**1. Introduction**

The ability of pumps to operate efficiently in reverse mode as turbines was first established by Thoma [1] in 1931, while mapping the full operating characteristic of a centrifugal pump. In recent decades, there has been renewed interest in the use of pumps as turbines (PATs). It has been significantly used in power supply installations in remote areas, both on- and off-grid. A comprehensive overview of the current state of knowledge and experience in this area was provided by Carravetta et al. [2]. In addition to small hydropower plants, PAT is also used for energy recovery to cover the need for pressure reduction in water distribution networks (WDN) [3]. Besides power generation, PAT also acts as a throttle valve for flow control in this case. Experience with these applications was described by Venturini [4]. A case study of a specific installation (including an economic evaluation) was presented by Stefanizzi [5].

A pump design for turbine mode is a separate issue, which has been addressed many times. A chronological overview of the individual methods used for a solution was given by Ballaco [6]. An analysis of the models used for designing PAT and its experimental verification can be found in Stefanizzi [7], Derakhshan [8], and Barabareli [9]. It should be added that experimental investigations are still indispensable when an exact knowledge of turbine characteristics is required [10]. An example of a method used for determining such characteristics and their subsequent use for parameter conversion in the case of the hydrotechnical potential changing was given by Polák [11].

Various authors have provided several relatively simple modifications with positive results (such as modifications consisting of the impeller tip and hub/shroud rounding) in

**Citation:** Polák, M. Innovation of Pump as Turbine According to Calculation Model for Francis Turbine Design. *Energies* **2021**, *14*, 2698. https://doi.org/10.3390/en14092698

Academic Editors: Adam Adamkowski and Anton Bergant

Received: 3 April 2021 Accepted: 26 April 2021 Published: 8 May 2021

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**Copyright:** © 2021 by the author. 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/).

order to increase overall PAT efficiency. Specific example can be found in Singh [12,13], Doshi [14], and others. Capurso [15] dealt with the issue of the impact of blade geometry modification. More technically demanding modification of the pump (consisting of the installation of guide vanes in front of the impeller) was described by Giosio [16]. Some authors dealing with PAT design and modifications (such as Frosina [17]) followed the path of numerical flow modelling. However, such procedures already require specialized software, which is not available to a wide range of users. The aim of this study is to create a user-friendly design of a Francis turbine impeller and to experimentally verify its results as applied in the PAT modification.

#### **2. Calculation Model**

This section presents a calculation model, which was originally used to design the impellers of low specific speed Francis turbines; it is based on a method detailed in [18]. However, a modified model can also be used to great effect for the design of the geometry modification of an impeller for PAT. For experimental verification of the model results, the test impeller was manufactured according to the calculation model used for a particular PAT. The impeller was then tested on a hydraulic circuit. The test results are presented in the second part of the article. The model is designed as a mathematical algorithm, for which any software that has mathematical functions can be used. In this case, MS Excel software was used to ensure maximum clarity of the results and simple operation. The user then worked with the MS Excel calculation protocol. The input variables of the calculation model are the hydrotechnic potential of the turbine installation site and the size (diameter) of the impeller. The potential is given by the net head *<sup>H</sup>* (m) and the flow rate *<sup>Q</sup>* (m3·s−1). Based on these values, the specific speed of the turbine (with regard to the power *Ns* (min<sup>−</sup>1)) is estimated from the following equation:

$$\text{Ns} = \text{N} \cdot \sqrt{\text{g}} \cdot \frac{Q^{1/2}}{H^{3/4}} \tag{1}$$

where *<sup>N</sup>* (min−1) is the assumed turbine shaft speed and *<sup>g</sup>* (m·s−2) is the gravitational acceleration [19]. The value of *Ns* is entered into the green-coloured cell on the 1st line in the calculation protocol on page 6. The net head of the site *H* (m) is entered in line 8. Another necessary input value is the outer diameter of the impeller *D*<sup>1</sup> (m), which is entered in line 9. All key input variables are thus given.

To design the impeller, the calculation model uses the theory of hydraulic similarity, based on the geometric similarity of velocity triangles. Velocity triangles are related to performance parameters by means of Euler's equation [20]:

$$Y\_T = \mu\_1 \cdot \mathfrak{c}\_{u1} - \mu\_2 \cdot \mathfrak{c}\_{u2} \tag{2}$$

or:

$$
\eta \cdot \eta \cdot \varrho \cdot \mathcal{Q} \cdot \mathcal{H} = \rho \cdot \mathcal{Q} (\mathfrak{u}\_1 \cdot \mathfrak{c}\_{\mathfrak{u}1} - \mathfrak{u}\_2 \cdot \mathfrak{c}\_{\mathfrak{u}2}) \tag{3}
$$

where *YT* (J·kg<sup>−</sup>1) is the turbine specific energy, *<sup>u</sup>*1, *cu*<sup>1</sup> and *<sup>u</sup>*2, *cu*<sup>2</sup> (m·s<sup>−</sup>1) are the velocity triangles vectors at the impeller inlet and outlet, respectively (see Figure 1), *η<sup>T</sup>* (-) is the turbine efficiency, and *<sup>ρ</sup>* (kg·m<sup>−</sup>3) is the fluid density.

The assumed total efficiency *η<sup>T</sup>* is based on the size of the turbine here (i.e., on the outer diameter of the impeller *D*<sup>1</sup> according to Moody's relation [21]):

$$\eta\_T = 1 - (1 - \eta\_M) \sqrt[4]{\frac{D\_M}{D\_1}} \tag{4}$$

where *η<sup>M</sup>* (1) is the efficiency of the corresponding turbine with the impeller diameter *DM* (m).

The described calculation model allows for the designing of turbine impellers' geometry with specific speed values *Ns* = 80 min−<sup>1</sup> and higher [18]. Figure 2 shows a diagram

of the simplified overview of its algorithm. The background colours in the diagram correspond to the colours of the cells in the calculation protocol.

**Figure 1.** Velocity triangles at the inlet and outlet of the Francis impeller blade.

**Figure 2.** Algorithm of the impeller design in the calculation model.

The procedure stemming from the original design of the model required the entry of some dimensional characteristics directly from the drawing of the impeller meridional cross-section (see Figure 3). It had to be drawn at a certain stage of the impeller design. For greater user comfort, this phase was converted by the author into a calculation algorithm by means of mathematical functions, which is then used by the model for further designs. However, this "service" can be used only for limited range of specific speed *Ns* = 80 to 100 min−1. The model can be also used for designing an impeller with a higher specific speed, but the required geometric characteristics need to be entered manually (lines 28, 29 and 31, 32) based on a self-made drawing. The procedure of this drawing is to divide the flow area of the impeller meridional cross-section into partial streams (two streams are sufficient in the case of a low specific speed narrow impeller, as shown in Figure 3). The border streamline is drawn at the inlet in the middle of the channel height. Inside the channel, the course of the streamlines is determined on the orthogonal trajectory using circles inscribed between the border streamline and the impeller contour (see Figure 3). At

the same time, the multiplication of the diameters of these circles and the distances of their centres from the turbine axis must be approximately the same for all of them [18].

$$d\_{AB} \cdot r\_{AB} = d\_{BC} \cdot r\_{BC} = const.\tag{5}$$

Based on this requirement, the impeller flow area is divided and the values *dAB*, *rAB* and *dBC*, *rBC* gained from the drawing are entered into the above-mentioned lines.

**Figure 3.** Meridional cross-section of the turbine impeller.

The values in the yellow-coloured cells in the calculation protocol are determined on the basis of mathematical functions, which the author created from the curves of the nomograms of the original Francis turbine design. To illustrate this, Figure 4 shows an example of the transformation of the curve *B/D*1*<sup>e</sup>* = *f(Ns)* from a nomogram to a mathematical function. The default original nomogram is at the bottom left, and a graphical representation of the transformation result can be seen at the top right. The black dashed line here corresponds to the original curve and the red line is calculated from the polynomial function shown below the graph. This equation is then used in the calculation model, namely in line 10.

**Figure 4.** An example of transformation of curve from nomogram to mathematical function. Reproduced and modified from [18], SNTL Prague: 1962.

Some curves in the nomograms may differ according to different authors. The calculation model also takes this fact into account and allows a more experienced user to intervene in the calculation and change the values in the yellow cells as needed.

The next section of the impeller design (lines 35 to 39) is a combination of the previous results and the graphic construction of the velocity triangles. Again, in the original calculation design, making the drawing of triangles manual and measuring the values from the drawn construction for further calculation were required at this stage. Regarding maximum user comfort, these "manual" operations (presented in the diagram in Figure 2 by

the dashed line) were transformed into mathematical functions and used by the calculation model in further operations.

The final outcomes of the model are the basic geometry characteristics for impeller construction, summarized in the form of the calculation protocol (see Figure 5). Besides the main impeller dimensions, the values of the angles (*α*1, *β*1) for the geometry of velocity triangles (or the blade at the inlet) are presented here. The shape of the blade at the outlet is determined by angles at three points—on the outer (*β*<sup>2</sup> *<sup>A</sup>*), mean (*β*<sup>2</sup> *<sup>B</sup>*), and inner streamline (*β*<sup>2</sup> *<sup>C</sup>*). The number of impeller blades z is presented at the very end of the protocol in line 48. In addition, the model also indicates the shaft speed N (line 12) and the flow rate *Q<sup>η</sup>* (line 14) corresponding to the optimum operation (BEP) at a given net head H.


**Figure 5.** Calculation protocol of the Francis impeller design based on [18].
