*Garrison et al., (1976)* presented a numerical model for the prediction of laminar compressible flow between Tesla turbines. Particularly, the model efficiency prediction reached values as high as nearly 90%, with low mass flow rates and an optimal

**1970**

**1971**

**1972**

**1974**

fluid.

results. **1975**

**1976**

faithfully predicted by the numerical model.

it was stated that these improvements could be easily achieved.

Reynolds numbers. Efficiency as high as 80% was predicted.

the optimal rotor radius ratio is between 0.25 and 0.3.

rotor inlet/outlet diameter ratio between 0.2 and 0.4.

### **1970**

*Adams R and Rice W, (1970)* performed an experimental investigation of the flow characteristics inside a Tesla turbine rotor. The focus on the analysis was the comparison between experimental measurements of pressure throughout the disks and the numerical model developed in *(Boyd KE and Rice W, 1968)*. Experimental results were faithfully predicted by the numerical model.

### **1971**

*Boyak BE and Rice W, (1971)* developed an integral solution for the three–dimensional laminar flow inside a Tesla turbine. The developed model was validated against experimental data and the matching with static pressure distribution was found to be very good. The model did not take into account compressibility nor heat transfer, but it was stated that these improvements could be easily achieved.

### **1972**

*Lawn MJ, (1972)* exploited the numerical models developed in (*Boyak BE and Rice W, 1971)* in order to draw the performance maps of a Tesla disk turbine. The performance maps were realized for various dimensionless tangential velocities at several Reynolds numbers. Efficiency as high as 80% was predicted.

### **1974**

*Lawn MJ and Rice W, (1974)* presented the performance maps of Tesla turbines highlighting how efficiency deeply depends on geometry and rotational speed. Particularly, nozzle direction was assessed as a fundamental parameter. It was also claimed that the Tesla turbine could be designed with a high efficiency configuration for any fluid.

*Pater et al., (1974)* performed an experimental campaign in order to investigate the behaviour of the flow inside two corotating disks. Static pressure alongside the disks was measured, as well as flow visualization (through the utilization of a dye) was performed. Previous analytical models developed both for pumps and turbines were thus verified, showing excellent agreement between analytical and experimental results.

### **1975**

*Bassett CE, (1975)* developed an integral solution of the compressible flow of Tesla turbines. The developed model was validated against published experimental data, demonstrating a good agreement between numerical and experimental results. The model took into account several parameters, such as inlet Mach number, Reynolds number, rotor diameter, mass flow rates. The maximum efficiency was obtained for the minimum assessed value of flow rate between the disks. It was also found that the optimal rotor radius ratio is between 0.25 and 0.3.

### **1976**

36

He conducted a test campaign focusing on the identification of the critical issues of the machine, using steam as working fluid; furthermore, different nozzle configurations were analysed. A valuable result was the understanding of one of the causes of inefficiency. Indeed, it was found that the nozzle flow strongly affects the perfor-

*Beans EW, (1961)* performed both a numerical and experimental investigation on the performance of the friction disk turbine. The performance model developed was reliably predicting the performance of the turbine. The performed campaign of investigation allowed assessing a maximum turbine efficiency of over 24% and a power

*Rice W*, *(1965)* realized one of the first analytical/numerical models of the flow inside the Tesla turbine based on simplified Navier–Stokes equations, assuming a steady, incompressible and inviscid flow; moreover, he designed and tested six different disk turbines operating with air. Rice W discovered that the best efficiencies were achieved with small size turbines operating at low flow rates, in contrast with conventional bladed turbines, and suggested the application to small power range, exploiting the qualities of low cost, ease of manufacture, low noise level and reliabil-

*Beans EW, (1966)* performed a throughout investigation of a 0.15 m scale air Tesla turbine, following the work previously developed in 1961. The performance investigation included the assessment of various channel width (from 0.6 to 12.7 mm), various rotational speeds (from 4000 to 18000 rpm) and various turbine inlet pressure (from 1.7 to 3.75 bar). The assessed efficiencies ranged between 7 and 25%. The maximum obtained power was of 1.8 kW at 3.75 bar of supply pressure and 1.5 mm

*Matsch L and Rice W, (1967a)* developed a mathematical model of the potential flow of an incompressible fluid between two circular disks. The main feature of the analysis was the weighting of the partial admission effects on the flow developing inside the disks. A follow up of the previous work included the effect of rotation of

*Boyd KE and Rice W, (1968)* developed a numerical model on the laminar incompressible flow of a Tesla turbine rotor with full peripheral admission. Particularly, various conditions were investigated, changing the three main input parameters: flow

*Matsch L and Rice W, (1968)* formulated an asymptotic fully developed flow solution of the flow of an incompressible fluid inside two rotating disks. Particularly, the asymptotic solution is function of two main parameters, Reynolds number and

the disks with a flow at low Reynolds number *(Matsch L and Rice W, 1967b)*.

rate, Reynolds number and mean tangential velocity component at inlet.

mance of the turbine.

**1961**

near 1.8 kW. **1965**

ity.

**1966**

of channel width. **1967**

**1968**

mass flow rate.

*Garrison et al., (1976)* presented a numerical model for the prediction of laminar compressible flow between Tesla turbines. Particularly, the model efficiency prediction reached values as high as nearly 90%, with low mass flow rates and an optimal rotor inlet/outlet diameter ratio between 0.2 and 0.4.

*Steidel R and Weiss H, (1976)* performed an experimental investigation of a Tesla turbine for geothermal applications. The tests were performed in wet conditions, with a vapour fraction between 6 and 15%. The maximum turbine efficiency observed was of a 6.8% at 4000 rpm; the maximum assessed power was of about 2.8 kW at 4000 rpm.

### **1978**

*Truman CR et al., (1978)* developed a numerical model of a laminar flow between two rotating disks of a vapour containing liquid droplets. This type of flow condition is typical of geothermal applications. Vapour–liquid interaction was modelled on the assumption of a drag force coefficient.

### **1979**

*Truman CR et al., (1979)* modified the model developed in (*Truman CR et al., 1978)* in order to simulate flows containing solid particles. The results showed that for very small particle sizes the behaviour of the fluid would not change; but when the particle size increases (>4 μm) the solid particles would not follow the trajectory of the fluid and could also not enter the rotor.

### **1990**

*Allen JS, (1990)* exposed a closed–form model for velocity and pressure inside a Tesla turbine rotor, achieved through the assumption of fully developed boundary layer. A closed formulation for evaluation of torque was also formulated. The detailed solution of the Navier Stokes equation reduction was presented.

### **1991**

*Rice W, (1991)* presented a state of the art analysis of the Tesla turbine. Particularly, the available models in literature were collected and discussed; an assessment on the possible applications was also carried out. High efficiency, as high as 95%, were claimed for very small mass flow rate, as well as several field of application, from small shaft power application to use of very viscous fluids or geothermal applications.

### **2001**

*Sandilya P et al., (2001)* developed a numerical assessment of the fluid dynamic and mass transfer in a Tesla turbine rotor. The effect of rotation on the shape of velocity was highlighted and compared to the stationary case. Furthermore, the flow pattern, as well as its effect on mass transfer coefficient, was carefully assessed.

### **2002**

*Patel N & Schmidt DD., (2002)* carried out an experimental campaign on a boundary layer turbine using biomass combustion gases as working fluid. The 40 hours campaign was executed in order to verify the effect of deposition, erosion and corrosion of the Tesla turbine due to the substances present in the flue gases. The maximum obtained efficiency was of 11% with 3.2 kW of power output and a rotational speed of 6284 rpm. The same turbine was tested also with steam as working fluid, allowing a power production of 9.3 kW with an efficiency of 13.7% at 6500 rpm.

39

*Lezsovits F, (2003)* investigated the possibility of utilizing a Tesla turbine in a decentralized power generation system based on biomass. The selection of the turbine was justified by the requirements of reliability, robustness and cost effectiveness; and on the claim that turbine efficiency was not a fundamental parameter for the specific

*Ladino AFR, (2004a, 2004b)* developed a computational fluid dynamic assessment of a Tesla turbine of a 100 mm rotor outer radius, with air as working fluid. Efficiency of around 20% was obtained for low mass flow rate when both laminar and turbulent model were applied. The visualization of the path lines, as well as the highlighting of

*Couto et al., (2006)* presented a simple and straightforward model, which allowed assessing the optimal number of channels of a Tesla turbine in order to be applied to

*Batista M, (2007)* provided an analytical solution of a steady state, incompressible flow of fluid evolving between two rotating disks. The model was obtained through the application of asymptotical evolution of unknown components of velocity and

*Bloudicek P and Palousek D, (2007)* presented a simplified design procedure for the design of a Tesla turbine. The process design, as well as the realization of an experimental campaign with water as working fluid was assessed. The effectiveness (actual machine performance over theoretic machine performance) was evaluated and the maximum reached value was 55% with a 13.63 W power output. The maximum obtained power output was of about 58 W. A significant data, giving evidence to the most attractive feature of the turbine, was the total cost of the prototype, which was

*Deam et al., (2008)* evaluated the benefits of utilizing Tesla turbines instead of gas turbines for small–scale applications. The development of scaling laws for gas turbine was refined and the results from the model were compared with experimental data on Tesla turbine. The main result was the demonstration that for small–sized turbines, viscous turbines are more efficient than conventional bladed turbines, as the losses

*Lemma et al., (2008)* performed a comprehensive experimental and numerical study on a 50 mm rotor Tesla turbine. The assessed performance of the turbine was over 20%, claiming that the main causes of losses were mainly parasitic losses and specifically bearing losses. If these losses could be avoided, or at least reduced, the

are quite high. The assessed experimental turbine efficiency was 23.5%.

turbine efficiency could reach values close to 40%.

pressure losses in the nozzle–rotor interaction was reported.

**2003**

application. **2004**

**2006**

**2007**

pressure.

of about 124 €. **2008**

a specific application.

### **2003**

*Lezsovits F, (2003)* investigated the possibility of utilizing a Tesla turbine in a decentralized power generation system based on biomass. The selection of the turbine was justified by the requirements of reliability, robustness and cost effectiveness; and on the claim that turbine efficiency was not a fundamental parameter for the specific application.

### **2004**

*Ladino AFR, (2004a, 2004b)* developed a computational fluid dynamic assessment of a Tesla turbine of a 100 mm rotor outer radius, with air as working fluid. Efficiency of around 20% was obtained for low mass flow rate when both laminar and turbulent model were applied. The visualization of the path lines, as well as the highlighting of pressure losses in the nozzle–rotor interaction was reported.

### **2006**

*Couto et al., (2006)* presented a simple and straightforward model, which allowed assessing the optimal number of channels of a Tesla turbine in order to be applied to a specific application.

### **2007**

*Batista M, (2007)* provided an analytical solution of a steady state, incompressible flow of fluid evolving between two rotating disks. The model was obtained through the application of asymptotical evolution of unknown components of velocity and pressure.

*Bloudicek P and Palousek D, (2007)* presented a simplified design procedure for the design of a Tesla turbine. The process design, as well as the realization of an experimental campaign with water as working fluid was assessed. The effectiveness (actual machine performance over theoretic machine performance) was evaluated and the maximum reached value was 55% with a 13.63 W power output. The maximum obtained power output was of about 58 W. A significant data, giving evidence to the most attractive feature of the turbine, was the total cost of the prototype, which was of about 124 €.

### **2008**

38

a power production of 9.3 kW with an efficiency of 13.7% at 6500 rpm.

*Steidel R and Weiss H, (1976)* performed an experimental investigation of a Tesla turbine for geothermal applications. The tests were performed in wet conditions, with a vapour fraction between 6 and 15%. The maximum turbine efficiency observed was of a 6.8% at 4000 rpm; the maximum assessed power was of about 2.8 kW at 4000

*Truman CR et al., (1978)* developed a numerical model of a laminar flow between two rotating disks of a vapour containing liquid droplets. This type of flow condition is typical of geothermal applications. Vapour–liquid interaction was modelled on the

*Truman CR et al., (1979)* modified the model developed in (*Truman CR et al., 1978)* in order to simulate flows containing solid particles. The results showed that for very small particle sizes the behaviour of the fluid would not change; but when the particle size increases (>4 μm) the solid particles would not follow the trajectory of the fluid

*Allen JS, (1990)* exposed a closed–form model for velocity and pressure inside a Tesla turbine rotor, achieved through the assumption of fully developed boundary layer. A closed formulation for evaluation of torque was also formulated. The detailed

*Rice W, (1991)* presented a state of the art analysis of the Tesla turbine. Particularly, the available models in literature were collected and discussed; an assessment on the possible applications was also carried out. High efficiency, as high as 95%, were claimed for very small mass flow rate, as well as several field of application, from small shaft power application to use of very viscous fluids or geothermal appli-

*Sandilya P et al., (2001)* developed a numerical assessment of the fluid dynamic and mass transfer in a Tesla turbine rotor. The effect of rotation on the shape of velocity was highlighted and compared to the stationary case. Furthermore, the flow pattern, as well as its effect on mass transfer coefficient, was carefully assessed.

*Patel N & Schmidt DD., (2002)* carried out an experimental campaign on a boundary layer turbine using biomass combustion gases as working fluid. The 40 hours campaign was executed in order to verify the effect of deposition, erosion and corrosion of the Tesla turbine due to the substances present in the flue gases. The maximum obtained efficiency was of 11% with 3.2 kW of power output and a rotational speed of 6284 rpm. The same turbine was tested also with steam as working fluid, allowing

solution of the Navier Stokes equation reduction was presented.

rpm.

**1978**

**1979**

**1990**

**1991**

cations. **2001**

**2002**

assumption of a drag force coefficient.

and could also not enter the rotor.

*Deam et al., (2008)* evaluated the benefits of utilizing Tesla turbines instead of gas turbines for small–scale applications. The development of scaling laws for gas turbine was refined and the results from the model were compared with experimental data on Tesla turbine. The main result was the demonstration that for small–sized turbines, viscous turbines are more efficient than conventional bladed turbines, as the losses are quite high. The assessed experimental turbine efficiency was 23.5%.

*Lemma et al., (2008)* performed a comprehensive experimental and numerical study on a 50 mm rotor Tesla turbine. The assessed performance of the turbine was over 20%, claiming that the main causes of losses were mainly parasitic losses and specifically bearing losses. If these losses could be avoided, or at least reduced, the turbine efficiency could reach values close to 40%.

*Valente A., (2008)* applied the Tesla turbine concept as an equipment for pressure reduction of Hydrocarbon gases in a near isothermal process. Experimental tests confirmed the wanted characteristic of near isothermal expansion. This allows the expander to be used as a substitute of a Joule–Thomson valve in hydrocarbon pipelines.

### **2009**

*Crowell R, (2009)* is the first author to take Tesla turbine as a possible expander candidate for small ORC applications. Particularly, the study assessed an integrated system for residential applications, exploiting solar hot water collectors as heat source with the utilization of a micro ORC in order to produce electricity. The fluid utilized in the simulation was Care30 (a refrigerant blend of Isobutane and Propane with similar characteristics to R134a). The developed model of the turbine was quite simple, as the aim was to assess the power output at the shaft. A turbine efficiency of 30% was assumed, which guaranteed a daily power production of 3.2 kWh.

*Hoya GP and Guha A, (2009)* designed and manufactured a flexible test rig for Tesla turbines. They carried out several experimental analyses, comparing various measurement methods and developed a new, simple and cheap approach (angular acceleration method) for measuring torque and power output, which overcame the difficulties associated with the determination of very low torque at very high rotational speed.

*Guha A and Smiley B, (2009)* investigated the nozzle, recognizing it as the source of the major irreversibility, according to their test results; they demonstrated that a careful design of the nozzle could reduce the nozzle losses by 40–50%. They showed that utilizing a plenum chamber could result in a total pressure loss of less than 1%; furthermore, it allowed also a considerable enhancement in the uniformity of the jet and an improved overall efficiency of the Tesla turbine.

*Lampart P et al., (2009)* developed a throughout CFD investigation on different Tesla turbine dimensions with Solkatherm SES36 as working fluid. The assessed efficiency of the turbines ranged from 30 to 50% depending on turbine size and rotational velocity. For a 0.32 m shaft diameter the assessed efficiency was around 50% and the power output obtained was between 1.5 and 5 kW.

### **2010**

*Carey VP, (2010a)* realized a one–dimensional idealized model of momentum transfer in the rotor, and used it to predict the turbine efficiency in a 4 kWe solar Rankine cycle combined heat and power system, using steam as working fluid. His model resulted similar to the one developed by *Rice W, (1965)*, using body–forces to represent the wall shear effects, assuming laminar flow: according to it, under optimal design conditions, 75% turbine isentropic efficiency was predicted.

Carey VP*, (2010b)* exploited the two–dimensional model developed in *(Carey, 2010a)* and compared the possible achievable results that could be obtained with CFD analysis; concluding that 1D/2D model is a useful tool for preliminary design of Tesla turbines.

*Emran et al., (2010)* developed a simple model for estimating the proper sizing of a Tesla turbine dynamometer and generator. Tesla turbine static torque model was assessed, both analytically and experimentally.

41

*Puzyrewski R & Tesch K, (2010)* developed a 1D analytical model of the Tesla rotor calibrating it through 3D CFD analysis. The calibration through 3D CFD allowed to realize very simple and reliable 1D model, which provided the prediction of high efficiency (above 75%) when the gap between the disks is very tight (0.1 mm). *Romanin V et al., (2010)* investigated the possible power and efficiency enhancement that could be obtained by a commercial Tesla turbine. Particularly, the comparison between experimental data and numerical simulation where performed. The maximum assessed experimental efficiency was 10.1%, while the numerical optimization

*Batista M., (2011)* re–organised the work presented in *(Batista, 2007)*, providing an analytical solution of a steady state, incompressible flow of fluid evolving between

*Choon et al., (2011)* performed a computational fluid dynamics optimization analysis on a Tesla turbine with water as working fluid with the aim of exploiting the energy hold within the household water supply. The CFD analysis allowed the upgrade of an existing prototype, resulting in an increase in efficiency from 6.8 to 10.7%, with a pressure drop of 0.04 m. The main results were the possibility of applying this technology to such a niche application, as the expander is suited to produce power at

*Cirincione N, (2011)* designed and realized an ORC waste heat recovery system with a Tesla–hybrid turbine, which was claimed to be able to reach isentropic efficiency levels above 70% with steam; the last not reported tests were carried out with

*Emran TA, (2011)* exploited and further developed the results obtained in (*Emran TA, 2010)*, improving the torque model of a Tesla turbine, as well as comparing the analytical solution with experimental tests. As a further result, the developed model

*Ho–Yan BP, (2011)* applied the model developed in (*Rice W, 1965)* in order to establish a preliminary design approach for Tesla turbine for Pico Hydro applications. The obtained performance was over 80% for a 300 W power output turbine, working

*Krishnan V et al., (2011)* realized some micro–turbines (1–2 cm diameter rotors) using commercially available technologies, and tested them with different nozzles and

*Lampart P & Jedrzejewski L, (2011)* further developed the analysis presented (Lampart et al., 2009), considering the Tesla turbine as the expander for a micro– power plant of 20 kW heat capacity. The maximum efficiency obtained from a computational fluid dynamic analysis was 51% with a mass flow rate of Solkatherm

*Podergajs M, (2011)* provided a short report on Tesla turbine historical background, flow model and applications. In the report, the author highlighted the range

*Romanin VD and Carey VP, (2011)* developed an integral solution scheme of the fluid dynamics of incompressible, steady flow inside a Tesla turbine rotor. Enhanced

allowed the design of an improved Tesla turbine with optimal dimensions.

rotors configurations, achieving almost 40% shaft mechanical efficiency.

with a pressure head of 20 m and a water flow rate of 2.5 l/s.

of application where the turbine could be an attractive solution.

of the turbine predicted that efficiency higher than 75% could be reached.

**2011**

two rotating disks.

very low pressure drops.

SES36 of 0.13 kg/s.

R245fa.

*Puzyrewski R & Tesch K, (2010)* developed a 1D analytical model of the Tesla rotor calibrating it through 3D CFD analysis. The calibration through 3D CFD allowed to realize very simple and reliable 1D model, which provided the prediction of high efficiency (above 75%) when the gap between the disks is very tight (0.1 mm).

*Romanin V et al., (2010)* investigated the possible power and efficiency enhancement that could be obtained by a commercial Tesla turbine. Particularly, the comparison between experimental data and numerical simulation where performed. The maximum assessed experimental efficiency was 10.1%, while the numerical optimization of the turbine predicted that efficiency higher than 75% could be reached.

### **2011**

40

*Valente A., (2008)* applied the Tesla turbine concept as an equipment for pressure reduction of Hydrocarbon gases in a near isothermal process. Experimental tests confirmed the wanted characteristic of near isothermal expansion. This allows the expander to be used as a substitute of a Joule–Thomson valve in hydrocarbon pipelines.

*Crowell R, (2009)* is the first author to take Tesla turbine as a possible expander candidate for small ORC applications. Particularly, the study assessed an integrated system for residential applications, exploiting solar hot water collectors as heat source with the utilization of a micro ORC in order to produce electricity. The fluid utilized in the simulation was Care30 (a refrigerant blend of Isobutane and Propane with similar characteristics to R134a). The developed model of the turbine was quite simple, as the aim was to assess the power output at the shaft. A turbine efficiency of 30%

*Hoya GP and Guha A, (2009)* designed and manufactured a flexible test rig for Tesla turbines. They carried out several experimental analyses, comparing various measurement methods and developed a new, simple and cheap approach (angular acceleration method) for measuring torque and power output, which overcame the difficulties associated with the determination of very low torque at very high rotational

*Guha A and Smiley B, (2009)* investigated the nozzle, recognizing it as the source of the major irreversibility, according to their test results; they demonstrated that a careful design of the nozzle could reduce the nozzle losses by 40–50%. They showed that utilizing a plenum chamber could result in a total pressure loss of less than 1%; furthermore, it allowed also a considerable enhancement in the uniformity of the jet

*Lampart P et al., (2009)* developed a throughout CFD investigation on different Tesla turbine dimensions with Solkatherm SES36 as working fluid. The assessed efficiency of the turbines ranged from 30 to 50% depending on turbine size and rotational velocity. For a 0.32 m shaft diameter the assessed efficiency was around 50%

*Carey VP, (2010a)* realized a one–dimensional idealized model of momentum transfer in the rotor, and used it to predict the turbine efficiency in a 4 kWe solar Rankine cycle combined heat and power system, using steam as working fluid. His model resulted similar to the one developed by *Rice W, (1965)*, using body–forces to represent the wall shear effects, assuming laminar flow: according to it, under optimal

Carey VP*, (2010b)* exploited the two–dimensional model developed in *(Carey, 2010a)* and compared the possible achievable results that could be obtained with CFD analysis; concluding that 1D/2D model is a useful tool for preliminary design of Tesla

*Emran et al., (2010)* developed a simple model for estimating the proper sizing of a Tesla turbine dynamometer and generator. Tesla turbine static torque model was

was assumed, which guaranteed a daily power production of 3.2 kWh.

and an improved overall efficiency of the Tesla turbine.

and the power output obtained was between 1.5 and 5 kW.

assessed, both analytically and experimentally.

design conditions, 75% turbine isentropic efficiency was predicted.

**2009**

speed.

**2010**

turbines.

*Batista M., (2011)* re–organised the work presented in *(Batista, 2007)*, providing an analytical solution of a steady state, incompressible flow of fluid evolving between two rotating disks.

*Choon et al., (2011)* performed a computational fluid dynamics optimization analysis on a Tesla turbine with water as working fluid with the aim of exploiting the energy hold within the household water supply. The CFD analysis allowed the upgrade of an existing prototype, resulting in an increase in efficiency from 6.8 to 10.7%, with a pressure drop of 0.04 m. The main results were the possibility of applying this technology to such a niche application, as the expander is suited to produce power at very low pressure drops.

*Cirincione N, (2011)* designed and realized an ORC waste heat recovery system with a Tesla–hybrid turbine, which was claimed to be able to reach isentropic efficiency levels above 70% with steam; the last not reported tests were carried out with R245fa.

*Emran TA, (2011)* exploited and further developed the results obtained in (*Emran TA, 2010)*, improving the torque model of a Tesla turbine, as well as comparing the analytical solution with experimental tests. As a further result, the developed model allowed the design of an improved Tesla turbine with optimal dimensions.

*Ho–Yan BP, (2011)* applied the model developed in (*Rice W, 1965)* in order to establish a preliminary design approach for Tesla turbine for Pico Hydro applications. The obtained performance was over 80% for a 300 W power output turbine, working with a pressure head of 20 m and a water flow rate of 2.5 l/s.

*Krishnan V et al., (2011)* realized some micro–turbines (1–2 cm diameter rotors) using commercially available technologies, and tested them with different nozzles and rotors configurations, achieving almost 40% shaft mechanical efficiency.

*Lampart P & Jedrzejewski L, (2011)* further developed the analysis presented (Lampart et al., 2009), considering the Tesla turbine as the expander for a micro– power plant of 20 kW heat capacity. The maximum efficiency obtained from a computational fluid dynamic analysis was 51% with a mass flow rate of Solkatherm SES36 of 0.13 kg/s.

*Podergajs M, (2011)* provided a short report on Tesla turbine historical background, flow model and applications. In the report, the author highlighted the range of application where the turbine could be an attractive solution.

*Romanin VD and Carey VP, (2011)* developed an integral solution scheme of the fluid dynamics of incompressible, steady flow inside a Tesla turbine rotor. Enhanced

drag effect, due to surface micro structuring was also taken into account. The enhancement of efficiency due to surface micro grooving could be effectively relevant, as high as 9% enhancement when compared to smooth surface. The maximum predicted efficiencies of the turbine were over 80%.

### **2012**

*Borate HP and Misal ND, (2012)* carried out a performance analysis of the Tesla turbine, with a specific focus on the effect of surface finish and the space between the disks. The assessment was carried out considering water as working fluid. Experimental and theoretical analysis allowed the characterization of the major inefficiency (from the nozzle) and that, if direction grooving on disk surface is applied, an increase in efficiency between 5 to 6% can be achieved.

*Peshlakay A, (2012)* performed a numerical and experimental analysis of a Tesla turbine, comparing different nozzles using air, water and steam as working fluids; achieving a rotor efficiency of 95% (± 9.5% uncertainty) and a global turbine efficiency of 31%.

*Romanin VD et al., (2012)* modified the model developed in (*Romanin VD and Carey VP, 2011)* in order to match the results obtained in (*Krishnan V et al., 2011)*. The modified analytical model results were compared with CFD analyses and experimental results. The correlation of the results was fairly good, as, on average, the experimental results efficiency was 45% lower than the expected analytical model results. The discrepancy was mainly due to the not proper modelling of the nozzle.

Romanin VD*, (2012)* presented a comprehensive study, which collected analytical perturbation model results, computational fluid dynamic investigation and experimental analysis. The obtained results were presented in (*Romanin V et al., 2010, Krishnan V et al., 2011, Romanin VD and Carey VP, 2011, Romanin VD et al., 2012)*.

*Sengupta S and Guha A, (2012)* presented an analytical model resulting from the reduction of Navier–Stokes equations. They assumed a steady, incompressible and laminar flow, introducing the viscosity of fluid and a velocity gradient near the walls. This mathematical theory represents a simple but effective method of predicting the performance and efficiency of a Tesla turbine.

*Van Wageningen T, (2012)* presented a concept review of the Tesla turbine to be utilized as the engine for a flapping wing mechanism. Tesla turbine seemed not to be suitable for this application when compared to two stroke and heat engine.

### **2013**

*Bao G et al., (2013)* defined and validated a numerical model using CFD tools to describe the flow boundary layer; then the model was applied to different organic working fluids to obtain the related performance curves, concluding that best performance can be achieved with thin gap width, turbulent flow and fluids with high kinematic viscosity.

*Deng Q et al., (2013)* proposed and improved the model developed in (*Guha A and Smiley B, 2009)* and compared the analytical formulation with computational fluid dynamic simulations. Various nozzle configurations were taken into account. The highest efficiency (43.6%) of a 0.05 m rotor diameter turbine was reached when the lowest number of nozzle (2) was taken into account, as well as the highest rotational velocity (24,658 rpm) and the largest pressure ratio (0.8).

43

Guha A and Sengupta S*, (2013)* further polished the model presented in *(Sengupta S and Guha A, 2012)* and moreover they investigated, for the first time, the roles of each force affecting the power and pressure fields, physically explaining several fluid dynamics behaviours like flow reversal and the shape of complex relative path lines.

*Guimaraes LNF et al., (2013)* considered the Tesla turbine as a possible concept to be applied to an Emergency Core Cooling Systems of a standard power plant. They performed an experimental campaign with air as working fluid, reaching a 55 W

*Gupta HE and Kodali SP, (2013)* presented a state of the art review of the Tesla turbine. The assessed performance, as well as the other main results of investigation of previous research projects were highlighted, taking into account also the patents developed. A special focus has been put on the assessment of possible applications of

*Khan MUS et al., (2013a)* developed a modified configuration of a Tesla turbine, which merged both the aspect of a bladeless and a bladed turbine. An experimental campaign with water as working fluid was carried out. The average power output of the turbine was of 100 W. This new turbine concept was developed in order to exploit the potential energy of water flows present in canals or rivers. In a successive work, *Khan MUS et al., (2013b)* explored the possible applications where the Tesla turbine could be utilized. Particularly, the presented assessment of applications of a Tesla turbine was very wide: from power generation coupled with renewable energy sources, to the utilization in irrigation channels, up to the use in hybrid electric vehi-

*Krishnan VG et al., (2013)* described a method to scale Tesla turbines to millimetre dimensions, also providing the assessment of loss mechanisms. 40% efficiency was

*Placco GM et al., (2013)* considered a modified Tesla turbine as the expander for a passive thermal circuit as a part of an Emergency Core Cooling System of nuclear power plant. An experimental campaign with air as working fluid was conducted,

*Sengupta S and Guha A, (2013)* investigated the three–dimensional flow field and the flow path lines within a Tesla disk turbine, comparing the results obtained from the analytical theory and the computational fluid dynamics; moreover, they investigated the operating parameters that affect the shape of the path lines within the rotor

*Yang Z et al., (2013)* described an innovative method to measure and predict the mechanical power output of a Tesla turbine. The method is based on the rotational inertia of the turbine, as well as on the friction in the bearings. It is called "dynamic dynamometry" and allows the realization of power curves as function of rotational

*Zhao D and Khoo J, (2013)* designed and tested a 40 mm bladeless turbine for the harvesting of energy from air and rainwater applications. 0.5 W power output was

*Guha A and Sengupta S, (2014a)* developed a similitude study on the flow of the Tesla turbine. The scaling laws were obtained through the Buckingham Pi theorem,

expected even when millimetre scale turbine was considered.

achieving power higher than 100 W with low mass flow rates.

and the local balance of the various forces.

obtained with air as working fluid at 3300 rpm.

power output value.

Tesla turbines.

cles.

speed.

**2014**

Guha A and Sengupta S*, (2013)* further polished the model presented in *(Sengupta S and Guha A, 2012)* and moreover they investigated, for the first time, the roles of each force affecting the power and pressure fields, physically explaining several fluid dynamics behaviours like flow reversal and the shape of complex relative path lines.

*Guimaraes LNF et al., (2013)* considered the Tesla turbine as a possible concept to be applied to an Emergency Core Cooling Systems of a standard power plant. They performed an experimental campaign with air as working fluid, reaching a 55 W power output value.

*Gupta HE and Kodali SP, (2013)* presented a state of the art review of the Tesla turbine. The assessed performance, as well as the other main results of investigation of previous research projects were highlighted, taking into account also the patents developed. A special focus has been put on the assessment of possible applications of Tesla turbines.

*Khan MUS et al., (2013a)* developed a modified configuration of a Tesla turbine, which merged both the aspect of a bladeless and a bladed turbine. An experimental campaign with water as working fluid was carried out. The average power output of the turbine was of 100 W. This new turbine concept was developed in order to exploit the potential energy of water flows present in canals or rivers. In a successive work, *Khan MUS et al., (2013b)* explored the possible applications where the Tesla turbine could be utilized. Particularly, the presented assessment of applications of a Tesla turbine was very wide: from power generation coupled with renewable energy sources, to the utilization in irrigation channels, up to the use in hybrid electric vehicles.

*Krishnan VG et al., (2013)* described a method to scale Tesla turbines to millimetre dimensions, also providing the assessment of loss mechanisms. 40% efficiency was expected even when millimetre scale turbine was considered.

*Placco GM et al., (2013)* considered a modified Tesla turbine as the expander for a passive thermal circuit as a part of an Emergency Core Cooling System of nuclear power plant. An experimental campaign with air as working fluid was conducted, achieving power higher than 100 W with low mass flow rates.

*Sengupta S and Guha A, (2013)* investigated the three–dimensional flow field and the flow path lines within a Tesla disk turbine, comparing the results obtained from the analytical theory and the computational fluid dynamics; moreover, they investigated the operating parameters that affect the shape of the path lines within the rotor and the local balance of the various forces.

*Yang Z et al., (2013)* described an innovative method to measure and predict the mechanical power output of a Tesla turbine. The method is based on the rotational inertia of the turbine, as well as on the friction in the bearings. It is called "dynamic dynamometry" and allows the realization of power curves as function of rotational speed.

*Zhao D and Khoo J, (2013)* designed and tested a 40 mm bladeless turbine for the harvesting of energy from air and rainwater applications. 0.5 W power output was obtained with air as working fluid at 3300 rpm.

### **2014**

42

drag effect, due to surface micro structuring was also taken into account. The enhancement of efficiency due to surface micro grooving could be effectively relevant, as high as 9% enhancement when compared to smooth surface. The maximum predicted ef-

*Borate HP and Misal ND, (2012)* carried out a performance analysis of the Tesla turbine, with a specific focus on the effect of surface finish and the space between the disks. The assessment was carried out considering water as working fluid. Experimental and theoretical analysis allowed the characterization of the major inefficiency (from the nozzle) and that, if direction grooving on disk surface is applied, an increase

*Peshlakay A, (2012)* performed a numerical and experimental analysis of a Tesla turbine, comparing different nozzles using air, water and steam as working fluids; achieving a rotor efficiency of 95% (± 9.5% uncertainty) and a global turbine effi-

*Romanin VD et al., (2012)* modified the model developed in (*Romanin VD and Carey VP, 2011)* in order to match the results obtained in (*Krishnan V et al., 2011)*. The modified analytical model results were compared with CFD analyses and experimental results. The correlation of the results was fairly good, as, on average, the experimental results efficiency was 45% lower than the expected analytical model results. The dis-

Romanin VD*, (2012)* presented a comprehensive study, which collected analytical perturbation model results, computational fluid dynamic investigation and experimental analysis. The obtained results were presented in (*Romanin V et al., 2010, Krish-*

*Sengupta S and Guha A, (2012)* presented an analytical model resulting from the reduction of Navier–Stokes equations. They assumed a steady, incompressible and laminar flow, introducing the viscosity of fluid and a velocity gradient near the walls. This mathematical theory represents a simple but effective method of predicting the

*Van Wageningen T, (2012)* presented a concept review of the Tesla turbine to be utilized as the engine for a flapping wing mechanism. Tesla turbine seemed not to be

*Bao G et al., (2013)* defined and validated a numerical model using CFD tools to describe the flow boundary layer; then the model was applied to different organic working fluids to obtain the related performance curves, concluding that best performance can be achieved with thin gap width, turbulent flow and fluids with high kine-

*Deng Q et al., (2013)* proposed and improved the model developed in (*Guha A and Smiley B, 2009)* and compared the analytical formulation with computational fluid dynamic simulations. Various nozzle configurations were taken into account. The highest efficiency (43.6%) of a 0.05 m rotor diameter turbine was reached when the lowest number of nozzle (2) was taken into account, as well as the highest rotational velocity

crepancy was mainly due to the not proper modelling of the nozzle.

*nan V et al., 2011, Romanin VD and Carey VP, 2011, Romanin VD et al., 2012)*.

suitable for this application when compared to two stroke and heat engine.

ficiencies of the turbine were over 80%.

in efficiency between 5 to 6% can be achieved.

performance and efficiency of a Tesla turbine.

(24,658 rpm) and the largest pressure ratio (0.8).

**2012**

ciency of 31%.

**2013**

matic viscosity.

*Guha A and Sengupta S, (2014a)* developed a similitude study on the flow of the Tesla turbine. The scaling laws were obtained through the Buckingham Pi theorem,

which lead to the definition of 7 fundamental non–dimensional numbers (rotor radius ratio, aspect ratio, tangential speed ratio at rotor inlet, flow angle at rotor inlet, dynamic similarity number, power coefficient and pressure drop coefficient). A further study of *Guha A and Sengupta S, (2014b)* demonstrated that the application of the Euler turbomachinery equation is consistent only if local velocity mass–averaged values are considered.

*Hasan A and Benzamia A, (2014)* investigated the effects of temperature and mass flow rate variations on the performance of a Tesla turbine of 100 mm outer rotor diameter, through the means of CFD analysis. The performance of the turbine was found to be dependent on the temperature, as the boundary layer thickness changes due to temperature effects.

*Pandey et al., (2014)* developed a computational analysis of a 1 kW Tesla turbine for pico hydropower applications. A 127 mm rotor outer radius was considered for the calculations, obtaining a total of 777 W for a 9 disks configuration of the turbine, reaching an efficiency of 77.7%.

*Schosser et al., (2014)* carried out an assessment on the design and optimization of a test facility for 3D tomographic PIV measurements of the flow field inside a Tesla turbine rotor. A throughout CFD optimization analysis was carried out on the test rig components and specifically on the Tesla turbine rotor.

*Shimeles S, (2014)* performed a design an optimization assessment of a Tesla disk turbine, applying an incompressible steady state computational fluid dynamic analysis. Various rotor configurations were simulated, changing both geometric and fluid dynamic conditions. The maximum predicted rotor efficiency was of about 27%, utilizing air as working fluid. The maximum total power output obtained was of over 6.5 kW at 11500 rpm.

*Siddiqui MS et al., (2014)* presented a computational fluid dynamic assessment of an air Tesla turbine. The turbine dimensions were selected in order to compare the results with experimental data provided in *(Rice, 1965)*. Different flow configurations were analysed both applying laminar and k–ε turbulence schemes. The maximum efficiency obtained was over 20% with the lowest flow parameter assessed. Efficiencies of numerical and experimental results were comparable.

*Singh A, (2014)* derived a closed form expression of the flow field and pressure distribution inside a Tesla turbine rotor by simplifying Navier–Stokes equations. A careful assessment of the flow behaviour as function of the Reynolds number was carried out.

*Thawichsri K and Nilnont W, (2014)* performed an experimental investigation on two Tesla turbines, differing in rotor outlet diameter (the first one of 120 mm, the second one of 75 mm). The working fluid expanding through the turbine was Isopentane. Three different hot source temperatures were analysed (70, 80 and 90 °C). For a 0.05 kg/s mass flow rate and a rotational speed of 3000 rpm, the total effectiveness was of 36%, with higher power output given by the bigger turbine.

*Zhao D et al., (2014)* performed an experimental campaign, as well as a computational design assessment, of three Tesla turbines for the harnessing of energy from rainwater. Firstly, an experimental campaign with air as working fluid was conducted; finally, a rainwater application was assessed. The overall energy conversion assessed was approximately between 2 and 3%.

45

*Baginski P and Jedrzejewski L, (2015)* focused their studies on the assessment of the dynamic analysis of the Tesla turbine. Both strength and modal analysis were carried out through the means of two commercial software (Abaqus and Ansys). The results obtained for the assessed geometry (the same one used in *(Lampart P and Jedrzejewski L, 2011)*, which consisted of a rotor composed by 11 disks, with 100 mm outer diameter and 2 mm thickness), showed that no resonance effect would present

Guimaraes LNF et al.*, (2015)* deepened the work presented in *(Guimaraes et al., 2013, Placco et al., 2013),* developing a Tesla turbine for space application, claiming that this technology could be utilized both in Brayton and Rankine cycle, with the optimal characteristics which make it suitable for space application, such as low pressure drop and almost no maintenance required, as well as the possibility of using it

*Holland K, (2015)* designed, built and tested a 92 mm rotor diameter Tesla turbine. The tests were carried out with air as working fluid and the maximum obtained effi-

*Kölling A et al., (2015)* applied the Tesla turbine concept to small–distributed power generation from biomass resources. Particularly, in the study, the design, construction and tests of a 175 mm rotor diameter Tesla turbine were presented. The maximum power obtained was of 1.4 kW when a saturated steam at 9 bar and 0.11 kg/s mass flow rate was utilized. The assessed isentropic efficiency was of about 30%.

*Krishnan V, (2015)* exploited the work presented in *(Krishnan V et al., 2011, Romanin VD and Carey VP, 2011, Krishnan V et al., 2013 )* providing a Matlab tool for generating the design of micro to small Tesla turbines. Analytical, computational and experimental results were obtained, demonstrating the soundness of concept of Tesla turbine

*Neckel AL and Godinho M, (2015)* realized and tested ten convergent–divergent nozzles to improve the injection efficiency of the working fluid; different geometries were first investigated with one and two–dimensional approaches and then an exper-

*Nedelcu D et al, (2015)* performed a numerical simulation both on stress evaluation and the flow calculation of the Tesla turbine rotor. The aim was to develop and

*Raje a et al., (2015)* presented a review of the principle of operation, design consideration, performance parameters and possible applications of the Tesla turbine. *Ruiz M, (2015)* applied the Tesla turbine concept to a completely different application, which does not involve power production, but instead utilized as a heat sink. Numerical as well experimental results were obtained both with single phase and two– phase conditions (water as working fluid). The milestone to be exploited for Tesla turbine power applications is the development of an accurate heat transfer model in

*Ruiz M and Carey VP, (2015)* presented the experimental results obtained when utilizing a Tesla turbine as a heat sink, employing the work developed in *(Ruiz, 2015)*. Heat transfer, as well as pressure drop of a micro tesla turbine with water as working

at start–up and that the first mode shape presented at 387 Hz.

ciency was of 8.5% at 3 bar inlet total pressure and 10,000 rpm.

imental test campaign with air as working fluid was carried out.

manufacture a turbine working with air for teaching application.

for sub–Watt and Watt range applications.

two–phase flow conditions.

fluid was analysed.

**2015**

with any kind of fluid.

### **2015**

44

which lead to the definition of 7 fundamental non–dimensional numbers (rotor radius ratio, aspect ratio, tangential speed ratio at rotor inlet, flow angle at rotor inlet, dynamic similarity number, power coefficient and pressure drop coefficient). A further study of *Guha A and Sengupta S, (2014b)* demonstrated that the application of the Euler turbomachinery equation is consistent only if local velocity mass–averaged val-

*Hasan A and Benzamia A, (2014)* investigated the effects of temperature and mass flow rate variations on the performance of a Tesla turbine of 100 mm outer rotor diameter, through the means of CFD analysis. The performance of the turbine was found to be dependent on the temperature, as the boundary layer thickness changes due to

*Pandey et al., (2014)* developed a computational analysis of a 1 kW Tesla turbine for pico hydropower applications. A 127 mm rotor outer radius was considered for the calculations, obtaining a total of 777 W for a 9 disks configuration of the turbine,

*Schosser et al., (2014)* carried out an assessment on the design and optimization of a test facility for 3D tomographic PIV measurements of the flow field inside a Tesla turbine rotor. A throughout CFD optimization analysis was carried out on the test rig

*Shimeles S, (2014)* performed a design an optimization assessment of a Tesla disk turbine, applying an incompressible steady state computational fluid dynamic analysis. Various rotor configurations were simulated, changing both geometric and fluid dynamic conditions. The maximum predicted rotor efficiency was of about 27%, utilizing air as working fluid. The maximum total power output obtained was of over 6.5

*Siddiqui MS et al., (2014)* presented a computational fluid dynamic assessment of an air Tesla turbine. The turbine dimensions were selected in order to compare the results with experimental data provided in *(Rice, 1965)*. Different flow configurations were analysed both applying laminar and k–ε turbulence schemes. The maximum efficiency obtained was over 20% with the lowest flow parameter assessed. Efficiencies

*Singh A, (2014)* derived a closed form expression of the flow field and pressure distribution inside a Tesla turbine rotor by simplifying Navier–Stokes equations. A careful assessment of the flow behaviour as function of the Reynolds number was

*Thawichsri K and Nilnont W, (2014)* performed an experimental investigation on two Tesla turbines, differing in rotor outlet diameter (the first one of 120 mm, the second one of 75 mm). The working fluid expanding through the turbine was Isopentane. Three different hot source temperatures were analysed (70, 80 and 90 °C). For a 0.05 kg/s mass flow rate and a rotational speed of 3000 rpm, the total effectiveness

*Zhao D et al., (2014)* performed an experimental campaign, as well as a computational design assessment, of three Tesla turbines for the harnessing of energy from rainwater. Firstly, an experimental campaign with air as working fluid was conducted; finally, a rainwater application was assessed. The overall energy conversion assessed

ues are considered.

temperature effects.

kW at 11500 rpm.

carried out.

reaching an efficiency of 77.7%.

components and specifically on the Tesla turbine rotor.

of numerical and experimental results were comparable.

was of 36%, with higher power output given by the bigger turbine.

was approximately between 2 and 3%.

*Baginski P and Jedrzejewski L, (2015)* focused their studies on the assessment of the dynamic analysis of the Tesla turbine. Both strength and modal analysis were carried out through the means of two commercial software (Abaqus and Ansys). The results obtained for the assessed geometry (the same one used in *(Lampart P and Jedrzejewski L, 2011)*, which consisted of a rotor composed by 11 disks, with 100 mm outer diameter and 2 mm thickness), showed that no resonance effect would present at start–up and that the first mode shape presented at 387 Hz.

Guimaraes LNF et al.*, (2015)* deepened the work presented in *(Guimaraes et al., 2013, Placco et al., 2013),* developing a Tesla turbine for space application, claiming that this technology could be utilized both in Brayton and Rankine cycle, with the optimal characteristics which make it suitable for space application, such as low pressure drop and almost no maintenance required, as well as the possibility of using it with any kind of fluid.

*Holland K, (2015)* designed, built and tested a 92 mm rotor diameter Tesla turbine. The tests were carried out with air as working fluid and the maximum obtained efficiency was of 8.5% at 3 bar inlet total pressure and 10,000 rpm.

*Kölling A et al., (2015)* applied the Tesla turbine concept to small–distributed power generation from biomass resources. Particularly, in the study, the design, construction and tests of a 175 mm rotor diameter Tesla turbine were presented. The maximum power obtained was of 1.4 kW when a saturated steam at 9 bar and 0.11 kg/s mass flow rate was utilized. The assessed isentropic efficiency was of about 30%.

*Krishnan V, (2015)* exploited the work presented in *(Krishnan V et al., 2011, Romanin VD and Carey VP, 2011, Krishnan V et al., 2013 )* providing a Matlab tool for generating the design of micro to small Tesla turbines. Analytical, computational and experimental results were obtained, demonstrating the soundness of concept of Tesla turbine for sub–Watt and Watt range applications.

*Neckel AL and Godinho M, (2015)* realized and tested ten convergent–divergent nozzles to improve the injection efficiency of the working fluid; different geometries were first investigated with one and two–dimensional approaches and then an experimental test campaign with air as working fluid was carried out.

*Nedelcu D et al, (2015)* performed a numerical simulation both on stress evaluation and the flow calculation of the Tesla turbine rotor. The aim was to develop and manufacture a turbine working with air for teaching application.

*Raje a et al., (2015)* presented a review of the principle of operation, design consideration, performance parameters and possible applications of the Tesla turbine.

*Ruiz M, (2015)* applied the Tesla turbine concept to a completely different application, which does not involve power production, but instead utilized as a heat sink. Numerical as well experimental results were obtained both with single phase and two– phase conditions (water as working fluid). The milestone to be exploited for Tesla turbine power applications is the development of an accurate heat transfer model in two–phase flow conditions.

*Ruiz M and Carey VP, (2015)* presented the experimental results obtained when utilizing a Tesla turbine as a heat sink, employing the work developed in *(Ruiz, 2015)*. Heat transfer, as well as pressure drop of a micro tesla turbine with water as working fluid was analysed.

*Schosser C and Pfitzner M, (2015)* developed a throughout CFD analysis, focusing on the velocity profile of the flow inside a Tesla turbine with air as working fluids. Particularly, they stated that laminar CFD results were better approximated by a fourth order polynomial function, compared to parabolic profile.

*Thawichsri K and Nilnont W, (2015)* performed an experimental comparison between a centrifugal turbine and a Tesla turbine working with Isopentane at low temperature (70–90°C) heat resources. The experimental assessment demonstrated, from one point that the centrifugal turbine was on average 30% more performing than the Tesla turbine, but also that the Tesla turbine is a very cheap and easily to manufacture turbine. The maximum reached efficiency of the Tesla turbine was of about 12%, the specific power output of 35 kJ/kg and the cycle efficiency of 8.6%, when the heat source was 90°C.

### **2016**

*Bankar N et al., (2016)* designed a Tesla turbine for micro power application. Particularly, they called the new design approach as "hybrid Tesla turbine", due to the introduction of some changes, such as the grooving of the disks and utilization of polycarbonate disks of thin thickness.

*Herrmann–Priesnitz B et al., (2016)* developed a fluid dynamics model on the structure of the boundary layer of a Tesla disk turbine. Inviscid core region was coupled to boundary layer formulation in order to develop a sound analytical model, which was compared to numerical simulations run in OpenFoam environment.

*Jose R et al., (2016a, 2016b)* realized a comprehensive project on the design, theoretical and experimental analysis of a Tesla turbine utilizing water as working fluid. The effect of surface roughness, as well as disk spacing and number of disks was analysed. The experimental results highlighted that a spiral grooving on disks could enhance the turbine efficiency of about 5–6%.

*Joshi KN et al., (2016)* coupled the Tesla turbine design configuration with a Pelton turbine, obtaining therefore a "hybrid Tesla wheel turbine". A computational fluid dynamic assessment on Tesla and hybrid Tesla Pelton turbine was carried out. The Hybrid turbine allowed a 5% increase in efficiency compared to the bladeless turbine. A rapid prototyping technique was also showed as a possible solution for Tesla–Pelton turbine fabrication.

*Qi W et al., (2016)* performed a numerical investigation on the influence of disk spacing on the performance of Tesla turbines. The analysis conducted considered 7 different spacing (from 0.1 to 1 mm) highlighting the changes in performance (power and efficiency), as well as on the mass flow rate of the turbine. Several rotational speeds were also considered and optimal values of disk spacing, as well dimensionless tangential velocities were obtained when air as working fluid was considered.

*Schosser C et al., (2016)* performed a throughout investigation of the flow field inside a Tesla turbine with air as working fluid, through the means of a stereoscopic 3D–PTV measurement technique. The main result was the assessment of the laminar velocity profile inside the channels, which slightly differs from parabolic distribution and is better described by a 4th order polynomial.

*Schosser C, (2016a)* exploited the study presented in *(Schosser C and Pfitzner M, 2015, Schosser C 2016b)* and carried out a throughout experimental and numerical in-

47

vestigation of a Tesla turbine working with air as working fluid. Analytical and numerical results were also compared and an efficiency prediction of over 50% was

based nanofluid with ferro–particles were utilized.

bine working with Ethylene Glycol.

a corresponding efficiency of 20.97%.

tional speed at rotor inlet) of 1.45 was considered.

rotational speeds between 700 and 1000 rpm.

utilized as working fluid.

ameter of 100 mm, run with an inlet pressure of 272 kPa.

Sengupta S and Guha A*, (2016)* investigated the performance of a Tesla turbine when nanofluids were utilized as working fluid; an increase of power output of 30% appears to be possible when the volume fraction of nanoparticles is increased from 0 to 0.05. Therefore, the maximum predicted efficiency was of about 56% when water

Zahid I et al.*, (2016)* analysed the performance of an air driven Tesla turbine. The claimed reached efficiency was of about 58% for a turbine with an external rotor di-

*Alrabie MS et al., (2017)* applied an analytical solution in order to assess the performance of a small–scale hydro Tesla turbine. The design of experiment was carried out and the definition of the optimal prototype geometry was obtained, assessing various geometric parameters (inlet, outlet rotor diameters, disks spacing, nozzle number and location), as well as the most significant thermodynamic parameters (density, viscosity). Finally, an 80 mm rotor outer diameter turbine was selected as the proper design in order to obtain 0.47 W power output at 1197 rpm. Similarity scaling law equations were applied to assess the performance of a 3 mm rotor outer diameter tur-

*Damodhar R et al., (2017)* designed and fabricated a cm–scale Tesla turbine utilizing water as working fluid. Design specifications were given; as well experimental tests were conducted. The maximum power obtained was of 0.12 W at 140 rpm, with

*Guha A and Sengupta S, (2017)* performed an optimization assessment of the Tesla turbine based on the non–dimensional scaling laws presented in *(Guha and Sengupta, 2014).* The role of each non–dimensional parameter was assessed and the physical explanation of its effects on the performance of the turbine is given. As an illustrative example of the methodology applied, a turbine efficiency of 68.8% was designed when a tangential inlet velocity ratio (the ratio between tangential velocity and rota-

*Li et al., (2017)* explored the performance characteristics of the Tesla turbine through experimental and numerical assessment. An 11 mm outer rotor diameter, with 1 mm disk spacing and a total of 13 disks, Tesla turbine was fabricated and tested with water as working fluid. The experimental assessment was carried out at various, rotational speeds, inlet pressures and mass flow rates; a maximum efficiency of about 8% was obtained. The numerical simulation assessed efficiencies higher than 40% for

*Lisker R et al., (2017)* developed an analytical model, which takes into account thin film condensation in a Tesla turbine rotor. The influence of thin film condensation on velocity fields and mass flow rate was highlighted when condensing steam was

*Mandal A and Saha S, (2017)* investigated the performance of a cm–scale Tesla turbine for micro–air vehicles application. Numerical simulations on 20 disks, 0.1 m outer rotor diameter air Tesla turbine were performed. Different rotational velocities

obtained.

**2017**

vestigation of a Tesla turbine working with air as working fluid. Analytical and numerical results were also compared and an efficiency prediction of over 50% was obtained.

Sengupta S and Guha A*, (2016)* investigated the performance of a Tesla turbine when nanofluids were utilized as working fluid; an increase of power output of 30% appears to be possible when the volume fraction of nanoparticles is increased from 0 to 0.05. Therefore, the maximum predicted efficiency was of about 56% when water based nanofluid with ferro–particles were utilized.

Zahid I et al.*, (2016)* analysed the performance of an air driven Tesla turbine. The claimed reached efficiency was of about 58% for a turbine with an external rotor diameter of 100 mm, run with an inlet pressure of 272 kPa.

### **2017**

46

*Schosser C and Pfitzner M, (2015)* developed a throughout CFD analysis, focusing on the velocity profile of the flow inside a Tesla turbine with air as working fluids. Particularly, they stated that laminar CFD results were better approximated by a fourth

*Thawichsri K and Nilnont W, (2015)* performed an experimental comparison between a centrifugal turbine and a Tesla turbine working with Isopentane at low temperature (70–90°C) heat resources. The experimental assessment demonstrated, from one point that the centrifugal turbine was on average 30% more performing than the Tesla turbine, but also that the Tesla turbine is a very cheap and easily to manufacture turbine. The maximum reached efficiency of the Tesla turbine was of about 12%, the specific power output of 35 kJ/kg and the cycle efficiency of 8.6%, when the heat

*Bankar N et al., (2016)* designed a Tesla turbine for micro power application. Particularly, they called the new design approach as "hybrid Tesla turbine", due to the introduction of some changes, such as the grooving of the disks and utilization of

*Herrmann–Priesnitz B et al., (2016)* developed a fluid dynamics model on the structure of the boundary layer of a Tesla disk turbine. Inviscid core region was coupled to boundary layer formulation in order to develop a sound analytical model, which was compared to numerical simulations run in OpenFoam environment.

*Jose R et al., (2016a, 2016b)* realized a comprehensive project on the design, theoretical and experimental analysis of a Tesla turbine utilizing water as working fluid. The effect of surface roughness, as well as disk spacing and number of disks was analysed. The experimental results highlighted that a spiral grooving on disks could

*Joshi KN et al., (2016)* coupled the Tesla turbine design configuration with a Pelton turbine, obtaining therefore a "hybrid Tesla wheel turbine". A computational fluid dynamic assessment on Tesla and hybrid Tesla Pelton turbine was carried out. The Hybrid turbine allowed a 5% increase in efficiency compared to the bladeless turbine. A rapid prototyping technique was also showed as a possible solution for Tesla–Pelton

*Qi W et al., (2016)* performed a numerical investigation on the influence of disk spacing on the performance of Tesla turbines. The analysis conducted considered 7 different spacing (from 0.1 to 1 mm) highlighting the changes in performance (power and efficiency), as well as on the mass flow rate of the turbine. Several rotational speeds were also considered and optimal values of disk spacing, as well dimensionless

*Schosser C et al., (2016)* performed a throughout investigation of the flow field inside a Tesla turbine with air as working fluid, through the means of a stereoscopic 3D–PTV measurement technique. The main result was the assessment of the laminar velocity profile inside the channels, which slightly differs from parabolic distribution

*Schosser C, (2016a)* exploited the study presented in *(Schosser C and Pfitzner M, 2015, Schosser C 2016b)* and carried out a throughout experimental and numerical in-

tangential velocities were obtained when air as working fluid was considered.

order polynomial function, compared to parabolic profile.

source was 90°C. **2016**

turbine fabrication.

polycarbonate disks of thin thickness.

enhance the turbine efficiency of about 5–6%.

and is better described by a 4th order polynomial.

*Alrabie MS et al., (2017)* applied an analytical solution in order to assess the performance of a small–scale hydro Tesla turbine. The design of experiment was carried out and the definition of the optimal prototype geometry was obtained, assessing various geometric parameters (inlet, outlet rotor diameters, disks spacing, nozzle number and location), as well as the most significant thermodynamic parameters (density, viscosity). Finally, an 80 mm rotor outer diameter turbine was selected as the proper design in order to obtain 0.47 W power output at 1197 rpm. Similarity scaling law equations were applied to assess the performance of a 3 mm rotor outer diameter turbine working with Ethylene Glycol.

*Damodhar R et al., (2017)* designed and fabricated a cm–scale Tesla turbine utilizing water as working fluid. Design specifications were given; as well experimental tests were conducted. The maximum power obtained was of 0.12 W at 140 rpm, with a corresponding efficiency of 20.97%.

*Guha A and Sengupta S, (2017)* performed an optimization assessment of the Tesla turbine based on the non–dimensional scaling laws presented in *(Guha and Sengupta, 2014).* The role of each non–dimensional parameter was assessed and the physical explanation of its effects on the performance of the turbine is given. As an illustrative example of the methodology applied, a turbine efficiency of 68.8% was designed when a tangential inlet velocity ratio (the ratio between tangential velocity and rotational speed at rotor inlet) of 1.45 was considered.

*Li et al., (2017)* explored the performance characteristics of the Tesla turbine through experimental and numerical assessment. An 11 mm outer rotor diameter, with 1 mm disk spacing and a total of 13 disks, Tesla turbine was fabricated and tested with water as working fluid. The experimental assessment was carried out at various, rotational speeds, inlet pressures and mass flow rates; a maximum efficiency of about 8% was obtained. The numerical simulation assessed efficiencies higher than 40% for rotational speeds between 700 and 1000 rpm.

*Lisker R et al., (2017)* developed an analytical model, which takes into account thin film condensation in a Tesla turbine rotor. The influence of thin film condensation on velocity fields and mass flow rate was highlighted when condensing steam was utilized as working fluid.

*Mandal A and Saha S, (2017)* investigated the performance of a cm–scale Tesla turbine for micro–air vehicles application. Numerical simulations on 20 disks, 0.1 m outer rotor diameter air Tesla turbine were performed. Different rotational velocities

were assessed, obtaining maximum rotor efficiency of 89.56% for an inlet velocity of 10 m/s and a maximum power output of 21.3 W for an inlet velocity of 20 m/s at 1500 rpm.

*Polisetti S et al., (2017)* fabricated and conducted a performance study on an air driven Tesla turbine. A 95 mm outer rotor diameter, 1.27 mm thick disks Tesla turbine was fabricated and various configurations were tested, varying number of disks, distance between the disks, number of nozzles and pressure at rotor inlet. The results were presented in the form of rotational speeds. The maximum reached rotational speed was 25,324 rpm for a 4 disks configuration at 6 bar inlet pressure.

*Schosser C et al., (2017)* presented a comparison between analytical and numerical models on the flow field developing inside a Tesla turbine single channel. Several analyses were conducted, showing the influence of non–dimensional parameters, such as dimensionless friction factor, inlet conditions and geometry parameters. Inflow effect was analysed, as the discrepancy between theoretical and numerical analyses was found to be relevant.

*Shah V and Dhokai S, (2017)* carried out experimental tests of an air driven Tesla turbine. The turbine was composed by 8 disks of 95 mm rotor outer diameter, with disk spacing of 1 mm. The maximum power obtained was of 137.83 W at 4500 rpm, with 4 bar of total inlet pressure.

*Shah V et al., (2017)* conducted a literature review on the principle of operation and on the parameters affecting the performance of the Tesla turbine.

*Song J and Gu CW, (2017)* developed a 1D model of a Tesla turbine working with organic fluids. Furthermore, a thermodynamic analysis of an organic Rankine cycle utilizing R245fa as working fluid, exploiting the Tesla turbine as the expander was carried out. The assessed isentropic efficiency of the turbine reached a maximum value of 45% when a 15° oriented nozzle was utilized and an evaporation temperature of 370 K was taken into account. The maximum power output of 1.24 kW was obtained when the heat source temperature was of 355 K. The maximum thermal cycle efficiency obtained was of about 8%.

*Song J et al., (2017)* improved and used the one–dimensional Tesla turbine model presented in *(Song J and Gu CW, 2017)* to predict the efficiency of a small scale ORC power plant adopting various working fluids and operating conditions: at design point, the ORC with R245ca released 1.25 kW power output at 4% thermodynamic efficiency.

*Thiyagarajan V et al., (2017)* applied the Tesla turbine concept to solar power refrigeration. A prototype of 50 mm outer rotor radius was fabricated and some preliminary tests carried out. Based on the data obtained, the theoretical time to convert 1 kilogram of water in ice was calculated (6.5 hr).

*Umashankar M et al., (2017)* assessed the possible application of the Tesla turbine in cogeneration of heat and power systems (CHP). Cogeneration concept and turbine analytical approach were assessed.

*Variava JM and Bhavsar AS, (2017)* performed an experimental and numerical analysis on Tesla turbine utilizing water as working fluid. The experimental investigation allowed assessing turbine efficiency as high as 60% with a 0.56 kg/s mass flow rate and a power output of 15.18 W.

49

*Sengupta S and Guha A, (2018)* investigated the effect on non–uniform admission inside a Tesla turbine rotor, taking into account discrete inflows effect, finite disks thickness and radial clearance. An extensive computational fluid dynamic assessment was carried out assessing both the variation in power and efficiency as function of the number of nozzles, clearance, rotational speed and disks thickness. Small disk thickness, straight disk edge, optimum radial clearance and high number of nozzle were

*Sengupta S and Guha A, (2018)* improved the one–dimensional model developed in *(Song J and Gu CW, 2017, Song et al., 2017)* and compared the predicted performances with the experimental results obtained by *Rice, (1965)* with air as working fluid.

Traum MJ et al.*, (2018)* extended the work developed by *Carey VP, (2010)* in order to assess the sensitivity of shaft power to a selected design variable. A differentiable closed form analytical model was applied, starting from the one developed in (*Carey VP, 2010)* and applying some substantial changes, such as: identifying model Reynolds number limitation; defining closed–form torque and power output expressions; introducing simple nozzle geometric height parameter and showing the importance of

**SERG (Sustainable Energy Research Group) contribution to Tesla turbine**

The SERG group of the department of Industrial Engineering of University of Florence contributed to the state of the art on Tesla turbines with the following re-

*Manfrida et al., (2017)* revisited the original concept presented by Tesla, improving the stator layout and applying a modular design to the turbine, increasing its flexibility. The evaluation of the turbine performance with two different organic working fluids (R245fa and n–Hexane) was also carried out, analysing the influence of the most significant turbomachinery non–dimensional parameters (ns, ds, ϕ, ψ). A preliminary design of a prototype was also presented, and the expected efficiency with the assessed working fluid highlighted. Maximum efficiency (about 40%) was obtained

*Manfrida G and Talluri L, (2018*) revised the mathematical model of the Tesla turbine, upgrading the model presented in (*Carey VP, 2010, Sengupta and Guha, 2012)*, taking into account compressible real fluid properties. A performance analysis on the Tesla turbine rotor was presented, allowing the understanding of the principal parameters influencing turbine efficiency and reaching a design of an air Tesla turbine rotor

*Manfrida G et al., (2018)* further polished the work conducted in (*Manfrida et al., 2017)*, presenting an upgraded concept of the Tesla turbine for Organic Rankine cycle applications. Concept design and performance assessment of a manufactured prototype were performed. The fluid assessment comprehended 5 different organic working fluids: R245fa, R134a, SES36, n–Pentane, n–Hexane. High turbine efficiency (over 55%) can be reached for n–Hexane as working fluid. Higher power production

**2018**

**state of the art**

**2018**

with over 90% efficiency.

search. **2017**

recommended as an optimal design solution.

checking maximum turbine rotational speed.

for n–Hexane as working fluid, for low mass flow rates.

### **2018**

*Sengupta S and Guha A, (2018)* investigated the effect on non–uniform admission inside a Tesla turbine rotor, taking into account discrete inflows effect, finite disks thickness and radial clearance. An extensive computational fluid dynamic assessment was carried out assessing both the variation in power and efficiency as function of the number of nozzles, clearance, rotational speed and disks thickness. Small disk thickness, straight disk edge, optimum radial clearance and high number of nozzle were recommended as an optimal design solution.

*Sengupta S and Guha A, (2018)* improved the one–dimensional model developed in *(Song J and Gu CW, 2017, Song et al., 2017)* and compared the predicted performances with the experimental results obtained by *Rice, (1965)* with air as working fluid.

Traum MJ et al.*, (2018)* extended the work developed by *Carey VP, (2010)* in order to assess the sensitivity of shaft power to a selected design variable. A differentiable closed form analytical model was applied, starting from the one developed in (*Carey VP, 2010)* and applying some substantial changes, such as: identifying model Reynolds number limitation; defining closed–form torque and power output expressions; introducing simple nozzle geometric height parameter and showing the importance of checking maximum turbine rotational speed.

### **SERG (Sustainable Energy Research Group) contribution to Tesla turbine state of the art**

The SERG group of the department of Industrial Engineering of University of Florence contributed to the state of the art on Tesla turbines with the following research.

### **2017**

*Manfrida et al., (2017)* revisited the original concept presented by Tesla, improving the stator layout and applying a modular design to the turbine, increasing its flexibility. The evaluation of the turbine performance with two different organic working fluids (R245fa and n–Hexane) was also carried out, analysing the influence of the most significant turbomachinery non–dimensional parameters (ns, ds, ϕ, ψ). A preliminary design of a prototype was also presented, and the expected efficiency with the assessed working fluid highlighted. Maximum efficiency (about 40%) was obtained for n–Hexane as working fluid, for low mass flow rates.

### **2018**

48

were assessed, obtaining maximum rotor efficiency of 89.56% for an inlet velocity of 10 m/s and a maximum power output of 21.3 W for an inlet velocity of 20 m/s at 1500

speed was 25,324 rpm for a 4 disks configuration at 6 bar inlet pressure.

and on the parameters affecting the performance of the Tesla turbine.

*Polisetti S et al., (2017)* fabricated and conducted a performance study on an air driven Tesla turbine. A 95 mm outer rotor diameter, 1.27 mm thick disks Tesla turbine was fabricated and various configurations were tested, varying number of disks, distance between the disks, number of nozzles and pressure at rotor inlet. The results were presented in the form of rotational speeds. The maximum reached rotational

*Schosser C et al., (2017)* presented a comparison between analytical and numerical models on the flow field developing inside a Tesla turbine single channel. Several analyses were conducted, showing the influence of non–dimensional parameters, such as dimensionless friction factor, inlet conditions and geometry parameters. Inflow effect was analysed, as the discrepancy between theoretical and numerical analyses was

*Shah V and Dhokai S, (2017)* carried out experimental tests of an air driven Tesla turbine. The turbine was composed by 8 disks of 95 mm rotor outer diameter, with disk spacing of 1 mm. The maximum power obtained was of 137.83 W at 4500 rpm,

*Shah V et al., (2017)* conducted a literature review on the principle of operation

*Song J and Gu CW, (2017)* developed a 1D model of a Tesla turbine working with organic fluids. Furthermore, a thermodynamic analysis of an organic Rankine cycle utilizing R245fa as working fluid, exploiting the Tesla turbine as the expander was carried out. The assessed isentropic efficiency of the turbine reached a maximum value of 45% when a 15° oriented nozzle was utilized and an evaporation temperature of 370 K was taken into account. The maximum power output of 1.24 kW was obtained when the heat source temperature was of 355 K. The maximum thermal cycle

*Song J et al., (2017)* improved and used the one–dimensional Tesla turbine model presented in *(Song J and Gu CW, 2017)* to predict the efficiency of a small scale ORC power plant adopting various working fluids and operating conditions: at design point, the ORC with R245ca released 1.25 kW power output at 4% thermodynamic effi-

*Thiyagarajan V et al., (2017)* applied the Tesla turbine concept to solar power refrigeration. A prototype of 50 mm outer rotor radius was fabricated and some preliminary tests carried out. Based on the data obtained, the theoretical time to convert

*Umashankar M et al., (2017)* assessed the possible application of the Tesla turbine in cogeneration of heat and power systems (CHP). Cogeneration concept and turbine

*Variava JM and Bhavsar AS, (2017)* performed an experimental and numerical analysis on Tesla turbine utilizing water as working fluid. The experimental investigation allowed assessing turbine efficiency as high as 60% with a 0.56 kg/s mass flow

rpm.

found to be relevant.

ciency.

with 4 bar of total inlet pressure.

efficiency obtained was of about 8%.

analytical approach were assessed.

rate and a power output of 15.18 W.

1 kilogram of water in ice was calculated (6.5 hr).

*Manfrida G and Talluri L, (2018*) revised the mathematical model of the Tesla turbine, upgrading the model presented in (*Carey VP, 2010, Sengupta and Guha, 2012)*, taking into account compressible real fluid properties. A performance analysis on the Tesla turbine rotor was presented, allowing the understanding of the principal parameters influencing turbine efficiency and reaching a design of an air Tesla turbine rotor with over 90% efficiency.

*Manfrida G et al., (2018)* further polished the work conducted in (*Manfrida et al., 2017)*, presenting an upgraded concept of the Tesla turbine for Organic Rankine cycle applications. Concept design and performance assessment of a manufactured prototype were performed. The fluid assessment comprehended 5 different organic working fluids: R245fa, R134a, SES36, n–Pentane, n–Hexane. High turbine efficiency (over 55%) can be reached for n–Hexane as working fluid. Higher power production

was reached with refrigerant fluids, and SES36 resulted as the best compromise between power production and efficiency.

*Talluri et al., (2018)* performed a design and optimization of Tesla turbine for ORC applications. An innovative rotor model was presented, as well as the assessment of each component losses was carried out. Not only the rotor, but also stator, still chamber and diffuser model were designed. As an outcome of the methodology process, three different turbine configurations were assessed; the maximum total to static efficiency assessed was of 64% when a 0.5 m rotor diameter was considered and n– Hexane was utilized as working fluid. The power production per single channel of such configuration was of 58 W, which allowed a very compact axial design. **Statistics**

To conclude the literature review, a resume of the available literature is presented in Tab. 2.6. The previously described works were resumed highlighting the type of analysis that was carried out and which kind of fluid was considered.

Fig. 2.18 displays the number of publication on Tesla turbines (updated in August 2018). It has to be remarked how the trend is drastically increasing in the last few years; this is directly linked to the high attention that micro power generation gained on the energy market. Fig. 2.19 collects the most active authors assessing the Tesla turbine. Rice W (Arizona State University, Tempe) is the author that most investigated the Tesla turbine, both from analytical and experimental point of view. Of particular relevance are also the work developed by Guha A and Sengupta S (Indian Institute of Technology, Kharagpur) and Carey VP (University of California, Berkeley), who developed analytical, computational and experimental research, properly explaining the flow behaviour inside a Tesla disk turbine. Finally, Fig. 2.20 resumes the working fluids that have mainly been considered when developing a study on Tesla turbine. Air is the principal fluid investigated, but of particular importance is the emerging interest in recent years of the application of Tesla turbines in organic Rankine cycles.

**Fig. 2.18** Number of publications (Journal paper, conference paper, Ph.D. and M.Sc. available thesis) from 1950 to nowadays (updated in August 2018)

51

**work Ref. Fluid Focus of**

perimental *Rice W*, *(1965)* Air Analytical/

*(1966)* Air Experi-

*Beans EW,*

*Matsch L and Rice W, (1967a)*

**work**

experimental

mental

– Analytical

**Fig. 2.19** Most active authors on Tesla turbine research

**Fig. 2.20** Working fluid taken into account in literature studies

perimental

experimental

**Table 2.6** Literature Review summary

*Leaman AB,*

*Armstrong*

*Beans EW,*

**Ref. Fluid Focus of**

*(1950)* Air Design/ ex-

*JH, (1952)* Steam Design/ ex-

*(1961)* Air Numerical/

**Fig. 2.19** Most active authors on Tesla turbine research

**Fig. 2.20** Working fluid taken into account in literature studies

**Table 2.6** Literature Review summary

50

**Fig. 2.18** Number of publications (Journal paper, conference paper, Ph.D. and M.Sc. available

thesis) from 1950 to nowadays (updated in August 2018)

was reached with refrigerant fluids, and SES36 resulted as the best compromise be-

such configuration was of 58 W, which allowed a very compact axial design.

analysis that was carried out and which kind of fluid was considered.

*Talluri et al., (2018)* performed a design and optimization of Tesla turbine for ORC applications. An innovative rotor model was presented, as well as the assessment of each component losses was carried out. Not only the rotor, but also stator, still chamber and diffuser model were designed. As an outcome of the methodology process, three different turbine configurations were assessed; the maximum total to static efficiency assessed was of 64% when a 0.5 m rotor diameter was considered and n– Hexane was utilized as working fluid. The power production per single channel of

To conclude the literature review, a resume of the available literature is presented in Tab. 2.6. The previously described works were resumed highlighting the type of

Fig. 2.18 displays the number of publication on Tesla turbines (updated in August 2018). It has to be remarked how the trend is drastically increasing in the last few years; this is directly linked to the high attention that micro power generation gained on the energy market. Fig. 2.19 collects the most active authors assessing the Tesla turbine. Rice W (Arizona State University, Tempe) is the author that most investigated the Tesla turbine, both from analytical and experimental point of view. Of particular relevance are also the work developed by Guha A and Sengupta S (Indian Institute of Technology, Kharagpur) and Carey VP (University of California, Berkeley), who developed analytical, computational and experimental research, properly explaining the flow behaviour inside a Tesla disk turbine. Finally, Fig. 2.20 resumes the working fluids that have mainly been considered when developing a study on Tesla turbine. Air is the principal fluid investigated, but of particular importance is the emerging interest in recent years of the application of Tesla turbines in organic Rankine cycles.

tween power production and efficiency.

**Statistics**

**Ref. Fluid Focus of**

*(2011)* Water Numerical

*(2011)* R245fa/Steam Design/ ex-

*al., (2011)* Water Experi-

*(2011)* – State of the

perimental

Experimental/ Mechanical

Performance assessment

mental

art

mental

Numerical/ Experimental

Analytical/ numerical/ experimental

assessment

R245fa/ R123 Numerical *Schosser et*

Numerical

Air Analytical

Water Experi-

Air Analytical

Air Practical

SES36 Numerical *Krishnan VG*

*Choon et al.,*

*Cirincione N,*

*Emran TA,*

*Ho–Yan BP,*

*Krishnan V et*

*Lampart P & Jedrzejewski L, (2011)*

*Podergajs M,*

*Romanin VD and Carey VP, (2011)*

*Borate HP and Misal ND, (2012)*

*Peshlakay A,*

*Romanin VD*

Romanin VD*,*

*Sengupta S and Guha A, (2012)*

*Van Wageningen T, (2012)*

*Bao G et al., (2013)*

*Deng Q et al.,*

*(2012)* Air/ Steam

*(2012)* Air / Water

*et al., (2012)* Water Analytical

Air/ R600/

*(2013)* Air Analytical/

*(2011)* Air

*(2011)* Water

**work Ref. Fluid Focus of**

Guha A and Sengupta S*, (2013)*

*Guimaraes LNF et al., (2013)*

*Gupta HE and Kodali SP, (2013)*

*Khan MUS et*

*Khan MUS et al., (2013b)*

*Placco GM et al., (2013)*

*Sengupta S and Guha A, (2013)*

*Yang Z et al.,*

*Zhao D and Khoo J, (2013)*

*Guha A and Sengupta S, (2014a)*

*Guha A and Sengupta S, (2014b)*

*Hasan A and Benzamia A, (2014)*

*Pandey et al.,*

*Shimeles S,*

*al., (2014)* Air

*(2013)* Air

**work**

Experimental

mental

Application assessment

Experimental

Analytical/ numerical

Experimental/ mechanical

Design/ experimental

ter Analytical

Air Analytical

Air Numeri-

cal

cal

Experimental/ numerical

numerical

Air Analytical

– State of the art

Air /Steam

*al., (2013a)* Water Experi-

*et al., (2013)* Water Design

Air/ Water

Air /Steam

Air

Air/ Water

Air/Wa-

*(2014)* Water Numeri-

*(2014)* Air Design/

Lorenzo Talluri

**Ref. Fluid Focus of**

Water – Ethylene Glycol

Hydrogen/ Sodium/ Glycerine

*(1974)* Water Experi-

*al., (1976)* Air Numerical

Steam with

Air with solid

*(1991)* – State of

*et al., (2001)* Air + SO2 Numerical

– Analytical *Ladino AFR,*

Air Numerical *Ladino AFR,*

– Analytical *Couto et al.,*

Experimental

Performance assessment

mental

mental

droplets Numerical *Lampart P et*

particles Numerical *Carey VP,*

the art

mental

Practical assessment

Water Numerical *Lemma et al.,*

– Analytical

*(1975)* Air Analytical *Crowell R,*

Water/Steam Experi-

*(1990)* Air Analytical Carey VP*,*

Air/Steam Experi-

*Matsch L and Rice W, (1967b)*

*Boyd KE and Rice W, (1968)*

*Boyd KE and Rice W, (1968)*

*Adams R and Rice W, (1970)*

*Boyak BE and Rice W, (1971)*

*Lawn MJ, (1972)*

*Lawn MJ and Rice W, (1974)*

*Pater et al.,*

*Bassett CE,*

*Garrison et*

*Steidel R and Weiss H, (1976)*

*Truman CR et al., (1978)*

*Truman CR et al., (1979)*

*Allen JS,*

*Rice W,*

*Sandilya P*

*Patel N & Schmidt DD., (2002)*

*Lezsovits F,*

*(2003)* Steam

**work Ref. Fluid Focus of**

*Batista M,*

*Bloudicek P and Palousek D, (2007)*

*Deam et al.,*

*Valente A., (2008)*

*Hoya GP and Guha A, (2009)*

*Guha A and Smiley B, (2009)*

*Emran et al.,*

*Puzyrewski R & Tesch K, (2010)*

*Romanin V et al., (2010)*

*Batista M.,*

*(2010)* Air

*(2008)* Air

*(2004a)* Air Numerical

*(2004b)* Air Numerical

*(2006)* Air/Water Analytical

*(2007)* – Analytical

Water

*(2008)* Air Practical

Hydrocarbon gases

*(2009)* Care30 Practical

*al., (2009)* SES36 Numerical

*(2010a)* Steam Analytical

*(2010b)* Steam Analytical/

Water/Steam

*(2011)* – Analytical

**work**

Design/ Experimental

assessment

Experimental/ Numerical

> Experimental

assessment

mental

mental

Numerical

Experimental/ Mechanical

Numerical

Numerical/ Experimental

Air Experi-

Air Experi-

Water Analytical/

**Ref. Fluid Focus of**

*(2017)* Air State of

R123/ R600/ R600a/ R236ea/ R236fa/ R245ca/ R245fa

Steam

–

– Experi-

R245fa Analyti-

Water Experi-

mental

the art

cal

Analytical

Practical assessment

Practical assessment

mental

*Shah V and Dhokai S, (2017)*

*Shah V et al.,*

*Song J and Gu CW, (2017)*

*Song J et al., (2017)*

*Thiyagarajan V et al., (2017)*

*Umashankar M et al., (2017)*

*Variava JM and Bhavsar AS, (2017)*

**work Ref. Fluid Focus of**

al.*, (2018)* Water Analyti-

R245fa/ n– Hexane

R245fa/ R134a/ SES36/ n– Hexane/ n– Pentane

*Sengupta S and Guha A, (2018)*

*Sengupta S and Guha A, (2018)*

Traum MJ et

*Manfrida et al., (2017)*

*Manfrida G and Talluri L, (2018*)

*Manfrida G et al., (2018)*

*Talluri et al.,*

*(2018)* <sup>n</sup>–Hexane

**work**

cal

cal

cal

Design/ analytical

cal

Design/ Analytical

Design/ analytical/ numerical

Air Numeri-

Air Analyti-

Air Analyti-

Lorenzo Talluri

**Ref. Fluid Focus of**

*al., (2014)* Air Numerical

Isopentane

Air/ Water

–

Air/ Steam

*Singh A, (2014)* – Analytical *Jose R et al.,*

Experimental

Experimental/ numerical

Mechanical/ numerical

> Experimental

Design/ experimental

Design/ experimental

Numerical/ experimental

mental

Mechanical/ numerical

the art

Numerical/ experimental

mental

Experimental

*(2016)* Air Design *Schosser C et*

Air Numerical *Mandal A and*

Air Experi-

Water Experi-

*Siddiqui MS et*

*Thawichsri K and Nilnont W, (2014)*

*Zhao D et al., (2014)*

*Baginski P and Jedrzejewski L, (2015)*

Guimaraes LNF et al.*, (2015)*

*Holland K,*

*Kölling A et al.,*

*Krishnan V,*

*Neckel AL and Godinho M, (2015)*

*Nedelcu D et al,*

*Raje a et al.,*

*Ruiz M and Carey VP, (2015)*

*Schosser C and Pfitzner M, (2015)*

*Thawichsri K and Nilnont W, (2015)*

*Bankar N et al.,*

*(2015)* Air

*(2015)* Steam

*(2015)* Water

*(2015)* Air

*Ruiz M, (2015)* Wa-

*(2015)* – State of

ter/Steam

Isopentane

**work Ref. Fluid Focus of**

*(2016b)* Water

*(2016b)* Air

*(2016a)* Water Numerical

*(2016)* Water Numerical

*(2016)* Air Numerical

Water + solid ferro–particles

*(2016)* Air Experi-

Ethylene

Water/

*Saha S, (2017)* Air Numerical

*(2017)* Air Experi-

*al., (2017)* Air Analytical/

*al., (2016a)* Air Experi-

*Herrmann– Priesnitz B et al., (2016)*

*Jose R et al.,*

*Joshi KN et al.,*

*Qi W et al.,*

*Schosser C et*

*Schosser C*

Sengupta S and Guha A*, (2016)*

Zahid I et al.*,*

*Alrabie MS et al., (2017)*

*Damodhar R et*

*Guha A and Sengupta S, (2017)*

*Lisker R et al., (2017)*

*Polisetti S et al.,*

*al., (2017)* Water

*Li et al., (2017)* Water

**work**

Design/ experimental

mental

Numerical/ experimental

Analytical/ numerical

mental

Design/ experimental

Experimental/ numerical

mental

numerical

Glycol Analytical

Air Analytical/ numerical

Steam Analytical

– Analytical/ numerical

57

**Fig. 3. 1** Typical configuration of Tesla turbine a) *(Armstrong, 1952)*, b) *(Rice, 1965)*, c) *(Stei-*

As discussed in Section 2.4.1, the first concept of the Tesla turbine was developed by Tesla in 1913 *(Tesla, 1913)*. Conversely, to traditional turbomachines, which exploit the pressure difference that is produced when a fluid flows around a row of blades; the Tesla turbine generates power through the frictional interaction between the evolving fluid and the bladeless rotor. This peculiar rotor, characterized by the absence of blades, is the main feature of the Tesla turbine. Indeed, due to the configuration of the rotor, as well as to the principle of operation, the Tesla turbine is also called bladeless

Differently from conventional turbines, the rotor is composed by a series of parallel flat disks with a very small gap between them. The admission of the flow in the rotor occurs through one or more nozzles, which allows the fluid to enter from the external radius of the disks and to exit from the openings made on the disks at the inner radius. Inside the rotor, the fluid depicts a spiral centrifugal path, due to the interaction between the viscous forces and exchange of momentum. Typical configurations of the Tesla turbine consider only the rotor as part of the turbine, and often, nozzles are just converging pipes positioned tangentially to the rotor outer radius, as

turbine, viscous turbine, boundary layer turbine or friction turbine.

**a) b)**

**c) d)**

**Chapter 3**

displayed in Fig. 3.1.

*del and Weiss, 1976)*, d) *(Carey, 2010)*

**Methodology and Models**

### **Chapter 3 Methodology and Models**

As discussed in Section 2.4.1, the first concept of the Tesla turbine was developed by Tesla in 1913 *(Tesla, 1913)*. Conversely, to traditional turbomachines, which exploit the pressure difference that is produced when a fluid flows around a row of blades; the Tesla turbine generates power through the frictional interaction between the evolving fluid and the bladeless rotor. This peculiar rotor, characterized by the absence of blades, is the main feature of the Tesla turbine. Indeed, due to the configuration of the rotor, as well as to the principle of operation, the Tesla turbine is also called bladeless turbine, viscous turbine, boundary layer turbine or friction turbine.

Differently from conventional turbines, the rotor is composed by a series of parallel flat disks with a very small gap between them. The admission of the flow in the rotor occurs through one or more nozzles, which allows the fluid to enter from the external radius of the disks and to exit from the openings made on the disks at the inner radius. Inside the rotor, the fluid depicts a spiral centrifugal path, due to the interaction between the viscous forces and exchange of momentum. Typical configurations of the Tesla turbine consider only the rotor as part of the turbine, and often, nozzles are just converging pipes positioned tangentially to the rotor outer radius, as displayed in Fig. 3.1.

**Fig. 3. 1** Typical configuration of Tesla turbine a) *(Armstrong, 1952)*, b) *(Rice, 1965)*, c) *(Steidel and Weiss, 1976)*, d) *(Carey, 2010)*

57 Lorenzo Talluri, *Micro turbo expander design for small scale ORC. Tesla turbine*, © 2020 Author(s), content CC BY 4.0 International, metadata CC0 1.0 Universal, published by Firenze University Press (www.fupress.com), ISSN 2612-8020 (online), ISBN 978-88-5518-061-0 (PDF), DOI 10.36253/978-88-5518-061-0

Conversely to the typical configurations, the Tesla turbine described and analysed in this work consists of several components: an external toroidal plenum chamber, a stator with fixed nozzles and a bladeless rotor composed by parallel thin disks fixed to the rotating shaft; Fig. 3.2 displays the schematic of the ORC prototype, nonetheless the same nomenclature is used for the air prototype.

**Fig. 3. 2** Schematic of Tesla turbine

59

Once all the inputs are fixed, the code draws blades on the circular crown, defining the main parameters such as width of the throat section, chord, pressure and suction side exit angles). The maximum feasible number of passages is then calculated (*Zmax*) and partialized (*Zs*). The partialization consists of the occlusion of several volumes present between two blades, due to the low mass flow rate required by the turbine

A 2D thermo–fluid dynamic model was developed in Engineering Equation

Blade design is a procedure, which enables the definition of the stator geometry. The stator is designed like a circular crown, of a certain thickness "*Hs*" in the axial direction, on which a series of channels ("or nozzles") are obtained, defined by

 Total length of the camber (*mtot*), which can be determined imposing the inlet/outlet diameter stator ratio at 1.25 as suggested in *(Glassman, 1976, Ventura*

Another feature of the code is that the camber angle distribution can be adjusted

Solver *(Klein and Nellis, 2012)*. Each procedure is thus explained.

The required inputs for the procedure are: Outlet diameter of the stator (*r1*); Inlet and outlet angles (*α0*, *α1*);

changing the exponent *n* of a predefined power law.

**1. Objectives and structure**

**1.1 Nozzle Design**

the blades profile.

*et al., 2012).*

**Fig. 3. 3** Blade design code interface

### **1. Objectives and structure**

A 2D thermo–fluid dynamic model was developed in Engineering Equation Solver *(Klein and Nellis, 2012)*. Each procedure is thus explained.

### **1.1 Nozzle Design**

Blade design is a procedure, which enables the definition of the stator geometry.

The stator is designed like a circular crown, of a certain thickness "*Hs*" in the axial direction, on which a series of channels ("or nozzles") are obtained, defined by the blades profile.

The required inputs for the procedure are:

Another feature of the code is that the camber angle distribution can be adjusted changing the exponent *n* of a predefined power law.

### **Fig. 3. 3** Blade design code interface

58

**Fig. 3. 2** Schematic of Tesla turbine

Conversely to the typical configurations, the Tesla turbine described and analysed in this work consists of several components: an external toroidal plenum chamber, a stator with fixed nozzles and a bladeless rotor composed by parallel thin disks fixed to the rotating shaft; Fig. 3.2 displays the schematic of the ORC prototype, nonetheless

the same nomenclature is used for the air prototype.

Once all the inputs are fixed, the code draws blades on the circular crown, defining the main parameters such as width of the throat section, chord, pressure and suction side exit angles). The maximum feasible number of passages is then calculated (*Zmax*) and partialized (*Zs*). The partialization consists of the occlusion of several volumes present between two blades, due to the low mass flow rate required by the turbine

rotor. The final geometry of the stator is then obtained, composed by a small number of real channels, located inside a thick volume.

Specifically, the program can include the coordinates of NACA or any other blade profile, allowing the construction of pressure and suction sidewalls. The user can adapt the geometric inputs and the drawn profile will be visible both in the configuration of a single blade and as section of stator, as shown in Fig. 3.3.

### **Blade design main equations**

In the lookup table, the coordinate of the selected NACA profile with chord value equal to 1 are stored. Absolute camber line values are thus determined:

$$\mathbf{m}\_{\rm i} = \left(\mathbf{m}\_{\rm r(i)} - \mathbf{m}\_{\rm r(i-1)}\right) \cdot \mathbf{m}\_{\rm tot} \tag{3.1}$$

Where:

mi – absolute discretised value of camber line

mr(i) – relative discretised value of camber line

mtot – total length of camber line

The camber line is then curved through an exponential law, which sets the local camber angle.

$$\alpha = \alpha\_0 + (\alpha\_0 - \alpha\_1) \cdot \mathbf{m}\_{\mathbf{r}}^{\mathbf{n}} \tag{3.2}$$

In order to find the coordinate of each point of the camber line, trigonometry analysis is applied. From the law of cosines, it is possible to determine the radius and from the law of sines the wrap angle (Fig 3.4).

$$\mathbf{r}\_{\mathrm{i}} = (\mathbf{r}\_{\mathrm{i}-1})^2 + (\mathbf{m}\_{\mathrm{i}})^2 - (2 \cdot \mathbf{m}\_{\mathrm{i}} \cdot \mathbf{r}\_{\mathrm{i}-1} \cdot \cos(\alpha\_{\mathrm{i}})) \tag{3.3}$$

$$\theta\_{\mathbf{i}} = \theta\_{\mathbf{i}-1} + \arcsin\left(\frac{\mathbf{m}\_{\mathbf{i}}}{\mathbf{r}\_{\mathbf{i}}} \cdot \sin(\mathbf{a}\_{\mathbf{i}})\right) \tag{3.4}$$

**Fig. 3. 4** Trigonometric representation of stator blade

61

In the present study, the stator design follows the approach derived from radial expanders vaned stators (*Fiaschi et al., 2012, Fiaschi et al., 2014, Fiaschi et al., 2016,*

It is then possible to find the Cartesian coordinate of the camber point with a sim-

The coordinate of suction and pressure side can be easily determined knowing the

Once the coordinates of suction and pressure side of each blade are known, it is possible to determine the distance between two blades at each point and therefore to

As the most original component of a Tesla turbine is the bladeless rotor, simple nozzles are commonly used instead of properly designed stator vanes. The Tesla turbine stator purpose is to generate the necessary tangential flow stream at rotor inlet and to convert the pressure energy of the flow in the plenum chamber to kinetic energy at nozzles output. The reduction of cross sectional area for a subsonic flow produces a favourable pressure gradient and an acceleration of the fluid *(Anderson, 2001)*, avoiding wall separation; as a result, the efficiency of nozzles is usually very high, often exceeding 96% *(Shepherd, 1956, Shames, 2002)*. Anyway, for small size nozzles, where the throat width is lower than 3 mm (as is the case for Tesla turbine), the boundary layer might occupy a significant portion of the cross sectional area *(Butenko et al., 1976)*, generating increased viscous losses. In these cases, the flow is laminar

) and the total pressure losses decrease with increasing the Reynolds number

As discussed in several papers *(Armstrong, 1952, Rice, 1965, Hoya and Guha, 2009, Guha and Smiley, 2009, Neckel and Godinho, 2015)*, the stator is commonly acknowledged to be one of the critical components of the Tesla turbine. It is indeed recognized among the main reasons for the low efficiency of the machine obtained in several experimental tests, as it is the source of high total pressure losses. Specifically, *Guha and Smiley, (2009)* carried out an experimental and computational study of the Tesla turbine nozzles and concluded that the stator is responsible for total pressure losses

) (3.7)

<sup>∙</sup> cos(αi)) (3.9)

<sup>∙</sup> sin(αi)) (3.10)

(3.8)

Xi = ri ∙ sin(θi) (3.5) Yi = ri ∙ cos(θi) (3.6) The chord and stagger angle (the angle between the chord and radial direction)

ple coordinate transformation.

(X0−X1) (Y0−Y1)

> ( 2 )

( 2 )

blade thickness (which is a function of the chord):

values are thus found:

sin(β)

determine the throat length.

β = arctan (

Chord <sup>=</sup> X0−X1

Xi PS/SS = Xi + (

Yi PS/SS = Yi + (

**1.2 Stator model**

(*Re<105*

*(Butenko et al., 1976)*.

up to 35% of the total inlet conditions.

It is then possible to find the Cartesian coordinate of the camber point with a simple coordinate transformation.

$$\mathbf{X}\_{\mathrm{i}} = \mathbf{r}\_{\mathrm{i}} \cdot \sin(\theta\_{\mathrm{i}}) \tag{3.5}$$

$$\mathbf{Y}\_{\mathbf{i}} = \mathbf{r}\_{\mathbf{i}} \cdot \cos(\theta\_{\mathbf{i}}) \tag{3.6}$$

The chord and stagger angle (the angle between the chord and radial direction) values are thus found:

$$\begin{aligned} \beta &= \arctan\left(\frac{(\mathbf{X}\_0 - \mathbf{X}\_1)}{(\mathbf{Y}\_0 - \mathbf{Y}\_1)}\right) \\ \text{Gord} &= \frac{\mathbf{X}\_0 - \mathbf{X}\_1}{} \end{aligned} \tag{3.7}$$

Chord <sup>=</sup> X0−X1 sin(β)

The coordinate of suction and pressure side can be easily determined knowing the blade thickness (which is a function of the chord):

$$\mathbf{X\_{iPS/SS}} = \mathbf{X\_i} + \left(\frac{\left(\frac{\mathbf{r}}{2}\right)\_{PS}}{\mathbf{ss}} \cdot \cos(\mathbf{a\_i})\right) \tag{3.9}$$

$$\mathbf{Y}\_{\rm i \, PS/SS} = \mathbf{Y}\_{\rm i} + \left(\frac{\binom{\mathbf{t}}{2}\_{\rm pS}}{\rm ss} \cdot \sin(\mathbf{q}\_{\rm i})\right) \tag{3.10}$$

Once the coordinates of suction and pressure side of each blade are known, it is possible to determine the distance between two blades at each point and therefore to determine the throat length.

### **1.2 Stator model**

60

rotor. The final geometry of the stator is then obtained, composed by a small number

ration of a single blade and as section of stator, as shown in Fig. 3.3.

equal to 1 are stored. Absolute camber line values are thus determined:

mi – absolute discretised value of camber line mr(i) – relative discretised value of camber line

mtot – total length of camber line

the law of sines the wrap angle (Fig 3.4).

mi ri

**Fig. 3. 4** Trigonometric representation of stator blade

Specifically, the program can include the coordinates of NACA or any other blade profile, allowing the construction of pressure and suction sidewalls. The user can adapt the geometric inputs and the drawn profile will be visible both in the configu-

In the lookup table, the coordinate of the selected NACA profile with chord value

The camber line is then curved through an exponential law, which sets the local

In order to find the coordinate of each point of the camber line, trigonometry analysis is applied. From the law of cosines, it is possible to determine the radius and from

ri = (ri−1)2 + (mi)<sup>2</sup> − (2 ∙ mi ∙ ri−1 ∙ cos(αi)) (3.3)

n (3.2)

<sup>∙</sup> sin(αi)) (3.4)

mi = (mr(i) − mr(i−1)) ∙ mtot (3.1)

of real channels, located inside a thick volume.

**Blade design main equations**

Where:

camber angle.

α = α0 + (α0 − α1) ∙ mr

θi = θi−1 + arcsin (

As the most original component of a Tesla turbine is the bladeless rotor, simple nozzles are commonly used instead of properly designed stator vanes. The Tesla turbine stator purpose is to generate the necessary tangential flow stream at rotor inlet and to convert the pressure energy of the flow in the plenum chamber to kinetic energy at nozzles output. The reduction of cross sectional area for a subsonic flow produces a favourable pressure gradient and an acceleration of the fluid *(Anderson, 2001)*, avoiding wall separation; as a result, the efficiency of nozzles is usually very high, often exceeding 96% *(Shepherd, 1956, Shames, 2002)*. Anyway, for small size nozzles, where the throat width is lower than 3 mm (as is the case for Tesla turbine), the boundary layer might occupy a significant portion of the cross sectional area *(Butenko et al., 1976)*, generating increased viscous losses. In these cases, the flow is laminar (*Re<105* ) and the total pressure losses decrease with increasing the Reynolds number *(Butenko et al., 1976)*.

As discussed in several papers *(Armstrong, 1952, Rice, 1965, Hoya and Guha, 2009, Guha and Smiley, 2009, Neckel and Godinho, 2015)*, the stator is commonly acknowledged to be one of the critical components of the Tesla turbine. It is indeed recognized among the main reasons for the low efficiency of the machine obtained in several experimental tests, as it is the source of high total pressure losses. Specifically, *Guha and Smiley, (2009)* carried out an experimental and computational study of the Tesla turbine nozzles and concluded that the stator is responsible for total pressure losses up to 35% of the total inlet conditions.

In the present study, the stator design follows the approach derived from radial expanders vaned stators (*Fiaschi et al., 2012, Fiaschi et al., 2014, Fiaschi et al., 2016,*

*Rohlik, 1975, Whitfield and Baines, 1990)*, also accounting for partial admission due to the reduced flow rates. In *(Fiaschi et al., 2014)*, the design guidelines for a radial ORC turboexpander are defined using a zero dimensional model: an input dataset (thermo– fluid dynamic variables, dimensional and non–dimensional parameters) is provided by the designer, while the outputs of calculations are the basic geometry, velocity triangles and nozzle efficiency.

In order to carry out a parametric analysis, a set of geometric and thermodynamic parameters, needs to be defined. The formers are: stator blade angles, number of nozzles, length and height of the channel and nozzle geometry profile; which allow the definition of the full geometry (in particular the throat width and the chord length). The main thermodynamic conditions to be fixed are the inlet total pressure and temperature and the mass flow rate.

Two iterative processes were implemented, the former one on mass flow rate, the latter one on two stator loss coefficients. The static pressure at throat section is taken as the iterative parameter for the matching of the mass flow rate, which will ultimately assume the imposed input mass flow rate.

The first loss coefficient (*ζN*) was taken from (*Fiaschi et al., 2016, Whitfield and Baines, 1990, Dixon, 2005)* and it is only dependent on *Φn*, which is the ratio between the real and isentropic velocity at stator exit. The range of this parameter is between 0.9 and 0.97. The second loss coefficient (*ζR*) *(Ventura et al., 2012)* is on the other hand dependent on geometry values, such as chord, pitch, stator height and stator exit angle, as well as on the Reynolds number of the flow. The two coefficients are compared and until they are not equal, an iterative process on flow exit velocity runs.

### **Stator main equations**

The thermo–fluid dynamic model for the calculation of the fluid behaviour into the Tesla turbine stator assumes real fluid Equations of State (EoS). For this reason, all the thermodynamic properties were evaluated as functions of couples of local variables (typically, *p* and *T*) using the Engineering Equation Solver (EES) EoS library data. Fig. 3.5 displays the enthalpy–entropy diagram of the Tesla expander stator and its nomenclature.

The thermodynamic of point 00 is defined from input values *p00* and *T00*. Point 0 is defined through iteration on density (first guess value is 00) and application of mass flow rate definition in order to find *v0*.

ṁ = v0 ∙ ρ0 ∙ A0 (3.11)

63

An iterative process was also implemented on two loss coefficients *ζ<sup>N</sup>* and *ζ<sup>R</sup>* in order to assess the right velocity at stator outlet and therefore the efficiency of the

> cos(α1) Hs

After the two iterative process point 1 and 01 are defined and, finally, stator effi-

(h01 <sup>−</sup> h1s) (3.16)

<sup>2</sup> − 1 (3.14)

) (3.15)

nozzle. Eqns. (3.14) and (3.15) display the calculation of the loss coefficients.

– Blade height at stator outlet ; 1– Absolute angle at stator outlet; *Re* – Reynolds number (based on blade height and absolute velocity at stator out-

+ pitch ∙

P1(down) <sup>=</sup> P1(up) <sup>+</sup> (ṁ <sup>−</sup> muṗ )

**Fig. 3. 5** Stator Enthalpy-Entropy diagram

<sup>=</sup> <sup>1</sup> ϕn

Re0.2 <sup>∙</sup> (3 <sup>∙</sup> tan(α1) pitch

chord <sup>⁄</sup>

ζN <sup>=</sup> h1 <sup>−</sup> h1s 1 <sup>2</sup> v1 2

ζR <sup>=</sup> 0.05

Where:

ciency is calculated as: ηst <sup>=</sup> (h01 <sup>−</sup> h1)

let)

(ṁ down − ṁ up)

∙ (P1(down) − P1(up)) (3.13)

The root finding interpolation method was applied in order to find the correct expansion ratio in the stator. *P1* is taken as the variable for the interpolation method; convergence is reached when the imposed mass flow rate and the calculated one are equal, as shown in the diagram of Fig. 3.6. Eqns. (3.12) and (3.13) are main equations of the applied scheme. In order to set the right starting point, the upper bound for *P1* was assumed as equal to *P0 – 100* Pa; the lower bound was set assuming an incompressible flow expansion.

$$\text{P}\_1\text{(up)} = \text{P}\_1\text{(up)} + \frac{\left(\dot{\text{m}} - \dot{\text{m}}\_{\text{up}}\right)}{\left(\dot{\text{m}}\_{\text{down}} - \dot{\text{m}}\_{\text{up}}\right)} \cdot \left\{\text{P}\_1\{\text{down}\} - \text{P}\_1\{\text{up}\}\right\} \tag{3.12}$$

An iterative process was also implemented on two loss coefficients *ζ<sup>N</sup>* and *ζ<sup>R</sup>* in order to assess the right velocity at stator outlet and therefore the efficiency of the nozzle. Eqns. (3.14) and (3.15) display the calculation of the loss coefficients.

$$\mathbf{V\_N} = \frac{\mathbf{h\_1} - \mathbf{h\_{1s}}}{\frac{1}{2}\mathbf{v\_1^2}} = \frac{1}{\Phi\_\mathbf{n}^2} - \mathbf{1} \tag{3.14}$$

$$\begin{array}{llll} \text{or} & \text{tran}(\alpha) & & \\ \text{0.05} & & \cos(\alpha) \end{array}$$

$$\mathbf{Q}\_{\rm R} = \frac{0.05}{\mathrm{Re}^{0.2}} \cdot \left( 3 \cdot \frac{\tan(\alpha\_1)}{\mathrm{pitch}} + \mathrm{pitch} \cdot \frac{\cos(\alpha\_1)}{\mathrm{H\_s}} \right) \tag{3.15}$$

Where:

62

∙ (P1(down) − P1(up)) (3.12)

*Rohlik, 1975, Whitfield and Baines, 1990)*, also accounting for partial admission due to the reduced flow rates. In *(Fiaschi et al., 2014)*, the design guidelines for a radial ORC turboexpander are defined using a zero dimensional model: an input dataset (thermo– fluid dynamic variables, dimensional and non–dimensional parameters) is provided by the designer, while the outputs of calculations are the basic geometry, velocity

In order to carry out a parametric analysis, a set of geometric and thermodynamic parameters, needs to be defined. The formers are: stator blade angles, number of nozzles, length and height of the channel and nozzle geometry profile; which allow the definition of the full geometry (in particular the throat width and the chord length). The main thermodynamic conditions to be fixed are the inlet total pressure and tem-

Two iterative processes were implemented, the former one on mass flow rate, the latter one on two stator loss coefficients. The static pressure at throat section is taken as the iterative parameter for the matching of the mass flow rate, which will ultimately

The first loss coefficient (*ζN*) was taken from (*Fiaschi et al., 2016, Whitfield and Baines, 1990, Dixon, 2005)* and it is only dependent on *Φn*, which is the ratio between the real and isentropic velocity at stator exit. The range of this parameter is between 0.9 and 0.97. The second loss coefficient (*ζR*) *(Ventura et al., 2012)* is on the other hand dependent on geometry values, such as chord, pitch, stator height and stator exit angle, as well as on the Reynolds number of the flow. The two coefficients are compared

The thermo–fluid dynamic model for the calculation of the fluid behaviour into the Tesla turbine stator assumes real fluid Equations of State (EoS). For this reason, all the thermodynamic properties were evaluated as functions of couples of local variables (typically, *p* and *T*) using the Engineering Equation Solver (EES) EoS library data. Fig. 3.5 displays the enthalpy–entropy diagram of the Tesla expander stator and

The thermodynamic of point 00 is defined from input values *p00* and *T00*. Point 0 is defined through iteration on density (first guess value is 00) and application of

ṁ = v0 ∙ ρ0 ∙ A0 (3.11) The root finding interpolation method was applied in order to find the correct expansion ratio in the stator. *P1* is taken as the variable for the interpolation method; convergence is reached when the imposed mass flow rate and the calculated one are equal, as shown in the diagram of Fig. 3.6. Eqns. (3.12) and (3.13) are main equations of the applied scheme. In order to set the right starting point, the upper bound for *P1* was assumed as equal to *P0 – 100* Pa; the lower bound was set assuming an incom-

and until they are not equal, an iterative process on flow exit velocity runs.

triangles and nozzle efficiency.

perature and the mass flow rate.

**Stator main equations**

its nomenclature.

pressible flow expansion.

P1(up) <sup>=</sup> P1(up) <sup>+</sup> (ṁ <sup>−</sup> muṗ )

(mdown ̇ − muṗ )

assume the imposed input mass flow rate.

mass flow rate definition in order to find *v0*.

– Blade height at stator outlet ; 1– Absolute angle at stator outlet;

*Re* – Reynolds number (based on blade height and absolute velocity at stator outlet)

After the two iterative process point 1 and 01 are defined and, finally, stator efficiency is calculated as:

$$\eta\_{\rm st} = \frac{(\mathbf{h}\_{01} - \mathbf{h}\_{1})}{(\mathbf{h}\_{01} - \mathbf{h}\_{1\rm s})} \tag{3.16}$$

**Fig. 3. 6** Flow Diagram of Stator model

65

Ain = Lt ∙ Hs (3.20) *Aout* is the cross section on the disks (transversal) covered by the flow jet:

When connecting the three–separated developed flow models (stator, rotor and

 Conservation of total enthalpy between the stator outlet and the rotor inlet, because the transformation may be considered adiabatic and without any

 Invariable static enthalpy, and consequently constant absolute velocity *v1*, while the flow direction changes during the gap crossing. It means that the pressure losses may be treated like an isenthalpic throttling process into a

The stator and rotor models only take into account of the distributed pressure losses inside the components, but they do not consider those concentrated into the stator–rotor gap. The passage of the flow from the throat of the nozzle to the gap first and to the rotor channel later, involve an abrupt cross–section enlargement followed

Both loss coefficient (abrupt enlargement and abrupt contraction) were obtained from standard incompressible flow loss treatment theory (Borda Carnot coefficient and polynomial fitting of empirical data *(Idel'chik, 1960)*). In order to take into account compressibility effects, an iterative procedure was implemented in EES to compute

For an incompressible fluid, the concentrated pressure losses are calculated by the definition of a loss coefficient, depending on the system geometry and the flow con-

<sup>2</sup> conρw2

Where ∆ is the pressure loss occurring immediately after the throat section (abrupt enlargement) and ∆ is the pressure loss related to the flow entering the rotor micro–channels (relative flow contraction), while *en* and *con* are their respective

The loss coefficient for abrupt enlargement (*en*) was modelled as an incompress-

)<sup>2</sup> (3.19)

<sup>2</sup> ρv<sup>2</sup> (3.17)

2 (3.18)

gap pressure losses), the following assumptions were applied:

by a contraction and generates large flow pressure losses.

average values of density between inlet and outlet sections.

For the case of Tesla turbine, it can be written as:

<sup>2</sup> enρv1

ditions, which reduces the kinetic energy of the fluid *(Idel'chik, 1960)*:

<sup>2</sup> + 1

ible Borda–Carnot coefficient *(Idel'chik, 1960)*, according to Eq. (3.19):

Where *Ao* is the throat cross section, here having rectangular shape:

**1.3 Stator-rotor coupling model**

work transfer.

**Stator–rotor losses main equations**

1

∆p <sup>=</sup> ∆pen <sup>+</sup> ∆pcon <sup>=</sup> <sup>1</sup>

Aout

Δp0 = ∙

loss coefficients.

en <sup>=</sup> (1 <sup>−</sup> Ain

valve. **Model characteristic**

**Assumptions**

### **1.3 Stator-rotor coupling model**

### **Assumptions**

When connecting the three–separated developed flow models (stator, rotor and gap pressure losses), the following assumptions were applied:

### **Model characteristic**

64

**Fig. 3. 6** Flow Diagram of Stator model

The stator and rotor models only take into account of the distributed pressure losses inside the components, but they do not consider those concentrated into the stator–rotor gap. The passage of the flow from the throat of the nozzle to the gap first and to the rotor channel later, involve an abrupt cross–section enlargement followed by a contraction and generates large flow pressure losses.

Both loss coefficient (abrupt enlargement and abrupt contraction) were obtained from standard incompressible flow loss treatment theory (Borda Carnot coefficient and polynomial fitting of empirical data *(Idel'chik, 1960)*). In order to take into account compressibility effects, an iterative procedure was implemented in EES to compute average values of density between inlet and outlet sections.

### **Stator–rotor losses main equations**

For an incompressible fluid, the concentrated pressure losses are calculated by the definition of a loss coefficient, depending on the system geometry and the flow conditions, which reduces the kinetic energy of the fluid *(Idel'chik, 1960)*:

$$
\Delta \mathbf{p}\_0 = \boldsymbol{\zeta} \cdot \frac{1}{2} \,\mathrm{\rho} \mathbf{v}^2 \tag{3.17}
$$

For the case of Tesla turbine, it can be written as:

$$\Delta \mathbf{p} = \Delta \mathbf{p}\_{\rm en} + \Delta \mathbf{p}\_{\rm con} = \frac{1}{2} \zeta\_{\rm en} \rho \mathbf{v}\_1^2 + \frac{1}{2} \zeta\_{\rm con} \rho \mathbf{w}\_2^2 \tag{3.18}$$

Where ∆ is the pressure loss occurring immediately after the throat section (abrupt enlargement) and ∆ is the pressure loss related to the flow entering the rotor micro–channels (relative flow contraction), while *en* and *con* are their respective loss coefficients.

The loss coefficient for abrupt enlargement (*en*) was modelled as an incompressible Borda–Carnot coefficient *(Idel'chik, 1960)*, according to Eq. (3.19):

$$\zeta\_{\rm en} = (1 - \frac{\mathbf{A}\_{\rm in}}{\mathbf{A}\_{\rm out}})^2 \tag{3.19}$$

Where *Ao* is the throat cross section, here having rectangular shape:

Ain = Lt ∙ Hs (3.20)

*Aout* is the cross section on the disks (transversal) covered by the flow jet:

$$\mathbf{A}\_{\rm out} = \left\{ \left[ \frac{\mathbf{L}\_{\rm t}}{\tan \alpha\_1} + \frac{\mathcal{G}}{\sin \alpha\_1} \right] / \cos \alpha\_1 - \mathcal{G} \cdot \tan \alpha\_1 - \frac{\mathcal{G}}{\tan \alpha\_{\rm PS}} \right\} \cdot \mathbf{H}\_{\rm S} \tag{3.21}$$

Eq. (3.19) can be used for a turbulent flow with a uniform velocity profile; when these assumptions aren't checked, some numerical and graphical correlations available in *(Idel'chik, 1960)* can be used.

The total pressure loss for abrupt expansion can be calculated using the velocity immediately upstream the enlargement. A parametric analysis showed that the pressure loss is mainly influenced by the velocity *v1* (second order law), followed by the density and, finally, by geometric parameters (throat width *TW* and gap extension ).

The loss coefficient for abrupt contraction (*con*) was obtained through a polynomial fitting of empirical data *(Idel'chik, 1960)*:

$$\begin{aligned} \zeta\_{\text{con}} &= -0.126 \left( \frac{\text{A}\_{\text{out}}}{\text{A}\_{\text{in}}} \right)^{4} + 1.0296 \left( \frac{\text{A}\_{\text{out}}}{\text{A}\_{\text{in}}} \right)^{3} - 1.279 \left( \frac{\text{A}\_{\text{out}}}{\text{A}\_{\text{in}}} \right)^{2} \\ &- 0.1209 (\frac{\text{A}\_{\text{out}}}{\text{A}\_{\text{in}}}) + 0.5 \end{aligned} \tag{3.22}$$

Where *Ain* is the total cylindrical surface, including disks and channel surfaces: Ain = 2πr2 ∙ Hs (3.23)

While *Aout* only takes into account the cylindrical surface of the channel: Aout = 2πr2 ∙ b ∙ nch (3.24)