A Generalized Multi-Stage Actuator Disc Model for characterizing unconventional horizontal axis wind turbine geometries

Abstract

This study presents a unified modeling framework for unconventional horizontal axis wind turbines (HAWTs) that utilize axially extended, non-radial blade geometries. Recent bio-inspired and helical designs have reported performance levels that are difficult to interpret within the standard single-plane actuator-disc framework: they overcome the classical Betz limit which is derived for an ideal single rotor operating under one-dimensional axial-flow assumptions. To address this gap in the technology description, we introduce the Generalized Multi-Stage Actuator Disc Model, which discretizes the rotor into a sequence of interacting aerodynamic stages aligned with the flow axis. We rigorously evaluate four topological configurations, Solid Divergent, Hollow Divergent, Solid Convergent, and Hollow Convergent, using numerical optimization to determine their theoretical performance limits. Our results show that all multi-stage configurations converge to a maximum theoretical efficiency of $η \approx 2 / 3$ (approximately 66.7%), exceeding the single-stage Betz limit of $16 / 27$ (59.3%), while remaining consistent with tandem-disc extensions of momentum theory. Furthermore, the analysis reveals that while constant-radius geometries are optimal under standard area normalization, divergent geometries evaluated under mass-conserving conditions behave as ideal diffusers, theoretically capable of recovering nearly all wake energy. We also quantify the impact of rotor solidity, showing that hollow configurations require maximized annular thickness to fully exploit the multi-stage effect. These findings provide a physical interpretation of non-conventional horizontal axis wind turbines, as geometries that lie outside the classical single-disc assumptions and establish a theoretical framework for designing turbines suitable for low-wind-speed environments.

Keywords

actuator disc theory axialized HAWT multi-stage aerodynamics Betz limit extension bio-inspired wind energy

Introduction

The urgency of climate action has transformed renewable energy from an alternative to a necessity (Rizzi et al. 2013). Wind energy, in particular, has captured global attention for its ability to deliver large-scale power with a minimal environmental footprint (Cullen 2013; National Research Council 2007). A key factor in maximizing the value of this resource lies in the efficiency of horizontal axis wind turbines (HAWTs), which dictates how effectively kinetic wind energy is converted into usable mechanical power (Yang et al., 2020).

The theoretical efficiency limit for a conventional HAWT is given by the well-known Betz limit. Using the classical one-dimensional actuator disc model (neglecting rotational wake losses), Betz and others showed that the maximum power coefficient is $C_{p} = 16 / 27 \approx 59.3 %$ for an ideal single rotor in free flow (Betz 1920; Glauert 1935). This result, derived from momentum balance across an idealized “actuator disc,” assumes a rotor with radial blades that uniformly decelerates the incoming air, thereby establishing an upper bound on the efficiency of traditional HAWTs under steady, axial-flow conditions. In practice, however, overall efficiency is significantly reduced by a combination of aerodynamic, drivetrain, and electrical conversion losses (Gorban et al. 2001), bringing the performance of most conventional designs down to approximately 35% (Jenkins et al., 2021; Sørensen 2015).

This gap between the theoretical aerodynamic limit and real performance of wind turbines has motivated broader efforts to improve Wind Energy Conversion Systems (WECS). Recent studies have explored system-level strategies such as control-layer optimization, power tracking, and operational regulation to enhance energy extraction under variable wind conditions (Brahmi et al., 2024, 2025). However, even with such improvements, losses associated with physical constraints remain largely unavoidable. As a result, an alternative line of research has focused on broadening the operational range of wind turbines, particularly by lowering the cut-in speed to unlock energy in regions with moderate wind resources (typically averaging below 7.5 m/s) (Ajayi et al. 2019; International Electrotechnical Commission 2019; Tummala et al., 2015). Conventional commercial models, optimized for high wind speeds (Dupont et al. 2017), often fail to capture the potential of these vast geographic areas (Antonini et al., 2024). Consequently, unconventional turbine designs have emerged that seek to extend the interaction between the turbine and the wind through complex, non-radial geometries.

Four notable examples are summarized in Table 1. These designs generally employ two main strategies: introducing multiple stages of energy capture or reshaping blade geometry to sustain wind–blade interaction along the flow axis. The first strategy is exemplified by Nugroho’s multi-stage wind turbine, which employs rotors in tandem to extract residual kinetic energy from the wake (Nugroho et al. 2019). The second strategy involves curved or helical designs, such as the Triplaris-Americana-inspired turbine (Herrera et al., 2019), the Rotulus turbine (Rotolus GmbH 2024), and the Archimedes F1 (RESPECT Co, Ltd, 2023). These turbines depart from the “thin disc” concept, creating a three-dimensional interaction volume that guides airflow along extended axial paths. Experimental measurements of these designs have reported efficiencies that appear to rival or exceed the classical Betz limit (Herrera et al., 2019), a phenomenon that requires rigorous analytical explanation rather than dismissal.

Table 1.

Notable examples of unconventional small-scale wind turbine designs.

Name	Diagram	Description/Type
Nugroho’s multi-stage wind turbine (Nugroho et al. 2019)		Solid Constant Cone (special case of convergent and divergent case)
Archimedes F1 (RESPECT Co, Ltd, 2023)		Solid Divergent Cone
Bio-inspired by the Triplaris Americana Seed (Herrera et al., 2019)		Hollow Divergent Cone
Rotulus Wind Turbine (Rotolus GmbH 2024)		Hollow Divergent and Convergent Cone

The analysis of such machines presents a significant theoretical challenge. The traditional Actuator Disc Theory (ADT) conceptualizes the rotor as an infinitesimally thin, planar disc with uniform loading (Betz 1921). While effective for conventional radial blades, this model is inadequate for geometries where the “disc” is actually a cone, a helix, or a multi-stage assembly. Thus, a gap exists in wind turbine theory: there is no general analytical framework that directly accommodates arbitrary blade geometries or multiple aerodynamic stages beyond the idealized cases of a single disc or a few tandem discs.

Significant extensions to the classical theory have been proposed to address real-world complexities. A general momentum theory for energy-extracting actuator discs that explicitly accounts for wake rotation and expansion was developed by Sharpe (2004). By analyzing the pressure drop associated with wake swirl, this framework provides a more sophisticated understanding of energy extraction in turbines with significant wake rotation, applicable to both wind and tidal systems. Similarly, Conway (1998) provided analytical solutions for actuator discs with variable radial load distributions. Drawing on analogies with finite wing theory, these solutions allow for the calculation of induced velocities for polynomial and elliptic load profiles, moving beyond the assumption of uniform thrust. Furthermore, Kuik (2017) explored the “Joukowsky” actuator disc—a disc with constant circulation—demonstrating that under specific conditions (vanishing vortex core radius), the momentum balance requires careful treatment of singularities to avoid physical paradoxes at low tip-speed ratios.

While these works have refined the physics of the planar actuator disc, a gap remains in analytically modeling the geometry of non-planar rotors. Most existing literature relies on numerical Blade Element Momentum Theory (BEMT) or Computational Fluid Dynamics (CFD) (Crawford 2006; Li et al., 2022a, 2022b) to handle coning or dihedral blades. BEMT-based methods (e.g. Hansen (2008)) can handle moderate design changes but still rely on the classical momentum theory within each annular streamtube, often requiring empirical corrections for heavily non-planar or multi-rotor arrangements. On the other hand, attempts at analytical modeling typically reduce complex designs to combinations of idealized actuator discs or other simplified constructs. There is no general analytical framework that directly accommodates arbitrary blade geometries (convergent/divergent) or multiple aerodynamic stages beyond the idealized tandem discs. Newman (1986) showed that an infinite number of stages could theoretically approach a power coefficient of $\approx 66.7 %$ .

In this paper, we examine how unconventional HAWTs with non-radial blade geometries can be analyzed by extending the classical actuator disc frameworks. We do not focus on the aerodynamic forces of specific airfoils as in Li et al. (2022a), but rather on the fundamental momentum balance of the actuator disc geometry itself. Our approach mirrors the original Betz research but introduces a Generalized Multi-Stage Actuator Disc model. In particular, we develop a model (termed Actuator Disc) that extends classical momentum theory to turbines with several axially spaced rotor stages and blade elements that may diverge or converge along the flow axis (i.e., not confined to a single radial plane). The formulation accounts for both solid and hollow rotor configurations—covering a full solid disc as well as annular (ring-shaped) rotor arrangements—and it rigorously captures axial flow interactions between stages.

Specifically, we introduce four idealized configurations composed of infinitesimally thin actuator discs arranged in series with radii that progressively increase or decrease along the flow axis: solid divergent, hollow divergent, solid convergent, and hollow convergent. These configurations serve as actuator-based analogs for bio-inspired and helical blade geometries. Using numerical techniques, we analyze the convergence of results to estimate the theoretical maximum efficiency attainable by these architectures. Using this generalized model, we demonstrate a theoretical efficiency up to about $66.6 %$ (approximately two-thirds) for certain multi-stage, axial-flow turbine geometries, consistent with the extended Betz-limit predicted by tandem disc theory. This result validates that even highly non-traditional HAWT designs cannot exceed the fundamental momentum constraints of axial flow, yet they can approach the improved limit of $\sim 2 / 3$ efficiency by utilizing multiple aerodynamic stages or curved blade geometries.

The article is organized as follows: We begin by revisiting the actuator disc model and the derivation of the Betz limit. We then introduce our proposed configurations of successive actuator discs and demonstrate their relevance to real-world turbines featuring non-radial blade geometries. Using numerical techniques, we estimate the theoretical maximum efficiency and compare it to experimental results. Finally, we discuss the implications and limitations of our findings, particularly regarding the trade-offs between ideal 1D momentum theory and the 3D radial expansion effects required by mass conservation (Conway 1998), proposing future directions for modeling unconventional HAWTs.

Maximum efficiency for radial-blade HAWTs: Actuator disc theory

In this section, we derive the theoretical maximum efficiency of a conventional HAWT using the classical ADT. This baseline calculation highlights the specific geometric and aerodynamic assumptions—such as radial blade configuration and strictly axial flow—that limit the validity of the Betz limit when applied to the unconventional architectures discussed in this paper.

The ADT idealizes the rotor as an infinitesimally thin, permeable disc of area $A$ that extracts energy by creating a pressure discontinuity, thereby slowing the wind from a free-stream velocity $v_{i n}$ to a far-wake velocity $v_{o u t}$ (see Figure 1). The model assumes steady, one-dimensional, incompressible, and inviscid flow, with $v_{d}$ denoting the velocity at the disc (Betz 1920; Hansen 2008).

Under these assumptions, the momentum balance across the control volume gives the thrust as the change in axial momentum flux,

T = \dot{m} (v_{i n} - v_{o u t}) = ρ A v_{d} (v_{i n} - v_{o u t}) .

(1)

The same energy extraction can also be evaluated from the kinetic-energy difference between the upstream and downstream flow. Applying Bernoulli’s equation upstream and downstream of the disc, and equating the work done by the pressure drop to the change in kinetic energy flux, gives

P = \frac{1}{2} \dot{m} (v_{i n}^{2} - v_{o u t}^{2}) .

(2)

Since the extracted power is also $P = T v_{d}$ , comparing equations (1) and (2) leads to the standard actuator-disc result that the disc velocity is the arithmetic mean of the upstream and downstream velocities:

v_{d} = \frac{v_{i n} + v_{o u t}}{2} .

(3)

Introducing the axial induction factor $a$ , defined as the fractional decrease in wind velocity at the rotor plane, the disc velocity is written as

v_{d} = (1 - a) v_{i n} .

(4)

Substituting this definition into the previous relation also gives the corresponding far-wake velocity, $v_{o u t} = (1 - 2 a) v_{i n}$ . The extracted power can therefore be expressed only in terms of $a$ as

P = 2 ρ A v_{i n}^{3} a {(1 - a)}^{2} .

(5)

The efficiency of the turbine is described by the power coefficient $C_{p}$ , which is the ratio of extracted power to the total available kinetic power in the wind stream, $P_{w i n d} = \frac{1}{2} ρ A v_{i n}^{3}$ . Thus,

C_{p} (a) = \frac{P}{P_{w i n d}} = 4 a {(1 - a)}^{2} .

(6)

The maximum value of this expression is obtained by differentiating with respect to $a$ ,

\frac{d C_{p}}{d a} = 4 (1 - a) (1 - 3 a) = 0 .

The physically relevant optimum is therefore $a = 1 / 3$ . Substituting this value back into equation (6) gives the Betz limit

C_{p, \max} = \frac{16}{27} \approx 59.3 % .

This limit represents the upper bound for any single actuator disc under the assumptions of 1D axial momentum theory. However, as noted by Sharpe (2004) and others (Kuik 2017; Newman 1986), systems that lie outside these geometric assumptions—such as those creating significant wake rotation, radial flows, or utilizing multiple interaction stages—may not be strictly bound by this specific derivation.

Extending the actuator disc model

Having established the original derivation for the maximum efficiency of radial-blade HAWTs, we now introduce a generalized actuator disc model tailored for unconventional geometries. Rather than replacing the classical formulation, this work aims to extend it to represent the complex flow behavior associated with non-radial blade designs. As discussed in the introduction, a primary strategy of these technologies is to prolong the blade-flow interaction along the axis of rotation. The model presented herein explicitly incorporates this effect by discretizing the rotor into a sequence of actuator discs, each representing a differential element of an axialized blade geometry.

Motivation and theoretical background

The classical Betz model demonstrates that, under ideal conditions, a single wind turbine can extract up to $16 / 27 \approx 59.3 %$ of the kinetic energy from a uniform wind stream (Betz 1920). However, this leaves over 40% of the available energy unexploited in the downstream wake. Theoretical extensions of the actuator disc theory have long suggested that this limit is not an absolute ceiling for all configurations. Notably, Newman (1986) analyzed a system of $n$ actuator discs arranged in tandem (series) and demonstrated that the maximum achievable power coefficient ( $C_{p}$ ) increases with the number of stages. For two discs, the theoretical optimum rises to $C_{p} = 16 / 25 = 64 %$ , and in the limit of an infinite number of stages, it approaches $C_{p} = 2 / 3 \approx 66.7 %$ (Newman 1986).

Despite these insights, standard multi-disc models typically assume widely spaced, constant-diameter rotors in strictly axial flow. They do not account for the continuous geometric variations found in modern bio-inspired or helical turbines (e.g. Herrera et al. (2019); RESPECT Co, Ltd (2023)). To address this limitation, we propose a multi-stage model in which the “stages” are not restricted to separate machines, but may also represent sequential cross-sections of a single continuous axialized rotor.

The physical basis for this approximation is that many unconventional axialized rotors do not extract momentum at a single plane, but distribute the wind–blade interaction over a finite axial distance. In this sense, a continuous geometry may be interpreted as a sequence of closely spaced energy-extracting cross-sections. The present formulation therefore does not treat a continuous blade as literally equivalent to a set of isolated turbines; rather, it uses discrete actuator discs as a momentum-theory discretization of a distributed axial interaction. As the number of stages increases and the spacing between them decreases, this discrete representation approaches the continuous limit in the same spirit that a segmented numerical model approaches a continuous distribution. This approximation is most appropriate when the dominant effect is progressive axial deceleration of the flow, while radial transport, azimuthal non-uniformity, viscous effects, and detailed blade-loading variations remain of secondary importance.

The Generalized Multi-Stage Actuator Disc Model

We represent the turbine as a series of $n + 1$ infinitesimally thin actuator discs indexed by $j = 0, 1, \dots, n$ and aligned along the flow direction, as illustrated in Figure 2.

Each successive disc interacts with the wake generated by the previous one. As a result, the velocity available at stage $j$ depends on the energy extracted upstream, particularly by the preceding stages. For the purpose of the geometric decomposition, we distinguish between two flow regions entering each stage:

1. Overlapping Region: The portion of the flow that has already passed through the previous disc. Its velocity is therefore reduced by the upstream energy extraction. With the present notation, this reduced velocity can be written as

v_{overlap}^{j} = v_{in} {(1 - a)}^{j - 1} .

2. Non-Overlapping Region: The portion of the flow entering the disc from the free stream or from the expanding wake boundary, which has not yet been slowed by the previous stage.

Wake expansion and geometric configurations

A critical phenomenon in this analysis is the downstream expansion of the wake. As the wind slows down across a stage, the streamtube must expand to conserve mass, as shown in Figure 3.

In conventional single-disc theory, this expansion is lost to the wake. However, unconventional designs can exploit this effect by varying the rotor diameter along the downstream axis. If the rotor diameter increases along with the wake expansion, subsequent stages can capture the kinetic energy contained in the expanding streamtube—energy that a constant-diameter tandem rotor would miss. Based on this principle, we define four idealized geometric configurations for our analysis:

1. Solid Divergent: A sequence of solid discs with increasing radii (cone-like), designed to capture the expanding wake.

2. Hollow Divergent: A sequence of annular (ring) elements with increasing radii, modeling helical blades that follow the wake periphery.

3. Solid Convergent: A sequence of solid discs with decreasing radii.

4. Hollow Convergent: A sequence of annular elements with decreasing radii.

To keep the model analytically tractable, the following derivations focus on the axial component of the velocity field, using $v_{i n}$ as the reference wind velocity. A full three-dimensional treatment would require solving for the radial wake velocity $v_{r a d i a l}$ , as illustrated in Figure 3. Here, however, the one-dimensional approximation is used as a first-order momentum model for estimating the upper efficiency limits of the proposed configurations.

Before treating each geometry separately, it is useful to collect the aerodynamic terms that remain common to all cases. For all cases, we factorize a common power coefficient term, $Γ$ , which groups the aerodynamic constants and the base kinetic energy flux relative to the first-stage radius $r_{0}$ :

Γ = \frac{π}{4} ρ v_{i n}^{3} (2 - a) (1 - {(1 - a)}^{2}) .

(7)

Solid divergent

The first actuator-disc configuration, hereafter referred to as the solid divergent case, represents a geometry in which the rotor expands along the flow direction. The turbine is modeled as a sequence of solid actuator discs (i.e., discs with no central opening), whose diameters increase progressively downstream. This “solid” assumption distinguishes the present configuration from the hollow cases introduced later. Conceptually, the setup emulates multi-stage designs in which an increasing blade span extracts energy over successive, expanding stages (see Figure 4). A representative example is the Archimedes F1 turbine, whose helical blades wrap outward around the rotor, increasing their radius in the direction of the flow.

We treat this configuration as a sequence of solid discs whose radius increases downstream. Specifically, the radius of the $j$ -th stage is scaled from the initial radius $r_{0}$ by a geometric factor $β > 1$ , so that

r_{j} = r_{0} β^{j} .

(8)

As the discs expand according to equation (8), the airflow interacting with downstream stages becomes non-uniform. The outer annular region of a stage intercepts “fresh” undisturbed wind, while the inner regions intercept flow that has already passed through one or more upstream stages. This allows the inflow at stage $j$ to be decomposed into $j + 1$ concentric circular layers (depicted with different colors in Figure 4), indexed by $k$ , where $k$ identifies the stage at which that annular layer is first introduced.

For a specific stage $j$ , each layer is therefore characterized by both its radial position and its interaction history:

• The layer corresponding to index $k$ spans the radial gap between $r_{k - 1}$ and $r_{k}$ , with $r_{- 1} = 0$ .

• The wind passing through this layer has already traversed the discs from stage $k$ up to stage $j - 1$ .

• Therefore, the wind has undergone $j - k$ previous interactions, and the incident velocity for layer $k$ at stage $j$ is $v_{i n} {(1 - a)}^{j - k}$ .

With this radial decomposition, the annular area associated with the $k$ -th layer follows from the difference between two consecutive radii. Using equation (8), this gives

A_{k} = π (r_{k}^{2} - r_{k - 1}^{2}) = π r_{0}^{2} (β^{2 k} - β^{2 (k - 1)}) .

(9)

for the central core layer $(k = 0)$ , equation (9) is replaced by the circular area $π r_{0}^{2}$ .

Since each layer may enter stage $j$ with a different velocity, the total power extracted at the $j$ -th stage is obtained by summing the contribution of all layers $k = 0, \dots, j$ :

{\dot{W}}^{j} = \sum_{k = 0}^{j} \frac{π}{4} ρ (2 - a) (1 - {(1 - a)}^{2}) {(v_{i n} {(1 - a)}^{j - k})}^{3} r_{0}^{2} (β^{2 k} - β^{2 (k - 1)}) .

(10)

After factorizing the common term $Γ$ , defined in equation (7), and the reference area factor $r_{0}^{2}$ from equation (10), the total power of an $n$ -stage turbine is obtained by summing over all stages $j = 0, \dots, n$ such as equation (11).

{\dot{W}}_{t o t a l}^{n} = Γ r_{0}^{2} \sum_{j = 0}^{n} [{(1 - a)}^{3 j} + \sum_{k = 1}^{j} (β^{2 k} - β^{2 (k - 1)}) {(1 - a)}^{3 (j - k)}] .

(11)

Hollow divergent

The next configuration, referred to as hollow divergent, also models outward-curving blades, but in this case, each actuator disc is hollow rather than solid. The geometry consists of a sequence of annular discs that expand in diameter along the axial direction, excluding the central region from wind interaction. This concept can be visualized by imagining a curved blade that expands outward and is segmented axially. When rotated about the central axis, each segment sweeps out an annular area; effectively forming a ring-shaped actuator (depicted in Figure 5). Several bio-inspired turbine designs exemplify this configuration, including those modeled after the seeds of Triplaris Americana (Herrera et al., 2019), Dryobalanops Aromatica (Omidvarnia and Sarhadi 2024), and Petrea Volubilis (Gaitan-Aroca et al. 2020), whose outward-curving surfaces enhance passive aerodynamic performance.

The hollow divergent configuration consists of expanding annular discs, modeling outward-curving blades that form a spiral or helical shape. Each stage is represented as a ring whose outer radius grows according to equation (12).

R_{o u t}^{j} = r_{0} β^{j} .

(12)

With a constant radial thickness $t$ , the corresponding inner radius is developed such as in 13.

R_{i n}^{j} = r_{0} β^{j} - t .

(13)

To keep the analytical model tractable, we describe the interaction at each stage using two active regions and one inactive region:

1. Outer Zone: The portion of the ring that extends radially beyond the previous stage. This region intercepts undisturbed wind, $v_{i n}$ .

2. Inner Zone: The portion of the ring that overlaps with the active area of the previous stage. This region intercepts wind that has been slowed by one interaction, $(1 - a) v_{i n}$ .

3. Hollow Center: The central region, which does not interact with the flow and therefore extracts no energy.

With this decomposition, the fresh-wind contribution at stage $j$ is associated with the radial area added between two consecutive outer radii. Using equation (12), this area is

A_{o u t e r}^{j} = π [{(β^{j} r_{0})}^{2} - {(β^{j - 1} r_{0})}^{2}] .

(14)

The wake-wind contribution comes from the portion of the ring that overlaps with the previous stage. Assuming that the thickness $t$ is sufficiently small for this overlap to be defined by the previous outer radius, the corresponding area is

A_{i n n e r}^{j} = π [{(β^{j - 1} r_{0})}^{2} - {(β^{j - 1} r_{0} - t)}^{2}] = π (2 β^{j - 1} r_{0} t - t^{2}) .

(15)

The total power is then obtained by adding the initial solid-disc contribution at $j = 0$ to the fresh-wind and wake-wind contributions of the subsequent annular stages. Substituting the area contributions from equations (14) and (15) gives the following result:

{\dot{W}}^{n} = Γ {r_{0}^{2} + \sum_{j = 1}^{n} [(β^{2 j} r_{0}^{2} - β^{2 (j - 1)} r_{0}^{2}) + {(1 - a)}^{3} (2 β^{j - 1} r_{0} t - t^{2})]} .

(16)

Hollow convergent

The third configuration, referred to as hollow convergent, represents inward-curving blades modeled as a series of progressively smaller annular discs. The radius of each stage scales by a contraction factor $ζ < 1$ : $r_{j} = r_{0} ζ^{j}$ (see Figure 6). Unlike the divergent cases, this setup forms a converging flow pathway, with blade surfaces that decrease in radius along the flow direction.

A representative real-world geometry that reflects this configuration is one half of the axial length of the Rotulus Wind Turbine (Rotolus GmbH 2024).

In this model, each annular stage includes an outer radius, an inner radius, and a hollow center. Compared with the divergent case, however, the flow regions are inverted:

1. Inner Zone: This zone aligns with the hollow center of the previous stage. Therefore, it intercepts fresh wind, $v_{i n}$ , that has passed through the hole.

2. Outer Zone: This zone sits directly behind the active ring of the previous stage. Therefore, it intercepts wake wind, $(1 - a) v_{i n}$ .

Using the same constant thickness $t$ as in the previous hollow case, the fresh-wind area associated with the Inner Zone is obtained from the ring thickness around the previous radius:

A_{i n n e r}^{j} = π [{(ζ^{j - 1} r_{0})}^{2} - {(ζ^{j - 1} r_{0} - t)}^{2}] \approx π (2 ζ^{j - 1} r_{0} t - t^{2}) .

(17)

The remaining swept area corresponds to the Outer Zone, which receives wake wind from the previous active ring:

A_{o u t e r}^{j} = π |{(ζ^{j} r_{0})}^{2} - {(ζ^{j - 1} r_{0})}^{2}| .

(18)

Combining the fresh-wind and wake-wind contributions from equations (17) and (18) over all stages gives the total extracted power as given in equation (19).

{\dot{W}}^{n} = Γ \sum_{j = 1}^{n} [(2 ζ^{j - 1} r_{0} t - t^{2}) + {(1 - a)}^{3} |ζ^{2 j} r_{0}^{2} - ζ^{2 (j - 1)} r_{0}^{2}|] .

(19)

Geometric Constraint: This model remains valid only while the annular geometry is preserved; specifically, the inner radius must remain positive, so

ζ^{n - 1} r_{0} > 2 t .

Solid convergent

The final configuration, hereafter referred to as the solid convergent case, represents inward-curving blades modeled through a fully solid, multi-stage actuator-disc sequence. In contrast to the hollow convergent geometry, each stage here is a solid disc. The turbine is discretized into progressively smaller discs aligned with the flow axis, thereby emulating a rotor whose effective radius tapers downstream (see Figure 7).

Among the four geometries considered, the solid convergent configuration is the least frequently encountered in practical turbine designs. Nevertheless, as discussed for the hollow convergent case, this topology remains conceptually relevant because it represents the limiting case of a rotor whose effective area continuously contracts downstream.

The solid convergent configuration is the simplest convergent geometry considered here. It models a rotor that tapers downstream, similar to a cone pointing away from the incoming wind. Each stage is represented by a solid disc whose radius decreases geometrically according to

r_{j} = r_{0} ζ^{j}, ζ < 1 .

(20)

Because the radius decreases at every step, stage $j$ is completely “shadowed” by stage $j - 1$ . Consequently, the entire surface area of stage $j$ interacts with wind that has already been processed by the previous stages. Using the stage-to-stage velocity reduction defined previously, the incident velocity at stage $j$ is

v_{j} = v_{i n} {(1 - a)}^{j} .

(21)

Combining the reduced velocity from equation (21) with the stage radius from equation (20), the power contribution of stage $j$ is

{\dot{W}}^{j} = \frac{π}{4} ρ (2 - a) (1 - {(1 - a)}^{2}) {(v_{i n} {(1 - a)}^{j})}^{3} r_{0}^{2} ζ^{2 j} .

(22)

Summing the stage contributions from equation (22) gives the total power extracted by the solid convergent turbine:

{\dot{W}}^{n} = Γ r_{0}^{2} \sum_{j = 0}^{n} {[ζ^{2} {(1 - a)}^{3}]}^{j} .

(23)

Results and discussion

This section presents the theoretical maximum efficiencies obtained for the multi-stage actuator disc configurations. Unlike classical single-disc theory, where the optimum is obtained from a relatively simple algebraic derivative, the multi-stage models lead to higher-order functions that depend on the number of stages $n$ , the velocity induction factor $a$ , and the geometric scaling factors $β$ or $ζ$ .

To evaluate these limits consistently, we use a numerical optimization framework. For each configuration, the efficiency $η$ is defined as the ratio between the total extracted power and the available kinetic power evaluated over a reference area:

η (a, geom) = \frac{{\dot{W}}_{t o t a l} (a, geom)}{\frac{1}{2} ρ A_{r e f} v_{i n}^{3}} .

(24)

here,

A_{r e f}

is the maximum cross-sectional area of the device and is used to normalize the extracted power.

The optimum is then obtained by seeking stationary points of the efficiency with respect to both the aerodynamic and geometric variables. In compact form, this condition can be written as

\nabla η = (\frac{\partial η}{\partial a}, \frac{\partial η}{\partial geom}) = 0 .

Since analytical closed-form solutions for the roots of these high-degree expressions are not practical for $n > 1$ , we use a numerical trust-region algorithm implemented in MATLAB to identify the global maxima within the physical constraints

0 < a < 1, β \geq 1, 0 < ζ \leq 1 .

For the hollow configurations, the normalized annular thickness is additionally constrained as $0 < μ \leq 1$ , where $μ = t / r_{0}$ . The limit $μ \to 0$ represents a very thin annular rotor, while $μ = 1$ represents the solid-disc limit.

Solid configurations

We first analyze the solid configurations. A key finding is the symmetry between the divergent and convergent solid models: although the geometries evolve in opposite directions, both converge to the same optimal behavior under the chosen efficiency metric.

Optimization formulation

For the solid divergent case, the total power can be written as the product of a common aerodynamic factor and a geometric summation term, as given in equation (25):

{\dot{W}}^{n} = \frac{π}{4} ρ v_{i n}^{3} (2 - a) (1 - {(1 - a)}^{2}) r_{0}^{2} \sum_{j = 0}^{n} [{(1 - a)}^{3 j} + \sum_{k = 1}^{j} (β^{2 k} - β^{2 (k - 1)}) {(1 - a)}^{3 (j - k)}] .

(25)

The reference area corresponds to the largest stage, $A_{r e f} = π {(r_{0} β^{n})}^{2}$ . Substituting equation (25) into equation (24) gives the efficiency function

η (a, β) = \frac{1}{2} (2 - a) (1 - {(1 - a)}^{2}) \frac{Φ (a, β)}{β^{2 n}} .

(26)

The role of the geometric scaling factor $β$ becomes clear from equation (26). Although increasing $β$ allows downstream stages to intercept additional wake expansion, the reference area also increases through the factor $β^{2 n}$ . Therefore, the optimum occurs at the lower boundary,

β = 1 .

Under this condition, the annular expansion terms vanish because

β^{2 k} - β^{2 (k - 1)} = 0,

and the geometric factor reduces to a finite geometric series:

Φ (a, 1) = \sum_{j = 0}^{n} {(1 - a)}^{3 j} = \frac{1 - {(1 - a)}^{3 (n + 1)}}{1 - {(1 - a)}^{3}} .

(27)

Consequently, the optimization problem for the solid multi-stage turbine reduces to a single-variable maximization with respect to $a$ :

η (a) = \frac{1}{2} (2 - a) (1 - {(1 - a)}^{2}) (\frac{1 - {(1 - a)}^{3 (n + 1)}}{1 - {(1 - a)}^{3}}) .

(28)

The optimal value of $a$ is then obtained by differentiating equation (28):

\frac{d η}{d a} = \frac{1}{2} [\frac{d}{d a} ((2 - a) (1 - {(1 - a)}^{2})) (\frac{1 - {(1 - a)}^{3 (n + 1)}}{1 - {(1 - a)}^{3}}) + (2 - a) (1 - {(1 - a)}^{2}) \frac{d}{d a} (\frac{1 - {(1 - a)}^{3 (n + 1)}}{1 - {(1 - a)}^{3}})) = 0 .

This condition is solved numerically for each value of $n$ .

The solid convergent case is treated analogously, but the geometric parameter is the contraction factor $ζ$ , with $0 < ζ \leq 1$ . When $ζ < 1$ , downstream stages contract and interact with a progressively smaller portion of the streamtube. Under the same reference-area normalization, the maximum normalized efficiency occurs at the boundary $ζ = 1$ . At this boundary, the convergent geometry reduces to the same constant-radius multi-stage configuration obtained for the divergent case with $β = 1$ .

Optimized solid results

Solving the optimality condition shows that, as the number of stages increases, the optimal induction factor $a$ decreases. Physically, this means that the most efficient multi-stage system extracts energy through many small velocity reductions rather than through one large deceleration. At the same time, the total efficiency approaches a limiting value.

Table 2 summarizes the results for the solid divergent case with

β = 1

and the solid convergent case with

ζ = 1

. Under these optimal geometric conditions, the two models are mathematically identical.

Table 2.

Optimal parameters for solid multi-stage configurations $(β = 1, ζ = 1)$ .

Stages	Optimum induction $a$	Max efficiency $η$	Convergence
1 (Betz)	$2 / 3 \approx 0.667$	0.5926 (16/27)	-
10	0.1881	0.6636	99.5% of limit
20	0.1193	0.6656	99.8% of limit
50	0.0608	0.6664	$\sim$ 100% of limit
$\infty$	$\to 0$	$2 / 3 \approx 0.6667$	Limit

The results confirm that the theoretical limit for an infinite series of actuator discs is $η = 2 / 3$ , consistent with Newman’s tandem-disc prediction. Importantly, within this 1D framework, the optimum occurs for a constant-radius geometry. In other words, the increase in captured wake energy associated with conical expansion does not compensate for the corresponding increase in reference area.

This conclusion applies specifically to the normalized peak efficiency defined in equation (24). It does not imply that divergent or convergent geometries lack technological value. Instead, it shows that if the objective function is normalized by the maximum cross-sectional area, the ideal 1D optimum occurs at the constant-radius boundary.

Hollow configurations

The hollow configurations introduce an additional geometric parameter: the thickness-to-radius ratio $μ = t / r_{0}$ . This parameter controls the effective solidity of the annular rotor and determines how much of the streamtube is actively involved in energy extraction, as illustrated in Figures 5 and 6.

Optimization formulation

For the hollow divergent case, the efficiency function includes the effect of the hollow center through the annular area terms. As in the solid case, the numerical optimization indicates that $β = 1$ remains optimal because geometric expansion increases the reference area faster than it increases the recoverable power.

With $β = 1$ , the remaining optimization depends primarily on $a$ and $μ$ . The efficiency can therefore be expressed abstractly as

η (a, μ) = \frac{{\dot{W}}^{n} (a, μ)}{\frac{1}{2} ρ A_{r e f} v_{i n}^{3}} .

The derivative $\partial η / \partial μ$ is positive over the interval $0 < μ \leq 1$ . Thus, increasing the annular thickness always increases the theoretical efficiency in this idealized framework. The maximum is obtained at $μ = 1$ , where the hollow disc becomes equivalent to the solid disc.

This behavior has a direct physical interpretation. A hollow rotor with small $μ$ interacts only with a thin annular portion of the streamtube, so the multi-stage coupling between consecutive actuator regions is weak. As $μ$ increases, a larger fraction of the flow participates in energy extraction. The hollow configuration therefore approaches the solid multi-stage limit as $μ \to 1$ .

The hollow convergent case follows the same principle. The contraction factor $ζ$ controls the axial reduction of radius, while $μ$ controls the active annular thickness. Under the present normalization, decreasing $ζ$ reduces the active downstream area, while increasing $μ$ strengthens the interaction with the streamtube. Thus, the optimized hollow configurations are controlled primarily by the active annular area and recover the solid multi-stage behavior in the limit $μ = 1$ .

Optimized hollow results

To illustrate the effect of solidity, we compute the optimum for a fixed stage count of

n = 20

, which is already close to the asymptotic multi-stage limit. The thickness ratio

μ

is then varied from a thin-ring configuration to the solid-disc limit. The results are shown in Table 3.

Table 3.

Impact of solidity $μ$ on efficiency $(n = 20, β = 1)$ .

Thickness ratio $μ$	Optimum induction $a$	Max efficiency $η$	Description
0.10 (thin ring)	0.6438	0.5974	Near Betz limit
0.25	0.6022	0.6056	Slight gain
0.50	0.5020	0.6228	Moderate gain
0.75	0.3193	0.6468	High gain
1.00 (solid disc)	0.1193	0.6656	Max limit ( $2 / 3$ )

The results show a clear trend: as $μ \to 1$ , the hollow configuration approaches the solid configuration, and the efficiency tends toward the $2 / 3$ multi-stage limit. Conversely, for very thin rings $(μ \to 0)$ , the overlap area between stages becomes small, so the interaction between stages is weak. In this limit, the system behaves closer to a single-stage actuator disc and remains near the classical Betz value.

Sensitivity analysis and general findings

To complement the optimized results, a sensitivity analysis evaluates the impact of the primary aerodynamic and geometric parameters: induction factor $a$ , number of stages $n$ , divergent scaling $β$ , convergent scaling $ζ$ , and hollow thickness ratio $μ = t / r_{0}$ . Consolidating these sensitivities outcomes in Figure 8 yields three principal findings:

1. Global Convergence and Stage Number (Figure 8(a)): As the number of stages increases, all configurations converge toward a maximum theoretical efficiency of $η \approx 2 / 3$ , consistent with the infinite tandem-disc limit of 1D multi-stage momentum theory. Concurrently, the optimal induction factor shifts toward smaller values, confirming that multi-stage efficiency relies on progressive energy extraction through distributed, smaller velocity reductions.

2. Optimal Geometry under Area Normalization (Figure 8(b)–(c)): Within this model’s assumptions—where reference area penalizes efficiency—the optimal geometry remains a constant-radius cylinder ( $β = 1, ζ = 1$ ). For divergent rotors ( $β > 1$ ), the expanding reference area dominates the additional recovered power, reducing normalized peak efficiency. Conversely, convergent rotors ( $ζ < 1$ ) interact with a progressively smaller streamtube portion, similarly reducing efficiency.

3. Impact of Solidity (Figure 8(d)): For hollow configurations, aerodynamic performance scales monotonically with the thickness ratio $μ$ . Thin rings interact weakly with the streamtube, behaving closer to single-stage systems, whereas thicker annular rotors fully exploit the multi-stage effect, approaching the theoretical maximum as $μ \to 1$ .

Ultimately, these results indicate that while bio-inspired expanding blade geometries (e.g., Archimedes- or Triplaris-inspired rotors) may provide practical benefits in wake mixing, starting torque, or structural integration, their peak 1D aerodynamic efficiency remains strictly bounded by the conventional multi-stage tandem limit.

Mass-conserving wake expansion model

In the previous geometric models, efficiency is optimized using the largest cross-sectional area of the turbine as the reference area. This choice penalizes geometric expansion. However, a more physically constrained treatment for divergent turbines is to prescribe the geometry so that it follows the expansion required by mass conservation within the streamtube.

In this framework, we assume a “closed” streamtube in which no fresh wind enters the turbine between stages. Instead, each subsequent stage is sized to capture the expanded wake of the preceding stage. This represents a limiting case: it is conservative in terms of fresh momentum entrainment, but idealized in terms of wake recovery.

Theoretical derivation

Consider a control volume enclosing the complete multi-stage system. For steady, incompressible flow, mass conservation requires the mass flow rate $\dot{m}$ to remain constant across every stage $j$ :

\dot{m} = ρ A^{j} v^{j} = constant,

(29)

where

A^{j}

and

v^{j}

are the area and axial velocity at stage

j

, respectively.

Using the stage-to-stage velocity relation introduced previously, the velocity at stage $j$ is written as $v^{j} = v_{i n} {(1 - a)}^{j}$ . Substituting this expression into equation (29) gives the required area expansion between two consecutive stages:

ρ A^{j} [v_{i n} {(1 - a)}^{j}] = ρ A^{j + 1} [v_{i n} {(1 - a)}^{j + 1}] \Rightarrow \frac{A^{j + 1}}{A^{j}} = \frac{1}{1 - a} .

Thus, to preserve continuity, the area of each stage must increase geometrically by the factor

ϕ = \frac{1}{1 - a} .

Starting from the inlet area $A_{0}$ , the area of the $j$ -th stage is therefore

A^{j} = A_{0} {(\frac{1}{1 - a})}^{j} .

(30)

Power and efficiency formulation

We now evaluate the power extracted by this mass-conserving geometry. At stage $j$ , the actuator-disc power expression is applied using the local area $A^{j}$ and the local incident velocity $v^{j}$ :

{\dot{W}}^{j} = \frac{1}{4} ρ A^{j} {(v^{j})}^{3} (2 - a) (1 - {(1 - a)}^{2}) .

(31)

Substituting the mass-conserving area from equation (30) and the velocity relation $v^{j} = v_{i n} {(1 - a)}^{j}$ into equation (31) gives

{\dot{W}}^{j} = \frac{1}{4} ρ [A_{0} {(\frac{1}{1 - a})}^{j} {[v_{i n} {(1 - a)}^{j}]}^{3} (2 - a) (1 - {(1 - a)}^{2}) .

after simplifying the powers of $(1 - a)$ , the contribution of each stage becomes

{\dot{W}}^{j} = \frac{1}{4} ρ A_{0} v_{i n}^{3} (2 - a) (1 - {(1 - a)}^{2}) {(1 - a)}^{2 j} .

The total power extracted by the turbine is obtained by summing the contributions from $j = 0$ to $j = n$ :

{\dot{W}}_{t o t a l}^{n} = \frac{1}{4} ρ A_{0} v_{i n}^{3} (2 - a) (1 - {(1 - a)}^{2}) \sum_{j = 0}^{n} {(1 - a)}^{2 j} .

(32)

The remaining summation is a finite geometric series. Using

\sum_{j = 0}^{n} x^{j} = \frac{1 - x^{n + 1}}{1 - x}, and, x = {(1 - a)}^{2},

we obtain

\sum_{j = 0}^{n} {(1 - a)}^{2 j} = \frac{1 - {(1 - a)}^{2 (n + 1)}}{1 - {(1 - a)}^{2}} .

Substituting this previous result into equation (32) the factor $1 - {(1 - a)}^{2}$ cancels, yielding the compact total-power expression

{\dot{W}}_{t o t a l}^{n} = \frac{1}{4} ρ A_{0} v_{i n}^{3} (2 - a) [1 - {(1 - a)}^{2 (n + 1)}] .

(33)

finally, the efficiency is defined relative to the available wind power entering the inlet area $A_{0}$ ,

{\dot{W}}_{i n} = \frac{1}{2} ρ A_{0} v_{i n}^{3} .

This is the critical distinction from the previous models: the normalization is based on the frontal intake area rather than on the largest downstream cross-section. Therefore,

η (a, n) = \frac{{\dot{W}}_{t o t a l}^{n}}{\frac{1}{2} ρ A_{0} v_{i n}^{3}} = \frac{1}{2} (2 - a) [1 - ({1 - a)}^{2 (n + 1)}] .

(34)

Optimization and numerical results

Equation (34) shows that the mass-conserving model behaves differently from the constant-reference-area models. In the classical single-disc case, the efficiency reduces to the Betz result. As additional stages are added, however, the model allows the expanded wake to remain within the active geometry, so the total extractable energy increases with $n$ .

For a finite system, the optimal induction factor is obtained by differentiating equation (34) with respect to $a$ . This gives the condition

\frac{\partial η (a, n)}{\partial a} = \frac{1}{2} [- (1 - {(1 - a)}^{2 (n + 1)}) + (2 - a) (2 n + 2) {(1 - a)}^{2 n + 1}] = 0 .

(35)

Solving equation (35) numerically gives the trends shown in Figure 9. Three main conclusions follow:

1. Single-Stage Recovery: For $n = 0$ , the derivative reduces to the single-disc optimum, giving $a = 2 / 3$ and $η = 16 / 27$ . Thus, the model recovers the Betz limit in the single-stage case.

2. Optimal Induction Shift: As $n$ increases, the optimal value of $a$ shifts toward zero. Physically, this means that when more stages are available, it becomes more efficient to extract energy through smaller velocity reductions distributed over many interactions.

3. Ideal Diffuser Limit: In the limit $n \to \infty$ , the term ${(1 - a)}^{2 (n + 1)}$ tends to zero for $0 < a < 1$ . Equation (34) then approaches

\lim_{n \to \infty} η = \frac{1}{2} (2 - a) .

Since the optimal $a$ also tends toward zero as the number of stages increases, the idealized efficiency tends toward unity. This limit should be interpreted as an ideal mass-conserving diffuser result rather than a practically attainable turbine efficiency.

Technological implications

The mass-conserving result shows that a divergent geometry can, in principle, behave as an ideal diffuser when its expansion matches the wake expansion required by continuity. This result identifies one of the central technological implications of the model. When the active rotor geometry expands with the wake, the turbine is no longer limited to extracting energy from a single planar cross-section. Instead, the rotor can continue interacting with the decelerated and expanding flow over multiple axial locations. Divergent geometries therefore provide a physical route for increasing the effective wind–rotor interaction beyond that of a conventional single-disc HAWT.

In practice, however, stagnation, viscous losses, flow separation, fresh air impingement, finite structural size, and manufacturing constraints prevent the ideal $100 %$ limit from being reached. For this reason, the mass-conserving limit should not be interpreted as a directly attainable turbine performance value. Its value is that it identifies the mechanism by which divergent topologies can recover part of the kinetic energy that would otherwise remain in the expanding wake downstream of a classical planar rotor.

To connect this interpretation with existing experimental data, we consider the Triplaris Americana-inspired turbine designed and tested by Herrera et al. (2019) (see Figure 10). That wind turbine is used here as a representative example of a non-conventional hollow divergent geometry. This case is especially relevant because, unlike a conventional HAWT, its blades are not confined to a single radial plane. They extend axially and radially in a form inspired by the autorotating Triplaris seed. This geometry increases the streamwise distance over which the rotor interacts with the incoming wind and provides a physical example of the type of distributed energy extraction represented by the proposed multi-stage actuator-disc model. This interpretation is also relevant to other axially extended bio-inspired turbines, including Archimedes-type designs (RESPECT Co, Ltd, 2023), where the rotor geometry departs from the classical single-plane actuator-disc assumption and interacts with the flow over a finite axial distance.

Figure 11 contrasts the reconstructed experimental $C_{p}$ curve reported by Herrera et al. (2019) for the Triplaris americana-inspired turbine against two theoretical references: the classical single-disc Betz limit ( $16 / 27 \approx 59.3 %$ ) and the ideal multi-stage upper bound ( $2 / 3 \approx 66.7 %$ ). It is imperative to note that the proposed 1D analytical framework is not conceived to reproduce the experimental dynamic curve of $C_{p}$ versus incident wind speed $u_{i n}$ . Instead, the proposed model in the present work evaluates theoretical efficiency limits as a function of the induction factor $a$ and geometric scaling parameters ( $β, ζ, μ$ ). Consequently, it explains the fundamental aerodynamic ceiling and the physical mechanism behind the efficiency gains, rather than predicting specific operational points.

The relevant trends in Figure 11 highlight this mechanism. First, the Triplaris wind turbine curve exhibits a pronounced peak at low-to-moderate wind speeds. Second, a statistically accountable cluster of experimental measurements lies close to, and locally above, the classical single-disc Betz reference. In a conventional radial HAWT, baseline efficiencies typically peak well below 50% due to inevitable aerodynamic and mechanical losses, safely below the 59.3% boundary. The fact that this axially extended rotor yields multiple measurements exceeding the classical Betz limit indicates a fundamental shift in its energy extraction capability. Third, these peak values safely remain below the ideal multi-stage upper bound. These trends validate our physical interpretation: hollow divergent rotors transcend the capabilities of a single planar actuator disc by distributing the wind–rotor interaction over multiple effective stages, thereby elevating the machine’s baseline efficiency.

Therefore, the present model clarifies that these unconventional machines do not violate the Betz limit: they simply operate outside the restrictive geometric assumptions of its classical derivation. The Betz limit remains strictly valid for the idealized system it describes: a single, planar, uniformly loaded actuator disc in one-dimensional axial flow. The hollow divergent Triplaris Americana-inspired rotor fundamentally departs from this paradigm. By creating an extended interaction volume, it accesses a higher theoretical power-coefficient ceiling, demonstrating that experimental performance enhancements previously deemed anomalous are, in fact, entirely consistent with an extended multi-stage momentum theory.

This interpretation suggests two technological implications:

• Low-Wind Optimization: Enhanced multi-stage energy extraction may be especially valuable in low-wind-speed regions $(< 7.5 m / s)$ , where maximizing the power coefficient $C_{p}$ is important for economic viability. The reconstructed Triplaris Americana curve supports this implication because its highest $C_{p}$ values occur in the low-to-moderate wind-speed range. This suggests that bio-inspired, axially extended geometries may be particularly useful for small-scale turbines operating in regions where conventional HAWTs are less effective. In this context, the proposed model provides a theoretical basis for considering axial extension and hollow divergent topology as design variables for low-wind applications.

• Wake Engineering: Divergent, mass-conserving geometries offer a passive strategy for reducing wake losses and may contribute to faster wake recovery in compact wind-energy systems. By expanding with the wake, these geometries can interact with flow regions that would otherwise pass downstream without further energy extraction. This provides a design principle for unconventional HAWTs: rather than treating wake expansion as an unavoidable loss, the rotor geometry can be shaped to exploit it as part of the energy-capture process. The practical implementation of this principle will still depend on losses associated with separation, viscous effects, structural supports, and manufacturability, but the model identifies the mechanism that makes such geometries aerodynamically relevant.

Conclusions

This study introduces a novel analytical framework—the Generalized Multi-Stage Actuator Disc Model—to evaluate the theoretical performance of unconventional HAWTs featuring non-radial, axialized blade geometries. By discretizing the rotor into a sequence of interacting actuator discs, we bridge the gap between classical single-stage momentum theory and the complex multi-stage physics exhibited by modern bio-inspired and helical turbine designs.

Summary of findings

Our rigorous mathematical and numerical analysis of four distinct configurations (solid/hollow, divergent/convergent) yields three fundamental conclusions:

1. The Extended Efficiency Limit: All multi-stage configurations converge to a theoretical maximum efficiency of $η \approx 2 / 3$ ( $66.7 %$ ) as the number of stages increases. This surpasses the classical Betz limit of $16 / 27$ ( $59.3 %$ ) for single-stage rotors, confirming that axially extended energy extraction can theoretically outperform single-plane extraction.

2. Geometric Optimality: Under standard efficiency definitions (normalizing by the largest swept area), the optimal geometry is a constant-radius cylinder ( $β = 1, ζ = 1$ ). However, when evaluated under a mass-conserving framework—where performance is referenced to the initial capture area $A_{0}$ —divergent geometries prove superior. By expanding to match the natural wake dilation, such designs can theoretically approach the behavior of ideal diffusers, recovering nearly all available kinetic energy from the captured streamtube.

3. Role of Solidity: The comparison between solid and hollow configurations demonstrates that aerodynamic performance scales with rotor solidity. For hollow designs (e.g., helical loops), efficiency improves as the annular thickness increases ( $μ \to 1$ ), suggesting that maximizing the active interaction area is critical for capitalizing on the multi-stage effect.

Limitations and future research directions

The Geometric Actuator Disc Model developed in this work is intended as an idealized framework for examining the theoretical performance trends of unconventional HAWT geometries with axially distributed energy extraction. Its main purpose is to extend the classical actuator-disc perspective so that non-radial and multi-stage configurations can be analyzed within a unified momentum-based formulation. Accordingly, the results presented here should be interpreted as theoretical estimates under one-dimensional axial-flow assumptions.

While this 1D momentum model captures the dominant effect of progressive axial energy extraction and establishes the fundamental limits, several additional physical aspects remain outside the present formulation. Realizing these gains in practice requires advancing to higher-fidelity descriptions and addressing the following areas:

1. Analytical Refinement: The current formulation retains the classical axial-momentum idealization, meaning it does not include tangential induction or wake-rotation effects. Furthermore, the discretization of a continuous geometry into a sequence of actuator discs serves as a useful momentum-level approximation to identify limiting behavior, but it ignores local stage interaction. Future work should extend this framework to 2D models that explicitly solve for the radial velocity components ( $v_{r a d i a l}$ ) and non-uniform pressure gradients within the expanding streamtube that influence wake development, replacing the discretized approximation with continuous integral formulations.

2. Advanced Simulation: The “mass-conserving” wake assumption must be validated via multi-dimensional Computational Fluid Dynamics and Blade Element Momentum Theory adapted for non-planar rotors. Because practical performance will also depend on local aerodynamic interference, viscous mixing, and other geometry-related losses not explicitly modeled here, researching how specific airfoil profiles can maintain attached flow in highly divergent, low-Reynolds-number conditions is critical.

3. Experimental Validation: Targeted wind tunnel experiments are needed to isolate the “multi-stage effect” from other aerodynamic phenomena, verifying the correlation between axial length and efficiency gains predicted by our model.

4. Structural and Manufacturing Assessments: Finally, the present aerodynamic study does not attempt to assess manufacturing or structural considerations in detail. While the model is inspired by concepts that have shown practical realizability in existing designs, future work must evaluate factors such as added mass, mechanical support requirements, and structural complexity associated with the physical implementation of these geometries.

In summary, by shifting the design perspective from purely radial expansion toward axial extension and distributed rotor–flow interaction, this work identifies an additional aerodynamic design dimension for unconventional HAWTs. While the present model does not establish real-world performance, resilience, or adaptability, it provides a theoretical foundation for further multi-dimensional CFD, non-planar BEMT, experimental techniques, and bio-inspired design studies of axially extended turbines intended for low-wind-speed energy harvesting.

Footnotes

Acknowledgements

The authors acknowledge the academic and institutional support of the Departamento de Ingeniería Industrial, Facultad de Ingeniería, Pontificia Universidad Javeriana, Bogotá, Colombia.

ORCID iD

Camilo Bayona-Roa

Author contributions

Sofía Vega-Sanchez: Conceptualization, Methodology, Formal analysis, Investigation, Validation, Writing–original draft, Writing–review and editing.

Juan David Roa-Camargo: Conceptualization, Methodology, Formal analysis, Investigation, Visualization, Writing–original draft, Writing–review and editing.

Camilo Bayona-Roa: Conceptualization, Methodology, Supervision, Writing–review and editing.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Data Availability Statement

The data supporting the findings of this study are available from the corresponding author upon reasonable request.*

Appendix

References

Ajayi

Ojo

Vasel

(2019) On the need for the development of low wind speed turbine generator system. IOP Conference Series: Earth and Environmental Science 331(1): 012062. https://doi.org/10.1088/1755-1315/331/1/012062

Antonini

EGA

Virgüez

Ashfaq

, et al. (2024) Identification of reliable locations for wind power generation through a global analysis of wind droughts. Communications Earth & Environment 5(1): 103. https://doi.org/10.1038/s43247-024-01260-7

Betz

(1920) Das Maximum Der Theoretisch möglichen Ausnützung Des Windes Durch Windmotoren. Zeitschrift für Das Gesamte Turbinenwesen 26: 307–309. Available at: https://ci.nii.ac.jp/naid/10005373900/en/.

Betz

(1921) The theory of the screw propeller. Naturwissenschaften 18: 19930080854. https://ntrs.nasa.gov/citations/19930080854

Brahmi

Bouchemha

Kahla

(2024) Improved on–Off control for electromagnetic torque ripple minimization of wind energy conversion system. In: Ziani

Chadli

Bououden

, et al. (eds) Proceedings of the 5th International Conference on Electrical Engineering and Control Applications–Volume 2. Springer Nature Singapore, pp. 193–201.

Brahmi

Bouchemha

Kahla

(2025) New optimal robust control for maximum power point tracking and electromagnetic torque ripple minimization of wind energy conversion system. Wind Engineering 49(2): 356–374. https://doi.org/10.1177/0309524X241268908

Conway

(1998) Analytical solutions for the actuator disk with variable radial distribution of load. Journal of Fluid Mechanics 365: 235–267. https://doi.org/10.1017/S0022112098001383

Crawford

(2006) Re-Examining the precepts of the blade-element-momentum theory for coning rotors. Wind Energy 9: 457–478. https://doi.org/10.1002/we.197

Cullen

(2013) Measuring the environmental benefits of wind-generated electricity. American Economic Journal: Economic Policy 5(4): 107–133. https://doi.org/10.1257/pol.5.4.107

10.

Dupont

Koppelaar

Jeanmart

(2017) Global available wind energy with physical and energy return on investment constraints. Applied Energy 209(October): 322–338. https://doi.org/10.1016/j.apenergy.2017.09.085

11.

Gaitan-Aroca

Sierra

Jose Ulises Castellanos Contreras (2020) Bio-inspired rotor design characterization of a horizontal axis wind turbine. Energies 13(14): 3515. https://doi.org/10.3390/en13143515

12.

Glauert

(1935) Windmills and fans. In: Durand

(ed) Aerodynamic Theory. Springer, Vol. IV, pp. 324–330.

13.

Gorban

Gorlov

Silantyev

(2001) Limits of the turbine efficiency for free fluid flow. Journal of Energy Resources Technology 123(4): 311–317. https://doi.org/10.1115/1.1414137

14.

Hansen

MOL

(2008) Aerodynamics of Wind Turbines. 2nd edition. Earthscan.

15.

Herrera

Correa

Villada

, et al. (2019) Structural design and manufacturing process of a low scale bio-inspired wind turbine blades. Composite Structures 208: 1–12. https://doi.org/10.1016/j.compstruct.2018.08.061

16.

International Electrotechnical Commission (2019) IEC 61400-1: Wind Turbines – Part 1: Design Requirements. 4th edition. IEC.

17.

Jenkins

Burton

Bossanyi

, et al. (2021) Wind Energy Handbook, 3rd ed. Wiley. https://doi.org/10.1002/9781119451143

18.

Kuik

GAMvan.

(2017) Joukowsky actuator disc momentum theory. Wind Energy Science 2(1): 307–316. https://doi.org/10.5194/wes-2-307-2017

19.

Gaunaa

Pirrung

, et al. (2022a) A computationally efficient engineering aerodynamic model for non-planar wind-turbine rotors. Wind Energy Science 7: 75–104. https://doi.org/10.5194/wes-7-75-2022

20.

Gaunaa

Pirrung

, et al. (2022b) How should the lift and drag forces be calculated from 2-d air-foil data for dihedral or coned wind-turbine blades? Wind Energy Science 7(July): 1341–1365. https://doi.org/10.5194/wes-7-1341-2022

21.

National Research Council (2007) Environmental Impacts of Wind-Energy Projects. National Academies Press. https://doi.org/10.17226/11935

22.

Newman

(1986) Multiple actuator-disc theory for wind turbines. Journal of Wind Engineering and Industrial Aerodynamics 24(3): 215–225. https://doi.org/10.1016/0167-6105(86)90023-1

23.

Nugroho

Diana

Ariyanti

(2019) Multistage wind turbine capability to the rotor axial distance changes In: 2019 International Electronics Symposium (IES). IEEE, pp. 359–363. https://doi.org/10.1109/elecsym.2019.8901579

24.

Omidvarnia

Sarhadi

(2024) Nature-inspired designs in wind energy: a review. Biomimetics 9(2): 90. https://doi.org/10.3390/biomimetics9020090

25.

RESPECT Co, Ltd (2023) Archimedes F1 Brochure 2023. https://thearchimedes.com/images/pdf/AWM/brochure/Respect/2023/HQ/compressed.pdf

26.

Rizzi

Van Eck

Frey

(2013) The production of scientific knowledge on renewable energies: worldwide trends, dynamics and challenges and implications for management. Renewable Energy 62(September): 657–671. https://doi.org/10.1016/j.renene.2013.08.030

27.

Rotolus GmbH (2024) Produktinformation - Rotolus. https://rotolus.de/produktinformation/.

28.

Sharpe

(2004) A general momentum theory applied to an energy-extracting actuator disc. Wind Energy 7(3): 177–188. https://doi.org/10.1002/we.118

29.

Sørensen

(2015) General Momentum Theory for Horizontal Axis Wind Turbines. Springer International Publishing. https://doi.org/10.1007/978-3-319-22114-4

30.

Tummala

Velamati

Sinha

, et al. (2015) A review on small scale wind turbines. Renewable and Sustainable Energy Reviews 56(December): 1351–1371. https://doi.org/10.1016/j.rser.2015.12.027

31.

Yang

Fang

Song

, et al. (2020) Review of control strategy of large horizontal-axis wind turbines yaw system. Wind Energy 24(2): 97–115. https://doi.org/10.1002/we.2564