Disc Thickness and Spacing Distance Impacts on Flow Characteristics of Multichannel Tesla Turbines

energies

Article Disc Thickness and Spacing Distance Impacts on Flow Characteristics of Multichannel Tesla Turbines

Wenjiao Qi, Qinghua Deng * , Yu Jiang, Qi Yuan and Zhenping Feng

Shaanxi Engineering Laboratory of Turbomachinery and Power Equipment, Institute of Turbomachinery, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China; [email protected] (W.Q.); [email protected] (Y.J.); [email protected] (Q.Y.); [email protected] (Z.F.) * Correspondence: [email protected]

Received: 9 November 2018; Accepted: 20 December 2018; Published: 24 December 2018

Abstract: Tesla turbines are a kind of unconventional bladeless turbines, which utilize the viscosity of working fluid to rotate the rotor and realize energy conversion. They offer an attractive substitution for small and micro conventional bladed turbines due to two major advantages. In this study, the effects of two influential geometrical parameters, disc thickness and disc spacing distance, on the aerodynamic performance and flow characteristics for two kinds of multichannel Tesla turbines (one-to-one turbine and one-to-many turbine) were investigated and analyzed numerically. The results show that, with increasing disc thickness, the isentropic efficiency of the one-to-one turbine decreases a little and that of the one-to-many turbine reduces significantly. For example, for turbine cases with 0.5 mm disc spacing distance, the former drops less than 7% and the latter decreases by about 45% of their original values as disc thickness increases from 1 mm to 2 mm. With increasing disc spacing distance, the isentropic efficiency of both kinds of turbines increases first and then decreases, and an optimal value and a high efficiency range exist to make the isentropic efficiency reach its maximum and maintain at a high level, respectively. The optimal disc spacing distance for the one-to-one turbine is less than that for the one-to-many turbine (0.5 mm and 1 mm, respectively, for turbine cases with disc thickness of 1 mm). To sum up, for designing a multichannel Tesla turbine, the disc spacing distance should be among its high efficiency range, and the determination of disc thickness should be balanced between its impacts on the aerodynamic performance and mechanical stress.

Keywords: Tesla turbine; ﬂuid dynamics; disc thickness; disc spacing distance; isentropic efﬁciency

1. Introduction In the last two decades, one of the interests is to develop small-scale turbomachinery due to market demands driving and manufacture techniques making progress [1], which is mainly applied to small power generation unit, such as distributed energy systems using low-grade energy and increasing energy utilization efficiency. In addition, small-scale turbomachinery is also applied as the mobile or small power device, which can be used in unmanned aircraft, miniaturized radio-controlled vehicles [2,3], and so on. Several research groups designed, manufactured and experimented some small or micro gas turbines to study their feasibility and effectiveness [4–9]. These studies show that two major problems are faced when conventional turbines are scaled down. One is the rapidly increasing rotational speed, and the other is the significantly decreasing flow efficiency. Moreover, Brayton cycle system cannot be realized when the flow efficiency of conventional turbines goes down to a certain amount due to their microminiaturization [10]. However, a Tesla turbine is considered as one of the best alternatives [11].

Energies 2019, 12, 44; doi:10.3390/en12010044 www.mdpi.com/journal/energies Energies 2019, 12, x FOR PEER REVIEW 2 of 26

Energies 2019, 12, 44 2 of 25 their microminiaturization [10]. However, a Tesla turbine is considered as one of the best alternatives [11]. InIn 1913,1913, thethe firstfirst TeslaTesla turbineturbine waswas designed,designed, manufacturedmanufactured andand patentedpatented byby thethe famousfamous scholarscholar NikolaNikola TeslaTesla [[12].12]. ItIt isis anan unconventionalunconventional bladelessbladeless turbomachinery,turbomachinery, whichwhich usesuses thethe viscousviscous stressstress actingacting onon rotating rotating disc disc walls walls to dragto drag the the rotor rotor to rotate, to rotate, as shown as shown in Figure in 1 Figure. The rotor 1. The of Teslarotor turbinesof Tesla isturbines composed is composed by several by flat, several parallel, flat, rigid, parallel, co-rotating rigid, co discs,-rotating which discs, are which placed are closely placed and closely mounted and onmounted a central on shaft. a central The shaft. narrow The spaces narrow between spaces thebetween adjacent thediscs adjacent are discdiscs channels. are disc channels. In Tesla turbines, In Tesla theturbines, working the fluidworking accelerates fluid accelerates in nozzles in or nozzles volute andor volute obtains and its obtains highest its flow highest velocity flow at velocity the stator at outlet;the stator then outlet; injects then into injectsthe disc into channels the disc nearly channels tangentially; nearly tangentiall and finallyy; flows and finally spirally flows towards spirally the disctowards holes the or disc slots holes around or theslots shaft. around the shaft.

(a) (b)

FigureFigure 1.1. Tesla turbineturbine schematic:schematic: ((aa)) 2-D2-D sketch;sketch; andand ((bb)) 3-D3-D rotor.rotor.

DuringDuring 1913–1950,1913–1950, littlelittle researchresearch waswas conductedconducted on Tesla turbinesturbines duedue toto thethe inventioninvention ofof gasgas turbinesturbines and and its its high high efficiency. efficiency. Thereafter, Thereafter, the theoretical the theoretical and experimental and experimental analysis analysis on Tesla on turbines Tesla hasturbines been restarted has been due restarted to its advantages; due to its advantages; for example, for itis example, simple to it manufacture is simple to and manufacture maintain with and lowmaintain cost and with it canlow usecost kinds and it of can working use kinds fluid of [13 working–16]. However, fluid [13 the–16 study]. However, progress the of study Tesla turbinesprogress wasof Tesla still slow.turbines In the was last still two slow. decades, In the much last research two decades, on Tesla much turbines research has been on Tesla reported turbines using has not been only theoreticalreported using and experimentalnot only theoretical methods and but experimental also numerical methods simulations but also due numerical to the rapid simulations development due ofto computationalthe rapid development fluid dynamics of computational (CFD) and fluid advanced dynamics computer (CFD) techniques. and advanced computer techniques. AccordingAccording to to the the inlet inlet geometry, geometry, Tesla Tesla turbines turbines are classified are classified into two into types: two voluted types: voluted Tesla turbine Tesla andturbine nozzled and Tesla nozzled turbine. Tesla There turbine. are few There reports are few on the reports voluted on Tesla the voluted turbine. Tesla The flow turbine. characteristics The flow andcharacteristics loss mechanism and loss of volutedmechanism Tesla of turbines voluted wereTesla investigated turbines were using investigated experimental using and experimental numerical methodsand numerical [17]. The methods results [17]. show The that results the theoretical show that limit the oftheor theetical isentropic limit efficiencyof the isentropic is about efficiency 40%, if the is parasiticabout 40%, losses, if the mainly parasitic including losses, bearingmainly including loss, viscous bearing losson loss, end viscous walls, loss and on eddy end loss walls, in the and volute, eddy canloss be in minimized.the volute, can be minimized. TheThe main main focus focus of of Tesla Tesla turbines turbines is on is investigating on investigating flow characteristics flow characterist in nozzledics in nozzledTesla turbines. Tesla Aturbines. one-dimensional A one-dimensional analysis method analysis of flowmethod field of in flow the discfield channel in the disc is proposed channel is with proposed some hypotheses, with some andhypotheses, the model and agrees the model well withagrees previous well with experimental previous experimental performance performance data [18]. Sengupta data [18]. and Sengupta Guha formulatedand Guha formulated a mathematical a mathematical theory by simplifying theory by simplifying the Navier–Stokes the Navier equations–Stokes using equations a magnitude using a analysismagnitude method, analysis which method, can assess which the can turbine assess performance the turbine [19 performance–21]. Talluri et[19 al.–21 developed]. Talluri a et new al. methoddeveloped for designing a new method of a Tesla for turbine designing for Organic of a Tesla Rankine turbine Cycle for (ORC) Organic applications, Rankine in whichCycle almost (ORC) allapplications, losses are consideredin which almost using all real losses gas are physical considered properties using [real22]. gas It can physical calculate properties the aerodynamic [22]. It can performancecalculate the aerodynamic of the multichannel performance Tesla of turbine; the multichannel however, theTesla results turbine; have however, not been the validated results have by numericalnot been validated and experimental by numerical analysis. and experimental To sum up, most analysis. theoretical To sum analysis up, most can theoretical only be applied analysis to onecan channelonly be Teslaapplied turbines, to one inchannel which Tesla the influence turbines, of in disc which thickness the influence has not of been disc taken thickness into account has not yet.been takenSome into researchersaccount yet. investigated the total aerodynamic performance and detailed flow fields using experimental method. The research group of Guha set up a test rig of Tesla turbines, and put forward

Energies 2019, 12, 44 3 of 25 several methods for measuring power [23]. In addition, they improved the inlet and nozzle to enhance the turbine efficiency [24]. Schosser et al. designed and set up a test rig to reveal the detailed flow field in the disc spacing of one channel Tesla turbine using 3D tomographic Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) measurements, however its rotational speed is restricted because of the mechanical stress [25]. From the above research, the detailed flow field in multichannel Tesla turbines cannot be investigated experimentally. The complicated turbine model cannot be analyzed theoretically, although it can be simulated by CFD method to reveal the detailed flow characteristics. The effects of some parameters, such as the rotational speed and nozzle number, have been studied for a Tesla turbine with a low-boiling medium [11,26]. The effects of disc spacing distance and rotational speed on a Tesla turbine were analyzed numerically by our research group, and the results show that an optimal value of both parameters exists for the turbine to obtain its highest efficiency, respectively [27–29]. The flow fields in the disc channel of one channel Tesla turbine (no stator) with different inlet conditions, including flow coefficient and inlet geometries, were simulated numerically; the efficiency decreases dramatically with increasing flow coefficient, and the inlet non-uniformity is also a factor that makes turbine efficiency decrease [30]. Sengupta et al. studied the influence of four parameters, namely the nozzle number, disc thickness, rotational speed and nozzle-rotor radial clearance, on the performance of a Tesla turbine, and the results show that in the turbine design, thin discs with flat disc tip edge, more nuzzles and an optimal radial clearance are suggested [31]. In the authors’ previous study, the nozzled multichannel Tesla turbine is classified into two categories according to nozzle geometry: one nozzle channel to one disc channel (one-to-one Tesla turbines) and one nozzle channel to several disc channels (one-to-many Tesla turbines), as shown in Figure2[ 29]. The one-to-one Tesla turbine and the one-to-many turbine with the same geometry operating under the same conditions were studied numerically and the only difference between two kinds of Tesla turbines is their nozzle geometries. The objective was to reveal and compare their fluid dynamic, and the results show that the flow mechanism of the two Tesla turbines are totally different.

(a) (b)

Figure 2. Sketch maps of multichannel Tesla turbines: (a) one-to-one Tesla turbine; and (b) one-to-many

Tesla turbine.

For multichannel Tesla turbines, the affecting factors include the geometrical parameters (such as nozzle number, nozzle geometry, disc outer diameter, disc spacing distance and disc thickness) and

Energies 2019, 12, 44 4 of 25 operating parameters (such as rotational speed mass flow rate and turbine pressure ratio). The above reference review shows that the influence of most affecting parameters has been investigated, although some research is on one channel Tesla turbine. However, up to now, few studies have been conducted on the impacts of disc thickness and disc spacing distance on the performance of multichannel Tesla turbines. The effect of disc thickness is studied in Ref. [31], but only the flow in the nozzle-rotor chamber and the rotor are included in numerical calculations. In addition, it has been analyzed experimentally for the one-to-many turbine, while, for the one-to-one turbine, its influence and the fluid mechanism have not been investigated [32]. The impacts of disc spacing distance on the one channel Tesla turbine were analyzed by our group, and the results show that an optimal disc spacing distance exists to enable the Tesla turbine to obtain its best performance [28]. In the turbine with narrower disc spacing distance, two boundary layers on two disc walls overlap, resulting in a decrease in frictional force and torque; in the turbine with wider disc spacing distance, more working fluid outside the boundary layer flows out of the disc channels resulting in less momentum exchange to drive a rotor. Because disc thickness has to be taken into consideration in multichannel Tesla turbines, the influence of disc spacing distance on flow fields must be different from that in one channel Tesla turbines. In this paper, two kinds of multichannel Tesla turbines (one-to-one turbines and one-to-many turbines) with different disc spacing distance and disc thickness are simulated numerically to investigate their influence on the aerodynamic performance and flow characteristics. This paper provides theoretical and engineering reference for designing a multichannel Tesla turbine.

2. Numerical Approach

2.1. Geometry Model and Boundary Conditions In this study, two groups of multichannel Tesla turbines were calculated numerically. One group is the one-to-one Tesla turbine, and the other is the one-to-many Tesla turbine. Each group has six cases of Tesla turbines, and the parameters of disc spacing distance and disc thickness are given in Table1, in which the case is named as “disc thickness-disc spacing distance”. In this research, the working ﬂuid is compressed air.

Table 1. Geometrical parameters of each case.

Group 1 (One-To-One Multichannel Tesla Turbines) Case th (mm) b (mm) b/th (-) Case 1-0.3 1 0.3 0.3 Case 1-0.5 1 0.5 0.5 Case 1-1 1 1 1 Case 2-0.3 2 0.3 0.15 Case 2-0.5 2 0.5 0.25 Case 2-1 2 1 0.5 Group 2 (One-To-Many Multichannel Tesla Turbines) Case th (mm) b (mm) b/th (-) Case 1-0.3 1 0.3 0.3 Case 1-0.5 1 0.5 0.5 Case 1-1 1 1 1 Case 2-0.5 2 0.5 0.25 Case 2-1 2 1 0.5 Case 2-2 2 2 1

Each turbine case consists of ﬁve discs and six disc channels. The other geometrical parameters and aerodynamic parameters are all the same for the two groups of the Tesla turbine, which are the same as those in the authors’ previous study [29] (Table2). Energies 2019, 12, 44 5 of 25

To save computing time and resources, the calculation domain was reduced based on a rotational symmetry and a symmetry about a middle plane that is normal to the axis, which ﬁnally became a quarter of the whole turbine. It is shown in Figure2 with red color.

Table 2. Geometrical and aerodynamic parameters.

Symbol Unit Value

Nn (-) 2 do,d (mm) 100 di,d (mm) 38.4 c (mm) 0.25 Nd (-) 5 Ndc (-) 6 α (◦) 10 pnt/pi (-) 3.42 Tnt (K) 373

The boundary conditions are given in Table2. In addition, because of the calculation domain reduction, the symmetry and rotational periodicity boundary conditions were set up at corresponding surfaces. The adiabatic and no-slip boundary condition was adopted for all walls. The frozen rotor method was applied to deal with the rotor-stator interface.

2.2. Numerical Solver and Mesh Sensitivity In this research, all numerical calculations were conducted using commercial software ANSYS CFX (ANSYS Inc., Canonsburg, PA, USA), in which the Reynolds Averaged Navier–Stokes (RANS) equations in the calculation domain were solved for turbulent ﬂow. The RANS equation groups are as follows. Continuity equation: ∂ρ ∂ + ρUj = 0 (1) ∂t ∂xj Momentum equations:

∂ρUi ∂ ∂p ∂ + ρUiUj = − + τij − ρuiuj + SM (2) ∂t ∂xj ∂xi ∂xj

Energy equations:

∂(ρhtot) ∂p ∂ ∂ ∂T ∂ − + (ρUjhtot) = (λ − ρujh) + [Ui(τij − ρuiuj)] + SE (3) ∂t ∂t ∂xj ∂xj ∂xj ∂xj in which xj are coordinates in three directions, Uj are average velocity in three directions, p is the static pressure, T is the temperature, ρ is the density, and τ is the molecular stress tensor. In Equations (2) and (3), SM and SE are source terms, ρuiuj are the Reynolds stresses, ρujh is an ∂ additional turbulence ﬂux term, [U (τ − ρu u )] represents the viscous work term. htot is the mean ∂xj i ij i j total enthalpy, h is the static enthalpy, and htot = h + UiUj/2 + k, where k is the turbulent kinetic 2 energy and k = ui /2. In addition, the density can be solved according to the state equation for ideal air,

ρ = pabs/RT (4) where pabs is the absolute pressure and R is the speciﬁc gas constant of air. Energies 2019, 12, 44 6 of 25

According to the above discussion, it can be found that RANS equations are not closed due to two additional terms, which are Reynolds stress and Reynolds flux. Therefore, some turbulence models in ANSYS CFX are provided to calculate turbulent flow. For Tesla turbines, the detailed experimental data cannot be found in public references, therefore the flow model verification could not be conducted. The critical Reynolds number for the plane Poiseuille flow is about 1000 [33], and is defined by the half disc spacing distance and the average velocity in Tesla turbines. In most Tesla turbines, the Reynolds number is always above the critical values, and the minimum Reynolds number was about 1500 in our computational cases. Therefore, the Shear Stress Transport (SST) turbulence model was adopted in this research, which is also used in numerical research by other researchers [11,25,26]. SST turbulence model is a kind of eddy viscosity model, and, according to eddy viscosity hypothesis, the Reynolds stresses are considered to be proportional to mean velocity gradients. They can be expressed as follows.

∂Ui ∂Uj 2 ∂Uk − ρuiuj = µt( + ) − δij(ρk + µt ) (5) ∂xj ∂xi 3 ∂xk where µt is the eddy viscosity. In addition, the Reynolds ﬂuxes of a scalar is stated as a linear relationship to the mean scalar gradient based on the eddy diffusivity hypothesis:

∂Φ − ρuiφ = Γt (6) ∂xi where ρuiφ is the Reynolds ﬂux of a scalar Φ, Γt is the eddy diffusivity, written as Γt = µt/Prt, and Prt is the turbulent Prandtl number. Equations (5) and (6) give expressions of turbulent ﬂuctuations as functions of the mean variables when the turbulent viscosity is known. In ANSYS CFX solver, SST turbulence model, which derives from the k − ε and k − ω turbulence models, uses this variable and expresses the turbulent viscosity as,

k µ = ρ (7) t ω where ω is turbulent frequency. The Reynolds averaged transport equations for k and ω in SST turbulence model are as follows.

" # ∂(ρω) ∂ ∂ µt ∂ω 1 ∂k ∂ω ω 2 + ρUjω = µ + + (1 − F1)2ρ + α3 Pk − β3ρω + Pkb (8) ∂t ∂xj ∂xj σω3 ∂xj σω2ω ∂xj ∂xj k

" # ∂(ρk) ∂ ∂ µt ∂k 0 + ρUj k = µ + + Pk − β ρkω + Pkb (9) ∂t ∂xj ∂xj σk2 ∂xj For the detailed coefficients in Equations (8) and (9), refer to Ref. [34]. Up to now, the RANS equations are closed, and can be solved to obtain the turbulent flow fields. For all numerical simulations in this research, spatial discretization for the above flow governing equations was conducted with high-resolution second-order central difference scheme and time discretization was with second-order backward Euler scheme. In addition, an empirical automatic wall function was adopted to guarantee the solution accuracy. The accuracy of second order was obtained in all numerical simulations. In addition, the y+ should be less than 2 for all walls as the SST turbulence model requires. In this study, a structured grid was generated for fluid domain, using ANSYS mesh generation software ICEM CFD (ANSYS Inc., Canonsburg, PA, USA). The mesh independence analysis for the one-to-many turbine Case 1-0.5 was conducted to ensure the accuracy of the results. Energies 2019, 12, 44 7 of 25

Three sets of mesh for this turbine were generated and the detailed information are given in Table3. The N-R chamber stands for the chamber between the nozzle and the rotor.

Table 3. Mesh information.

Stator (Nozzle/N-R Chamber) Rotor (Each Disc Channel) Case No. Number of Nodes Total Node Number of Nodes Total Node (r,θ,z Directions) Number (r,θ,z Directions) Number Case 1 (55/13) × (36/269) × 99 526,516 65 × 288 × 23 400,660 Case 2 (67/17) × (45/335) × 107 923,517 81 × 333 × 29 782,217 Case 3 (87/21) × (57/417) × 135 1,830,306 102 × 417 × 37 1,581,306

The aerodynamic performance parameters and their relative variation values (based on the results of Case 3) are given in Table4. The isentropic efﬁciency η is deﬁned as the actual shaft power P divided by the isentropic power across the whole turbine m∆hisen, as follows:

P MΩ η = = (10) m∆h h (γ−1)/γi isen mcpTnt 1 − (pi/pnt) where P equals the whole torque M multiplied by the angular speed of the rotor Ω. cp and γ are the physical properties of the working fluid, which are specific heat at constant pressure and the specific heat ratio. The subscript “nt” represent the total parameters at the turbine inlet, while “i” indicates the static parameter at the turbine outlet. Figure3 presents the Mach number contours on the mid-section of DC1 (“DC” is short for “disc channel”) for the one-to-many turbine Case 1-0.5 with three different node cases. The detailed locations of the three DCs are indicated in Figure2. Note that DC1 and DC2 each consists of two rotating disc walls, and DC3 has one rotating disc wall and one motionless casing wall. Table4 indicates that the variation value of each aerodynamic performance decreases with node number. Figure3 shows that the Mach number distribution for each case is similar, but the difference between Cases 2 and 3 is much less than that between Cases 1 and 2. To sum up, the results of Case 2 fulfill the requirements for mesh independence, therefore the mesh of Case 2 was applied for the this Energies 2019, 12, x FOR PEER REVIEW 8 of 8 turbine in numerical simulations. Moreover, for other turbine cases with different geometry, the grid distribution changes accordingly to get a reliable result.

(a) (b) (c)

FigureFigure 3. 3.Mach Mach number number contourscontours on the mid-sections of of DC1 DC1 for for the the one-to-many one-to-many turbine turbine Case Case 1-0.5 1-0.5 withwith three three different different node node cases: cases: ((aa)) CaseCase 1; ( b)) Case Case 2; 2; and and (c (c) )Case Case 3. 3.

Table 4. Mesh independence.

Node Number m δ m Case No. P (kW) δ P (%) η (-) δη (%) (million) (kg/s) (%) Case 1 1.72 0.03576 0.619 0.5960 1.568 0.1504 0.940 Case 2 3.27 0.03562 0.225 0.5886 0.307 0.1491 0.067 Case 3 6.57 0.03554 0 0.5868 0 0.1490 0

3. Results and Discussions

3.1. Total Aerodynamic Performance of Two Kinds of Multichannel Tesla Turbines Two kinds of multichannel Tesla turbines with disc thickness of 1 mm and 2 mm were calculated numerically, respectively. It is obvious that, if the disc thickness is too large, the air flowing into disc channels will be more difficult, therefore, the aerodynamic performance becomes worse. However, the disc thickness should not be too small, which is mainly confined to its mechanical stress and processing problem. As disc thickness decreases, the stiffness of the discs decreases rapidly, and accordingly the mechanical stress increases, which should be lower than material allowable stress. Previous study on the computational solid mechanical (CSM) analysis indicates that the disc thickness should not be less than 1 mm for the Tesla turbine with disc outer diameter of 100 mm.

3.1.1. Isentropic Efficiency Figure 4 shows the variation curves of the isentropic efficiency for each turbine case with the rotational speed for the two kinds of multichannel Tesla turbines. In Figure 4, note that the negative isentropic efficiency has no physical meaning and only indicates that under this condition the Tesla turbine consumes external energy without power outputting. It can be seen that the isentropic efficiency of the one-to-one turbine is much higher than that of the one-to-many Tesla turbine, and it changes much faster with rotational speed. For the one-to-one turbine, the isentropic efficiency of the turbine cases with same disc spacing distance but different disc thickness is almost equal, although the turbine cases with smaller disc thickness are slightly more efficient, which is more obvious at higher rotational speeds. For example, the isentropic efficiency of Case 2-0.5 has a less than 7% reduction of that of Case 1-0.5 at all rotational

Energies 2019, 12, 44 8 of 25

Table 4. Mesh independence.

Node Number m Case No. δm(%) P (kW) δP (%) η (-) δη (%) (million) (kg/s) Case 1 1.72 0.03576 0.619 0.5960 1.568 0.1504 0.940 Case 2 3.27 0.03562 0.225 0.5886 0.307 0.1491 0.067 Case 3 6.57 0.03554 0 0.5868 0 0.1490 0

3. Results and Discussions

3.1. Total Aerodynamic Performance of Two Kinds of Multichannel Tesla Turbines Two kinds of multichannel Tesla turbines with disc thickness of 1 mm and 2 mm were calculated numerically, respectively. It is obvious that, if the disc thickness is too large, the air flowing into disc channels will be more difficult, therefore, the aerodynamic performance becomes worse. However, the disc thickness should not be too small, which is mainly confined to its mechanical stress and processing problem. As disc thickness decreases, the stiffness of the discs decreases rapidly, and accordingly the mechanical stress increases, which should be lower than material allowable stress. Previous study on the computational solid mechanical (CSM) analysis indicates that the disc thickness should not be less than 1 mm for the Tesla turbine with disc outer diameter of 100 mm.

3.1.1. Isentropic Efficiency Figure4 shows the variation curves of the isentropic efficiency for each turbine case with the rotational speed for the two kinds of multichannel Tesla turbines. In Figure4, note that the negative isentropic efficiency has no physical meaning and only indicates that under this condition the Tesla turbine consumes external energy without power outputting. It can be seen that the isentropic efficiency of the one-to-one turbine is much higher than that of the one-to-many Tesla turbine, and it changes much faster with rotational speed. For the one-to-one turbine, the isentropic efficiency of the turbine cases with same disc spacing distance but different disc thickness is almost equal, although the turbine cases with smaller disc thickness are slightly more efficient, which is more obvious at higher rotational speeds. For example, the isentropic efficiency of Case 2-0.5 has a less than 7% reduction of that of Case 1-0.5 at all rotational speeds lower than 45,000 r/min. The turbine case with disc spacing distance of 0.5 mm obtains the highest efficiency at lower rotational speeds. That is, an optimal disc spacing distance for the one-to-one multichannel Tesla turbine exists to obtain its best performance, which is in accordance with that for one channel Tesla turbines [28]. In fact, a high efficiency range of disc spacing distance exists, among which the isentropic efficiency can remain at a high level. However, the turbine with disc spacing distance of 1 mm performs best at higher rotational speeds, because the isentropic efficiency changes more rapidly with rotational speed for the turbine case with narrower disc spacing distance. It can be predicted that an optimal disc spacing distance must exist at higher rotational speeds, which must be higher than 0.5 mm. For the one-to-many turbine, obviously, the isentropic efficiency of the turbine cases with greater disc thickness is much lower than that of the cases with smaller disc thickness, which agrees well with the experimental results in Ref. [32]. In detail, the turbine isentropic efficiency of Case 2-0.5 is about 55% of that of Case 1-0.5. Meanwhile, the one-to-many turbine also has an optimal disc spacing distance. To sum up, for designing multichannel Tesla turbines, a one-to-one Tesla turbine is recommended. Moreover, the disc spacing distance should be in its high efficiency range, and the disc thickness should be as small as possible while satisfying the requirement of allowable material stress. Usually, the disc thickness can be determined by the results of CSM, and it should be larger than 1 mm for discs of 100 mm diameter. Energies 2019, 12, x FOR PEER REVIEW 9 of 9 speeds lower than 45,000 r/min. The turbine case with disc spacing distance of 0.5 mm obtains the highest efficiency at lower rotational speeds. That is, an optimal disc spacing distance for the one-to- one multichannel Tesla turbine exists to obtain its best performance, which is in accordance with that for one channel Tesla turbines [28]. In fact, a high efficiency range of disc spacing distance exists, among which the isentropic efficiency can remain at a high level. However, the turbine with disc spacing distance of 1 mm performs best at higher rotational speeds, because the isentropic efficiency changes more rapidly with rotational speed for the turbine case with narrower disc spacing distance. It can be predicted that an optimal disc spacing distance must exist at higher rotational speeds, which must be higher than 0.5 mm. For the one-to-many turbine, obviously, the isentropic efficiency of the turbine cases with greater disc thickness is much lower than that of the cases with smaller disc thickness, which agrees well with the experimental results in Ref. [32]. In detail, the turbine isentropic efficiency of Case 2-0.5 is about 55% of that of Case 1-0.5. Meanwhile, the one-to-many turbine also has an optimal disc spacing distance. To sum up, for designing multichannel Tesla turbines, a one-to-one Tesla turbine is recommended. Moreover, the disc spacing distance should be in its high efficiency range, and the disc thickness should be as small as possible while satisfying the requirement of allowable material stress.Energies 2019Usually,, 12, 44 the disc thickness can be determined by the results of CSM, and it should be larger9 of 25 than 1 mm for discs of 100 mm diameter.

(a) (b)

Figure 4.4.Isentropic Isentropic efﬁciency efficiency versus versus rotational rotational speed sp ofeed each of turbine each caseturbine for two case kinds for oftwo multichannel kinds of multichannelTesla turbines: Tesla (a) one-to-one turbines: ( Teslaa) one-to-one turbine; andTesla (b turbine;) one-to-many and (b) Tesla one-to-many turbine. Tesla turbine.

3.1.2. Flow Coefficient Coefficient Figure 55 presentspresents thethe curvescurves ofof thethe flowflow coefficientcoefficient forfor eacheach turbineturbine casecase versusversus thethe rotationalrotational speed. The flow coefficient Cm is defined as: speed. The flow coefficient Cm is defined as:

vo,d m Cm = vo,d = m (11) C =Ωr = 2 m Ωo,d 2πρπρo,dΩΩbr2 o,d (11) ro,d2 o,dbr o,d where vo,d and ωro,d represent the average radial velocity and the disc rotational linear velocity at the where v and ωr represent the average radial velocity and the disc rotational linear velocity at rotor inlet,o,d respectively.o,d the rotorThe inlet, flow coefficientrespectively. of the one-to-one Tesla turbine is much less than that of the one-to-many turbineThe with flow same coefficient disc spacing of the distanceone-to-one and Tesla disc thicknessturbine is (Figure much 5less). In than detail, that for of turbine the one-to-many Case 1-0.5, turbinethat of the with one-to-many same disc spacing is nearly distance double thatand ofdisc the thickness one-to-one (Figure turbine. 5). In Compared detail, for to turbine the one-to-one Case 1- 0.5,turbine, that of when the one-to-many the operating is parameters nearly double and that other of geometricalthe one-to-one parameters turbine. Compared are given, theto the mass one-to- flow onerate turbine, of the one-to-many when the operating turbine is parameters much higher and due ot toher much geometrical larger nozzle parameters throat are area given, (for Case the mass 1-0.5, flowthe one-to-many rate of the one-to-many turbine is about turbine 2.6 times);is much the higher density due at to the much rotor larger inlet nozzle changes throat much area less (for with Case the multichannel Tesla turbine type (for Case 1-0.5, the one-to-many turbine is about 1.3 times) than the mass flow rate; therefore, the flow coefficient of the one-to-many turbine is much higher. For the one-to-one turbine, the flow coefficient of the turbine cases with same disc spacing distance is almost the same (the relative variation between Cases 1-0.5 and 2-0.5 is less than 2%), although it has a little bit increase with disc thickness, especially at higher rotational speeds. Meanwhile, this coefficient of the turbine cases with disc spacing distance of 0.3 mm and 0.5 mm is almost the same, and clearly higher than that of 1 mm, which is because the air density at the rotor inlet goes up with disc spacing distance. For the one-to-many turbine, the variation relationships of the flow coefficient with disc thickness and spacing distance become complicated, and these two geometrical parameters affect significantly on this coefficient. As shown in Figure2, the axial length ratio of nozzle channel to disc channels of the one-to-many Tesla turbine is larger than 1 and it goes up with increasing disc thickness or decreasing disc spacing distance, thus the flow coefficient increases as disc thickness goes up and disc spacing distance goes down, respectively. Same with the one-to-one turbine, the air density at the rotor inlet mainly depends on disc spacing distance, and increases with it, which also makes the coefficient decrease as disc spacing distance increases. Energies 2019, 12, x FOR PEER REVIEW 10 of 10

1-0.5, the one-to-many turbine is about 2.6 times); the density at the rotor inlet changes much less with the multichannel Tesla turbine type (for Case 1-0.5, the one-to-many turbine is about 1.3 times) than the mass flow rate; therefore, the flow coefficient of the one-to-many turbine is much higher. For the one-to-one turbine, the flow coefficient of the turbine cases with same disc spacing distance is almost the same (the relative variation between Cases 1-0.5 and 2-0.5 is less than 2%), although it has a little bit increase with disc thickness, especially at higher rotational speeds. Meanwhile, this coefficient of the turbine cases with disc spacing distance of 0.3 mm and 0.5 mm is almost the same, and clearly higher than that of 1 mm, which is because the air density at the rotor inlet goes up with disc spacing distance. For the one-to-many turbine, the variation relationships of the flow coefficient with disc thickness and spacing distance become complicated, and these two geometrical parameters affect significantly on this coefficient. As shown in Figure 2, the axial length ratio of nozzle channel to disc channels of the one-to-many Tesla turbine is larger than 1 and it goes up with increasing disc thickness or decreasing disc spacing distance, thus the flow coefficient increases as disc thickness goes up and disc spacing distance goes down, respectively. Same with the one-to-one turbine, the air density at the rotor inlet mainly depends on disc spacing distance, and increases with it, which also Energiesmakes 2019the ,coefficient12, 44 decrease as disc spacing distance increases. 10 of 25

(a) (b)

Figure 5. Flow coefficient coefﬁcient versus rotational speed of each case for two kinds of multichannel Tesla turbines: ( a) one-to-one Tesla turbine;turbine; andand ((bb)) one-to-manyone-to-many TeslaTesla turbine.turbine.

3.1.3. Percentages of Mass Flow Rate and Torque inin DiscDisc ChannelsChannels Figure6 6 shows shows the the mass mass flow flow rate rate and and torque torque percentages percentages in in three three DCs DCs for for all all the the turbine turbine cases cases at 30,000at 30,000 r/min. r/min. In FigureIn Figure6, both 6, both percentages percentages in DCs in 1DCs and 21 areand nearly 2 are equalnearly for equal both for kinds both of turbines,kinds of whichturbines, indicates which thatindicates the flow that fields the flow in DCs fields 1 andin DCs 2 are 1 and similar. 2 are In similar. addition, In addition, DCs 1 and DCs 2 have 1 and more 2 have air thanmore DC3,air than while DC3, generating while generating more torque. more torque. For the one-to-one turbine, as disc thickness changes,changes, both percentages in each disc channel are almost thethe same same for for turbine turbine cases cases with with same discsame spacing disc spacing distance, distance, respectively. respectively. The torque The percentage torque ofpercentage DC3 for theof DC3 turbine for the cases turbine with 0.5cases mm with in disc0.5 mm spacing in disc distance spacing are distance almost theare same,almost about the same, 17%, whileabout that17%, for while turbine that with for 0.3turbine mm inwith disc 0.3 spacing mm in distance disc spacing decreases distance to 12%. decreases All these to flow 12%. phenomena All these areflow explained phenomena in the are next explained sections. in the next sections. For the one-to-many turbine, the variation relation relationshipship of mass flowflow rate percentage with disc spacing distance is not coincident. For the turbine cases with disc thickness of 1 mm, those in DCs 1 andEnergies 2 decrease 2019, 12, x asFOR disc PEER spacing REVIEW distance increases, whilewhile for the turbine cases with disc thickness11 of of 11 2 mm, they decrease firstfirst and then increase. In detail,detail, those of Cases 1-1 and 2-1 reach the minimum values respectively. Moreover, disc thickness and spacing distance have few effects on the torque percentage in each disc channel.

(a) (b)

Figure 6. 6. MassMass flow ﬂow rate rate and and torque torque percentages percentages in th inree three DCs DCsfor two for kinds two kinds of Tesla of turbines Tesla turbines at 30,000 at 30,000r/min: ( r/min:a) one-to-one (a) one-to-one Tesla turbine; Tesla turbine; and (b) andone-to-many (b) one-to-many Tesla turbine. Tesla turbine.

To be noted, the calculation domain has five disc rotating walls and the average torque percentage is 20%. However, for all the turbine cases, the torque percentage in DC3 is less than 20%. Meanwhile, DC3 flows more working fluid than DCs 1 and 2. That means DC3 uses much more working fluid and generates less torque and power. Thus, the disc spacing distance of DC3 should be smaller to get less working fluid. To sum up, disc thickness slightly influences the aerodynamic performance of the one-to-one turbine. The isentropic efficiency of the turbine cases with disc thickness of 2 mm is a little lower than that of 1 mm. Specifically, for turbine cases with 0.5 mm in disc spacing distance, it decreases by only 7% at most rotational speeds. Moreover, the disc spacing distance affects greatly the aerodynamic performance of the one-to-one turbine, and an optimal disc spacing distance exists to make the turbine obtain the highest isentropic efficiency. Both the disc thickness and spacing distance have significant impacts on the aerodynamic performance of the one-to-many turbine. The one-to-many turbine also has an optimal disc spacing distance, which is larger than that for the one-to-one turbine (1 mm and 0.5 mm, respectively, for turbine with disc thickness of 1 mm). In addition, the performance becomes remarkably worse with an increase in disc thickness (a reduction of about 45%).

3.2. Flow Status of One-To-One Multichannel Tesla Turbines From the above discussion, the isentropic efficiency of the one-to-one Tesla turbine drops a little as disc thickness increases. It increases first and then decreases with increasing disc spacing distance, and reaches its maximum value for the turbine case with disc spacing distance of 0.5 mm at lower rotational speeds including the optimal rotational speed of 30,000 r/min. To reveal the reasons that the disc thickness and spacing distance affect the turbine performance, the flow characteristics in the Tesla turbine were analyzed.

Energies 2019, 12, 44 11 of 25

To be noted, the calculation domain has five disc rotating walls and the average torque percentage is 20%. However, for all the turbine cases, the torque percentage in DC3 is less than 20%. Meanwhile, DC3 flows more working fluid than DCs 1 and 2. That means DC3 uses much more working fluid and generates less torque and power. Thus, the disc spacing distance of DC3 should be smaller to get less working fluid. To sum up, disc thickness slightly influences the aerodynamic performance of the one-to-one turbine. The isentropic efficiency of the turbine cases with disc thickness of 2 mm is a little lower than that of 1 mm. Specifically, for turbine cases with 0.5 mm in disc spacing distance, it decreases by only 7% at most rotational speeds. Moreover, the disc spacing distance affects greatly the aerodynamic performance of the one-to-one turbine, and an optimal disc spacing distance exists to make the turbine obtain the highest isentropic efficiency. Both the disc thickness and spacing distance have significant impacts on the aerodynamic performance of the one-to-many turbine. The one-to-many turbine also has an optimal disc spacing distance, which is larger than that for the one-to-one turbine (1 mm and 0.5 mm, respectively, for turbine with disc thickness of 1 mm). In addition, the performance becomes remarkably worse with an increase in disc thickness (a reduction of about 45%).

3.2. Flow Status of One-To-One Multichannel Tesla Turbines From the above discussion, the isentropic efficiency of the one-to-one Tesla turbine drops a little as disc thickness increases. It increases first and then decreases with increasing disc spacing distance, and reaches its maximum value for the turbine case with disc spacing distance of 0.5 mm at lower rotational speeds including the optimal rotational speed of 30,000 r/min. To reveal the reasons that the disc thickness and spacing distance affect the turbine performance, the flow characteristics in the Tesla turbine were analyzed.

3.2.1. One-To-One Turbine with Different Disc Thickness Figures7 and8 show Mach number contours and streamlines and on the mid-sections of three DCs for the one-to-one turbine Cases 1-0.5 and 2-0.5 at 30,000 r/min, respectively. The streamlines and Mach number contours of DCs 1 and 2 are almost the same, but different from that of DC3, which agrees well with the above analysis. Therefore, in the next analysis, only the flow fields of DCs 1 and 3 are given. Obviously, in flow channels 1 and 2, part of the air that just injects nearly tangentially into the disc channels flows into the N-R chamber due to centrifugal force, and finally flows into DC3. In flow channel 3, the air from the nozzle together with that from the N-R chamber flows into DC3 at a larger flow angle (relative to the tangential direction). Thus, it is easy to know that for the one-to-one turbine less air flows through DCs 1 and 2 than that through DC3. In addition, it can be observed that the Mach number on the mid-sections of DCs 1 and 2 is much higher than that of DC3, and the flow angle is much less than that in DC3, which leads to much higher relative tangential velocity and longer path lines in DCs 1 and 2. Therefore, the torque percentage in DC3 is lower than its average value of 20%. Comparing Figures7 and8, the flow fields in each disc channel of the turbine cases with same disc spacing distance but different disc thickness have little difference. This indicates that their aerodynamic performance also has little difference. Figure9 illustrates the contours of relative tangential velocity and vector distributions on the radial cross-section of 0◦, whose location is marked in Figure1. In this figure, only a small part of the disc channels is presented and the axial size is doubled to show the flow field clearly. The reference vector is given near the text of “rotor”. In the disc channels, the contour represents the relative tangential velocity, and the vector indicates the radial and axial velocities. In the stator, the contour shows the velocity normal to this section, and the vector describes the axial velocity and another velocity on this section. In the disc channels, the positive relative tangential velocity indicates that the air at this region generates usable torque; otherwise, it consumes external mechanical energy, most of which appears near the casing wall. Energies 2019, 12, x FOR PEER REVIEW 12 of 12

channel 3, the air from the nozzle together with that from the N-R chamber flows into DC3 at a larger flow angle (relative to the tangential direction). Thus, it is easy to know that for the one-to-one turbine less air flows through DCs 1 and 2 than that through DC3. In addition, it can be observed that the Mach number on the mid-sections of DCs 1 and 2 is much higher than that of DC3, and the flow angle is much less than that in DC3, which leads to much higher relative tangential velocity and longer path lines in DCs 1 and 2. Therefore, the torque percentage in DC3 is lower than its average value of 20%. Comparing Figures 7 and 8, the flow fields in each disc channel of the turbine cases with same disc spacing distance but different disc thickness have little difference. This indicates that their aerodynamic performance also has little difference. Figure 9 illustrates the contours of relative tangential velocity and vector distributions on the radial cross-section of 0°, whose location is marked in Figure 1. In this figure, only a small part of the disc channels is presented and the axial size is doubled to show the flow field clearly. The reference vector is given near the text of “rotor”. In the disc channels, the contour represents the relative tangential velocity, and the vector indicates the radial and axial velocities. In the stator, the contour shows the velocity normal to this section, and the vector describes the axial velocity and another velocity on this section. In the disc channels, the positive relative tangential velocity indicates that the air at this region generates usable Energies 2019, 12, 44 12 of 25 torque; otherwise, it consumes external mechanical energy, most of which appears near the casing wall. FigureFigure9 9 indicates indicates thatthat onon thisthis sectionsection thethe axialaxial velocity is close to zero, zero, and and the the radial radial velocity velocity is is muchmuch lowerlower thanthan thethe relative relative tangential tangential velocity. velocity. In In addition, addition, the the air air with with high high velocity velocity flows flows from from the nozzlethe nozzle through through the N-R the chamber N-R chamber and finally and intofinally the in discto channelsthe disc rapidlychannels with rapidly little flowwith deflectionlittle flow to thedeflection axial direction. to the axial Some direction. vortex Some generates vortex at somegenerates region at some of the region N-R chamber. of the N-R chamber.

Energies 2019, 12, x FOR(a) PEER REVIEW (b) (c) 13 of 13 FigureFigure 7.7. Mach number contours and streamlines streamlines on on the the mid-sections mid-sections of of three three DCs DCs for for the the one-to-one one-to-one turbine turbine CaseCase 1-0.51-0.5 atat 30,00030,000 r/min: (a (a) )DC1; DC1; (b (b) )DC2; DC2; and and (c ()c )DC3. DC3.

(a) (b) (c)

FigureFigure 8.8. Mach number contours and and streamlines streamlines on on the the mid-sections mid-sections of of three three DCs DCs for for the the one-to-one one-to-one turbineturbine CaseCase 2-0.52-0.5 atat 30,000 r/min: (a (a) )DC1; DC1; (b (b) )DC2; DC2; and and (c ()c )DC3. DC3.

TheThe flow flow fields fields for for turbine turbine Cases Cases 1-0.5 1-0.5 and and 2-0.5 2-0.5 are almost are almost the same the samein most in regions most regions on the radial on the radialcross-section. cross-section. However, However, the flow the velocity flow velocity of the working of the working fluid in fluid the N-R in the chamber N-R chamber for turbine for turbine Case Case2-0.5 2-0.5is much is much lower lower than than that thatfor forCase Case 1-0.5, 1-0.5, which which consumes consumes more more extra extra energy energy to tomove move them. them. Moreover,Moreover, itit cancan bebe observed from the relative relative tangen tangentialtial velocity velocity contours contours that that the the tangential tangential velocity velocity gradientgradient nearnear thethe rotatingrotating discdisc wallswalls forfor CaseCase 2-0.5 2-0.5 is is a a little little less less than than that that for for Case Case 1-0.5, 1-0.5, which which leads leads to lowerto lower torque. torque. All All these these factors factors lead lead to to the the isentropic isentropic efficiency efficiency of of Case Case 2-0.5 2-0.5 a a little little lower lower than than that that of 1-0.5of 1-0.5 (23.92% (23.92% and and 23.04%, 23.04%, respectively). respectively). ToTo sumsum up,up, thethe flowflow field field and aerodynamic aerodynamic performance performance of of the the one-to-one one-to-one Tesla Tesla turbine turbine changes changes aa littlelittle withwith discdisc thickness.thickness. In detail, the the isentropic isentropic efficiency efficiency of of the the turbine turbine with with the the thicker thicker disc disc is is little lower, which results from lower velocity in the N-R chamber and a little lower tangential velocity gradient close to the disc walls.

(a) (b)

Figure 9. Contours of relative tangential velocity and vector distributions on radial cross-sections of 0° for the one-to-one Tesla turbine cases at 30,000 r/min: (a) Case 1-0.5; and (b) Case 2-0.5.

3.2.2. One-To-One Turbine with Different Disc Spacing Distance To reveal the detailed flow characteristics in the disc channels of the one-to-one turbine with different disc spacing distance, Figure 10 shows the variation curves of circumferential mass flow

Energies 2019, 12, x FOR PEER REVIEW 13 of 13

(a) (b) (c)

Figure 8. Mach number contours and streamlines on the mid-sections of three DCs for the one-to-one turbine Case 2-0.5 at 30,000 r/min: (a) DC1; (b) DC2; and (c) DC3.

The flow fields for turbine Cases 1-0.5 and 2-0.5 are almost the same in most regions on the radial cross-section. However, the flow velocity of the working fluid in the N-R chamber for turbine Case 2-0.5 is much lower than that for Case 1-0.5, which consumes more extra energy to move them. Moreover, it can be observed from the relative tangential velocity contours that the tangential velocity gradient near the rotating disc walls for Case 2-0.5 is a little less than that for Case 1-0.5, which leads to lower torque. All these factors lead to the isentropic efficiency of Case 2-0.5 a little lower than that Energiesof 1-0.52019 (23.92%, 12, 44 and 23.04%, respectively). 13 of 25 To sum up, the flow field and aerodynamic performance of the one-to-one Tesla turbine changes a little with disc thickness. In detail, the isentropic efficiency of the turbine with the thicker disc is littlelittle lower,lower, which which results results from from lower lower velocity velocity in the in N-Rthe N-R chamber chamber and aand little a lowerlittle tangentiallower tangential velocity gradientvelocity gradient close to the close disc to walls.the disc walls.

(a) (b)

FigureFigure 9.9. ContoursContours ofof relativerelative tangentialtangential velocityvelocity and and vector vector distributions distributions on on radial radial cross-sections cross-sections of of 0 ◦ for0° for the the one-to-one one-to-one Tesla Tesla turbine turbine cases cases at 30,000at 30,000 r/min: r/min: (a ()a Case) Case 1-0.5; 1-0.5; and and (b ()b Case) Case 2-0.5. 2-0.5.

3.2.2.3.2.2. One-To-OneOne-To-One TurbineTurbine withwith DifferentDifferent DiscDisc SpacingSpacing Distance ToTo revealreveal thethe detaileddetailed flowflow characteristicscharacteristics in the disc disc channels channels of of the the one-to-one one-to-one turbine turbine with with differentdifferent discdisc spacingspacing distance, Figure 10 shows th thee variation variation curves curves of of circumferential circumferential mass mass flow flow average velocity, including relative tangential velocity and radial velocity, versus the radius ratio (local radius to rotor inlet radius) for different one-to-one turbine cases. In this figure, the abscissa values of 1 and 0.384 represent the rotor inlet and outlet, respectively. As shown in Figure 10a, for all turbine cases, the average relative tangential velocity in DC1 obtains the highest value at the rotor inlet, and decreases with radius ratio due to the contribution to the output power. For turbine Cases 1-0.5 and 1-1, it then increases as radius ratio continues to decrease (the air flows towards the turbine outlet), and the stationary point of velocity occurs much earlier with increasing disc spacing distance. In addition, the average relative tangential velocity decreases at the region near the rotor inlet and increases at the region near the rotor outlet with increasing disc spacing distance. Similarly, the average relative tangential velocity in DC3 for all turbine cases decreases first and then increases as the air flows towards the turbine outlet. However, it is much higher for the turbine cases with larger disc spacing distance. In general, the average relative tangential velocity difference between DCs 1 and 3 decreases with increasing disc spacing distance. As shown in Figure 10b, for all turbine cases, the average radial velocity in DC1 decreases with radius ratio, which seems to violate the continuity equation of working fluid. The flow area (cylindrical surface of the disc channel) decreases as the air flows through the disc channels to the rotor outlet (radius ratio decreases), while the density also decreases, thus the radial velocity should increase based on the continuity equation. However, it is completely opposite to the results in Figure 10, which is due to the partial admission of the nozzled Tesla turbine. These discrete nozzles are installed symmetrically at the disc tip, and most air is injected into DCs 1 and 2 by the nozzles. Although the flow area decreases as radius ratio decreases, the effective flow area increases due to the air diffusion when the air flows from the rotor inlet to outlet, which leads to a decrease in average radial velocity in DC1. Energies 2019, 12, x FOR PEER REVIEW 14 of 14

average velocity, including relative tangential velocity and radial velocity, versus the radius ratio (local radius to rotor inlet radius) for different one-to-one turbine cases. In this figure, the abscissa values of 1 and 0.384 represent the rotor inlet and outlet, respectively. As shown in Figure 10a, for all turbine cases, the average relative tangential velocity in DC1 obtains the highest value at the rotor inlet, and decreases with radius ratio due to the contribution to the output power. For turbine Cases 1-0.5 and 1-1, it then increases as radius ratio continues to decrease (the air flows towards the turbine outlet), and the stationary point of velocity occurs much earlier with increasing disc spacing distance. In addition, the average relative tangential velocity decreases at the region near the rotor inlet and increases at the region near the rotor outlet with increasing disc spacing distance. Similarly, the average relative tangential velocity in DC3 for all turbine cases decreases first and then increases as the air flows towards the turbine outlet. However, it is much higher for the turbine cases with larger disc spacing distance. In general, the average relative tangential velocity difference between DCs 1 and 3 decreases with increasing disc spacing distance. As shown in Figure 10b, for all turbine cases, the average radial velocity in DC1 decreases with radius ratio, which seems to violate the continuity equation of working fluid. The flow area (cylindrical surface of the disc channel) decreases as the air flows through the disc channels to the rotor outlet (radius ratio decreases), while the density also decreases, thus the radial velocity should increase based on the continuity equation. However, it is completely opposite to the results in Figure 10, which is due to the partial admission of the nozzled Tesla turbine. These discrete nozzles are installed symmetrically at the disc tip, and most air is injected into DCs 1 and 2 by the nozzles. Although the flow area decreases as radius ratio decreases, the effective flow area increases due to the air diffusion when the air flows from the rotor inlet to outlet, which leads to a decrease in average Energiesradial velocity2019, 12, 44 in DC1. 14 of 25

(a) (b)

Figure 10.10. Variation curvescurves ofof circumferentialcircumferential massmass ﬂowflow averageaverage velocityvelocity versusversus the radiusradius ratio for threethree one-to-oneone-to-one turbineturbine cases:cases: ((aa)) relativerelative tangentialtangential velocity;velocity; andand ((bb)) radialradial velocity.velocity.

InIn addition, the average radial velocity in DC3 decreasesdecreases firstfirst and then increases; however, itit has much lessless variation variation than than that that in DC1.in DC1. Some Some air fromair from the N-R the chamberN-R chamber flows intoflows DC3, into and DC3, the effectiveand the floweffective area flow of DC3 area is muchof DC3 larger is much than larger that of than DC1. that However, of DC1. the However, flow velocity the flow at the velocity rotor inlet at the from rotor the N-Rinlet chamberfrom the isN-R much chamber less, and is much the average less, and radial the velocityaverage stillradial decreases velocity slightly still decreases due to airslightly diffusion. due Asto air the diffusion. air flows towardsAs the air the flows rotor towards outlet, the the average rotor outlet, radial the velocity average has radial a slight velocity increase has due a toslight the pressureincrease dropdue to the the rotor pressure and the drop decrease the rotor in effective and the flow decrease area. Moreover,in effective the flow average area. radialMoreover, velocity the inaverage DCs 1 radial and 3 bothvelocity decreases in DCs with 1 and increasing 3 both decreases disc spacing with distance, increasing and disc the spacing average distance, radial velocity and the at theaverage rotor radial inlet in velocity DC1 is at much the rotor higher inlet than in thatDC1 in is DC3.much higher than that in DC3. InIn summary,summary, thethe relativerelative tangentialtangential velocity,velocity, determiningdetermining thethe torque and power, decreasesdecreases with radiusradius ratio due to outputting power,power, and then increases slightly, resultingresulting fromfrom thethe decreasedecrease inin discdisc rotational linear velocity and the pressure in the drop. Furthermore, its difference between DCs 1 and 3 decreases with increasing disc spacing distance. Moreover, the radial velocity, depending on the mass flow rate, has a strong relationship with the effective flow area, and it decreases with increasing disc spacing distance. Figure 11 shows Mach number contours and streamlines and on the mid-sections of DCs 1 and 3 for the one-to-one turbine cases with different disc spacing distance at 30,000 r/min. As shown in Figure 11, the Mach number at the nozzle outlet decreases as disc spacing distance increases because of the higher pressure drop in the disc channels of the turbine cases with wider disc spacing. With increasing flow velocity at the nozzle outlet, the flow angle at the rotor inlet should decrease if the nozzle outlet flow angle of different turbine cases is the same. However, it is totally opposite to the flow fields in Figure 11, which is because the nozzle outlet flow angle changes with disc spacing distance despite their same nozzle outlet geometrical angles. The air in the scarfed part of the nozzle tends to speed up and changes its direction to the side without the nozzle wall, when the pressure ratio of the nozzle (nozzle outlet pressure to nozzle inlet pressure) is under the critical pressure ratio of air. The deflection angle goes up with an increase in difference value between the nozzle pressure ratio and the critical pressure ratio, as shown in turbine Cases 1-0.3 and 1-0.5 in Figure 11. When the nozzle pressure ratio is over the critical pressure ratio, the flow will not deflect in the scarfed part and the flow angle at the rotor inlet is almost equal to the nozzle outlet geometrical angle, shown in turbine Case 1-1. Energies 2019, 12, x FOR PEER REVIEW 15 of 15 rotational linear velocity and the pressure in the drop. Furthermore, its difference between DCs 1 and 3 decreases with increasing disc spacing distance. Moreover, the radial velocity, depending on the mass flow rate, has a strong relationship with the effective flow area, and it decreases with increasing disc spacing distance.

EnergiesFigure2019, 1211, 44shows Mach number contours and streamlines and on the mid-sections of DCs 151 and of 25 3 for the one-to-one turbine cases with different disc spacing distance at 30,000 r/min.

(a) (b) (c)

(d) (e) (f)

FigureFigure 11. 11. MachMach number number contours andand streamlinesstreamlines on on the the mid-sections mid-sections of of DCs DCs 1 and1 and 3 for3 for the the one-to-one one-to- oneturbine turbine cases cases with with different different disc disc spacing spacing distance distance at 30,000 at 30,000 r/min: r/min: (a) ( DC1a) DC1 for for Case Case 1-0.3; 1-0.3; (b) ( DC1b) DC1 for forCase Case 1-0.5; 1-0.5; (c) DC1(c) DC1 for Casefor Case 1-1; (1-1;d) DC3 (d) forDC3 Case for 1-0.3;Case (1-0.3;e) DC3 (e for) DC3 Case for 1-0.5; Case and 1-0.5; (f) DC3and for(f) CaseDC3 1-1.for Case 1-1. Comparing the flow fields in DCs 1 and 3, it can be found that more air flows into DC3 from both the nozzleAs shown and in the Figure N-R chamber, 11, the Mach and thenumber velocity at the of thenozzle working outlet fluid decreases from the as disc N-R spacing chamber distance is much increaseslower. Therefore, because theof the average higher relative pressure tangential drop in velocity the disc and channels radial of velocity the turbine at the cases rotor inletwith ofwider DC3 discare underspacing. those With of increasing DC1. Apart flow from velocity DC1 for at turbine the nozzle Case outlet, 1-0.3, the the flow flow angle velocity at the in other rotor discinlet channels should decreasedecreases if first the andnozzle then outlet increases flow asangle the radiusof differen decreases.t turbine Moreover, cases is the flowsame. field However, difference it is between totally oppositeDCs 1 and to 3the decreases flow fields with in increasing Figure 11, discwhich spacing is because distance. the nozzle All the outlet above flow flow angle phenomena changes are with the discsame spacing as those distance in Figure despite 10. their same nozzle outlet geometrical angles. The air in the scarfed part of theBased nozzle on tends the aboveto speed discussion, up and changes the isentropic its direction efficiency to the of side the one-to-onewithout the turbine nozzle goes wall, up when first theand pressure then down ratio with of the increasing nozzle (nozzle disc spacing outlet distance, pressure that to nozzle is to say, inlet an pressure) optimal disc is under spacing the distance critical pressureexists to allowratio of a Teslaair. The turbine deflection to obtain angle the goes highest up with isentropic an increase efficiency. in difference To explain value this phenomenon,between the nozzleTable5 givespressure the coefficientsratio and the of componentcritical pressure energy ratio, loss foras theshown one-to-one in turbine turbine Cases cases 1-0.3 at 30,000and 1-0.5 r/min, in Figureincluding 11. When the nozzle the nozzle loss, pressure the disc lossratio andis over the th leaving-velocitye critical pressure loss, ratio, which the flow are definedwill not asdeflect each component energy loss divided by the isentropic enthalpy drop across the whole turbine. Energies 2019, 12, 44 16 of 25

Table 5. Coefﬁcients of component energy loss for one-to-one Tesla turbine cases.

Nozzle Loss Disc Loss Leaving-Velocity Isentropic Case Name Coefficient Coefficient Loss Coefficient Efficiency Case 1-0.3 0.2857 0.4196 0.0760 0.2187 Case 1-0.5 0.1849 0.4405 0.1303 0.2443 Case 1-1 0.1085 0.4358 0.2440 0.2117

With increasing disc spacing distance, the nozzle loss coefficient goes down and the leaving velocity loss coefficient goes up remarkably. The disc loss coefficient increases first and then decreases, and its variation values are relatively lower than those of the nozzle loss and the leaving-velocity loss. The variation rules of the component energy loss for the one-to-one turbine with disc spacing distance are analyzed in detail based on the flow fields. In the multichannel Tesla turbine, the nozzle energy loss includes the loss resulting from the viscous friction (called “friction loss in the nozzle”), and the loss occurring in the nozzle and the N-R chamber caused by the sudden variation of flow area, when the air flows through the nozzle and the N-R chamber and finally to the disc channels (called “local loss in the nozzle”). For the one-to-one turbine, the friction loss in the nozzle decreases with increasing disc spacing distance, caused by a decrease in proportion of the boundary layer in the nozzle channel and a decrease in flow velocity (Figure 11). In addition, most air flows out of the nozzle to the disc channels quickly (Figure9), and the N-R chamber has little influence on the flow field. Thus, the local loss in the nozzle slightly influences the nozzle loss. In detail, the flow area ratio of the N-R chamber to the disc channels decreases with increasing disc spacing distance, therefore, the local loss for the one-to-one turbine decreases. As a result, the nozzle loss, including the friction loss and the local loss, decreases remarkably with increasing disc spacing distance. With an increase in disc spacing distance, more air outside the boundary layer has no use in momentum exchange, which indicates more energy loss in the rotor. This is in agreement with the result in one channel Tesla turbines [28]. The local loss occurring in the disc channels due to the flow area variation has a decrease with increasing disc spacing distance, although the local loss also has little influence on the disc loss. Therefore, the combined actions of the two factors lead to increasing first and then decreasing disc loss as disc spacing distance increases. The Mach number at the turbine outlet rises with disc spacing distance, resulting in increasing leaving-velocity loss coefficient, as shown in Figure 11. These results are the same as those in Table5.

3.3. Flow Status of One-To-Many Multichannel Tesla Turbines As discussed in the above section, the isentropic efficiency of the one-to-many turbine goes up first and then down with increasing disc spacing distance. It decreases remarkably as disc thickness increases. Table6 gives the coefficients of component energy loss for the one-to-many turbine cases at 30,000 r/min. With increasing disc spacing distance, the nozzle loss coefficient and disc loss coefficient decrease, and the leaving-velocity loss coefficient rises. With increasing disc thickness, the nozzle loss has a slight increase, the disc loss decreases, and the leaving-velocity loss increases remarkably. The variation rules of these coefficients are explained in the following text.

Table 6. Coefﬁcients of component energy loss for one-to-many Tesla turbine cases.

Nozzle Loss Disc Loss Leaving-Velocity Isentropic Case Name Coefficient Coefficient Loss Coefficient Efficiency Case 1-0.3 0.0724 0.5539 0.2387 0.1350 Case 1-0.5 0.0578 0.4574 0.3310 0.1538 Case 1-1 0.0417 0.3936 0.4055 0.1592 Case 2-0.5 0.0699 0.4324 0.4040 0.0937 Case 2-1 0.0471 0.3504 0.4925 0.1100 Case 2-2 0.0375 0.3083 0.5561 0.0981 Energies 2019, 12, 44 17 of 25 Energies 2019, 12, x FOR PEER REVIEW 18 of 18

3.3.1. One-To-Many Turbine withwith DifferentDifferent DiscDisc ThicknessThickness To explain the variation rules of the aerodynamicaerodynamic performance of the one-to-many turbine with disc thickness, Figure 12 presents Mach number contourscontours and streamlines on the mid-sections of DCs 1 and 3 for turbine Cases 1-0.51-0.5 andand 2-0.5.2-0.5. The Mach number at the nozzle outlet for Case 1-0.5 is slightly higher than than that that for for Case Case 2-0.5. 2-0.5. Therefore, the flowflow angle at the rotor inlet is a littlelittle lower. Moreover, the flowflow angle in most regions of the disc channels for Case 1-0.5 is lower than that for Case 2-0.5, leading to longer path lines and more momentum exchange. Different from the oneone-to-oneto-one turbine, Mach number increases suddenly at the rotor inlet directly facing the nozzle outlet, which is due to a sudden increase in flow flow area when the air passes the N-R chamberchamber to thethe discdisc channels.channels. The flow flow velocity at that place increases more significantlysignificantly for Case 2-0.5 than that forfor CaseCase 1-0.5.1-0.5.

(a) (b)

Figure 12. 12. MachMach number number contours contours and and streamlines streamlines on the on mid-sections the mid-sections of DCs of 1 DCs and 13 for and the 3 one-to- for the one-to-manymany turbine turbinecases with cases different with differentdisc thickness disc thickness at 30,000 atr/min: 30,000 (a) r/min:DC1 for (Casea) DC1 1-0.5; for (b Case) DC1 1-0.5; for (Caseb) DC1 2-0.5; for ( Casec) DC3 2-0.5; for Case (c) DC3 1-0.5; for and Case (d 1-0.5;) DC3 and for (Cased) DC3 2-0.5. for Case 2-0.5.

Energies 2019, 12, 44 18 of 25

Energies 2019, 12, x FOR PEER REVIEW 19 of 19 Figure 13 presents the contours of the pressure ratio on the mid-sections of DC1 for the Figure 13 presents the contours of the pressure ratio on the mid-sections of DC1 for the one-to- one-to-many turbine cases, and the pressure ratio equals the ratio of the local pressure to the turbine many turbine cases, and the pressure ratio equals the ratio of the local pressure to the turbine inlet inlet total pressure. The pressure ratio at the nozzle outlet for Case 1-0.5 is lower than that for Case 2-0.5, total pressure. The pressure ratio at the nozzle outlet for Case 1-0.5 is lower than that for Case 2-0.5, due to lower flow area ratio of the nozzle channel to the disc channels. It indicates a higher pressure due to lower flow area ratio of the nozzle channel to the disc channels. It indicates a higher pressure drop in the nozzle and higher flow velocity at the nozzle outlet for Case 1-0.5 (Figures 12 and 13). drop in the nozzle and higher flow velocity at the nozzle outlet for Case 1-0.5 (Figures 12 and 13). A A sudden decrease in pressure ratio also occurs at the same region of the sudden increase in Mach sudden decrease in pressure ratio also occurs at the same region of the sudden increase in Mach number. The pressure ratio decreases more remarkably for Case 2-0.5 than that for Case 1-0.5, because number. The pressure ratio decreases more remarkably for Case 2-0.5 than that for Case 1-0.5, because of much more decrease in flow area when the air flows through the N-R chamber to the disc channels. of much more decrease in flow area when the air flows through the N-R chamber to the disc channels. The following is the detailed analysis of the component energy loss for the one-to-many The following is the detailed analysis of the component energy loss for the one-to-many turbine turbine cases with different disc thickness based on the flow fields. With increasing disc thickness, cases with different disc thickness based on the flow fields. With increasing disc thickness, the theproportion proportion of the of boundary the boundary layer in layer the nozzle in the cha nozzlennel channeldecreases decreases and the flow and velocity the flow also velocity decreases, also decreases,which leads which to lower leads friction to lower loss friction in the nozzle. loss in Ac thetually, nozzle. compared Actually, with compared the one-to-one with the turbine, one-to-one the turbine,nozzle friction the nozzle loss friction of the lossone-to-many of the one-to-many turbine is turbinemuch lower is much due lower to much due toless much proportion less proportion of the ofboundary the boundary layer in layer the innozzle the nozzle channel. channel. It should It shouldbe quite be little quite and little has and a little has influence a little influence on the nozzle on the nozzleloss. The loss. local The loss local in lossthe nozzle in thenozzle for thefor turbine the turbine case with case thicker with thicker discs is discs much is higher muchhigher than that than with that withthinner thinner discs discs because because the flow the flowarea areareduces reduces more more when when the air the flows air flows out of out the of N-R the N-Rchamber chamber to the to thedisc disc channels channels They They lead lead to toa little a little increase increase in innozzle nozzle loss loss coefficient coefficient with with increasing increasing disc disc thickness thickness (see(see Table Table6 ).6). AsAs discdisc thickness thickness increases, increases, the the working working fluid fluid becomes becomes much much more more due due to the to largerthe larger nozzle nozzle throat area,throat and area, the and flow the velocity flow invelocity the disc inchannels the disc becomeschannels muchbecomes higher much (see higher Figure (see 12). Figure This leads 12). toThis less boundaryleads to less layer boundary thickness, layer leading thickness, to less leading energy to loss less in energy the disc loss channels. in the disc The channels. local loss The at thelocal inlet loss of theat the disc inlet channels of the goes disc upchannels with disc goes thickness, up with disc due tothickness, the higher due flow to the area higher ratio offlow the area N-R ratio chamber of the to theN-R disc chamber channels. to the As disc a result, channels. the disc As loss a result, decreases the disc with loss increasing decreases disc with thickness increasing (Table disc6). thickness The Mach number(Table 6). at The the turbineMach number outlet isat much the turbine higher outlet for Case is much 2-0.5 higher than that for forCase Case 2-0.5 1-0.5 than (Figure that for 12 Case), which 1- leads0.5 (Figure to higher 12), leaving-velocitywhich leads to higher loss. leaving-velocity loss.

(a) (b)

FigureFigure 13.13. ContoursContours ofof pressurepressure ratio on the mid-sectionsmid-sections of of DC1 DC1 for for the the one-to-many one-to-many Tesla Tesla turbine turbine casescases with with differentdifferent discdisc thicknessthickness atat 30,00030,000 r/min:r/min: ( (aa)) Case Case 1-0.5; 1-0.5; and and (b (b) )Case Case 2-0.5. 2-0.5.

FigureFigure 1414 givesgives contourscontours ofof relativerelative tangential velocity and and vector vector distributions distributions on on the the radial radial cross-sectionscross-sections ofof 00°◦ forfor thethe one-to-manyone-to-many Tesla turbineturbine cases. Unlike Unlike the the one-to-one one-to-one turbine, turbine, the the air air at at thethe nozzle nozzle outlet outlet inin thethe one-to-manyone-to-many turbine bends to to the the axial axial direction direction to to flow ﬂow into into the the disc disc channels, channels, especiallyespecially atat thethe nozzle of which which downstream downstream is is a awall. wall. The The vortex vortex occurs occurs on onthe the downstream downstream wall wall of ofthe the N-R N-R chamber chamber and and thethe inlet inlet of the of thedisc disc channels, channels, leading leading to the to energy the energy loss at loss those at thoseplaces. places. The Theenergy energy loss loss on the on thedownstream downstream wall wall of the of theN-R N-R chamber chamber wall wall is the is themain main source source of the of thelocal local loss lossin inthe the nozzle. nozzle. That That at atthe the inlet inlet of of the the disc disc channels channels is isa apart part of of the the disc disc loss, loss, which which also also includes includes the the frictionfriction loss. loss.

Energies 2019, 12, 44 19 of 25 Energies 2019, 12, x FOR PEER REVIEW 20 of 20

(a) (b)

FigureFigure 14.14. Contours of relative tangentialtangential velocityvelocity andand vectorvector distributionsdistributions onon radial radial cross-sections cross-sections of of 0 ◦ for0° for the the one-to-many one-to-many Tesla Tesla turbine turbine cases cases with with different different disc disc thickness: thickness: (a) Case(a) Case 1-0.5; 1-0.5; and and (b) Case(b) Case 2-0.5. 2-0.5. Compared with turbine Case 1-0.5, the flow velocity at the nozzle outlet for turbine Case 2-0.5 is lower,Compared thus leading with turbine to a larger Case flow 1-0.5, angle the flow at the velocity rotor inlet.at the Thenozzle radial outlet velocity for turbine at the Case rotor 2-0.5 inlet is is higherlower, andthus the leading relative to tangentiala larger flow velocity angle isat lower. the roto Inr addition, inlet. The the radial vortex velocity on the at downstream the rotor inlet wall is of thehigher N-R and chamber the relative and at tangential the inlet of velocity the disc is channels lower. In for addition, turbine Casethe vortex 2-0.5 is on much the downstream larger than that wall for turbineof the N-R Case chamber 1-0.5, therefore and at the the inlet energy of the loss disc caused channels by the for variation turbine ofCase the 2-0.5 flow is area much for larger Case 2-0.5than is muchthat for higher. turbine Case 1-0.5, therefore the energy loss caused by the variation of the flow area for Case 2-0.5Combining is much higher. Figures 12 and 14, the relationship of the mass flow rate percentages in each disc channelCombining of the one-to-many Figures 12 and turbine 14, the with relationship disc thickness of the ismass analyzed. flow rate Compared percentages with in Caseeach disc 1-0.5, thechannel flow of angle the one-to-many in DC1 for Caseturbine 2-0.5 with is disc much thickness larger, is causing analyzed. more Compared air to flow with through Case 1-0.5, DC1 the for Caseflow angle 2-0.5. in The DC1 mass for flowCase rate2-0.5 of is themuch air larger, flowing causing into each more disc air to channel flow through of the one-to-many DC1 for Case turbine2-0.5. hasThe amass significant flow rate difference of the from air flowing that of the into one-to-one each disc turbine. channel Figure of the 14 one-to-many shows that the turbine nozzle has outlet a areasignificant of the difference air flowing from into that DCs of 1the and one-to-one 2 is the same,turbine. and Figure equals 14 shows to the that corresponding the nozzle outlet area ofarea one discof the spacing air flowing distance into and DCs one 1 and disc 2 thickness. is the same, It is an largerd equals than to that the flowing corresponding into DC3, area which of one equals disc to thespacing corresponding distance and area one of disc one thickness. disc spacing It is distance larger than and that half flowing of disc thickness.into DC3, which The area equals ratio to of the the workingcorresponding fluid flowing area of intoone DC1disc tospacing DC3 increases distance with and dischalf thickness,of disc thickness. leading toThe increasing area ratio mass of the flow working fluid flowing into DC1 to DC3 increases with disc thickness, leading to increasing mass flow rate percentage in DC1. Based on the two factors, the mass flow rate percentage in DC1 for Case 2-0.5 rate percentage in DC1. Based on the two factors, the mass flow rate percentage in DC1 for Case 2- is higher than that for Case 1-0.5, which agrees with the results in Figure6. 0.5 is higher than that for Case 1-0.5, which agrees with the results in Figure 6. 3.3.2. One-To-Many Turbine with Different Disc Spacing Distance 3.3.2. One-To-Many Turbine with Different Disc Spacing Distance Mach number contours and streamlines on the mid-sections of DCs 1 and 3 for the one-to-many turbineMach cases number with differentcontours and disc streamlines spacing distance on the aremid-sections presented of in DCs Figure 1 and15 .3 Withfor the increasing one-to-many disc spacingturbine cases distance, with the different Mach numberdisc spacing increases distance at the are nozzle presented outlet, in Figure which leads15. With to lowerincreasing flow disc angle atspacing the rotor distance, inlet. Thus,the Mach the number turbine casesincreases with at larger the nozzle disc spacingoutlet, which have leads longer to path lower lines. flow Moreover, angle at the rotor inlet. Thus, the turbine cases with larger disc spacing have longer path lines. Moreover, the the sudden increase in Mach number at the rotor inlet decreases significantly with increasing disc sudden increase in Mach number at the rotor inlet decreases significantly with increasing disc spacing spacing distance. distance. As discussed above, some air in DC1 passes through the N-R chamber and finally flows into DC3, As discussed above, some air in DC1 passes through the N-R chamber and finally flows into which leads to more air in DC3 than that in DC1. For the turbine cases with disc thickness of 1 mm, DC3, which leads to more air in DC3 than that in DC1. For the turbine cases with disc thickness of 1 with increasing disc spacing distance, the working fluid in DC1 flowing into the N-R chamber becomes mm, with increasing disc spacing distance, the working fluid in DC1 flowing into the N-R chamber much less due to less flow angle; with increasing disc spacing distance, the nozzle outlet area ratio becomes much less due to less flow angle; with increasing disc spacing distance, the nozzle outlet of the air flowing into DC1 to that into DC3 decreases, causing less air in DC1. Under the combined area ratio of the air flowing into DC1 to that into DC3 decreases, causing less air in DC1. Under the actions of the two factors, the air in DC1 becomes less with increasing disc spacing distance. combined actions of the two factors, the air in DC1 becomes less with increasing disc spacing distance.

Energies 2019, 12, 44 20 of 25 Energies 2019, 12, x FOR PEER REVIEW 21 of 21

(a) (b) (c)

(d) (e) (f)

FigureFigure 15. 15. MachMach number number contours contours and andstreamlines streamlines on the on mid-sections the mid-sections of DCs of 1 DCsand 3 1 for and the 3 one-to- for the manyone-to-many turbine turbinecases with cases different with different disc spacing disc spacing distance distance at 30,000 at 30,000 r/min: r/min: (a) DC1 (a) for DC1 Case for Case1-0.3; 1-0.3; (b) DC1(b) DC1 for Case for Case 1-0.5; 1-0.5; (c) DC1 (c) DC1 for forCase Case 1-1; 1-1; (d) ( dDC3) DC3 for for Case Case 1-0.3; 1-0.3; (e ()e DC3) DC3 for for Case Case 1-0.5; 1-0.5; and and (f ()f )DC3 DC3 forfor Case Case 1-1. 1-1.

WithWith increasing discdisc spacingspacing distance, distance, the the local local loss loss in thein the N-R N-R chamber chamber decreases, decreases, resulting resulting from froma decrease a decrease in flow in areaflow ratio area ofratio the N-Rof the chamber N-R ch toamber the discto the channels. disc channels. Based on Based the aboveon the analysis, above analysis,the friction the loss friction in the nozzleloss in ofthe the nozzle one-to-many of the turbineone-to-many is quite turbine little and is has quite few little contributions and has tofew the contributionsnozzle loss. Thus, to the the nozzle nozzle loss. loss Thus, decreases the nozzle with loss increasing decreases disc with spacing increasing distance disc (Table spacing6). distance (TableFor 6). the one-to-many turbine, the energy loss in the disc channels also increases with disc spacing distance,For the which one-to-many is the same turbine, as the the one-to-one energy loss turbine. in the In disc addition, channels the also local increases loss occurring with disc at spacing the inlet distance,of the disc which channels is the decreases same as the with one-to-one increasing turbine. disc spacing In addition, distance the significantly. local loss occurring Thus, the at the disc inlet loss ofof the the disc one-to-many channels decreases turbine decreases with increasing significantly disc spacing with increasing distance discsignificantly. spacingdistance. Thus, the Moreover, disc loss ofthe the leaving-velocity one-to-many turbine loss goes decreases up with significantly disc spacing with distance, increasing resulting disc from spacing increasing distance. Mach Moreover, number theat theleaving-velocity turbine outlet loss (Figure goes 15 up). with disc spacing distance, resulting from increasing Mach number at theFigure turbine 16 outlet shows (Figure the contours 15). of pressure ratio on the mid-sections of DC1 for the one-to-many turbineFigure cases 16 withshows different the contours disc spacing of pressure distance. ratio Theon the pressure mid-sections ratio variation of DC1 for with the increasing one-to-many disc turbinespacing cases distance with is different same as disc that spacing with decreasing distance. discThe pressure thickness, ratio which variation is because with bothincreasing decreasing disc spacingdisc thickness distance and is same increasing as that disc with spacing decreasing distance disc leadthickness, to a decrease which is in because flow area both ratio decreasing of the nozzle disc thicknesschannel toand the increasing disc channels disc forspacing the one-to-many distance lead turbine. to a decrease In general, in flow an increase area ratio in flowof the area nozzle ratio channelof the nozzle to the todisc disc channels channels for leadsthe one-to-many to an increase turbine. in pressure In general, at the an nozzle increase outlet in flow and area a significant ratio of the nozzle to disc channels leads to an increase in pressure at the nozzle outlet and a significant

Energies 2019, 12, 44 21 of 25 Energies 2019, 12, x FOR PEER REVIEW 22 of 22 decrease inin pressurepressure at at the the rotor rotor inlet inlet facing facing the the nozzle nozzle outlet. outlet. Therefore, Therefore, the ﬂow the velocityflow velocity at the nozzleat the nozzleoutlet decreasesoutlet decreases and that and at thethat rotor at the inlet rotor increases inlet increases much more. much more.

(a) (b)

Figure 16. ContoursContours of of pressure pressure ratio on the mid-sections of DC1 DC1 for for one-to-many one-to-many Tesla Tesla turbine cases with different disc spacing distance at 30,000 r/min: ( (aa)) Case Case 1-0.3; 1-0.3; and and ( (bb)) Case Case 1-1. 1-1.

3.4. Influence Influence of Disc Thickness and Spacing Distance on Energy Loss As discussed above, the disc thickness has a li littlettle influence influence on the aerodynamic performance of the one-to-one Tesla Tesla turbine. In detail, the isentrop isentropicic efficiency efficiency has a slight decrease with increasing disc thickness (less than 7% reduction for the turbineturbine with 0.5 mm in disc spacing distance). That of the one-to-many turbine turbine decreases decreases significantly significantly with with disc disc thickness thickness (decrease (decrease of of about about 45% 45% for the turbine with 0.5 mm in disc spacing distance), whic whichh is due to higher leaving-velocity. For both the one-to-one turbine and the one-to-many turbine, the is isentropicentropic efficiency efficiency goes up first first and then down with increasing disc spacing distance. For the one-to-one turbine, the flow flow coefficient coefficient remains unchanged with the variation of disc thickness, and and it increases with disc thickness for the one-to-manyone-to-many turbine. In addition, it increases with decreasing discdisc spacingspacing distance distance for for both both two two kinds kinds of of turbines. turbines. This This coefficient coefficient of theof the one-to-one one-to- oneturbine turbine is much is much lower lower than than that that of the of one-to-manythe one-to-many turbine turbine with with same same disc disc spacing spacing distance distance and discand discthickness thickness (The (The former former is about is about half ofhalf the of latter the latter for turbine for turbine Case Case 1-0.5). 1-0.5). The nozzle loss of the multichannel Tesla Tesla turbine includes the the friction friction loss loss and and the the local local loss. loss. That ofof thethe one-to-oneone-to-one turbine turbine decreases decreases with with increasing increasing disc disc spacing spacing distance; distance; that that of the of one-to-many the one-to- manyturbine turbine decreases decreases with increasingwith increasing disc spacing disc spacin distanceg distance and increasesand increases slightly slightly with with increasing increasing disc discthickness. thickness. In general, In general, the the one-to-one one-to-one turbine turbine has has higher higher nozzle nozzle loss loss than than thethe one-to-manyone-to-many turbine (18.49% and 5.78%, respectively, for Case 1-0.5) due to much higher friction loss of the one-to-one turbine, although although the the local local loss loss of of the the one-to-one one-to-one is is lower lower than than that that of of the the one-to-many one-to-many turbine. turbine. Combining the the discussion discussion in in the the previous previous sections, sections, it it can can be be concluded concluded that that most most of ofthe the nozzle nozzle loss loss is theis the friction friction loss loss for for the the one-to-one one-to-one turbine, turbine, while, while, for for the the one-to-many one-to-many turbine, turbine, the the local local loss loss in in the nozzle is competitive competitive with with the the friction friction loss loss in in the the nozzle. nozzle. The The friction friction loss loss decreases decreases rapidly rapidly with with a decreasea decrease in inproportion proportion ratio ratio of of the the boundary boundary layer layer in in the the nozzle nozzle channel. channel. In In detail, detail, it it drops with increasing discdisc spacingspacing distance distance for for the the one-to-one one-to-one turbine turbine and and increasing increasing disc disc spacing spacing distance distance or disc or discthickness thickness for the for one-to-manythe one-to-many turbine, turbine, and and the frictionthe friction loss loss of the of one-to-manythe one-to-many turbine turbine is quite is quite low. low.The localThe local loss inloss the in nozzle the nozzle decreases decreases with with decreasing decreasing flow flow area area ratio ratio of the of N-R the N-R chamber chamber to the todisc the discchannels, channels, which which is increasing is increasing disc spacingdisc spac distanceing distance or decreasing or decreasing disc thickness.disc thickness. For the the one-to-one one-to-one turbine, turbine, the the disc disc loss loss goes goes up first up and first then and down then with down increasing with increasing disc spacing disc distance,spacing distance, and its andvariation its variation range is range much is muchless (abo lessut (about 2%), 2%),while, while, for the for theone-to-many one-to-many turbine, turbine, it decreasesit decreases significantly significantly with with increasing increasing disc disc spacin spacingg distance distance and and decreases decreases a a little little with with increasing disc thickness (the maximum variation is 24.56%). For both the one-to-one turbine and the one-to- many turbine, the disc energy loss in the disc channels rises with disc spacing distance. In addition,

Energies 2019, 12, 44 22 of 25 disc thickness (the maximum variation is 24.56%). For both the one-to-one turbine and the one-to-many turbine, the disc energy loss in the disc channels rises with disc spacing distance. In addition, the local loss at the inlet of the disc channels decreases with increasing disc spacing distance. According to the variation rule of the local loss with disc spacing distance, it can be concluded that the local loss of the one-to-one turbine is less than that of the one-to-many turbine. The leaving-velocity loss of the one-to-one turbine rises with disc spacing distance, while that of the one-to-many turbine goes up with both disc thickness and spacing distance. Moreover, compared with the one-to-one turbine, the leaving-velocity loss of the one-to-many turbine with same disc thickness and spacing distance is much higher (33.1% and 13.03%, respectively, for Case 1-0.5).

4. Conclusions As one of the best alternative options for small-scale conventional bladed turbines, Tesla turbines can be comparable with conventional turbines when they are well designed. Two kinds of multichannel Tesla turbines (one-to-one Tesla turbine and one-to-many Tesla turbine) with different disc spacing distance and disc thickness were studied numerically to analyze the influence of the two rotor geometrical parameters on their aerodynamic performance and flow characteristics, which offers a theoretical guidance in the design method. The main conclusions are summarized as follows. (1) For both the one-to-one turbine and the one-to-many turbine, the isentropic efficiency goes up first and then down with increasing disc spacing distance, thus an optimal disc spacing distance exists to allow the multichannel Tesla turbine to obtain the highest efficiency. With increasing disc thickness, the isentropic efficiency of the one-to-many turbine decreases dramatically, while that of the one-to-one turbine decreases slightly. For turbine cases with disc spacing distance of 0.5 mm, as disc thickness increases from 1 mm to 2 mm, the isentropic efficiency of the one-to-one turbine was lowered by less than 7% (relative variation), while that of the one-to-many turbine drops by about 45% (relative variation). In addition, the isentropic efficiency of the one-to-one turbine is more sensitive to rotational speed than the one-to-many turbine. (2) For the one-to-one turbine, with increasing disc thickness, the flow field is similar for turbine cases with same disc spacing distance; the flow velocity in the N-R chamber and the relative tangential velocity gradient close to the disc walls decrease slightly, causing the isentropic efficiency to decrease slightly. With increasing disc spacing distance, the flow velocity at the nozzle outlet decreases, and the flow angle at the rotor inlet increases, leading to an increase in mass flow rate percentage of DCs 1 and 2. (3) For the one-to-many turbine, with increasing disc thickness, the flow velocity at the nozzle outlet decreases and its sudden increase at the rotor inlet facing the nozzle outlet is greater; the flow angle in the disc channels increases slightly and the flow velocity at the turbine outlet also increases. With increasing disc spacing distance, the flow velocity at the nozzle outlet goes up and its sudden increase becomes less; the flow angle in the disc channels decreases and the velocity at the rotor outlet increases.

Author Contributions: Conceptualization, W.Q. and Q.D.; Methodology, W.Q. and Q.D.; Investigation, W.Q. and Y.J.; Data Curation, W.Q. and Y.J.; Writing—Original Draft Preparation, W.Q.; Writing—Review and Editing, Q.D., Q.Y. and Z.F.; Visualization, W.Q. and Y.J.; and Supervision, Q.Y. and Z.F. Funding: This research was supported financially by National Natural Science Foundation of China (grant number 51376143). Conflicts of Interest: The authors declare no conflict of interest. Energies 2019, 12, 44 23 of 25

Nomenclature b disc spacing distance, mm c radial clearance of nozzle-rotor chamber, mm cp specific heat at constant pressure, J/kg·K Cm flow coefficient d diameter, mm h static enthalpy, m2/s2 2 2 htot mean total enthalpy, m /s k turbulent kinetic energy, m mass flow rate, kg/s M torque, N·m Ma Mach number n rotational speed of the rotor, r/min N number pabs absolute pressure pnt/pi ratio of total pressure at the nozzle inlet to pressure at the turbine outlet P power, kW Prt turbulent Prandtl number r radial coordinate or radius, mm R specific gas constant of air, 3 SE energy source, kg/(m·s ) 2 2 SM momentum source, kg/(m ·s ) t time, s th disc thickness, mm T temperature, K Tnt total temperature at the nozzle inlet, K Uj averaged velocity in three directions, m/s v average radial velocity, m/s W relative tangential velocity, m/s W average relative tangential velocity, m/s xj coordinates in three directions, m z axial coordinate, m α nozzle exit geometrical angle (relative to the tangential direction), ◦ γ specific heat ratio Γt eddy diffusivity, kg/(m·s) δ relative variation of parameters ∆hisen isentropic enthalpy drop of the whole turbine, J/kg η isentropic efficiency θ circumferential coordinate, rad µt eddy viscosity or turbulent viscosity, kg/(m·s) ρ density, kg/m3 ρuiuj Reynolds stresses ρujh turbulence flux ρuiφ Reynolds flux τ molecular stress tensor, kg/(m·s2) ω turbulent frequency, s-1 Ω rotational angular speed, rad/s Subscripts d disc dc disc channel i inner n nozzle o outer Energies 2019, 12, 44 24 of 25

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