# An Efficient Method for Computing the Power Potential of Bypass Hydropower Installations

Olivier Cleynen, Dennis Powalla, Stefan Hoerner, Dominique Thévenin

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Olivier Cleynen, Dennis Powalla, Stefan Hoerner, Dominique Thévenin. An Efficient Method for Computing the Power Potential of Bypass Hydropower Installations. Energies, 2022, 15 (9), pp.3228. 10.3390/en15093228. [hal-04206128]

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Article

# An Efficient Method for Computing the Power Potential of Bypass Hydropower Installations

Olivier Cleynen* ID, Dennis Powalla ID, Stefan Hoerner ID and Dominique Thévenin ID

Laboratory of Fluid Dynamics and Technical Flows, Institute of Fluid Dynamics and Thermodynamics, University Otto von Guericke of Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany; dennis.powalla@ovgu.de (D.P.); hoerner@ovgu.de (S.H.); thevenin@ovgu.de (D.T.)

* Correspondence: olivier.cleynen@ovgu.de

**Abstract:** Small-scale hydropower installations make possible a transition towards decentralized electrical power production with very low ecological footprint. However, the prediction of their power potential is difficult, because the incoming flow velocity and the inlet and outlet water heights are often outside of the control of the operator. This leads to a need for a method capable of calculating an installation's power potential and efficiency rapidly, in order to cover for many possible load cases. In this article, the use of a previously-published theoretical framework is demonstrated with the case of a mid-scale hydropower device, a 26 m long water vortex power plant. It is shown that a simplified CFD simulation with a single output (the mass flow rate) is sufficient to obtain values for the two coefficients in the model. Once this is done, it becomes possible to evaluate the device's real-life performance, benchmarking it against reference values anchored in physical principles. The method can be used to provide design guidance and rapidly compare different load cases, providing answers that are not easily obtained using intuition or even experiments. These results are obtained for a computing cost several orders of magnitude smaller than those associated with a full description of the flow using CFD methods.

**Keywords:** micro-hydro; pico-hydro; efficiency; computational fluid dynamics

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**Citation:** Cleynen, O.; Powalla, D.; Hoerner, S.; Thévenin, D. An Efficient Method for Computing the Power Potential of Bypass Hydropower Installations. *Energies* **2022**, *15*, 3228. https://doi.org/10.3390/en15093228

Academic Editors: Zhengwei Wang and Yongguang Cheng

Received: 25 March 2022

Accepted: 24 April 2022

Published: 28 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

# 1. Introduction

## 1.1. Scientific Context

Ecological considerations are driving new interest in the development of small-scale hydropower devices. These make possible a transition towards electrical power production that is not only decentralized, bringing significant socioeconomic benefits [1,2], but also better able to abide by increasingly constraining ecological regulations regarding river fauna.

In this way, recent work has gone towards optimizing devices such as Savonius turbines, Darrieus turbines, water wheels, and water vortex power plants, the last of which are further described below. In traditional installations, fish passage is often impeded (thus breaking river continuity [3–5]), there is high risk of lethal fish impact with turbomachine parts [6–9], as well as high risk of barotrauma [10–12]. All of those problematic aspects add up to a high ecological footprint, and all are reduced or avoided with these new types of small-scale machines.

From a turbomachine engineering perspective however, the development of these devices is challenging because they operate in environments that are much less controlled than those of typical large-scale hydropower turbines [13,14]. In particular, the incoming flow velocity, as well as the inlet and outlet water heights, are not under full control and continuously vary along with the natural discharge over the year [15]. This makes the prediction of their power potential difficult, since many different cases have to be studied, as, e.g., in [16].
In this article, the case of a water vortex power plant (WVPP) will be studied. The WVPP is a device that aims to combine both ecology and hydropower, providing river continuity for fish migration while exploiting the discharge of this bypass facility. Here we show why the quantification of its power potential is so challenging. The practical problem encountered on a full-scale laboratory installation will be described. A simple theoretical model will then be applied to the case, demonstrating the use of an efficient method for computing the power potential of bypass hydropower installations.

## 1.2. Operation of a Water Vortex Power Plant

Water vortex power plants are hydropower devices which use a type of Francis turbine (radial inlet, vertical outlet), but without the use of any guide vanes; instead, their turbine inlet consists of an open-air, spiral-shaped basin. This type of device is suited for combinations of modest flow rates (in the order of 1 m³ s⁻¹) and modest hydraulic head (in the order of 1 m). Recent research interest in this type of device has focused on their geometry [17,18] and their compatibility with fish river migration [19].

As part of the research project Fluss-Strom, the dynamics of a 26 m long WVPP were investigated. The constructor of the device markets it as the “Fisch-Freundliches Wehr”, for use as fish migration corridors bypassing dams and large weirs. Work was conducted towards erecting a full-scale laboratory installation at the Technische Universität Dresden (Figure 1, for the purpose of carrying out biological investigations with live fish [20–23]. Numerical investigations have also been carried out, as presented further below.

<figure><img src="image_4.png" /><figcaption>**Figure 1.** (Left): computer drawing of the device with its main dimensions. In both images, the flow is from top left to bottom right, with the turbine basin in the center. (Right): photograph of the WVPP installation in the laboratory of the Technische Universität Dresden. (drawing reproduced from Powalla et al. 2021 [24], with geometry provided by Ecoligent GmbH).</figcaption></figure>

In the Dresden laboratory, water is picked up by pumps whose discharge goes through a settling tank before being fed to the WVPP. Water then flows through the free-surface installation, before overflowing an outlet weir, back into the inlet of the pumps. A casual, bystander observation of the dynamics of the installation in Dresden during testing reveals that its behavior is not simple. The system is controlled by prescribing the volume flow (provided by the pumps) and turbine rotation speed; the water levels in the inlet and outlet channels are part of the response.

The need for a model by which to analyze the installation's behavior can be prompted with the following observation. The powerplant is first operated with a given volume flow and turbine rotation speed. Then, the turbine speed is reduced. This increased restriction to the flow causes the water in the upstream channel to “back up”, and within the following minute, the water level there has increased significantly. As a result, the turbine's load increases, which, together with the change in rotation speed, causes the power output to change. From the point of view of the experimental scientist, and of the device's future operator, is this change desirable? Is more hydraulic power now available to the turbine?
## 1.3. Purpose of Article

The present article answers the question above by providing a framework with which to analyze the situation. Here, the former power output (which occurred for a higher turbine speed and identical volume flow) must be meaningfully compared with the new one. Is the turbine more efficient in one of the two cases? Should the expression for efficiency take the increase in inlet cross-sectional area into account?

In the literature, the turbine power of such devices is sometimes non-dimensionalized as an "efficiency", formulated as $\dot{W}_{shaft}/(\dot{m}g\Delta h)$, as would be done with a high-head cross-flow installation. In this case, however, the device is intended for use as a *bypass* to the main river flow (see Figure 2); the mass flow that is not captured by the WVPP is lost and will pass through the main river flow instead. The denominator in the expression above (the power that corresponds to an "efficiency" of 100%) is therefore a fleeting amount that is affected by the device's operation, making comparison of different operating points difficult.

<figure><img src="image_5.png" /><figcaption>**Figure 2.** Drawing of the WVPP in the type of installation it has been designed for: as the bypass to the main flow of a river. The heights $h_1$ and $h_2$, formally described in ref. [25], remain unaffected by its operation.</figcaption></figure>

These introductory questions prompted by observation of the Dresden laboratory installation can be addressed by defining a maximum by which the actual measured performance of the installation can be compared. To this purpose, we must express power and efficiency in a meaningful way, so they may be compared across different scales and operating conditions.

First, a review of the key parameters that describe the physical processes at hand will be carried out. In a second step, a demonstrative computational fluid dynamics (CFD) simulation will be presented, showing how numerical values for these parameters may be obtained at a relatively low computational cost. Finally, the questions above will be answered with a quantitative example.

## 2. Full Flow Simulations and Their Limitations

An important tool for fluid flow analysis is of course numerical simulation through computational fluid dynamics (CFD). As part of the same *Fluss-Strom* research project, a series of numerical investigations of the Dresden WVPP installation have been carried out at the University of Magdeburg [24,26–28]. The main focus of those investigations is the device's compatibility with fish migration, with further work currently underway in the laboratory to develop increasingly capable fish behavior models using these simulations [19], but a secondary objective, of concern in the present work, is the quantification of its hydropower potential.

A family of numerical fluid flow simulations is now available to reproduce the flow in the WVPP (Figure 3). These simulations are based on a Reynolds-averaged Navier-Stokes (RANS) approach and have been validated with experimental measurements in the Dresden laboratory. The properties of the simulations are detailed in [24,28]: an unsteady solver
is used with volume-of-fluid and $k-\omega$ SST models on a structured, 4.7-million-cell grid which accounts for the turbine movement. These are not of concern for the present work; instead, the emphasis is here placed on the computational costs involved in running those simulations: obtaining a single reading for the turbine power in [24] uses up 19,000 CPU-hours. The reasons for this high cost are, firstly, that the inherent physics of the flow are challenging to describe numerically, and secondly, that the simulations must be run for long periods of simulated time (of the order of 45 s) in order to account for the device's long response times. Indeed, the large open-air channels upstream and downstream of the turbine basin act as large mass buffers whenever boundary conditions or turbine speed is changed, while large amounts of momentum are stored in the rotating motion of the water in the basin itself.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Views from complete simulations of the flow in the WVPP. Top: the device has been sectioned longitudinally so as to display the water level and turbine position in the WVPP. A few streamlines colored according to velocity are visible in the outlet of the turbine. Bottom: view of the mesh structure in the turbine basin, colored according to velocity. Full details of this simulation are published in Powalla et al. 2021 [24].</figcaption></figure>

It therefore follows that a three-dimensional, two-phase CFD simulation accounting for the plant's complete geometry cannot currently be used to map out the behavior of the WVPP across a large range of volume flows, inlet heights, and outlet heights. Instead, a simpler model is needed in order to evaluate the potential power available to the plant, by which its efficiency can be quantified across many conditions.

**3. Model for the Energy Budget of a Water Vortex Power Plant**

In order to obtain a fast, flexible quantification of the available power across various regimes, the WVPP is now analyzed with the lens of the model presented in Cleynen et al. 2017 [25]. The underlying hypothesis is that the machine is installed as a bypass to a weir or traditional hydraulic dam, as depicted in Figure 2; it is assumed that the inlet and outlet water heights $h_1$ and $h_2$ are unaffected by its operation. The mass flow rate $\dot{m}$ is a priori unknown.

In order to quantify and non-dimensionalize power, reference values are chosen. The cross-section of the WVPP's inlet is used as a reference area $A_f = L_{width \ inlet}h_1$, and a
representative upstream river velocity $U_∞$ is used as a reference velocity. The turbine power is non-dimensionalized as the power coefficient from [25], becoming:

$$C_P \text{ hydraulic} = \frac{\dot{W}_{\text{hydraulic}}}{\frac{1}{2} \rho A_f U_\infty^3} \qquad (1)$$

$$C_P \text{ shaft} = \frac{\dot{W}_{\text{shaft}}}{\frac{1}{2} \rho A_f U_\infty^3} \qquad (2)$$

In these Equations (1) and (2), the reference power in the denominator, $\frac{1}{2}\rho A_f U_\infty^3$, is a partly arbitrary quantity, so that $C_P$ is not expected to reach any value in particular. Nevertheless, for any given inlet boundary condition, higher power coefficient values unambiguously indicate higher power. When $h_1$ is increased, the reference power grows in proportion, reflecting the device’s increased ability to capture mass flow $\dot{m}$ in the inlet.

The device’s performance is quantified separately, using the load efficiency $\eta_{load}$ (the fraction of available hydraulic power that is actually being provided to the turbine) and $\eta_{hydraulic}$ (the fraction of the hydraulic power provided to the turbine that is actually being converted to shaft power), as per ref. [25], so that the power coefficients can be rewritten as:

$$C_P \text{ hydraulic} = \frac{1}{\frac{1}{2} \rho A_f U_\infty^3} \eta_{\text{load}} \dot{W}_{\text{hydraulic, max}} \qquad (3)$$

$$C_P \text{ shaft} = \frac{1}{\frac{1}{2} \rho A_f U_\infty^3} \eta_{\text{hydraulic}} \eta_{\text{load}} \dot{W}_{\text{hydraulic, max}} \qquad (4)$$

In [25], a model for the hydraulic power available to the device was developed, based on the actuator theories from Betz, Lanchester and Zhukovsky, and accounting for the altitude drop $\Delta(z + h)$. This power is quantified using a single, relatively simple equation, where the only variable is the velocity through the actuator $u_A$:

$$C_P \text{ hydraulic} = \left[ 4 + \frac{K_{D2}}{R^2} \right] \left( R \frac{u_A}{U_\infty} \right)^3 - 4 \left( R \frac{u_A}{U_\infty} \right)^2 - K_{D0} \left( R \frac{u_A}{U_\infty} \right) \qquad (5)$$

In this equation, the size ratio $R \equiv A_A/A_f$ compares the actuator area with the inlet area, as per ref. [25]. The two parameters which must be quantified to obtain a power curve are the static drop coefficient $K_{D0}$ (determined entirely by the device’s environment), and the loss coefficient $K_{D2}$, which quantifies all of the hydraulic losses occurring through the device. These two parameters are defined as per ref. [25] as:

$$K_{D0} = \frac{-\rho g \Delta(z + h)}{\frac{1}{2} \rho u_\infty^2} \qquad (6)$$

$$K_{D2} = \frac{-F_{\text{loss}}}{\frac{1}{2} \rho u_A^2 A_f} \qquad (7)$$

In the present case, the static drop coefficient $K_{D0}$ is already known, since it is determined entirely by the device’s installation settings. The loss coefficient $K_{D2}$, however, is the result of the flow (it is expressed as a function of $F_{loss}$, the force representing all momentum losses occurring in the flow): some initial reference measurement or numerical simulation is needed in order to estimate its value, and thus predict the device’s internal losses during operation. This is achieved subsequently, using a strongly-reduced CFD model.

## 4. Low-Resource CFD Model of the Power Plant

For the purpose of rapidly quantifying the drop coefficient of the device, a CFD simulation is prepared, based on the simulations presented in Powalla et al. 2021 [24]. The full details of the flow, for example the intricacies of the water movement within the turbine, are not of interest here. Instead, what is needed is only a measure of the device’s internal energy losses: dissipation incurred through movement of the water in the inlet
and outlet channels, as well as through the rotation occurring in the turbine basin. Those losses can be correctly evaluated when the turbine is abstracted away, i.e., replaced by a pair of linked surfaces. To achieve this, the simulation is therefore greatly simplified, as shown in Figure 4. The global inlet and outlet are set to total pressure boundaries with a static pressure distribution. The modeling of two-phase flow is abandoned, prescribing instead a perfectly flat slip wall for each of the upstream and downstream channel ceilings. The turbine is removed and is replaced with a cylindrical outlet in the upstream channel, and a disc-shaped inlet in the downstream channel. A coupling of the mass flow and total pressure between these two surfaces is implemented. Water exits the top part of the weir in the yellow cylinder-shaped "top outlet", a surface with a prescribed mass flow boundary condition. The mass-flow-averaged total pressure is read out from this surface, and in turn, this value is prescribed as a boundary condition for the "bottom inlet" (purple disc in Figure 4). The resulting mass flow in this "bottom inlet" is read out and serves to prescribe the mass flow in the "top outlet", with some under-relaxation for increased stability.

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Main features of a simplified simulation for the WVPP, based on the family of simulations depicted in Figure 3, but with much of the sophistication removed. (Top): a cross-section of the weir is shown, with plane sections colored according to velocity. (Bottom): coupling mechanism between the upper and lower parts of the weir, seen from below the turbine basin.</figcaption></figure>

The resulting, highly-simplified simulation features 2.6 million cells; after a crude initialization, a stable flow field is obtained after 52 s of simulated time, marching with a time step of 0.02 s. This is obtained at the expense of only 550 CPU-hours (less than 3 % of the computational costs of the reference two-phase CFD simulation with moving rotor), making the computation well within reach of an ordinary desktop computer.

In this simulation, since there is no turbine to obstruct the flow, the mass flow is governed by the dissipation losses associated with the transit of water through the complete installation. The mass flow $\dot{m}$ is the only output from the simulation required to quantify $F_{\text{losses}}$, the force representative of all such losses in the model, calculated from the difference between the fluid's momentum at inlet and outlet (see [25]). With $F_{\text{losses}}$, the loss coefficient $K_{D2}$ is quantified, and with it, the power curve of the installation can be drawn, quantifying the hydraulic power available to the turbine as a function of the mass flow.
For the case of the WVPP installed in the laboratory in Dresden, the performance is quantified as follows. The machine features $Δz = -0.875$ m and $L_{\text{width inlet}} = L_{\text{width outlet}} = 2$ m.

The upstream reference velocity is chosen as $U_∞ = 1.2 \text{ m s}^{-1}$. In the outlet of the turbine chamber leading to the outlet channel, a cross-section is selected arbitrarily to represent an actuator surface, with area $A_A = 0.817 \text{ m}^2$. The choice of that “abstracted turbine” cross-section is a simple matter of convenience and does not affect results. Here, it is expected that this section will likely remain unaffected by modifications of interest in later studies, such as changes to the turbine basin geometry or to the diameter of the throat.

The operating boundary conditions are chosen as $h_1 = 0.825$ m and $h_2 = 0.7$ m. These values result in an inlet area $A_f = 1.65 \text{ m}^2$ and a corresponding actuator-to-inlet area ratio $R = 0.495$. The static drop coefficient is then given, before the simulation is run, as $K_{D0} = 13.625$.

The simulation is run until the mass flow has stabilized to a satisfactory level (changing by less than $1 \text{ kg s}^{-2}$). The mass flow then reaches a value of $953 \text{ kg s}^{-1}$ (this is the sole output of the simulation).

The effect of all momentum losses in the system is summed up as the single force $F_{\text{loss}} = -16.09 \text{ kN}$. In this manner, the loss coefficient (Equation (7)) is finally obtained as $K_{D2} = 14.329$. Using the mass flow, the relative actuator velocity (adjusted for relative area) is computed as $R u_A / U_∞ = 0.481$ (the fastest the water can ever flow through the device given these boundary conditions). The available power curve, plotted using Equation (5) with the obtained $K_{D0}$ and $K_{D2}$ values, is plotted in Figure 5.

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Power coefficient of the WVPP when $h_1 = 0.83$ m & $h_2 = 0.7$ m, as predicted using a single, simplified CFD simulation. The vertical axis is the power coefficient $C_p$ based on the frontal area $A_f$ (Equation (1)), and the horizontal axis is the adjusted actuator velocity (non-dimensional inlet velocity, directly proportional to the mass flow). The large dot on the zero-y axis stands for the value read out from the simplified CFD simulation, while the power curve is the solution to Equation (5), calibrated with the $K_{D2}$ value obtained from the simulation.</figcaption></figure>

In this Figure 5, a single point is shown for the single simulation used to generate the curve. A discrepancy between that point and the corresponding prediction according to the power curve is observed. This difference is attributed to non-uniformities in the flow (particularly at the outlet), which are neglected in the intentionally simple post-processing of the simulation. The power curve in Figure 5 features a maximum of $C_{p \text{ hydraulic}} = 2.76$ at an adjusted speed ($R u_A / U_\infty$)$_{\text{opt}} = 0.29$ (corresponding to $\dot{m}_{C_{p \text{ hydraulic max}}} = 574 \text{ kg s}^{-1}$). This information, available before any detail about the turbine is specified, already provides information useful for its design, for example in determining velocity triangles or sizing mechanical components.
In order to check the validity of the model, further simplified simulations are run, in which the coupling between the upper and lower regions of the simulations is modified: each time, only a specified fraction of the total pressure read out in the top outlet is prescribed in the bottom outlet. The mass flow is therefore reduced by the (fictive) force exerted by the actuator. The result of these simulations are displayed in Figure 6 together with the previously-obtained power curve. Additionally, the turbine shaft power coefficient obtained from a complete, moving-turbine simulation of a corresponding case is displayed as a single, orange-colored point.

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Modeled power curve for the WVPP. The right-most data point and the blue curve are these from Figure 5. The other green points are generated using further simplified simulations from which energy is extracted summarily in the main basin. The orange cross indicates the turbine shaft power coefficient obtained in a complete CFD simulation for conditions very close ($h_1 = 0.85$ m, $h_2 = 0.74$ m) to those used when plotting the power curve.</figcaption></figure>

In Figure 6, the agreement between the values obtained in the actuator simulations and the curve prediction built on a single, unobstructed-flow simulation is, for the purposes of this work, deemed excellent. In the top portion of the power curve, the model underpredicts power by 6% on average. The curve plotted by is not a best-fit model, but indeed the solution of Equation (5), built on the assumption that the losses internal to the WVPP can be well-described using a single, constant loss coefficient.

In this same figure, the single data point corresponding to a complete simulation (two-phase CFD with rotating turbine) falls well below the curve, at 71% of $C_p^{\text{hydraulic max}}$. In this case, the product of the load and hydraulic efficiencies, which account together for the not-quite-optimal mass flow, free-surface effects in the device, and dissipation losses within and around the turbine, is $\eta_{\text{load}}\eta_{\text{hydraulic}} = 71\%$.

**5. Results**

The model presented above can be used to evaluate performance in different situations. For example, in recent work, a series of four complete simulations of the WVPP was run, with boundary conditions adjusted so that the inlet and outlet heights would be set to respectively $h_1 = 0.64$ m and $h_2 = 0.56$ m. Each time, the rotation speed $\omega$ of the turbine was varied, and the power of the turbine was extracted. These two properties are presented together in Figure 7.
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Four turbine shaft power readings carried out in full CFD calculations of the WVPPT (individual crosses with a trend curve added as a thin line), with $h_1$ = 0.64 m and $h_2$ = 0.56 m. Both axes are dimensional (power in W and rotation speed in revolutions per minute). In these four simulations, the mass flow decreased steadily as the turbine speed was increased.</figcaption></figure>

The model presented in this article allows for the non-dimensionalization of these values, and their comparison to a maximum theoretical reference point. For this, a new, simplified (single-phase, turbine-less) simulation is run. Compared to the first case studied, the changed inlet height modifies the value of R to 0.641 and the boundary conditions to $K_{D0}$ = 13.03. Using the mass flow obtained in the simulation, the loss coefficient is quantified as $K_{D2}$ = 16.23. These results make it possible to transform the Figure 7 into the two Figures 8 and 9, which present the power coefficient and product of efficiencies for each of the simulations, together with the theoretical limit for these.

<figure><img src="image_11.png" /><figcaption>**Figure 8.** The turbine shaft power readings from Figure 7, this time non-dimensionalized as per the model developed in this chapter (orange crosses). In addition, a single simplified simulation has been run (rightmost circle data point), and the corresponding power curve has been drawn.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 9.** The data from Figure 8, this time presented so that the efficiency of the power extraction (the product of the load efficiency ηload and hydraulic efficiency ηhydraulic) is quantified; the reference power for this is the maximum of the power curve.</figcaption></figure>

Finally, the model can be used to answer the questions formulated in the introduction in a quantitative manner. The simulations in Figure 8 were subjected to an inlet height of 0.825 m; what happens if this height is raised to 1.2 m? The answer is obtained by running one additional simplified CFD simulation. Once the property fields have been initialized with values from the previous case, a converged flow state is obtained after 12 s of simulated time and an expense of 190 CPU-hours (which is 1 % of the computational cost of the reference two-phase CFD simulation with moving turbine). The resulting available power curve is plotted together with that from Figure 8 in Figure 10.

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Power curves of the WVPP for two cases: the blue curve, already plotted in Figure 8, is for when $h_1$ = 0.825 m; while the green curve is for the same installation when $h_1$ = 1.2 m. Each curve is based on a single reading from a simplified CFD simulation.</figcaption></figure>
Using the information in Figure 10, the answers are summarized as follows:

* The inlet height has been increased by 45%, and the available power has increased almost in proportion by 38% (the maximum power coefficient is decreased by 5%).

* In order to have access to this additional power, the turbine must operate with a strongly reduced inlet velocity (from $0.35U_∞$ down to $0.21U_∞$).

A quick re-dimensionalization of these results indicates that the maximum available power $\dot{W}_{\text{hydraulic max}}$ goes from 4.54 to 6.25 kW, while the corresponding mass flow $\dot{m}_{\text{opt}}$ reduces from 571 to 512 kg s$^{-1}$; these new values become the reference point by which to quantify the installation's efficiency.

It is therefore seen that the model developed in the first part of this chapter can be used to answer questions that, although simple, have no obvious immediate answer. In this case, increasing the inlet height increases the load on the turbine, but also the internal dissipative losses in the plant; if the inlet velocity were to be kept constant, these losses and the increased momentum abandoned in the outlet (where velocity would increase by virtue of mass conservation) would reduce the power available to the turbine. From the point of view of the operator, this change in operating conditions is only desirable if the turbine and generator are able to operate efficiently at a reduced flow rate.

If the power characteristics of the WVPP were of further interest, models better suited to predicting its losses could naturally be developed, using more sophistication. Here, the approach was voluntarily kept as simple as possible, focusing on illustrating the capabilities of a model which was developed to cover a much more general class of devices, including turbomachines operating in floating installations.

## 6. Conclusions

This article demonstrates that a theoretical analysis of the achievable performance in floating or bypass hydropower installations allows designers and operators to answer questions of practical importance in a computationally-efficient way.

A one-dimensional model describing the fluid flow through hydraulic devices had formerly been presented: a tool able to characterize the performance of machines operated in conditions where the mass flow rate is a control variable and the outlet water height cannot be controlled. This corresponds for instance to machines operating in a cascading flow alongside a dam. In the model, the device operating speed required to attain full load efficiency, and the corresponding maximum hydraulic power, can be quantified independently of the hydraulic efficiency.

Usage of the model was here demonstrated with the case of a mid-scale hydropower device, a 26 m long water vortex power plant (WVPP). In order to characterize the installation's power budget, only two properties need to be quantified: the static drop coefficient $K_{D0}$, a function of the boundary conditions only, and the loss coefficient $K_{D2}$, a measure of the device's hydraulic flow resistance. It is shown that a simplified CFD simulation with a single output (the mass flow rate) is sufficient to obtain a useful value for $K_{D2}$.

In this manner, the power available to the machine can be quantified given any set of boundary conditions. It becomes possible to evaluate the device's real-life performance, benchmarking it against reference values anchored in physical principles. Our model can be used to provide design guidance, and compare sets of boundary conditions one against the other easily, providing answers that are not easily obtained using intuition or even experiments. These results are obtained with a computing cost reduction of 97–99 % compared to those associated with a full description of the flow using CFD methods.

**Author Contributions:** Conceptualization, methodology, writing—original draft preparation, O.C.; software, investigation, validation, visualization, O.C. and D.P.; writing—review and editing, O.C. D.P., S.H. and D.T.; supervision, project administration, funding acquisition, S.H. and D.T. All authors have read and agreed to the published version of the manuscript.
**Funding:** The authors are grateful for the financial support provided by the Fluss-Strom "VP4: Kaskade Fischfreundliches Wehr" project financed by the Bundesministerium für Bildung und Forschung (German Federal Ministry of Education and Research) under the project number 03WKC04.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The work of Sergei Sukhorukov in running CFD simulations on the Neumann cluster of the University of Magdeburg, as part of his Master's Thesis under the mentorship of the first author, as well as the experimental work of the Technische Universität Dresden, in particular of Nadine Müller, are gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

## Nomenclature

$A_A$ actuator frontal area [m²]

$A_f$ device frontal area [m²]

$C_P$ power coefficient [—]

$g$ gravitational acceleration [m s⁻²]

$h$ height from bed to water surface, positive upwards [m]

$K_{D0}$ static drop coefficient (def. (6) as per ref. [25]) [—]

$K_{D2}$ loss coefficient (def. (7) as per ref. [25]) [—]

$\dot{m}$ mass flow [kg s⁻¹]

$R$ size ratio $R \equiv A_A / A_f$, as per ref. [25] [—]

$u_A$ fluid velocity at the actuator [m s⁻¹]

$U_\infty$ free-stream velocity [m s⁻¹]

$\dot{W}$ mechanical or electrical power [W]

$z$ altitude of bed, positive upwards [m]

$\Delta$ net difference

$\eta$ efficiency (from 0 to 1) [—]

$\rho$ density [kg m⁻³]

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