δ P = 0.5 ρ {(V_{0})}^{2} - 0.5 ρ {(V_{3})}^{2} + P_{b}

(1)

where P_b is the added total pressure. The modification relates to the role of the added energy associated with the increased force by the wake mixing as well as the enhanced augmentation (as the duct–rotor interaction) in the duct turbine which is not considered in the disc model.

From equation (1), the thrust coefficient $(= Δ P / 0.5 ρ V_{0}^{2})$ can be written as

C_{T} = (1 - \frac{V_{3}^{2}}{V_{0}^{2}}) + C p_{b}

(2)

where $C p_{b}$ is the added pressure coefficient associated with the added energy.

By introducing a wake induction factor, $f (a)$ as the deceleration factor towards the wake region is formulated by

f (a) = 1 - \frac{V_{3}}{V_{0}}

(3)

The thrust coefficient in equation (2) is then written as

C_{T} = 2 f (a) - f (a)^{2} - C p_{b}

(4)

To determine $f (a)$ , the far wake axial induction factor in the model by Jamieson (2008) is employed with a slight modification, such that

f (a) = k (a - a_{0}) + f'

(5)

where k is a constant, $a_{0}$ is an axial induction factor by duct in the zero power extraction, a is an axial induction factor at the rotor and $f'$ is a wake induction factor adjustment. The reason for adopting the $f'$ is to satisfy the zero thrust coefficient under zero loading $(a = a_{0})$ as no energy extraction takes place. Without $f'$ , at the zero loadings $(a = a_{0} and f (a) = 0)$ , mathematically, the thrust equation will be not zero as the equation contains the parameter of $C p_{b}$ . The detail in finding $f'$ is presented in section ‘The method to obtain the added pressure coefficient $C p_{b}$ and the wake induction factor adjustment $f'$ ’.

The parameter of k is found by considering fully blocked condition, that is, V₃ = 0. Accordingly, using equation (3), the following relation can be obtained

f (a_{stop}) = 1

(6)

where $a_{stop}$ is the induction factor when the flow stops. This variable can be assumed analytically to be equal to 1 at the fully blocked condition. Using these boundary conditions $(a_{stop} = 1 and f (a_{stop}) = 1)$ and applying them into equation (5), then k can be written as

k = \frac{1 - f'}{(1 - a_{0})}

(7)

Using equation (7), then $f (a)$ in equation (5) becomes

f (a) = \frac{1 - f'}{(1 - a_{0})} (a - a_{0}) + f'

(8)

The equation of the coefficient of performance is derived by multiplying the velocity ratio at the rotor plane relative to the ambient velocity $(V_{1} / V_{0})$ to the thrust coefficient $(C_{T})$ . The velocity ratio is calculated in the same way as in the actuator disc model, that is, $V_{1} / V_{0} = (1 - a)$ , however, as was also done by Jamieson (2008), the value of a is permitted to be negative to account for the enhanced speed on the rotor plane due to the flow acceleration induced by the duct. By combining the thrust equation (equation (4)), the wake induction equation (equation (8)) and the velocity ratio $(V_{1} / V_{0} = (1 - a))$ , the coefficient of performance is written as

C_{P} = (1 - a) (2 (\frac{1 - f'}{(1 - a_{0})} (a - a_{0}) + f') - {(\frac{1 - f'}{(1 - a_{0})} (a - a_{0}) + f')}^{2} - C p_{b})

(9)

The method to obtain the added pressure coefficient $C p_{b}$ and the wake induction factor adjustment $f'$

To find $C p_{b}$ , the thrust equation (equation (4)) is considered when the flow is fully blocked, that is, $f (a_{STOP}) = 1$ and $C_{T} = C_{TBLOCK}$ , which yields

C p_{b} = 1 - C_{TBLOCK}

(10)

where $C_{TBLOCK}$ is the thrust coefficient associated exclusively with the rotor disc at full blockage condition and is obtained through experiment or computation (computational fluid dynamics (CFD)). To determine this parameter, initially, the force acted on a solid disc (to represent the fully blocked rotor) which is placed in the corresponding duct from the experiment or CFD is measured. The measured force then is divided by the disc frontal area to obtain the $C_{TBLOCK}$ . The value of $C_{TBLOCK}$ can depend on the interaction factor between the rotor and the duct, that is, the rotor–duct wall gap and the wake mixing under loading condition.

To find the wake induction factor adjustment $f'$ , the thrust equation (equation (4)) is considered when the power extraction is zero, that is, $C_{T} = 0$ and $f (a_{0}) = f'$ , yielding to

C p_{b} = f'^{2} - 2 f'

(11)

By rearranging the equation (11), the wake induction factor by the duct $(f')$ can be written as

f' = 1 \pm \sqrt{1 - C p_{b}}

(12)

There are two possible values; however, only the negative value accords to the wake induction factor $f (a)$ less than one. Wake induction factor $f (a)$ values greater than 1 would correspond to non-physical negative downstream velocity $(i . e . V_{3} < 0)$ .

The implementation in blade element momentum model

The following section presents the modified blade element model (MBEM) for the duct turbine that includes the blade parameters and the wake rotation. The principle of the MBEM is to combine the momentum model and the blade element model for the duct turbine to correct the flow magnitude and direction on the blade element, which then are utilised to calculate the performance.

In the momentum model for the duct turbine, the single disc in the previous model is replaced by multi-annular discs $(dA = 2 Π rdr)$ . By modifying equation (4), the thrust equation for each annular rotor area $(dT = 0.5 ρ {V_{0}}^{2} C_{T} dA)$ can be written as

dT = ρ {V_{0}}^{2} r Π dr [(2 (k (a - a_{0}) + f')) - {(k (a - a_{0}) + f')}^{2} - C p_{b}]

(13)

Similar to the momentum model for the bare turbine, the torque on the annular area is written by

dQ = 4 Π ρ Ω a' r^{3} V_{0} (1 - a) dr

(14)

In the blade element model for the duct turbine, the previous multi-annular discs are assumed as multi-blade elements where the thrust and torque are characterised by the flow magnitude and direction acting on each blade element (Figure 4), formulated by

dT = B \frac{1}{2} ρ {V_{REL}}^{2} (C_{L} \cos φ + C_{D} \sin φ) c dr

(15)

dQ = B \frac{1}{2} ρ {V_{REL}}^{2} (C_{L} \sin φ - C_{D} \cos φ) cr dr

(16)

V_REL is a sum vector of the axial velocity at the rotor $(V_{0} (1 - a))$ and the rotational flow velocity $(Ω r (1 + a'))$ formulated by

V_{REL} = \sqrt{{(V_{0} (1 - a))}^{2} + {(Ω r (1 + a'))}^{2}}

(17)

C_L and C_D are lift and the drag coefficients which depend on the attack angle (α) and Reynolds number. The attack angle can be obtained from the angle difference between the blade pitch (θ) and the fluid relative angle $(φ)$ (as shown in Figure 4). The relative angle $(φ)$ , indicating the flow direction relative to the blade rotational plane, can be obtained from the following equation

φ = arc \tan (\frac{V_{0} (1 - a)}{Ω r (1 + a')})

(18)

To determine the flow direction and the magnitude $(φ and V_{REL})$ , the axial induction factor, $a$ , and rotational induction factors, $a'$ , are required to be determined initially using an iterative approach through following equations

a_{(i + 1)} = a_{0} + \frac{[0.5 (F^{*} + C p_{b} + {(k (a - a_{0}) + f')}^{2}) - f']}{k}

(19)

and

a'_{i + 1} = \frac{Q^{*}}{(1 - a)}

(20)

where

F^{*} = \frac{Bc {V_{REL}}^{2} (C_{L} \cos φ + C_{D} \sin φ)}{4 {V_{0}}^{2} Π r}

(21)

and

Q^{*} = \frac{B {V_{REL}}^{2} (C_{L} \sin φ - C_{D} \cos φ) c}{8 Π Ω r^{2} V_{0}}

(22)

Similar to the conventional BEM, equations (19) and (20) are obtained by equating the thrust in equations (13) and (15), and the torque in equations (14) and (16).

Figure 4.

The blade element of the duct turbine.

As the relative velocity and the relative angle on the blade are known (from the known a and a′ obtained from the iterative approach), the torque can be determined, and then, the power is calculated by

P_{w} = \int_{rh}^{R} Ω dQ

(23)

The coefficient of the performance is determined by

C_{P} = \frac{2 \int_{rh}^{R} Ω dQ}{ρ A {(V_{0})}^{3}}

(24)

where R is the turbine radius (m) and rh is the turbine hub radius (m).

This MBEM model considers the Prandtl tip loss factor, F(r), formulated by Prandtl (1927)

F (r) = \frac{2}{π} acos (\exp \frac{- B V_{0} (1 - a) (R - r)}{2 R V_{o}})

(25)

As shown in equation (25), since factor depends on the velocity on the rotor plane, $V_{0} (1 - a)$ , $F (r)$ can be obtained until a and a′ are known from the iterative approach. The tip loss factor F(r) from the corrected a and a′ is then utilised to correct the torque in equation (16).

The calculation procedure of MBEM

This section details the calculation of the performance of the duct turbine using the MBEM. The aim of this method is to calculate the power generated by the duct turbine from the inputted blade parameters and rotation and the experimental data of axial induction factor at zero power extraction and the thrust coefficient of the rotor at the full blockage condition. To calculate the power, initially, it determines the flow magnitude and direction on each blade element. An iterative method is utilised to assist in determining the flow magnitude and direction on each blade element through the a and a′ corrections using equations (19) and (20). As the flow magnitude and direction are known, the torque for each element can be determined (equation (16)). Then the torque in each element is summed, and the results are multiplied with the rotation, to determine the power (equation (23)). After the power is known, the performance coefficient can be determined (using equation (24)). Figure 5 details the steps to calculate the performance of the duct turbine.

Figure 5.

The steps to determine the performance of the duct turbine using MBEM.

Validation of the model

In this section, for the validation purpose, both modified disc model for the duct turbine and the MBEM are compared to the outcomes of experimental and computation work in the literature.

Comparison of the modified disc model for the duct turbine to the actuator disc model

In this section, the developed model is applied to the bare turbine case and the predicted performance is compared to that of the disc model. A CFD data of the thrust coefficient at full blockage condition are employed as the inputted parameter of $C_{TBLOCK}$ in the developed model. To determine $C_{TBLOCK}$ , a cylinder solid disc is simulated into a wind/water stream in CFD. Then, the coefficient is determined by dividing the measured force on the disc to the frontal disc area. The measured thrust coefficient from the solid disc in our CFD work is 1.41. This value is higher than the thrust coefficient when the flow stops in the ideal disc model (which is 1) as the CFD includes the effect of the added force associated with the wake mixing. Using the data of $C_{TBLOCK}$ (1.41) and the equations (10) and (12), $C p_{b}$ will be – 0.26 and $f'$ will be −0.12. It assumes the axial induction factor at full blockade condition $(a_{STOP})$ to be one $(as (1 - a_{STOP}) = 0)$ . As $a_{stop}$ and $f'$ are known, using equation (7), k is 1.18. Now, the thrust coefficient in equation (4) can be written as

C_{T} = 2.8 a - 1.4 a^{2}

(26)

and the performance coefficient (equation (9)) can be written as

C_{p} = (2.8 a - 1.4 a^{2}) (1 - a)

(27)

Figure 6(a) shows the comparison of the performance coefficients, $C_{p}$ for the variation of thrust coefficients $C_{T}$ predicted by the model in the bare turbine case and by the disc model. The results show a good agreement between the developed model and the disc model. The performance coefficient predicted by the developed model peaks at around 0.55, nearly the Betz limit (0.59) at the thrust coefficient close to 8/9, in coincidence with optimum thrust for the bare turbine in the disc model.

Figure 6.

(a) The performance characteristics of the model in bare turbine case and the disc model and (b) the thrust coefficients in various axial induction factor.

The developed model can calculate the force generated at high axial induction factors, where the uncorrected actuator disc model fails to perform (see Figure 6(b)). One of the limits of the disc model is that the calculation is no longer valid when the axial induction factors greater than 0.5, as the velocity in the far wake would be negative. In reality, axial induction factor values greater than 0.5 correspond to circulation of the flow in the downstream region (wake mixing), further increasing the drag on the rotor plane. As the disc theory makes no attempt to account for this phenomenon, the empirical relationships developed by Glauert (1926) are often used to predict wind/water turbine behaviour at the large axial induction factors. The empirical relationships are based on a fitting on the experimental data by Lock et al. (1926). As shown in Figure 6(b), the developed model predicts behaviour of the thrust generated at high axial induction factor and the curve generated is in good agreement with the experimental data.

Comparison modified disc model for the duct turbine to the experimental and computational works on the duct turbine

To validate the model in the duct turbine case, the performance characteristic is compared to the experimental and computational works by Abe and Ohya (2004) and Mansour and Meskinkhod (2014). The method to find the values of $C_{TBLOCK}$ is similar to the previous case in the bare turbine; however, now the disc is enclosed into the duct model in Abe and Ohya (2004) and Mansour and Meskinkhod (2014). The axial induction factor at zero loadings $(a_{0})$ is obtained by employing velocity field data obtained from the CFD for the corresponding duct (under unloaded condition), formulated by

a_{0} = 1 - γ_{0}

(28)

The details of the duct model and the input parameters are presented in Figure 7(a). Instead of thrust coefficient, the parameter of loading coefficient $(K = C_{T} (V_{o} / V_{1})^{2})$ is employed by following the experimental and computational works in Abe and Ohya (2004) and Mansour and Meskinkhod (2014).

Figure 7.

(a) The duct model and (b) the performance characteristics.

This comparison includes the curve of CP versus K generated by the model in Jamieson (2008) where the boundary condition for the fully blocked rotor is set to be zero to follow the inviscid model. From Figure 7(b), the developed model shows a good agreement to those in Abe and Ohya (2004) and Mansour and Meskinkhod (2014), both at low and high loading coefficients (K). A notable feature of the curve generated by the Jamieson (2008) model is a large deviation at the high loadings. The reason relates to the assumption of zero thrust at full blockade condition (to follow the inviscid assumption) that neglects the added force associated with the duct–rotor interaction and the wake mixing.

Comparison of MBEM to the experimental and computational works

The following section presents the comparison of the performance characteristic by the developed MBEM model to those generated by experimental and computational works by Jafari and Kosasih (2014a). The turbine in these works is 0.19 m diameter and has three blades with the NACA 63210 395 airfoil and with constant pitch of 25° and constant chord of 0.025 m. The duct model has the ratio of length to the diameter of 0.6. The turbine was placed in a wind tunnel with the working section of 2.25D (width) × 2.25D (height) × 7.5D (length). The details of the dimension of the duct turbine model and the computational domain are shown in Figure 8(a).

Figure 8.

(a) The duct turbine model in Jafari and Kosasih (2014a) and the computational set up to obtain $a_{o}$ (1) and $C_{TBLOCK}$ (2), (b) the distribution of the $a_{0}$ in the radial direction of the rotor plane obtained from the computation, (c) the thrust coefficient under full blockage condition $(C_{TBLOCK})$ obtained from the computation and (d) the comparison of the performance characteristics between the model and the computational and experimental works in Jafari and Kosasih (2014a).

To obtain the inputted duct parameters for the model, that is, the axial induction factor at zero loading $(a_{0})$ and the thrust coefficient at full blockage condition $(C_{TBLOCK})$ , a CFD simulation is employed. To determine $a_{0}$ , initially, an empty duct model is made and then included in the computational domain as shown in Figure 8(a) (2). The wind is simulated to enter the domain with the velocity of 5 m/s (as in Jafari and Kosasih, 2014a). Then it measures the velocity near the duct entrance (where the rotor is located) that is further utilised to determine $a_{0}$ using equation (28). The value of $a_{0}$ is based on the averaged values of the axial induction factor at zero loading in the radial direction (averaged $a_{0}$ (r)). To determine the $C_{TBLOCK}$ , a solid disc is placed near the duct entrance (see Figure 8(a) (3)), and then similar simulation with the previous work (in determining $a_{0}$ ) is performed. Then the force on the disc is measured, before being divided by the disc frontal area to obtain $C_{TBLOCK}$ .

Figure 8(d) shows the comparison of the performance coefficients from the developed MBEM to the computational and experimental works. The curve predicted by the model is in good agreement with the literature data; the performance increases initially as the tip speed ratio (TSR) increases and subsequently reaches peak performance at the TSR around three prior to decreasing at larger TSRs.

The effect of the duct and rotor parameters on the performance of the duct turbine

The following section presents the findings of a parametric study on the influence of the geometric parameters of the duct on the duct turbine performance using the proposed MBEM model. The parameters investigated are the duct angle, the duct length, the existence of a flange, the rotor location inside and the use of bi-directional duct. The baseline turbine model and its channel, including the shape and the size, follow those of Jafari and Kosasih (2014a).

The expansion duct angle

In this section, the effect of the duct angle is investigated using the MBEM model. Figure 9(d) shows the performance characteristics of the turbine for duct angles 8°, 12° and 16°. It is shown that an increase in the duct angle from 8° to 12° results in increased performance coefficient at all TSRs. The maximum performance by the rotor with the duct angle of 8° is around 0.32, while that with a higher angle (12°) is around 0.35. One of the reasons relates to the higher flow augmentation under unloaded condition by the 12° duct than that of 8° duct. As shown in Figure 9(b), the average induction factor $(a_{0})$ by the 12° duct is around −0.44, more negative (more flow augmentation factor, $γ_{0}$ ) than that of the duct with 8° $(a_{0} - 0.42)$ . The larger the augmentation, the larger the available energy. Thus, a turbine operating under high available energy (by the duct assistance in augmenting the flow) potentially generates a high performance.

Figure 9.

(a) The duct turbine models under different expansion angles, (b) the distribution of the $a_{0}$ in the radial direction of the rotor plane obtained from the computation, (c) the thrust coefficient under full blockage condition $(C_{TBLOCK})$ obtained from the computation and (d) the comparison of the performance characteristics under various duct angles.

In contrast to change in duct angle from 8° to 12°, for an increase in duct angle from 12° to 16° the model predicts no significant change in performance. The reason is that no additional augmentation occurs as the duct angle increases from 12° to 16°.

It is interesting to note that even the augmentation factor under empty duct is lower (as shown in Figure 9(b)), the performance by the duct turbine with the 16° expansion angle is still higher than that with 8° expansion angle (as shown in Figure 9(d)). The reason could be related to the higher $C_{TBLOCK}$ by the duct with 16° expansion angle than that of 8°. The $C_{TBLOCK}$ by the 16° duct is 2.03 while that of 8° duct is just round 1.67. This indicates the 16° duct can generate the larger augmentation when the rotor is present, compared to the 8° duct. This is the reason why the CPs of the duct turbine with 16° expansion angle is higher than that of the 8° duct turbine. These results indicate the high augmentation under unloaded condition is not the only factor determining the high performance of the duct turbine, but also the $C_{TBLOCK}$ , which can relate to the phenomenon of the enhanced augmentation by the duct–rotor interaction.

This phenomenon is almost similar to that found in the experimental works by Abe and Ohya (2004) that the augmentation under unloaded condition does not guarantee to obtain high performance at the loaded condition. A simulation on a duct turbine with 15° expansion angle in this work shows initially a poorer augmentation under unloaded condition, compared to those with lower angles, due to large flow separation. However, with the presence of rotor, the separated flow is suppressed resulting in more effective flow expansion as well as the flow concentration at the duct entrance than those in the unloaded condition. This causes the higher maximum performance obtained by the 15° duct than the those with the lower angle when the rotor is present.

The effect of duct length

In the following section, the effect of the duct length is investigated. The baseline duct is shortened and extended to provide various duct lengths (see Figure 10(a)). The method to obtain the data of the $a_{0}$ and $C_{TBLOCK}$ is similar to those of the previous section which is utilised CFD simulation under the empty and full blocked ducts. These data then are inputted into the MBEM model.

Figure 10.

(a) The duct turbine model under various duct lengths, (b) the distribution of the $a_{0}$ in the radial direction of the rotor plane obtained from the computation, (c) the thrust coefficient under full blockage condition $(C_{TBLOCK})$ obtained from the computation and (d) the comparison of the performance characteristics under various duct lengths.

Figure 10(d) shows the performance characteristics of the turbine under different duct lengths. The longer duct results in higher performance at any TSR than shorter ducts. The duct with 1D length reaches the maximum performance coefficient at around 0.43, while for 0.6D and 0.25D peak performances are 0.35 and 0.27, respectively. Overall, a longer duct is found to have a higher augmentation factor at any TSR than shorter duct cases (see Figure 10(b)). For example, the duct with 1D length has $a_{0}$ of −0.51 (corresponding to the augmentation factor of 1.51), while those with 0.6D and 0.25D length have $a_{0}$ of −0.44 and −0.29, respectively. The longer duct enables the flow inside to be significantly deflected outward downstream, thereby concentrating more flow at the entrance which results in the higher energy density than the shorter duct cases.

The effect of flange

In the following section, the effect of the additional flange is investigated. A flange with 0.1-D is attached at the rear of the baseline duct (see Figure 11(a)) and its performance is compared to that in the baseline turbine (without flange). Similar methods to the previous sections are applied in obtaining $a_{0}$ and $C_{TBLOCK}$ .

Figure 11.

(a) The duct models; with and without a flange, (b) the distribution of the $a_{0}$ in the radial direction of the rotor plane obtained from the computation, (c) the thrust coefficient under full blockage condition $(C_{TBLOCK})$ obtained from the computation and (d) the comparison of the performance characteristics between the duct turbine with and without flange.

Figure 11(d) shows the comparison of the performance characteristics of the duct turbine with and without flange. The model predicts that the addition of the simple flange increases the performance coefficient across all TSRs. With flange attached the maximum performance is increased by around 35% and the TSR corresponding to the peak performance shifts to a higher value relative to those of the baseline turbine. The flange generates a large flow circulation behind that is often associated with a low-pressure zone development, providing a vacuuming effect (Mansour and Meskinkhod, 2014; Ohya et al., 2008; Ohya and Karasudani, 2010). The vacuuming zone causes the downstream flow to expand beyond its duct area ratio resulting in the high augmentation. The results of this study show that the flange generates additional augmentation factor under unloaded condition (as indicated by lower $a_{0}$ ) by around 30%, relative to the baseline duct. The flange also shows a more effective in creating the added force associated with the enhanced augmentation by the duct–rotor interaction as being indicated by the higher $C_{TBLOCK}$ over the baseline one (as shown in Figure 11(c)). As the flange is attached, the $C_{TBLOCK}$ increases by 40%, relative to the no flange case. This phenomenon also can be related to the additional blockage effect. The flange adds the device area perpendicular to the flow direction to increase the blockage ratio for such confinement domain. The blockage effect is more pronounced when the rotor exists, as its dependence on the wake size. Thereby when the rotor is present in the flanged turbine, the wake size enlarges more resulting in more added force, relative to that of the no flange case.

The effect of rotor location inside the duct

The following section presents a parametric study on the effect of the rotor location inside the duct on the performance of the turbine. The importance of conducting this study is that the location can determine the duct–rotor interaction. By changing the location, the gap between the rotor and the duct wall can vary thereby affecting the flow augmentation. The gap plays an important role in the flow separation suppression near the duct wall which affects the augmentation. As the gap is present, the deflected flow near the tip enables to be concentrated near the wall, to enhance momentum to the boundary layer in this region (Hansen et al., 2000; Shives and Crawford, 2011). As the typical duct shape is a low area at the entrance, then gradually expanding towards its exit, to locate the rotor at the entrance results in a lower gap, while to locate the rotor at the duct exit enlarges the gap. In this study, three-rotor location scenarios are proposed; at the entrance, at near the entrance (baseline) and at the exit (see Figure 12(a)).

Figure 12.

(a) The duct model in the various rotor locations, (b) the distribution of the $a_{0}$ in the radial direction of the rotor plane obtained from the computation, (c) the thrust coefficient under full blockage condition $(C_{TBLOCK})$ obtained from the computation and (d) the comparison of the performance characteristics in various rotor locations.

Figure 12(d) shows the performance characteristics under different rotor locations. The result shows that either locating the rotor at the duct entrance or at the duct exit weakens the performance, relative to that with the rotor located near the entrance. Locating the rotor at the entrance reduces the peak performance by around 12% while locating at the duct exit reduces the peak performance by around 20%, relative to that near the entrance.

As seen in Figure 12(b), locating the rotor at the entrance does not significantly change the duct induction factor unloaded condition $(a_{0})$ but does reduce the thrust coefficient for the blockage rotor $(C_{TBLOCK})$ relative to the case when the rotor is located near the entrance (baseline). This is suspected to be due to the small rotor–duct gap working to limit the flow concentration near the duct wall, thereby weakening the suppression of flow separation along the inner duct wall.

With the rotor located at the duct exit, there is found to be a reduction in both the duct augmentation under unloaded condition and $C_{TBLOCK}$ (see Figure 12(a)). The flow throughout the latter part of the duct is in a generally decelerating phase (to recover to the ambient). Accordingly, locating the rotor in the decelerating zone should be anticipated to weaken the augmentation. The reduced $C_{TBLOCK}$ at the duct exit is due to too large a gap between the rotor and the duct leading to reduction in the flow concentration. Consequently, there is a weakening of the additional axial momentum acting on the internal duct wall boundary layer and therefore increased likelihood of flow separation from the duct wall with an associated reduction in flow augmentation. Similar dependence of the performance of the duct-augmented turbine on the duct–rotor gap has been reported (Song et al., 2019).

The duct turbine performance in the bi-directional duct

The following section presents a parametric study on the effect of a bi-directional duct on the performance of the turbine. The use of the bi-directional duct configuration not only enhances the power gain of the turbine but also accommodates a different flow direction in extracting the kinetic energy, such as in the tidal environment. A bi-directional duct which adopts the design in Belloni et al. (2018) is used in this study and its performance characteristic is compared to the baseline turbine. The design has L/D of 1 with a curved inside profile and a straight outside profile (as shown in Figure 13(a)).

Figure 13.

(a) The baseline and bi-directional duct models, (b) the distribution of the $a_{0}$ in the radial direction of the rotor plane obtained from the computation, (c) the thrust coefficient under full blockage condition $(C_{TBLOCK})$ obtained from the computation and (d) the comparison of the performance characteristics in the various bi-directional duct configurations.

As can be seen in Figure 13(d), the performance of the bi-directional duct turbine is found to be comparable to that of the baseline turbine. Even the flow augmentation under the empty condition is higher (as indicated by the lower $a_{0}$ ), the maximum performance of the bi-directional duct turbine is similar to that of the baseline configuration (at CP around 0.35). The greater effectiveness of the duct–rotor interaction in adding the force to the baseline turbine could be the reason. When the rotor is present, the flow force could increase by the mechanism of the enhanced flow acceleration on the gap between the blade tip and the duct wall that improves the flow attachment near the duct wall, thereby increasing the flow augmentation. The same effect is possible for the bi-directional duct. However, the added force effect could be diminished by the energy loses within the bi-directional duct. The longer design of the bi-directional duct compared to the baseline case could have larger energy losses associated with high viscous loss between the flow and the duct wall thereby decreasing the force on the rotor plane. The different results on the effectiveness of the enhanced force by the duct–rotor interaction between the two duct cases can be explained by the parameter $C_{TBLOCK}$ . The measured $C_{TBLOCK}$ of the bi-directional duct is around 1.49, lower than that of the short flanged duct which is around 1.74. This means the additional energy under loading condition (when the rotor is present in the duct) by the baseline duct turbine is higher than that of the bi-directional duct.

Conclusion

The effects of the duct shape, the addition of flange and the location of rotor within a duct have been parametrically studied. A modified disc model for the duct turbine and blade element model which accounts for duct acceleration factor and rotor thrust under full blockage condition is developed and used in the study. The results show the duct length plays a more influential role in enhancing the power than the duct expansion angle. Increasing the duct length by the factor of two, relative to the baseline case, has an increase in the maximum performance of around 23%. From the perspective of minimising duct length for cost and practical reasons, the addition of the flange is found to enhance the performance of the shorter turbine. The results show adding the flange by 0.1 diameter enhances the maximum performance by around 35%. Finally, the maximum power gain is predicted to be obtained when the rotor location is not exactly at the inlet plane of the duct (or minimum area) but slightly inside the duct. The results from this study can be utilised as a design reference of duct-augmented turbine to maximise its enhanced power output.

Footnotes

Appendix 1 Author’s note

Aditya Rachman is also affiliated with the Mechanical Engineering Department of Halu Oleo University, Indonesia.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Aditya Rachman

References

Abe

Ohya

(2004) An investigation of flow fields around flanged ducts using CFD. Wind Engineering and Industrial Aerodynamics 92: 315–330.

Allsop

Peyrard

Thies

, et al. (2017) Hydrodynamic analysis of a, ducted open centre tidal stream turbine using blade element momentum theory. Ocean Engineering 141: 531–542.

Belloni

CSK

Willden

RHJ

Houlsby

, et al. (2018) An investigation of shrouded and open-centre tidal turbines employing CFD-embedded BEM. Renewable Energy 108: 622–634.

Bussel

GJW

(2007) The science of making more torque from wind: Duct experiments and theory revisited. Journal of Physics: Conference Series75: 012010.

Foreman

Gilbert

Oman

, et al. (1978) Diffuser augmentation of wind turbines. Solar Energy 20(4): 305–311.

Gaden

Bibeau

(2010) A numerical investigation into the effect of diffusers on the performance of hydro kinetic turbines using a validated momentum source turbine model. Renewable Energy 35(6): 1152–1158.

Glauert

(1926) The analysis of experimental results in the Windmill Brake and Vortex Ring states of an airscrew. Aeronautical Research Committee Reports and Memoranda, Report 1026. London: Her Majesty’s Stationery Office.

Hansen

MOL

Sørensen

Flay

RGJ

, et al. (2000) Effect of placing a duct around a wind turbine. Wind Energy 3(4): 207–213.

Hjort

Larsen

(2014) A multi-element duct augmented wind turbine. Energies 7: 3256–3281.

10.

Igra

(1981) Research and development for shrouded wind turbines. Energy Conversion Management 21: 13–48.

11.

Jafari

Kosasih

(2014a) Analysis of the power augmentation mechanisms of duct shrouded micro turbine with computational fluid dynamics simulations. Wind and Structures an International Journal 19(2): 199–217.

12.

Jafari

SAH

Kosasih

(2014b) Flow analysis of shrouded small wind turbine with a simple frustum diffuser with computational fluid dynamics simulations. Wind Engineering and Industrial Aerodynamics 125: 102–110.

13.

Jamieson

(2008) Generalized limits for energy, extraction in a linear constant. Wind Energy 11: 445–457.

14.

Khamlaj

Rumpfkeil

(2017) Theoretical analysis of shrouded horizontal axis wind turbines. Energies 10(1): 1–19.

15.

Lawn

(2003) Optimization of the power output from shrouded turbines. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 217(1): 107–117.

16.

Lock

CNH

Batemen

Townsend

HCH

, et al. (1926) An extension of the vortex theory of airscrews with applications to airscrews of small pitch, including experimental results. Aeronautical Research Committee Reports and Memoranda, no: 1014. London: Her Majesty’s Stationery Office.

17.

Mansour

Meskinkhod

(2014) Computational analysis of flow fields around flanged ducts. Journal of Wind Engineering and Industrial Aerodynamics 124: 109–120.

18.

Matsushima

Takagi

Muroyama

, et al. (2006) Characteristics of a highly efficient propeller type small wind turbine with a duct. Renewable Energy 31: 1343–1354.

19.

Ohya

Karasudani

(2010) Shrouded wind turbine generating high output power with wind-lens technology. Energies 3: 634–649.

20.

Ohya

Karasudani

Sakurai

, et al. (2008) Development of a shrouded wind turbine with a flanged duct. Journal of Wind Engineering and Industrial Aerodynamics 96(5): 524–539.

21.

Oman

Foreman

Gilbert

, et al. (1977) Investigation of duct augmented wind turbines, parts I. Grumman Research Dept Report RE-534, ERDA Report C00-2616-2. New York, pp. 118.

22.

Owis

Badawy

MTS

Abed

, et al. (2015) Numerical investigation of loaded and unloaded duct equipped with a flange. International Journal of Scientific & Engineering Research 6(11): 312–341.

23.

Palmer

Andres

Ayuso

, et al. (2017) Mixed CFD-1D wind turbine duct design optimization. Renewable Energy 105: 386–399.

24.

Phillips

(2003) An investigation on duct augmented wind turbine design. PhD Thesis.

25.

Prandtl

(1927) Vier Abhandlungen zur Hydrodynamik und Aerodynamik. Göttingen: Göttinger Nachr, pp. 88–92.

26.

Prasad

Kumar

Swaminathan

, et al. (2020) Computational investigation and design optimization of a duct augmented wind turbine (DAWT). Materials Today: Proceedings 22: 1186–1191.

27.

Roshan

Alimirzazadeh

Rad

(2015) RANS simulations of the stepped shroud effect on the performance of shrouded wind turbine. Journal of Wind Engineering and Industrial Aerodynamics 145: 270–279.

28.

Shives

Crawford

(2011) Developing an empirical model for ducted tidal turbine performance using numerical simulation results. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 226: 112–125.

29.

Siavash

Najafi

Hashjin

, et al. (2020) Mathematical modeling of a horizontal axis shrouded wind turbine. Renewable Energy 146: 856–866.

30.

Song

Wang

Yan

, et al. (2019) Numerical and experimental analysis of a duct-augmented micro-hydro turbine. Ocean Engineering 171: 1590–1602.

31.

Vaz

TDR

Aline

Mesquita

, et al. (2014) An extension of the blade element momentum method applied to duct augmented wind turbines. Energy Conversion and Management 87: 1116–1123.

The effects of shape and size on duct-augmented horizontal axis turbine performance

Abstract

Keywords

Introduction

Derivation of the model

The method to obtain the added pressure coefficient C p b and the wake induction factor adjustment f ′

The implementation in blade element momentum model

The calculation procedure of MBEM

Validation of the model

Comparison of the modified disc model for the duct turbine to the actuator disc model

Comparison modified disc model for the duct turbine to the experimental and computational works on the duct turbine

Comparison of MBEM to the experimental and computational works

The effect of the duct and rotor parameters on the performance of the duct turbine

The expansion duct angle

The effect of duct length

The effect of flange

The effect of rotor location inside the duct

The duct turbine performance in the bi-directional duct

Conclusion

Footnotes

Appendix 1

Author’s note

Declaration of conflicting interests

Funding

ORCID iD

References

The method to obtain the added pressure coefficient $C p_{b}$ and the wake induction factor adjustment $f'$