Performance Study of a Bladeless Microturbine

energies

Article Performance Study of a Bladeless Microturbine

Krzysztof Rusin *, Włodzimierz Wróblewski, Sebastian Rulik, Mirosław Majkut and Michał Strozik

Department of Power Engineering and Turbomachinery, Silesian University of Technology, 44-100 Gliwice, Poland; [email protected] (W.W.); [email protected] (S.R.); [email protected] (M.M.); [email protected] (M.S.) * Correspondence: [email protected]

Abstract: The paper presents a comprehensive numerical and experimental analysis of the Tesla turbine. The turbine rotor had 5 discs with 160 mm in diameter and inter-disc gap equal to 0.75 mm. The nozzle apparatus consisted of 4 diverging nozzles with 2.85 mm in height of minimal cross- section. The investigations were carried out on air in subsonic flow regime for three pressure ratios: 1.4, 1.6 and 1.88. Maximal generated power was equal to 126 W and all power characteristics were in good agreement with numerical calculations. For each pressure ratio, maximal efficiency was approximately the same in the experiment, although numerical methods proved that efficiency slightly dropped with the increase of pressure ratio. Measurements included pressure distribution in the plenum chamber and tip clearance and temperature drop between the turbine’s inlet and the outlet. For each pressure ratio, the lowest value of the total temperature marked the highest efficiency of the turbine, although the lowest static temperature was shifted towards higher rotational speeds. The turbine efficiency could surpass 20% assuming the elimination of the impact of the lateral gaps between the discs and the casing. The presented data can be used as a benchmark for the validation of analytical and numerical models. Keywords: microturbine; bladeless turbine; radial turbine; Tesla turbine; experimental data; numeri- Citation: Rusin, K.; Wróblewski, W.; cal simulation Rulik, S.; Majkut, M.; Strozik, M. Performance Study of a Bladeless Microturbine. Energies 2021, 14, 3794. https://doi.org/10.3390/en14133794 1. Introduction

Academic Editors: Chirag Trivedi and One of the biggest challenges in power engineering is to provide a continuous power Francesco Castellani supply to all-electric grid users. The power units have to be on the one hand reliable and efficient, but on the other hand must not pollute the environment. The growing share Received: 9 June 2021 of renewable energy units in the energy mix is a positive trend, but certain limitations Accepted: 22 June 2021 (e.g., a limit of the share of wind energy in the total amount of electricity generated in the Published: 24 June 2021 system [1] and the unpredictability of photovoltaics and wind turbines [2]) do not allow to fully replace power plants burning fossil fuels. One of the ways of improving this situation Publisher’s Note: MDPI stays neutral is a recuperation of waste heat. More than 50% of the energy used in the world is wasted with regard to jurisdictional claims in as low-temperature heat [3] (e.g., 60% of fuel energy in internal combustion engines is published maps and institutional affil- lost as waste heat [4]), so retrieving a part of this energy is vital to increase the efficiency iations. of a process and thereby decrease the fuel consumption. The organic Rankine cycle is a technology suitable for this aim. It is similar to a classic Rankine cycle, but the working medium is an organic fluid characterized by the low temperature of boiling. One of the most important parts of such a unit is the expander. Dumont et al. [5] and Zywica et al. [6] Copyright: © 2021 by the authors. conducted a comparative analysis of expanders suitable for small-scale ORC systems. The Licensee MDPI, Basel, Switzerland. Tesla turbine seems to be an interesting alternative to the expanders used so far. This article is an open access article The Tesla turbine is a bladeless radial turbine invented by Nicola Tesla in 1913 [7]. distributed under the terms and Contrary to conventional turbines, its principle of operation is based on viscous effects conditions of the Creative Commons in a fluid. Despite its unsophisticated construction, phenomena in the turbine are quite Attribution (CC BY) license (https:// complex. The supply system is responsible for the increase of the kinetic energy of the creativecommons.org/licenses/by/ working medium and its orientation in relation to the discs. It may consist of a plenum 4.0/).

Energies 2021, 14, 3794. https://doi.org/10.3390/en14133794 https://www.mdpi.com/journal/energies Energies 2021, 14, 3794 2 of 18

chamber and a set of nozzles or guide vanes [8]. The supply system is the major reason for low turbine efficiency [9,10]. Despite the fact that nozzles can work efficiently, there is a problem with phenomena like shock waves or the interaction between the inlet apparatus and rotor [11]. More details regarding the selection of the inlet shape can be found in [9,12]. The rotor of the Tesla turbine is its most characteristic component. It is comprised of a set of thin discs mounted co-axially on a shaft. Fluid entering the rotor creates stresses on the discs’ walls. The circumferential component of the stress can be considered as a driving factor of the turbine, contrary to conventional turbines, where the wall stresses contribute to the efficiency drop. It can be utilized to formulate an alternative approach to efficiency determination. Allen [13] proposed the turbine performance estimation as a ratio of circumferential stresses generated on the disc wall to the sum of radial and circumferential components. The rotor theoretical efficiency calculated based on the adiabatic enthalpy drop under specific conditions (laminar flow and small mass flow rate) can be higher than 95% [10]. However, due to the non-uniformity of the flow and kinetic energy loss in the outlet system, it is not an easy task to reach that level of efficiency. Many researchers have carried out numerical and analytical investigations concerning flow in the rotor. Boyd and Rice [14] derived a 2D analytical model of incompressible flow. Carey [15] developed a one-dimensional model of the rotor, where viscous shear forces were substituted by body forces. The research proved that the Tesla turbine applied in a small-scale Rankine system can achieve isentropic efficiency above 75%. The model was confirmed to give results in line with numerical simulations of a real turbine [16]. Further improvement of this model was presented in [17,18], where the flow was treated as turbulent and compressible with the consideration of the radial pressure gradient. The model agreed well with experimental data within the low and medium range of rotational speeds but differed for high velocities. Guha and Sengupta [19] developed a three-dimensional model of the flow in the rotor, in which the impact of inertial, viscous, Coriolis and centrifugal forces on the fluid dynamics and generated power was explained. Based on the models presented in [15,19,20], Manfrida and Talluri [21] presented a two-dimensional analytical model, which took into account real fluid properties depending on local variables. In [22,23], the model was upgraded with the stator impact and detailed treatment of the losses and tests on refrigerant and hydrocarbon fluids were carried out. The model was also validated using Computational Fluid Dynamics CFD analysis and gave a correct distribution of kinematic and thermodynamic parameters [24]. Talluri et al. [25] performed an analysis concerning the possibility of the application of a Tesla turbine in ORC systems. One of the most important parameters is the quality and shape of the disc surface. For micro-channels, where a relative roughness is higher than 0.05, the constriction of flow becomes an important factor [26]. Rusin et al. [11] proved by means of numerical simulation that the appropriate value of roughness can lead to a rise in generated power by 35%. Romanin and Carey presented in [27] an analytical model which took into account roughness. Another important factor influencing the turbine efficiency is disc spacing. Qi et al. [28] investigated an optimal spacing for two sets of inlet system configurations: one-to-one, where one nozzle supplies exactly one gap and one-to-many where one nozzle supplies all gaps. Research presented in [29] concerns the impact of the shape of the disc tips (blunt, sharp, circular and elliptic tips) and inlet nozzle configuration on the turbine performance. According to this research, a blunt tip is recommended for one-to-many inlet while a sharp tip for one-to-many tips. A similar scientific problem was a topic of investigation of Sengupta and Guha [30]. They concluded that small disc thickness, blunt disc tips, full nozzle opening, optimal radial clearance and uniform inlet condition were necessary factors to achieve high turbine performance. An important part of the Tesla turbine analysis is the experimental investigations. One of the early works concerning this subject was the thesis of Armstrong [31]. He performed a series of investigations on a Tesla turbine with steam as the working medium. Steidel and Weiss [32] performed an experimental campaign on a 0.32 m Tesla turbine utilizing geothermal fluids. They obtained an efficiency below 7%. Leaman [33] investigated a Energies 2021, 14, 3794 3 of 18

turbine whose rotor consisted of 4 discs with 10 cm in diameter and obtained 8.6% maximal efficiency. Davydov and Sherstyuk [34] carried out investigations on a turbine with 40 mm in diameter and air as a working medium. In the comprehensive study, they determined the influence of admission degree, gap spacing, number of discs and type of outlet on turbine performance. Works [35,36] provided design recommendations for scaling Tesla turbines to the millimetre scale and an experimental campaign. The maximal efficiency of such a turbine with water as the medium was equal to 36%. A new technique for measurement turbine net torque called the angular acceleration method was presented in [37]. Lemma et al. [38] investigated a 50 mm turbine whose plenum chamber was 100 mm in diameter. Maximum obtained power and efficiency were equal to around 220 W and 25%, respectively. It was concluded that the turbine could reach over 38% of efficiency if parasitic losses, like losses in the bearing, viscous losses in the end walls, and other dissipative losses in the plenum chamber, were eliminated. Talluri [39] investigated two turbines: the first one with 125 mm rotor diameter and air as the medium and the second one with 216 mm rotor diameter intended for ORC unit with R1233zd(E) as the medium. The maximum achieved efficiency of the air turbine was 11.2% and 9.62% for the ORC turbine. Rusin et al. [40] presented an investigation of a Tesla turbine, whose rotor had 73 mm in diameter. The maximum power obtained for a pressure ratio equal to 4 was 71.5 W and the maximum efficiency totalled 8.9%. Renuke et al. [41] manufactured two turbines: 64.5 mm and 120 mm in diameter and investigated the influence of inter-disc gap, disc thickness and exhaust area. Maximal efficiency was 23% for the larger turbine. Renuke et al. [42] investigated also 3 kW air Tesla expander with the focus on the influence of the number of nozzles. The authors concluded that the turbine worked better with two nozzles rather than with one or four nozzles. Experimental and numerical study concerning the impact of the disc thickness, gap between discs, radius ratio, and the outlet area of the exhaust are shown in [43]. Another important field of research concerning the Tesla turbine is its dynamic behaviour. Due to extremely high rotational speeds, often exceeding 20,000 rpm, the arising vibrations may constitute a serious problem. Bagińskiand J˛edrzejewski[44] carried out the strength and modal analysis of the rotor model of the Tesla turbine. Silvestri et al. [45] carried out extensive numerical and experimental investigations devoted to the dynamic behaviour of the rotor. A single disc and the whole assembled turbine rotor were examined. The developed models are of great utility in the design process of the turbine. Research presented in this paper pertains to the experimental analysis of the 160 mm air Tesla turbine. The study fills in the gap of the lack of experimental data concerning the thermodynamic parameters within the turbine components. The investigations provide the performance characteristics, but also the pressure in the plenum chamber, in the radial rotor clearance and in the outlet, for different operating conditions, which makes it possible to divide the pressure drop between the turbine components. Measurement of the temperature drop between the inlet and the outlet is carried out as well. The experimental results are also used for the validation of the numerical model created using Ansys CFX software. The paper is organized as follows: a new Tesla turbine prototype with the test stand is presented in the first part. The second part concerns the set-up of the numerical model and the last part is devoted to a comprehensive analysis of the results obtained.

2. Materials and Methods 2.1. Tesla Turbine Prototype The object of investigation was the new construction of a Tesla turbine (Figures1 and2 ), which was designed in the Department of Power Engineering and Turbomachinery at the Silesian University of Technology. Its rotor in nominal conﬁguration consisted of ﬁve disks of 160 mm in diameter with inter-disc gaps equal to 0.75 mm. The surfaces of the discs were polished (arithmetical mean deviation of roughness heights Ra < 1) to avoid consideration of the roughness in numerical calculations. The lateral clearance between Energies 2021, 14, x FOR PEER REVIEW 4 of 18

disks of 160 mm in diameter with inter-disc gaps equal to 0.75 mm. The surfaces of the discs were polished (arithmetical mean deviation of roughness heights Ra < 1) to avoid consideration of the roughness in numerical calculations. The lateral clearance between the rotor and the turbine casing was equal to 0.75 mm and the disc tip clearance totalled 0.5 mm. The flexible construction made it possible to change the inter-disc gap between 0.3 mm to 1.2 mm while keeping constant lateral clearance by the addition of stationary discs with appropriate thickness mounted to the casing. The rotor discs were made out of aluminium alloy and their thickness was equal to 2 mm, which prevented the discs from

Energies 2021, 14, 3794 vibration at high rotational speeds. The inlet apparatus4 of consisted 18 of a plenum chamber and 4 converging inlet nozzles. The inlet nozzles were in configuration one-to-many [28], which means that the nozzles spread across the rotor width. This approach is less effi- the rotor and the turbine casing was equal to 0.75 mm and the disc tip clearance totalled 0.5cient mm. Thecompared flexible construction to the made one-to-one it possible to changeapproach the inter-disc solution gap between as the fluid hitting the disc tips 0.3generates mm to 1.2 mm losses. while keeping Nevertheless, constant lateral it clearance was bynecessary the addition ofto stationary maintain the flexibility of the rotor discs with appropriate thickness mounted to the casing. The rotor discs were made out of aluminiumconstruction alloy and and their thicknessto avoid was equalmisalignment to 2 mm, which preventedof the nozzles the discs from and the inter-disc gaps. The di- vibrationmensions at high of rotational the nozzle’s speeds. The minimal inlet apparatus cross-se consistedction of a plenum were chamber 2.85 mm in height and width de- and 4 converging inlet nozzles. The inlet nozzles were in configuration one-to-many [28], whichpended means thaton thethe nozzles rotor spread configuration across the rotor width. (i.e., This inter-disc approach is lessgap efficient width and the number of discs). comparedThe aim to the of one-to-one the plenum approach chamber solution as the was fluid to hitting decrease the disc tips the generates total pressure losses [9] and to or- losses. Nevertheless, it was necessary to maintain the flexibility of the rotor construction andganize to avoid the misalignment tangential of the nozzlesinflow and of the the inter-disc medium gaps. The at dimensions the nozzles. of the The working medium was nozzle’s minimal cross-section were 2.85 mm in height and width depended on the rotor configurationsupplied (i.e., to inter-discthe chamber gap width by and two the number pipes, of discs). each The 12.4 aim ofmm the plenum in diameter and 200 mm in length. chamberThere was were to decrease 8 taps the in total the pressure rotor losses casing [9] and intended to organize the to tangential measure inflow flow parameters in the plenum of the medium at the nozzles. The working medium was supplied to the chamber by two pipes,chamber each 12.4 and mm inthe diameter rotor and tip 200 clearance. mm in length. The There wereoutlet 8 taps comprised in the rotor 5 ellipsoidal holes in the ro- casingtor’s intended discs toclose measure to flowthe parametersshaft and in thethe plenum collecting chamber chamber. and the rotor tipThe total area of the set of holes clearance. The outlet comprised 5 ellipsoidal holes in the rotor’s discs close2 to the shaft andin thethe collecting disc was chamber. approximately The total area of the setequal of holes to in 350 the disc mm was, approximately which was enough to prevent the flow equalfrom to 350choking mm2, which at wasthe enough outlet to prevent for the the flowinvestigat from chokinged atpressure the outlet for ratios. The collecting chamber the investigated pressure ratios. The collecting chamber gathers the working medium and enablesgathers outflow. the working medium and enables outflow.

FigureFigure 1. Internal 1. Internal ﬂow system flow of the system turbine (blue of arrows–ﬂowthe turbine direction). (blue arrows–flow direction).

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FigureFigure 2. Cross-section 2. Cross-section of the turbine. of the turbine. 2.2. Test Stand 2.2.The Test test Stand stand was located in the Laboratory of Flow Machines of the Department of Power Engineering and Turbomachinery at the Silesian University of Technology. It consistedThe of thetest air stand installation, was the located measurement in the equipment Laboratory and the of Tesla Flow turbine. Machines The air of the Department of installationPower Engineering (Figure3) works inand under-pressure Turbomachinery and its main at parts the are Silesian the roots blower,University air of Technology. It tankconsisted and piping. of Thethe rootsair installation, blower generates the under measur pressureement (up to equipment 50 kPa) in the tankand to the Tesla turbine. The which the Tesla turbine was connected by the piping of 110 mm in diameter. Maximum outputair installation of the roots blower (Figure totalled 3) 12.3works m3/s, in which under-pressure was enough to work and in its the main continuous parts are the roots blow- mode.er, air The tank aim ofand the piping. 3.4 m3 tank The was roots to stabilize blower the pressuregenerates pulsation under and pressure to make it (up to 50 kPa) in the possibletank to to regulatewhich pressurethe Tesla at the turbine turbine outlet was by connected the control valves. by the Closing piping the valves of 110 mm in diameter. made airflow only through the turbine, thereby decreasing its operating3 pressure. Fully openMaximum valves made output the roots of blowerthe roots suck airblower from the totalled ambient, 12.3 omitting m /s, the which turbine. Duewas to enough to work in the pressurecontinuous losses inmode. the piping, The the aim static of pressurethe 3.4 at m the3 tank turbine was outlet to wasstabilize slightly the higher pressure pulsation and thanto make the pressure it possible in the tank. to regulate For this reason pressure and the at fact the that turbine there was outlet a pressure by the drop control valves. Closing in the turbine’s rotor which increased pressure at the outlet section of the nozzles, those conditionsthe valves were notmade enough airflow to produce only transonic through flow atthe the nozzle’sturbine, outlet. thereby The turbine’s decreasing its operating inletpressure. pipes were Fully equipped open with valves two confusors. made Theirthe roots aim was bl toower minimize suck pressure air from losses, the ambient, omitting preventthe turbine. vena contracta Due fromto pressure occurring andlosses level in out the the velocitypiping, profile. the static The measurement pressure at the turbine outlet equipment (Figure4) made it possible to acquire most of the important turbine parameters. Figurewas5 slightly presents the higher location than of the the selected pressure measurement in the sensors. tank. There For werethis six reason pressure and the fact that there transducers:was a pressure two in the drop middle in of the the plenumturbine’s chamber rotor (separated which byincreased 90◦), two in pressure the rotor at the outlet section tipof clearance, the nozzles, one in thethose outlet conditions section of the turbine,were not and oneenough at the inlet.to produce The temperature transonic flow at the noz- was measured utilizing J-type thermocouples with the sensors located in the ambient andzle’s at theoutlet. outlet The section. turbine’s The turbine inlet was pipes loaded were by means equipped of a 1 kWwith brushless two confusors. motor Their aim was to actingminimize as the generator pressure connected losses, to prevent the electric vena load. Thiscontracta configuration from allowed occurring setting and level out the ve- thelocity generator profile. load The at a demanded measurement level. Torque equipment transducer (Figur modele HBM 4) made T21WN it placedpossible to acquire most of between the turbine shaft and the motor enabled the measurement of the rotational speed upthe to 20,000important rpm and turbine torque up parameters. to 2 nm with the accuracyFigure of5 ±presents0.2%. It gave the an opportunitylocation of the selected meas- tourement directly measure sensors. the shaftThere power were of thesix turbine pressure which tr couldansducers: be later compared two in tothe the middle of the plenum chamber (separated by 90°), two in the rotor tip clearance, one in the outlet section of the turbine, and one at the inlet. The temperature was measured utilizing J-type thermocouples with the sensors located in the ambient and at the outlet section. The turbine was loaded by means of a 1 kW brushless motor acting as the generator connected to the electric load. This configuration allowed setting the generator load at a demanded level. Torque transducer model HBM T21WN placed between the turbine shaft and the motor enabled the measurement of the rotational speed up to 20,000 rpm and torque up to 2 nm with the accuracy of ±0.2%. It gave an opportunity to directly measure the shaft power of the turbine which could be later compared to the results from the numerical modelling. All signals were gathered by the data acquisition system and processed using in-house build software programed in LabView.

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Energies 2021, 14, x FOR PEER REVIEWresults from the numerical modelling. All signals were gathered by the data acquisition6 of 18 Energies 2021, 14, x FOR PEER REVIEW 6 of 18 system and processed using in-house build software programed in LabView.

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Figure 3. Scheme of air installation. Figure 3. Scheme of air installation.

Figure 3. Scheme of air installation.

Figure 4. Scheme of measurement unit. Figure 4. Scheme of measurement unit. Figure 4. Scheme of measurement unit. Figure 4. Scheme of measurement unit.

Figure 5. Location of sensors: temperature (red dots) and pressure (blue dots). Figure 5. Location of sensors: temperature (red dots) and pressure (blue dots). Figure 5. Location of sensors: temperature (red dots) and pressure (blue dots). FigureThe 5. measurement Location of sensors: procedure temperature started with (red setting dots) anda demanded pressure load(blue on dots). the generator. Then,The the measurementpressure level was procedure adjusted started by mean withs of the setting control a demanded valves in the load air tank.on the Due generator. Then, Thethe pressuremeasurement level procedurewas adjusted started by mean withs setting of the controla demanded valves load in the on airthe tank.generator. Due Then, the pressure level was adjusted by means of the control valves in the air tank. Due

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The measurement procedure started with setting a demanded load on the generator. Then,to the the high pressure inertia level of the was rotor, adjusted the bytime means to obtain of the steady-state, control valves i.e., in thesmall air variability tank. Due toof thepower high and inertia rotational of the rotor,speed, the for time a given to obtain load steady-state,was between i.e., 3–5 small min. variability Data were of sampled power andwith rotational a frequency speed, of 100 for Hz a given and a load single was me betweenasurement 3–5 point min. was Data obtained were sampled by averaging with athe frequency samples ofover 100 30 Hz s. and a single measurement point was obtained by averaging the samples over 30 s. 2.3. Numerical Model 2.3. NumericalNumerical Model simulations were performed using Ansys CFX commercial software, whichNumerical is based simulationson the finite were volume performed method. using The Ansyssoftware CFX uses commercial an implicit software, finite volume which isscheme based onto solve the finite discretized volume set method. of governing The software equations, uses anwhich implicit are mass, finite volumemomentum scheme and toenergy solve conservation discretized set laws of governing[46]: equations, which are mass, momentum and energy conservation laws [46]: ∂ρ +∇∙ =0 (1) + ∇·(ρU) = 0 (1) ∂t ∂(ρU) +∇∙ ⊗ =∇p+∇∙ (2) + ∇·(ρU ⊗ U) = −∇p + ∇·τ (2) ∂t ∂(ρhtot) ∂p (3) − + +∇∙∇·(ρUh )= ∇·=∇∙(λ∇∇T) ++∇∙∇·(U∙·τ) (3) ∂t ∂t tot TermTerm ∇∙(U∇·∙τ)( Urepresents·τ) represents the internal the internal heating heating by viscosity by viscosity in the fluid in the and fluid due and to duethe tur- to thebine’s turbine’s principle principle of operation, of operation, it cannot it cannot be neglected. be neglected. Spatial Spatial discretization discretization of the of equa- the equationstions was wascarried carried out outusing using the thehigh-resolution high-resolution scheme. scheme. Shear Shear stress stress transport transport with with in- intermittencytermittency was was selected selected as as the the turbulence turbulence closure. closure. TheThe geometrygeometry of the the 3D 3D model model reflected reflected all all details details of ofthe the turbine turbine prototype. prototype. The Theonly onlydifference difference was wasthe outlet the outlet ducts ducts from from the theturbine, turbine, which which were were prolonged prolonged to obtain to obtain un- undisturbeddisturbed parameters parameters at atthe the outlet. outlet. Details Details of ofthe the boundary boundary conditions conditions are are highlighted highlighted in inFigure Figure 6.6 .

Figure 6. Details of boundary conditions. Figure 6. Details of boundary conditions. Ambient pressure and total temperature equal to 296 K were set at the inlet sections with theAmbient medium pressure (5%) turbulence and total intensity.temperature Static equal pressure to 296 averaged K were set over at thethe surfacesinlet sections was appliedwith the at medium the outlet. (5%) These turbulence boundary intensity. conditions Static corresponded pressure averaged to the quantities over the measured surfaces inwas the applied experiment, at the which outlet. made These it easierboundary to compare conditions the results.corresponded The upper to the parts quantities of the inter-discmeasured gaps in the were experiment, modelled aswhich stationary, made but it easier with adiabatic, to compare smooth the andresults. rotating The walls.upper Theparts bottom of the partinter-disc of the gaps rotor, were which modelled contained as stationary, the spacers, but was with a adiabatic, rotating domain smooth and and wasrotating connected walls. toThe the bottom upper part part of using the rotor, the frozen which rotor contained interface. the spacers, The no-slip wascondition a rotating wasdomain applied and onwas all connected walls of the to the inter-disc upper part gaps, using the plenum the frozen chamber rotor interface. and inlets The to letno-slip the boundarycondition layerwas applied develop on and all account walls of for the pressure inter-disc losses. gaps, The the free plenum slip was chamber adopted and on inlets the outletto let ducts’the boundary walls to attenuate layer develop swirling and and accoun preventt for air frompressure re-entering losses. theThe computational free slip was adopted on the outlet ducts’ walls to attenuate swirling and prevent air from re-entering

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the computational domain at the outlet’s surface. All analyses were undertaken in a steady state. A single calculation was considered as converged when the standard devi- Energies 2021, 14, 3794 ation of power, mass flow rate and efficiency of the last 200 iterations did8 of 18not exceed 0.01 W, 10−4 kg/s and 10−3, respectively. Mesh study was performed to assure that the obtained results were independent of thedomain grid at size. the outlet’sAll meshes surface. were All analyses generated were using undertaken Ansys in aMeshing steady state. tool. A singleThe study was di- calculation was considered as converged when the standard deviation of power, mass videdflow rate into and two efficiency parts. ofIn thethe last first 200 part, iterations a global did notmesh exceed independence 0.01 W, 10−4 kg/sstudy and was performed and10−3 ,the respectively. second part pertained to boundary layer discretization. Power was used as an indicatorMesh of study the wasmesh performed qualityto as assure it was that calc theulated obtained based results on were the global independent thermodynamic of and kinematicthe grid size. flow All meshes parameters were generated [40]: using Ansys Meshing tool. The study was divided into two parts. In the first part, a global mesh independence study was performed and the second part pertained to boundary layer discretization.=∙ Power was used as an indicator of (4) the mesh quality as it was calculated based on the global thermodynamic and kinematic flowTorque parameters was [40 calculated]: using the integral of tangential stresses over the disc surface, which in turn was based on dynamicN = ω and·M eddy viscosities and velocity(4) gradients. The studyTorque was carried was calculated out for using inlet the total integral pressure of tangential 3 bar, stresses outlet over static the discpressure surface, 1 bar and rota- tionalwhich inspeed turn was20,000 based rpm. on dynamicFigure and7 presents eddy viscosities a global and mesh velocity independence gradients. The study. Six unstructured,study was carried hexahedral out for inlet meshes total pressure were 3created bar, outlet and static in pressure each 1the bar maxi and rotationalmal value of the di- speed 20,000 rpm. Figure7 presents a global mesh independence study. Six unstructured, mensionless distance y+ = 1. Power obtained for the finest one was considered as the hexahedral meshes were created and in each the maximal value of the dimensionless benchmarkdistance y+ = for 1. Power other obtained cases. It for can the be finest seen one th wasat consideredall results asasymptotically the benchmark forapproached the benchmarkother cases. It value. can be seenThe that relative all results difference asymptotically between approached the coarsest the benchmark and the value. finest mesh was 3.5%.The relative Increasing difference the betweennode number the coarsest over and 6 M the did finest not mesh improve was 3.5%. the Increasing results, thebut to make sure thatnode the number increase over 6of M the did flow not improve parameters the results, would but not to make force sure additional that the increase refinement, of the mesh the flow parameters would not force additional refinement, the mesh counting ~8 M nodes countingwas selected ~8 forM thenodes second was stage selected of the study. for the second stage of the study.

FigureFigure 7.7.Global Global mesh mesh study. study. Six meshes differing on the height of the first element were created in the second stage of theSix study. meshes In all cases,differing the growth on the rate height between of consecutive the first element cells was equalwere to created 1.25 and in the second stagea variable of the number study. of layersIn all provided cases, the that growth the same rate distance between from the consecutive wall was covered cells was equal to 1.25with and an inflating a variable mesh. number The heights of layers were: 5provi·10−5 dedmm, that10−5 themm ,same5·10− 4distancemm, 10−3 frommm, the wall was · −3 · −3 + ≤ covered4 10 mm withand an8 10 inflatingmm. The mesh. four finestThe gridsheights ensured were: maximal 5∙10−5 ymm,1. 10 It−5 is mm, shown 5∙10−4 mm, 10−3 in Figure8 that results within 99.8% of accuracy were obtained for these meshes. Power −3 −3 + mm,obtained 4∙10 from mm the and last 8 two,∙10 for mm. which The y+ four> 16 andfinest y+ >grids 30 differed ensured substantially. maximal Fory ≤ the 1. It is shown in Figurecoarsest 8 mesh, that theresults power within was more 99.8% than two of timesaccuracy underestimated, were obtained which proved for these a great meshes. Power obtainedsignificance from of boundary the last layer two, discretization for which y for+ > the 16 Tesla and turbine y+ > 30 modelling. differed The substantially. mesh For the −3 coarsestwith the firstmesh, element the heightpower of 10wasmm more was than chosen. two times underestimated, which proved a great significance of boundary layer discretization for the Tesla turbine modelling. The mesh with the first element height of 10−3 mm was chosen.

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Figure 8. Boundary layer mesh study.

The last performed analysis concerned the growth rate between cells in the inﬂationinflation layer. ItIt turned turned out out that that there there was was no difference no difference between between 1.05, 1.15 1.05, and 1.15 1.25 and values, 1.25 therefore, values, therefore,the latter wasthe latter selected. was selected. Finally, the mesh selected selected for for the the numerical numerical test test campaign campaign counted counted approximately approximately 8 M8 M nodes. nodes. 3. Results and Discussion 3. Results and Discussion Investigations were carried out on the inter-disc gap δ = 0.75 mm for three different Investigations were carried out on the inter-disc gap δ = 0.75 mm for three different pressure ratios: π1 = 1.88, π2 = 1.6 and π3 = 1.4 (labelled on charts also in “ratio_source” pressure ratios: π1 = 1.88, π2 = 1.6 and π3 = 1.4 (labelled on charts also in “ratio_source” convention). The results were compared with data obtained from the numerical calcula- convention). The results were compared with data obtained from the numerical calculations. tions Figure9 presents the power characteristics as a function of rotational speed. For CFD Figure 9 presents the power characteristics as a function of rotational speed. For calculations, curves were created out of 5 sampling points as it was enough to create a CFD calculations, curves were created out of 5 sampling points as it was enough to create precise approximation due to the lack of random error. In the case of the experiment, all a precise approximation due to the lack of random error. In the case of the experiment, all data points gathered during the investigations were placed on the chart. It can be seen data points gathered during the investigations were placed on the chart. It can be seen that the higher the pressure ratio, the higher the generated power. Pressure ratios also thatshifted the the higher peak the values pressure toward ratio, higher the rotational higher the speeds. generated In the power. case of Pressure the experiment, ratios also the shiftedmaximal the values peak were: values 126 toward W (at 10,800higher rpm), rotati 93.5onal W speeds. (at 9300 In rpm) the case and 59of Wthe (at experiment, 8160 rpm), the maximal values were: 126 W (at 10,800 rpm), 93.5 W (at 9,300 rpm) and 59 W (at 8,160 respectively for π1, π2 and π3. Due to the occurrence of high vibrations at 9650 rpm, which rpm),disturbed respectively torque measurement, for π1, π2 and itπ was3. Due impossible to the occurrence to gather of reliable high vibrations data in the at vicinity9650 rpm, of whichthis rotational disturbed speed. torque For measurement, this reason, the it power was impossible peak was notto gather measured reliable directly data in in the the case vicinity of this rotational speed. For this reason, the power peak was not measured directly of π2, but it was interpolated instead. All curves had high coefﬁcients of determination in the case of π2, but it was interpolated instead. All curves had high coefficients of de- and the lowest one occurred for π3. One of the possible explanations can be the fact that terminationsmall power and values the were lowest measured one occurred with lower for π precision3. One of duethe topossible low torque explanations (in the range can be of the0.02–0.1 fact that nm) small and,therefore, power values the standard were meas deviationured with was lower higher precision than in other due to cases. low Fortorque the (insame the reason, range theof 0.02–0.1 measuring nm) precision and, therefore, dropped the with standard the rotational deviation speed was in higher the case than of allin othercurves cases. as the For torque the decreased.same reason, Nevertheless, the measuring all approximation precision dropped curves with were the symmetrical rotational speedabout thein the peak case values. of all curves Comparison as the withtorque the decreased. numerical Ne datavertheless, shows that all approximation the agreement wascurves quite were good symmetrical for low revolutions, about the butpeak the values. results Comparison gradually diverged with the with numerical the increase data showsof rotational that the speed. agreement Peaks werewas quite slightly good higher for low than revolutions, in the case ofbut the the experiment; results gradually for π1: 140diverged W (11,700 with rpm),the increase for π2 :of 105.5 rotational W (10,400 speed. rpm) Peaks and were for slightlyπ3: 71.1 higher W (8800 than rpm). in the It case can ofbe the said experiment; that the maximal for π1: values140 W (11,700 of power rpm), were for overestimated π2: 105.5 W (10,400 by approximately rpm) and for 10% π3: and71.1 Woptimal (8800 rotationalrpm). It can speed be wassaid also that higher the maximal by about values 10% in of relation power towere the experimentaloverestimated ones. by Thisapproximately trend indicates 10% thatand oneoptimal of the rotational possible explanationsspeed was also regarding higher by the about discrepancies 10% in rela- was tionthe different to the experimental pressure conditions ones. This in thetrend turbine indicates components. that one of the possible explanations regarding the discrepancies was the different pressure conditions in the turbine components.

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FigureFigure 9. 9. PowerPower as as a a function function of of rotational rotational speed speed for for different different pressurepressure ratios.ratios.

FigureFigure 10 presents pressure in inin the the plenum plenum chamber chamber as as a a function function of of rotational rotational speed. speed. DataData obtained obtained from from CFD CFD were were calculated calculated as as an an area area averageaverage overover surfacessurfaces locatedlocated in in thethe middlemiddle of of the the plenumplenum chamber.chamber. InIn the the case case of of the the experiment,experiment, one one sample sample pointpoint waswas calculatedcalculated as as an an average average of of two two valuesvalues obtainedobtaobtainedined from from pressurepressure transducerstransducers locatedlocated inin thesethese locations locations (see (see Figure Figure 5 5).5).). Second-orderSecond-orderSecond-order polynomialspolynomialspolynomials werewerewere used usedused for forfor approximation approximationapproximation of ofof thethe data data with with the the determinationdetermination coefficients coefﬁcientscoefficients on on thethe levellevel ofof 0.9.0.9.0.9. As As thethe rotationalrotational speedspeed increased,increased, pressure pressure in inin the the plenum plenum chamberchamber rose. rose. This This tendency tendency was was visible visible in in bothboth CFDCFD andand experiment. experiment. TheThe differencedifference in in CFDCFD betweenbetween valuesvalues forfor 5,0005,000 5000 rpmrpm andand 12,50012,500 rpmrpm was was equalequal to to 3 3 kPakPa forfor eacheach pressurepressure ratio.ratio.ratio. ExperimentalExpeExperimentalrimental curves curves werewere approximatelyapproximately 10 10 kPakPa π π π ((ππ33),), 11 11 kPa kPa ( (ππ22)) andand 16 16 kPa kPa ( (ππ11)) lowerlower thanthan theirtheir numericalnumerical counterparts. counterparts. Moreover,Moreover, their their slopeslope anglesangles werewerewere higher, higher,higher, which whichwhich means meansmeans that thatthat the thethe pressure pressurepressure increment incrementincrement for aforfor unit aa unitunit of rotational ofof rota-rota- tionalspeedtional speed wasspeed higher, waswas higher,higher, e.g., between e.g.,e.g., betweenbetween 5000 rpm 5,0005,000 and rpmrpm 12,500 andand rpm 12,50012,500 it was: rpmrpm 5 itit kPa, was:was: 6 kPa55 kPa,kPa, and 66 kPa 6.75kPa and kPaand for π , π and π , respectively. 6.756.75 kPakPa3 2 forfor ππ33,, π1π22 andand ππ11,, respectively.respectively.

FigureFigure 10.10. 10. PressurePressurePressure inin the inthe theplenumplenum plenum chamberchamber chamber asas aa functionfunction as a function ofof rotationalrotational of rotational speedspeed forfor speed differentdifferent for differentpressurepressure ratios.pressureratios. ratios.

FigureFigure 11 showsshows pressurepressure distributiondistribution inin theththee tiptip clearance.clearance. It It can be seenseen thatthat thethe differencesdifferences between between CFD CFD and and the the experiment experiment we werewerere much much smaller smaller than than in in the the case case ofof thethe plenumplenum chamber,chamber, but butbut the thethe general generalgeneral tendencies tendenciestendencies were wewerere the thethe same. same.same. The TheThe static staticstatic pressure pressurepressure drop dropdrop in the inin thenozzlesthe nozzlesnozzles became becamebecame smaller smallersmaller as the asaspressure thethe pressurepressure drop drop indrop the inin rotor thethe rose.rotorrotor This rose.rose. can ThisThis be cancan explained bebe explainedexplained by the byfactby thethe that factfact the thatthat outwardly thethe outwardlyoutwardly acting centrifugal actingacting centricentri forcefugalfugal resulting forceforce resultingresulting from the frfr rotationalomom thethe rotationalrotational speed rose speedspeed and rosehadrose toandand be hadhad countered toto bebe counteredcountered by the inwardly byby thethe inwardlyinwardly acting pressure actingacting force.pressurepressure For force.force. thisreason, ForFor thisthis the reason,reason, pressure thethe pressuregradientpressure ingradientgradient the rotor inin rosethethe rotorrotor as well roserose and asas pressure wellwell andand at pressurepressure the rotor atat entrance thethe rotorrotor had entranceentrance to be sufficiently hadhad toto bebe sufficientlyhigh.sufficiently Curves high.high. obtained CurvesCurves from obtainedobtained CFD and fromfrom experiment CFDCFD andand were experimentexperiment parallel were forwere each parallelparallel pressure forfor ratio.eacheach pressurePressurepressure inratio.ratio. the Pressure tipPressure clearance inin thethe can tiptip be clearanceclearance considered cancan as bebe the consideredconsidered outlet pressure asas thethe from outletoutlet the pressurepressure inlet nozzles, fromfrom thethereforethe inletinlet nozzles, itsnozzles, increase thereforetherefore would resultitsits increaseincrease in a smaller woulwould pressured resultresult in dropin aa smallersmaller in the nozzles. pressurepressure As dropdrop none inin of the the nozzles.investigatednozzles. AsAs none pressurenone ofof thethe ratios investigatedinvestigated was critical, pressurepressure the nozzles ratiosratios were waswas not critical,critical, choked thethe and nozzlesnozzles the change werewere in not thenot chokedbackpressurechoked andand thethe influenced changechange inin the thethe mass backpressurebackpressure flow rate. Itinflinfl canuenceduenced be seen thethe in massmass Figure flowflow 12, rate. whichrate. ItIt presentscancan bebe seenseen the inmassin FigureFigure flow 12,12, rate whichwhich for different presentspresents pressure thethe massmass ratios flowflow as raterate a function forfor differentdifferent of rotational pressurepressure speed. ratiosratios asas aa functionfunction ofof rotationalrotational speed.speed.

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Figure 11. Pressure in the tip clearance as a function of rotational speed for different pressure rati- Figure 11. Pressure in the tip clearance as a function of rotational speed for different pressure ratios.Figure 11. Pressure in the tip clearance as a function of rotational speed for different pressure ratios. os.

Figure 12. Mass flow rate as a function of rotational speed for different pressure ratios. FigureFigure 12. 12. MassMass flow flow rate rate as as a a function function of of rotati rotationalonal speed speed for for different different pressure pressure ratios. ratios. TheThe highest highest values values of of mass mass flow flow rate rate equal to about 0.033 kg/s were were obtained obtained for for the the The highest values of mass flow rate equal to about 0.033 kg/s were obtained for the biggestbiggest pressurepressure ratio ratio and and in allin casesall cases decreased decreased gradually gradually in the rangein the of range5000 rpm–12,500 of 5,000 rpm– rpm biggest pressure ratio and in all cases decreased gradually in the range of 5,000 rpm– 12,500by about rpm 0.003 by about kg/s. Another0.003 kg/s. way Another of explaining way of theexplaining drop of massthe drop flow of rate mass with flow the rota-rate 12,500 rpm by about 0.003 kg/s. Another way of explaining the drop of mass flow rate withtional the speed rotational concerns speed the concerns rotor. As the it was rotor. already As it mentioned,was already fluid mentioned, particles fluid are subjectedparticles with the rotational speed concerns the rotor. As it was already mentioned, fluid particles areto thesubjected action ofto thethe centrifugalaction of the forces, centrifuga whichl forces, decreases which the radialdecreases component the radial of component velocity. As are subjected to the action of the centrifugal forces, which decreases the radial component ofa consequence,velocity. As a theconsequence, mass flow ratethe mass has to flow drop rate to conservehas to drop the to continuity conserve equation.the continuity In all of velocity. As a consequence, the mass flow rate has to drop to conserve the continuity equation.cases, the In mass all cases, flow ratethe wasmass underpredicted flow rate was underpredicted in the numerical in computations, the numerical whichcomputa- can equation. In all cases, the mass flow rate was underpredicted in the numerical computations,be elucidated which can by be the elucidated higher pressure by the higher in the tippressure clearance in the in CFDtip clearance than in thein CFD experiment, than in tions, which can be elucidated by the higher pressure in the tip clearance in CFD than in thei.e., experiment, higher nozzle’s i.e., backpressurehigher nozzle’s and backpres minor leakagessure and in theminor test leakages stand working in the intest negative stand the experiment, i.e., higher nozzle’s backpressure and minor leakages in the test stand workinggauge pressure. in negative Nevertheless, gauge pressure. as was Nevertheless, already shown, as a was bigger already mass flowshown, rate a didbigger not mass trans- working in negative gauge pressure. Nevertheless, as was already shown, a bigger mass flowlate intorate higherdid not generated translate into power. higher This generated leads to thepower. conclusion This leads that to the the turbine conclusion efficiency that flow rate did not translate into higher generated power. This leads to the conclusion that thediffered turbine between efficiency thenumerical differed between calculations the andnumerical the experiment. calculations The and efficiency the experiment. was based the turbine efficiency differed between the numerical calculations and the experiment. Theon aefficiency total to static was enthalpybased on dropa total in to the static turbine enthalpy [40]: drop in the turbine [40]: The efficiency was based on a total to static enthalpy drop in the turbine [40]: N i ηi == κ−1 (5) = . pout,stat κ (5) mcpT 1 − , in,,tot 1 p,in,tot (5) , 1 , , TheThe efficiency efficiency as as a a function function of of rotational rotational speed speed can can be be seen seen in in Figure Figure 13.13. Data Data points points The efficiency as a function of rotational speed can be seen in Figure 13. Data points forfor the the experiment experiment were were calculated calculated based based on on the the approximation approximation of of the the mass mass flow flow rate rate and and for the experiment were calculated based on the approximation of the mass flow rate and power.power. Estimation Estimation of of an an error error was was perform performeded using using the propagation of uncertainty [47]: [47]: power. Estimation of an error was performed using the propagation of uncertainty [47]: s 2 2 ∂η(m,, N ) ∂η(m,, N ) u(η)= = , ·∙σ2(m)++, ∙·σ2(N) (6)(6) = ∂m ∙ + ∂N ∙ (6)

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Figure 13. Efficiency as a function of rotational speed for CFD and experiment.

Numerical calculations lack random error, therefore no error analysis was per- FigureformedFigure 13.13. for EfficiencyEfficiency CFD data. as as a a function function of of rotati rotationalonal speed speed for for CFD CFD and and experiment. experiment. It can be seen that the numerical calculations proved that the smaller the pressure NumericalNumerical calculations lacklack randomrandom error, error, therefore therefore no errorno error analysis analysis was performedwas per- ratio, the higher the efficiency. This can be explained by the fact that a consequence of the formedfor CFD for data. CFD data. low pressure ratio is also low mass flow rate which in turn impacts the flow regime. ItIt can can be be seen seen that that the the numerical numerical calculations calculations proved proved that that the the smaller smaller the the pressure pressure Shear-force turbomachinery works efficiently only in the laminar flow regime, therefore a ratio,ratio, the the higher higher the the efficiency. efficiency. This This can can be beexplained explained by bythe thefact fact that that a consequence a consequence of the of decrease in mass flow rate and thereby Reynolds number will increase efficiency. The lowthe lowpressure pressure ratio ratio is also is also low low mass mass flow flow rate rate which which in inturn turn impacts impacts the the flow flow regime. regime. highest predicted efficiencies in CFD were equal to 10.30% (for π3), 9.95% (for π2) and Shear-forceShear-force turbomachinery turbomachinery works works efficiently efficiently only only in in the the laminar laminar flow flow regime, regime, therefore therefore a 9.60% (for π1) obtained for the same optimal rotational speeds as in the case of peak val- decreasea decrease in inmass mass flow flow rate rate and and thereby thereby Reyn Reynoldsolds number number will will increase increase efficiency. efficiency. The The ueshighest of power. predicted The efficienciescurves were in getting CFD were flatte equalned as to the 10.30% pressure (for πratio), 9.95% increased. (for π In) andthe highest predicted efficiencies in CFD were equal to 10.30% (for π33), 9.95% (for π22) and case9.60% of (forthe experimentalπ ) obtained forinvestigations, the same optimal the tren rotationalds were speeds different. as in For the each case pressure of peak valuesratio, 9.60% (for π11) obtained for the same optimal rotational speeds as in the case of peak val- the highest efficiency was on the same level of about 8.3%. The peak values were located uesof power.of power. The The curves curves were were getting getting flattened flattened as the as pressurethe pressure ratio ratio increased. increased. In the In casethe for the same rotational speeds as in the case of the peak values of power. Data gathered caseof the of experimentalthe experimental investigations, investigations, the the trends tren wereds were different. different. For For each each pressure pressure ratio, ratio, the for π3 were burdened with the highest uncertainty and the highest precision was ob- thehighest highest efficiency efficiency was was on on the the same same level level of aboutof about 8.3%. 8.3%. The The peak peak values values were were located located for tainedthe same for rotationalπ2. speeds as in the case of the peak values of power. Data gathered for π for the same rotational speeds as in the case of the peak values of power. Data gathered3 wereLaminar burdened flow with is thenot highest the only uncertainty condition and of high the highest Tesla turbine precision efficiency. was obtained As can for beπ2 . for π3 were burdened with the highest uncertainty and the highest precision was ob- seen inLaminar Figure flow14, which is not presents the only the condition efficienc ofy as high a function Tesla turbine of mass efficiency. flow rate, As for can each be tained for π2. pressureseen in Figureratio there 14, which was an presents optimal the value efficiency of throughput. as a function Moreover, of mass the flow efficiency rate, for eachwas Laminar flow is not the only condition of high Tesla turbine efficiency. As can be stronglypressure sensitive ratio there to the was change an optimal of the value mass of flow throughput. rate. The optimal Moreover, values the efficiencyin the case was of seen in Figure 14, which presents the efficiency as a function of mass flow rate, for each CFDstrongly were: sensitive 0.0254 kg/s to the (for change π3), 0.0282 of the masskg/s (for flow π rate.2) and The 0.0305 optimal kg/s values (for π in1). theA 10% case change of CFD pressure ratio there was an optimal value of throughput. Moreover, the efficiency was inwere: mass 0.0254 flow rate kg/s would (for π result), 0.0282 in kg/sapproxim (for πately) and 6 times, 0.0305 3 kg/s times (for andπ 2). times A 10% lower change effi- in strongly sensitive to the 3change of the mass flow2 rate. The optimal values1 in the case of ciency,mass flow respectively rate would for result pressure in approximately ratios 1.88, 1.6 6 times, and 1.4. 3 times In the and case 2 times of the lower experiment, efficiency, CFD were: 0.0254 kg/s (for π3), 0.0282 kg/s (for π2) and 0.0305 kg/s (for π1). A 10% change optimalrespectively values for of pressure mass flow ratios rate 1.88,totaled 1.6 0.0264 and 1.4. kg/s In (for the π case3), 0.03 of thekg/s experiment, (for π2) and optimal 0.0319 in mass flow rate would result in approximately 6 times, 3 times and 2 times lower effi- kg/svalues (for of π mass1). Sensitivity flow rate to totaled the change 0.0264 was kg/s also (for strongπ3), 0.03as 10% kg/s alteration (for π2) andwould 0.0319 decrease kg/s ciency, respectively for pressure ratios 1.88, 1.6 and 1.4. In the case of the experiment, the(for efficiencyπ1). Sensitivity 2.5 times, to the5 times change and was 3.5 times also strong for pressure as 10% ratios alteration 1.88, would1.6 and decrease 1.4, respec- the optimal values of mass flow rate totaled 0.0264 kg/s (for π3), 0.03 kg/s (for π2) and 0.0319 tively.efficiency 2.5 times, 5 times and 3.5 times for pressure ratios 1.88, 1.6 and 1.4, respectively. kg/s (for π1). Sensitivity to the change was also strong as 10% alteration would decrease the efficiency 2.5 times, 5 times and 3.5 times for pressure ratios 1.88, 1.6 and 1.4, respectively.

FigureFigure 14. 14. EfficiencyEfﬁciency as as a a function function of of mass mass flow ﬂow rate rate for for CFD CFD and and experiment. experiment.

TheThe explanation explanation of of such such a a strong strong sensitivit sensitivityy could could be be a a variation variation of of the the tangential tangential Figurevelocityvelocity 14. ratio. ratio.Efficiency This This as kinematic kinematica function quantityof quantity mass flow is is defined definedrate for CFDas as a a andratio ratio experiment. between between the the circumferential circumferential fluid velocity and a disc velocity at a given radius. According to the Tesla turbomachinery theory,The it explanation should tend of to such 1. In a this strong ideal sensitivit situation,y thecould fluid be neithera variation possesses of the an tangential excess of velocityunconverted ratio. kineticThis kinematic energy nor quantity energy is is defined transmitted as a toratio the fluid,between like the in acircumferential compressor. In

Energies 2021, 14, 3794 13 of 18

reality, the optimal value of this parameter may vary between 1.1–1.5, depending on several factors, e.g., the number of nozzles or the uniformity of flow admittance. An increase in the mass flow rate will result in the change of the velocity components in the rotor, the rotational speed of the discs and, thereby, the tangential velocity ratio in the whole rotor. This information is essential from the design point of view as the usual operational constraints involve rotational speed (depends on the generator) and throughput. In this case, the mass flow rate has to be divided between a sufficiently high number of inter-disc gaps in such a way that the tangential velocity ratio will be kept at an optimal level and the Reynolds number as low as possible. Interesting insights can be obtained by analyzing CFD results for individual gaps. Table1 shows the power and mass flow rate in the individual gaps related to the total mass flow rate and power generated by the whole turbine for different pressure ratios and rotational speed n = 10,000 rpm. Gaps between the rotor and the turbine’s casing are labelled as G0 and G00 (c.f. Figure1). G 1,G2,G3 and G4 symbolize inter-disc gaps of the rotor. It can be seen that a substantial portion of the mass flow rate flowed through the lateral gaps, but they were responsible for the generation of a small fraction of the total power. In the case of π3 = 1.4, 57.3% of mass flow rate generated 12.2% of the total power, for π2 = 1.6, 54.5% of the total throughput contributed only to 11.4% of total power and for the highest pressure ratio π1 = 1.88, 51% of the flow generated merely 11% of the power. Those numbers allowed it to be stated that the flow in the lateral gaps can be considered as a loss and should be avoided. There are several ways of dealing with this effect. As can be seen, this effect was getting slightly weaker with the increase in the pressure ratio. Another way is to increase the number of inter-disc gaps to decrease their resistance to flow. The last method concerns minimizing the size of the lateral gaps, so the tendency of flowing that way will drop.

Table 1. Power and mass ﬂow rate for individual gaps for 10,000 rpm.

π Parameter G0 G1 G2 G3 G4 G00

Ngap/N 0.112 0.244 0.223 0.209 0.202 0.01 1.4 total mgap/mtotal 0.282 0.103 0.109 0.11 0.105 0.291 N /N 0.109 0.245 0.227 0.211 0.202 0.005 1.6 gap total mgap/mtotal 0.267 0.109 0.113 0.119 0.113 0.278 N /N 0.108 0.248 0.232 0.216 0.194 0.002 1.88 gap total mgap/mtotal 0.252 0.118 0.121 0.127 0.12 0.263

The difference in power generation was also visible between the inter-disc gaps. It can be said that the closer the gap from the outlet, the bigger the share in the power generation. The maximal differences between the individual gaps were on the level of 5 percentage points (e.g., 24.8% vs. 19.4% for π1 = 1.88). Gaps G2 and G3 were characterized by a higher total share in the mass flow rate than gaps G1 and G4. It can be explained by the presence of gaps G00 and G0. The working medium flowing in the tip clearance was sucked into the lateral gaps and prevented from entering G1 and G4 gaps. Table2 shows the efficiency for the individual gaps obtained for 10,000 rpm by the two methods. The first method, which utilizes an enthalpy drop, was already described by Equation (5). The second method compares circumferential and radial stresses generated on the disc walls, as proposed in [37]: R τθdA ηs = R R (7) τθdA + τrdA EnergiesEnergies 20212021, 14,,14 x ,FOR 3794 PEER REVIEW 14 of14 18 of 18

TableTable 2. Efficiency 2. Efﬁciency for for individual individual gaps gaps for for 10,000 10,000 rpm rpm obtained obtained from from two two methods. methods.

π Parameter G0 G1 G2 G3 G4 G00 π Parameter G0 G1 G2 G3 G4 G00 ηi 0.041 0.245 0.212 0.196 0.199 0.030 1.4 ηi 0.041 0.245 0.212 0.196 0.199 0.030 1.4 s ηη s 0.1180.118 0.561 0.561 0.519 0.519 0.517 0.517 0.526 0.526 0.110 0.110 ηiη 0.0400.040 0.221 0.221 0.199 0.199 0.175 0.175 0.177 0.177 0.010 0.010 1.6 1.6 i ηsη s 0.1410.141 0.568 0.568 0.551 0.551 0.547 0.547 0.556 0.556 0.098 0.098 ηiη 0.0400.040 0.195 0.195 0.177 0.177 0.158 0.158 0.140 0.140 0.010 0.010 1.881.88 i ηsη s 0.1550.155 0.571 0.571 0.565 0.565 0.559 0.559 0.558 0.558 0.096 0.096

ThisThis equation equation has has a strong a strong foundation foundation in inthe the Tesla Tesla turbine turbine principle principle of ofoperation. operation. RadialRadial stresses stresses are are considered considered as as a aloss loss whereas whereas circumferential stressesstresses propelpropel the the turbine. turbine.Efficiency Efficiency equal equal to 0% to would 0% would be obtained be obtained only if thereonly areif there no circumferential are no circumferential stresses, ergo stresses,the flow ergo is carriedthe flow out is onlycarried in the out radial only directionin the radial so the direction turbine so is notthe revolving.turbine is Efficiencynot revolving.equal Efficiency to 100% is equal impossible to 100% to is obtain impossible as the to radial obtain stresses as the wouldradial stresses have to would be equal have to 0, to whichbe equal implies to 0, which the lack implies of the the radial lack component of the radial and component thereby flow. and thereby flow. It canIt can be beseen seen that that both both methods methods yielded yielded sign significantlyificantly different different resu results.lts. In Inthe the case case of of thethe enthalpy enthalpy method, method, the the efficiency efficiency was was at the at the level level of of20%, 20%, but but in inthe the case case of ofthe the stress stress method,method, it itwas was equal equal to ~50%~50% onon average. average. The The efficiency efficiency was was getting getting smaller smaller as the as pressure the pressureratio increased ratio increased in the enthalpyin the enthalpy method, me butthod, the but trend the wastrend reversed was reversed in the in stress the method.stress method.However, However, the relation the relation between between lateral gaps lateral and gaps inter-disc and inter-disc gaps was gaps sustained: was sustained: lateral gaps lateralhad gaps approximately had approximately five times five smaller times efficiency smaller efficiency than the inter-disc than the gaps.inter-disc gaps. FigureFigure 15 15presents presents the the temperature temperature drop drop in the in the turbine turbine in the in the case case of the of the experiment. experiment. It wasIt was calculated calculated as the as thedifference difference between between the thetemperature temperature at the at inlet the inlet and andthe outlet. the outlet. It is It importantis important to note to that note the that chart the chartshould should be trea beted treated rather rather as a qualitat as a qualitativeive assessment assessment of the of thermalthe thermal processes processes taking takingplace in place the inturbine the turbine and cannot and cannot be used be to used calculate, to calculate, e.g., the e.g., en- the thalpyenthalpy drop dropor other or other thermodynamic thermodynamic quantities. quantities. The Thereason reason for forthis this is the is the temperature temperature measurementmeasurement at the at theoutlet. outlet. The probe The probe was located was located in the axis in the of the axis outlet of the duct outlet (see ductFigure (see 5), Figurein the 5stream), in the of streamflowing ofair flowing and measurem air andent measurement of static temperature of static temperature was not possible. was not Onpossible. the other On hand, the otherthe total hand, temperature the total temperature cannot be measured cannot be precisely measured because precisely the becausefluid comingthe fluid to rest coming at the toprobe’s rest at wall the would probe’s not wall convert would the not kinetic convert energy the kineticinto temperature energy into adiabaticallytemperature due adiabatically to frictional due heating. to frictional For these heating. reasons, For theseit was reasons, not possible it was to not directly possible compareto directly the experiment compare the with experiment numerical with calculations. numerical calculations.

FigureFigure 15. 15.TemperatureTemperature drop drop in the in the turbine turbine from from the the experiment. experiment.

TakingTaking it all it allinto into account, account, it can it can be besaid said that that there there was was a relation a relation between between the the turbine turbine performanceperformance and and the the temperature temperature drop. drop. The The curves curves were were of parabolic shapeshape andand resembled resem- bledthe the efficiency efficiency and and power power characteristics, characteristics, with with peaks peaks obtained obtained for the for same the same rotational rotational speeds. speeds.The highest The highest drop drop occurred occurred for the for ratio the πratio1 = 1.88π1 = and1.88 totalledand totalled roughly roughly 3.5 K. 3.5 Peak K. Peak values valuesfor other for other ratios ratios were equalwere equal to 2.55 to K 2.55 for π K2 =for 1.6 π and2 = 1.6 1.75 and K for1.75π 3K= for 1.4. π The3 = 1.4. determination The de- 2 terminationcoefficients coefficients were between were Rbetween= 0.72–0.76, R2 = 0.72–0.76, which was which slightly waslower slightly than lower in the than case in of theprevious case of previous measurements. measurements. The explanation The explanation can be the can very be longthe very time long that time is required that is to requiredobtain theto obtain thermal the equilibrium thermal equilibrium between the between fluid and the the fluid probe, and greatly the probe, surpassing greatly the

Energies 2021, 14, x FOR PEER REVIEW 15 of 18

Energies 2021, 14, 3794 15 of 18 surpassing the time to reach the steady-state for power and rotational speed. An additional difficulty was a small variation of the temperature at the level of a few kelvin. It can be concluded that in the case of a lack of precise tools measuring efficiency, a pair of time to reach the steady-state for power and rotational speed. An additional difficulty was athermo-probes small variation would of the be temperature enough to at locate the level the ofoptimal a few kelvin.conditions It can of bethe concluded turbine’s thatopera- in thetion. case of a lack of precise tools measuring efficiency, a pair of thermo-probes would be enoughFigure to locate 16 presents the optimal the total conditions and static of the temperatures turbine’s operation. at the outlet section of the turbine Figureas a function 16 presents of the the rotational total and speed static obta temperaturesined from atCFD. the outletIt can be section seen ofthat the the turbine static astemperatures a function ofwere the decreasing rotational speedwith rotational obtained speed from CFD.up to Ita cancertain be seenpoint that and the then static the temperaturestrend reversed were as a decreasingresult of two with contradi rotationalctive thermodynamic speed up to a certain phenomena point andtaking then place. the trendOn the reversed one hand, as a the result static of twoenthalpy contradictive of the flow thermodynamic was being converted phenomena into taking mechanical place. Onenergy the in one the hand, rotor, the which static decreased enthalpy the of thestatic flow temperature. was being convertedOn the other into hand, mechanical viscous energyforces were in the dissipating rotor, which energy decreased into heat the staticand thereby temperature. increasing On the the other static hand, temperature. viscous forcesThe heating were dissipating effect was energygetting into stronger heat and with thereby rotational increasing speed theas the static higher temperature. rotation, The the heatinglonger fluid effect particles’ was getting path stronger from the with inlet rotational to the speedoutlet asand the the higher contact rotation, between the longer discs’ fluidwalls particles’and the fluid. path These from thetwo inlet effects to the were outlet competing and the and contact for each between pressure discs’ ratio, walls it andwas thepossible fluid. to These determine two effects a point were at competingwhich the trend and for reversed. each pressure In the ratio,case of it wasπ1 it possiblewas n = to16,000 determine rpm, for a point π2 it atwas which n = 15,000 the trend rpm reversed. and for π In3 it the was case 12,300 of π1 rpm.it was Inn all= 16,000cases, how- rpm, forever,π2 theit was staticn =enthalpy 15,000 rpm drop and was for positiveπ3 it was and 12,300 contributed rpm. In to all the cases, power however, generation. the static To- enthalpytal temperature drop was curves positive had andsimilar contributed trends. It to can the be power seen generation. that the lowest Total points temperature of the curves hadcorresponded similar trends. to the It can optimal be seen working that the condition lowest points of the of theturbine. curves The corresponded difference tobetween the optimal the total working and conditionstatic curves of the for turbine. a given The pressure difference ratio between and rotational the total speed and static was curvesdue to forthe afluid given velocity, pressure which ratio at and this rotational point can speed be considered was due to as the a kinematic fluid velocity, loss. which Rota- attional this speed point canat which be considered the total outlet as a kinematic temperature loss. was Rotational predicted speed to be at equal which to thethe totaltotal outletinlet temperature temperature coincided was predicted for each to be pressure equal to ratio thetotal with inlet the temperatureidle turbine operation coincided ap- for eachproximated pressure from ratio power with thecurves idle turbine(c.f. Figure operation 9). In approximatedthis case, the total from temperature power curves drop (c.f. Figurewould9 be). Inequal this case,to 0, thealthough total temperature the static te dropmperatures would at be the equal inlet to 0,and although the outlet the would static temperaturesdiffer. at the inlet and the outlet would differ.

Figure 16. Total andand staticstatic temperaturetemperature atat thethe outletoutlet fromfrom CFD.CFD.

4. Conclusions The manuscript presents an experimental and numerical analysis of a 160 mm Tesla turbineturbine propelledpropelled byby air.air. Experimental investigationsinvestigations were conducted on airair forfor threethree π π π pressurepressure ratios π11 = 1.88, 1.88, π22 == 1.6 1.6 and and π3 3== 1.4. 1.4. The The highest highest obtained obtained power power was was equal equal to to126 126 W Win the in the experiment experiment and and 140 140W in W numerical in numerical calculations. calculations. Comparison Comparison between between CFD CFDand the and experiment the experiment showed showed good good agreement agreement for low for lowrotational rotational speed. speed. The Thehighest highest effi- efficiencyciency was was 10.3% 10.3% in CFD in CFD and and8.3% 8.3% in the in experiment. the experiment. For each For pressure each pressure ratio, ratio,the maxi- the maximalmal efficiency efficiency was was approximately approximately the the same same in in the the experiment, experiment, although numericalnumerical methods proved that the efficiencyefficiency slightly droppeddropped with the increase of pressure ratio. TheThe efficiencyefficiency ofof anan individualindividual gapgap waswas betweenbetween 14–24%,14–24%, butbut thethe efficiencyefficiency ofof thethe laterallateral gaps between the rotorrotor andand thethe casingcasing waswas withinwithin 1–4%,1–4%, whichwhich ultimatelyultimately decideddecided aboutabout the overall turbine efficiency since almost 50% of the total throughput flew through those the overall turbine efficiency since almost 50% of the total throughput flew through those gaps. Moreover, the efficiency was highly sensitive to the change in mass flow rate: a gaps. Moreover, the efficiency was highly sensitive to the change in mass flow rate: a 10% 10% alteration of flow could result in a 6 fold decrease in efficiency. The mass flow rate distribution over the gaps pointed out the important role of the highly efficient sealing in

Energies 2021, 14, 3794 16 of 18

the external gaps. This problem will be less significant when the number of discs in the turbine increases. Pressure distributions in the plenum chamber and rotor tip clearance changed linearly with rotational speed. Measurement of the temperature drop between the inlet and the outlet can be helpful in search of optimal working conditions of the turbine, although the determination of the precise values might be difficult in the case when the dynamic temperature drop is of the same order of magnitude as the static temperature drop. All data gathered during the experimental campaign were used for the validation of the numerical model. The mass flow rate was predicted within 10% accuracy for all examined pressure ratios. Power was slightly overestimated in CFD with respect to the experiment and the difference rose with the rotational speed. Pressure in the rotor radial clearance calculated in CFD was in line with the experimental values; however, the differences in the plenum chamber were bigger. It allows concluding that the numerical model captured the most important flow phenomena, but high rotational speeds require special treatment, especially in turbulence modelling. In conclusion, it can be said that to work efficiently, a Tesla turbine has to have a sufficiently high number of discs so it works in a laminar flow regime and the mass flow rate is divided between inter-disc gaps optimally. Experimental data presented in the paper may be used for the validation of the analytical and numerical models of the turbine components. Research also gives insights on the mutual relationships between the turbine parameters and their importance to the turbine performance. The presented conclusions may be also useful in the design process of a Tesla turbine. Future works will concern the influence of the supply system parameters (number of nozzles, angle of admittance, nozzle cross-section) on the turbine performance.

Author Contributions: Conceptualization, K.R. and W.W.; methodology K.R. and W.W.; software, K.R., S.R. and M.M.; validation, S.R. and W.W.; formal analysis K.R., W.W. and S.R.; investigation, K.R., S.R., M.M. and M.S.; data curation, M.M. and M.S.; writing—original draft preparation, K.R.; writing—review and editing, W.W., S.R. and M.M.; visualization, K.R.; supervision, W.W.; funding acquisition, W.W. All authors have read and agreed to the published version of the manuscript. Funding: The presented research was performed within the Silesian University of Technology statutory research funds. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

−1 −1 cp Heat capacity at constant pressure J kg K h Enthalpy kJ M Torque Nm m Mass ﬂow rate kg s−1 N Power W n Rotational speed rad s−1 p Pressure Pa T Temperature K U Velocity m s−1 Greek symbols η Efﬁciency - λ Heat conductivity W m−1 K−1 κ Heat capacity ratio - π Pressure ratio - ρ Density kg m−3 σ Standard deviation - τ Tangential stress Pa ω Angular velocity s−1 Energies 2021, 14, 3794 17 of 18

Subscripts in Inlet out outlet r Radial direction ref Reference case value s Stress tot Total

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